A Multi-criteria Decision Making Approach for Location Planning

8/12/2019 A Multi-criteria Decision Making Approach for Location Planning

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Mathematical and Computer Modelling 53 (2011) 98109

Contents lists available atScienceDirect

Mathematical and Computer Modelling

journal homepage:www.elsevier.com/locate/mcm

A multi-criteria decision making approach for location planning forurban distribution centers under uncertainty

Anjali Awasthi a,, S.S. Chauhan b, S.K. Goyal b

a CIISE, Concordia University, Montreal, Canadab Decision Sciences, JMSB, Concordia University, Montreal, Canada

a r t i c l e i n f o

Article history:

Received 9 December 2009

Received in revised form 25 July 2010

Accepted 26 July 2010

Keywords:

Location planning

Urban distribution center

Fuzzy TOPSIS

Multi-criteria decision making

a b s t r a c t

Location planning for urban distribution centers is vital in saving distribution costs and

minimizing traffic congestion arising from goods movement in urban areas. In this paper,

we present a multi-criteria decision making approach for location planning for urban

distribution centers under uncertainty. The proposed approach involves identification of

potential locations, selection of evaluation criteria, use of fuzzy theory to quantify criteria

values under uncertainty and application of fuzzy TOPSIS to evaluate and select the best

location for implementing an urban distribution center. Sensitivity analysis is performed

to determine the influence of criteria weights on location planning decisions for urban

distribution centers.

The strength of the proposed work is the ability to deal with uncertainty arising due

to a lack of real data in location planning for new urban distribution centers. The proposed

approach canbe practically applied by logisticsoperators in deciding on thelocationof new

distribution centers considering the sustainable freight regulations proposed by municipaladministrations. A numerical application is provided to illustrate the approach.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

Location planning for urban distribution centers is vital in minimizing traffic congestion arising from goods movementin urban areas. In recent years, transport activity has grown tremendously and this has undoubtedly affected the traveland living conditions in urban areas. Considering this growth in the number of urban freight movements and theirnegative impacts on city residents and the environment, municipal administrations are implementing sustainable freightregulations like restricted delivery timing, dedicated delivery zones, congestion charging etc. With the implementationof these regulations, the logistics operators are facing new challenges in location planning for distribution centers. For

example, if distribution centers are located close to customer locations, then it will increase traffic congestion in the urbanareas. If they are located far from customer locations, then the distribution costs for the operators will be very high. Underthese circumstances, it is clear that the location planning for distribution centers in urban areas is a complex decision thatinvolves consideration of multiple criteria like maximum customer coverage, minimum distribution costs, least impacts oncity residents and the environment, and conformance to freight regulations of the city.

In this paper, we are focusing on a multi-criteria decision making approach for location planning for urban distributioncenters under uncertainty. We will use fuzzy theory to model decision making parameters. Fuzzy theory was introducedby Zadeh[1] to deal with uncertainty and vagueness in decision making. It is used to model systems that are difficult todefine precisely [2,3]. In such systems, parameters are defined using linguistic terms instead of exact numerical values [4].

Corresponding author. Tel.: +1 514 848 2424x5622; fax: +1 514 848 3171.E-mail addresses:[email protected](A. Awasthi),[email protected](S.S. Chauhan),[email protected](S.K. Goyal).

0895-7177/$ see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2010.07.023
http://dx.doi.org/10.1016/j.mcm.2010.07.023http://www.elsevier.com/locate/mcmhttp://www.elsevier.com/locate/mcmmailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.mcm.2010.07.023http://dx.doi.org/10.1016/j.mcm.2010.07.023mailto:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/mcmhttp://www.elsevier.com/locate/mcmhttp://dx.doi.org/10.1016/j.mcm.2010.07.023


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A. Awasthi et al. / Mathematical and Computer Modelling 53 (2011) 981 09 99

For example, the impact of motor traffic on city residents can be expressed as high, very high, low etc. using linguistic terms.Then, conversion scales are applied to transform the linguistic scales into fuzzy numbers.

The problem of location planning for urban distribution centers can be classified as a special case of the more generalfacility location problems. The facility location problem usually involves a set of locations (alternatives) which are evaluatedagainst a set of weighted criteria independent from each other. The alternative that performs best with respect to all criteriais chosen for implementation. The distinct feature in location planning for urban distribution centers is the consideration ofinterests of other stakeholders like city residents, municipal administrators etc. The goal is not only to minimize distribution

costs but also to conform to sustainable freight regulations of the city and create least negative effects on city residentsand their environment. Several approaches have been reported in the literature for solving the facility location problems[58]. Agrawal[9] present a hybrid Taguchi-immune approach to optimize an integrated supply chain design problem withmultiple shipping. Sun et al. [10]present a bilevel programming model for the location of logistics distribution centers.Multi-criteria facility location models have been investigated by researchers in [1114].

The most commonly used approaches can be classified as continuous location models, network location models andinteger programming models. In continuous location models, every point on the plane is a candidate for facility locationand a suitable distance metric is used for selecting the locations. In network location models, distances are computed asshortest paths in a graph. Nodes represent demand points and potential facility sites correspond to a subset of the nodesand to points on arcs. The integer programming models start with a given set of potential facility sites and use integerprogramming to identify best locations for facilities. The objectives may be either of the minsum type or of the minmax type.Minsum models are designed to minimize average distances while minmax models have to minimize maximum distances.A detailed overview of facility location models for distribution network design can be found in[8].

It is worth pointing out that most of the above papers study the location problem under a certain environment, thatis, the parameters in the problem are fixed numbers and known in advance. In reality, often the parameters cannot beobtained with certainty. To deal with such situations, fuzzy theory is used [ 15,16]. Application of fuzzy theory for locationplanning for facilities has been investigated by researchers in[1722]. Liang and Wang [20] developed an algorithm forfacility site selection based on fuzzy theory and hierarchical structure analysis. Chen [21] developed a fuzzy multi-attributedecision making approach for the distribution center location selection problem. Chu [ 23]developed a fuzzy TOPSIS modelto solve the facility location selection problem under group decision making. Kahraman et al. [19] used four fuzzy multi-attribute group decision makingapproaches in evaluating facility locations. Chou et al.[22] presented a fuzzy simple additiveweighting system under group decision making for facility location selection with objective and subjective attributes. Leeand Lin [24] presented a fuzzy quantified SWOT procedure for environmental evaluation of an international distributioncenter.

The rest of the paper is organized as follows. In Section2,we present the preliminaries of fuzzy set theory. Section3presents the proposed framework for multi-criteria location planning for urban distribution centers under a fuzzy

environment. In Section4,we present a numerical application to illustrate the proposed approach. Finally, the conclusionsand future work are presented in Section5.

2. Fuzzy set theory

Some related definitions of fuzzy set theory adapted from [1,3,25,26]are presented as follows.

Definition 1. A fuzzy set a in a universe of discourseX is characterized by a membership function a(x)that maps eachelementxinXto a real number in the interval [0, 1]. The function value a(x)is termed the grade of membership ofxin a[27]. The nearer the value ofa(x)to unity, the higher the grade of membership ofxin a.

Definition 2. A triangular fuzzy number (Fig. 1)is represented as a triplet a= (a1, a2, a3). The membership functiona(x)of triangular fuzzy number ais given by

a(x)=

0, x a1,xa1

a2 a1, a1 x a2,

a3 x

a3 a2, a2 x a3,

0, x> a3

(1)

wherea1, a2, a3are real numbers and a1 < a2 < a3. The value ofxat a2 gives the maximal grade ofa(x), i.e.,a(x) = 1;it is the most probable value of the evaluation data. The value ofxata1gives the minimal grade ofa(x), i.e.,a(x) = 0; itis the least probable value of the evaluation data. Constants a1and a3are the lower and upper bounds of the available areafor the evaluation data. These constants reflect the fuzziness of the evaluation data [ 20]. The narrower the interval [a1, a3],the lower the fuzziness of the evaluation data.

For a detailed study of fuzzy set theory, please refer to [ 1,3,25,26].


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100 A. Awasthi et al. / Mathematical and Computer Modelling 53 (2011) 98109

Fig. 1. Triangular fuzzy numbera.

Fig. 2. Two triangular fuzzy numbers.

Table 1

Linguistic terms for alternative ratings.

Linguistic term Membership function

Very poor (VP) (1, 1, 3)

Poor (P) (1, 3, 5)

Fair (F) (3, 5, 7)

Good (G) (5, 7, 9)

Very good (VG) (7, 9, 9)

Table 2

Linguistic terms for criteria ratings.

Linguistic term Membership function

Very low (1, 1, 3)

Low (1, 3, 5)

Medium (3, 5, 7)

High (5, 7, 9)

Very high (7, 9, 9)

2.1. The distance between fuzzy triangular numbers

Let a = (a1, a2, a3)and b= (b1, b2, b3)be two triangular fuzzy numbers(Fig. 2).

The distance between them is given using the vertex method by

d(a,b) =

1

3[(a1 b1)

2 +(a2 b2)2 +(a3 b3)

2]. (2)

2.2. Linguistic variables

In fuzzy set theory, conversion scales are applied to transform the linguistic terms into fuzzy numbers. In this paper, wewill apply a scale of 19 for rating the criteria and the alternatives. Table 1 presents the linguistic variables and fuzzy ratingsfor the alternatives andTable 2presents the linguistic variables and fuzzy ratings for the criteria.

3. The proposed framework for location planning for urban distribution centers

The proposed framework for location planning for urban distribution centers comprises four steps. These steps arepresented in detail as follows:


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Table 3

Criteria for location selection.

Criteria Definition Criteria type

Accessibility (C1) Access by public and private transport modes to the location Benefit (the more the better)

Security (C2) Security of the location from accidents, theft and vandalism Benefit (the more the better)

Connectivity to multimodal

transport (C3)

Connectivity of the location with other modes of transport,

e.g. highways, railways, seaport, airport etc.

Benefit (the more the better)

Costs (C4) Costs in acquiring land, vehicle resources, drivers, and taxesetc. for the location

Cost (the less the better)

Environmental impact (C5) Impact of location on the environment, for example, air

pollution, noise

Cost (the less the better)

Proximity to customers (C6) Distance of location to customer locations Benefit (the more the better)

Proximity to suppliers (C7) Distance of location to supplier locations Benefit (the more the better)

Resource availability (C8) Availability of raw material and labor resources in the

location


Conformance to sustainable

freight regulations (C9)

Ability to conform to sustainable freight regulations imposed

by municipal administrations for e.g. restricted delivery

hours, special delivery zones


Possibility of expansion (C10) Ability to increase size to accommodate growing demands Benefit (the more the better)

Quality of service (C11) Ability to assure timely and reliable service to clients Benefit (the more the better)

3.1. Selection of location criteria

Step 1 involves the selection of location criteria for evaluating potential locations for urban distribution centers.These criteria are obtained from literature review, and discussion with transportation experts and members of the citytransportation group. 11 criteria are finally chosen to determine the best location for implementing urban distributioncenters. These criteria are shown inTable 3.

It can be seen inTable 3that criterion C3 and criterion C4 belong to the cost category, that is, the lower the value, themore preferable the alternative for the best location. The remaining criteria are benefit type criteria, that means the higherthe value, the more preferable the alternative for the best location.

3.2. Selection of potential locations

Step 2 involves selection of potential locations for implementing urban distribution centers. The decision makers usetheir knowledge, prior experience with the transportation conditions of the city and the presence of sustainable freightregulations in the city to identify candidate locations for implementing urban distribution centers. For example, if certainareas are restricted for delivery by municipal administration, then these areas are barred from being considered as potentiallocations for implementing urban distribution centers. Ideally, the potential locations are those that cater to the interest ofall city stakeholders, that is city residents, logistics operators, municipal administrations etc.

3.3. Locations evaluation using fuzzy TOPSIS

The third step involves evaluation of potential locations against the selected criteria ( Table 3) using the technique calledfuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Situation). The TOPSIS approach chooses the alternativethat is closest to thepositive ideal solution andfarthestfrom thenegative ideal solution.A positive ideal solution is composedof the best performance values for each attribute whereas the negative ideal solution consists of the worst performancevalues. Fuzzy TOPSIS has been applied to facility location problems by researchers in[23,2831]. The various steps of fuzzyTOPSIS are presented as follows:

Step1. Assignment of ratings to the criteria and the alternatives.Let us assume there are J possible candidates called A = {A1,A2, . . . ,Aj} which are to evaluated against m criteria,

C = {C1, C2, . . . , Cm}. The criteria weights are denoted bywi(i = 1, 2, . . . , m). The performance ratings of each decisionmaker Dk(k= 1, 2, . . . , K) for each alternativeAj(j= 1, 2, . . . , n) with respect to criteria Ci(i = 1, 2, . . . , m) are denoted

by Rk =xijk(i = 1, 2, . . . , m; j= 1, 2, . . . , n; k= 1, 2, . . . , K)with membership functionRk (x).

Step2. Compute aggregate fuzzy ratings for the criteria and the alternatives.

If the fuzzy ratings of all decision makers are described as triangular fuzzy numbers Rk = (ak, bk, ck), k = 1, 2, . . . , K,

then the aggregated fuzzy rating is given by R = (a, b, c), k= 1, 2, . . . , K, where

a = mink

{ak}, b=1

K

Kk=1

bk, c=maxk

{ck}. (3)

If the fuzzy rating and importance weight of the kth decision maker are xijk = (aijk, bijk, cijk)and wijk = (wjk1, wjk2, wjk3),i = 1, 2, . . . , m,j = 1, 2, . . . , n, respectively, then the aggregated fuzzy ratings ( xij)of alternatives with respect to each


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criterion are given by xij =(aij, bij, cij)where

aij =mink

{aijk}, bij =1

K

Kk=1

bijk, cij =maxk

{cijk}. (4)

The aggregated fuzzy weights(wij)of each criterion are calculated as wj =(wj1, wj2, wj3)where

wj1 =mink {wjk1}, wj2 =

1

K

Kk=1

wjk2, wj3 =maxk {cjk3}. (5)

Step3. Compute the fuzzy decision matrix.

The fuzzy decision matrix for the alternatives(D)and the criteria( W)is constructed as follows:

D=

C1 C2 Cn

A1 x11 x12 . . . x1nA2 x21 x22 . . . x2nA3 . . . . . . . . . . . .A4 xm1 xm2 . . . xmn

, i = 1, 2, . . . , m; j= 1, 2, . . . , n (6)

W =( w1, w2, . . . , wn). (7)

Step4. Normalize the fuzzy decision matrix.

The raw data are normalized using a linear scale transformation to bring the various criteria scales onto a comparablescale. The normalized fuzzy decision matrix Ris given by

R = [rij]mxn, i = 1, 2, . . . , m; j= 1, 2, . . . , n (8)

where

rij =

aij

cj,

bij

cj,

cij

cj

and cj =max

icij (benefit criteria) (9)

rij =

aj

cij,

aj

bij,

aj

aij

and aj =min

iaij (cost criteria). (10)

Step5. Compute the weighted normalized matrix.

The weighted normalized matrix Vfor criteria is computed by multiplying the weights ( wj)of evaluation criteria withthe normalized fuzzy decision matrixrij:

V = [vij]mxn, i = 1, 2, . . . , m;j= 1, 2, . . . , n where vij =rij(.)wj. (11)

Step6. Compute the fuzzy ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS).The FPIS and FNIS of the alternatives are computed as follows:

A =(v1 ,v2 , . . . , v

n ) where v

j =max

i{vij3}, i = 1, 2 . . . , m; j= 1, 2, . . . , n (12)

A =(v1,v2, . . . , vn)where v

j =mini

{vij1}, i = 1, 2 . . . , m; j= 1, 2, . . . , n. (13)

Step7. Compute the distance of each alternative from FPIS and FNIS.The distance(di, d

i )of each weighted alternativei = 1, 2, . . . , mfrom the FPIS and the FNIS is computed as follows:

di =

nj=1

dv (vij,v

j), i = 1, 2, . . . , m (14)

di =

nj=1

dv (vij,v

j ), i = 1, 2, . . . , m (15)

wheredv (a,b)is the distance measurement between two fuzzy numbers aand b.

Step8. Compute the closeness coefficient(CCi)of each alternative.The closeness coefficient CCirepresents the distances to the fuzzy positive ideal solution (A

) and the fuzzynegative idealsolution (A) simultaneously. The closeness coefficient of each alternative is calculated as

CCi =di

d

i

+d

i

, i = 1, 2, . . . , m. (16)

Step9. Rank the alternatives.


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A2A3

A1

Highway

Distribution Center location

Customer locations

Restricted delivery zones

Fig. 3. Potential locations for the urban distribution center.

Rank the alternatives according to the closeness coefficient(CCi)in decreasing order and select the alternative with thehighest closeness coefficient for final implementation. The best alternative is closest to the FPIS and farthest from the FNIS.

3.4. Sensitivity analysis

Sensitivity analysis addresses the question How sensitive is the overall decision to small changes in the individualweights assigned during the pairwise comparison process? This question can be answered by slightly varying the values ofthe weights and observing the effects on the decision. This is useful in situations where uncertainties exist in the definitionof the importance for different factors. In our case, we will conduct sensitivity analysis in order to see the importance ofcriteria weights in deciding the best location for implementing the urban distribution center.

4. Numerical illustration

Let us assume that a logistics company is interested in implementing a new urban distribution center. A committee ofthree decision makers D1, D2 and D3 is formed to select the best selection. The alternatives available for location selectionare A1, A2 and A3(Fig. 3).

It can be seen that location A1 is situated outside the city close to a highway while locations A2 and A3 are located insidethe city. Location A2 is situated on the outskirts inside the city close to highways and to the customer locations whereaslocation A3 is situated in the city center far from highways.

The criteria used for evaluation of locations for urban distribution centers are same as those presented in Table 3, that is:Accessibility (C1), Security (C2), Connectivity to multimodal transport (C3), Costs (C4), Environmental impact (C5), Proximityto customers (C6), Proximity to suppliers (C7), Resource availability (C8), Conformance to sustainable freight regulations(C9), Possibility of expansion (C10), and Quality of service (C11). Among these, Criteria C3 and C4 are the cost categorycriteria (the less the better) while the remaining ones are the benefit category (the more the better) criteria.

The committee provided linguistic assessments for the eleven criteria using rating scales given in Table 1 and to the three

alternatives (locations) for each of the 11 location criteria using rating scales ofTable 2. Tables 4 and 5 present the linguisticassessments for the criteria and the alternatives.

Then, the aggregated fuzzy weights (wij) for each criterion are calculated using Eq. (5).For example, for criterion C1Accessibility, the aggregated fuzzy weight is given by wj =(wj1, wj2, wj3)where

wj1 =mink

{7, 7, 7}, wj2 =1

3

3k=1

(9+9+9), wj3 =maxk

{9, 9, 9} = wj =(7, 9, 9).

Likewise, we compute the aggregate weights for the remaining 10 criteria. The aggregate weights of the 11 criteria arepresented inTable 6.

Then, the aggregate fuzzy weights of the alternatives are computed using Eq. (4).For example, the aggregate rating foralternative A1 for criterion C1 given by the three decision makers is computed as follows:

aij =mink

{7, 5, 7}, bij =1

3

3k=1

(9+7+9), cij =maxk

{9, 9, 9} =(5, 8.33, 9).


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Table 4

Linguistic assessments for the 11 criteria.

Criteria Decision makers

D1 D2 D3

Accessibility (C1) VH VH VH

Security (C2) VH H VH

Connectivity to multimodal transport (C3) VH H VH

Costs (C4) H VH VH

Environmental impact (C5) M H MProximity to customers (C6) VH H VH

Proximity to suppliers (C7) H VH H

Resource availability (C8) M H H

Conformance to sustainable freight regulations (C9) H H VH

Possibility of expansion (C10) VH H VH

Quality of service (C11) VH VH VH

Table 5

Linguistic assessments for the three alternatives.

Criteria Alternatives Decision makers

D1 D2 D3

C1

A1 VG G VG

A2 G VG VG

A3 VG VG VG

C2

A1 G VG G

A2 M G M

A3 P P M

C3

A1 M G M

A2 M M M

A3 M G G

C4

A1 G G M

A2 M G M

A3 M G G

C5

A1 G M G

A2 G M M

A3 G P G

C6

A1 G VG VG

A2 VG VG G

A3 VG VG G

C7

A1 M VG M

A2 G G M

A3 G G G

C8

A1 M M G

A2 G M M

A3 M M M

C9

A1 M M M

A2 M P P

A3 P VP VP

C10

A1 G G VG

A2 VG G VG

A3 G G G

C11

A1 VG G VG

A2 VG G G

A3 G VG VG

Likewise, the aggregate ratings for the three alternatives (A1, A2, A3) with respect to the 11 criteria are computed. Theaggregate fuzzy decision matrix for the alternatives is presented inTable 7.

In the next step, we perform normalization of the fuzzy decision matrix of alternatives using Eqs. (8)(10).For example,the normalized rating for alternative A1 for criterion C1 (Accessibility) is given by

cj =maxi

{9, 9, 9} =9

rij =

59

, 8.339

, 99

=(0.56, 0.926, 1).


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Table 6

Aggregate fuzzy weights for criteria.

Criteria Decision makers Aggregated fuzzy weights

D1 D2 D3

C1 (7, 9, 9) (7, 9, 9) (7, 9, 9) (7, 9, 9)

C2 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)

C3 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)

C4 (5, 7, 9) (7, 9, 9) (7, 9, 9) (5, 8.33, 9)

C5 (3, 5, 7) (5, 7, 9) (3, 5, 7) (3, 5.66, 9)C6 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)

C7 (5, 7, 9) (7, 9, 9) (5, 7, 9) (5, 7.66, 9)

C8 (3, 5, 7) (5, 7, 9) (5, 7, 9) (3, 6.33, 9)

C9 (5, 7, 9) (5, 7, 9) (7, 9, 9) (5, 7.66, 9)

C10 (7, 9, 9) (5, 7, 9) (7, 9, 9) (7, 8.33, 9)

C11 (7, 9, 9) (7, 9, 9) (7, 9, 9) (7, 9, 9)

Table 7

Aggregate fuzzy weights for alternatives.

Criteria Alternatives Decision makers Aggregate ratings

D1 D2 D3

C1

A1 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)

A2 (5, 7, 9) (7, 9, 9) (7, 9, 9) (5, 8.33, 9)A3 (7, 9, 9) (7, 9, 9) (7, 9, 9) (7, 9, 9)

C2

A1 (5, 7, 9) (7, 9, 9) (5, 7, 9) (5, 7.66, 9)

A2 (3, 5, 7) (5, 7, 9) (3, 5, 7) (3, 5.66, 9)

A3 (1, 3, 5) (1, 3, 5) (3, 5, 7) (1, 3.66, 7)

C3

A1 (3, 5, 7) (5, 7, 9) (3, 5, 7) (3, 5.66, 9)

A2 (3, 5, 7) (3, 5, 7) (3, 5, 7) (3, 5, 7)

A3 (3, 5, 7) (5, 7, 9) (5, 7, 9) (3, 6.33, 9)

C4

A1 (5, 7, 9) (5, 7, 9) (3, 5, 7) (3, 6.33, 9)

A2 (3, 5, 7) (5, 7, 9) (3, 5, 7) (3, 5.66, 9)

A3 (3, 5, 7) (5, 7, 9) (5, 7, 9) (3, 6.33, 9)

C5

A1 (5, 7, 9) (3, 5, 7) (5, 7, 9) (3, 6.33, 9)

A2 (5, 7, 9) (3, 5, 7) (3, 5, 7) (3, 5.66, 9)

A3 (5, 7, 9) (1, 3, 5) (5, 7, 9) (1, 5.66, 9)

C6 A1 (5, 7, 9) (7, 9, 9) (7, 9, 9) (5, 8.33, 9)A2 (7, 9, 9) (7, 9, 9) (5, 7, 9) (5, 8.33, 9)

A3 (7, 9, 9) (7, 9, 9) (5, 7, 9) (5, 8.33, 9)

C7

A1 (3, 5, 7) (7, 9, 9) (3, 5, 7) (3, 6.33, 9)

A2 (5, 7, 9) (5, 7, 9) (3, 5, 7) (3, 6.33, 9)

A3 (5, 7, 9) (5, 7, 9) (5, 7, 9) (5, 7, 9)

C8

A1 (3, 5, 7) (3, 5, 7) (5, 7, 9) (3, 5.66, 9)

A2 (5, 7, 9) (3, 5, 7) (3, 5, 7) (3, 5.66, 9)

A3 (3, 5, 7) (3, 5, 7) (3, 5, 7) (3, 5, 7)

C9

A1 (3, 5, 7) (3, 5, 7) (3, 5, 7) (3, 5, 7)

A2 (3, 5, 7) (1, 3, 5) (1, 3, 5) (1, 3.66, 7)

A3 (1, 3, 5) (1, 1, 3) (1, 1, 3) (1, 1.66, 5)

C10

A1 (5, 7, 9) (5, 7, 9) (7, 9, 9) (5, 7.66, 9)

A2 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)

A3 (5, 7, 9) (5, 7, 9) (5, 7, 9) (5, 7, 9)

C11

A1 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)

A2 (7, 9, 9) (5, 7, 9) (5, 7, 9) (5, 7.66, 9)

A3 (5, 7, 9) (7, 9, 9) (7, 9, 9) (5, 8.33, 9)

The normalized value of alternative A1 for criterion C4 (Costs) is given by

aj =mini

(3, 3, 3)= 3

rij =

3

9,

3

6.33,

3

3

=(0.33, 0.473, 1).

Likewise, we compute the normalized values of the alternatives for the remaining criteria. The normalized fuzzy decisionmatrix for the three alternatives is presented inTable 8.

Then, thefuzzy weighted decision matrix forthe three alternatives is constructed using Eq.(11). The rijvalues from Table 8and wj values fromTable 6are used to compute the fuzzy weighted decision matrix for the alternatives. For example, for


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Table 8

Normalized fuzzy decision matrix for alternatives.

Criteria aj c

j Normalized ratings

A1 A2 A3

C1 5 9 (0.56, 0.926, 1) (0.56, 0.926, 1) (0.78, 1, 1)

C2 1 9 (0.56, 0.852, 1) (0.33, 0.629, 1) (0.11, 0.407, 0.78)

C3 3 9 (0.33, 0.629, 1) (0.33, 0.56, 0.78) (0.33, 0.703, 1)

C4 3 9 (0.33, 0.473, 1) (0.33, 0.53, 1) (0.33, 0.473, 1)

C5 1 9 (0.11, 0.157, 0.333) (0.11, 0.177, 0.33) (0.11, 0.177, 1)

C6 5 9 (0.56, 0.926, 1) (0.56, 0.926, 1) (0.56, 0.96, 1)

C7 3 9 (0.33, 0.703, 1) (0.33, 0.703, 1) (0.56, 0.78, 1)

C8 3 9 (0.33, 0.629, 1) (0.33, 0.629, 1) (0.33, 0.56, 0.78)

C9 1 7 (0.428, 0.714, 1) (0.14, 0.528, 1) (0.14, 0.24,0.71)

C10 5 9 (0.56, 0.85, 1) (0.56, 0.926, 1) (0.56, 0.78, 1)

C11 5 9 (0.56, 0.93, 1) (0.56, 0.85, 1) (0.56, 0.926, 1)

Table 9

Weighted normalized alternatives, FPIS and FNIS.

Criteria Alternatives FPIS (A) FPNS (A)

A1 A2 A3

C1 (3.89, 8.33, 9) (3.89, 8.33, 9) (5.44, 9, 9) (9, 9, 9) (3.89, 3.89, 3.89)

C2 (2.78, 7.09, 9) (1.67, 5.24, 9) (0.56, 3.38, 7) (9, 9, 9) (0.56, 0.56, 0.56)C3 (1.67, 5.238, 9) (1.67, 4.63, 7) (1.67, 5.86, 9) (9, 9, 9) (1.67, 1.67, 1.67)

C4 (1.67, 3.947, 9) (1.67, 4.41, 9) (1.67, 3.95, 9) (9, 9, 9) (1.67, 1.67, 1.67)

C5 (0.33, 0.894, 3) (0.33, 1, 3) (0.33, 1, 9) (9, 9, 9) (0.33, 0.33, 0.33)

C6 (2.78, 7.709, 9) (2.77, 7.7, 9) (2.78, 7.71, 9) (9, 9, 9) (2.78, 2.78, 2.78)

C7 (1.67, 5.38, 9) (1.67, 5.38, 9) (2.78, 5.96, 9) (9, 9, 9) (1.67, 1.67, 1.67)

C8 (1, 3.98, 9) (1, 3.987, 9) (1, 3.52, 7) (9, 9, 9) (1, 1, 1)

C9 (2.14, 5.47, 9) (0.714, 4, 9) (0.7, 1.8, 6.4) (9, 9, 9) (0.71, 0.714, 0.714)

C10 (2.778, 7.09, 9) (2.778, 7.7, 9) (2.78, 6.48, 9) (9, 9, 9) (2.78, 2.78, 2.78)

C11 (3.89, 8.33, 9) (3.89, 7.6, 9) (3.89, 8.33, 9) (9, 9, 9) (3.89, 3.89, 3.89)

Table 10

Distancesdv (Ai,A)anddv (Ai.,A

)for alternatives.

Criteria dv (A1,A) dv (A2,A

) dv (A3,A) dv (A1,A

) dv (A2,A) dv (A3,A

)

C1 3.908 3.908 4.266 2.974 2.974 2.051C2 6.296 5.611 4.063 3.754 4.753 5.961

C3 4.709 3.520 4.875 4.75 5.0596 4.602

C4 4.431 4.518 4.431 5.138 4.991 5.138

C5 1.572 1.586 5.015 7.673 7.636 6.806

C6 4.584 4.584 4.584 3.666 3.666 3.666

C7 4.747 4.747 4.946 4.715 4.715 3.995

C8 4.927 4.927 3.755 5.447 5.447 5.713

C9 5.576 5.146 3.359 4.448 5.579 6.496

C10 4.371 4.584 4.178 3.754 3.666 3.873

C11 3.908 3.667 3.908 2.974 3.048 2.974

alternative A1, the fuzzy weight for criterion C1 (Accessibility) is given by

vij =(0.56, 0.926, 1)(.)(7, 9, 9) = (3.89, 8.33, 9).

Likewise, we compute the fuzzy weights of the three alternatives for the remaining criteria (Table 9). Then, we compute thefuzzy positive ideal solution (A) and the fuzzy negative ideal solutions (A) usingEqs. (12)(13)for the three alternatives.For example, for criterion C1 (Accessibility),A =(9, 9, 9) andA =(3.89, 3.89, 3.89). Similar computations are performedfor the remaining criteria. The results are presented in the last two columns ofTable 9.

Then, we compute the distance dv (.) for each alternative from the fuzzy positive ideal matrix (A) and fuzzy negative

ideal matrix (A) using Eqs.(2),(12)and(13).For example, for alternative A1 and criterion C1, the distances dv (A1,A)and

dv (A1,A)are computed as follows:

dv (A1,A) =

1

3[(3.899)2 +(8.339)2 +(99)2] =2.974

dv (A1,A)=

1

3[(3.893.89)2 +(8.333.89)2 +(93.89)2] =3.907.

Likewise, we compute the distances for the remaining criteria for the three alternatives. The results are shown in Table 10.


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Table 11

Closeness coefficients (CCi) of the three alternatives.

A1 A2 A3 Ranking order

di 49.033 46.803 47.385

A1> A3> A2d+i 49.302 51.539 51.281

CCi 0.498 0.475 0.4802

Then, we compute the distances di anddi usingEqs. (14)(15).For example, for alternative A1 and criterion C1, thedistancesdi andd

i are given by

di =

1

3[(3.899)2 +(8.339)2 +(99)2] +

1

3[(2.789)2 +(7.099)2 +(99)2] +

+

1

3[(3.899)2 +(8.339)2 +(99)2] =49.3

di =

1

3[(3.893.89)2 +(8.333.89)2 +(93.89)2] +

1

3[(2.780.56)2 +(7.090.56)2 +(90.56)2]

+ +

1

3[(3.893.89)2 +(8.333.89)2 +(93.89)2] =49.03.

Using the distancesdi anddi , we compute the closeness coefficient for the three alternatives using Eq.(16).For example,for alternative A1, the closeness coefficient is given by

CCi =di /(d

i +d

+i ) = 49.03/(49.3+49.03) = 0.499.

Likewise,CCiis computed for the other two alternatives. The final results are shown in Table 11.

By comparing theCCi values of the three alternatives (Table 11), we find that A1 > A3 > A2. Therefore, location A1 isselected as the best location for implementing the new urban distribution center for the logistics company.

4.1. Sensitivity analysis

To investigate the impact of criteria weights on the location selection of an urban distribution center, we conducted asensitivity analysis. 16 experiments were conducted. The details of the 16 experiments are presented inTable 12.

It can be seen inTable 12that in the first five experiments, the weights of all criteria are set equal to (1, 1, 3), (1, 3, 5),(3, 5, 7), (5, 7, 9) and (7, 9, 9). In experiments 615, the weight of one criterion is set as the highest and the remaining are setto the lowest value. For example, in experiment 6, the criterion C1 has the highest weight = (7, 9, 9) whereas the remainingcriteria have weight=(1, 1, 3). In experiment 17, the weights of all the cost category criteria are set as the highest, that is,criteria C3 and C4 at the highest weight = (7,9, 9), and the weights of benefit category criteria (C1C2, C5C11) are set as thelowest weight=(1, 1, 3). In experiment 18, the weight of the cost category criteria (C3, C4) is set as the lowest weight =(1,1, 3) and the weights of the benefit category criteria (C1C2, C5C11) are set as the highest weight =(7, 9, 9).

The results of the sensitivity analysis are presented in Fig. 4. It can be seen that for all 18 experiments, location A1has emerged as the best location in 14 experiments. In experiments 6, 10, 12 and 17, the location A3 has emerged as thewinner. Therefore, we can say that the location decision is relatively insensitive to benefit criteria weights; however whenthe weights of cost criteria (C3, C4) are set as the highest, then the best solution is changed from A1 to A3.

5. Conclusion

In this paper, we present a multi-criteria decision making framework for location planning for urban distributioncenters under a fuzzy environment. The proposed approach comprises four steps. In step 1, we identify the criteria forevaluating potential locations for distribution centers. These criteria are: Accessibility, Security, Connectivity to multimodaltransport, Costs, Environmental impact, Proximity to customers, Proximity to suppliers, Resource availability, Conformanceto sustainable freight regulations, Possibility of expansion, and Quality of service. In step 2, the potential locations forimplementing urban distribution centers are identified. In step 3, the decision makers provide ratings for the criteria andthe potential locations. Fuzzy TOPSIS is used to determine aggregate scores for all potential locations and the one with thehighest score is finally chosenfor implementation. Sensitivity analysis is performed to assessthe influenceof criteria weightson the decision making process.

The strength of our work is the ability to deal with multiple criteria and model uncertainty in location planning for urbandistribution centers through the use of linguistic parameters and fuzzy theory. The proposed approach can be practically

applied by logistics operators in implementing new distribution centers, considering the sustainable freight regulationsproposed by municipal administrations.


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Table 12

Experiments for sensitivity analysis.

Experiment number CCi Description

A1 A2 A3

1 0.416 0.405 0.410 All criteria weights= (1, 1, 3)





6 0.440 0.429 0.447 Weight of criteria 1=(7, 9, 9)

Weight of remaining criteria=(1, 1, 3)





















17 0.393 0.387 0.421 Weight of criteria 3 & 4=(7, 9, 9)


18 0.540 0.504 0.491 Weight of criteria 3 & 4=(1, 1, 3)Weight of remaining criteria=(7, 9, 9)

0

0.1

0.2

0.3

0.4

0.5

0.6

Experiment 1

Experiment 2

Experiment 3

Experiment 4

Experiment 5

Experiment 6

Experiment 7

Experiment 8

Experiment 9

Experiment 10

Experiment 11

Experiment 12

Experiment 13

Experiment 14

Experiment 15

Experiment 16

Experiment 17

Experiment 18

Sensitivity Analysis

A1

A2

A3

Fig. 4. Results of sensitivity analysis.


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A Multi-criteria Decision Making Approach for Location Planning

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