Research ArticleCustomer-Oriented Vehicle Routing Problem withEnvironment Consideration Two-Phase OptimizationApproach and Heuristic Solution

Fanting Meng12 Yong Ding 2 Wenjie Li3 and Rongge Guo 2

1State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing 100044 China2School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China3Sid and Reva Department of Civil Environmental and Infrastructure Engineering GeorgeMason University Fairfax VA 22030 USA

Correspondence should be addressed to Yong Ding ydingbjtueducn

Received 15 January 2019 Accepted 10 March 2019 Published 26 March 2019

Academic Editor A M Bastos Pereira

Copyright copy 2019 Fanting Meng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With the fastest consumer demand growth the increasing customerrsquos demands trend to multivarieties and small-batch andthe customer requires an efficient distribution planning How to plan the vehicle route to meet customer satisfaction of massdistribution as well as reduce the fuel consumption and emission has become a hot topic This paper proposes a two-phaseoptimization method to handle the vehicle routing problem considering the customer demands and time windows coupled withmultivehicles The first phase of the optimization method provides a fuzzy hierarchical clustering method for customer groupingThe second phase formulates the optimization en-group vehicle routing problem model and a genetic algorithm to account forvehicle routing optimization within each group so that fuel consumption and emissions are minimized Finally we provide somenumerical examples Results show that the two-phase optimization method and the designed algorithm are efficient

1 Introduction

The vehicle routing problem (VRP) is a well-known NP-hard problem which has introduced by Dantzig and Ramser[1] Generally the VRP is to generate sets of vehicle routesto reduce travel cost It is worth noting that the VRPhas important practical significance of social benefits andenterprisers benefits Especially with the development of theeconomic and electrical business the logistic delivery acts asa key role From the survey data we can find out that thecarbon emission of logistic accounts for 80 of total roadtransport With the continuous development of commodityeconomy it becomes important to process customer ordersand deliver them via suitable freight vehicles Furthermorecustomers need to be quickly responded for good deliverywhich also become a hot topic

Previous researches on VRP mostly take into account theeconomic factors such as distance cost and time windowsAnd varieties of heuristic algorithm are designed to solvethese problems [2 3] In recent years how to reduce the

energy consumption and carbon emissions has become ahot topic [4 5] Especially with the rapid development ofonline shopping and express delivery there is a sharplyincasement of tail gas emissions from fuel logistics vehiclesTherefore reducing the energy consumption and carbonemission has great significance for the ecological system Forlogistics transportation enterprisers the fuel consumptioncost is one of the main operating costs which is directlyrelated to the carbon platoon which will affect the operatorsrsquoprofits and social benefits Enterprises can reduce the cost offuel consumption and greenhouse gas emission to achievethe goal of energy conservation and emission reductionTherefore how to optimize the routes of distribution vehiclesand how to reduce energy consumption and carbon emissionin the premise of meeting customers demand are the focus ofresearch for logistics transportation enterprisers

However with the continuous development of the com-modity economy customers have put forward higher require-ments of service quality of logistics enterprises So how toimprove the quality of distribution service has become the

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1073609 19 pageshttpsdoiorg10115520191073609

2 Mathematical Problems in Engineering

focus of enterprises In particular customer demands tendto be diversified which proposes higher requirements fordelivery service and will lead to increased costs Providingappropriate service for each customer can improve thesatisfaction gain customers recognition and loyalty to theenterprisers as well as improving the market competitive-ness of enterprise Deliver the required quantity of goodsby reasonable arrangement of distribution vehicle routesThat is the last ring of the logistics service and the lastpart of the customer orders which will directly affect thecustomersrsquo satisfaction Inappropriate distribution servicesresult in customersrsquo terrible experience and maybe lead toloss of customers and even worse increase the hidden costsof the enterprises So it is equally important to reducetransportation costs as well as improve the service qualityTo achieve a win-win situation for the vehicle distributionpath designing an appropriate clusteringmethod is proposedto divide the customers into reasonable groups That caneffectively improve the efficiency of the people on both sidesof the enterprisers and customers

Therefore this paper proposes a two-phase optimizationmethod to design the vehicle delivery paths with comprehen-sive consideration of customer demand energy conservationand emission reduction For the first phase the fuzzy systemclustering method (FSCM) is applied to reasonably groupthe customers according to the qualitative and quantitativeindicators The second phase is en-group path optimizationan en-group vehicle routing optimization model is estab-lished with the target of reducing energy consumption andemission reduction To solve the problem a genetic algorithm(GA) is designed Based on the proposed method we canimprove the quality of distribution service as well as reducingenergy consumption and carbon emission in the distributionprocess

Our paper makes a number of contributions to theliterature(1) We consider the customer requirements for vehiclerouting problem On this basis a fuzzy hierarchical clustermethod is designed which can process quantitative andqualitative data In order to guarantee the delivery quality thecustomers can be clustered into different groups based on theorder information Then the problem can be converted intoan en-group path optimization problem based on clusteringresults so that can guarantee the delivery quality(2) Energy saving and emission reduction are also thetargets of our study To characterize the problem we formu-late the model under both deterministic and dynamic trafficconditions separately As we know the energy consumptionis affected by the different vehicle speed So considering thevariety of the vehicle speed is more meaning for the study ofreduced energy consumption On this basis we design a GAto solve the problem and generate an approximate optimalvehicle routing which can gain better solution results(3) A two-phase model is proposed to meet the customerdemands as well as reduce the fuel consumption Contrastingto the commonmodel the two-phase model can improve thesolving efficiency and more suitable to deal with customerrequirements The first phase model is a customer-orientedmathematics model which can process customer demands

and at this phase we group the customers into appropriategroup which can improve customersrsquo satisfaction The sec-ond phase model is an en-group VRP that aims to minimizethe comprehensive costs including fuel and carbon costs andtime penalty costs

The remainder of this paper consists of five sectionsSection 2 discusses previous related work and highlights thecontributions of this paper Section 3 provides the descriptionof the proposed model In Section 4 we provide the fuzzyhierarchical cluster method and en-group GA algorithm tosolve the problem We describe the design of experimentsand numerical results in Section 5 Finally we concludeour work and provide the propositions for further study inSection 6

2 Literature Review

TheVRP was proposed by Dantzig and Ramser and has beenused to solve the optimal distribution route plan of trucksAs the question is raised it has received wide attention fromexperts and scholars especially in the field of operationsresearch Our work is closing to the multivehicles and greenVRP which considers fuel consumption and time windowsAccording to the research of different problems varioustypes of VRP are generated and corresponding mathematicalmodels are established [6 7] Brandao studied the MultitypesVRP and used a tabu search algorithm (TS to solve theproblem analyzed the influence of the vehicle type on thesolution effect) To gain optimal results types of algorithmswere designed to solve the problems [8ndash10] SpecificallyYazgi designed a greedy stochastic adaptive memory searchalgorithm to solve mixed fixed vehicle routing problemsDemir et al [11] designed a large neighborhood searchalgorithm (VLSN) to solve the problem In this literature wereview the existing research from three perspectives

In order to reduce the influence of the environment inrecent years some researchers turn on to study the vehiclerouting problems with consideration of fuel consumption(VRPFC) Bektas et al [12] studied the carbon emissionsfrom VRP issues in several articles summed up the relevantcalculation model for vehicle energy consumption and gavethe effective application scope of the model Palmer [13]established an integrated vehicle routing model and proposeda measurement model to calculate the 1198621198742 emission Miguel[3] aimed to reduce carbon emissions and fuel consumptionand defined these problems as emissions vehicle routingproblem (EVRP) Demir et al [14] proposed a biobjectivemodel to formulate the pollution routing problem (PRP)with the objective ofminimizing the fuel consumption as wellas travel time Kwon et al [15] studied the problem of vehiclepath optimization with consideration of fuel consumptionand carbon emissions based on time-varying velocity Theauthors used the time-insertion algorithm to solve the singlepath problem and designed a column generation algorithmbased on tabu search algorithm to solve the multipathproblem Huang [4] studied the vehicle routing problem ofenergy conservation and emission reduction and proposedthe concept of green vehicle routing problem Qian et al[16] established a heterogeneous vehicle routing model by

Mathematical Problems in Engineering 3

considering the carbon emissions during the vehicle oper-ation a tabu search algorithm was designed to solve theproblem Most of the VRP were formulated as mixed inter-linear models and heuristic algorithms are designed to solvethe problem The existing studies use the emission modelwhich include MEET and National Atmospheric EmissionInventory [5 17]

The vehicle routing problem with time windows wasproposed by Avelsbergh [18] and widely studied by theresearchers Most of the studies focus on design better algo-rithms to solve the problem such as GA TS etc [8 19 20]Specifically Calvete [21] established a multiobjective opti-mization model for the vehicle routing problem (VRPSTW)which considered soft time windows and designed a goalplanning method to solve the problem Miguel [3] pro-posed an iterative route construction and improvementalgorithm (IRCI) based on path generation algorithm to solvethe VRPTW problem Cihan et al [2] proposed a time-dependent dual-target VRP In the established model it wasassumed that the path selection will be influenced by the timewindows

While many contributions have been made to the VRPwith time windows and the green vehicle problem toour knowledge fewer studies have examined the customerdemands However processing the customer orders andquick response to varieties of customer demands for deliver-ing the goods via suitable freight vehicles appears to be a keyelement for efficient logistics service In previous literature[22] it has been pointed out that customer-based orderclustering as well as the vehicle routing strategies should becoordinated to solve the vehicle problem With the develop-ment of the VRP a lot of studies focus on mathematicallyformulating the corresponding problems In our research wefirst aim to reasonably group the customers based on thedifferences of customer demand On this basis we target tooptimize the vehiclesrsquo travel cost including fixed cost energyconsumption carbon emission and punishment for timewindows For simplicity the problem in this study is calledthe customer-oriented vehicle routing problem consideringenergy conservation and emission reduction (C-VRPESER)

In this research we point out seven factors which willaffect the customer satisfaction based on the studies [23] suchas incorporating location analysis service quality servicetime external similarity factor etc Furthermore we establishthe fuzzy similar matrix to cluster the customers with similarproduct demand On the basis of the clustering results weformulate the en-group optimization model and design a GAto solve the model This study aims to make the followingtheoretical contribution to VRP which considers customerdemand as well as energy conservation and emission reduc-tion Then the following two-phase optimization model isproposed to solve the C-VRPESER

3 Problem Description andModel Formulation

In this paper we consider a transportation network that isrepresented by a directed complete graph 119872 = (119873119860) Thevertex set119873 is composed by a depot 0 and a set of customers

1198730 = 119873 0 = 1 2 3 119899 The number of homogeneousvehicles is a deterministic exogenous parameter and the setof vehicles is represented by 119870 = 1 2 3 119898 The capacityof each vehicle is equal to 119876119890 Each vehicle 119896 has the earliestdeparture time 119890119896 from the depot after unloaded and thelatest arrival time 119897119896 at the depot after finishing the services ofcustomers In practically we can get the customer 119894 demandfor goods from the previous order denoted by 1205931119894 and we alsocan get the other demand for different attributes1205932119894 1205933119894 1205934119894 1205935119894 1205936119894 and 1205937119894 And to simplify the problem we divide the sevenattributes into two types quantitative and qualitative For thequalitative demand we use triangle fuzzy number (1205721199011119905 1205721199012119905and 1205721199013119905) to present five demand levels (ldquovery highrdquo ldquohighrdquoldquomediumrdquo ldquolowrdquo and ldquovery lowrdquo) in which 119905 presents thecustomer number and 119901 On the other hand mathematicalmethod is used to eliminate the dimensional and evaluatethe similarity On this basis we cluster the customers intodifferent groups as shown in Figure 1 The customers canbe clustered into three groups which can be presented bydifferent shapes triangle quadrilateral and pentagon Anden-route goods delivery paths are designed according to thecustomersrsquo location and time windows which decide thedelivery sequence And then we connect each customer witha line with arrowheads And in each group the vehicle shouldstart from the depot (the vehicle remained at the depot) andreturn to the depot after serving all the en-group customers

The optimization goal of the problem is to minimize thetotal cost on the basis of customersrsquo group that realizes the en-group distribution vehicle routing optimization By reason-ably scheduling different types of vehicles for each customergroup the utilization rate of vehicles can be improved Andit also can meet the restriction of vehicle load and time limitThe optimal objective includes the vehicle fixed costs timepenalty cost fuel cost and carbon emissions On this waywe can realize minimizing the total cost considering energyconservation and emissions reduction and reducing the costof comprehensive optimization

In the next sections this paper focuses on formulatinga two-phase model to solve the problem The first model isgenerated to cluster the customers into groups and the secondphase model aims to give out an optimal route en-groupwhich can reduce the energy consumption and emission

31 First Phase Model The customers demand attributesconsist of several indicators part of them is quantitativeindicators such as time and location Others are qualitativeindexes such as external and internal characteristics ofcommodity customer satisfaction etc Through consideringthe related factors of distribution enterprisers service qualityand customer satisfaction we choose seven indicators asdecision variables (1) physical properties (2) geographicalposition (3) service quality (4) product value (5) type ofgoods (6) security of goods and (7) time windows Inorder to quantify the decision variables we define the sevendecision variables in Table 1 And the above indicators canrepresent most of the customerrsquos demand attributes

However the decision variables should eliminate dimen-sion before using For modeling convenience this paper

4 Mathematical Problems in Engineering

Table 1 Definition of customerrsquos demand attributes

Variables Variabledefinitions Variable explanation

1205931119894 Physicalproperties

Number of delivery of goods weight volume this article take the quality asreference the vehiclersquos rated load determines the number of delivery vehicles1205932119894 Geographical

positionCustomersrsquo location customers with similar geography have larger possibility be

divided into same groupso that realize the joint distribution

1205933119894 Service qualityIncluding reaction time service attitude and service ability Customer demands forservice quality are differentand customers with similar service can be divided into

same groups1205934119894 Product value Value of products generally refers to the market valueThe high value cargo can beseparated from others this article define the value of the market price of five levels

1205935119894 Type of goodsTypes of goods and external similarity with other goods If the external similarity is

higher that it convenient to load and unload the cargo which will improve thedelivery efficient1205936119894 Security of

goodsThis property is mainly denote the customer security requirements(eg fragile)

Especially for the dangerous cargothe safety is very important1205937119894 Pressed for time Present the customer request delivery time limit of time the latest deadline we canarrange the customer according to their time windows

Depot

1

6

14

2

4

12

3

513

11

810

9

7

Figure 1 Illustration of the customer groups and distribution routes

uses the fuzzy clustering method to cluster the customersinto appropriate groups which have the similar demandattributes Taking into considering the above seven decisionvariables the qualitative indicators include 1205933119894 1205934119894 1205935119894 and 1205936119894and the other three variables are quantitative indicators

In order to calculate the qualitative data the linguisticterms ldquovery highrdquo ldquohighrdquo ldquomediumrdquo ldquolowrdquo and ldquovery lowrdquoare introduced to represent the qualitative decision variablesSpecifically language ldquovery highrdquo can be represented by

triangular fuzzy number (075 1 1) which indicates that thecustomerrsquos demand of the service level is much higher thanother customers Similarly the language ldquohighrdquo ldquomediumrdquoldquolowrdquo and ldquovery lowrdquo can be represented by (05 075 1)(025 05 075) (0 025 05) and (00025) Based on thelanguage description we translate the linguistic variables intotriangular fuzzy numbers so that we can calculate the com-prehensive similarity between the customers The linguisticvariables represent the correlation between this indicator and

Mathematical Problems in Engineering 5

customer satisfaction In this way each customerrsquos demandindicators can be denoted by triangular fuzzy number (12057211990111989411205721199011198942 1205721199011198943) which can be denoted as follows120593119901119894 = [1205721199011198941 1205721199011198942 1205721199011198943] (1)

We can get the demand attributers from the orderswhich will be pregiven by the customers However we shouldtransfer the linguistic variables and the quantitative data asstandardized data so that we can use them to calculate thesimilarity And the procedure is shown in below

(1)e Processing of Qualitative DecisionVariables In the firststep we use the standard deviation transformation to processthe qualitative data which can guarantee the data availabilityAnd for the second step we introduce the range conversion tostandardize the quantitative variables The constraint (1) canbe firstly transformed as follows120593119901119894 = [1199011198941 1199011198942 1199011198943] 119894 isin 1198730 (2)

where

120593119901119894 = 100381610038161003816100381610038161003816120572119901119894119905 minus 120572119901119894119905100381610038161003816100381610038161003816119878119901119905 119894 isin 1198730 (3)

120572119901119894119905 = sum119873119894=1 120572119901119894119905119873 119905 = 1 2 3 119894 isin 1198730 (4)

However the indicator 120572119901119894119905 is the mean value 119878119901119905 is thestandard deviation and |119873| is the total customer numberThevalue range of 119901 = 3 4 5 6119878119901119905 = [[[

sum119873119894=1 (120572119901119894119905 minus 120572119901119894119905)2|119873| minus 1 ]]]12 119905 = 1 2 3 119894 isin 1198730 (5)

Then we can calculate the similarity of qualitativedecision-making variables with Hemingwayrsquos distancemethod as follows

119883119901119894119895 = 1 minus (sum3119905=1 10038161003816100381610038161003816119901119894119905 minus 11990111989511990510038161003816100381610038161003816)3 119894 119895 isin 1198730 (6)

119883119901119894119895 represents the qualitative correlation between thedifferent customers and the range of the value is during [0 1]119883119901119894119895 indicates relationship of attributors 120593119901119894 (119896) and 120593119901119895 (119896)(2)e Processing of Quantitative Decision VariablesQuanti-tative decision variables are determined values However thedimension of the index is different In order to eliminate thedimension the quantitative data are needed to be normalizedand converted into standardized data Then the normalizedvariables are treated as a kind of special fuzzy number whichcan be used to calculate the similarity of the correspondingquantitative decision variables between two customers

120593119901119894 = 120593119901119894 minusmin1le119899le|119873| 120593119901119899 max1le119899le|119873| 120593119901119899 minusmin1le119899le|119873| 120593119901119899

for 119901 isin 2 7 119894 isin 1198730 (7)

120593119901119894 and 120593119901119899 are the actual value of the quantitative decisionvariables corresponding to customers 119894 and customers 119899119873 isthe number of customers On the basis of the standardizationof the quantitative decision the similarity can be calculatedby

119883119901119894119895 = 1 minus 10038161003816100381610038161003816120593119901119894 minus 120593119901119895 10038161003816100381610038161003816max1le119899le|119873| 120593119901119899 minusmin1le119899le|119873| 120593119901119899

for 119901 isin 2 7 119894 isin 1198730(8)

Additionally 1205931119894 represents the weight of the goods itcan determine the number of customer groups And it isused as the termination condition of the customer clusteringalgorithm in this article So we do nothing for this variable

However the comprehensive similarity for customersis not simply adding up the similarity for each decisionvariable The variables present different influence for thedelivery quality In this paper we consider the importanceof each decision variable and use a mathematical method tocalculate the weight coefficient for each parameter A fuzzymathematics method is introduced to determine the weightof each decision variable

(3) Weight Calculation of Decision Variables After obtainingthe order information the managers (experts) of the dis-tribution center are required to submit evaluation reportsabout the delivery service index The evaluation value of theimpact of each decision variable for the customers is alsogiven out by the managers The expertsrsquo evaluation languageis generally described as the influence of decision variableson customer satisfaction We divide the areas of influenceinto five evaluations specifically largest large medium lowand lowest To calculate the numerical value the languageevaluation is converted to triangular fuzzy numbers andshown in Table 2

Based on the importance of the decision variables for cus-tomer satisfaction the weight of the corresponding decisionvariables can be calculated For convenience we denote somekey parameters as follows119882119906119901(119906 = 1 2 119903 119901 = 1 2 119898) represents theevaluation of decision variables 119901 for decision maker 119906where 119903 is the total number of decision makers and 119898 isthe number of decision variables And it can be denotedby triangular fuzzy number 119882119906119901 = 119886119906119901 119887119906119901 119888119906119901 120596119901 isthe weight value for decision 119901 119885119894119901 = 119860 119894119901 119861119894119901 119862119894119901(119905 =1 2 |119873| 119901 = 1 2 119898) where |119873| is the total numberof the covering customers119885119894119901 represents the comprehensiveevaluation index of decision variables 119901 by all decisionmakers for customer 119894

119860 119894119901 = 1119903 otimes 119903sum119906=1119886119906119901 (9)

119861119894119901 = 1119903 otimes 119903sum119906=1119887119906119901 (10)

119862119894119901 = 1119903 otimes 119903sum119906=1119888119906119901 (11)

6 Mathematical Problems in Engineering

Table 2 Membership function of evaluation linguistic variables

Language terminology Triangular fuzzy number Judgment scalelargest (07511) 1large (050751) 075medium (02505075) 05small (002505) 025smallest (00025) 0

The comprehensive evaluation index of decision variables119901 for customer 119894 can be calculated as

119885119894119901 = 1119903 otimes 119903sum119906=1119882119906119901 (12)

To simplify we define the comprehensive membership ofdecision variable 119901 for customer 119894 as follows

119901119894 = 15 (119860 119894119901 + 2119861119894119901 + 2119862119894119901) (13)

The comprehensive evaluation value of a decision variable119901 denotes the average value of all customers such as

119901 = 1119873 119873sum119894=1119901119894 for 120596119901 gt 0 (14)

Considering that 120596119901 should satisfy the constraint sum119898119901=1 120596119901 =1 so we calculate 120596119901 as follows120596119901 = 119901sum119898119901=1 119901 119901 = 1 2 119898 (15)

Then the similarity between customer 119894 and customer 119895can be denoted as follows

119878119894119895 = 119898sum119901=1

120596119901119883119901119894119895 119894 119895 isin 1198730 (16)

Finally we get the comprehensive similarity betweencustomers which can be represented by a similarity matrixjust as follows

119865 = (119878119894119895)119873times119873 =(((

1 11987812 sdot sdot sdot 119878111987311987821 sdot sdot sdot 1198782119873minus1 1198782119873 d

1198781198731 119878119873119873minus1 1)))

(17)

32 Second Phase Model For the second phase model weshould assign the appropriate route for the customer en-groups In this part we consider the energy conservation andemission reduction and target to reduce the comprehensivecosts There we use the comprehensive emission measure-ment model to calculate the energy consumption and carbonemission [24] At first we regard the travel speed as fixedvariables and establish a conventional model On this basiswe consider the speed as time variation variables and reformthe model which is consistent with the actual situation Tosolve the problem we design a genetic algorithm

To provide a precise statement of this problem we definethe parameters and indices shown in Table 3

For the en-group route optimization we consider twodecision variables as follows

119909119897119896119894119895 = 1 if the vehicle 119897119896 travel on the link (119894 119895)0 otherwise

(18)

119911119897119896119894 = 1 if the customer 119894 need to be serviced by the 119896th vehicle for type 1198970 otherwise

(19)

The model can be formulated as follows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(20)

Mathematical Problems in Engineering 7

Table 3 Sets indices and parameters in the C-VRPESER

Notations Detailed Definition119873 Where119873 = 0 1 2 sdot sdot sdot 119899 0 represents the depot1198730 = 119873 0 = 1 2 sdot sdot sdot 119899is the set of the customers which should be delivered119860 119860 = (119894 119895) 119894 119895 isin 119873 represents the arcs119879119877 Set of of customers in the same customer group 119877119897 Type of the vehicle119897119896 119896th vehicle with 119897 type119876119897119890 Maximum capacity of each type vehicles119894 isin 1198730 Number of customer119902119894 Demand of customers 119894119904119894 Time of vehicle arrive at customer 119894[119886119894 119887119894] Desired preferred time window for customer 119894 to be serviced119905119894 Service time of customer 119894 need119889119894119895 Distance between the customer 119894 and 119895

V119894119895 Speed of the vehicle travel on the link (119894 119895)119902119897119896119894119895 Load of the vehicle 119897119896 travel on the link (119894 119895)119876119897119888 Light weight of the 119897 type vehicle119888119900 Per unit cost of fuel119888119890 Per unit cost of carbon emissions (carbon tax)120575119888 Fuel emission factor119888119897 Fixed costs of the vehicle used119890119896 Departure time of the vehicle k from the depot119897119896 Latest arrival time at the depot after services the customers1198881 Penalty coefficient for earliness arrival1198882 Penalty coefficient for delay delivery119878119894 Service time of customer 119894st sum

119897119896isin119871119870

sum119895isin119879119877

1199091198971198960119895 = 1 (21)

sum119897119896isin119871119870

sum119895isin119879119877

1199091198971198961198950 = 1 (22)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 = 1 forall119894 isin 119879119877 (23)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119895ℎ minus sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896ℎ119895 = 0 forallℎ isin 119879119877 (24)

sum119894isin119879119877

119902119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119876119890 (25)

119886119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119904119897119896119894 le 119887119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 forall119894 isin 119879119877 (26)

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119889119894119895V119894119895 ) le 119904119897119896119894 (27)

119909119897119896119894119895 isin 0 1 (28)

119910119897119896119894 ge 0 (29)

The objective function (20) is to minimize the totalrouting cost including fixed cost energy consumption costcarbon emission cost and punishment for time windowsConstraints (21) and (22) guarantee that all service vehiclesstart from the depot and finally return to the depot foronce Constraint (23) ensures each vehicle travels on the arc(V119894 V119895) nomore than once Constraint (24) is the flow balanceand shows that the number of the vehicles which arrive atthe customer is the same as those which depart Constraint(25) ensures the loading customer demands no more thanthe vehicle capacity Inequality (26) is the time windowconstraint Constraint (27) represents the time limit forvehicle travel horizon Constraint (28) is the binary variableconstraint Constraint (29) represents the nonnegative oftime

33 Model Reformulation The above model is generatedbased on the assumption that the drive speed is invariableHowever in reality the speed is influenced by a lot of elementsTo make the model more practical we assume that thespeed is a ladder type change variable This paper supposesthe vehicles traveling speed as shown in Figure 2 On thisbasis we build the time-varying velocity distribution vehiclerouting optimization model considering the velocity change

The speed is assumed as a step function of time asFigure 2 And during different time intervals the vehiclesrsquo

8 Mathematical Problems in Engineering

20253035404550556065707580

50 10 15 20 24

Figure 2 The speed trend diagram

travel speed is different To model convince the runningtime is divided into 119868 interval according different extentsspeed which can be represented by 1198791 1198792 119879119868 The 119898thtime interval is indicated as [119905119898 119905119898+1] in each time intervalvelocity is constant

The time of vehicles running on the path (119894 119895) is a functionof the time 119897119905119894 that is the vehiclesrsquo departing time from thecustomer 119894 The speed of vehicles running on the path oftencrosses more than one speed range It is assumed that thevehicle will cross 119901 + 1 time intervals It is clear that thevehicle has 119901 different speed Assume the vehicle derived thecustomer 119894 in the119898 time interval with the speed V119898119894119895 (119897119905119894) thenthe vehicle speed turns V119898+1119894119895 (119897119905119894) in the next time intervaland we denote the vehicle speed as V119898+119901119894119895 (119897119905119894)when the vehiclederives the link (119894 119895) And the distance for the link (119894 119895) isdenoted by 119889119898+119901119894119895 (119897119905119894) And the total time 119905119905119894 for the vehicledrived across the link (119894 119895) can be represented by the timefunction 119897119905119894 as follows

119905119905119894119895 (119897119905119894) = 119889119898119894119895V119898119894119895 (119897119905119894) + 119889119898+1119894119895

V119898+1119894119895 (119897119905119894) + sdot sdot sdot + 119889119898+119901119894119895V119898+119901119894119895 (119897119905119894) (30)

Considering the variety of the vehicle traveling speed wecan represent the fuel consumption function as follows

119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)= 119868sum119902=1

119874119865119894119895 (119876119897119888 + 119902119894119895 V119898+119902119894119895 119889119898+119902119894119895 ) (31)

If we assumed that the 119896th vehicle of the 119897 kind beginsthe task at time 119910119897119896 the objective function can be denoted asfollows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]

+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(32)

The value of V119894119895 depends on the time interval The timeconstraint is represented as follows

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119905119905119894119895 (119904119897119896119894 + 119905119894)) le 119904119897119896119895 (33)

The vehicle service time cannot be earlier than the starttime of the distribution center not later than the time of thedelivery center

1199051 le 119910119897119896 le 119905119868 (34)

4 Algorithm

To present the customers-oriented vehicle routing problemwith environment consideration a two-phase model is for-mulated We design two algorithms to solve the problemFirst we design a fuzzy system clustering algorithm to realizethe customers grouping That algorithm is mainly used toconsider the customers demand attributes including quan-titative and qualitative attributes The second phase model isan en-group VRP that need to design a heuristic algorithm togain the en-group optimal delivery routes Here we comparethe performance of several heuristic algorithms as shown inTable 4 In view of the robustness and global optimization ofgenetic algorithm we design a genetic algorithm to solve theproblem

41 Fuzzy System Clustering Algorithm On the basis ofthe similarity matrix we design a fuzzy system clusteringalgorithm to cluster the customer into different groups whichare often used to process the data with quantitative andqualitative data [22] And the specific algorithm procedureis summarized as in Algorithm 1

To obtain the cluster solutions we apply the procedure inFigure 3

Mathematical Problems in Engineering 9

Table 4 Heuristic algorithm performance comparison

Algorithm Parameters Features

Tabu search algorithm(TS)

Tabu sizeNeighborhood structureNumber of solutions

Local optimumHigher solving efficiency

Ant colony algorithm(ACO)

Initial pheromoneInformation increase

Information accumulationDisappearance factorInspiration factor

Good parallelismGlobal optimization features

Time-consuming

Simulated annealingalgorithm (SA)

Initial temperatureTemperature return functionMode of state generation

Sampling principle

Simple implementationFlexible application

The algorithm converges slowly

Genetic algorithm (GA)

Population numberSelection operatorCrossover operatorMutation operator

Strong global optimization abilityParallelism and strong robustness

1 Step 1 Data initialization input the fuzzy equivalent matrix 119865(119896) Let iterations 120587=12 Step 2 Find out the largest elements in the matrix 119865119904(119896)119879 and chose the customer with the smaller number as the targetcustomer and denote it with 1199043 Step 3Mark the selected target customer and delete the corresponding row 119865119904(119896)119879 Specifically once a customer is chosen asthe target customer its column is marked by 119865119904(119896) mark the corresponding elements in row 119865119904(119896)119879 as 04 Step 4 Find out the largest element of matrix 119865119904(119896)119879 denoted the similarity value as 1205961199031199045 (1) If 120596119903119904 le 120582(pre-given threshold value) then go to Step 5 Otherwise calculate the total weight of the customers1198766 (2)Make judgement if 119876 ge 119876119897119890 go to Step 5 Otherwise put customers 119904 and 119903 in same group then delete the correspondingrow 119865119903(119896)119879 of matrix 119865(119896) return to Step 47 Step 5 Set the customer clustering termination conditions if the matrix is empty then clustering process stops else let120587= 120587 + 1 and got to Step 2

Algorithm 1 Fuzzy system clustering algorithm

42 En-Group Genetic Algorithm The proposed model is amixed-integer linear programming model When the scale ofthe problem is expended it is difficult to solve the problemwith accurate algorithm However the GA is suitable to solvemultivariables complex problem with multiple parameters Itiswidely applied in solvingVRPdue to its extensive generalityhigh efficiency and strong robustnessTherefore we deign GAto solve the problemThe key steps are designed as follows

Encoding Natural number cod approach is used tocharacterize chromosomes Chromosome genesrsquo encoding isbased on the results of the customer groups with discontin-uous customer numbers There 0 expresses the distributioncenter and customers are expressed by numbers 1 2 3 sdot sdot sdot 119873Different groups are corresponding different chromosomalgenes The beginning and the end of the genes in the chro-mosome cod sequence must be rdquo0rdquo on behalf of the vehiclewhich departs from the distribution center and finally returnsto the distribution center after completing the deliverytasks

Initial Solution Construct initial population chromo-some cod using MATLAB programming software under

the premise of customer grouping Specific we randomlygive the initial solution as 119901119900119901(119894 2 119899 minus 1) =119870119867119911(rand(119899119906119898119890119897(119870119867119911)))119901119900119901(119894 1) = 0 and 119901119900119901(119894 119899) =0 such as 119870119867119911(rand(119899119906119898119890119897(119870119867119911)))=[0 1 4 5 9 12 18 0]And generate the initial population as a matrix with thedimension 119901119900119901119904119894119911119890 lowast (119899 + 2)

Fitness Function In this paper the objective functionvalue is used to measure the fitness value as follows

119891119896 = 1 minus (120583119896 minusmin120583)(max 120583 minusmin 120583) (35)

Selection Operator It is individual selection according toroulette wheel if the fitness of an individual is denoted by 119891119896then the probability of being selected can be represented bythe following

119901119896 = 119891119896sum119870119896=1 119891119896 (36)

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 3

considering the carbon emissions during the vehicle oper-ation a tabu search algorithm was designed to solve theproblem Most of the VRP were formulated as mixed inter-linear models and heuristic algorithms are designed to solvethe problem The existing studies use the emission modelwhich include MEET and National Atmospheric EmissionInventory [5 17]

The vehicle routing problem with time windows wasproposed by Avelsbergh [18] and widely studied by theresearchers Most of the studies focus on design better algo-rithms to solve the problem such as GA TS etc [8 19 20]Specifically Calvete [21] established a multiobjective opti-mization model for the vehicle routing problem (VRPSTW)which considered soft time windows and designed a goalplanning method to solve the problem Miguel [3] pro-posed an iterative route construction and improvementalgorithm (IRCI) based on path generation algorithm to solvethe VRPTW problem Cihan et al [2] proposed a time-dependent dual-target VRP In the established model it wasassumed that the path selection will be influenced by the timewindows

While many contributions have been made to the VRPwith time windows and the green vehicle problem toour knowledge fewer studies have examined the customerdemands However processing the customer orders andquick response to varieties of customer demands for deliver-ing the goods via suitable freight vehicles appears to be a keyelement for efficient logistics service In previous literature[22] it has been pointed out that customer-based orderclustering as well as the vehicle routing strategies should becoordinated to solve the vehicle problem With the develop-ment of the VRP a lot of studies focus on mathematicallyformulating the corresponding problems In our research wefirst aim to reasonably group the customers based on thedifferences of customer demand On this basis we target tooptimize the vehiclesrsquo travel cost including fixed cost energyconsumption carbon emission and punishment for timewindows For simplicity the problem in this study is calledthe customer-oriented vehicle routing problem consideringenergy conservation and emission reduction (C-VRPESER)

In this research we point out seven factors which willaffect the customer satisfaction based on the studies [23] suchas incorporating location analysis service quality servicetime external similarity factor etc Furthermore we establishthe fuzzy similar matrix to cluster the customers with similarproduct demand On the basis of the clustering results weformulate the en-group optimization model and design a GAto solve the model This study aims to make the followingtheoretical contribution to VRP which considers customerdemand as well as energy conservation and emission reduc-tion Then the following two-phase optimization model isproposed to solve the C-VRPESER

3 Problem Description andModel Formulation

In this paper we consider a transportation network that isrepresented by a directed complete graph 119872 = (119873119860) Thevertex set119873 is composed by a depot 0 and a set of customers

1198730 = 119873 0 = 1 2 3 119899 The number of homogeneousvehicles is a deterministic exogenous parameter and the setof vehicles is represented by 119870 = 1 2 3 119898 The capacityof each vehicle is equal to 119876119890 Each vehicle 119896 has the earliestdeparture time 119890119896 from the depot after unloaded and thelatest arrival time 119897119896 at the depot after finishing the services ofcustomers In practically we can get the customer 119894 demandfor goods from the previous order denoted by 1205931119894 and we alsocan get the other demand for different attributes1205932119894 1205933119894 1205934119894 1205935119894 1205936119894 and 1205937119894 And to simplify the problem we divide the sevenattributes into two types quantitative and qualitative For thequalitative demand we use triangle fuzzy number (1205721199011119905 1205721199012119905and 1205721199013119905) to present five demand levels (ldquovery highrdquo ldquohighrdquoldquomediumrdquo ldquolowrdquo and ldquovery lowrdquo) in which 119905 presents thecustomer number and 119901 On the other hand mathematicalmethod is used to eliminate the dimensional and evaluatethe similarity On this basis we cluster the customers intodifferent groups as shown in Figure 1 The customers canbe clustered into three groups which can be presented bydifferent shapes triangle quadrilateral and pentagon Anden-route goods delivery paths are designed according to thecustomersrsquo location and time windows which decide thedelivery sequence And then we connect each customer witha line with arrowheads And in each group the vehicle shouldstart from the depot (the vehicle remained at the depot) andreturn to the depot after serving all the en-group customers

The optimization goal of the problem is to minimize thetotal cost on the basis of customersrsquo group that realizes the en-group distribution vehicle routing optimization By reason-ably scheduling different types of vehicles for each customergroup the utilization rate of vehicles can be improved Andit also can meet the restriction of vehicle load and time limitThe optimal objective includes the vehicle fixed costs timepenalty cost fuel cost and carbon emissions On this waywe can realize minimizing the total cost considering energyconservation and emissions reduction and reducing the costof comprehensive optimization

In the next sections this paper focuses on formulatinga two-phase model to solve the problem The first model isgenerated to cluster the customers into groups and the secondphase model aims to give out an optimal route en-groupwhich can reduce the energy consumption and emission

31 First Phase Model The customers demand attributesconsist of several indicators part of them is quantitativeindicators such as time and location Others are qualitativeindexes such as external and internal characteristics ofcommodity customer satisfaction etc Through consideringthe related factors of distribution enterprisers service qualityand customer satisfaction we choose seven indicators asdecision variables (1) physical properties (2) geographicalposition (3) service quality (4) product value (5) type ofgoods (6) security of goods and (7) time windows Inorder to quantify the decision variables we define the sevendecision variables in Table 1 And the above indicators canrepresent most of the customerrsquos demand attributes

However the decision variables should eliminate dimen-sion before using For modeling convenience this paper

4 Mathematical Problems in Engineering

Table 1 Definition of customerrsquos demand attributes

Variables Variabledefinitions Variable explanation

1205931119894 Physicalproperties

Number of delivery of goods weight volume this article take the quality asreference the vehiclersquos rated load determines the number of delivery vehicles1205932119894 Geographical

positionCustomersrsquo location customers with similar geography have larger possibility be

divided into same groupso that realize the joint distribution

1205933119894 Service qualityIncluding reaction time service attitude and service ability Customer demands forservice quality are differentand customers with similar service can be divided into

same groups1205934119894 Product value Value of products generally refers to the market valueThe high value cargo can beseparated from others this article define the value of the market price of five levels

1205935119894 Type of goodsTypes of goods and external similarity with other goods If the external similarity is

higher that it convenient to load and unload the cargo which will improve thedelivery efficient1205936119894 Security of

goodsThis property is mainly denote the customer security requirements(eg fragile)

Especially for the dangerous cargothe safety is very important1205937119894 Pressed for time Present the customer request delivery time limit of time the latest deadline we canarrange the customer according to their time windows

Depot

1

6

14

2

4

12

3

513

11

810

9

7

Figure 1 Illustration of the customer groups and distribution routes

uses the fuzzy clustering method to cluster the customersinto appropriate groups which have the similar demandattributes Taking into considering the above seven decisionvariables the qualitative indicators include 1205933119894 1205934119894 1205935119894 and 1205936119894and the other three variables are quantitative indicators

In order to calculate the qualitative data the linguisticterms ldquovery highrdquo ldquohighrdquo ldquomediumrdquo ldquolowrdquo and ldquovery lowrdquoare introduced to represent the qualitative decision variablesSpecifically language ldquovery highrdquo can be represented by

triangular fuzzy number (075 1 1) which indicates that thecustomerrsquos demand of the service level is much higher thanother customers Similarly the language ldquohighrdquo ldquomediumrdquoldquolowrdquo and ldquovery lowrdquo can be represented by (05 075 1)(025 05 075) (0 025 05) and (00025) Based on thelanguage description we translate the linguistic variables intotriangular fuzzy numbers so that we can calculate the com-prehensive similarity between the customers The linguisticvariables represent the correlation between this indicator and

Mathematical Problems in Engineering 5

customer satisfaction In this way each customerrsquos demandindicators can be denoted by triangular fuzzy number (12057211990111989411205721199011198942 1205721199011198943) which can be denoted as follows120593119901119894 = [1205721199011198941 1205721199011198942 1205721199011198943] (1)

We can get the demand attributers from the orderswhich will be pregiven by the customers However we shouldtransfer the linguistic variables and the quantitative data asstandardized data so that we can use them to calculate thesimilarity And the procedure is shown in below

(1)e Processing of Qualitative DecisionVariables In the firststep we use the standard deviation transformation to processthe qualitative data which can guarantee the data availabilityAnd for the second step we introduce the range conversion tostandardize the quantitative variables The constraint (1) canbe firstly transformed as follows120593119901119894 = [1199011198941 1199011198942 1199011198943] 119894 isin 1198730 (2)

where

120593119901119894 = 100381610038161003816100381610038161003816120572119901119894119905 minus 120572119901119894119905100381610038161003816100381610038161003816119878119901119905 119894 isin 1198730 (3)

120572119901119894119905 = sum119873119894=1 120572119901119894119905119873 119905 = 1 2 3 119894 isin 1198730 (4)

However the indicator 120572119901119894119905 is the mean value 119878119901119905 is thestandard deviation and |119873| is the total customer numberThevalue range of 119901 = 3 4 5 6119878119901119905 = [[[

sum119873119894=1 (120572119901119894119905 minus 120572119901119894119905)2|119873| minus 1 ]]]12 119905 = 1 2 3 119894 isin 1198730 (5)

Then we can calculate the similarity of qualitativedecision-making variables with Hemingwayrsquos distancemethod as follows

119883119901119894119895 = 1 minus (sum3119905=1 10038161003816100381610038161003816119901119894119905 minus 11990111989511990510038161003816100381610038161003816)3 119894 119895 isin 1198730 (6)

119883119901119894119895 represents the qualitative correlation between thedifferent customers and the range of the value is during [0 1]119883119901119894119895 indicates relationship of attributors 120593119901119894 (119896) and 120593119901119895 (119896)(2)e Processing of Quantitative Decision VariablesQuanti-tative decision variables are determined values However thedimension of the index is different In order to eliminate thedimension the quantitative data are needed to be normalizedand converted into standardized data Then the normalizedvariables are treated as a kind of special fuzzy number whichcan be used to calculate the similarity of the correspondingquantitative decision variables between two customers

120593119901119894 = 120593119901119894 minusmin1le119899le|119873| 120593119901119899 max1le119899le|119873| 120593119901119899 minusmin1le119899le|119873| 120593119901119899

for 119901 isin 2 7 119894 isin 1198730 (7)

120593119901119894 and 120593119901119899 are the actual value of the quantitative decisionvariables corresponding to customers 119894 and customers 119899119873 isthe number of customers On the basis of the standardizationof the quantitative decision the similarity can be calculatedby

119883119901119894119895 = 1 minus 10038161003816100381610038161003816120593119901119894 minus 120593119901119895 10038161003816100381610038161003816max1le119899le|119873| 120593119901119899 minusmin1le119899le|119873| 120593119901119899

for 119901 isin 2 7 119894 isin 1198730(8)

Additionally 1205931119894 represents the weight of the goods itcan determine the number of customer groups And it isused as the termination condition of the customer clusteringalgorithm in this article So we do nothing for this variable

However the comprehensive similarity for customersis not simply adding up the similarity for each decisionvariable The variables present different influence for thedelivery quality In this paper we consider the importanceof each decision variable and use a mathematical method tocalculate the weight coefficient for each parameter A fuzzymathematics method is introduced to determine the weightof each decision variable

(3) Weight Calculation of Decision Variables After obtainingthe order information the managers (experts) of the dis-tribution center are required to submit evaluation reportsabout the delivery service index The evaluation value of theimpact of each decision variable for the customers is alsogiven out by the managers The expertsrsquo evaluation languageis generally described as the influence of decision variableson customer satisfaction We divide the areas of influenceinto five evaluations specifically largest large medium lowand lowest To calculate the numerical value the languageevaluation is converted to triangular fuzzy numbers andshown in Table 2

Based on the importance of the decision variables for cus-tomer satisfaction the weight of the corresponding decisionvariables can be calculated For convenience we denote somekey parameters as follows119882119906119901(119906 = 1 2 119903 119901 = 1 2 119898) represents theevaluation of decision variables 119901 for decision maker 119906where 119903 is the total number of decision makers and 119898 isthe number of decision variables And it can be denotedby triangular fuzzy number 119882119906119901 = 119886119906119901 119887119906119901 119888119906119901 120596119901 isthe weight value for decision 119901 119885119894119901 = 119860 119894119901 119861119894119901 119862119894119901(119905 =1 2 |119873| 119901 = 1 2 119898) where |119873| is the total numberof the covering customers119885119894119901 represents the comprehensiveevaluation index of decision variables 119901 by all decisionmakers for customer 119894

119860 119894119901 = 1119903 otimes 119903sum119906=1119886119906119901 (9)

119861119894119901 = 1119903 otimes 119903sum119906=1119887119906119901 (10)

119862119894119901 = 1119903 otimes 119903sum119906=1119888119906119901 (11)

6 Mathematical Problems in Engineering

Table 2 Membership function of evaluation linguistic variables

Language terminology Triangular fuzzy number Judgment scalelargest (07511) 1large (050751) 075medium (02505075) 05small (002505) 025smallest (00025) 0

The comprehensive evaluation index of decision variables119901 for customer 119894 can be calculated as

119885119894119901 = 1119903 otimes 119903sum119906=1119882119906119901 (12)

To simplify we define the comprehensive membership ofdecision variable 119901 for customer 119894 as follows

119901119894 = 15 (119860 119894119901 + 2119861119894119901 + 2119862119894119901) (13)

The comprehensive evaluation value of a decision variable119901 denotes the average value of all customers such as

119901 = 1119873 119873sum119894=1119901119894 for 120596119901 gt 0 (14)

Considering that 120596119901 should satisfy the constraint sum119898119901=1 120596119901 =1 so we calculate 120596119901 as follows120596119901 = 119901sum119898119901=1 119901 119901 = 1 2 119898 (15)

Then the similarity between customer 119894 and customer 119895can be denoted as follows

119878119894119895 = 119898sum119901=1

120596119901119883119901119894119895 119894 119895 isin 1198730 (16)

Finally we get the comprehensive similarity betweencustomers which can be represented by a similarity matrixjust as follows

119865 = (119878119894119895)119873times119873 =(((

1 11987812 sdot sdot sdot 119878111987311987821 sdot sdot sdot 1198782119873minus1 1198782119873 d

1198781198731 119878119873119873minus1 1)))

(17)

32 Second Phase Model For the second phase model weshould assign the appropriate route for the customer en-groups In this part we consider the energy conservation andemission reduction and target to reduce the comprehensivecosts There we use the comprehensive emission measure-ment model to calculate the energy consumption and carbonemission [24] At first we regard the travel speed as fixedvariables and establish a conventional model On this basiswe consider the speed as time variation variables and reformthe model which is consistent with the actual situation Tosolve the problem we design a genetic algorithm

To provide a precise statement of this problem we definethe parameters and indices shown in Table 3

For the en-group route optimization we consider twodecision variables as follows

119909119897119896119894119895 = 1 if the vehicle 119897119896 travel on the link (119894 119895)0 otherwise

(18)

119911119897119896119894 = 1 if the customer 119894 need to be serviced by the 119896th vehicle for type 1198970 otherwise

(19)

The model can be formulated as follows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(20)

Mathematical Problems in Engineering 7

Table 3 Sets indices and parameters in the C-VRPESER

Notations Detailed Definition119873 Where119873 = 0 1 2 sdot sdot sdot 119899 0 represents the depot1198730 = 119873 0 = 1 2 sdot sdot sdot 119899is the set of the customers which should be delivered119860 119860 = (119894 119895) 119894 119895 isin 119873 represents the arcs119879119877 Set of of customers in the same customer group 119877119897 Type of the vehicle119897119896 119896th vehicle with 119897 type119876119897119890 Maximum capacity of each type vehicles119894 isin 1198730 Number of customer119902119894 Demand of customers 119894119904119894 Time of vehicle arrive at customer 119894[119886119894 119887119894] Desired preferred time window for customer 119894 to be serviced119905119894 Service time of customer 119894 need119889119894119895 Distance between the customer 119894 and 119895

V119894119895 Speed of the vehicle travel on the link (119894 119895)119902119897119896119894119895 Load of the vehicle 119897119896 travel on the link (119894 119895)119876119897119888 Light weight of the 119897 type vehicle119888119900 Per unit cost of fuel119888119890 Per unit cost of carbon emissions (carbon tax)120575119888 Fuel emission factor119888119897 Fixed costs of the vehicle used119890119896 Departure time of the vehicle k from the depot119897119896 Latest arrival time at the depot after services the customers1198881 Penalty coefficient for earliness arrival1198882 Penalty coefficient for delay delivery119878119894 Service time of customer 119894st sum

119897119896isin119871119870

sum119895isin119879119877

1199091198971198960119895 = 1 (21)

sum119897119896isin119871119870

sum119895isin119879119877

1199091198971198961198950 = 1 (22)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 = 1 forall119894 isin 119879119877 (23)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119895ℎ minus sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896ℎ119895 = 0 forallℎ isin 119879119877 (24)

sum119894isin119879119877

119902119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119876119890 (25)

119886119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119904119897119896119894 le 119887119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 forall119894 isin 119879119877 (26)

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119889119894119895V119894119895 ) le 119904119897119896119894 (27)

119909119897119896119894119895 isin 0 1 (28)

119910119897119896119894 ge 0 (29)

The objective function (20) is to minimize the totalrouting cost including fixed cost energy consumption costcarbon emission cost and punishment for time windowsConstraints (21) and (22) guarantee that all service vehiclesstart from the depot and finally return to the depot foronce Constraint (23) ensures each vehicle travels on the arc(V119894 V119895) nomore than once Constraint (24) is the flow balanceand shows that the number of the vehicles which arrive atthe customer is the same as those which depart Constraint(25) ensures the loading customer demands no more thanthe vehicle capacity Inequality (26) is the time windowconstraint Constraint (27) represents the time limit forvehicle travel horizon Constraint (28) is the binary variableconstraint Constraint (29) represents the nonnegative oftime

33 Model Reformulation The above model is generatedbased on the assumption that the drive speed is invariableHowever in reality the speed is influenced by a lot of elementsTo make the model more practical we assume that thespeed is a ladder type change variable This paper supposesthe vehicles traveling speed as shown in Figure 2 On thisbasis we build the time-varying velocity distribution vehiclerouting optimization model considering the velocity change

The speed is assumed as a step function of time asFigure 2 And during different time intervals the vehiclesrsquo

8 Mathematical Problems in Engineering

20253035404550556065707580

50 10 15 20 24

Figure 2 The speed trend diagram

travel speed is different To model convince the runningtime is divided into 119868 interval according different extentsspeed which can be represented by 1198791 1198792 119879119868 The 119898thtime interval is indicated as [119905119898 119905119898+1] in each time intervalvelocity is constant

The time of vehicles running on the path (119894 119895) is a functionof the time 119897119905119894 that is the vehiclesrsquo departing time from thecustomer 119894 The speed of vehicles running on the path oftencrosses more than one speed range It is assumed that thevehicle will cross 119901 + 1 time intervals It is clear that thevehicle has 119901 different speed Assume the vehicle derived thecustomer 119894 in the119898 time interval with the speed V119898119894119895 (119897119905119894) thenthe vehicle speed turns V119898+1119894119895 (119897119905119894) in the next time intervaland we denote the vehicle speed as V119898+119901119894119895 (119897119905119894)when the vehiclederives the link (119894 119895) And the distance for the link (119894 119895) isdenoted by 119889119898+119901119894119895 (119897119905119894) And the total time 119905119905119894 for the vehicledrived across the link (119894 119895) can be represented by the timefunction 119897119905119894 as follows

119905119905119894119895 (119897119905119894) = 119889119898119894119895V119898119894119895 (119897119905119894) + 119889119898+1119894119895

V119898+1119894119895 (119897119905119894) + sdot sdot sdot + 119889119898+119901119894119895V119898+119901119894119895 (119897119905119894) (30)

Considering the variety of the vehicle traveling speed wecan represent the fuel consumption function as follows

119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)= 119868sum119902=1

119874119865119894119895 (119876119897119888 + 119902119894119895 V119898+119902119894119895 119889119898+119902119894119895 ) (31)

If we assumed that the 119896th vehicle of the 119897 kind beginsthe task at time 119910119897119896 the objective function can be denoted asfollows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]

+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(32)

The value of V119894119895 depends on the time interval The timeconstraint is represented as follows

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119905119905119894119895 (119904119897119896119894 + 119905119894)) le 119904119897119896119895 (33)

The vehicle service time cannot be earlier than the starttime of the distribution center not later than the time of thedelivery center

1199051 le 119910119897119896 le 119905119868 (34)

4 Algorithm

To present the customers-oriented vehicle routing problemwith environment consideration a two-phase model is for-mulated We design two algorithms to solve the problemFirst we design a fuzzy system clustering algorithm to realizethe customers grouping That algorithm is mainly used toconsider the customers demand attributes including quan-titative and qualitative attributes The second phase model isan en-group VRP that need to design a heuristic algorithm togain the en-group optimal delivery routes Here we comparethe performance of several heuristic algorithms as shown inTable 4 In view of the robustness and global optimization ofgenetic algorithm we design a genetic algorithm to solve theproblem

41 Fuzzy System Clustering Algorithm On the basis ofthe similarity matrix we design a fuzzy system clusteringalgorithm to cluster the customer into different groups whichare often used to process the data with quantitative andqualitative data [22] And the specific algorithm procedureis summarized as in Algorithm 1

To obtain the cluster solutions we apply the procedure inFigure 3

Mathematical Problems in Engineering 9

Table 4 Heuristic algorithm performance comparison

Algorithm Parameters Features

Tabu search algorithm(TS)

Tabu sizeNeighborhood structureNumber of solutions

Local optimumHigher solving efficiency

Ant colony algorithm(ACO)

Initial pheromoneInformation increase

Information accumulationDisappearance factorInspiration factor

Good parallelismGlobal optimization features

Time-consuming

Simulated annealingalgorithm (SA)

Initial temperatureTemperature return functionMode of state generation

Sampling principle

Simple implementationFlexible application

The algorithm converges slowly

Genetic algorithm (GA)

Population numberSelection operatorCrossover operatorMutation operator

Strong global optimization abilityParallelism and strong robustness

1 Step 1 Data initialization input the fuzzy equivalent matrix 119865(119896) Let iterations 120587=12 Step 2 Find out the largest elements in the matrix 119865119904(119896)119879 and chose the customer with the smaller number as the targetcustomer and denote it with 1199043 Step 3Mark the selected target customer and delete the corresponding row 119865119904(119896)119879 Specifically once a customer is chosen asthe target customer its column is marked by 119865119904(119896) mark the corresponding elements in row 119865119904(119896)119879 as 04 Step 4 Find out the largest element of matrix 119865119904(119896)119879 denoted the similarity value as 1205961199031199045 (1) If 120596119903119904 le 120582(pre-given threshold value) then go to Step 5 Otherwise calculate the total weight of the customers1198766 (2)Make judgement if 119876 ge 119876119897119890 go to Step 5 Otherwise put customers 119904 and 119903 in same group then delete the correspondingrow 119865119903(119896)119879 of matrix 119865(119896) return to Step 47 Step 5 Set the customer clustering termination conditions if the matrix is empty then clustering process stops else let120587= 120587 + 1 and got to Step 2

Algorithm 1 Fuzzy system clustering algorithm

42 En-Group Genetic Algorithm The proposed model is amixed-integer linear programming model When the scale ofthe problem is expended it is difficult to solve the problemwith accurate algorithm However the GA is suitable to solvemultivariables complex problem with multiple parameters Itiswidely applied in solvingVRPdue to its extensive generalityhigh efficiency and strong robustnessTherefore we deign GAto solve the problemThe key steps are designed as follows

Encoding Natural number cod approach is used tocharacterize chromosomes Chromosome genesrsquo encoding isbased on the results of the customer groups with discontin-uous customer numbers There 0 expresses the distributioncenter and customers are expressed by numbers 1 2 3 sdot sdot sdot 119873Different groups are corresponding different chromosomalgenes The beginning and the end of the genes in the chro-mosome cod sequence must be rdquo0rdquo on behalf of the vehiclewhich departs from the distribution center and finally returnsto the distribution center after completing the deliverytasks

Initial Solution Construct initial population chromo-some cod using MATLAB programming software under

the premise of customer grouping Specific we randomlygive the initial solution as 119901119900119901(119894 2 119899 minus 1) =119870119867119911(rand(119899119906119898119890119897(119870119867119911)))119901119900119901(119894 1) = 0 and 119901119900119901(119894 119899) =0 such as 119870119867119911(rand(119899119906119898119890119897(119870119867119911)))=[0 1 4 5 9 12 18 0]And generate the initial population as a matrix with thedimension 119901119900119901119904119894119911119890 lowast (119899 + 2)

Fitness Function In this paper the objective functionvalue is used to measure the fitness value as follows

119891119896 = 1 minus (120583119896 minusmin120583)(max 120583 minusmin 120583) (35)

Selection Operator It is individual selection according toroulette wheel if the fitness of an individual is denoted by 119891119896then the probability of being selected can be represented bythe following

119901119896 = 119891119896sum119870119896=1 119891119896 (36)

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

4 Mathematical Problems in Engineering

Table 1 Definition of customerrsquos demand attributes

Variables Variabledefinitions Variable explanation

1205931119894 Physicalproperties

Number of delivery of goods weight volume this article take the quality asreference the vehiclersquos rated load determines the number of delivery vehicles1205932119894 Geographical

positionCustomersrsquo location customers with similar geography have larger possibility be

divided into same groupso that realize the joint distribution

1205933119894 Service qualityIncluding reaction time service attitude and service ability Customer demands forservice quality are differentand customers with similar service can be divided into

same groups1205934119894 Product value Value of products generally refers to the market valueThe high value cargo can beseparated from others this article define the value of the market price of five levels

1205935119894 Type of goodsTypes of goods and external similarity with other goods If the external similarity is

higher that it convenient to load and unload the cargo which will improve thedelivery efficient1205936119894 Security of

goodsThis property is mainly denote the customer security requirements(eg fragile)

Especially for the dangerous cargothe safety is very important1205937119894 Pressed for time Present the customer request delivery time limit of time the latest deadline we canarrange the customer according to their time windows

Depot

1

6

14

2

4

12

3

513

11

810

9

7

Figure 1 Illustration of the customer groups and distribution routes

uses the fuzzy clustering method to cluster the customersinto appropriate groups which have the similar demandattributes Taking into considering the above seven decisionvariables the qualitative indicators include 1205933119894 1205934119894 1205935119894 and 1205936119894and the other three variables are quantitative indicators

In order to calculate the qualitative data the linguisticterms ldquovery highrdquo ldquohighrdquo ldquomediumrdquo ldquolowrdquo and ldquovery lowrdquoare introduced to represent the qualitative decision variablesSpecifically language ldquovery highrdquo can be represented by

triangular fuzzy number (075 1 1) which indicates that thecustomerrsquos demand of the service level is much higher thanother customers Similarly the language ldquohighrdquo ldquomediumrdquoldquolowrdquo and ldquovery lowrdquo can be represented by (05 075 1)(025 05 075) (0 025 05) and (00025) Based on thelanguage description we translate the linguistic variables intotriangular fuzzy numbers so that we can calculate the com-prehensive similarity between the customers The linguisticvariables represent the correlation between this indicator and

Mathematical Problems in Engineering 5

customer satisfaction In this way each customerrsquos demandindicators can be denoted by triangular fuzzy number (12057211990111989411205721199011198942 1205721199011198943) which can be denoted as follows120593119901119894 = [1205721199011198941 1205721199011198942 1205721199011198943] (1)

We can get the demand attributers from the orderswhich will be pregiven by the customers However we shouldtransfer the linguistic variables and the quantitative data asstandardized data so that we can use them to calculate thesimilarity And the procedure is shown in below

(1)e Processing of Qualitative DecisionVariables In the firststep we use the standard deviation transformation to processthe qualitative data which can guarantee the data availabilityAnd for the second step we introduce the range conversion tostandardize the quantitative variables The constraint (1) canbe firstly transformed as follows120593119901119894 = [1199011198941 1199011198942 1199011198943] 119894 isin 1198730 (2)

where

120593119901119894 = 100381610038161003816100381610038161003816120572119901119894119905 minus 120572119901119894119905100381610038161003816100381610038161003816119878119901119905 119894 isin 1198730 (3)

120572119901119894119905 = sum119873119894=1 120572119901119894119905119873 119905 = 1 2 3 119894 isin 1198730 (4)

However the indicator 120572119901119894119905 is the mean value 119878119901119905 is thestandard deviation and |119873| is the total customer numberThevalue range of 119901 = 3 4 5 6119878119901119905 = [[[

sum119873119894=1 (120572119901119894119905 minus 120572119901119894119905)2|119873| minus 1 ]]]12 119905 = 1 2 3 119894 isin 1198730 (5)

Then we can calculate the similarity of qualitativedecision-making variables with Hemingwayrsquos distancemethod as follows

119883119901119894119895 = 1 minus (sum3119905=1 10038161003816100381610038161003816119901119894119905 minus 11990111989511990510038161003816100381610038161003816)3 119894 119895 isin 1198730 (6)

119883119901119894119895 represents the qualitative correlation between thedifferent customers and the range of the value is during [0 1]119883119901119894119895 indicates relationship of attributors 120593119901119894 (119896) and 120593119901119895 (119896)(2)e Processing of Quantitative Decision VariablesQuanti-tative decision variables are determined values However thedimension of the index is different In order to eliminate thedimension the quantitative data are needed to be normalizedand converted into standardized data Then the normalizedvariables are treated as a kind of special fuzzy number whichcan be used to calculate the similarity of the correspondingquantitative decision variables between two customers

120593119901119894 = 120593119901119894 minusmin1le119899le|119873| 120593119901119899 max1le119899le|119873| 120593119901119899 minusmin1le119899le|119873| 120593119901119899

for 119901 isin 2 7 119894 isin 1198730 (7)

120593119901119894 and 120593119901119899 are the actual value of the quantitative decisionvariables corresponding to customers 119894 and customers 119899119873 isthe number of customers On the basis of the standardizationof the quantitative decision the similarity can be calculatedby

119883119901119894119895 = 1 minus 10038161003816100381610038161003816120593119901119894 minus 120593119901119895 10038161003816100381610038161003816max1le119899le|119873| 120593119901119899 minusmin1le119899le|119873| 120593119901119899

for 119901 isin 2 7 119894 isin 1198730(8)

Additionally 1205931119894 represents the weight of the goods itcan determine the number of customer groups And it isused as the termination condition of the customer clusteringalgorithm in this article So we do nothing for this variable

However the comprehensive similarity for customersis not simply adding up the similarity for each decisionvariable The variables present different influence for thedelivery quality In this paper we consider the importanceof each decision variable and use a mathematical method tocalculate the weight coefficient for each parameter A fuzzymathematics method is introduced to determine the weightof each decision variable

(3) Weight Calculation of Decision Variables After obtainingthe order information the managers (experts) of the dis-tribution center are required to submit evaluation reportsabout the delivery service index The evaluation value of theimpact of each decision variable for the customers is alsogiven out by the managers The expertsrsquo evaluation languageis generally described as the influence of decision variableson customer satisfaction We divide the areas of influenceinto five evaluations specifically largest large medium lowand lowest To calculate the numerical value the languageevaluation is converted to triangular fuzzy numbers andshown in Table 2

Based on the importance of the decision variables for cus-tomer satisfaction the weight of the corresponding decisionvariables can be calculated For convenience we denote somekey parameters as follows119882119906119901(119906 = 1 2 119903 119901 = 1 2 119898) represents theevaluation of decision variables 119901 for decision maker 119906where 119903 is the total number of decision makers and 119898 isthe number of decision variables And it can be denotedby triangular fuzzy number 119882119906119901 = 119886119906119901 119887119906119901 119888119906119901 120596119901 isthe weight value for decision 119901 119885119894119901 = 119860 119894119901 119861119894119901 119862119894119901(119905 =1 2 |119873| 119901 = 1 2 119898) where |119873| is the total numberof the covering customers119885119894119901 represents the comprehensiveevaluation index of decision variables 119901 by all decisionmakers for customer 119894

119860 119894119901 = 1119903 otimes 119903sum119906=1119886119906119901 (9)

119861119894119901 = 1119903 otimes 119903sum119906=1119887119906119901 (10)

119862119894119901 = 1119903 otimes 119903sum119906=1119888119906119901 (11)

6 Mathematical Problems in Engineering

Table 2 Membership function of evaluation linguistic variables

Language terminology Triangular fuzzy number Judgment scalelargest (07511) 1large (050751) 075medium (02505075) 05small (002505) 025smallest (00025) 0

The comprehensive evaluation index of decision variables119901 for customer 119894 can be calculated as

119885119894119901 = 1119903 otimes 119903sum119906=1119882119906119901 (12)

To simplify we define the comprehensive membership ofdecision variable 119901 for customer 119894 as follows

119901119894 = 15 (119860 119894119901 + 2119861119894119901 + 2119862119894119901) (13)

The comprehensive evaluation value of a decision variable119901 denotes the average value of all customers such as

119901 = 1119873 119873sum119894=1119901119894 for 120596119901 gt 0 (14)

Considering that 120596119901 should satisfy the constraint sum119898119901=1 120596119901 =1 so we calculate 120596119901 as follows120596119901 = 119901sum119898119901=1 119901 119901 = 1 2 119898 (15)

Then the similarity between customer 119894 and customer 119895can be denoted as follows

119878119894119895 = 119898sum119901=1

120596119901119883119901119894119895 119894 119895 isin 1198730 (16)

Finally we get the comprehensive similarity betweencustomers which can be represented by a similarity matrixjust as follows

119865 = (119878119894119895)119873times119873 =(((

1 11987812 sdot sdot sdot 119878111987311987821 sdot sdot sdot 1198782119873minus1 1198782119873 d

1198781198731 119878119873119873minus1 1)))

(17)

32 Second Phase Model For the second phase model weshould assign the appropriate route for the customer en-groups In this part we consider the energy conservation andemission reduction and target to reduce the comprehensivecosts There we use the comprehensive emission measure-ment model to calculate the energy consumption and carbonemission [24] At first we regard the travel speed as fixedvariables and establish a conventional model On this basiswe consider the speed as time variation variables and reformthe model which is consistent with the actual situation Tosolve the problem we design a genetic algorithm

To provide a precise statement of this problem we definethe parameters and indices shown in Table 3

For the en-group route optimization we consider twodecision variables as follows

119909119897119896119894119895 = 1 if the vehicle 119897119896 travel on the link (119894 119895)0 otherwise

(18)

119911119897119896119894 = 1 if the customer 119894 need to be serviced by the 119896th vehicle for type 1198970 otherwise

(19)

The model can be formulated as follows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(20)

Mathematical Problems in Engineering 7

Table 3 Sets indices and parameters in the C-VRPESER

Notations Detailed Definition119873 Where119873 = 0 1 2 sdot sdot sdot 119899 0 represents the depot1198730 = 119873 0 = 1 2 sdot sdot sdot 119899is the set of the customers which should be delivered119860 119860 = (119894 119895) 119894 119895 isin 119873 represents the arcs119879119877 Set of of customers in the same customer group 119877119897 Type of the vehicle119897119896 119896th vehicle with 119897 type119876119897119890 Maximum capacity of each type vehicles119894 isin 1198730 Number of customer119902119894 Demand of customers 119894119904119894 Time of vehicle arrive at customer 119894[119886119894 119887119894] Desired preferred time window for customer 119894 to be serviced119905119894 Service time of customer 119894 need119889119894119895 Distance between the customer 119894 and 119895

V119894119895 Speed of the vehicle travel on the link (119894 119895)119902119897119896119894119895 Load of the vehicle 119897119896 travel on the link (119894 119895)119876119897119888 Light weight of the 119897 type vehicle119888119900 Per unit cost of fuel119888119890 Per unit cost of carbon emissions (carbon tax)120575119888 Fuel emission factor119888119897 Fixed costs of the vehicle used119890119896 Departure time of the vehicle k from the depot119897119896 Latest arrival time at the depot after services the customers1198881 Penalty coefficient for earliness arrival1198882 Penalty coefficient for delay delivery119878119894 Service time of customer 119894st sum

119897119896isin119871119870

sum119895isin119879119877

1199091198971198960119895 = 1 (21)

sum119897119896isin119871119870

sum119895isin119879119877

1199091198971198961198950 = 1 (22)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 = 1 forall119894 isin 119879119877 (23)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119895ℎ minus sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896ℎ119895 = 0 forallℎ isin 119879119877 (24)

sum119894isin119879119877

119902119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119876119890 (25)

119886119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119904119897119896119894 le 119887119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 forall119894 isin 119879119877 (26)

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119889119894119895V119894119895 ) le 119904119897119896119894 (27)

119909119897119896119894119895 isin 0 1 (28)

119910119897119896119894 ge 0 (29)

The objective function (20) is to minimize the totalrouting cost including fixed cost energy consumption costcarbon emission cost and punishment for time windowsConstraints (21) and (22) guarantee that all service vehiclesstart from the depot and finally return to the depot foronce Constraint (23) ensures each vehicle travels on the arc(V119894 V119895) nomore than once Constraint (24) is the flow balanceand shows that the number of the vehicles which arrive atthe customer is the same as those which depart Constraint(25) ensures the loading customer demands no more thanthe vehicle capacity Inequality (26) is the time windowconstraint Constraint (27) represents the time limit forvehicle travel horizon Constraint (28) is the binary variableconstraint Constraint (29) represents the nonnegative oftime

33 Model Reformulation The above model is generatedbased on the assumption that the drive speed is invariableHowever in reality the speed is influenced by a lot of elementsTo make the model more practical we assume that thespeed is a ladder type change variable This paper supposesthe vehicles traveling speed as shown in Figure 2 On thisbasis we build the time-varying velocity distribution vehiclerouting optimization model considering the velocity change

The speed is assumed as a step function of time asFigure 2 And during different time intervals the vehiclesrsquo

8 Mathematical Problems in Engineering

20253035404550556065707580

50 10 15 20 24

Figure 2 The speed trend diagram

travel speed is different To model convince the runningtime is divided into 119868 interval according different extentsspeed which can be represented by 1198791 1198792 119879119868 The 119898thtime interval is indicated as [119905119898 119905119898+1] in each time intervalvelocity is constant

The time of vehicles running on the path (119894 119895) is a functionof the time 119897119905119894 that is the vehiclesrsquo departing time from thecustomer 119894 The speed of vehicles running on the path oftencrosses more than one speed range It is assumed that thevehicle will cross 119901 + 1 time intervals It is clear that thevehicle has 119901 different speed Assume the vehicle derived thecustomer 119894 in the119898 time interval with the speed V119898119894119895 (119897119905119894) thenthe vehicle speed turns V119898+1119894119895 (119897119905119894) in the next time intervaland we denote the vehicle speed as V119898+119901119894119895 (119897119905119894)when the vehiclederives the link (119894 119895) And the distance for the link (119894 119895) isdenoted by 119889119898+119901119894119895 (119897119905119894) And the total time 119905119905119894 for the vehicledrived across the link (119894 119895) can be represented by the timefunction 119897119905119894 as follows

119905119905119894119895 (119897119905119894) = 119889119898119894119895V119898119894119895 (119897119905119894) + 119889119898+1119894119895

V119898+1119894119895 (119897119905119894) + sdot sdot sdot + 119889119898+119901119894119895V119898+119901119894119895 (119897119905119894) (30)

Considering the variety of the vehicle traveling speed wecan represent the fuel consumption function as follows

119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)= 119868sum119902=1

119874119865119894119895 (119876119897119888 + 119902119894119895 V119898+119902119894119895 119889119898+119902119894119895 ) (31)

If we assumed that the 119896th vehicle of the 119897 kind beginsthe task at time 119910119897119896 the objective function can be denoted asfollows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]

+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(32)

The value of V119894119895 depends on the time interval The timeconstraint is represented as follows

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119905119905119894119895 (119904119897119896119894 + 119905119894)) le 119904119897119896119895 (33)

The vehicle service time cannot be earlier than the starttime of the distribution center not later than the time of thedelivery center

1199051 le 119910119897119896 le 119905119868 (34)

4 Algorithm

To present the customers-oriented vehicle routing problemwith environment consideration a two-phase model is for-mulated We design two algorithms to solve the problemFirst we design a fuzzy system clustering algorithm to realizethe customers grouping That algorithm is mainly used toconsider the customers demand attributes including quan-titative and qualitative attributes The second phase model isan en-group VRP that need to design a heuristic algorithm togain the en-group optimal delivery routes Here we comparethe performance of several heuristic algorithms as shown inTable 4 In view of the robustness and global optimization ofgenetic algorithm we design a genetic algorithm to solve theproblem

41 Fuzzy System Clustering Algorithm On the basis ofthe similarity matrix we design a fuzzy system clusteringalgorithm to cluster the customer into different groups whichare often used to process the data with quantitative andqualitative data [22] And the specific algorithm procedureis summarized as in Algorithm 1

To obtain the cluster solutions we apply the procedure inFigure 3

Mathematical Problems in Engineering 9

Table 4 Heuristic algorithm performance comparison

Algorithm Parameters Features

Tabu search algorithm(TS)

Tabu sizeNeighborhood structureNumber of solutions

Local optimumHigher solving efficiency

Ant colony algorithm(ACO)

Initial pheromoneInformation increase

Information accumulationDisappearance factorInspiration factor

Good parallelismGlobal optimization features

Time-consuming

Simulated annealingalgorithm (SA)

Initial temperatureTemperature return functionMode of state generation

Sampling principle

Simple implementationFlexible application

The algorithm converges slowly

Genetic algorithm (GA)

Population numberSelection operatorCrossover operatorMutation operator

Strong global optimization abilityParallelism and strong robustness

1 Step 1 Data initialization input the fuzzy equivalent matrix 119865(119896) Let iterations 120587=12 Step 2 Find out the largest elements in the matrix 119865119904(119896)119879 and chose the customer with the smaller number as the targetcustomer and denote it with 1199043 Step 3Mark the selected target customer and delete the corresponding row 119865119904(119896)119879 Specifically once a customer is chosen asthe target customer its column is marked by 119865119904(119896) mark the corresponding elements in row 119865119904(119896)119879 as 04 Step 4 Find out the largest element of matrix 119865119904(119896)119879 denoted the similarity value as 1205961199031199045 (1) If 120596119903119904 le 120582(pre-given threshold value) then go to Step 5 Otherwise calculate the total weight of the customers1198766 (2)Make judgement if 119876 ge 119876119897119890 go to Step 5 Otherwise put customers 119904 and 119903 in same group then delete the correspondingrow 119865119903(119896)119879 of matrix 119865(119896) return to Step 47 Step 5 Set the customer clustering termination conditions if the matrix is empty then clustering process stops else let120587= 120587 + 1 and got to Step 2

Algorithm 1 Fuzzy system clustering algorithm

42 En-Group Genetic Algorithm The proposed model is amixed-integer linear programming model When the scale ofthe problem is expended it is difficult to solve the problemwith accurate algorithm However the GA is suitable to solvemultivariables complex problem with multiple parameters Itiswidely applied in solvingVRPdue to its extensive generalityhigh efficiency and strong robustnessTherefore we deign GAto solve the problemThe key steps are designed as follows

Encoding Natural number cod approach is used tocharacterize chromosomes Chromosome genesrsquo encoding isbased on the results of the customer groups with discontin-uous customer numbers There 0 expresses the distributioncenter and customers are expressed by numbers 1 2 3 sdot sdot sdot 119873Different groups are corresponding different chromosomalgenes The beginning and the end of the genes in the chro-mosome cod sequence must be rdquo0rdquo on behalf of the vehiclewhich departs from the distribution center and finally returnsto the distribution center after completing the deliverytasks

Initial Solution Construct initial population chromo-some cod using MATLAB programming software under

the premise of customer grouping Specific we randomlygive the initial solution as 119901119900119901(119894 2 119899 minus 1) =119870119867119911(rand(119899119906119898119890119897(119870119867119911)))119901119900119901(119894 1) = 0 and 119901119900119901(119894 119899) =0 such as 119870119867119911(rand(119899119906119898119890119897(119870119867119911)))=[0 1 4 5 9 12 18 0]And generate the initial population as a matrix with thedimension 119901119900119901119904119894119911119890 lowast (119899 + 2)

Fitness Function In this paper the objective functionvalue is used to measure the fitness value as follows

119891119896 = 1 minus (120583119896 minusmin120583)(max 120583 minusmin 120583) (35)

Selection Operator It is individual selection according toroulette wheel if the fitness of an individual is denoted by 119891119896then the probability of being selected can be represented bythe following

119901119896 = 119891119896sum119870119896=1 119891119896 (36)

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 5

customer satisfaction In this way each customerrsquos demandindicators can be denoted by triangular fuzzy number (12057211990111989411205721199011198942 1205721199011198943) which can be denoted as follows120593119901119894 = [1205721199011198941 1205721199011198942 1205721199011198943] (1)

We can get the demand attributers from the orderswhich will be pregiven by the customers However we shouldtransfer the linguistic variables and the quantitative data asstandardized data so that we can use them to calculate thesimilarity And the procedure is shown in below

(1)e Processing of Qualitative DecisionVariables In the firststep we use the standard deviation transformation to processthe qualitative data which can guarantee the data availabilityAnd for the second step we introduce the range conversion tostandardize the quantitative variables The constraint (1) canbe firstly transformed as follows120593119901119894 = [1199011198941 1199011198942 1199011198943] 119894 isin 1198730 (2)

where

120593119901119894 = 100381610038161003816100381610038161003816120572119901119894119905 minus 120572119901119894119905100381610038161003816100381610038161003816119878119901119905 119894 isin 1198730 (3)

120572119901119894119905 = sum119873119894=1 120572119901119894119905119873 119905 = 1 2 3 119894 isin 1198730 (4)

However the indicator 120572119901119894119905 is the mean value 119878119901119905 is thestandard deviation and |119873| is the total customer numberThevalue range of 119901 = 3 4 5 6119878119901119905 = [[[

sum119873119894=1 (120572119901119894119905 minus 120572119901119894119905)2|119873| minus 1 ]]]12 119905 = 1 2 3 119894 isin 1198730 (5)

Then we can calculate the similarity of qualitativedecision-making variables with Hemingwayrsquos distancemethod as follows

119883119901119894119895 = 1 minus (sum3119905=1 10038161003816100381610038161003816119901119894119905 minus 11990111989511990510038161003816100381610038161003816)3 119894 119895 isin 1198730 (6)

119883119901119894119895 represents the qualitative correlation between thedifferent customers and the range of the value is during [0 1]119883119901119894119895 indicates relationship of attributors 120593119901119894 (119896) and 120593119901119895 (119896)(2)e Processing of Quantitative Decision VariablesQuanti-tative decision variables are determined values However thedimension of the index is different In order to eliminate thedimension the quantitative data are needed to be normalizedand converted into standardized data Then the normalizedvariables are treated as a kind of special fuzzy number whichcan be used to calculate the similarity of the correspondingquantitative decision variables between two customers

120593119901119894 = 120593119901119894 minusmin1le119899le|119873| 120593119901119899 max1le119899le|119873| 120593119901119899 minusmin1le119899le|119873| 120593119901119899

for 119901 isin 2 7 119894 isin 1198730 (7)

120593119901119894 and 120593119901119899 are the actual value of the quantitative decisionvariables corresponding to customers 119894 and customers 119899119873 isthe number of customers On the basis of the standardizationof the quantitative decision the similarity can be calculatedby

119883119901119894119895 = 1 minus 10038161003816100381610038161003816120593119901119894 minus 120593119901119895 10038161003816100381610038161003816max1le119899le|119873| 120593119901119899 minusmin1le119899le|119873| 120593119901119899

for 119901 isin 2 7 119894 isin 1198730(8)

Additionally 1205931119894 represents the weight of the goods itcan determine the number of customer groups And it isused as the termination condition of the customer clusteringalgorithm in this article So we do nothing for this variable

However the comprehensive similarity for customersis not simply adding up the similarity for each decisionvariable The variables present different influence for thedelivery quality In this paper we consider the importanceof each decision variable and use a mathematical method tocalculate the weight coefficient for each parameter A fuzzymathematics method is introduced to determine the weightof each decision variable

(3) Weight Calculation of Decision Variables After obtainingthe order information the managers (experts) of the dis-tribution center are required to submit evaluation reportsabout the delivery service index The evaluation value of theimpact of each decision variable for the customers is alsogiven out by the managers The expertsrsquo evaluation languageis generally described as the influence of decision variableson customer satisfaction We divide the areas of influenceinto five evaluations specifically largest large medium lowand lowest To calculate the numerical value the languageevaluation is converted to triangular fuzzy numbers andshown in Table 2

Based on the importance of the decision variables for cus-tomer satisfaction the weight of the corresponding decisionvariables can be calculated For convenience we denote somekey parameters as follows119882119906119901(119906 = 1 2 119903 119901 = 1 2 119898) represents theevaluation of decision variables 119901 for decision maker 119906where 119903 is the total number of decision makers and 119898 isthe number of decision variables And it can be denotedby triangular fuzzy number 119882119906119901 = 119886119906119901 119887119906119901 119888119906119901 120596119901 isthe weight value for decision 119901 119885119894119901 = 119860 119894119901 119861119894119901 119862119894119901(119905 =1 2 |119873| 119901 = 1 2 119898) where |119873| is the total numberof the covering customers119885119894119901 represents the comprehensiveevaluation index of decision variables 119901 by all decisionmakers for customer 119894

119860 119894119901 = 1119903 otimes 119903sum119906=1119886119906119901 (9)

119861119894119901 = 1119903 otimes 119903sum119906=1119887119906119901 (10)

119862119894119901 = 1119903 otimes 119903sum119906=1119888119906119901 (11)

6 Mathematical Problems in Engineering

Table 2 Membership function of evaluation linguistic variables

Language terminology Triangular fuzzy number Judgment scalelargest (07511) 1large (050751) 075medium (02505075) 05small (002505) 025smallest (00025) 0

The comprehensive evaluation index of decision variables119901 for customer 119894 can be calculated as

119885119894119901 = 1119903 otimes 119903sum119906=1119882119906119901 (12)

To simplify we define the comprehensive membership ofdecision variable 119901 for customer 119894 as follows

119901119894 = 15 (119860 119894119901 + 2119861119894119901 + 2119862119894119901) (13)

The comprehensive evaluation value of a decision variable119901 denotes the average value of all customers such as

119901 = 1119873 119873sum119894=1119901119894 for 120596119901 gt 0 (14)

Considering that 120596119901 should satisfy the constraint sum119898119901=1 120596119901 =1 so we calculate 120596119901 as follows120596119901 = 119901sum119898119901=1 119901 119901 = 1 2 119898 (15)

Then the similarity between customer 119894 and customer 119895can be denoted as follows

119878119894119895 = 119898sum119901=1

120596119901119883119901119894119895 119894 119895 isin 1198730 (16)

Finally we get the comprehensive similarity betweencustomers which can be represented by a similarity matrixjust as follows

119865 = (119878119894119895)119873times119873 =(((

1 11987812 sdot sdot sdot 119878111987311987821 sdot sdot sdot 1198782119873minus1 1198782119873 d

1198781198731 119878119873119873minus1 1)))

(17)

32 Second Phase Model For the second phase model weshould assign the appropriate route for the customer en-groups In this part we consider the energy conservation andemission reduction and target to reduce the comprehensivecosts There we use the comprehensive emission measure-ment model to calculate the energy consumption and carbonemission [24] At first we regard the travel speed as fixedvariables and establish a conventional model On this basiswe consider the speed as time variation variables and reformthe model which is consistent with the actual situation Tosolve the problem we design a genetic algorithm

To provide a precise statement of this problem we definethe parameters and indices shown in Table 3

For the en-group route optimization we consider twodecision variables as follows

119909119897119896119894119895 = 1 if the vehicle 119897119896 travel on the link (119894 119895)0 otherwise

(18)

119911119897119896119894 = 1 if the customer 119894 need to be serviced by the 119896th vehicle for type 1198970 otherwise

(19)

The model can be formulated as follows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(20)

Mathematical Problems in Engineering 7

Table 3 Sets indices and parameters in the C-VRPESER

Notations Detailed Definition119873 Where119873 = 0 1 2 sdot sdot sdot 119899 0 represents the depot1198730 = 119873 0 = 1 2 sdot sdot sdot 119899is the set of the customers which should be delivered119860 119860 = (119894 119895) 119894 119895 isin 119873 represents the arcs119879119877 Set of of customers in the same customer group 119877119897 Type of the vehicle119897119896 119896th vehicle with 119897 type119876119897119890 Maximum capacity of each type vehicles119894 isin 1198730 Number of customer119902119894 Demand of customers 119894119904119894 Time of vehicle arrive at customer 119894[119886119894 119887119894] Desired preferred time window for customer 119894 to be serviced119905119894 Service time of customer 119894 need119889119894119895 Distance between the customer 119894 and 119895

V119894119895 Speed of the vehicle travel on the link (119894 119895)119902119897119896119894119895 Load of the vehicle 119897119896 travel on the link (119894 119895)119876119897119888 Light weight of the 119897 type vehicle119888119900 Per unit cost of fuel119888119890 Per unit cost of carbon emissions (carbon tax)120575119888 Fuel emission factor119888119897 Fixed costs of the vehicle used119890119896 Departure time of the vehicle k from the depot119897119896 Latest arrival time at the depot after services the customers1198881 Penalty coefficient for earliness arrival1198882 Penalty coefficient for delay delivery119878119894 Service time of customer 119894st sum

119897119896isin119871119870

sum119895isin119879119877

1199091198971198960119895 = 1 (21)

sum119897119896isin119871119870

sum119895isin119879119877

1199091198971198961198950 = 1 (22)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 = 1 forall119894 isin 119879119877 (23)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119895ℎ minus sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896ℎ119895 = 0 forallℎ isin 119879119877 (24)

sum119894isin119879119877

119902119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119876119890 (25)

119886119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119904119897119896119894 le 119887119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 forall119894 isin 119879119877 (26)

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119889119894119895V119894119895 ) le 119904119897119896119894 (27)

119909119897119896119894119895 isin 0 1 (28)

119910119897119896119894 ge 0 (29)

The objective function (20) is to minimize the totalrouting cost including fixed cost energy consumption costcarbon emission cost and punishment for time windowsConstraints (21) and (22) guarantee that all service vehiclesstart from the depot and finally return to the depot foronce Constraint (23) ensures each vehicle travels on the arc(V119894 V119895) nomore than once Constraint (24) is the flow balanceand shows that the number of the vehicles which arrive atthe customer is the same as those which depart Constraint(25) ensures the loading customer demands no more thanthe vehicle capacity Inequality (26) is the time windowconstraint Constraint (27) represents the time limit forvehicle travel horizon Constraint (28) is the binary variableconstraint Constraint (29) represents the nonnegative oftime

33 Model Reformulation The above model is generatedbased on the assumption that the drive speed is invariableHowever in reality the speed is influenced by a lot of elementsTo make the model more practical we assume that thespeed is a ladder type change variable This paper supposesthe vehicles traveling speed as shown in Figure 2 On thisbasis we build the time-varying velocity distribution vehiclerouting optimization model considering the velocity change

The speed is assumed as a step function of time asFigure 2 And during different time intervals the vehiclesrsquo

8 Mathematical Problems in Engineering

20253035404550556065707580

50 10 15 20 24

Figure 2 The speed trend diagram

travel speed is different To model convince the runningtime is divided into 119868 interval according different extentsspeed which can be represented by 1198791 1198792 119879119868 The 119898thtime interval is indicated as [119905119898 119905119898+1] in each time intervalvelocity is constant

The time of vehicles running on the path (119894 119895) is a functionof the time 119897119905119894 that is the vehiclesrsquo departing time from thecustomer 119894 The speed of vehicles running on the path oftencrosses more than one speed range It is assumed that thevehicle will cross 119901 + 1 time intervals It is clear that thevehicle has 119901 different speed Assume the vehicle derived thecustomer 119894 in the119898 time interval with the speed V119898119894119895 (119897119905119894) thenthe vehicle speed turns V119898+1119894119895 (119897119905119894) in the next time intervaland we denote the vehicle speed as V119898+119901119894119895 (119897119905119894)when the vehiclederives the link (119894 119895) And the distance for the link (119894 119895) isdenoted by 119889119898+119901119894119895 (119897119905119894) And the total time 119905119905119894 for the vehicledrived across the link (119894 119895) can be represented by the timefunction 119897119905119894 as follows

119905119905119894119895 (119897119905119894) = 119889119898119894119895V119898119894119895 (119897119905119894) + 119889119898+1119894119895

V119898+1119894119895 (119897119905119894) + sdot sdot sdot + 119889119898+119901119894119895V119898+119901119894119895 (119897119905119894) (30)

Considering the variety of the vehicle traveling speed wecan represent the fuel consumption function as follows

119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)= 119868sum119902=1

119874119865119894119895 (119876119897119888 + 119902119894119895 V119898+119902119894119895 119889119898+119902119894119895 ) (31)

If we assumed that the 119896th vehicle of the 119897 kind beginsthe task at time 119910119897119896 the objective function can be denoted asfollows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]

+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(32)

The value of V119894119895 depends on the time interval The timeconstraint is represented as follows

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119905119905119894119895 (119904119897119896119894 + 119905119894)) le 119904119897119896119895 (33)

The vehicle service time cannot be earlier than the starttime of the distribution center not later than the time of thedelivery center

1199051 le 119910119897119896 le 119905119868 (34)

4 Algorithm

To present the customers-oriented vehicle routing problemwith environment consideration a two-phase model is for-mulated We design two algorithms to solve the problemFirst we design a fuzzy system clustering algorithm to realizethe customers grouping That algorithm is mainly used toconsider the customers demand attributes including quan-titative and qualitative attributes The second phase model isan en-group VRP that need to design a heuristic algorithm togain the en-group optimal delivery routes Here we comparethe performance of several heuristic algorithms as shown inTable 4 In view of the robustness and global optimization ofgenetic algorithm we design a genetic algorithm to solve theproblem

41 Fuzzy System Clustering Algorithm On the basis ofthe similarity matrix we design a fuzzy system clusteringalgorithm to cluster the customer into different groups whichare often used to process the data with quantitative andqualitative data [22] And the specific algorithm procedureis summarized as in Algorithm 1

To obtain the cluster solutions we apply the procedure inFigure 3

Mathematical Problems in Engineering 9

Table 4 Heuristic algorithm performance comparison

Algorithm Parameters Features

Tabu search algorithm(TS)

Tabu sizeNeighborhood structureNumber of solutions

Local optimumHigher solving efficiency

Ant colony algorithm(ACO)

Initial pheromoneInformation increase

Information accumulationDisappearance factorInspiration factor

Good parallelismGlobal optimization features

Time-consuming

Simulated annealingalgorithm (SA)

Initial temperatureTemperature return functionMode of state generation

Sampling principle

Simple implementationFlexible application

The algorithm converges slowly

Genetic algorithm (GA)

Population numberSelection operatorCrossover operatorMutation operator

Strong global optimization abilityParallelism and strong robustness

1 Step 1 Data initialization input the fuzzy equivalent matrix 119865(119896) Let iterations 120587=12 Step 2 Find out the largest elements in the matrix 119865119904(119896)119879 and chose the customer with the smaller number as the targetcustomer and denote it with 1199043 Step 3Mark the selected target customer and delete the corresponding row 119865119904(119896)119879 Specifically once a customer is chosen asthe target customer its column is marked by 119865119904(119896) mark the corresponding elements in row 119865119904(119896)119879 as 04 Step 4 Find out the largest element of matrix 119865119904(119896)119879 denoted the similarity value as 1205961199031199045 (1) If 120596119903119904 le 120582(pre-given threshold value) then go to Step 5 Otherwise calculate the total weight of the customers1198766 (2)Make judgement if 119876 ge 119876119897119890 go to Step 5 Otherwise put customers 119904 and 119903 in same group then delete the correspondingrow 119865119903(119896)119879 of matrix 119865(119896) return to Step 47 Step 5 Set the customer clustering termination conditions if the matrix is empty then clustering process stops else let120587= 120587 + 1 and got to Step 2

Algorithm 1 Fuzzy system clustering algorithm

42 En-Group Genetic Algorithm The proposed model is amixed-integer linear programming model When the scale ofthe problem is expended it is difficult to solve the problemwith accurate algorithm However the GA is suitable to solvemultivariables complex problem with multiple parameters Itiswidely applied in solvingVRPdue to its extensive generalityhigh efficiency and strong robustnessTherefore we deign GAto solve the problemThe key steps are designed as follows

Encoding Natural number cod approach is used tocharacterize chromosomes Chromosome genesrsquo encoding isbased on the results of the customer groups with discontin-uous customer numbers There 0 expresses the distributioncenter and customers are expressed by numbers 1 2 3 sdot sdot sdot 119873Different groups are corresponding different chromosomalgenes The beginning and the end of the genes in the chro-mosome cod sequence must be rdquo0rdquo on behalf of the vehiclewhich departs from the distribution center and finally returnsto the distribution center after completing the deliverytasks

Initial Solution Construct initial population chromo-some cod using MATLAB programming software under

the premise of customer grouping Specific we randomlygive the initial solution as 119901119900119901(119894 2 119899 minus 1) =119870119867119911(rand(119899119906119898119890119897(119870119867119911)))119901119900119901(119894 1) = 0 and 119901119900119901(119894 119899) =0 such as 119870119867119911(rand(119899119906119898119890119897(119870119867119911)))=[0 1 4 5 9 12 18 0]And generate the initial population as a matrix with thedimension 119901119900119901119904119894119911119890 lowast (119899 + 2)

Fitness Function In this paper the objective functionvalue is used to measure the fitness value as follows

119891119896 = 1 minus (120583119896 minusmin120583)(max 120583 minusmin 120583) (35)

Selection Operator It is individual selection according toroulette wheel if the fitness of an individual is denoted by 119891119896then the probability of being selected can be represented bythe following

119901119896 = 119891119896sum119870119896=1 119891119896 (36)

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

6 Mathematical Problems in Engineering

Table 2 Membership function of evaluation linguistic variables

Language terminology Triangular fuzzy number Judgment scalelargest (07511) 1large (050751) 075medium (02505075) 05small (002505) 025smallest (00025) 0

The comprehensive evaluation index of decision variables119901 for customer 119894 can be calculated as

119885119894119901 = 1119903 otimes 119903sum119906=1119882119906119901 (12)

To simplify we define the comprehensive membership ofdecision variable 119901 for customer 119894 as follows

119901119894 = 15 (119860 119894119901 + 2119861119894119901 + 2119862119894119901) (13)

The comprehensive evaluation value of a decision variable119901 denotes the average value of all customers such as

119901 = 1119873 119873sum119894=1119901119894 for 120596119901 gt 0 (14)

Considering that 120596119901 should satisfy the constraint sum119898119901=1 120596119901 =1 so we calculate 120596119901 as follows120596119901 = 119901sum119898119901=1 119901 119901 = 1 2 119898 (15)

Then the similarity between customer 119894 and customer 119895can be denoted as follows

119878119894119895 = 119898sum119901=1

120596119901119883119901119894119895 119894 119895 isin 1198730 (16)

Finally we get the comprehensive similarity betweencustomers which can be represented by a similarity matrixjust as follows

119865 = (119878119894119895)119873times119873 =(((

1 11987812 sdot sdot sdot 119878111987311987821 sdot sdot sdot 1198782119873minus1 1198782119873 d

1198781198731 119878119873119873minus1 1)))

(17)

32 Second Phase Model For the second phase model weshould assign the appropriate route for the customer en-groups In this part we consider the energy conservation andemission reduction and target to reduce the comprehensivecosts There we use the comprehensive emission measure-ment model to calculate the energy consumption and carbonemission [24] At first we regard the travel speed as fixedvariables and establish a conventional model On this basiswe consider the speed as time variation variables and reformthe model which is consistent with the actual situation Tosolve the problem we design a genetic algorithm

To provide a precise statement of this problem we definethe parameters and indices shown in Table 3

For the en-group route optimization we consider twodecision variables as follows

119909119897119896119894119895 = 1 if the vehicle 119897119896 travel on the link (119894 119895)0 otherwise

(18)

119911119897119896119894 = 1 if the customer 119894 need to be serviced by the 119896th vehicle for type 1198970 otherwise

(19)

The model can be formulated as follows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(20)

Mathematical Problems in Engineering 7

Table 3 Sets indices and parameters in the C-VRPESER

Notations Detailed Definition119873 Where119873 = 0 1 2 sdot sdot sdot 119899 0 represents the depot1198730 = 119873 0 = 1 2 sdot sdot sdot 119899is the set of the customers which should be delivered119860 119860 = (119894 119895) 119894 119895 isin 119873 represents the arcs119879119877 Set of of customers in the same customer group 119877119897 Type of the vehicle119897119896 119896th vehicle with 119897 type119876119897119890 Maximum capacity of each type vehicles119894 isin 1198730 Number of customer119902119894 Demand of customers 119894119904119894 Time of vehicle arrive at customer 119894[119886119894 119887119894] Desired preferred time window for customer 119894 to be serviced119905119894 Service time of customer 119894 need119889119894119895 Distance between the customer 119894 and 119895

V119894119895 Speed of the vehicle travel on the link (119894 119895)119902119897119896119894119895 Load of the vehicle 119897119896 travel on the link (119894 119895)119876119897119888 Light weight of the 119897 type vehicle119888119900 Per unit cost of fuel119888119890 Per unit cost of carbon emissions (carbon tax)120575119888 Fuel emission factor119888119897 Fixed costs of the vehicle used119890119896 Departure time of the vehicle k from the depot119897119896 Latest arrival time at the depot after services the customers1198881 Penalty coefficient for earliness arrival1198882 Penalty coefficient for delay delivery119878119894 Service time of customer 119894st sum

119897119896isin119871119870

sum119895isin119879119877

1199091198971198960119895 = 1 (21)

sum119897119896isin119871119870

sum119895isin119879119877

1199091198971198961198950 = 1 (22)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 = 1 forall119894 isin 119879119877 (23)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119895ℎ minus sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896ℎ119895 = 0 forallℎ isin 119879119877 (24)

sum119894isin119879119877

119902119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119876119890 (25)

119886119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119904119897119896119894 le 119887119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 forall119894 isin 119879119877 (26)

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119889119894119895V119894119895 ) le 119904119897119896119894 (27)

119909119897119896119894119895 isin 0 1 (28)

119910119897119896119894 ge 0 (29)

The objective function (20) is to minimize the totalrouting cost including fixed cost energy consumption costcarbon emission cost and punishment for time windowsConstraints (21) and (22) guarantee that all service vehiclesstart from the depot and finally return to the depot foronce Constraint (23) ensures each vehicle travels on the arc(V119894 V119895) nomore than once Constraint (24) is the flow balanceand shows that the number of the vehicles which arrive atthe customer is the same as those which depart Constraint(25) ensures the loading customer demands no more thanthe vehicle capacity Inequality (26) is the time windowconstraint Constraint (27) represents the time limit forvehicle travel horizon Constraint (28) is the binary variableconstraint Constraint (29) represents the nonnegative oftime

33 Model Reformulation The above model is generatedbased on the assumption that the drive speed is invariableHowever in reality the speed is influenced by a lot of elementsTo make the model more practical we assume that thespeed is a ladder type change variable This paper supposesthe vehicles traveling speed as shown in Figure 2 On thisbasis we build the time-varying velocity distribution vehiclerouting optimization model considering the velocity change

The speed is assumed as a step function of time asFigure 2 And during different time intervals the vehiclesrsquo

8 Mathematical Problems in Engineering

20253035404550556065707580

50 10 15 20 24

Figure 2 The speed trend diagram

travel speed is different To model convince the runningtime is divided into 119868 interval according different extentsspeed which can be represented by 1198791 1198792 119879119868 The 119898thtime interval is indicated as [119905119898 119905119898+1] in each time intervalvelocity is constant

The time of vehicles running on the path (119894 119895) is a functionof the time 119897119905119894 that is the vehiclesrsquo departing time from thecustomer 119894 The speed of vehicles running on the path oftencrosses more than one speed range It is assumed that thevehicle will cross 119901 + 1 time intervals It is clear that thevehicle has 119901 different speed Assume the vehicle derived thecustomer 119894 in the119898 time interval with the speed V119898119894119895 (119897119905119894) thenthe vehicle speed turns V119898+1119894119895 (119897119905119894) in the next time intervaland we denote the vehicle speed as V119898+119901119894119895 (119897119905119894)when the vehiclederives the link (119894 119895) And the distance for the link (119894 119895) isdenoted by 119889119898+119901119894119895 (119897119905119894) And the total time 119905119905119894 for the vehicledrived across the link (119894 119895) can be represented by the timefunction 119897119905119894 as follows

119905119905119894119895 (119897119905119894) = 119889119898119894119895V119898119894119895 (119897119905119894) + 119889119898+1119894119895

V119898+1119894119895 (119897119905119894) + sdot sdot sdot + 119889119898+119901119894119895V119898+119901119894119895 (119897119905119894) (30)

Considering the variety of the vehicle traveling speed wecan represent the fuel consumption function as follows

119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)= 119868sum119902=1

119874119865119894119895 (119876119897119888 + 119902119894119895 V119898+119902119894119895 119889119898+119902119894119895 ) (31)

If we assumed that the 119896th vehicle of the 119897 kind beginsthe task at time 119910119897119896 the objective function can be denoted asfollows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]

+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(32)

The value of V119894119895 depends on the time interval The timeconstraint is represented as follows

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119905119905119894119895 (119904119897119896119894 + 119905119894)) le 119904119897119896119895 (33)

The vehicle service time cannot be earlier than the starttime of the distribution center not later than the time of thedelivery center

1199051 le 119910119897119896 le 119905119868 (34)

4 Algorithm

To present the customers-oriented vehicle routing problemwith environment consideration a two-phase model is for-mulated We design two algorithms to solve the problemFirst we design a fuzzy system clustering algorithm to realizethe customers grouping That algorithm is mainly used toconsider the customers demand attributes including quan-titative and qualitative attributes The second phase model isan en-group VRP that need to design a heuristic algorithm togain the en-group optimal delivery routes Here we comparethe performance of several heuristic algorithms as shown inTable 4 In view of the robustness and global optimization ofgenetic algorithm we design a genetic algorithm to solve theproblem

41 Fuzzy System Clustering Algorithm On the basis ofthe similarity matrix we design a fuzzy system clusteringalgorithm to cluster the customer into different groups whichare often used to process the data with quantitative andqualitative data [22] And the specific algorithm procedureis summarized as in Algorithm 1

To obtain the cluster solutions we apply the procedure inFigure 3

Mathematical Problems in Engineering 9

Table 4 Heuristic algorithm performance comparison

Algorithm Parameters Features

Tabu search algorithm(TS)

Tabu sizeNeighborhood structureNumber of solutions

Local optimumHigher solving efficiency

Ant colony algorithm(ACO)

Initial pheromoneInformation increase

Information accumulationDisappearance factorInspiration factor

Good parallelismGlobal optimization features

Time-consuming

Simulated annealingalgorithm (SA)

Initial temperatureTemperature return functionMode of state generation

Sampling principle

Simple implementationFlexible application

The algorithm converges slowly

Genetic algorithm (GA)

Population numberSelection operatorCrossover operatorMutation operator

Strong global optimization abilityParallelism and strong robustness

1 Step 1 Data initialization input the fuzzy equivalent matrix 119865(119896) Let iterations 120587=12 Step 2 Find out the largest elements in the matrix 119865119904(119896)119879 and chose the customer with the smaller number as the targetcustomer and denote it with 1199043 Step 3Mark the selected target customer and delete the corresponding row 119865119904(119896)119879 Specifically once a customer is chosen asthe target customer its column is marked by 119865119904(119896) mark the corresponding elements in row 119865119904(119896)119879 as 04 Step 4 Find out the largest element of matrix 119865119904(119896)119879 denoted the similarity value as 1205961199031199045 (1) If 120596119903119904 le 120582(pre-given threshold value) then go to Step 5 Otherwise calculate the total weight of the customers1198766 (2)Make judgement if 119876 ge 119876119897119890 go to Step 5 Otherwise put customers 119904 and 119903 in same group then delete the correspondingrow 119865119903(119896)119879 of matrix 119865(119896) return to Step 47 Step 5 Set the customer clustering termination conditions if the matrix is empty then clustering process stops else let120587= 120587 + 1 and got to Step 2

Algorithm 1 Fuzzy system clustering algorithm

42 En-Group Genetic Algorithm The proposed model is amixed-integer linear programming model When the scale ofthe problem is expended it is difficult to solve the problemwith accurate algorithm However the GA is suitable to solvemultivariables complex problem with multiple parameters Itiswidely applied in solvingVRPdue to its extensive generalityhigh efficiency and strong robustnessTherefore we deign GAto solve the problemThe key steps are designed as follows

Encoding Natural number cod approach is used tocharacterize chromosomes Chromosome genesrsquo encoding isbased on the results of the customer groups with discontin-uous customer numbers There 0 expresses the distributioncenter and customers are expressed by numbers 1 2 3 sdot sdot sdot 119873Different groups are corresponding different chromosomalgenes The beginning and the end of the genes in the chro-mosome cod sequence must be rdquo0rdquo on behalf of the vehiclewhich departs from the distribution center and finally returnsto the distribution center after completing the deliverytasks

Initial Solution Construct initial population chromo-some cod using MATLAB programming software under

the premise of customer grouping Specific we randomlygive the initial solution as 119901119900119901(119894 2 119899 minus 1) =119870119867119911(rand(119899119906119898119890119897(119870119867119911)))119901119900119901(119894 1) = 0 and 119901119900119901(119894 119899) =0 such as 119870119867119911(rand(119899119906119898119890119897(119870119867119911)))=[0 1 4 5 9 12 18 0]And generate the initial population as a matrix with thedimension 119901119900119901119904119894119911119890 lowast (119899 + 2)

Fitness Function In this paper the objective functionvalue is used to measure the fitness value as follows

119891119896 = 1 minus (120583119896 minusmin120583)(max 120583 minusmin 120583) (35)

Selection Operator It is individual selection according toroulette wheel if the fitness of an individual is denoted by 119891119896then the probability of being selected can be represented bythe following

119901119896 = 119891119896sum119870119896=1 119891119896 (36)

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 7

Table 3 Sets indices and parameters in the C-VRPESER

Notations Detailed Definition119873 Where119873 = 0 1 2 sdot sdot sdot 119899 0 represents the depot1198730 = 119873 0 = 1 2 sdot sdot sdot 119899is the set of the customers which should be delivered119860 119860 = (119894 119895) 119894 119895 isin 119873 represents the arcs119879119877 Set of of customers in the same customer group 119877119897 Type of the vehicle119897119896 119896th vehicle with 119897 type119876119897119890 Maximum capacity of each type vehicles119894 isin 1198730 Number of customer119902119894 Demand of customers 119894119904119894 Time of vehicle arrive at customer 119894[119886119894 119887119894] Desired preferred time window for customer 119894 to be serviced119905119894 Service time of customer 119894 need119889119894119895 Distance between the customer 119894 and 119895

V119894119895 Speed of the vehicle travel on the link (119894 119895)119902119897119896119894119895 Load of the vehicle 119897119896 travel on the link (119894 119895)119876119897119888 Light weight of the 119897 type vehicle119888119900 Per unit cost of fuel119888119890 Per unit cost of carbon emissions (carbon tax)120575119888 Fuel emission factor119888119897 Fixed costs of the vehicle used119890119896 Departure time of the vehicle k from the depot119897119896 Latest arrival time at the depot after services the customers1198881 Penalty coefficient for earliness arrival1198882 Penalty coefficient for delay delivery119878119894 Service time of customer 119894st sum

119897119896isin119871119870

sum119895isin119879119877

1199091198971198960119895 = 1 (21)

sum119897119896isin119871119870

sum119895isin119879119877

1199091198971198961198950 = 1 (22)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 = 1 forall119894 isin 119879119877 (23)

sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119895ℎ minus sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896ℎ119895 = 0 forallℎ isin 119879119877 (24)

sum119894isin119879119877

119902119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119876119890 (25)

119886119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 le 119904119897119896119894 le 119887119894 sum119897119896isin119871119870

sum119895isin119879119877

119909119897119896119894119895 forall119894 isin 119879119877 (26)

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119889119894119895V119894119895 ) le 119904119897119896119894 (27)

119909119897119896119894119895 isin 0 1 (28)

119910119897119896119894 ge 0 (29)

The objective function (20) is to minimize the totalrouting cost including fixed cost energy consumption costcarbon emission cost and punishment for time windowsConstraints (21) and (22) guarantee that all service vehiclesstart from the depot and finally return to the depot foronce Constraint (23) ensures each vehicle travels on the arc(V119894 V119895) nomore than once Constraint (24) is the flow balanceand shows that the number of the vehicles which arrive atthe customer is the same as those which depart Constraint(25) ensures the loading customer demands no more thanthe vehicle capacity Inequality (26) is the time windowconstraint Constraint (27) represents the time limit forvehicle travel horizon Constraint (28) is the binary variableconstraint Constraint (29) represents the nonnegative oftime

33 Model Reformulation The above model is generatedbased on the assumption that the drive speed is invariableHowever in reality the speed is influenced by a lot of elementsTo make the model more practical we assume that thespeed is a ladder type change variable This paper supposesthe vehicles traveling speed as shown in Figure 2 On thisbasis we build the time-varying velocity distribution vehiclerouting optimization model considering the velocity change

The speed is assumed as a step function of time asFigure 2 And during different time intervals the vehiclesrsquo

8 Mathematical Problems in Engineering

20253035404550556065707580

50 10 15 20 24

Figure 2 The speed trend diagram

travel speed is different To model convince the runningtime is divided into 119868 interval according different extentsspeed which can be represented by 1198791 1198792 119879119868 The 119898thtime interval is indicated as [119905119898 119905119898+1] in each time intervalvelocity is constant

The time of vehicles running on the path (119894 119895) is a functionof the time 119897119905119894 that is the vehiclesrsquo departing time from thecustomer 119894 The speed of vehicles running on the path oftencrosses more than one speed range It is assumed that thevehicle will cross 119901 + 1 time intervals It is clear that thevehicle has 119901 different speed Assume the vehicle derived thecustomer 119894 in the119898 time interval with the speed V119898119894119895 (119897119905119894) thenthe vehicle speed turns V119898+1119894119895 (119897119905119894) in the next time intervaland we denote the vehicle speed as V119898+119901119894119895 (119897119905119894)when the vehiclederives the link (119894 119895) And the distance for the link (119894 119895) isdenoted by 119889119898+119901119894119895 (119897119905119894) And the total time 119905119905119894 for the vehicledrived across the link (119894 119895) can be represented by the timefunction 119897119905119894 as follows

119905119905119894119895 (119897119905119894) = 119889119898119894119895V119898119894119895 (119897119905119894) + 119889119898+1119894119895

V119898+1119894119895 (119897119905119894) + sdot sdot sdot + 119889119898+119901119894119895V119898+119901119894119895 (119897119905119894) (30)

Considering the variety of the vehicle traveling speed wecan represent the fuel consumption function as follows

119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)= 119868sum119902=1

119874119865119894119895 (119876119897119888 + 119902119894119895 V119898+119902119894119895 119889119898+119902119894119895 ) (31)

If we assumed that the 119896th vehicle of the 119897 kind beginsthe task at time 119910119897119896 the objective function can be denoted asfollows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]

+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(32)

The value of V119894119895 depends on the time interval The timeconstraint is represented as follows

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119905119905119894119895 (119904119897119896119894 + 119905119894)) le 119904119897119896119895 (33)

The vehicle service time cannot be earlier than the starttime of the distribution center not later than the time of thedelivery center

1199051 le 119910119897119896 le 119905119868 (34)

4 Algorithm

To present the customers-oriented vehicle routing problemwith environment consideration a two-phase model is for-mulated We design two algorithms to solve the problemFirst we design a fuzzy system clustering algorithm to realizethe customers grouping That algorithm is mainly used toconsider the customers demand attributes including quan-titative and qualitative attributes The second phase model isan en-group VRP that need to design a heuristic algorithm togain the en-group optimal delivery routes Here we comparethe performance of several heuristic algorithms as shown inTable 4 In view of the robustness and global optimization ofgenetic algorithm we design a genetic algorithm to solve theproblem

41 Fuzzy System Clustering Algorithm On the basis ofthe similarity matrix we design a fuzzy system clusteringalgorithm to cluster the customer into different groups whichare often used to process the data with quantitative andqualitative data [22] And the specific algorithm procedureis summarized as in Algorithm 1

To obtain the cluster solutions we apply the procedure inFigure 3

Mathematical Problems in Engineering 9

Table 4 Heuristic algorithm performance comparison

Algorithm Parameters Features

Tabu search algorithm(TS)

Tabu sizeNeighborhood structureNumber of solutions

Local optimumHigher solving efficiency

Ant colony algorithm(ACO)

Initial pheromoneInformation increase

Information accumulationDisappearance factorInspiration factor

Good parallelismGlobal optimization features

Time-consuming

Simulated annealingalgorithm (SA)

Initial temperatureTemperature return functionMode of state generation

Sampling principle

Simple implementationFlexible application

The algorithm converges slowly

Genetic algorithm (GA)

Population numberSelection operatorCrossover operatorMutation operator

Strong global optimization abilityParallelism and strong robustness

1 Step 1 Data initialization input the fuzzy equivalent matrix 119865(119896) Let iterations 120587=12 Step 2 Find out the largest elements in the matrix 119865119904(119896)119879 and chose the customer with the smaller number as the targetcustomer and denote it with 1199043 Step 3Mark the selected target customer and delete the corresponding row 119865119904(119896)119879 Specifically once a customer is chosen asthe target customer its column is marked by 119865119904(119896) mark the corresponding elements in row 119865119904(119896)119879 as 04 Step 4 Find out the largest element of matrix 119865119904(119896)119879 denoted the similarity value as 1205961199031199045 (1) If 120596119903119904 le 120582(pre-given threshold value) then go to Step 5 Otherwise calculate the total weight of the customers1198766 (2)Make judgement if 119876 ge 119876119897119890 go to Step 5 Otherwise put customers 119904 and 119903 in same group then delete the correspondingrow 119865119903(119896)119879 of matrix 119865(119896) return to Step 47 Step 5 Set the customer clustering termination conditions if the matrix is empty then clustering process stops else let120587= 120587 + 1 and got to Step 2

Algorithm 1 Fuzzy system clustering algorithm

42 En-Group Genetic Algorithm The proposed model is amixed-integer linear programming model When the scale ofthe problem is expended it is difficult to solve the problemwith accurate algorithm However the GA is suitable to solvemultivariables complex problem with multiple parameters Itiswidely applied in solvingVRPdue to its extensive generalityhigh efficiency and strong robustnessTherefore we deign GAto solve the problemThe key steps are designed as follows

Encoding Natural number cod approach is used tocharacterize chromosomes Chromosome genesrsquo encoding isbased on the results of the customer groups with discontin-uous customer numbers There 0 expresses the distributioncenter and customers are expressed by numbers 1 2 3 sdot sdot sdot 119873Different groups are corresponding different chromosomalgenes The beginning and the end of the genes in the chro-mosome cod sequence must be rdquo0rdquo on behalf of the vehiclewhich departs from the distribution center and finally returnsto the distribution center after completing the deliverytasks

Initial Solution Construct initial population chromo-some cod using MATLAB programming software under

the premise of customer grouping Specific we randomlygive the initial solution as 119901119900119901(119894 2 119899 minus 1) =119870119867119911(rand(119899119906119898119890119897(119870119867119911)))119901119900119901(119894 1) = 0 and 119901119900119901(119894 119899) =0 such as 119870119867119911(rand(119899119906119898119890119897(119870119867119911)))=[0 1 4 5 9 12 18 0]And generate the initial population as a matrix with thedimension 119901119900119901119904119894119911119890 lowast (119899 + 2)

Fitness Function In this paper the objective functionvalue is used to measure the fitness value as follows

119891119896 = 1 minus (120583119896 minusmin120583)(max 120583 minusmin 120583) (35)

Selection Operator It is individual selection according toroulette wheel if the fitness of an individual is denoted by 119891119896then the probability of being selected can be represented bythe following

119901119896 = 119891119896sum119870119896=1 119891119896 (36)

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

8 Mathematical Problems in Engineering

20253035404550556065707580

50 10 15 20 24

Figure 2 The speed trend diagram

travel speed is different To model convince the runningtime is divided into 119868 interval according different extentsspeed which can be represented by 1198791 1198792 119879119868 The 119898thtime interval is indicated as [119905119898 119905119898+1] in each time intervalvelocity is constant

The time of vehicles running on the path (119894 119895) is a functionof the time 119897119905119894 that is the vehiclesrsquo departing time from thecustomer 119894 The speed of vehicles running on the path oftencrosses more than one speed range It is assumed that thevehicle will cross 119901 + 1 time intervals It is clear that thevehicle has 119901 different speed Assume the vehicle derived thecustomer 119894 in the119898 time interval with the speed V119898119894119895 (119897119905119894) thenthe vehicle speed turns V119898+1119894119895 (119897119905119894) in the next time intervaland we denote the vehicle speed as V119898+119901119894119895 (119897119905119894)when the vehiclederives the link (119894 119895) And the distance for the link (119894 119895) isdenoted by 119889119898+119901119894119895 (119897119905119894) And the total time 119905119905119894 for the vehicledrived across the link (119894 119895) can be represented by the timefunction 119897119905119894 as follows

119905119905119894119895 (119897119905119894) = 119889119898119894119895V119898119894119895 (119897119905119894) + 119889119898+1119894119895

V119898+1119894119895 (119897119905119894) + sdot sdot sdot + 119889119898+119901119894119895V119898+119901119894119895 (119897119905119894) (30)

Considering the variety of the vehicle traveling speed wecan represent the fuel consumption function as follows

119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)= 119868sum119902=1

119874119865119894119895 (119876119897119888 + 119902119894119895 V119898+119902119894119895 119889119898+119902119894119895 ) (31)

If we assumed that the 119896th vehicle of the 119897 kind beginsthe task at time 119910119897119896 the objective function can be denoted asfollows

min 119906119877 = 1198881sum119894isin119879119877

max [(119886119894 minus 119904119894) 0]+ 1198882sum119894isin119879119877

max [(119904119894 minus 119887119894) 0]

+ sum(119894119895)isin119879119877

119888119900119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895)+ sum(119894119895)isin119879119877

119888119890120575119888119874119865119894119895 (119876119897119888 + 119902119897119896119894119895 V119894119895 119889119894119895) + 119888119897(32)

The value of V119894119895 depends on the time interval The timeconstraint is represented as follows

119909119897119896119894119895 (119904119897119896119894 + 119905119894 + 119905119905119894119895 (119904119897119896119894 + 119905119894)) le 119904119897119896119895 (33)

The vehicle service time cannot be earlier than the starttime of the distribution center not later than the time of thedelivery center

1199051 le 119910119897119896 le 119905119868 (34)

4 Algorithm

To present the customers-oriented vehicle routing problemwith environment consideration a two-phase model is for-mulated We design two algorithms to solve the problemFirst we design a fuzzy system clustering algorithm to realizethe customers grouping That algorithm is mainly used toconsider the customers demand attributes including quan-titative and qualitative attributes The second phase model isan en-group VRP that need to design a heuristic algorithm togain the en-group optimal delivery routes Here we comparethe performance of several heuristic algorithms as shown inTable 4 In view of the robustness and global optimization ofgenetic algorithm we design a genetic algorithm to solve theproblem

41 Fuzzy System Clustering Algorithm On the basis ofthe similarity matrix we design a fuzzy system clusteringalgorithm to cluster the customer into different groups whichare often used to process the data with quantitative andqualitative data [22] And the specific algorithm procedureis summarized as in Algorithm 1

To obtain the cluster solutions we apply the procedure inFigure 3

Mathematical Problems in Engineering 9

Table 4 Heuristic algorithm performance comparison

Algorithm Parameters Features

Tabu search algorithm(TS)

Tabu sizeNeighborhood structureNumber of solutions

Local optimumHigher solving efficiency

Ant colony algorithm(ACO)

Initial pheromoneInformation increase

Information accumulationDisappearance factorInspiration factor

Good parallelismGlobal optimization features

Time-consuming

Simulated annealingalgorithm (SA)

Initial temperatureTemperature return functionMode of state generation

Sampling principle

Simple implementationFlexible application

The algorithm converges slowly

Genetic algorithm (GA)

Population numberSelection operatorCrossover operatorMutation operator

Strong global optimization abilityParallelism and strong robustness

1 Step 1 Data initialization input the fuzzy equivalent matrix 119865(119896) Let iterations 120587=12 Step 2 Find out the largest elements in the matrix 119865119904(119896)119879 and chose the customer with the smaller number as the targetcustomer and denote it with 1199043 Step 3Mark the selected target customer and delete the corresponding row 119865119904(119896)119879 Specifically once a customer is chosen asthe target customer its column is marked by 119865119904(119896) mark the corresponding elements in row 119865119904(119896)119879 as 04 Step 4 Find out the largest element of matrix 119865119904(119896)119879 denoted the similarity value as 1205961199031199045 (1) If 120596119903119904 le 120582(pre-given threshold value) then go to Step 5 Otherwise calculate the total weight of the customers1198766 (2)Make judgement if 119876 ge 119876119897119890 go to Step 5 Otherwise put customers 119904 and 119903 in same group then delete the correspondingrow 119865119903(119896)119879 of matrix 119865(119896) return to Step 47 Step 5 Set the customer clustering termination conditions if the matrix is empty then clustering process stops else let120587= 120587 + 1 and got to Step 2

Algorithm 1 Fuzzy system clustering algorithm

42 En-Group Genetic Algorithm The proposed model is amixed-integer linear programming model When the scale ofthe problem is expended it is difficult to solve the problemwith accurate algorithm However the GA is suitable to solvemultivariables complex problem with multiple parameters Itiswidely applied in solvingVRPdue to its extensive generalityhigh efficiency and strong robustnessTherefore we deign GAto solve the problemThe key steps are designed as follows

Encoding Natural number cod approach is used tocharacterize chromosomes Chromosome genesrsquo encoding isbased on the results of the customer groups with discontin-uous customer numbers There 0 expresses the distributioncenter and customers are expressed by numbers 1 2 3 sdot sdot sdot 119873Different groups are corresponding different chromosomalgenes The beginning and the end of the genes in the chro-mosome cod sequence must be rdquo0rdquo on behalf of the vehiclewhich departs from the distribution center and finally returnsto the distribution center after completing the deliverytasks

Initial Solution Construct initial population chromo-some cod using MATLAB programming software under

the premise of customer grouping Specific we randomlygive the initial solution as 119901119900119901(119894 2 119899 minus 1) =119870119867119911(rand(119899119906119898119890119897(119870119867119911)))119901119900119901(119894 1) = 0 and 119901119900119901(119894 119899) =0 such as 119870119867119911(rand(119899119906119898119890119897(119870119867119911)))=[0 1 4 5 9 12 18 0]And generate the initial population as a matrix with thedimension 119901119900119901119904119894119911119890 lowast (119899 + 2)

Fitness Function In this paper the objective functionvalue is used to measure the fitness value as follows

119891119896 = 1 minus (120583119896 minusmin120583)(max 120583 minusmin 120583) (35)

Selection Operator It is individual selection according toroulette wheel if the fitness of an individual is denoted by 119891119896then the probability of being selected can be represented bythe following

119901119896 = 119891119896sum119870119896=1 119891119896 (36)

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 9

Table 4 Heuristic algorithm performance comparison

Algorithm Parameters Features

Tabu search algorithm(TS)

Tabu sizeNeighborhood structureNumber of solutions

Local optimumHigher solving efficiency

Ant colony algorithm(ACO)

Initial pheromoneInformation increase

Information accumulationDisappearance factorInspiration factor

Good parallelismGlobal optimization features

Time-consuming

Simulated annealingalgorithm (SA)

Initial temperatureTemperature return functionMode of state generation

Sampling principle

Simple implementationFlexible application

The algorithm converges slowly

Genetic algorithm (GA)

Population numberSelection operatorCrossover operatorMutation operator

Strong global optimization abilityParallelism and strong robustness

1 Step 1 Data initialization input the fuzzy equivalent matrix 119865(119896) Let iterations 120587=12 Step 2 Find out the largest elements in the matrix 119865119904(119896)119879 and chose the customer with the smaller number as the targetcustomer and denote it with 1199043 Step 3Mark the selected target customer and delete the corresponding row 119865119904(119896)119879 Specifically once a customer is chosen asthe target customer its column is marked by 119865119904(119896) mark the corresponding elements in row 119865119904(119896)119879 as 04 Step 4 Find out the largest element of matrix 119865119904(119896)119879 denoted the similarity value as 1205961199031199045 (1) If 120596119903119904 le 120582(pre-given threshold value) then go to Step 5 Otherwise calculate the total weight of the customers1198766 (2)Make judgement if 119876 ge 119876119897119890 go to Step 5 Otherwise put customers 119904 and 119903 in same group then delete the correspondingrow 119865119903(119896)119879 of matrix 119865(119896) return to Step 47 Step 5 Set the customer clustering termination conditions if the matrix is empty then clustering process stops else let120587= 120587 + 1 and got to Step 2

Algorithm 1 Fuzzy system clustering algorithm

42 En-Group Genetic Algorithm The proposed model is amixed-integer linear programming model When the scale ofthe problem is expended it is difficult to solve the problemwith accurate algorithm However the GA is suitable to solvemultivariables complex problem with multiple parameters Itiswidely applied in solvingVRPdue to its extensive generalityhigh efficiency and strong robustnessTherefore we deign GAto solve the problemThe key steps are designed as follows

Encoding Natural number cod approach is used tocharacterize chromosomes Chromosome genesrsquo encoding isbased on the results of the customer groups with discontin-uous customer numbers There 0 expresses the distributioncenter and customers are expressed by numbers 1 2 3 sdot sdot sdot 119873Different groups are corresponding different chromosomalgenes The beginning and the end of the genes in the chro-mosome cod sequence must be rdquo0rdquo on behalf of the vehiclewhich departs from the distribution center and finally returnsto the distribution center after completing the deliverytasks

Initial Solution Construct initial population chromo-some cod using MATLAB programming software under

the premise of customer grouping Specific we randomlygive the initial solution as 119901119900119901(119894 2 119899 minus 1) =119870119867119911(rand(119899119906119898119890119897(119870119867119911)))119901119900119901(119894 1) = 0 and 119901119900119901(119894 119899) =0 such as 119870119867119911(rand(119899119906119898119890119897(119870119867119911)))=[0 1 4 5 9 12 18 0]And generate the initial population as a matrix with thedimension 119901119900119901119904119894119911119890 lowast (119899 + 2)

Fitness Function In this paper the objective functionvalue is used to measure the fitness value as follows

119891119896 = 1 minus (120583119896 minusmin120583)(max 120583 minusmin 120583) (35)

Selection Operator It is individual selection according toroulette wheel if the fitness of an individual is denoted by 119891119896then the probability of being selected can be represented bythe following

119901119896 = 119891119896sum119870119896=1 119891119896 (36)

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

10 Mathematical Problems in Engineering

Input data

Chose decisionvariables

Generate fuzzysimilarity matrix

Add the total weightof the customers and

denoted by Q

Put the customer s and r intoa group and then delete the

row of the correspondingsimilarity matrix

If there have noelement in the row

Delete therow

If the matrix isempty

End

Yes

Yes

Yes

NO

Denote the othercustomer with the

largest element

NO

No

Yes

Q ⩽ Qe

Srs ⩾

Find out the largestelement in the matrix

customerFsk

T and chose the target

Figure 3 The flowchart of the fuzzy hierarchical clustering

Crossover Operator In order to improve the convergencespeed of the algorithm we chose adaptive crossover proba-bility as follows

119875119888 = 119901 minus 119896 times 119891119898119886119909 minus 119891119891119898119886119909 minus 119891 (119891 ge 119891)119901 otherwise

(37)

The values of the parameters 119901 are between the values08 sim 1 and the value of the parameter 119896 is between the valueof 02sim04 In this paper a partial matching crossover strategyis selected which means that the genes of two crossingpoints are interchanged by ldquotwo points crossingrdquo but the firstldquo0rdquo does not participate in the intersection as described inFigure 4

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 11

original

mark

Change

0 4 6 5 8 9 3 2 7 01

Chromosome

0 8 3 4 7 5 1 9 2 06

A46||58932||71B83||47519||26

A46||47519||71B83||58932||26

A56||47519||83B83||58932||26

A26||47519||83B71||58932||46

Substitute

Figure 4 The specific crossover operator

Mutation OperatorThis paper uses the basic bit variationof random exchange which means random exchange of thegenes encoding of chromosomes to produce new chromo-somes

Termination In this paper the termination criterionis considered as only if the maximum number of itera-tions is reached end the algorithm and output the optimalsolution

Based on the aforementioned analysis the procedureof this heuristic algorithm is summarized By algorithmrepresented as follows

Step 1 Obtain the customer group results and determine thecustomer number corresponding to the genetic code

Step 2 Set the number of initial populations 119899 adaptivecrossover probability 119875119888 mutation probability 119875119898 and maxi-mum number of iterations 119873

Step 3 Randomly generate the initial solutions119883 and let iter119898=1Step 4 By selecting crossing and mutating generatenew strategies 119883119898 calculate the objective and let119898 = 119898 + 1Step 5 If the algorithm terminates the condition (eg reachesthe set maximum iteration number) the algorithm is termi-nated otherwise returned to Step 4

To obtain the optimal solutions we applied the procedureillustrated in Figure 5

5 Case Study

In this section in order to verify the effectiveness of theproposed model and algorithms we conducted experimentswith real data The proposed cluster algorithm and GAalgorithm are coded in the MATLAB 2012a All numerical

tests are carried out on a Windows Server 2007 with IntelCPU 64 G RAM to improve the solving efficiency

51 Experiment Description and Parameter Settings Theexperimenting setting provides the experimental results andanalysis which are shown below As we know the distributioncenter is equipped with several Dong Feng vehicles with therated load for 2 tons and 3 tons and the parameters are shownin Table 5 There are 24 customers that should be servicedwhose demands can be known in advance and the customersare supermarkets and the markets in the distribution areaThe demand information and location of the customer areshown in Table 6 To calculate the cost we assume that thedriving speed of the vehicle is 40kmh for the fixed speed andthe distance between each customer and distribution centeris calculated according to their coordinates The fuel price is555 YuanL and the cost of carbon emission is 5152 YuanTThe fixed cost of using a vehicle is estimated at 100 RMB pervisit

52 Customers Group For simplicity we choose five param-eters as the main variables to cluster the customers intodifferent groups in which include geographical positionpressed for time physical properties product value andservice qualities Furthermore we use the customer demandservice time windows and the latest delivery time as theinput data to decide the service authorities and generate thedelivery sequenceThe corresponding fuzzy evaluation valuesof the five parameters are shown in Table 7

In this paper different types of vehicles are used fordistribution services which will lead to the fact that groupresults are different and thus different vehicle schedulingschemes are available just as shown in Tables 8 and 9 Takeconsideration of the two tables we can point out that thevehicle capacity influences the cluster results The overallutilization rates of the two programs are 9125 Howeverthe two programs produce different vehicle fixed costs anddifferent resources In contrast to the two programs takinginto account the fixed costs of labor costs we chose the first

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

12 Mathematical Problems in Engineering

Decimal encoding

Generate initialpopulation

Obtain the customergroup chromosome coding

Determine encodingcontent Randperm ()

Calculate the fitness value

Whether meet terminationcondition

No

Yes Output optimalsolution

Choosing

RouletteElite retention strategy

Crossing

Self-adaptive crossover probability

VariationRandom interchange operation

(SWAP)

End

Algorithmterminal12

453

678

Generate new population

Iter=iter+1 fk = 1- (k-min )(max minus Ｇin )

Max fk

pc =

Ｊ minus k timesfＧＲ-ffＧＲ-f

(f ge f)

Ｊ (f le f)

Figure 5 The specific flowchart of GA

Table 5 The parameters of the covered vehicles

parameters 2T vehicle 3T vehicleGross vehicle weight 119908119911 2T 25TVehicle emissions 119881119897 32 38Type of fuel 119891 diesel dieselLoad capacity119908119890 2T 3TThe front surface area of a vehicle 119878119888 241198982 31198982program as the actual cluster result in which intragroup pathcan be obtained And we use Figure 6 as the demonstration

53 Delivery Vehicle Path Plan Combining the customergrouping results in Section 51 we use the designed geneticalgorithm to solve the intragroup delivery vehicle routingmodel Considering that the grouped customers only needone delivery vehicle to complete the delivery service theen-group path solution was converted to the TSP problemSet the initial population to 100 the mutation probability is009 and the maximum number of iterations was set to 200Finally we get the approximate optimal delivery vehicle pathplan with a short time consumption The convergence of thealgorithm is shown in Figure 7

Considering the energy cost carbon emission cost timepenalty cost and fixed cost in the transportation processas objective functions the distribution vehicle route withinthe group is optimized As a heuristic algorithm the geneticalgorithm is generally not an optimal solution but an approx-imately optimal solution Therefore it needs to execute theprogram multiple times to obtain a better approximately

optimal solution Finally we get the optimal solution shownin Table 10 To show the specific vehicle routing we give outthe delivery path of group 1 as shown in Figure 8

54 Group Optimization Effect Analysis The proposed mod-els consider the customer demand and group the customerson the basis of considering the diversity of customer needsHowever the traditional two-phase vehicle path optimizationmodel is generally based on the customerrsquos geographicallocation and time window and does not fully consider thecustomerrsquos other demand attributes Compared with othermethods the customer-oriented distribution vehicle routingmodel proposed in this paper has two advantages For onehand it can improve the efficiency of problem solvingFor the other hand customers with similar demands aredivided into same groups so that the distribution processcan provide more targeted services In order to prove theoptimization effect of the grouping method this paper solvesthe case by designing the genetic algorithm without group-ing and compares the solution results with the calculationresults under the grouping situation In the case of without

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 13

Table 6 The demand information of the covered customers

Number Demand Distance External similarity Product value Time window Quality of Service1 05 (-103 97) high medium [800 900] high2 02 (-44-66) low low [1000 1200] lowest3 06 (-85 63) medium high [1200 1230] high4 04 (27 93) low medium [1000 1100] medium5 03 (-53 -45) lowest high [1400 1430] low6 01 (-06 52) low low [800 930] medium7 05 (186 -132) high highest [1100 1300] highest8 03 (-119 37) highest medium [1500 1630] medium9 01 (33 78) medium low [1400 1530] high10 01 (-18 -111) medium medium [1300 1500] medium11 02 (86 25) lowest lowest [1400 1430] low12 03 (-69 16) medium high [1400 1500] high13 04 (78 -7) medium highest [900 1100] medium14 02 (168 -184) low high [1500 1600] high15 01 (127 -18) high medium [11001130] highest16 06 (143 -213) medium medium [800 1100] medium17 02 (82 -158) low lowest [1400 1700] lowest18 03 (115 104) high medium [1300 1430] high19 01 (11 -85) high highest [1000 1500] high20 04 (207 -227) medium low [1100 1230] high21 03 (64 -09) low medium [800 930] low22 02 (-73 116) highest high [1300 1430] medium23 04 (-122 -24) medium low [1000 1200] medium24 05 (104 217) high high [1200 1300] highest

Table 7 The qualitative data of triangle fuzzy evaluation

Customer number Product external similarity Product value Service quality1 (05 075 1) (025 05 075) (050751)2 (0 025 05) (0 025 05) (0 0 025)3 (025 05 075) (05 075 1) (05 075 1)4 (0 025 05) (025 05 075) (025 05 075)5 (0 0 025) (05 075 1) (0 025 05)6 (0 025 05) (0 025 05) (025 05 075)7 (05 075 1) (075 1 1) (075 1 1)8 (07511) (02505075) (002505)9 (025 05 075) (0 025 05) (05 075 1)10 (02505075) (02505075) (02505075)11 (0 0 025) (0 0 025) (0 025 05)12 (025 05 075) (05 075 1) (050751)13 (025 05 075) (075 1 1) (025 05 075)14 (05 075 1) (05 075 1) (05 075 1)15 (05 075 1) (025 05 075) (075 1 1)16 (02505075) (02505075) (02505075)17 (0 025 05) (0 0 025) (0 0 025)18 (050751) (025 05 075) (050751)19 (05 075 1) (075 1 1) (050751)20 (025 05 075) (0 025 05) (075 1 1)21 (0 025 05) (025 05 075) (0 025 05)22 (075 1 1) (05 075 1) (0 0 025)23 (025 05 075) (0 025 05) (02505075)24 (05 075 1) (05 075 1) (05 075 1)

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

14 Mathematical Problems in Engineering

Table 8 The clustering results of vehicle restrictions for 2 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=2tGroup 1 621241714822 19Group 2 724151195 20Group 3 9121031323 19Group 4 11181620 15

Table 9 The clustering results of vehicle restrictions for 3 tons

Clustercondition Group No Customer No Cargo volume (tons)

119876119890120582=3t Group 1 621241714822511 24Group 2 7241511918 20Group 3 91210313231620 19

Table 10 The optimization paths results with customer grouping

Group No The optimal path en-group Distance(Km) CostGroup 1 0 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr 22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr 17 997888rarr 14 997888rarr 11 997888rarr 0 10922 4033Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr 15 997888rarr 24 997888rarr 18 997888rarr 0 10456 32996Group 3 0 997888rarr 13 997888rarr 16 997888rarr 20 997888rarr 10 997888rarr 23 997888rarr 3 997888rarr 12 997888rarr 9 997888rarr 0 10644 35911

grouping the optimal path of delivery vehicles is shown inTable 11

(1) Improve Solution EfficiencyWithout grouping we shouldcalculate the number of vehicles used in the task In thispaper 119898 = [sum1205931119894 119876119890] + 1 is a generally chosen and thenumber of vehicles is given out 3 of 3t Dongfeng vehicleThe genetic algorithm coding method adopts the naturalnumber coding method and finally determine the chromo-some of length 28 as (0 2 4 13 0 7 0 16 23 0)Thecustomers demand between two zeros on the chromosomecannot exceed the vehicle load We can point out that thechromosome coding length is longer than en-group whichmakes the cross-variation complexity and easy to fall intothe local optimum However the position of 0 needs tobe adjusted after each iteration to ensure vehicle loadingconstraints The complexity of the algorithm is high and thesolution is difficult Through the solution it is found thatthe convergence effect of the algorithm is less than the en-group algorithm and it is easy to fall into the local optimumThe calculation results are quite different the solution timewithout grouping is about 537s however the total solutiontime after grouping is about 300s It easy to point out that thecomputational efficiency is improved by 79

(2) Improve the Quality of Delivery Services In order tomeasure the optimization effect of customer grouping ondistribution service We calculate the similarity standarddeviation of customer demand attribute values within eachgroup which are regarded as the comprehensive satisfactionevaluation value of this group The smaller the standarddeviation the greater the similarity of customer demandwhich helps to arrange targeted delivery services and improve

service quality This paper takes the parameter service qualityas an example to calculate the standard deviation of theevaluation value Calculate the similarity standard deviationfor two conditions and compare the results The calculationresult is shown in Figure 9

According to Figure 9 it can be seen that the customerhas a high similarity to the quality of service requirements oneach of the distribution vehicle routes based on the customergrouping and the variance is small On the contrary thesolution without grouping has lower similarity in servicequality requirements on each path

55 Analysis of Optimization Effect of Energy Saving andEmission Reduction The energy consumption and carbonemissions are affected by many factors the main influencingfactors are path length vehicle load and vehicle speedExisting research mainly aims to reduce transportation costsby shortening the length of vehicle path The optimizationgoal set in this paper is no longer simple consideringthe shortest path but comprehensively considering reduc-ing energy consumption and carbon emission costs andachieving comprehensive optimization goals based on energyconservation and emission reduction In order to prove thevalidity of the model the classic shortest path model isselected and the optimization effects of the two types ofmodels are analyzed through case comparison

The optimal path of each group is solved by the shortestpath model and the minimum energy consumption andcarbon emission model respectively The change of objectivefunction causes the change of fitness function On this basiswe obtain the optimal path for each evaluation mechanismFurthermore the corresponding path length energy con-sumption and carbon emission are calculated respectively

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 15

Group 1

Group 3

Group 2

Ungrouped

Group 1Group 2Group 3

Figure 6 The three-dimensional diagram of the customer grouping

Obj

ectiv

e val

ue

340

360

380

400

420

440

460

480Iteration change diagram

20 40 60 80 100 120 140 160 180 2000iterations

Figure 7 The algorithm convergence diagram

To simplify we denote the minimum energy consumptionand carbon emission path model as ECCM and the shortestpath model is remarked as SPM Calculate the two modelsseparately and get their corresponding distribution vehiclepath plan as shown in Tables 12 and 13

It can be seen that in the distribution vehicle path schemecalculated by the two models only the optimal paths of thesecond group are the same the other two groups are differentThis shows that the shortest path of the delivery vehicleis not necessarily the optimal route of the delivery vehiclecorresponding to the minimum energy consumption andcarbon emissions Taking the first group as an example thedemand of customer 4 is much larger than other customers

In order to reduce the overall cost the vehicle will serve thecustomer 4 as early as possible if the path length does notchange much Finally we get the total energy consumption ofthe vehicle distribution calculated by the optimization modelECCM which is 4164L and the carbon emission is 11367kgCompared with the optimization model SPM the energyconsumption and carbon emissions are reduced by 65

56 Sensitivity Analysis of Velocity Variations The impact ofspeed changes on vehicle energy consumption and carbonemissions was studied by analyzing the impact of peak hourson road vehicle speeds According to the investigation thedistribution area generally has traffic congestion from 800

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

16 Mathematical Problems in Engineering

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

22

23

24

0

Total cost 40348633Iter 200

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25cu

stom

ers l

ocat

ion

Y

15100 5 20 25minus10 minus5minus15customers location X

Figure 8 The optimal path of group 1

Group1Group2Group3

En-group Ungroup

021

013 013

026

035

024

040

035

030

025

020

015

010

005

000

Figure 9 The comparison chart of customer group service quality variance

to 900 am 1200 to 1400 and 1700 to 1900 during theseperiods the speed of vehicle is lower Assume that the speedof the vehicle varies in steps over one day as shown inTable 14

The distribution activity starts at 800 am taking theoptimization of the delivery vehicle path of customer group1 as an example the optimal path is obtained as shown inFigure 10 And we also give out the contrast of the threegroupsrsquo optimal path with and without speed variation justas Table 15 Affected by the speed change it may cause thevariation of the delivery order For example the delivery pathof the customer group 3 is changedThis is because the vehiclewill try to choose the distribution vehicle path with the lowesttotal cost If the vehicle travels according to the original route

the cost is 3739 Yuan which is higher than the new deliveryvehicle path In this example the total cost of distributionconsidering time-varying speed is 116036 Yuan which is63 higher than the cost of 109171 Yuan by fixed speed andthe energy consumption is increased by 54This shows thatthe speed change has a certain impact on the distribution costand fuel consumption and it needs to be considered whendesigning the distribution line

6 Conclusion and Further Study

The fastest consumer demand growth requires an efficientdistribution planning Nowadays special attention is givento routes in which fuel consumption and emissions can be

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 17

Table 11 The optimization paths results with no grouping

Group No The optimal path en-group Distribution quality(tons)Path 1 0 997888rarr 3 997888rarr 1 997888rarr 22 997888rarr 6 997888rarr 9 997888rarr 4 997888rarr 18 997888rarr 11 997888rarr 21 997888rarr 0 27Path 2 0 997888rarr 13 997888rarr 19 997888rarr 16 997888rarr 14 997888rarr 7 997888rarr 17 997888rarr 10 997888rarr 2 997888rarr 5 997888rarr 20 997888rarr 0 3path 3 0 997888rarr 12 997888rarr 8 997888rarr 23 997888rarr 24 997888rarr 15 997888rarr 0 16

Table 12 The data results of the ECCM

Group No The optimal pathen-group

Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 10 997888rarr 21 997888rarr 11 997888rarr4 997888rarr 6 997888rarr 22 997888rarr8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 0 10704 1464 3936

Group 20 997888rarr 1 997888rarr 24 997888rarr18 997888rarr 13 997888rarr 7 997888rarr19 997888rarr 0 10121 1253 3421

Group 30 997888rarr 13 997888rarr 20 997888rarr16 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 10576 1447 395

Table 13 The data results of the SPM

Group No The optimal path en-group Distance(Km)

Energyconsumption

(L)

Carbonemission(Kg)

Group 1 0 997888rarr 21 997888rarr 11 997888rarr 14 997888rarr 17 997888rarr 2 997888rarr 5 997888rarr 8 997888rarr 22 997888rarr 6 997888rarr 6 997888rarr 0 10326 1633 4456Group 2 0 997888rarr 1 997888rarr 24 997888rarr 18 997888rarr 13 997888rarr 7 997888rarr 19 997888rarr 0 10121 1253 3421Group 3 0 997888rarr 13 997888rarr 20 997888rarr 16 997888rarr 10 997888rarr 23 997888rarr 12 997888rarr 3 997888rarr 9 997888rarr 0 10194 1549 4230

Table 14 The relational table of velocity and time

Period [000600] [600900] [9001200] [12001400] [14001700] [17001900] [19002200]Speed(Kmh) 50 30 40 35 40 30 40

Table 15 The data results of the SM

Group No The optimal pathen-group

Whether the pathdirection changes

Total cost(Y)

Carbon emission(Kg)

Group 10 997888rarr 21 997888rarr 6 997888rarr 4 997888rarr22 997888rarr 8 997888rarr 5 997888rarr 2 997888rarr17 997888rarr 14 997888rarr 11 997888rarr 0 NO 44584 1589

Group 2 0 997888rarr 1 997888rarr 19 997888rarr 7 997888rarr15 997888rarr 24 997888rarr 18 997888rarr 0 NO 34352 1304

Group 30 997888rarr 13 997888rarr 16 997888rarr20 997888rarr 10 997888rarr 23 997888rarr3 997888rarr 12 997888rarr 9 997888rarr 0 Change 37100 1495

reduced The manuscript proposes a two-phase approachto handle vehicle routing problem with time windows andmultivehicles coupled with environmental concerns andcustomer demand First phase is concerned to establishcostumer clusters and second phase is devoted to formulatethe en-group mathematical formulation of the problem anddevelop a genetic algorithm to account for vehicle routingoptimization within each group so that fuel consumptionand emissions are minimized The comprehensive emission

measurement model from Barth and Scora (2004) is usedto calculate the energy consumption and carbon emissionThe algorithm accounts for similarity between customerswhich is represented in a fuzzy similarity matrix on thisbasis the customers can be cluster in appropriate groupsConsidering the energy carbon emission time penalty andfixed costs in the transport process incorporated in theobjective function the distribution vehicle route en-group isoptimized Numerical results are presented for real data from

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

18 Mathematical Problems in Engineering

The total cost 44584558

X coordinate

Y co

ordi

nate

25

20

15

10

5

0

minus5

minus10

minus15

minus20

minus252520151050minus5minus10minus15

Figure 10 The optimization paths of time-varying speed problem

Shandong province Sensitivity analysis of velocity variationsis also presented which verified that the proposed two-phaseapproach could improve the service quality and reduce thefuel consumption and carbon emission

Further studies can focus on the following aspects (1)Electric logistics vehicles are not considered in this paperwhich will be meaningful for the green vehicle routingproblem The electric vehicle is a critical topic for furtherstudy (2)This study designs a GA algorithm to generate thenear-optimal solution in the future designing more effectivealgorithms for larger scale case will be a direction

Data Availability

Data used in this manuscript are available on the Chinacarbon trading website at httpwwwtanpaifangcom andthe customer orders

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The work described in this paper was jointly supportedby the National Natural Science Foundation of China (no71571011) the Fundamental Research Funds for CentralUniversities (no 2018YJS191) and the State Key Laboratoryof Rail Traffic Control and Safety Beijing Jiaotong University(no RCS2017ZJ001)

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] C Cetinkaya I Karaoglan and H Gokcen ldquoTwo-stage vehiclerouting problem with arc time windows a mixed integerprogramming formulation and a heuristic approachrdquo European

Journal of Operational Research vol 230 no 3 pp 539ndash5502013

[3] F Miguel ldquoEmissions minimization vehicle routing problemrdquoe 89th Annual Meeting of the Transportation Research Boardpp 224ndash231 2010

[4] Y Huang L Zhao T Van Woensel and J-P Gross ldquoTime-dependent vehicle routing problem with path flexibilityrdquo Trans-portation Research Part B Methodological vol 95 pp 169ndash1952017

[5] J F Ehmke A M Campbell and B W Thomas ldquoVehiclerouting tominimize time-dependent emissions in urban areasrdquoEuropean Journal of Operational Research vol 251 no 2 pp478ndash494 2016

[6] B Golden A Assad L Levy and F Gheysens ldquoThe fleet sizeand mix vehicle routing problemrdquo Computers amp OperationsResearch vol 11 no 1 pp 49ndash66 1984

[7] J Brandao ldquoA tabu search algorithm for the heterogeneousfixed fleet vehicle routing problemrdquo Computers amp OperationsResearch vol 38 no 1 pp 140ndash151 2011

[8] B Ornbuki M Nakamura and M Osamu ldquoA hybrid searchbased on genetic algorithm and tabu search for vehicle routingrdquoin Proceedings of the 6th International Conference on ArtificialIntelligence and So Computing vol 7 pp 176ndash181 BanffCanada 2002

[9] J CordeauMGendreau andG Laporte ldquoA tabu search heuris-tic for periodic and multi-depot vehicle routing problemsrdquoNetworks vol 30 no 2 pp 105ndash119 1997

[10] G Y Tutuncu ldquoAn interactive GRAMPS algorithm for theheterogeneous fixed fleet vehicle routing problem with andwithout backhaulsrdquo European Journal of Operational Researchvol 201 no 2 pp 593ndash600 2010

[11] E Demir T Bektas and G Laporte ldquoAn adaptive large neigh-borhood search heuristic for the pollution-routing problemrdquoEuropean Journal of Operational Research vol 223 no 2 pp346ndash359 2012

[12] T Bektas and G Laporte ldquoThe pollution-routing problemrdquoTransportation Research Part B Methodological vol 45 no 8pp 1232ndash1250 2011

[13] A Palmere development of an integrated routing and carbondioxide emissions model for goods vehicles Cranfield University2007

Mathematical Problems in Engineering 19

[14] EDemir T Bektas andG Laporte ldquoThe bi-objective pollution-routing problemrdquo European Journal of Operational Researchvol 232 no 3 pp 464ndash478 2014

[15] Y-J Kwon Y-J Choi andD-H Lee ldquoHeterogeneous fixed fleetvehicle routing considering carbon emissionrdquo TransportationResearch Part D Transport and Environment vol 23 no 8 pp81ndash89 2013

[16] J Qian Fuel emission optimization in vehicle routing problemswith time-varying speeds Lancaster University ManagementSchool Department of Management Science 2011

[17] J Qian and R Eglese ldquoFuel emissions optimization in vehiclerouting problems with time-varying speedsrdquo European Journalof Operational Research vol 248 no 3 pp 840ndash848 2016

[18] M W P Savelsbergh ldquoLocal search in routing problems withtime windowsrdquo Annals of Operations Research vol 4 no 1ndash4pp 285ndash305 1985

[19] O Braysy and M Gendreau ldquoVehicle routing problem withtime windows part I route construction and local searchalgorithmsrdquo Transportation Science vol 39 no 1 pp 104ndash1182005

[20] O B Gendreau ldquoVehicle routing problem with time windowsPart II metaheuristicsrdquoTransportation Science vol 39 no 1 pp119ndash139 2005

[21] H I Calvete C Gale M-J Oliveros and B Sanchez-ValverdeldquoA goal programming approach to vehicle routing problemswith soft time windowsrdquo European Journal of OperationalResearch vol 177 no 3 pp 1720ndash1733 2007

[22] J B Sheu ldquoA hybrid fuzzy-optimization approach to customergrouping-based logistics distribution operatiordquo Applied Mathe-matical Modelling vol 31 no 2 pp 1048ndash1066 2007

[23] T Hu and J B Sheu ldquoA fuzzy-based customer classificationmethod for demand-responsive logistical distribution opera-tionsrdquo Fuzzy Sets and Systems vol 139 no 2 pp 431ndash450 2003

[24] M Barth G Scora and T Younglove ldquoModal emissions modelfor heavy-duty diesel vehiclesrdquoTransportation Research Recordno 1880 pp 10ndash20 2004

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom