Fast Local Searches for the Vehicle Routing Problem With Time Windows

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Fast local searches for the vehicle routing problem with time windowsOlli Braysy

INFOR; Nov 2002; 40, 4; A B M N F O R M Globalpg. 319

FAST L O C A L SEARCHES FOR THE V E H I C L E ROUTING

PROBLEM WITH TIME WINDOWS

O L L I B R Y S Y

SINTEF Applied Mathematics,

Department of Optimization,

P.O. Box 124 Blindent, N-0314 Oslo, Norway,

E-mail: Olli. Braysy@sintef. no

A B S T R A C T

The purpose of this paper is to present new deterministic local searches for solving the vehicle rout

ing problem with time windows. The proposed algorithms are based on a new three-phase approach.In the first phase an initial solution is created with one of the two proposed route construction heuristics. In the second phase a special local search operator based on ejection chains is used to reduce the

number of routes. Finally, in the third phase well-known Or-opt exchanges are used to minimize thetotal distance of the routes. The findings of computational experiments indicate that the proposed

methods arc competitive with the best approaches proposed earlier in the literature in terms of solution quality, while being much faster. Moreover, the proposed algorithms may easily be used to create initial solutions for a wide variety of vehicle routing algorithms.

RSUM

L'objectifde cet article est de

prsenterde nouvelles

rechercheslocales

dterministespour

rsoutirele problme de tournes de vhicules avec fentres de temps. Les algorithmes proposs sont basssur une nouvelle approche en trois tapes. Dans la premire tape, une solution de dpart est obtenueavec l'une des deux heuristiques de construction de route proposes. Dans la seconde tape, unerecherche spciale base sur des chanes d'jection est utilise de manire rduire le nombre deroutes. Finalement, dans une troisime tape, un voisinage classique Or-Opt est utilis pour minimiser la longueur totales des routes. Les rsultats des calculs exprimentaux indiquent que les mthodes proposes sont, en terme de qualit, concurrentielles avec les meilleures approches proposesauparavant dans la littrature tout en tant nettement plus rapides. En outre, les algorithmes proposs peuvent facilement tre utiliss pour obtenir des solutions initiales pour un large ventaild'algorithmes de tournes de vhicules.

\ . I N T R O D U C T I O N

In this paper we focus on the Vehicle Rout ing Prob lem wi th Time Windows (VRPTW), an

important problem occurring in many distribution systems. To describe the VRPTW, let G = (C,

A) be a directed graph, where C= {c0,ct,...,cn} is a vertex set, and/1 = {(c,,c 7 ) : i ^ /') is an arc set.

Vertex c0 denotes a depot and the remaining vertices ofC represent customer locations. Each arc

(c,,Cy) has an associated nonnegative distance d- and a nonnegative travel time r,y. The VRPTW

consists of designing a set of least cost vehicle routes such that:

a. Eve ry route starts and ends at the depot, c{).

b. Every customer ofC (exc luding the depot) is visited exactly once by exactly one vehicle.

c. The total demand of any vehicle route does not exceed the vehicle capacity.

d. The service of each customer c, begins wi th in a service time window (


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320 O. BRAYSY

is the primary objective, followed by the minimization of the total traveled distance. Some of the

most useful applications of the VR PTW include bank deliveries, postal deliveries, industr ial

refuse collection, national franchise restaurant services, school bus rout ing, and security patrol

services. Because of its high practical importance the VRPTW has been the focus of intensive

research efforts fo r both optimizat ion and heuristic approaches. Special purpose surveys for the

VRPTW can be found in Golden and Assad (1986), Desrochers ct al. (1988) , Golden and Assad

(1988), Solomon and Desrosiers ( 1988), Dcsrosiers et al. (1995), Cordeau et al. (2001) and Braysy

and Gendreau (2001a and 2001b). Because of the high comple xity of the VRPT W, the major ity of

research has focused on heuristics. These heuristics can be divided into three main categories:

construction heuristics, improvement heuristics (local searches) and metaheuristics.

A number o f route construc tion algorithms are proposed in Solomon (1987). These construc

t ion algorithms select and insert nodes sequentially unt i l a feasible solution has been created.

Nodes are chosen based on a specific cost mi nimiz ati on criter ion subject to the restri ction that

the insertion does not create a violation of vehicle capacity or time window constraints. Diffe r

ent variants of construction heuristics can be found also in Potvin and Rousseau (1993), Bramcland Si mch i-Lev i (1996), Dullaer t (2000) and Ioannou et al. (2001). For more details on these

approaches, we refer to Braysy and Gendreau (2001a)

Most of the recently publi shed V RPTW heuristics use a two-phase approach. First, a con

struction heuristic is used to generate a feasible ini t ial solution. During the second phase, an iter

ative improvement heuristic is applied to the ini t ial solution. These route improvement methods

iteratively modify the current solution by performing local searches for better neighboring solu

tions. Genera lly, a neighborhood comprises the set of solutions that can be reached from the

present one by swapping a subset ofk arcs between solutions. For the most successful applica

tions to the VR PT W, see Thompson and Psaraftis (1993) , Potvin and Rousseau (1995), Russell

(1995), Shaw (1997 and 1998), Cordone and Wolfl er- Calv o (1998) and Caseau and Laburthe

(1999). For more details, we refer to Braysy and Gendreau (2001a).

To escape from local opt ima, the improvement procedure can be embedded in a metahcuris tic

such as simulated annealing, tabu search or genetic algori thm. Genetic algorithms belong to the

classical local search framework where improvement is sought with each move in the neighbor

hood of the current solution. In contrast, tabu search and simulated annealing are part of a new

paradigm that allows the selection of worse solutions once a local optimum has been reached.

For most successful recent applications, see Rochat and Taillard (1995), Taillard et al. (1997),

Gambardella et al. (1999), Homberger and Gehring (1999), Cordeau et al. (2001), Gehring and

Flomberger (2001), Braysy (2001b), Berger et al. (2001) and a recent survey by Braysy and

Gendreau (2001b).

In the past, the focus in the routing research has been mainly on i mpro ving solution quality,

regarding computation speed as secondary. Nevertheless, there is an industry need for fast, high-

quality solution methods for rou ting problems. Because of the integration of routing problems in

other company activitie s, the corresponding extended routing problems require faster soluti on

methods to obtain results in an acceptable time span. Especially for dynamic routing problems,

where decisions on load acceptance and routing must be made in real-time, the computational

speed of a solution approach is cruc ial. The main co ntri buti on of this paper is the development

of several fast local search strategies for the VRPTW . The proposed algori thms are shown to be

very efficient approaches for solving the vehicle routing problem wi th time windows, producingresults that outperform recent local search approaches. They arc also competitive wi th the best

metaheuristics reported in the literature in terms of solution quality , while being much faster.

Flexible, the local search algorithms presented in this paper can also be combined wi th different

routing algori thms to further impr ove solution quality .

The remainder of this paper is organized as follows. An overv iew of the solution strategy is

first given, and then, the different components of the algorithms are described in Section 2. Firs t,

the construction heuristics used to create initial solutions are described. Second, the basic local

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V E H I C L E ROUTING PROBLEM W I T H T I M E WINDOWS 321

search techniques used in comput ing new solut ions are depicted. Then , the eject ion chain

method is introduced. The computational experiments assessing the value of the proposed

approach are presented in Section 3. Also, a comparative performance analysis involving vari

ous local searches and metaheuristics is briefly reported. Section 4 concludes the paper.

2. T H E P R O B L E M S O LV I N G M E T H O D O L O G Y

2.1. Overview

We suggest a new three-phase approach for solving vehicle routing problems. I n the first phase an

ini t ial solution is created using one of the two proposed construction heuris tics. In the second

phase an effort is made to reduce the number o f routes using an ejection chain-based approach.

The first two phases are repeated for different parameter settings defined for the construction heu

ristics. In the th i rd phase, well-known Or-opt exchanges (Or, 1976) are used to minimize the total

distance. The basic idea in Or-opt exchanges is to reinsert segments of consecutive customers to

other feasible locations, and accept the move i f the inser tion improves solution value. The Or-opt

is applied only to solutions having the smallest number of routes, as found during the first twophases. At the start, the routes of each solution are reordered such that the routes having some bad

features or weaknesses are considered first for improvement. Here we consider long waiting time

and long average distance wi th respect to the number of customers on the route as bad features.

Parameter p is used to control the weight ofthese two factors. The larger the value of[3 the more

long average traveled distance is emphasized in the reordering of the routes. To speed up the fea

sibil i ty checks on each insertion or move, we used push-forward and push-backward strategies, as

introduced in Solomon et al. (1988). The proposed algorithm is as follows:

Step 1. Use H y b r i d Construction or Merge heuristic to create an ini t ial solution.

Step 2. Repeat the route elimination procedure unt i l no more routes can be eliminated.Step 3. Repeat steps 1 and 2 using al l the parameter values wi th in the specified l imi ts . Store the

created solutions.

Step 4. Identify all the created solutions wi th the smallest number of routes and insert them

into set RB.

Step 5. Reorder the routes in solution 5,- according to parameter (3 and improve 5,- using the

Or-opt procedure.

Step 6. Repeat step 5 for all S: in RB and update the best solution found, Sh, if needed.

Step 7. Return Sh.

2.2. Route Constru ction and L o c a l Sear ch Heuristics

Two construction heuristics are proposed to create the ini t ial solutions, namely Hy b r id Con

struction heuristic (HC) and Merge Heuristic ( M H ) . They are described below, followed by a

description offive local search procedures B 1 , B2, B3, B4 and B5 that are based on HC and M H

heuristics, ejection chain-based route elimination procedure and Or-opt exchanges.

2.2.1. Hybrid Construction Heuristic

The hybrid cons truc tion heuristic is a cheapest-insertion-based heuris tic introduced in Braysy

(2001b). It borrows its basic ideas f rom the studies of Solomon (1987) and Russell (1995).

Routes arebui l t

one at a time in a sequential fashion and after k customers have been inserted in

the current partia l route, the route is reordered using Or -opt exchanges. Here /< is a constant

parameter value decided by the user. The routes are initialized by selecting the first initialization

(seed) customer ( in i t ia l iz ing the first route) among the 8 customers farthest away from the depot

and f rom each other. The subsequent seeds are selected among the 30% of the customers farthest

away from the depot in clockwise or counterclockwise sweep, or such that the next seed selected

is the closest unrouted customer to the previously selected seeds. For more details, we refer to

Braysy (2001a).



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322 O. BRA Y SY

Once the first customer is selected, the unrouted customers are examined one by one and the

customer that minimizes the weighted combination of additional detour and waiting time is

selected and inserted in the best feasible inse rtion place. Here we do not consider all customers

for insertion. Instead, we consider only customers that are geographically close to at least one of

the previously inserted customers on the route. The closeness d is a fixed constant determined

by the user. The customers farthest away from the depot are usually the most diff icult ones to

route, since there are often only a few feasible insertion places available for them. Therefore, we

favor the selection ofthese customers by subtracting from the insertion cost the distance of the

corresponding customer to the depot multiplied by a user defined parameter a 3 . More formally

the cost function for customer cu is

C

u =a \ *Du + a 2 ' ^ - 3 *4)> ( 1)

where

Du = diu + dUj-dip (2)

wu

=w;-wt, O )a i + a 2 = 1, a 3 > 0.

Notations dju, duJ and refer to the distance between the corresponding pair of customers (c,-,

c H), ( fH , c ;) and (ch Cj) and W* and W(", correspond to the total waiting time before and after the

insertion respectively. Finally d0ll

is the distance from the customer cu to the depot and a, , a 2

and ct3 are parameter values determined by the user. For more details, we refer to Briiysy (2001a

and 2001b). In the experimental part the combination of HC heuristic, route elimination proce

dure and Or-opt exchanges is marked by B l .

2.2.2. Merge Heuristic

The merge heuristic draws its basic concepts from the well-known savings heuristic of Clarke

and Wright (1964). We implemented a parallel version of the savings heuristic, as in Clarke and

Wright (1964). We begin by serving each customer by a separate route and then construct the

solution by repeatedly merging a pair of routes producing the greatest saving as long as the time

or capacity constraints are not v iola ted. The saving 5,-- incu rred by combi ni ng routes i and j is

calculated using the weighted combination of distance and waiting time to evaluate the total cost

o f each individual route k, TCk = a, TD k + a2TWk, where TDk and TWk are the total distance

and total waiting lime on route k. Then, the cost of the merged route {TCri) is deducted from the

joint cost of routes / a n d i . e . , S- = TCt + TC- - TC-. Moreover, an attempt is made to improve

the combined route by trying to reorder customers, to reduce TC^, before evaluating the saving

incurred by uniting the two routes. The improvement is tried when m more customers have been

added to the merged route, using the intra-route improvement heuristic O-opt, described in

Briiysy (2001b). The basic idea is to rebuild routes wi th the cheapest insertion heuristic by using

more than one customer to initialize the routes. M or e precisely, instead o f selecting jus t one seed

customer , a user-defined number , /, o f customers are selected from the current route such that

they are geographically as dispersed as possible. These seed customers are then put in all feasi

ble orders and each of the partial routes initialized by these / customers is rebu ilt wi th the hybrid

construction heuristic by inserti ng customers in increasing order o f their time window width.

After k customers have been inserted, the route is reordered using the Or-opt operator. To avoid

double work, the previously calculated savings are stored into a matrix. In the experimental part,

the combination ofM H , route elimination procedure and Or-opt exchanges is marked by B2.

In addition to testing B l and B2 heuristics independently, they were combined sim ply by run

ning both methods and taking the best result. In the experimental part we refer to this method by

notation B3. Acco rdi ng to our experience most approaches developed for the VRPTW have dif

ficulties in obtaining good output regarding the number of routes. To alleviate this problem we

suggest an implementation that is solely designed to create quickly initial solutions wi th a small

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number of routes. This approach can be combined for example wi th any metaheuristic or large

neighborhood search procedure to minimize the total traveled distance. The basic idea is the

same as that used in the implementat ion of B 1. However, here we use the knowledge of the best

solut ions reported in the literature and the search is stopped once the number o f routes is equal

to the best known. Also, the number of parameter combinations tried is greater than in the B1

implementation. In the experimental part we use notation B4 for this method. The usage of the

best known number of routes can be justified by the fact that in practice the number of vehicles

is often known, i.e., the company has a l im i te d number of vehicles (we assume that each route is

served by a separate vehicle). However, when the number of routes/vehicles is not known in

advance, there are also several ways to handle the problem algorithmically. For example one can

use some other heuristic or metaheuristic method to estimate the number of routes or alterna

t ively use different lower bounds such as solutions f rom different Bi n Packing Problems (K on -

toravdis et al., 1995). One possibility is also to use the information gathered during the search,

i.e., since we use very powerful ejection chains to reduce the number of routes, we often get

solutions where the number of routes is very close to the optimum. Thus, this information canalso be used to estimate the number of routes. Finally, we created a f if th implem entati on B5 by

examining all parameter values specified for B4, without prematurely stopping the search, i.e.,

without using the knowledge of the best-known solutions.

2.3. Route Elimination Procedure

After creating an ini t ial solution, the ejection chain-based technique introduced in Briiysy

(2001b) is used to reduce the number of routes. The basic idea in ejection chains is to combine a

series of simple moves into a compound move. In a VRP context, these simple moves refer to

removal of a customer from its route and reinsertion of the removed customer in another route.

The goal is to "make room" for a new customer in a route by first rem oving another customer

from the same route. In each phase wi th in the ejection chain, one customer remains unrouted.

The removal and insertion procedures are repeated unt i l one can insert a customer to another

route without the need to remove (eject) any customer. For more details on ejection chains, and

their application to VRP and VRPTW we refer to Glover (1991, 1992), Rego (1998, 2001)

Rousseau et al. (2000) and Caseau and Laburthe (1999), respectively.

The main innovation in the ejection chain approach described in Braysy (2001b) is the reor

dering of the routes wi th in the ejection chain. First, the customer c, originally served by route reis inserted at a location on another route r, that least increases the value of cost function ( 1 ) with

a, = 0.5 and a3 = 0. I f the time window constraints are violated, simple intra-route rinsert ions

are tried to make r, feasible after inserting c;. The basic idea ofthese reinsertions is to reduce

lateness in the infeasible route by trying to serve some customers i n alternate locations wi th in

the same route. We used the first-accept strategy, i.e., accept every move that reduces lateness.

A l l routes are considered for elimination, one at a time, starting from the routes having the

smallest number o f customers. I f direct reloca tion of a customer c, on the route re currently con

sidered for elimination to an alternate route is not possible, the ejection chain procedure wi th

breadth-first strategy is initialized using cr In the breadth-first search, we try first all possible

chains involving one ejection and two insertions, then chains w i t h two ejections and three inser

tions and so on. I f we are not able to relocate a ll customers on re, a special post-processing is

applied. Here we insert as many customers as possible from the nei ghboring routes to the justexamined route re to facilitate eliminating some other route. Since we must keep route re, it is

reasonable to utilize its resources as w e l l as possible. In addition, several strategies/restrictions

were used to speed up the algorithm:

- Th e maximum length of ejection chain and number ofwi th in route reinsertions are restricted

to c and n, respectively.

- To restrict the number of chains to explore, we set a l imi t 1 to the allowed increase in distance.

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324 O. BRAYSY

Thus, i f a certain insertion causes the total distance of the target route to become 7 times

longer, the corresponding insertion and/or chain is ignored.

- We used push-forward and push backward strategies int roduced in Sol omon et al. (1988) to

speed up the feasibility checks of each insertion or reinsertion wi th in the reordering operator.

In add iti on, to further increase the efficiency offeasibility checks, we maintained in memory

the arrival times and the latest possible arrival times at each customer.

- To minimize the number of insertions tried, we try to eliminate first routes having the smallest

number o f time-constrained1

customers, and we first try to insert the customers in each phase

to the routes that are geographically closest. The geographical closeness is defined as the

smallest distance between the cur rently considered customer c( and any customer on the desti

nation route. Moreover, since it is computationally prohibitive to try to eliminate a long route,

only the shortest route is considered for elimination in case of problems wi th over 15 custom

ers per route.

- I f the distance between the customer c; currently considered for insertion and all the custom

ers in a given route r, exceeds a user-defined l i m i t d, the insertion of customer c, is not even

tried in rt.

3. C O M P U T A T I O N A L E X P E R I M E N T

3 . 1 . Problem D ata and Para meter Values

To analyze the performance of our algorithm, an experiment was conducted over 56 VRPrW

benchmarks, partitioned in six data sets denoted R l , C I , R C I , R2, C2 and RC2 (Solomon,

1987). The problems vary in fleet size, vehicle capacity, travel time and spatial and temporal

customer distribution (position, time window density, time window w i d t h , and service time).

Each problem instance involves a centra l depot, 100 customers to be serviced, as w e l l as con

straints imposed on vehicle capacity and customer vis i t or delivery time (time windows). CI andC2 data sets are characterized by a clustered customer distribution, whereas R1 and R2 refer to

randomly distr ibuted customers. Data sets RCI and RC2 represent a combinat ion of random and

clustered customer distributions. The travel time between tw o locations corresponds to the

Euclidean distance. The algorithms were implemented in JAVA and the computational experi

ments were conducted using a Pentium 200 MH z computer.

We found it com put ati ona lly intractable to optimize the value for each parameter separately.

Therefore, we tuned the parameter values only once by selecting first a parameter setting based

on in tu i t ion . Then, a few (310) intuitively selected values were tried for each parameter, while

keeping the other parameter values f ixed . To reduce the workload, only a set offour problems

were used in the tuning. Each time, we selected a parameter value that gave the best average out

put for the selected four test problems.

The general parameters used by all our algori thms remained fixed during the computations.

However, in some cases the parameter values depend on the characteristics o f the problem. We

separate two groups of problems, based on the average number of customers on each route to

control the comp lex ity of the search. Mo re precisely, we include problem sets R l , C I and RC I

in group I and the other problem sets R2, C2 and RC2 in group I I . The parameters and their va l

ues are: d = 20 and c = 9 in group I , d= 30 and c = 4 in group I I , h = 5, (3 = 0.5, k = 3 ,1= 1.15, i

= 4, m = 7, a,: 0.7- 1.0 (in increments of 0.05 and 0.1 units for B2 and for B l , B4 and B5,

respectively), a 3 : 0.5-1 .7 (in increments of 0.2 units). In case of B l we used the fo l lowing va l

ues forot3: 0.5, 1.1, 1.72.

1 Wc define a customer as time-constrained if the width of the time window is less than 50% of the depot's time window.2 The values were selected by taking the lower and upper bound and the middle value of the range defined for B4 and

B5.

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In order to keep the memory requirements reasonable, a maximum l imi t of 300 is set to the

ejection chains stored in memory. In case the elimination procedure returns the same number of

routes, nf, each time, we compare n{.to the number of routes in the ini t ial solution, n'r. I f the va l

ues are the same, we assume that the problem is easy, and we have obtained the minimum num

ber of routes. Otherwise, the maximum length of ejection chains is increased to c = 10, the

geographical closeness range is increased to d = 0.4 and 7 = 1.65, and the elimination procedure

is repeated in order to introduce more power to the search.

The reader must note that even i f there are a lot of different parameters, in general the results

are not sensitive to changes in parameter values. In most cases the effect of the parameter values

on total distance was less than 1%, which made it possible to f ind a single robust value that is

not dependent on the prob lem. The only exceptions were parameters a, , a 3 and the seed selec

t ion schemes that had a clearly bigger impact on the results (in some cases over 50%) . We found

i t impossible to determine a robust value for them that would give good results for all test prob

lems. Therefore, we tried several values for these parameters wi th in specified ranges, as

depic ted above. The parameter sens itivity is discussed in more detai l in Braysy (2001a).

3.2. C omp utatio nal Results

In Table 1 we compare the results obtained w i t h the proposed five local search procedures wi th

the results of recent local searches and metaheuristics. We included in Table 1 only approaches

where sufficient information is provided by the authors. At least the computer, number of com

putational runs as well as the time consumption, number of vehicles and total traveled distance

for each problem group must be reported to make the comparison possible. The first column to

the left gives the authors and columns R l , R2, C I , C2, RC1 and RC2 present the average num

ber of vehicles and average total distance w i t h respect to the six problem groups of Solomon

(1987). The CNV/CTD col umn indicates the cumul ative number of vehicles ( CNV ) and cumu

lative total distance (C TD ) over all 56 test problems. The rightmost co lumn describes the com

puter, number of runs, and the CPU time used to obtain the reported results. Two CPU time

values are described: the one reported by the authors and in the parenthesis the modified CPU

time. These modified CPU times are scaled to equal the running times on a Sun Sparc 10/50

using Dongarra' s (1998) factors. I f several runs are required to get the reported results, the com

putational times are multiplied by this number to see the real computational effort. One must

note that these modified times are only indi cat ive and should be used only to get some kind of

picture of the magnitude of the running times. The Table is divided in three parts, presenting

results ofsome recent local searches and metaheuristics, and our local searches, respectively.

In most earlier papers the results are ranked according to a hierarchical objective function,

where the number o f routes is considered as the pr imary object ive , and for the same number of

routes, total traveled distance is considered as the secondary objective. Therefore, a solution

requiring fewer routes is always considered better than a solution wi th more routes, regardless of

the total distance. The rationale behind this approach is that each additional route requires very

costly investments in regard to new vehicles and drivers causing f ixed costs for a long time

period. Thus one assumes that each route is served by a separate vehicle. In real-life problems

this assumption does not always hold, and also other objectives are considered instead ofseek

ing just a solution w i t h the least number of vehicles. Here we used the hierarchical objective

function to make the comparison w i t h previous approaches possible.According to Table 1, it seems that our approaches are very compet itive regarding the number

of routes. Only Braysy (2001b) reports slightly low er C N V than B4 and B5. A l l other earlier

approaches except Homberger and Gehring (1999) and Gehring and Homberger (2001) report

greater CNV . The better results in the above mentioned three papers are, however, obtained at

the cost of clearly higher computational effort, which may make it prohibitive to use them in

large-scale or dynamic real-life problems. The reason for the good performance regarding the

number of routes can be found in the robust route elimination procedure. In general, the differ-



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328 O. BRAYSY

ence in cumulative number of vehicles between the approaches in Table 1 is about 6%, which is

in our opinion remarkable, considering the huge f ixed costs caused by buying an additional

vehicle and hiring a new driv er for it.

As one can observe from Table 1, all of our five approaches outperform the previous local

searches in terms of solution quality. As for the melaheuristics, only Homberger and Gehring

(1999), Gehring and Homberger (2001) and Braysy (2001b) report better results, while being

multiple times slower. The slight overestimation oftotal distance wi th respect to the best values

reported in the literature can be justified by the fact that our approaches arc much faster than

previous approaches repo rting equal CNV, especially i f one considers that our algorith ms are

implemented using JAVA, which is according to our experience much slower than C-program-

ming language3. The main advantage of using JAVA is the clearly increased development speed

due to automatic memory handling mechanisms. This makes it possible to quickly code and test

new ideas. Here one must note that due to the conflicting nature of route number and dis tance

optimization, the comparison over total distance values is reasonable only for the same number

of routes. We consider the given differences in tim e consumpti on signifi cant especially in

dynamic and very large-scale problems, where the speed of the solution method is of prime con

cern. Moreover, as our local searches are deterministic and simple, they can easily be used to

create in i t ia l solutions for more complex approaches to further improve the solution quality, i f

more time is available. Especially B4 was designed only to quickly create ini t ial solutions wi th a

small number of vehicles, and to be hybridized wi th some metaheuristic for distance minimiza

t ion . According to Table 1, B4 achieves its objective very well by giving solutions wi th the l ow

est known number of vehicles to all Solomon's benchmark problems, except one, while being

more than 18 times faster than any previous approach. Local searches by Russell (1995) and

Antes and Derigs (1995) seem to be faster than our approaches, but they fall clearly behind in

solution quality.

4. C O N C L U S I O N S

We proposed five variants of deterministic local searches that use a new three-phase approach.

In the first phase an ini t ial solut ion is created using one of the two proposed construct ion heuris

tics. Then, a special route-elimination operator, based on a new type of ejection chains is used to

minimize the number of routes. Finally, in the th i rd phase, the created solutions are i mpr ove d in

terms of distance using well-known Or-opt exchanges. The computational testing of the pro

posed methods was carried out on the 56 test problems of Solomon (1987).

The proposed local search methods were compared wi th the best previous local searches and

metaheuristics found in the litera ture . The results show that the methods proposed in this paper

arc very efficient approaches for solving the vehicle routing problems wi th time wind ows, pro

ducing results that outperform recent local search approaches and are competitive wi th the best

metaheuristics in terms of solution quality, while being much faster. Our local searches show

good performance especially regarding the number of routes that is often considered to be the

primary objective. The fact that only Braysy (2001b) is able to report lower cumulative number

of vehicles over al l test problems questions the performance of metaheuristics in optimizing the

number of routes. On the other hand, for distance minimization purposes, metaheuristics seem to

be necessary to avoid being trapped in local minima. As heuristics need to be especially effec

tive for dynamic and very large-scale problems, we consider our approaches practica lly relevant.Future research w i l l be conducted to evaluate the performance of different combinations wi th

metaheuristics to further improve the solution quality.

3 We implemented a modification of the approach described in this paper using both JAVA and C++, and found thatJAVA is approximately 10 times slower than C++ on the average. This is not taken into consideration in running timesin Table 1.

Reproduced wi th permiss ion of th e copyr ight owner. Fur ther reproduct ion proh ib i ted wi thout p e rm i ss i o n .


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V E H I C L E ROUTI NG PROBLE M W I T H T I M E W I N D O W S 329

A C K N O W L E D G E M E N T S

This work was partially supported by the E m i l Aaltonen Foundation, Liikesivistysrahasto Foun

dation and the TOP program funded by the Research Council of Norway. This support is grate

ful ly acknowledged.

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Olli Braysy is currently a Research Scientist at SINTEF Applied Mathematics, Department of Optimiza

tion in Oslo, Norway. He received the master, licentiate and doctoral degrees from Department of Mathematics and Statistics at the University of Vaasa, Finland in years 1998-2001. His research interests are

focused on different routing and scheduling problems and heuristic and mefaheuristic solution methods.

Fast Local Searches for the Vehicle Routing Problem With Time Windows

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