14
This work was supported by Grant No. 981-1014-075-1 from the Basic Research Program of the KOSEF. * Corresponding author. Tel.: #82-31-201-2553; fax: #82- 31-203-4004. E-mail address: ybpark@nms.kyunghee.ac.kr (Y.-B. Park). Int. J. Production Economics 73 (2001) 175}188 A hybrid genetic algorithm for the vehicle scheduling problem with due times and time deadlines Yang-Byung Park* Industrial Engineering Division, College of Mechanical and Industrial System Engineering, Kyung Hee University, 1 Seocheon, Kihung, Yongin-Si, Kyunggi-Do 449-701, South Korea Received 2 May 2000; accepted 9 December 2000 Abstract In this paper, I propose a hybrid genetic algorithm (HGAV) incorporating a greedy interchange local optimization algorithm for the vehicle scheduling problem with service due times and time deadlines where three con#icting objectives of the minimization of total vehicle travel time, total weighted tardiness, and #eet size are explicitly treated. The vehicles are allowed to visit the nodes exceeding their service due times with a penalty, but within their latest allowable times. The HGAV employs a mixed farming and migration strategy along with a variant of partially mapped crossover and several mutation operators newly developed while maintaining the solution feasibility during the whole evolutionary process. The HGAV is extensively evaluated on the various types of test problems, including a sensitivity analysis with varied genetic parameters on the weighted sum of three objectives of a solution. The results show that the HGAV always attains better solutions than the BC-saving algorithm, but with a much longer computation time. 2001 Elsevier Science B.V. All rights reserved. Keywords: Vehicle scheduling problem; Service due times; Service time deadlines; Hybrid genetic algorithm; Sensitivity analysis 1. Introduction The vehicle scheduling problem involves deter- mining minimum cost vehicle schedules for a #eet of vehicles originating and terminating from a cen- tral depot. The vehicles service a set of nodes with vehicle capacity and travel time constraints. All nodes must be assigned to vehicles such that each node is serviced exactly once and each vehicle can- not violate the given constraints. The vehicle sched- uling and routing problem has been extensively studied. Bodin et al. [1], Golden and Assad [2], Soloimon [3], Laporte [4], and Anily [5] provide a comprehensive survey of vehicle scheduling and routing problem and several variations. During the past few years, several studies have reported their experiences about applying genetic algorithms to the vehicle scheduling and routing problem with time and capacity constraints. The obvious advantages of genetic algorithms include ease of implementation, emphasis on global as well as local search, use of randomization in search process, and interactive feature. 0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 1 7 4 - 2

A hybrid genetic algorithm for the vehicle scheduling problem with due times and time deadlines

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�This work was supported by Grant No. 981-1014-075-1 fromthe Basic Research Program of the KOSEF.

*Corresponding author. Tel.: #82-31-201-2553; fax: #82-31-203-4004.E-mail address: [email protected] (Y.-B. Park).

Int. J. Production Economics 73 (2001) 175}188

A hybrid genetic algorithm for the vehicle scheduling problemwith due times and time deadlines�

Yang-Byung Park*

Industrial Engineering Division, College of Mechanical and Industrial System Engineering, Kyung Hee University, 1 Seocheon, Kihung,Yongin-Si, Kyunggi-Do 449-701, South Korea

Received 2 May 2000; accepted 9 December 2000

Abstract

In this paper, I propose a hybrid genetic algorithm (HGAV) incorporating a greedy interchange local optimizationalgorithm for the vehicle scheduling problem with service due times and time deadlines where three con#icting objectivesof the minimization of total vehicle travel time, total weighted tardiness, and #eet size are explicitly treated. The vehiclesare allowed to visit the nodes exceeding their service due times with a penalty, but within their latest allowable times. TheHGAV employs a mixed farming and migration strategy along with a variant of partially mapped crossover and severalmutation operators newly developed while maintaining the solution feasibility during the whole evolutionary process.The HGAV is extensively evaluated on the various types of test problems, including a sensitivity analysis with variedgenetic parameters on the weighted sum of three objectives of a solution. The results show that the HGAV always attainsbetter solutions than the BC-saving algorithm, but with a much longer computation time. � 2001 Elsevier Science B.V.All rights reserved.

Keywords: Vehicle scheduling problem; Service due times; Service time deadlines; Hybrid genetic algorithm; Sensitivity analysis

1. Introduction

The vehicle scheduling problem involves deter-mining minimum cost vehicle schedules for a #eetof vehicles originating and terminating from a cen-tral depot. The vehicles service a set of nodes withvehicle capacity and travel time constraints. Allnodes must be assigned to vehicles such that each

node is serviced exactly once and each vehicle can-not violate the given constraints. The vehicle sched-uling and routing problem has been extensivelystudied. Bodin et al. [1], Golden and Assad [2],Soloimon [3], Laporte [4], and Anily [5] providea comprehensive survey of vehicle scheduling androuting problem and several variations.

During the past few years, several studies havereported their experiences about applying geneticalgorithms to the vehicle scheduling and routingproblem with time and capacity constraints. Theobvious advantages of genetic algorithms includeease of implementation, emphasis on global as wellas local search, use of randomization in searchprocess, and interactive feature.

0925-5273/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 1 7 4 - 2

Gabbert et al. [6] presented a genetic algorithmapproach to learning low-cost routes and schedulesfor a large rail freight transportation network.Thangiah et al. [7] developed a kind of cluster-"rstroute-second heuristic for the vehicle routing prob-lem with time deadlines. They used a genetic algo-rithm to cluster nodes. Blanton and Wainwright[8] developed two new crossover operators to ap-ply an order-based genetic algorithm to the vehiclerouting problem with time windows. Thangiah [9]used a genetic algorithm to search for the attributesof a set of circles, corresponding to the number ofvehicles, for clustering nodes in the vehicle routingproblem with time windows.

Cheng and Gen [10] proposed a hybrid geneticalgorithm to solve the fuzzy vehicle routing andscheduling problem where fuzzy due time is used torepresent the grade of satisfaction for the servicetime at each node. They designed an insertion heu-ristic-based crossover operator to "nd out im-proved o!spring. Potvin et al. [11] used a geneticalgorithm to "nd good parameter settings in theroute construction phase of a parallel insertionheuristic for the vehicle routing problem with timewindows. Malmborg [12] applied a geneticalgorithm to a service-level-based vehicle routingproblem where vehicles serve the material #ow re-quirements between a collection of work centersover a "xed operating period and the objective is tominimize the delay associated with material trans-fer. Chen et al. [13] modeled the rolling batchproblem in iron and steel industry as the vehiclerouting problem with time windows and solved itusing genetic algorithm.

Genetic algorithms have proven to be a versatileand e!ective approach for solving the vehicle rout-ing and scheduling problem. Nevertheless, there aremany situations in which the simple genetic algo-rithm does not perform well particularly due toa premature convergence to a local optimization.One of the most common approaches to resolve thepremature convergence is to employ a hybrid gen-etic algorithm. In the hybrid approaches, geneticalgorithms are used to perform global explorationamong a population while heuristic methods areused to perform local exploitation around chromo-somes. Because of the complementary properties ofgenetic algorithms and conventional heuristics, the

hybrid approach often outperforms either methodoperating alone and so many applications arestrongly in favor of the hybrid approach [14,15].

In many vehicle scheduling, the vehicles areallowed to visit the nodes exceeding the given ser-vice due times with a penalty, but within timedeadlines (or latest allowable service times), and thevehicles that arrive at the nodes earlier than duetimes serve them immediately. Hence, optimizationof vehicle scheduling involves not only total vehicletravel time and the number of vehicles required, buttotal weighted tardiness occurring when the ve-hicles arrive at the nodes after due times. This typeof vehicle scheduling problem is termed a vehiclescheduling problem with due times and time dead-lines, or VSPDT. The VSPDT arises in a widearray of practical decision making problems. In-stances of the VSPDT occur in retail distribution,school bus routing, industrial refuse collection,after-service (A/S) visit, mail delivery, dial-a-rideservice, freight operations, etc. E$cient schedulingof vehicles can save a signi"cant amount of logisticscost and improve the customer service level.

In this paper, I propose a hybrid genetic algo-rithm (HGAV) incorporating a greedy interchangelocal optimization procedure to solve the VSPDTwhere three con#icting objectives of the minimiz-ation of total travel time, total weighted tardiness,and #eet size are explicitly considered. The hybridgenetic algorithm developed is extensively evalu-ated on various types of test problems, includinga sensitivity analysis with varied genetic parameterson the weighted sum of three objectives.

2. HGAV

2.1. Procedure of HGAV

The HGAV guarantees the generation of feasiblesolutions in its whole evolutionary process by usingthe special solution coding and applying the repairprocess to correct any infeasible solutions gener-ated. The HGAV also employs an elitist strategy toreport the best solution found during the evolution-ary process. The HGAV adopts a mixed farmingand migration strategy [16] that is being imple-mented in designing a `virtual parallel computera,

176 Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188

in order to avoid a premature convergence to a lo-cal optimum and, at the same time, to maintaina high survival probability for the chromosomeswith dominant elements (or genes). The basic ideaof this strategy is to divide initial population intoseveral sub-populations and apply a genetic algo-rithm to each of them while performing a localoptimization periodically to some best chromo-somes collected from them.

The procedure of HGAV is described as follows:Step 1: Divide the initial population into l sub-

populations with the same number of chromo-somes each.

Step 2: Apply the crossover and mutation to thechromosomes in each sub-population, and thencalculate their "tness values. If the current genera-tion number is a multiple of m, then move to Step 3.Otherwise, move to Step 4. m is an integer valuerepresenting an interval of local optimization to beperformed in Step 3.

Step 3: Select the best n di!erent chromosomesfrom each sub-population and apply a local optim-ization technique to all of them (a total of l ) nchromosomes). Then, substitute the current bestn di!erent chromosomes in each sub-population bythe best n di!erent chromosomes obtained by ap-plying a local optimization technique.

Step 4: If a termination condition is satis"ed, thenreport the best chromosome found so far and stop.Otherwise, apply a roulette wheel selection method[17] to each sub-population and return to Step 2.

2.2. HGAV Parameters

2.2.1. Solution codingA solution is coded through a diploid structure

(or two strings). The "rst string includes all nodes inany order except depot. The second string has thesame number of "elds as the "rst one and it denotesthe vehicle numbers assigned to the nodes at thecorresponding "elds in the "rst string. The examplebelow represents three vehicles whose routesare 0-3-9-1-10-7-0, 0-2-4-5-6-0 and 0-8-0, where0 denotes a central depot.

A�

3 9 2 1 4 10 8 5 6 7A

�1 1 2 1 2 1 3 2 2 1

2.2.2. InitializationBecause of the existence of time and capacity

constraints, it is impossible to use a simple randomprocedure to generate the initial population. In theproposed initial procedure, the "rst string ofa chromosome is constructed by generating nodenumbers randomly and the second string is con-structed by assigning vehicle numbers to the corre-sponding nodes sequentially from the leftmost "eld.Vehicle number 1 is always assigned to the "rstnode in the "rst string.

To determine the vehicle number for a node,a feasible insertion position for the node is searchedon the partial routes starting from vehicle number1 in ascending order. It is always required to exam-ine the route feasibility whenever a node insertion isconsidered. If the "rst feasible insertion position forthe node is found, then the node is moved to the"eld of its insertion position in the "rst string andthe corresponding vehicle number is assigned to thesame "eld in the second string. Then, the node andvehicle numbers after the insertion position aremoved backward by one "eld in each string. If nofeasible insertion position for the node is found,a new vehicle number is assigned to it at the current"eld.

2.2.3. Fitness evaluationFitness is calculated based on the three objective

functions. The weighted sum of three objectives fora chromosome is the combination of the normalizedtotal travel time, the normalized total weighted tar-diness, and the normalized #eet size. The weightedsum of three objectives is computed by

f (S�)"�w�

t�

t����

#w�

d�

d����

#w�

v�

v�����100,

�����

w�"1 and w

�*0, i"1, 2, 3, (1)

where w�is the weight of the minimization of the

total travel time, w�the weight of the minimization

of total weighted tardiness, w�

the weight of theminimization of the #eet size, t�

���the maximum

total travel time among the initial chromosomes,d����

the maximum total weighted tardiness amongthe initial chromosomes, v�

���the maximum #eet

size among initial chromosomes, t�the total travel

Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188 177

time of chromosome k, d�the total weighted tardi-

ness of chromosome k, and v�

the #eet size ofchromosome k.

Based on the weighted sum of three objectives,the relative "tness of a chromosome used in theselection process (Step 4 of HGAV) for reproduc-tion is calculated by

r(S�)"

f (S����

)!f (S�)

f (S����

)!f (S����

), (2)

where f (S����

) is the maximum weighted sum ofobjectives among chromosomes in sub-populationof chromosome k, and f (S�

���) is the minimum

weighted sum of objectives among chromosomes insub-population of chromosome k. Eq. (2) expressesthe relative di!erence among chromosomes in sub-population better than (1), particularly when thedi!erence between f (S�

���) and f (S�

���) is small.

2.2.4. Local optimizationFor a local optimization, a greedy interchange

algorithm introduced by Dror and Levy [18] isapplied with a minor modi"cation. The local op-timization procedure consists of two phases. In the"rst phase, a two-node and a two-arc interchangeare applied to each route in a chromosome. When-ever an interchange is considered, the route feasibil-ity is tested. The weighted saving from either thetwo-node or two-arc interchange is given by

sav"w�

�t

t����

#w�

�d

d����

, (3)

where �t is the saving in travel time from two-node(arc) interchange, and �d the saving in weightedtardiness from the two-node (arc) interchange.

Lemma 1. When an interchange of two arcs (i, j) and(k, l) where j(k is attempted, all nodes followingnode l on the route after interchange are servedbefore time deadlines if the new arrival times to thenodes between i and l on the route before interchangedo not violate time deadlines and the new arrival timeto l is decreased.

In the second phase, a one-node and a two-nodeinterchange are applied to all the pairs of routes ina chromosome. The candidate nodes for inter-

change must satisfy the individual route constraintsas well. The weighted saving from one-node inter-change is computed by adding w

��v/v�

���to Eq. (3)

where �v denotes the saving in #eet size from one-node interchange.

Lemma 2. When a one-node interchange betweena route of i and j, where i�j, and a route of k, r, and l,where k;r;l, is attempted with node r:

(i) all the nodes following node j on the route afterinterchange are served before time deadlines ift��

#t�!t

��)0, where t

��represents a travel

time from i to r and(ii) the new arrival times to the nodes following

j after interchange are given by a��"a

�#�,

where a�denotes the arrival time to node u be-

fore interchange and �"t��

#t�!t

��.

2.2.5. CrossoverA variant of the partially mapped crossover

(PMX) [19] is used. The variant is required toconsider the multiple vehicles and to include therepair process for infeasible solutions. The cross-over is initiated by choosing two random cut pointsof parents and swaps the segments before the "rstcut points of parents with the nodes and corre-sponding vehicle numbers at the same time. Next, itcopies the segments between two cut points of eachparent into each o!spring. Finally, it swaps thesegments after the second cut points of parents.

When there is a con#ict for node number in"lling the o!spring after the "rst cut point, the nodeand vehicle number under consideration are re-placed by other ones based on the prede"ned map-pings. When there is infeasible node in "lling theo!spring after the "rst cut point, its vehicle is re-placed by the "rst feasible one while considering thevehicles in ascending order.

2.2.6. MutationSince a chromosome represents multiple vehicle

routes simultaneously, normal mutation maycause random breaking, sub-tours, unnecessary in-crease of #eet size, and repetitions of nodes, result-ing in loss of the optimality information gained bygenetic algorithm over the previous generations.Thus, I propose a set of mutation operators that

178 Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188

overcomes the defects of normal mutation in gen-etic algorithm. The mutation operators utilize theroute information of a chromosome and they are asfollows.

(i) Merge mutation: Select one node randomly inthe route whose number of nodes is less than threeand change its vehicle number to one of the othervehicle numbers if the change does not violate theconstraints. If it is not possible to change the ve-hicle for any selected node due to the constraintviolations, give up a merge mutation. This muta-tion helps to reduce the #eet size of a solution.

(ii) Time mutation: Select the arrival node of thelongest path on the route with the longest traveltime and change its vehicle number to the one withthe shortest travel time if the change does notviolate the constraints. If the change violates theconstraints, attempt repeatedly to change to thevehicle number with the next shortest travel time. Ifit is not possible to change the vehicle for the se-lected node, give up a time mutation. This mutationhelps to reduce the total travel time of a solution.

(iii) Tardy mutation: Select a node whose weightedtardiness is the largest on the route with the largesttotal weighted tardiness and change its vehiclenumber to the one with the smallest total weightedtardiness if the change does not violate the con-straints. If the change violates the constraints, tryrepeatedly to change to the vehicle number with thenext smallest total weighted tardiness. If it is notpossible to change the vehicle for the selected node,give up a tardy mutation. This mutation helps toreduce the total weighted tardiness of a solution.

(iv) Random mutation: Select a node randomly ina chromosome and change its vehicle number toone of the other vehicle numbers if the change doesnot violate the constraints. If it is not possible tochange the vehicle for the selected node to anyother one, assign a new vehicle number to the node.

(v) Exchange mutation: Select randomly twonodes in a chromosome and exchange mutually thepositions of these nodes together with the corre-sponding vehicle numbers if the exchange does notviolate the constraints. If it is not possible to ex-change them, attempt repeatedly an exchange withanother two randomly selected nodes. If it is notpossible to exchange any two nodes, give up anexchange mutation.

The "rst three mutations try to "nd, from a givensolution, a new solution improved on one of threeobjectives. These mutations integrate the gradientdescent method implicitly into genetic algorithm,and thus the "tness of a chromosome may be im-proved by implementing one of the three mutationsselectively based on the route characteristics ofa chromosome to mutate. The last two mutationsincrease the population diversity in the geneticsearch.

In order to maximize the e!ectiveness of muta-tion, a simple mutation selection procedure bywhich the above variousmutations are integrated isdeveloped. In the selection procedure, the order ofconsidering the mutations is varied on the basis ofthe importance of three objectives. As an example,the following selection procedure is based on theassumption that the importance order of three ob-jectives is the minimization of #eet size, total traveltime, and total weighted tardiness. The multiple ofminimal values in Steps 2 and 3 can be adjusted.

Step 1: Select a merge mutation if its applicationis possible. Otherwise, move to Step 2.

Step 2: Select a time mutation if the maximaltravel time of routes is 1.3 times longer than theirminimal value and its application is possible.Otherwise, move to Step 3.

Step 3: Select a tardy mutation if the maximaltotal weighted tardiness of routes is 1.3 times largerthan their minimal value and its application ispossible. Otherwise, move to Step 4.

Step 4: Select a random mutation and an ex-change mutation alternately.

3. Example

In this section, I present result of solving anexample of VSPDT by HGAV. The data used forthe example are based on Solomon's R101 problem[3]. The data for the node coordinates and de-mands are the same as those used in R101 problem.The due times of nodes are deemed as the lowerlimits of time windows used in R101 problemand the time deadlines are determined by adding10 time units to the respective due times. Theimportant characteristics of the example namedMR101 are summarized in Table 1. The test

Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188 179

Fig. 1. Variation of three objective values and their weighted sum over 1000 generations in example.

problems of VSPDT are distinguished by a$xingan &M' before the original names of Solomon'sproblems. The actual data of the example are avail-able from the author.

The weights of the three objectives are set to 0.6for total travel time, 0.3 for total weighted tardi-ness, and 0.1 for #eet size. In applying the HGAV,the population size is set to 90, the number ofsub-populations to 3, the crossover rate to 0.4, themutation rate to 0.4, the interval of local optimiza-tion to 5, and the number of chromosomes for localoptimization in sub-population to 3.

Fig. 1 shows graphically three objective valuesand their weighted sum of the best chromosomesobtained after every 100 generations up to 1000generations during the HGAV run in the example,including the results obtained after the "rst genera-tion. The results show that the solution is improvedgradually until about 600 generations, but afterthen no signi"cant improvement is observed. Thebest solution obtained after 1000 generations hasa total travel time of 1586.4, total weighted tardi-ness of 11.7, #eet size of 21, and weighted sum ofthree objectives of 37.4. The vehicle routes of thebest solution are depicted in Fig. 2.

4. Computational study

In order to evaluate the performance of theHGAV, the computational study is performed in

three aspects: "rst, a comparison of the HGAVwithmodi"ed HGAV that allows infeasible solutions byassociating a penalty with all constraint violations;second, a sensitivity analysis with varied geneticparameters; third, a comparison of the HGAV withBC-saving algorithm [20]. The HGAV and BC-saving algorithm were programmed in Visual Basic5.0 and all computational experiments were con-ducted on an IBM-compatible PC 586 (32MRAM,Intel MMX 233).

4.1. Comparison of HGAV with modixed HGAV

The HGAV applies special representation map-pings and repair processes to correct any infeasiblesolutions generated, and so always guaranteesthe generation of feasible solutions. The HGAV canbe modi"ed to allow infeasible solutions by asso-ciating the penalties with all constraint violationsand including them in the "tness function of a solu-tion. The performances of these two di!erent ap-proaches for handling the given constraints arecompared with each other. In the modi"ed HGAV,the weighted sum of three objectives of a chromo-some is given by

f (S�)"�w�

t�

t����

#w�

d�

d����

#w�

v�

v����

#��l�#�

�r�#�

�q���100, (4)

180 Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188

Fig. 2. Solution of vehicle routes for example after 1000 generations.

where ��is the penalty coe$cient for the constraint

of time deadline, ��

the penalty coe$cient forthe constraint of vehicle return time to depot,��the penalty coe$cient for the constraint of ve-

hicle capacity, l�the number of nodes whose time

deadlines are violated in chromosome k, r�

thenumber of routes that violate the vehicle returntime to depot in chromosome k, q

�the number of

routes that violate the vehicle capacity in chromo-some k.

For experiments, two problems are constructedon the basis of the data used in Solomon's R101and RC201 problems [3]. The data for the duetimes and time deadlines of nodes are determinedby the same way as in the example in Section 3. Thecharacteristics of two test problems named MR101and MRC201 are summarized in Table 1. In ap-plying the HGAV and modi"ed HGAV, the popu-lation size is set to 90, the crossover and mutationrates are set to 0.4 each, and w

�, w

�and w

�are

arbitrarily assumed to be 0.8, 0.1 and 0.1 each. Inthe modi"ed HGAV, �

�, �

�and �

�are equally

set to 1.

Fig. 3 shows the comparison of the HGAV withmodi"ed HGAV with respect to the solution qual-ity and speed of searching for the best solution intwo test problems. It is seen from Fig. 3 that bothalgorithms improve the solutions on both problemsas the evolutionary process is continued, exceptbetween generations 100 and 200 when the modi-"ed HGAV is applied to MRC201 problem.However, the HGAV quickly "nds a very goodsolution right after the "rst round of evolution andconverges to the "nal solution much faster than themodi"ed HGAV on both problems. This outcomeseems to be due to the fact that the modi"edHGAVconcentrates on improving infeasible solutions be-ing continuously generated rather than on movingtoward an optimal feasible solution during theevolutionary process.

4.2. Sensitivity analysis with varied geneticparameters of HGAV

It is required to determine the values for geneticparameters in applying the HGAV. The important

Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188 181

Table 1Summary of characteristics of 20 test problems

Set Problem Vehicle capacity Vehicle return time Time deadline Service time

MR1 MR101 200 230 Due time#10 10MR102 200 230 Due time#10 10MR103 200 230 Due time#10 10MR105 200 230 Due time#30 10MR106 200 230 Due time#30 10

MR2 MR201 1000 1000 Due time#116 10MR202 1000 1000 Due time#57 10MR203 1000 1000 Due time#103 10MR205 1000 1000 Due time#240 10MR206 1000 1000 Due time#240 10

MRC1 MRC101 200 240 Due time#30 10MRC102 200 240 Due time#30 10MRC103 200 240 Due time#30 10MRC105 200 240 Due time#54 10MRC106 200 240 Due time#60 10

MRC2 MRC201 1000 960 Due time#120 10MRC202 1000 960 Due time#120 10MRC203 1000 960 Due time#240 10MRC205 1000 960 Due time#223 10MRC206 1000 960 Due time#240 10

Fig. 3. Comparison of HGAV with modi"ed HGAV in MR101 and MRC201 problems.

genetic parameters include the number of genera-tions (GEN), initial population size (pop}size),number of sub-populations (l), crossover rate (p

�),

mutation rate (p�), number of chromosomes for

local optimization in each sub-population (n), and

local optimization interval (m). It is known that thegenetic parameters a!ect signi"cantly the perfor-mance of genetic algorithm such as the solutionquality and search time for a solution. However,there exists no analytical method generally

182 Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188

Fig. 4. Results with varied population size.

applicable to "nd the best values for the parametersbecause they are heavily dependent upon the prob-lems to be solved.

In this section, I present the sensitivity analysiswith varied genetic parameters of the HGAV on theweighted sum of three objectives. It should be notedthat the experimental results are not conclusive,because each sensitivity analysis is performed withone varied parameter while the remaining para-meters are "xed to their basic settings and so theinteraction e!ects among the parameters are ignor-ed. A total of 4 sensitivity analyses are performed.Their outcomes will be used in determining thebest parameter values for the HGAV in Section 4.3.For experiments, MR101 problem is employed.The basic setting of parameters is GEN"1000,pop}size"90, l"3, p

�"0.4, p

�"0.4, n"3, and

m"5. The basic setting of weights of three objec-tives is arbitrarily assumed as w

�"0.6, w

�"0.3,

and w�"0.1.

Before discussing the sensitivity analysis onMR101 problem, it is noted that very similar resultswere obtained from the sensitivity analysis withvaried genetic parameters for the various settings ofthree objectives' weights in several problems listedin Table 1. That is, the results of the sensitivityanalysis were almost the same for the problems andsettings of three objectives' weights.

4.2.1. Variation of population sizeBased on the above settings, the population size

is varied from 30 to 390 by 60. Fig. 4 shows graphi-

cally the results for each setting. The results showthat when the population size is larger than 90, itsvalue has no signi"cant in#uence on the perfor-mance of HGAV.

4.2.2. Variation of local optimization intervalBased on the above basic settings, the local op-

timization interval is varied from 10 to 100 by 10including an interval of 1. Fig. 5 shows graphicallythe results for each setting. It is seen from the "gurethat the solution is improved as the local optimiza-tion interval is reduced. However, the computationtime is increased to about 1.7}2.5 times as theinterval is reduced to half. This means that thereexists a trade-o! between the solution quality andthe computation time.

4.2.3. Variations of crossover and mutation rateBased on the above settings, each of the cross-

over and mutation rates is varied from 0.0 to 1.0 by0.2 while the other is "xed to its basic setting. Fig. 6shows graphically the results for each setting. Theresults show that neither the varied crossover northe mutation rate has a signi"cant in#uence on theperformance of HGAV, though crossover andmutation with a probability of 0.4 yield marginallythe best solution.

4.3. Comparison of HGAV with BC-saving algorithm

The proposed HGAV is compared with the BC-saving algorithm [20] that is a heuristic developed

Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188 183

Fig. 5. Results with varied local optimization interval.

Fig. 6. Results with varied crossover and mutation rates.

for solving the bicriteria vehicle scheduling prob-lem with time- and area-dependent travel speedsand is proven to be excellent. The BC-saving algo-rithm constructs the routes in parallel by combin-ing two partially formed routes whose total savingis the largest while keeping the constraints given.The total saving is the weighted sum of the savingsin vehicle travel time (S¹

��) and weighted tardiness

(SD��) that are expected by connecting the last node

i to visit of one route to the "rst node j to visit of theother route. In applying the BC-saving algorithmto solve the VSPDT, I used the following savingequation, assuming that vehicle travel speed is al-ways constant. The third term in the equation rep-

resents the saving in #eet size.

BC��

"w�

S¹��

t����

#w�

SD��

d����

#w�

1

v����

. (5)

The procedural steps of the BC-saving algorithmare brie#y summarized as follows:

Step 1: Construct N!1 distinct routes in whicheach customer is served by a dedicated vehicle.Compute the vehicle travel time and total weightedtardiness for each route.

Step 2: Select two partially formed routes andperform the feasibility test for their combinedroute with respect to the restrictions for vehicle

184 Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188

and delivery time. Repeat the process of selectionand feasibility test for all the partially formedroutes, and compute BC

��of all the feasible pairs of

nodes for combination. If there exist no feasiblepairs of nodes to link, go to Step 4.

Step 3: Combine the two routes with the largestvalue of BC

��into a single route, and go to Step 2.

Step 4: Compute the improvement in theweighted sum of three objectives, Eq. (1) in Section2.2, by interchanging adjacent pairs of nodes ina route while keeping the restrictions. If there existno feasible adjacent pairs of nodes to interchangeor no improvement is expected from the feasibleinterchanges, then go to Step 5. Otherwise, con-struct a new route by switching the positions of theadjacent pair that shows the largest improvement.Repeat the interchange process to the newly con-structed route until no more improvement is ex-pected.

Step 5: Apply Step 4 to the rest of the routes andstop. The vehicles are scheduled on the basis of the"nal vehicle routes obtained.

For comparison test, "ve problems are selectedfrom each of Solomon's problem sets R1, R2, RC1,and RC2 [3]. A total of 20 test problems witha single depot and 100 nodes are constructed on thebasis of these problems. The geographical data arerandomly generated by a random uniform distribu-tion in the problem sets R1 and R2, and semi-clustered by a mix of randomly generated data andclusters in the problem sets RC1 and RC2. The setsR1 and RC1 have a short scheduling horizon; incontrast, the sets R2 and RC2 have a long schedul-ing horizon. In test problems, data for node coordi-nates and demands are the same as the one used inSolomon's problems. The due times of nodes aredeemed as the lower limits of the time windowsused in Solomon's problems and the time deadlinesare determined by adding a numerical value to therespective due times. The numerical value is ap-proximately the average travel time between nodesin each problem. The actual data for test problemsare available from the author. Some importantcharacteristics of test problems are summarized inTable 1.

In applying the HGAV to test problems, I setGEN"1000, pop}size"90, p

�"0.4, p

�"0.4,

n"3 and m"5, based on the experimental results

obtained in Section 4.2. Three settings for theweights of three objectives with the di!erent "rst-priority objectives are considered in all test prob-lems to investigate how they impact on the perfor-mance of HGAV. The computational results aregiven in Table 2. The reduction rate is computed by

(weighted sum of three objectives of the solutionobtained by BC-saving algorithm!weighted sumof three objectives of the solution obtained byHGAV)�100/weighted sum of three objectives ofthe solution obtained by BC-saving algorithm.

Thus, the positive rate denotes the percentage im-provement of solution by the HGAV over the BC-saving algorithm. The average reduction rates forfour problem sets and three settings of the objectiveweights are summarized in Table 3.

The computational results are very impressive. Itis known from Tables 2 and 3 that the HGAV isclearly superior to the BC-saving algorithm in alltest problems regardless of the problem types andobjective weights, showing shorter total travel time,less total weighted tardiness, and smaller #eet size.The overall average of reduction rate is computedas about 8.0%. In particular, it appears that theHGAV performs very well in the problem setsMR2and MRC2 with the weight setting of (0.3, 0.1, 0.6).This outcome may be explained by the fact that thesets MR2 and MRC2 have the characteristic ofrequiring smaller #eet size than the other two sets,so the reduction in #eet size with a high weight of0.6 leads to a signi"cant decrease in the weightedsum of objectives. From Table 2, it is seen that threeobjectives are traded o! well in the HGAV accord-ing to the changes of their weights. For instance,when the weight setting is (0.3, 0.6, 0.1), the totalweighted tardiness is by far the least among threeweight settings, owing to the increase in total traveltime and #eet size.

In terms of computation time, I did not performa precise and extensive comparison of two algo-rithms on the test problems. Based on the roughmeasurement, the HGAV and BC-saving algo-rithms require 6}30 CPU minutes and 10}20 CPUseconds, respectively, depending on the problemsto be solved. The HGAV requires much longerCPU time to obtain a high-quality solution thanthe BC-saving algorithm.

Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188 185

Tab

le2

Com

par

ison

ofHGAV

with

BC-sav

ingalgo

rith

m�

Pro

blem

(0.6,0

.3,0.1)

�(0.3,0.6,

0.1)

(0.3,0.1,

0.6)

HGAV

BC-sav

ing

RR

HGAV

BC-sav

ing

RR

HGAV

BC-sav

ing

RR

(%)

(%)

(%)

ST

VT

STVT

ST

VT

STVT

ST

VT

STVT

MR1

MR10

137

.4(158

6,12

,21

)�39

.0(164

2,19

,23

)5.2

22.8

(163

4,3,

21)

24.6

(169

5,5,

24)

7.3

56.7

(158

0,40

,20

)61

.3(162

2,45

,22

)7.5

MR10

242

.0(152

7,90

6,19

)44

.7(157

8,10

25,21

)6.1

33.5

(171

2,59

9,24

)35

.3(175

8,64

6,26

)5.6

56.6

(150

3,10

89,18

)61

.2(151

5,11

18,20

)7.6

MR10

343

.5(142

0,44

80,18

)46

.0(147

8,49

56,19

)5.5

39.5

(165

9,35

57,21

)42

.1(166

9,39

42,23

)6.1

59.7

(139

9,60

20,15

)65

.8(140

6,64

97,17

)9.3

MR10

537

.7(151

1,14

5,19

)39

.9(155

3,18

9,21

)5.4

25.2

(172

6,18

,22

)27

.7(179

0,31

,26

)8.9

60.0

(159

8,25

0,19

)66

.6(155

0,28

2,22

)9.9

MR10

639

.8(141

4,19

11,17

)42

.9(147

1,23

26,19

)7.3

32.6

(166

7,13

02,22

)34

.8(167

1,15

42,24

)6.4

56.0

(140

1,25

97,16

)61

.3(141

5,27

85,18

)8.7

MR2

MR20

127

.1(978

,115

8,8)

29.2

(102

3,13

19,9)

7.2

19.6

(102

7,55

4,9)

20.3

(108

8,58

2,9)

3.4

40.1

(960

,175

0,6)

50.2

(969

,17

69,8)

20.2

MR20

227

.2(912

,463

3,7)

29.3

(951

,50

07,8)

7.1

15.7

(658

,301

1,5)

16.6

(894

,62

60,7)

5.6

42.6

(889

,589

7,5)

56.2

(910

,63

49,7)

24.2

MR20

328

.5(929

,114

58,6)

31.2

(973

,12

919,

7)8.5

26.2

(105

8,96

74,7)

28.0

(107

7,10

063,

8)6.3

46.5

(870

,101

12,5)

55.1

(909

,15

169,

6)15

.5M

R20

527

.1(941

,212

2,7)

29.0

(979

,22

28,8)

6.2

21.8

(101

4,10

75,9)

23.1

(105

8,12

77,10

)7.3

37.7

(894

,798

5,4)

44.4

(911

,76

70,5)

15.2

MR20

627

.2(874

,572

1,7)

28.5

(929

,63

73,7)

4.7

23.0

(104

1,43

99,7)

25.0

(105

1,49

59,8)

8.7

40.3

(823

,159

87,4)

54.1

(851

,91

72,6)

25.6

MRC1

MRC10

140

.1(187

5,31

4,20

)42

.0(192

4,35

2,22

)4.6

25.5

(209

3,17

,25

)27

.2(216

1,49

,26

)6.2

56.8

(186

5,39

8,19

)61

.3(187

8,42

2,21

)7.3

MRC10

242

.6(171

0,34

78,19

)45

.5(177

2,37

96,19

)6.4

36.1

(210

9,21

31,23

)38

.5(209

6,25

18,24

)6.3

51.1

(169

0,32

16,17

)56

.1(172

7,37

87,19

)5.5

MRC10

342

.2(160

4,60

49,15

)44

.8(164

8,65

95,17

)5.9

42.1

(187

7,46

27,22

)43

.6(189

7,49

50,22

)3.4

53.9

(157

4,87

45,14

)58

.3(159

0,78

23,16

)8.0

MRC10

538

.2(164

5,99

1,18

)39

.4(167

7,10

33,19

)2.9

28.0

(206

8,20

3,25

)29

.7(214

8,26

2,26

)5.8

53.7

(162

2,10

22,17

)53

.9(163

7,10

48,17

)0.3

MRC10

635

.1(158

8,10

24,17

)36

.4(161

4,11

22,18

)3.6

26.0

(198

5,20

7,25

)27

.7(209

3,32

2,24

)6.1

49.1

(152

2,13

46,16

)49

.6(154

0,15

47,16

)1.0

MRC2

MRC20

120

.7(945

,573

,7)

22.8

(100

9,69

2,8)

9.2

16.0

(107

9,15

6,9)

17.7

(118

8,17

5,10

)9.7

39.1

(927

,248

7,6)

43.2

(102

9,10

14,7)

9.5

MRC20

222

.7(915

,614

5,7)

24.8

(982

,65

69,8)

8.5

22.1

(114

6,41

17,10

)23

.5(120

8,48

34,10

)5.7

40.6

(880

,887

3,6)

40.9

(891

,99

75,6)

0.8

MRC20

326

.0(934

,133

22,7)

27.9

(104

3,14

328,

7)6.3

25.8

(118

8,78

56,10

)27

.6(122

9,83

44,11

)6.6

41.2

(900

,125

42,5)

55.3

(969

,14

834,

7)25

.5M

RC20

521

.2(899

,258

9,5)

22.5

(923

,26

26,6)

5.8

17.1

(103

7,82

1,8)

18.1

(104

2,83

2,9)

5.4

37.3

(101

5,56

78,5)

40.8

(905

,31

77,6)

8.4

MRC20

620

.0(935

,150

4,6)

22.0

(964

,16

59,7)

6.9

16.2

(101

1,10

48,7)

17.7

(102

7,11

76,8)

7.9

35.1

(921

,315

6,5)

40.7

(914

,36

15,6)

13.8

�ST:weigh

ted

sum

ofthreeobjectives;V

T:v

alues

ofthreeobjectives;RR:r

eductionra

te.

�(W

eigh

tof

totaltrav

eltime,

weigh

tofto

talweigh

tedtard

iness,

weigh

tof#eetsize).

�(Totaltrav

eltime,

totalweigh

ted

tard

iness,#eetsize).

186 Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188

Table 3Summary of reduction rate (%)

(w�, w

�, w

�) Set

MR1SetMR2

SetMRC1

SetMRC2

Average

(0.6, 0.3, 0.1) 5.9 6.7 4.7 7.3 6.2(0.3, 0.6, 0.1) 6.9 6.3 5.6 7.0 6.4(0.3, 0.1, 0.6) 8.6 20.1 5.1 11.6 11.4Average 7.1 11.0 5.1 8.6 8.0

It is observed from the experiments that thecomputation time required by the HGAV is ap-proximately proportional to the number of genera-tions and local optimizations executed. So its longcomputation time is mainly due to the large num-ber of generations (GEN"1000) and relativelyshort interval of local optimization (m"5). There-fore, the computation time can be reduced to about3.6}18 CPU minutes (or 6�0.6}30�0.6 CPU min-utes) with a little sacri"ce of the solution quality, bystopping the evolution at 600 generations based onthe experimental outcome in the example. It is alsoobserved that the computation time required by theHGAV is varied depending on the number ofroutes in the problem. In general, the problem ofmore routes requires longer computation time. Asa matter of fact, the high quality solution obtainedby the HGAV is a result of a high computationcost.

5. Conclusions

In this paper, I have presented a hybrid geneticalgorithm incorporating a greedy interchange localoptimization procedure for the vehicle schedulingproblem with due times and time deadlines wherethree con#icting objectives of the minimization oftotal travel time, total weighted tardiness, and #eetsize are explicitly considered. The algorithm em-ploys a mixed farming and migration strategyalong with a variant of the partially mapped cross-over and several mutation operators newly de-veloped while maintaining the solution feasibilityduring the whole evolutionary process.

From the extensive computational experiments,three important observations are summarized

as follows:

1. A better solution is always obtained by theHGAV in much longer CPU time in comparisonwith the BC-saving algorithm. The reductionrate of the weighted sum of three objectives bythe HGAV is computed as 8.0% on the average.

2. The HGAV quickly "nds a very good solutionright after the "rst round of evolution and con-verges to the "nal solution much faster than themodi"ed HGAV that allows infeasible solutionsby associating the penalties with all constraintviolations.

3. In applying the HGAV, the population size overa certain level has no signi"cant impact on thesolution quality, a solution is improved as thelocal optimization interval is reduced while thecomputation time is increased to about doubleas the interval is reduced to half, and the vari-ation of the crossover or mutation rate has nosigni"cant in#uence on the solution quality.

In some practical situations of applying HGAM,the individual values of three objectives for di!erentweight vectors in the form of three-dimensionalvector might be helpful in examining the Pareto-optimal solutions. Finally, it is noted that furtherresearch should be focused on the reduction incomputation time required by the HGAV. Thereduction may be possible by developing a newmethod of generating better initial population anda new crossover operator that enables to introducemany good schemata (a template allowing explora-tion of similarities among chromosomes) [19] tothe sub-populations in subsequent generations.

References

[1] L. Bodin, B. Golden, A. Assad, M. Ball, Routing andscheduling of vehicles and crews: the state of the art,Computers and Operations Research 10 (1983) 62}212.

[2] B. Golden, A. Assad (Eds.), Vehicle routing with timewindow constraints: algorithmic solutions, AmericanJournal of Mathematical and Management Sciences, Vol.15, American Sciences Press, Columbia, OH, 1986.

[3] M.M. Solomon, Time window constrained routing andscheduling problems: A survey, Transportation Science 22(1) (1988) 1}13.

Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188 187

[4] G. Laporte, The vehicle routing problem: An overview ofexact and approximate algorithms, European Journalof Operational Research 59 (1992) 345}358.

[5] S. Anily, Vehicle routing and the supply chain, in: S. Tayur,R. Ganeshan, M. Magazine (Eds.), Quantitative Modelsfor Supply Chain Management, Kluwer Academic Pub-lishers, Dordrecht, 1999, pp. 149}196 (Chapter 6).

[6] P. Gabbert, D. Brown, C. Huntley, B. Markowicz, D.Sappington, A system for learning routes and scheduleswith genetic algorithms, Proceedings of Fourth Inter-national Conference on Genetic Algorithms, 1991,pp. 430}436.

[7] S.R. Thangiah, R. Vinayagamoorthy, A.V. Gubbi, Vehiclerouting with time deadlines using genetic and local algo-rithms, Proceedings of the Fifth International Conferenceon Genetic Algorithms, Los Altos, CA, Morgan Kauf-mann, Los Altos, CA, 1993, pp. 506}513.

[8] J.L. Blanton, R.L. Wainwright, Multiple vehicle routingwith time and capacity constraints using genetic algo-rithms, Proceedings of the Fifth International Conferenceon Genetic Algorithms, Los Altos, CA, Morgan Kauf-mann, Los Altos, CA, 1993, pp. 452}459.

[9] S.R. Thangiah, An adaptive clustering method using geo-metric shape for vehicle routing with time windows, Pro-ceedings of the Sixth International Conference on GeneticAlgorithms, Los Altos, CA, Morgan Kaufmann, 1995,pp. 536}543.

[10] R. Cheng, M. Gen, Fuzzy vehicle routing and schedulingproblem using genetic algorithms, in: F. Herrera,J. Verdegay (Eds.), Genetic Algorithms and Soft Comput-ing, Springer, Berlin, 1996, pp. 683}709.

[11] J. Potvin, D. Dube, C. Robillard, A hybrid approach tovehicle routing using neural networks and genetic algo-rithms, Applied Intelligence 6 (1996) 241}252.

[12] C. Malmborg, A genetic algorithm for service level basedvehicle scheduling, European Journal of Operational Re-search 93 (1996) 121}134.

[13] X. Chen, W. Wan, X. Xu, Modeling rolling batch planningas vehicle routing problem with time windows, Computersand Operations Research 25 (12) (1998) 1127}1136.

[14] M. Gen, R. Cheng, Genetic Algorithms & EngineeringDesign, Wiley, New York, 1997.

[15] C. Reeves, Genetic algorithms and neighborhoodsearch, Evolutionary Computing, Springer, Berlin, 1994,pp. 115}130.

[16] F. Marin, O. Trelles-Salazar, F. Sandoval, Genetic algo-rithm on LAN-based message passing architectures usingPVM: Application to the routing problem, Proceedings ofthe Third Conference on Parallel Problem Solving fromNature, Israel, 1994, pp. 534}543.

[17] D.E. Goldberg, Genetic Algorithms in Search, Optimiza-tion & Machine Learning, Addison-Wesley, Reading, MA,1989.

[18] M. Dror, L. Levy, Avehicle routing improvement algo-rithm comparison of a `GREEDYa and a matching imple-mentation for inventory routing, Computers andOperations Research 13 (1) (1986) 33}45.

[19] Z. Michalewicz, Genetic Algorithms#Data Structures"Evolution Programs, 2nd Edition, Springer, Berlin, 1994.

[20] Y.B. Park, Asolution of the bicriteria vehicle schedulingproblems with time and area-dependent travel speeds,Computers and Industrial Engineering 38 (2000) 173}187.

188 Y.-B. Park / Int. J. Production Economics 73 (2001) 175}188