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TRANSPORTATION SCIENCE

Vol. 39, No. 1, February 2005, pp. 119–139

issn 0041-1655 eissn 1526-5447 05 3901 0119

informs®

doi 10.1287/trsc.1030.0057

© 2005 INFORMS

Vehicle Routing Problem with Time Windows,

Part II: Metaheuristics

Olli Bräysy

Agora Innoroad Laboratory, University of Jyväskylä, P. O. Box 35, FIN-40014 Jyväskylä, Finland

olli.braysy@jyu.ﬁ

Michel Gendreau

Département d’informatique et de recherche opérationelle, and Centre de recherche sur les transports,

Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal, Canada H3C 3J7, michelg@crt.umontreal.ca

This paper surveys the research on the metaheuristics for the Vehicle Routing Problem with Time Windows

(VRPTW). The VRPTW can be described as the problem of designing least cost routes from one depot to

a set of geographically scattered points. The routes must be designed in such a way that each point is visited

only once by exactly one vehicle within a given time interval; all routes start and end at the depot, and the total

demands of all points on one particular route must not exceed the capacity of the vehicle. Metaheuristics are

general solution procedures that explore the solution space to identify good solutions and often embed some of

the standard route construction and improvement heuristics described in the ﬁrst part of this article. In addition

to describing basic features of each method, experimental results for Solomon’s benchmark test problems are

presented and analyzed.

Key words: vehicle routing; time windows; heuristics; metaheuristics; tabu search; genetic algorithms

History : Received: December 2001; revision received: December 2002; accepted: December 2002.

The motivation and background for researching

the vehicle routing problem with time windows

(VRPTW) as well as the problem description are

discussed in the ﬁrst part of this article (Bräysy and

Gendreau 2005). However, for the sake of complete-

ness, we recall the basic elements of the VRPTW

here. The VRPTW can be described as the problem of

designing least cost routes from one depot to a set of

geographically scattered points. The routes must be

designed in such a way that each point is visited only

once by exactly one vehicle within a given time inter-

val; all routes start and end at the depot, and the total

demands of all points on one particular route must

not exceed the capacity of the vehicle.

The VRPTW has multiple objectives in that the

goal is to minimize not only the number of vehi-

cles required, but also the total travel time or total

travel distance incurred by the ﬂeet of vehicles. A

hierarchical objective function is typically associated

with all procedures studied. That is, the number of

routes is ﬁrst minimized and then, for the same

number of routes, the total traveled distance or time

is minimized. The VRPTW is a basic distribution

management problem that can be used to model

many real-world problems and has been the subject of

intensive research efforts focused mainly on heuristic

and metaheuristic approaches. The heuristic solution

methods and previous survey papers are discussed

in the ﬁrst part of this article. Here, we focus on

metaheuristic approaches. Metaheuristics are general

solution procedures that explore the solution space to

identify good solutions and often embed some of the

standard route construction and improvement heuris-

tics. In a major departure from classical approaches,

metaheuristics allow deteriorating and even infea-

sible intermediate solutions in the course of the

search process. For the most well-known metaheuris-

tic approaches, a description of the basic principles is

given ﬁrst, followed by a description of applications

to the VRPTW. Most of the methods are compared

with other similar approaches based on the experi-

mental results obtained for the Solomon’s (1987) test

problems.

The remainder of this paper is organized as follows.

The tabu search algorithms for the VRPTW are

reviewed in §1. Section 2 focuses on genetic algorithms

and evolution strategies, as well as hybrids based on

them. Other metaheuristic approaches are discussed

in §3, including methods such as simulated annealing,

ant algorithms, guided local search, variable neighbor-

hood search, etc. In §4, we summarize the ﬁndings and

analyze the efﬁciency of the described metaheuristics.

Finally, §5 concludes the paper.

1. Tabu Search Algorithms

Tabu search (TS) is a local search metaheuristic intro-

duced by Glover (1986). Details about tabu search can

also be found in Glover (1989), Glover (1990), Hertz

119

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

120 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

et al. (1997), Glover and Laguna (1997), and Gendreau

(2003). TS explores the solution space by moving at

each iteration from a solution sto the best solution in

a subset of its neighborhood Ns. Contrary to clas-

sical descent methods, the current solution may dete-

riorate from one iteration to the next. New, poorer

solutions are accepted only to avoid paths already

investigated. This insures new regions of a prob-

lem’s solution space will be investigated with the goal

of avoiding local minima and ultimately ﬁnding the

desired solution. To avoid cycling, solutions possess-

ing some attributes of recently explored solutions are

temporarily declared tabu or forbidden. The duration

that an attribute remains tabu is called its tabu tenure,

and it can vary over different intervals of time. The

tabu status can be overridden if certain conditions are

met; this is called the aspiration criterion and it hap-

pens, for example, when a tabu solution is better than

any previously seen solution. Finally, various tech-

niques are often employed to diversify or to inten-

sify the search process. For theoretical aspects of tabu

search, see Faigle and Kern (1992) and Fox (1993).

Garcia et al. (1994) were the ﬁrst to apply tabu

search for VRPTW. The authors presented a parallel

implementation on a network of 16 Meiko T-800

transputers. The tabu search they developed is a

fairly simple one, involving Solomon’s I1 insertion

heuristic to create an initial solution and 2-opt∗

and Or-opt exchanges for improvement (for details,

see Part I of this survey). Many authors since that

time have presented numerous tabu search imple-

mentations involving sophisticated diversiﬁcation

and intensiﬁcation techniques, explicit strategies for

minimizing the number of routes, complex post-

optimization techniques, hybridizations with other

search techniques such as simulated annealing and

genetic algorithms, parallel implementations, and

allowance of infeasible solutions during the search.

The initial solution is typically created with some

cheapest insertion heuristic, described in the ﬁrst part

of this survey article. The most common is Solomon’s

(1987) I1 insertion heuristic. An exception can be

found in Chiang and Russell (1997), where a parallel

version of the insertion heuristic of Russell (1995)

is used. De Backer and Furnon (1997) and Schulze

and Fahle (1999) use the savings heuristic of Clarke

and Wright (1964); Tan et al. (2000) use a modiﬁed

version of Solomon’s insertion heuristic, proposed by

Thangiah (1994), and Cordeau et al. (2001) use a

modiﬁed version of the sweep heuristic developed by

Gillett and Miller (1974). Lau et al. (2003) introduce

the concept of a holding list, a data structure con-

taining the unserviced customers. In the beginning all

customers are in the holding list, and simple relocate

and exchange operators are then used to transfer cus-

tomers back and forth from the holding list.

After creating an initial solution, an attempt is

made to improve it using local search with one or

more neighborhood structures and the best-accept

strategy. Most of the neighborhoods used are well

known and were previously introduced in the context

of various construction and improvement heuristics.

Examples of such neighborhoods are 2-opt, Or-opt, 2-

opt∗, relocate, exchange, and CROSS-, GENI-, and -

exchanges, discussed in detail in the ﬁrst part of this

article.

To reduce the complexity of the search, some

authors propose special strategies for limiting the

neighborhood. For instance, Garcia et al. (1994) only

allow moves involving arcs that are close in distance.

Taillard et al. (1997) decompose solutions into a col-

lection of disjoint subsets of routes by using the polar

angle associated with the center of gravity of each

route. Tabu search is then applied to each subset sep-

arately. A complete solution is reconstructed by merg-

ing the new routes found by tabu search. Another

frequently used strategy to speed up the search is

to implement the proposed algorithm in parallel on

several processors. For instance, Badeau et al. (1997)

apply the solution approach of Taillard et al. (1997)

using a two-level parallel implementation. Results on

benchmark problems show that this parallelization

of the original sequential approach does not degrade

solution quality, for the same amount of computa-

tion, while providing substantial speed-ups. Other

studies describing parallel implementations can be

found in Garcia et al. (1994) and Schulze and Fahle

(1999). On the other hand, to cross the barriers of

the search space, created by time window constraints,

some authors allow infeasibilities during the search.

For instance, Brandão (1999), Cordeau et al. (2001),

and Lau et al. (2003) allow violation of each constraint

type (load, duration, and time windows constraints).

The violations of constraints are penalized in the cost

function, and the parameter values regarding each

type of violation are adjusted dynamically.

Because the number of routes is often considered

as the primary objective, some authors use different

explicit strategies for reducing the number of routes.

For example, the algorithms of Garcia et al. (1994)

and Potvin et al. (1996) try to move customers from

routes with a few customers into other routes using

Or-opt exchanges. Similarly, the method of Schulze

and Fahle (1999) tries to eliminate routes having at

most three customers by trying to move these cus-

tomers into other routes. In Lau et al. (2003) a limit is

set for the number of routes that cannot be exceeded

during the search.

Most of the proposed tabu searches use special-

ized diversiﬁcation and intensiﬁcation strategies to

guide the search. For example, Rochat and Taillard

(1995) propose using a so-called “adaptive memory.”

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 121

The adaptive memory is a pool of routes taken from

the best solutions visited during the search. Its pur-

pose is to provide new starting solutions for the tabu

search through selection and combination of routes

extracted from the memory. The selection of routes

from the memory is done probabilistically and the

probability of selecting a particular route depends on

the value of the solution to which the route belongs.

The selected tours are improved using tabu search

and inserted subsequently back into adaptive mem-

ory. Later, Taillard et al. (1997) used the same strat-

egy to tackle the VRP with soft time windows. In this

problem, lateness at customer locations is allowed,

although a penalty is incurred and added to the objec-

tive value. Taillard et al. (1997) also diversify the

search by penalizing frequently performed exchanges

and intensifying the search by reordering the cus-

tomers within the best routes using Solomon’s I1

insertion heuristic. Chiang and Russell (1997), Schulze

and Fahle (1999), and Cordeau et al. (2001) use a simi-

lar strategy for diversiﬁcation, but in Chiang and Rus-

sell (1997) the intensiﬁcation is used to reduce waiting

time by forbidding certain customers from moving

into another route. Schulze and Fahle (1999) also pro-

pose a strategy similar to adaptive memory, wherein

all routes generated by the tabu search heuristic are

collected in a pool. At the termination of the local

optimization steps, the worst solution is replaced by

a new one created by solving the set-covering prob-

lem on the routes in the pool using the Lagrangian

relaxation-based heuristic of Beasley (1990).

Carlton (1995) and Chiang and Russell (1997) test

a reactive tabu search that dynamically adjusts its

parameter values based on the current search status

to avoid both cycles as well as an overly constrained

search path. More precisely, the size of the tabu list

is managed by increasing the tabu list size if iden-

tical solutions occur too often, and reducing it if no

feasible solution can be found. Tan et al. (2000) diver-

sify the search each time a local minimum is found

by performing a series of random -interchange hops

combined with the 2-opt∗operator. A candidate list is

maintained to record elite solutions discovered dur-

ing the search process. These elite solutions are then

used as a starting point for intensiﬁcation. Lau et al.

(2001) present a generic, constraint-based diversiﬁca-

tion technique, where VRPTW is modeled as a lin-

ear constraint satisfaction problem that is solved by a

simple local search algorithm.

Finally, several authors report using various

post-optimization techniques. For instance, Rochat

and Taillard (1995) solve exactly a set-partitioning

problem at the end, using the routes in the adaptive

memory to return the best possible solution. Taillard

et al. (1997) apply an adaptation of the GENIUS

heuristic (Gendreau et al. 1992) for time windows

to each individual route of the ﬁnal solution. Simi-

larly, in Cordeau et al. (2001) the best solution iden-

tiﬁed after niterations is post-optimized by applying

to each individual route a specialized heuristic for

the traveling salesman problem with time windows

(Gendreau et al. 1998). The main features of the tabu

search heuristics just described are summarized in

Table 1, where we present the initial solution heuris-

tics, neighborhood operators used, as well as mention

whether the proposed approach uses explicit strate-

gies for reducing the number of routes. In the last col-

umn, some notes are given. Further details about tabu

search heuristics for VRPTW can be found in Bräysy

and Gendreau (2005).

The tabu search algorithms described in Table 1

are compared in Table 2, where the ﬁrst column

to the left gives the authors. Columns R1, R2, C1,

C2, RC1, and RC2 present the average number of

vehicles and average total distance with respect to

the six problem groups of Solomon (1987). Finally,

the rightmost column indicates the cumulative num-

ber of vehicles (CNV) and cumulative total distance

(CTD) over all 56 test problems. For more infor-

mation about Solomon’s benchmark problems, we

refer to the ﬁrst part of this article and the orig-

inal paper by Solomon (1987). Due to the lack of

exact information, we cannot consider all algorithms

here. Table 2 shows the best solutions attained with

each method without paying attention to the com-

putational effort. Even though Brandão (1999) uses

rounded distances during the execution of the algo-

rithm, we believe that the differences in ﬁnal solutions

remain small and the results are therefore compara-

ble. To our knowledge, only De Backer and Furnon

(1997) proposed a deterministic method. All other

procedures in Table 2 are stochastic, i.e., in practice

one gets different results with each run. All meth-

ods consider the number of vehicles as the primary

optimization criterion. The only exceptions are the

approaches of De Backer and Furnon (1997) and Tan

et al. (2000) that concentrate solely on minimization

of distance. The second objective is total traveled dis-

tance in Rochat and Taillard (1995), Taillard et al.

(1997), Chiang and Russell (1997), Brandão (1999),

Cordeau et al. (2001), and Lau et al. (2001, 2002). The

other procedures use total duration of routes as the

second objective, causing a slight over-estimation of

the reported total distance values. The CTD values

in Tables 1, 2, and 3 are rounded to integers due to

the usage of rounded distance measures reported by

other authors for calculation.

According to Table 2, the tabu search by Cordeau

et al. (2001) seems to produce the best results in

terms of solution quality. However, the difference

with regard to other well-performing approaches by

Taillard et al. (1997) and Chiang and Russell (1997)

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

122 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

Table 1 The Main Features of Tabu Search Heuristics for VRPTW

Authors Year Initial solution Neighborhood operators Route min. Notes

Garcia et al. 1994 Solomon’s I1 heuristic 2-opt∗, Or-opt Yes Neighborhood restricted to arcs

close in distance

Rochat and Taillard 1995 Modiﬁcation of Solomon’s I1, 2-opt, relocate No Adaptive memory

2-opt

Carlton 1995 Insertion heuristic Relocate No Reactive tabu search

Potvin and Bengio 1996 Solomon’s I1 heuristic 2-opt*, Or-opt Yes Neighborhood restricted to arcs

close in distance

Taillard et al. 1997 Solomon’s I1 heuristic CROSS No Soft time windows, adaptive

memory

Badeau et al. 1997 Solomon’s I1 heuristic CROSS No Soft time windows, adaptive

memory

Chiang et al. 1997 Modiﬁcation of Russell (1995) -interchange No Reactive tabu search

De Backer and Furnon 1997 Savings heuristic Exchange, relocate, 2-opt∗, No Constraint programming used

2-opt, Or-opt to check feasibility of moves

Brandão 1999 Insertion heuristics Relocate, exchange, GENI No Neighborhoods restricted to arcs

close in distance

Schulze and Fahle 1999 Solomon’s I1, parallel I1 Ejection chains, Or-opt Yes Generated routes stored in a pool

and savings heuristics

Tan et al. 2000 Insertion heuristic of -interchange, 2-opt∗No —

Thangiah et al. (1994)

Lau et al. 2001 Insertion heuristic Exchange, relocate No Constraint-based diversiﬁcation

Cordeau et al. 2001 Modiﬁcation of sweep heuristics Relocate, GENI No —

Lau et al. 2003 Relocation from a holding list Exchange, relocate Yes Holding list for unrouted nodes,

limit for number of routes

Table 2 Comparison of Tabu Search Algorithms

Authors R1 R2 C1 C2 RC1 RC2 CNV/CTD

Garcia et al. (1994) 1292 309 1000 300 1288 375 436

1317712226 8771 6023147351527065977

Rochat and Taillard (1995) 1225 291 1000 300 1188 338 415

120850 96172 82838 58986 137739 111959 57231

Potvin and Bengio (1996) 1250 309 1000 300 1263 338 426

1294511544 8502 5946145631404863530

Taillard et al. (1997) 1217 282 1000 300 1150 338 410

120935 98027 82838 58986 138922 111744 57523

Chiang and Russell (1997) 1217 273 1000 300 1188 325 411

120419 98632 82838 59142 139744 122954 58502

De Backer and Furnon (1997) 1417 527 1000 325 1425 625 508

121486 93018 82977 60484 138512 109996 56998

Brandão (1999) 1258 318 1000 300 1213 350 425

1205 995 829 591 1371 1250 58562

Schulze and Fahle (1999) 1225 282 1000 300 1175 338 414

123915 106668 82894 58993 140926 128605 60346

Tan et al. (2000) 1383 382 1000 325 1363 425 467

126637 108024 87087 63485 145816 129338 62008

Lau et al. (2001) 1400 355 1000 300 1363 425 464

121154 96043 83213 61225 138505 123265 58432

Cordeau et al. (2001) 1208 273 1000 300 1150 325 407

121014 96957 82838 58986 138978 113452 57556

Lau et al. (2003) 1217 300 1000 300 1225 338 418

121155 100112 83213 58986 141877 117093 58477

Note. For each algorithm, the average results with respect to Solomon’s benchmarks are depicted. The notations CNV and CTD in the rightmost column indicate

the cumulative number of vehicles and cumulative total distance over all 56 test problems.

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 123

Table 3 The Main Features of Genetic Algorithms and Evolution Strategies for the VRPTW

Authors Year Initial population Crossover Mutation

Blanton and Wainwright 1993 Random ordering Special order–based operators Random exchange of two genes

Thangiah 1995a Random clustering +insertion heuristic Two-point crossover Random change of bit values

Thangiah 1995b Random clustering +insertion heuristic Two-point crossover Random change of bit values

Thangiah et al. 1995 Random clustering +insertion heuristic Two-point crossover Random change of bit values

Potvin and Bengio 1996 Solomon’s insertion Reinsertion of a route or route Combinations of relocate operators to eliminate

segment into another parent routers, and Or-opt

Berger et al. 1998 Nearest neighbor Modiﬁcation of LNS of Shaw (1998), reinsertion Relocate to reduce the number of routes

with modiﬁed Solomon’s heuristic and nearest neighbor for within-route reordering

Homberger and Gehring 1999 Stochastic savings heuristic Uniform order-based to create sequence Or-opt, 2-opt∗,-interchanges, special

for controlling Or-opt Or-opt for route elimination

Gehring and Homberger 1999 Stochastic savings heuristic — Or-opt, 2-opt∗,-interchanges, special

Or-opt for route elimination

Gehring and Homberger 2001 Stochastic savings heuristic — Or-opt, 2-opt∗,-interchanges, special

Or-opt for route elimination

Tan et al. 2001a Solomon’s insertion, PMX Random swap of nodes

-interchange, random

Tan et al. 2001b Random ordering One-point —

Wee Kit et al. 2001 Not deﬁned Relocations based on second parent, modiﬁes seed Tabu search using 2-opt∗, exchange, relocate,

selection and cost function of Solomon’s I1 and 2-opt, applied only later generations

Berger et al. 2003 Random insertion heuristic Modiﬁcation of LNS of Shaw (1998), reinsertion with Modiﬁed LNS, -interchanges, relocate, insertion

modiﬁed Solomon’s heuristic and procedure heuristics of Liu et al. and Solomon

of Liu and Shen (1999)

Le Bouthillier and Crainic 2005 Construction heuristics combined with Order (OX) and edge recombination (ER) 2-opt, Or-opt, 3-opt, taburoute

2-opt, 3-opt, and Or-opt

Mester 2002 Cheapest insertion with varying criteria — Or-opt, 2-opt∗,-interchanges, GENIUS,

modiﬁed LNS

Jung and Moon 2002 Solomon’s insertion Selecting arcs based on 2D image of a Or-opt, 2-opt∗, relocation, splitting of routes

solution and nearest neighbor rule

Homberger and Gehring 2005 Stochastic savings heuristic — Or-opt, 2-opt∗,-interchanges, special

Or-opt for route elimination

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

124 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

is less than 1% in the number of vehicles. Regard-

ing the total traveled distance, the differences between

the three methods are also small. The differences in

CTD remain within 2%. The algorithm by De Backer

and Furnon (1997) seems to give the worst results

with respect to the number of vehicles. The reason

for this can be found in the optimization criterion

used. De Backer and Furnon (1997) and Tan et al.

(2000) consider only the total traveled distance, while

the other procedures minimize the number of vehicles

ﬁrst. However, in spite of this difference in objective

function, the method by De Backer and Furnon (1997)

produces better outcomes than the other approaches

in terms of total distance only for problem groups

R2 and RC2. Overall, the difference in cumulative

number of vehicles is about 14% if De Backer and

Furnon (1997) and Tan et al. (2000) are not consid-

ered. In our opinion, this difference is quite signiﬁ-

cant, and in terms of total distance, the differences

are even greater. For example, the difference between

the approaches of Garcia et al. (1994) and Rochat and

Taillard (1995) in CTD is about 15% and the difference

between Garcia et al. (1994) and Taillard et al. (1997)

in problem group RC2 is about 37%.

As far as the computational effort is concerned, con-

clusions are very difﬁcult to draw, because most of

the authors do not report the CPU time consumption

or the number of runs used to obtain the results in

Table 2. For example, it is impossible to compare the

best approaches by Cordeau et al. (2001), Taillard et al.

(1997), and Chiang and Russell (1997) in terms of CPU

time consumption. Another comparison with other

metaheuristics is presented in Table 6 and Figure 1,

where only results for which computational effort is

reported are considered.

2. Genetic Algorithms

The genetic algorithm (GA) is an adaptive heuris-

tic search method based on population genetics. The

basic concepts were developed by Holland (1975),

while the practicality of using the GA to solve com-

plex problems was demonstrated in De Jong (1975)

and Goldberg (1989). Details and references about

genetic algorithms can also be found in Mühlenbein

(1997) and Alander (2000), respectively. GA evolves

a population of individuals encoded as chromosomes

by creating new generations of offspring through an

iterative process until some convergence criteria are

met. Such criteria might, for instance, refer to a max-

imum number of generations, or the convergence

to a homogeneous population composed of similar

individuals. The best chromosome generated is then

decoded, providing the corresponding solution.

The creation of a new generation of individuals

involves four major steps or phases: representation,

selection, recombination, and mutation. The repre-

sentation of the solution space consists of encoding

signiﬁcant features of a solution as a chromosome,

deﬁning an individual member of a population. The

selection phase consists of randomly choosing two

parent individuals from the population for mating

purposes. The probability of selecting a population

member is generally proportional to its ﬁtness to

emphasize genetic quality while maintaining genetic

diversity. Here, ﬁtness refers to a measure of proﬁt,

utility, or goodness to be maximized while exploring

the solution space. The recombination or reproduc-

tion process makes use of genes of selected parents

to produce offspring that will form the next gen-

eration. As for mutation, it consists of randomly

modifying some gene(s) of a single individual at

a time to further explore the solution space and

ensure, or preserve, genetic diversity. The occur-

rence of mutation is generally associated with a low

probability. A new generation is created by repeating

the selection, reproduction, and mutation processes

until a speciﬁed set of new chromosomes have been

created and placed in the new population. The set

of chromosomes to be created and replaced depends

on the selection strategy and type of GA applied. In

some cases, all chromosomes in the old population

are replaced by new ones, and in some cases a set

of old chromosomes are preserved. A proper balance

between genetic quality and diversity is therefore

required within the population to support efﬁcient

search.

Thangiah et al. (1991) were the ﬁrst to apply a

genetic algorithm to VRPTW (the same method is

described in more detail in Thangiah 1995a and in

the following we refer only to the latter paper). This

ﬁrst paper describes an approach that uses a genetic

algorithm to ﬁnd good clusters of customers, within

a “cluster-ﬁrst, route-second” problem-solving strat-

egy. The routes within each cluster are then con-

structed with cheapest insertion heuristics, and also

-exchanges are applied to improve solution qual-

ity. During the past few years, numerous papers

have been written on generating good solutions for

VRPTW with GAs. Almost all these papers present

hybridizations of a GA with different construction

heuristics (Blanton and Wainwright 1993, Berger et al.

1998), local searches (Thangiah 1995a, b; Thangiah

et al. 1995; Potvin and Bengio 1996; Jung and Moon

2002) and other metaheuristics such as tabu search

(Wee Kit et al. 2001) and ant colony systems (Berger

et al. 2003).

Homberger and Gehring (1999) present two evolu-

tion strategies (Rechenberg 1973, Schwefel 1977) for

the VRPTW. Together with GAs and evolutionary pro-

gramming, the evolution strategies form the class of

evolutionary algorithms (Fogel 1995). By deﬁnition,

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 125

Russell (1995)

Bräysy (2002)

Garcia et al. (1994)

Rochat et al. (1995)

Potvin et al. (1996)

Taillard et al. (1997)

Schulze et al. (1999)

Homberger et

al. (1999)

Gehring et al. (1999)

Kontoravdis et al. (1995)

Liu et al. (1999)

Kilby et al.

(1999)

Gambardella et al.

(1999)

Brandão (1999)

Bräysy (2002)Bräysy (2003)

Berger et al. (2001) Homberger et

al. (2005)

Bräysy et al. (2004b)

Li et al. (2001)

Ibaraki et al. (2002)

405

410

415

420

425

430

435

440

0 50 100 150 200 250 300 350 400 450 500 550 600

Time in minutes

CNV

Figure 1 The Efﬁciency of the Described Methods

Note. The notation CNV refers to the cumulative number of vehicles over 56 test problems of Solomon (1987). Note that the computation times are normalized

to equal Sun Sparc 10, using Dongarra’s (1998) factors. Moreover, if the results are the best ones over multiple runs, the time consumption is multiplied by

this number to illustrate the real computational burden. Local searches of Russell (1995) and Bräysy (2002) are described in Part I of the survey.

the main differences between these three types of

algorithms lie in the representation and in the role

of mutation. For more details, we refer the reader

to Bräysy et al. (2004a). In evolution strategies of

Homberger and Gehring (1999) the individual rep-

resentation includes a vector of so-called “strategy

parameters” in addition to the solution vector and

both components are evolved by means of recombina-

tion and mutation operators. In the proposed applica-

tion to the VRPTW, these strategy parameters refer to

how often a randomly selected local search operator

is applied, and to a binary parameter used to alternate

the search between minimizing the number of vehi-

cles and total distance. Only one offspring is created

through the recombination of parents. In this way, a

number > of offspring is created, where is the

population size. At the end, ﬁtness values are used to

select offspring for the next population.

In Gehring and Homberger (1999) the evolution

strategies of Homberger and Gehring (1999) are

hybridized with tabu search to minimize the total

distance, and the approach is parallelized using

the concept of cooperative autonomy, i.e., several

autonomous sequential solution procedures cooperate

through the exchange of solutions. The authors also

develop a new set of larger benchmark problems that

are based on the benchmark problems of Solomon

(1987). Gehring and Homberger (2001) introduce three

different improvements to the parallel method of

Gehring and Homberger (1999). In the evaluation of

individuals, capacity related information is also used

to determine the routes for elimination. Additional

improvements include greater population size and

new termination criteria. In Homberger and Gehring

(2005), a single processor implementation of Gehring

and Homberger (2001) is presented. Another differ-

ence is that in Homberger and Gehring (2005) capac-

ity information is not used in the evaluation criterion.

Mester (2002) has also experimented with evolution

strategies similar to Homberger and Gehring (1999).

Le Bouthillier and Crainic (2005) present a parallel

cooperative methodology in which several agents

communicate through a pool of feasible solutions. The

agents consist of simple construction and local search

algorithms, GAs and adaptations of the taburoute

method of Gendreau et al. (1994).

Although theoretical results that characterize the

behavior of the GA have been obtained for bit-string

chromosomes, not all problems lend themselves easily

to this representation. This is the case, in particular,

for sequencing problems, such as the vehicle routing

problem, where an integer representation is more

often appropriate. Therefore, in most applications to

VRPTW, the genetic operators are applied directly to

solutions, represented as integer strings, thus avoid-

ing coding issues. In most cases the authors use

delimiters to separate customers served by different

routes. An exception is found in Tan et al. (2001a),

where the basic grouping is determined by the inser-

tion heuristic of Solomon (1987), and -interchanges

are used to create alternative groupings. A similar

study is reported also in Tan et al. (2001c). Jung and

Moon (2002) suggest using the 2D image of a solution

for chromosomal cutting. In Thangiah (1995a, b) and

Thangiah et al. (1995), traditional bit-string encod-

ing is used, and each chromosome represents a set

of possible clustering schemes within a cluster-ﬁrst,

route-second search strategy. Blanton and Wainwright

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126 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

(1993) used the so-called Davis encoding method,

where a chromosome represents a permutation of

ncustomers to be partitioned into mvehicles. The

ﬁrst mcustomers of a chromosome are placed into the

mdifferent vehicles. The remaining n−mcustomers

are examined individually, using a greedy insertion

heuristic. In the messy genetic algorithm (Goldberg

et al. 1989) of Tan et al. (2001b) the solution is encoded

using ordered pairs, consisting of customer and vehi-

cle identiﬁcation indexes.

The initial population is typically created either

randomly or using modiﬁcations of well-known con-

struction heuristics. A random strategy can be found

in Blanton et al. (1993), Tan et al. (2001b), and

Le Bouthillier and Crainic (2005), though the last

one applies also a set of construction heuristics com-

bined with 2-opt, 3-opt, and Or-opt improvement

heuristics. Thangiah (1995a, b) and Thangiah et al.

(1995) cluster the customers randomly into separate

groups and then use the cheapest insertion heuristic

of Golden and Stewart (1985) to route customers

within each group. Homberger and Gehring (1999,

2005) and Gehring and Homberger (1999, 2001) use

a modiﬁcation of the savings heuristic of Clarke and

Wright (1964), where the savings element is selected

randomly from the savings list. Berger et al. (2003)

modify a randomly generated initial population with

-exchanges and a reinitialization procedure based on

the insertion procedure of Liu and Shen (1999), to

create a population of solutions with the number of

vehicles equal to the lowest found. In Mester (2002),

all customers are ﬁrst served by separate routes.

Then, a set of six initial solutions is created using

cheapest reinsertions of single customers with vary-

ing insertion criteria, and the best solution obtained is

selected as starting point. Solomon’s insertion heuris-

tic is used in Potvin and Bengio (1996), Tan et al.

(2001a), and Jung and Moon (2002). The last two

authors also create a set of solutions randomly and by

modifying the heuristic solution with -interchanges.

Berger et al. (1998) use Solomon’s (1987) nearest

neighbor heuristic. To the best of our knowledge, only

Berger et al. (1998) and Berger et al. (2003) use more

than one population. For instance, the algorithm pro-

posed in Berger et al. (2003) evolves two populations

in parallel. The ﬁrst population is used to minimize

the total distance and the second population tries to

minimize violations of the time window constraints.

Fitness values are usually based on routing costs,

i.e., number of routes, total distance, and duration.

In addition, Le Bouthillier and Crainic (2005) con-

sider waiting and residual time at each customer. In

Blanton and Wainwright (1993) the ﬁtness value is

the number of unserviced customers in case of infea-

sible solutions. Homberger and Gehring (1999, 2005)

and Gehring and Homberger (1999, 2001) also con-

sider how easily the shortest route of the solution (in

terms of the number of customers on the route) can be

eliminated, in addition to the number of routes and

the total distance. In Berger et al. (2003), the evalua-

tion of the individuals is based on a weighted sum of

objectives related to violated constraints, number of

vehicles, and total distance.

The most typical selection scheme for selecting a

pair of individuals (parents) for recombination is the

well-known roulette-wheel scheme. In this stochastic

scheme, the probability of selecting an individual

is proportional to its ﬁtness value. For details, see

Goldberg (1989). Tan et al. (2001a) and Jung and

Moon (2002) use so-called tournament selection. The

basic idea is to perform the roulette-wheel scheme

twice and to select the better out of the two indi-

viduals identiﬁed by the roulette-wheel scheme. In

Wee Kit et al. (2001), Homberger and Gehring (1999,

2005), Gehring and Homberger (1999, 2001), and

Mester (2002), the parents are selected randomly.

Finally, Potvin and Bengio (1996) and Le Bouthillier

and Crainic (2005) use a ranking scheme, where the

probability of selecting an individual is based on its

rank.

The recombination is the most crucial part of a

genetic algorithm. The traditional two-point cross-

over, which exchanges a randomly selected portion

of the bit string between the chromosomes, is used

in Thangiah (1995a, b) and Thangiah et al. (1995),

while Tan et al. (2001a) and Tan et al. (2001b)

use the well-known PMX and one-point crossovers,

respectively. The basic idea in PMX crossover is to

choose two cut points at random and, based on these

cut points, to perform a series of swapping opera-

tions in the second parent. The one-point crossover

switches two sets of customers to be serviced by

two different routes. Traditional order–based opera-

tors, based on a precedence relationship among the

genes in a chromosome, are used in Blanton and

Wainwright (1993), Homberger and Gehring (1999),

and Le Bouthillier and Crainic (2005). The last authors

also use well-known edge recombination crossover,

and in Homberger and Gehring (1999), crossover is

used to modify the initially randomly created muta-

tion codes. The mutation code is used to control a set

of removal and insertion operators performed by the

Or-opt operator.

In the context of VRPTW, many authors have

proposed specialized heuristic crossover procedures,

instead of traditional operators. Potvin and Bengio

(1996) propose a sequence-based and a route-based

crossover. The sequence-based crossover ﬁrst selects

a link randomly from each parent solution. Then,

the customers that are serviced before the break-

point on the route of parent-solution, P1, are linked

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Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 127

to the customers that are serviced after the break-

point on the route of parent solution, P2. Finally, the

new route replaces the old one in parent solution,

P1. The route-based crossover replaces one route of

parent solution, P2, by a route of parent solution,

P1. In Berger et al. (1998) and Berger et al. (2003), a

removal procedure is ﬁrst carried out to remove some

key customer nodes in a similar fashion to the large

neighborhood search (LNS) of Shaw (1998). Then,

an insertion procedure (inspired from Solomon 1987,

Liu and Shen 1999, respectively) is locally applied

to reconstruct the partial solution. The ﬁrst opera-

tor of Wee Kit et al. (2001) tries to modify the order

of the customers in the ﬁrst parent by trying to cre-

ate consecutive pairs of customers according to the

second parent. The second crossover operator tries

to copy common characteristics of parent solutions

to offspring by modifying the seed selection proce-

dure and cost function of an insertion heuristic simi-

lar to Solomon’s (1987). In Jung and Moon (2002), the

recombination is based on selecting a set of inherited

arcs using different types of curves drawn on the 2D

space where customers are located, and including the

missing arcs in nearest neighbor manner.

The mutation is often considered as secondary

strategy, and its purpose in traditional genetic algo-

rithms is mainly to help escape from local minima.

However, in the evolution strategies of Homberger

and Gehring (1999, 2005), Gehring and Homberger

(1999, 2001), and Mester (2002), the search is mainly

driven by mutation, based on traditional local search

operators (2-opt∗, Or-opt, and -interchanges). Mester

(2002) also uses the GENIUS heuristic of Gendreau

et al. (1992) and so called multiparametric mutation

that consists of removing a set of customers from a

solution randomly, based on the distance to the depot

or by selecting one customer from each route. Then,

a cheapest insertion heuristic is used to reschedule

the removed customers. In Potvin and Bengio (1996),

Wee Kit et al. (2001), Le Bouthillier and Crainic (2005),

and Jung and Moon (2002), the mutation is entirely

or partially based on well-known local search oper-

ators (Or-opt, crossover, and relocation). In Potvin

and Bengio (1996), Berger et al. (1998), Homberger

and Gehring (1999, 2005), Gehring and Homberger

(1999, 2001), and Berger et al. (2003), mutation is also

used to reduce the number of routes by using Or-opt,

Solomon’s insertion heuristic, or by performing one

or several subsequent relocate moves. Berger et al.

(1998) use mutation to locally reorder routes with the

nearest neighbor heuristic of Solomon (1987).

Berger et al. (2003) present ﬁve mutation opera-

tors including the LNS of Shaw (1998), -exchanges,

exchange of customers served too late in the current

solution, elimination of the shortest route using the

procedure by Liu and Shen (1999), and within-route

reordering using Solomon’s (1987) heuristic.

In some recent papers, different intensiﬁcation

techniques are coupled to a GA. For instance,

Tan et al. (2001a) introduce a special hill-climbing

technique, where a randomly selected part of the

population is improved by partial -exchanges. In

Wee Kit et al. (2001), a simple tabu search based

on 2-opt∗, exchange, relocate, and 2-opt neighbor-

hoods is applied to individual solutions in the later

generations to intensify the search. Like the tabu

searches discussed in §1, many genetic algorithms

allow infeasibilities during the search to escape from

local minima. Examples of such strategies can be

found in Blanton and Wainwright (1993), Thangiah

(1995a, b), Thangiah et al. (1995), Berger et al. (2003),

and Le Bouthillier and Crainic (2005). Parallel imple-

mentations can be found in Gehring and Homberger

(1999, 2001), Le Bouthillier and Crainic (2005), and

Homberger and Gehring (2005). Mester (2002) pro-

poses a set of strategies for dividing a problem into

parts to speed up the search. The main features of the

various genetic algorithms and evolution strategies

described above are summarized in Table 3, where we

report the authors, strategies used to create the initial

population, as well as crossover and mutation oper-

ators used. Further details about genetic algorithms

and evolution strategies for the VRPTW can be found

in Bräysy et al. (2004a).

The genetic algorithms and evolution strategies

described above are compared in Table 4. First,

the best results averaged over each problem set of

Solomon (1987) are reported. The latter part of the

table describes the computer used, number of inde-

pendent runs, and average time consumption in min-

utes as reported by the authors. All algorithms in

Table 4 are stochastic and are implemented in C,

except Wee Kit et al. (2001) and Mester (2002) that are

coded in Java and Visual Basic, respectively. A hierar-

chical objective function is used in every case, except

in Tan et al. (2001a, b) and Jung and Moon (2002),

where the only objective is to minimize total distance.

The number of routes is considered as the primary

objective and, for the same number of routes, the sec-

ondary objective is to minimize the total traveled dis-

tance. An exception is found in Potvin and Bengio

(1996), where the second objective is to minimize

the total duration of routes. This may cause some

overestimation of traveled distance that should be

taken into account when comparing the total distance

values.

According to Table 4, the methods by Homberger

and Gehring (2005), Berger et al. (2003), and Mester

(2002) seem to produce the best results. The differ-

ences between these best methods in terms of solution

quality are small. When it comes to other approaches,

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

128 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

Table 4 Comparison of Evolutionary and Genetic Algorithms

Authors R1 R2 C1 C2 RC1 RC2 CNV/CTD

(1) Thangiah (1995a) 1275 318 1000 300 1250 338 429

130025 112428 89211 74913 147413 141113 65074

(2) Potvin and Bengio (1996) 1258 300 1000 300 1213 338 422

129683 111764 83811 59000 144625 136813 62634

(3) Berger et al. (1998) 1258 309 1000 300 1213 350 424

126158 103001 83461 59425 144135 128425 60539

(4) Homberger and Gehring (1999) 1192 273 1000 300 1163 325 406

122806 96995 82838 58986 139257 114443 57876

(5) Gehring and Homberger (1999) 1242 282 1000 300 1188 325 415

1198 947 829 590 1356 1140 56942

(6) Gehring and Homberger (2001) 1200 273 1000 300 1150 325 406

121757 96129 82863 59033 139513 113937 57641

(7) Berger et al. (2003) 1192 273 1000 300 1150 325 405

122110 97543 82848 58993 138989 115937 57952

(8) Tan et al. (2001a) 1317 500 1011 325 1350 500 478

1227 980 861 619 1427 1123 58605

(9) Tan et al. (2001b) 1291 500 1000 300 1260 580 471

12050 9296 84196 6112139231080156931

(10) Wee Kit et al. (2001) 1258 318 1000 300 1275 375 432

120332 95117 83332 59300 138206 113279 57265

(11) Mester (2002) 1200 273 1000 300 1150 325 406

1208 954 829 590 1387 1119 57219

(12) Jung and Moon (2002) 1325 536 1000 300 1300 625 486

117995 87841 82838 58986 134364 100421 54779

(13) Le Bouthillier and Crainic (2005) 1217 282 1000 300 1150 325 409

120927 96591 82838 58986 138922 114370 57574

(14) Homberger and Gehring (2005) 1192 273 1000 300 1150 325 405

121273 95503 82838 58986 138644 112317 57309

Note. For each algorithm, the average results with respect to Solomon’s benchmarks are reported. Notations CNV and CTD in the rightmost column indicate

the cumulative number of vehicles and cumulative total distance over all 56 test problems.

(1) Solbourne 5/802, –, 2.1 min.; (2) Sun Sparc 10, –, 25 min.; (3) Sun Sparc 10, –, 1–10 min.; (4) Pentium 200 MHz, 10 runs, 13 min.; (5) 4×Pentium

200 MHz, 1 run, 10 min.; (6) 4 ×Pentium 400 MHz, 5 runs, 13.5 min.; (7) Pentium 400 MHz, –, 30 min.; (8) Pentium II 330 MHz, –, 25 min.; (9) Pentium II

330 MHz, –, 25 min.; (10) Digital Personal Workstation 433a, –, 147.4 min.; (11) Pentium III 450 MHz, –, 150.2 min.; (12) Pentium III 1 GHz, 100 runs,

0.8 min.; (13) 5 ×Pentium 500 MHz, –, 60 min.; (14) Pentium 400 MHz, –, –.

the worst results regarding the CNV are produced by

Tan et al. (2000, 2001a) and Jung and Moon (2002) that

focus only on minimizing the total distance. Jung and

Moon (2002) seems to be clearly the best of the three

methods, producing results that are competitive even

with the best known in terms of distance. Problem

group RC2 seems to be the most problematic regard-

ing the total traveled distance. The difference between

Thangiah (1995a) and Gehring and Homberger (2001)

is about 25%, which can hardly be justiﬁed in practi-

cal settings.

Because only Homberger and Gehring (1999),

Gehring and Homberger (1999, 2001), and Jung and

Moon (2002) report the number of runs required

to obtain the results in Table 4, it is impossible to

draw any ﬁnal conclusions regarding which method

performs best. Considering only these best reported

results, methods proposed in Homberger and Gehring

(1999, 2005) and Gehring and Homberger (1999, 2001)

can be considered to be Pareto optimal in terms of

solution quality and time consumption. Another com-

parison with other metaheuristics can be found in

Table 6, where only results for which computational

effort is reported are considered.

3. Miscellaneous Metaheuristics

In addition to tabu search and genetic algorithms,

a variety of other metaheuristics have been applied

to the VRPTW. We now rapidly describe their most

important features.

Kontoravdis and Bard (1995) propose a two-

phase greedy randomized adaptive search procedure

(GRASP) for the VRPTW. The construction proce-

dure ﬁrst initializes a number of routes by select-

ing seed customers that are either geographically

most dispersed or the most time constrained. After

initialization, the algorithm ﬁnds the best feasible

insertion location in each route for every unrouted

customer and calculates a speciﬁc penalty value using

Solomon’s (1987) cost function. This penalty is the

sum of differences between the least insertion cost

for each route and the overall best cost. A list Lof

unassigned customers with the largest penalty value

is created and the next customer to be routed is ran-

domly selected from this list. Then local search is

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 129

applied to the best feasible solution found after every

ﬁve iterations. In the local search phase, each route is

considered for elimination, with routes having fewer

customers examined ﬁrst. If it is not possible to insert

a customer into another route, some customer is ﬁrst

removed from the target route rand inserted into

another route r, before inserting the current cus-

tomer into the target route r. Finally, a 2-opt exchange

procedure is applied after successful eliminations to

improve the solution in terms of distance. To estimate

the number of routes, the authors present three lower

bounds. The ﬁrst stems from the underlying bin pack-

ing structure created by the capacity constraints. The

second is derived from the maximum clique of the

associated incompatibility graph. An arc in this graph

corresponds to a pair of customers who cannot be on

the same route due to capacity or time window viola-

tions. The third also considers the bin packing prob-

lem generated by the time window constraints. Here,

bin capacity is the length of the scheduling horizon,

while the items are of a size equal to either the time

needed to go from a customer to its closest neighbor

or the depot, or the time from the depot to the ﬁrst

customer on each route.

Potvin and Robillard (1995) combine the parallel

cheapest insertion heuristic of Potvin and Rousseau

(1993) with a competitive neural network. The neural

network procedure is used to select the seed cus-

tomers for the insertion heuristic. The theory of neural

networks is beyond the scope of this paper, for

details see, for example, Hopﬁeld and Tank (1985)

and Kohonen (1988). A weight vector is deﬁned for

every vehicle. Initially, all weight vectors are placed

close to the depot in random fashion. Then one cus-

tomer at a time is selected and the distance to all

weight vectors is calculated. The closest weight vec-

tor is updated by moving it closer to the customer.

This process is repeated for all customers a number of

times; each time the process is restarted, the update of

the weight vector becomes less sensitive. At the end

of this phase, the seed customers are selected as the

customers closest to the weight vectors.

Potvin et al. (1996) use the same approach as

Potvin and Robillard (1995) to select the seed cus-

tomers for the parallel insertion heuristic of Potvin

and Rousseau (1993). The algorithm requires a value

for three parameters, 1,2, and . The ﬁrst two

constants determine the importance of distance and

travel time in the cost function for each unrouted cus-

tomer. The third factor is used to control the savings

in distance. A genetic algorithm is used to ﬁnd val-

ues for these three constants. A stochastic selection

procedure is applied to the ﬁtness values based on

the number of routes and total route time of the best

solution produced by the parallel insertion heuris-

tic. A classical two-point crossover operator is used

for recombination. It swaps a segment of consecutive

bits between the parents. The mutation changes with

very low probability a bit value from zero to one or

from one to zero. The results are slightly better com-

pared to using the original insertion heuristic without

preprocessing.

Benyahia and Potvin (1995) use a similar GA

approach to optimize the parameter values of the

sequential and parallel versions of Solomon’s (1987)

insertion heuristic. However, here seed customers

are selected as in original papers instead of neural

networks. Moreover, authors introduce additional

cost measures, involving slack and waiting times, sav-

ings of insertion compared to servicing the customer

by individual route, and ratio of additional distance

to original distance between the pair of consecutive

customers.

Bachem et al. (1996) describe an improvement heu-

ristic based on the mechanisms of trading. The par-

tition of customers into the tours is determined by

ﬁnding matches in a leveled bipartite graph that the

authors call a “trading graph.” The nodes correspond

to either an insertion (buy) of a customer into a

tour or a deletion (sell). The edges represent possible

exchanges and the weight of each edge is the gain

that is obtained by the corresponding action. Thus,

every matching of the trading graph corresponds to

a number of interchanges of customers. In each iter-

ation, tours are shufﬂed by choosing some permu-

tation at random. Then, for each tour either a sell

or buy action is selected and ﬁnally possible trading

matches are evaluated and the best one selected. The

approach allows infeasibilities against certain penalty

factors, as well as trading matchings with negative

weights causing deterioration. Because of this deteri-

oration, a tabu list is also added to prevent cycling.

The approach was implemented using two different

kinds of parallelizations. In the ﬁrst approach, each

tour was mapped into one processor that makes a sell

or buy decision. To reduce the idle time of the proces-

sors, the second approach partitions the current tour

plan such that each processor gets about the same

number of different tours.

Chiang and Russell (1996) develop a simulated

annealing approach for VRPTW. Simulated annealing

(SA) is a stochastic relaxation technique, which has

its origin in statistical mechanics. It is based on an

analogy from the annealing process of solids, where

a solid is heated to a high temperature and grad-

ually cooled for it to crystallize in a low energy

conﬁguration.

Simulated annealing guides the original local

search method in the following way. The solution Sis

accepted as the new current solution if ≤0, where

=CS−CS. To allow the search to escape a local

optimum, moves that increase the objective function

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

130 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

value are accepted with a probability e−/T if >0,

where Tis a parameter called the “temperature.” The

value of Tvaries from a relatively large value to a

small value close to zero. These values are controlled

by a cooling schedule, which speciﬁes the initial, and

temperature values at each stage of the algorithm. For

details, see Metropolis et al. (1953), Kirkpatrick et al.

(1983), and Aarts et al. (1997).

The authors combine the simulated annealing

process with the parallel construction approach of

Russell (1995) that incorporates improvement proce-

dures during the construction process. Two different

neighborhood structures, the -interchange mecha-

nism =1and the k-node interchange process

of Christoﬁdes and Beasley (1984) are implemented

using the ﬁrst-accept strategy. The enhancement of the

annealing process with a short-term memory function

via a dynamically varying tabu list is examined as a

basis for improving the metaheuristic approach.

Thangiah et al. (1994) develop a number of meta-

heuristics based on a two-phase approach. In the

ﬁrst phase, an initial solution is created by either

the cheapest insertion heuristic or the sectoring-based

genetic algorithm GIDEON by Thangiah (1995a). The

second phase applies one of the following search

procedures that use the -interchange mechanism: A

local search descent procedure using either ﬁrst-best

or a global-best strategy, a hybrid simulated anneal-

ing (SA) algorithm with nonmonotonic cooling sched-

ule, or a hybrid simulated annealing and tabu search.

Tabu search algorithm is combined with the SA-based

acceptance criterion to decide which moves to accept

from the candidate list. The main feature of the local

search procedures is that infeasible solutions with

penalties are allowed if considered attractive.

Tan et al. (2000) develop a fast simulated annealing

method based on two-interchanges with best-accept

strategy and a monotonously decreasing cooling

scheme. After the ﬁnal temperature is reached, special

temperature resets based on the initial temperature

and the temperature that produced the current best

solution are used to restart the procedure. The initial

solution is created using a modiﬁcation of the push-

forward insertion heuristic proposed by Thangiah

et al. (1994).

Li et al. (2003) propose a tabu-embedded simu-

lated annealing restart metaheuristic. Initial solutions

are created by the insertion and extended sweep

heuristics of Solomon (1987). Three neighborhood

operators based on shifting and exchanging customer

segments between and within routes are combined

with a simulated annealing procedure that is forced

to restart from the current best solution several times.

Solomon’s insertion procedure is used to reduce the

number of routes and to intensify the search by

reordering routes and trying to insert customers into

other routes. Finally, the search is diversiﬁed by per-

forming some random shifts and exchanges of cus-

tomer segments.

Tan et al. (2001c) hybridize a basic simulated

annealing with the tabu search of Tan et al. (2000).

The initial solution is created with Solomon’s I1 inser-

tion heuristic, and the neighborhood is searched with

-exchanges using the ﬁrst-accept strategy. A linear

cooling schedule is used, and the search is diversiﬁed

by randomly shifting and interchanging customers

between randomly selected routes.

Bent and Van Hentenryck (2004) present a two-

stage hybrid metaheuristic, where in the ﬁrst stage is a

basic simulated annealing used to minimize the num-

ber of routes, and the second stage focuses on distance

minimization using the large neighborhood search

(Shaw 1998). The simulated annealing randomly uses

the traditional move operators: 2-opt, Or-opt, reloca-

tion, exchange, and 2-opt∗(described in the ﬁrst part

of this survey), and a special evaluation criteria for

minimizing the number of routes. In addition to route

size and minimal delay introduced in Homberger and

Gehring (1999), the sum of squares of route sizes is

used to favor inserting customers from short to larger

routes. The large neighborhood search implementa-

tion differs from Shaw (1998), described in the ﬁrst

part of this survey, in including a restarting strategy

and a more precise lower bound. The initial solution

procedure is not described in the paper.

Czech and Czarnas (2002) describe a parallel sim-

ulated annealing to ﬁnd the best possible solutions

to a set of Solomon’s benchmark instances. The best

solutions reported in earlier studies are taken as ini-

tial solutions, and the neigborhood search is based on

random relocations of single customers with the best-

accept strategy. The temperature values are reduced

geometrically, and the procedure memorizes the best

solutions found during the entire search. Each of the

parallel processes carry out the annealing searches

using the same initial solution and cooling schedule,

and the processes cooperate at certain intervals by

passing their best solutions.

Kilby et al. (1999) introduced guided local search

(GLS) for VRPTW. GLS is a memory-based tech-

nique developed by Voudouris (1997) and Voudouris

and Tsang (1998). It operates by augmenting the cost

function with a penalty term based on how close

the search moves to previously visited local minima,

thus encouraging diversiﬁcation. GLS moves out of

local minima by penalizing particular solution fea-

tures (usually one) it considers should not occur in

a near-optimal solution weighted by the number of

times the feature has already been penalized. The

more often a feature appears and is penalized, the less

likely it is to be penalized further. The authors choose

arcs as the feature to penalize. In the initial solution, no

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 131

visits are allocated to any vehicle. A penalty is associ-

ated with not performing a visit, and so the search pro-

cess constructs a solution in the process of minimiz-

ing cost using four different local searches. The local

search operators used are 2-opt, relocate, exchange,

and 2-opt∗with best-accept strategy (for details, see

Part I of this article). The proposed approach is a deter-

ministic and entirely greedy one.

De Backer et al. (2000) test iterative improvement

techniques within a constraint programming (CP)

framework (for details, see Part I of this survey).

The improvement techniques are coupled to tabu

search and GLS to avoid the search being trapped in

local minima. The CP system is used only as back-

ground operator to check the validity of solutions

and to speed up legality checks of improvement pro-

cedures. Four simple local search operators, namely

2-opt, relocate, exchange, and 2-opt∗, are used with

the best-accept strategy to improve the solutions. The

tabu search implementation is a simple one, involving

only two tabu lists for storing recently removed and

inserted arcs. The GLS implementation is similar to

that of Kilby et al. (1999) described above. The initial

solution is produced by the Savings method of Clarke

and Wright (1964), followed by descent to a local min-

imum. GLS is found to be superior to the simple tabu

search and a combined method, guided tabu search,

is concluded to perform slightly better than the sim-

ple tabu search and GLS on the long-haul benchmark

problems (classes C2, R2, and RC2 of Solomon).

Liu and Shen (1999) propose a two-stage meta-

heuristic based on a new neighborhood structure

focusing on the relationship between routes and

nodes. In the construction phase, routes are con-

structed in a nested parallel manner by repeatedly

estimating the lower bound for the unrouted cus-

tomers. The seed customers are selected according to

differences in time windows or so that they are geo-

graphically as dispersed as possible with regard to

a previously chosen seed pair with the largest num-

ber of customers between them. The second step is to

use these partial routes to service unrouted customers

until no feasible insertion locations can be found. If

an unrouted customer cannot be inserted into any

route, ﬁve simple operations are used. These oper-

ations include insertion of one or two customers in

other routes in ejection chain manner (for details, see

Part I of this survey), and reordering the routes to

make it possible to insert a new customer. In addition,

a different set of unrouted customers is generated by

exchanging routed and unrouted customers. During

the route construction phase, a number of routes with

low capacity utilization rates are eliminated. Once a

feasible solution is constructed, -exchanges and sim-

ple reinsertions within the routes are used to improve

solution quality. Finally, intraroute reinsertions that

worsen the objective value are accepted to escape

from local minima.

Gambardella et al. (1999) use an ant colony opti-

mization (ACO) approach (Dorigo et al. 1999) with

a hierarchy of two cooperative artiﬁcial ant colonies.

The ﬁrst colony is used to minimize the number

of vehicles, while the second colony minimizes the

total traveled distance. The two colonies cooperate

through updating the best solution found, and in

case the new best solution contains fewer vehicles,

both colonies are reactivated with the reduced num-

ber of vehicles. The ACO is inspired by an anal-

ogy with real ant colonies foraging for food. In their

search for food, ants mark the paths they travel by

laying an aromatic essence called pheromone. The

quantity of pheromone laid on the path depends

on the length of the path and the quality of the

food source. This pheromone provides information to

other ants that are attracted to it. With time, paths

leading to the more interesting food sources, i.e.,

close to the nest and with large quantities of food,

become more frequented and are marked with larger

amounts of pheromone. In the described VRPTW

solution method, two measures are associated with

each arc, the attractiveness Nij and the pheromone

trail Tij . The tours are constructed using the nearest-

neighbor heuristic with probabilistic rules, i.e., the

next customer node to be inserted at the end of the

current tour is not always the best according to Nij

and Tij . During the search the pheromone trails are

updated both locally and globally. The effect of local

updating is to dynamically change the desirability of

arcs: Every time an ant uses an arc, the quantity of

pheromone associated with this arc is decreased and

the arc becomes less attractive. On the other hand,

global updating is used to intensify the search in the

neighborhood of the best solution computed. Each

artiﬁcial ant constructs a separate feasible solution,

and the attractiveness Nij is computed by taking into

account the distance between customer nodes, the

time window of the considered customer node, and

the number of times the considered customer node

has not been inserted into the solution. In addition,

the CROSS-exchanges of Taillard et al. (1997) are used

to improve the quality of the feasible solutions.

Caseau et al. (1999b) hybridize several techniques,

such as limited discrepancy search (LDS), LNS, ejec-

tion chains (for details, see Part I of this survey),

and ejection trees. The initial solution is ﬁrst con-

structed using the heuristic of Caseau and Laburthe

(1999a). Ejection chains and ejection trees are then

used to relocate most costful customers into other

routes. The basic difference of ejection trees compared

to ejection chains is to remove more than one cus-

tomer from a route while inserting only one customer.

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

132 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

The LNS procedure used is the same as that pro-

posed by Shaw (1998). The authors develop an alge-

bra for these operators and test various combinations

of different operators using several parameter val-

ues. A learning technique that automatically tunes an

existing combination or discovers a new one is pro-

posed. It is shown that automatic tuning yields a bet-

ter solutions than hand-tuning with considerably less

effort.

Rousseau et al. (2002) examine a variable neighbor-

hood descent scheme introduced by Mladenovic and

Hansen (1997) and new large neighborhood operators

within a CP framework. The ﬁrst operator introduced

is inspired by ideas from LNS of Shaw (1998). The

algorithm removes ﬁrst, randomly, a subset of cus-

tomers with a bias toward customers generating the

longest detour. A CP version of the GENI algorithm

for the traveling salesman problem with time win-

dows (TSPTW) by Gendreau et al. (1998) is used

to reinsert removed customers. The second opera-

tor introduced is called naive ejection chains (Glover

1991, 1992), and it is used to create the initial solution

and diversify the search. To limit search space and

prevent cycling, the ﬁrst completed ejection chain is

always accepted and each customer is allowed to

be moved only once. The third proposed operator,

SMART, removes a set of arcs instead of customers

from the solution, creating an incomplete solution.

The removed arcs can be either consecutive or ran-

domly selected with a bias toward the longer arcs.

This smaller routing problem is then solved either

exactly by using the modiﬁed TSPTW model devel-

oped by Pesant et al. (1998) or, in the case of

a larger neighborhood size, by using LDS with a

bounded number of discrepancies. The search oscil-

lates between the two suggested operators (ejection

chains are not considered here) to escape local min-

ima. In the end, routes are either exactly reordered

using the algorithm of Pesant et al. (1998) or, in case

of longer routes, using the postoptimization part of

the algorithm proposed by Pesant et al. (1997).

Bräysy (2003) presents a new four-phase determin-

istic metaheuristic algorithm based on a modiﬁca-

tion of the variable neighborhood search (VNS) of

Mladenovic and Hansen (1997). In the ﬁrst phase,

an initial solution is created using a construction

heuristic that borrows its basic ideas from the studies

of Solomon (1987) and Russell (1995). Routes are built

one at a time in sequential fashion and after kcus-

tomers have been inserted into the route, the route

is reordered using Or-opt exchanges. Then a special

route-elimination operator based on a new type of

ejection chains (for details, see Part I of this survey)

is used to minimize the number of routes. In the

third phase, the created solutions are improved in

terms of distance using VNS oscillating between

four new improvement procedures. These procedures

are based on modiﬁcations to CROSS-exchanges of

Taillard et al. (1997) and cheapest insertion heuristics.

In the fourth phase, the objective function used by

the local search operators is modiﬁed to also consider

waiting time to escape from local minima.

Bräysy et al. (2004b) continue the study of Bräysy

(2003) by introducing modiﬁcations to the construc-

tion and improvement heuristics, and by applying

a new postoptimization technique based on thresh-

old accepting (Dueck and Scheurer 1990) that can be

considered as deterministic modiﬁcation of the sim-

ulated annealing. To be more precise, the reordering

procedure is removed from the construction heuris-

tic, and an extension of ejection chains that allows

for infeasible solutions and removal of several cus-

tomers from each route is used in the second phase.

In the distance optimization phase, only modiﬁcations

of CROSS-exchanges are used. The modiﬁcations also

include considering inverting the order of the cus-

tomers in the selected segments, and more insertion

positions for segments. The postoptimization is based

on a new interroute exchange heuristic that combines

the ideas of CROSS- and GENIUS-exchanges.

Anderson et al. (2000) propose an interactive

scheme for VRPTW. The basic idea is to use a sim-

ple hill-climbing algorithm based on the relocation

of ncustomers to ﬁnd a local minimum. Then using

visualization and interaction techniques, the human

user identiﬁes promising regions of the search space

for the computer to explore, thus helping the search

to escape from local minima. For example, the eval-

uation function for the moves may be edited by the

user during the search and the user can assign pri-

orities to speciﬁc customers to force some moves and

choose between ﬁrst-accept and best-accept strategies.

Finally, a branch-and-bound algorithm is used to opti-

mize each tour separately.

Ibaraki et al. (2002) introduce three methods for

the vehicle routing problem with general time win-

dows. The time window constraint for each customer

is treated as a penalty function that can be noncon-

vex and discontinuous as long as it is piecewise lin-

ear. After ﬁxing the order of customers for a vehicle

to visit, a dynamic programming method is used

to determine the optimal start times to serve the

customers so that the total penalty is minimized.

The authors propose three metaheuristics to improve

randomly generated initial solutions. Multistart local

search (MLS) independently creates and improves a

number of initial solutions and in the end returns

the best solution obtained during the entire search.

Iterated local search (ILS) is a variant of MLS, in

which the initial solutions for local search proce-

dure are generated by perturbing good solutions

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 133

obtained during the search using random CROSS-

exchanges. Adaptive multistart local search (AMLS)

keeps a set, P, of good solutions found in the pre-

vious search and generates initial solutions for local

search by combining parts of the solutions in P. The

local search procedure uses four different neighbor-

hoods in a variable neighborhood search manner,

namely CROSS-exchange, 2-opt∗, cyclic exchange, and

intraroute exchange based on Or-opt. The original

cyclic transfer neighborhood and Or-opt exchanges

described in the ﬁrst part of this survey are modiﬁed

so that they also consider reversing the order of the

customers in the reinserted paths. The authors pro-

pose a number of strategies to search the neighbor-

hoods efﬁciently, including ideas such as evaluating

solutions in a speciﬁed order, time-oriented neighbor-

list, using memory to prevent searching unpromising

regions of solution space, and partial updating of the

improvement graph for cyclic transfers.

The metaheuristic algorithms described above are

compared in Table 5. We believe that Kontoravdis and

Bard (1995) used rounded distance values after one

decimal place. Thus, their results are slightly better,

for example, for the problem group C1 when com-

pared to other approaches using real distance val-

ues. Only Liu and Shen (1999), Kilby et al. (1999),

and Bräysy (2003) propose deterministic methods. All

other algorithms include some randomness in the

search process. Most of the algorithms in Table 5

are implemented in C. However, Chiang and Russell

(1996) and Liu and Shen (1999) use Fortran, Bräysy

(2003) uses Java, and Kilby et al. (1999) and Tan et al.

(2000) do not report the programming language used.

According to our experience, Java is approximately 5

to 10 times slower than C, and there are not signiﬁcant

differences between C and Fortran regarding speed.

Kilby et al. (1999) and Tan et al. (2000, 2001c) consider

only the total traveled distance in their objective

function. For all other algorithms in Table 5, the pri-

mary optimization criterion is the number of vehicles.

In most of the cases, the secondary criterion is the

total distance of the routes. Only Potvin and Robillard

(1995) and Chiang and Russell (1996) consider total

duration of the routes as a secondary criterion, which

must be taken into account in the comparison.

According to Table 5, the best performing ap-

proaches in terms of solution quality seem to be

Bent and Van Hentenryck (2004), Bräysy (2003), and

Ibaraki et al. (2002). Czech and Czarnas (2002) report

excellent results to problem groups RC1 and RC2. If

Kilby et al. (1999) and Tan et al. (2000, 2001c) who

focus on minimizing only total distance are not con-

sidered, the worst approach seems to be Potvin and

Robillard (1995), utilizing neural networks. Basically,

the method of Potvin and Robillard (1995) is the same

as the parallel construction heuristic of Potvin and

Rousseau (1993). The neural network metaheuristic is

only used to select the seed customers initializing the

routes.

Generally, the difference in the number of vehicles

is quite signiﬁcant: It is about 5%, even if the worst

approach by Potvin and Robillard (1995) is not con-

sidered. In terms of total traveled distance, the differ-

ences are much greater. For example, the difference

between the methods proposed in Kontoravdis and

Bard (1995) and Rousseau et al. (2002) in problem

group RC2 is even 28%. However, if only the best

performing approaches by Bent and Van Hentenryck

(2004), Bräysy (2003), and Ibaraki et al. (2002) are con-

sidered, the differences in CNV and CTD are only

≈0.8% and 0%, respectively. It is quite interesting to

note that even if Caseau and Laburthe (1999b) use

a very sophisticated and intuitively appealing hybrid

approach, its performance is not comparable with

the other methods. Finally, the results of Anderson

et al. (2000) suggest that the recent metaheuristic

approaches outperform humans in designing efﬁ-

cient routes, though the differences are small. Because

many of the authors do not report the number of runs

or the time consumption required to obtain the results

in Table 5, ﬁnal conclusions regarding which method

performs best are impossible to reach. Consider-

ing the information available, only methods of Kon-

toravdis and Bard (1995), Liu and Shen (1999), Bent

and Van Hentenryck (2004), Bräysy (2003), Ibaraki

et al. (2002), and Bräysy et al. (2004b) can be iden-

tiﬁed as Pareto optimal in terms of solution qual-

ity and time consumption. Another comparison with

tabu searches and genetic algorithms is presented in

Table 6 and Figure 1 of the next section, where only

results for which computational effort is reported, are

considered.

4. Discussion

To summarize the results presented in Tables 2, 4,

and 5, it appears that the 10 metaheuristics showing

the best performance are the ones by Gambardella

et al. (1999), Homberger and Gehring (1999, 2005),

Gehring and Homberger (2001), Bent and Van Hen-

tenryck (2004), Bräysy (2003), Berger et al. (2003),

Ibaraki et al. (2002), Bräysy et al. (2004b), and Mester

(2002). Six of these use so-called pools of solutions

to memorize the best solutions found during the

search. All the presented approaches use different

local search techniques within the search, and eight

of the methods oscillate between different neighbor-

hood structures. The neighborhood structures used

seem to be typically quite small. Seven algorithms

use modiﬁcations of traditional route construction

heuristics to create an initial solution, and nine meth-

ods create a set of several different initial solutions.

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

134 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

Table 5 Miscellaneous Metaheuristic Algorithms

Authors R1 R2 C1 C2 RC1 RC2 CNV/CTD

(1) Thangiah et al. (1994) 1233 300 1000 300 1200 338 418

122742 100500 83089 64086 139113 117338 58905

(2) Kontoravdis and Bard (1995) 1258 309 1000 300 1263 350 427

132544 116427 8273 58965 150094 141421 64196

(3) Potvin and Robillard (1995) 1358 309 1056 338 1363 363 457

153941325112372 8756182891578978451

(4) Bachem et al. (1996) 1258 300 — — 1213 338 —

13920119961501615001

(5) Chiang and Russell (1996) 1250 291 1000 300 1238 338 422

130882 116642 90980 66630 147390 139370 64996

(6) Liu and Shen (1999) 1217 282 1000 300 1188 325 412

124957 101658 83006 59103 141287 120487 59318

(7) Kilby et al. (1999) 1267 300 1000 300 1213 338 423

120033 96656 83075 59224 138815 113342 57423

(8) Caseau et al. (1999b) 1217 — — — 1200 — —

120727 135662

(9) Gambardella et al. (1999) 1200 273 1000 300 1163 325 407

121773 96775 82838 58986 138242 112919 57525

(10) Tan et al. (2000) 1450 364 1011 325 1475 425 483

142012 127897 95857 76646 164877 164189 72194

(11) Anderson et al. (2000) — — — — 1163 — —

1397

(12) Tan et al. (2001c) 1310 460 1000 330 1270 560 470

121316 95230 84192 61275 141562 112037 57799

(13) Bräysy (2003) 1192 273 1000 300 1150 325 405

122212 97512 82838 58986 138958 112838 57710

(14) Li and Lim (2001) 1208 291 1000 300 1175 325 411

121514 95343 82838 58986 138547 114248 57467

(15) Bent and Van Hentenryck (2004) 1192 273 1000 300 1150 325 405

121110 95427 82838 58986 138417 112446 57273

(16) Rousseau et al. (2002) 1208 300 1000 300 1163 338 412

121021 94108 82838 58986 138278 110522 56953

(17) Czech and Czarnas (2002) — — — — 1150 325 —

138417 111949

(18) Bräysy et al. (2004b) 1200 273 1000 300 1150 325 406

121469 96044 82838 58986 138920 112414 57422

(19) Ibaraki et al. (2002) 1192 273 1000 300 1150 325 405

121740 95911 82838 58986 139103 112279 57444

Note. For each algorithm, the average results with respect to Solomon’s benchmarks are reported. Notations CNV and CTD in the rightmost column indicate

the cumulative number of vehicles and cumulative total distance over all 56 test problems.

(1) NeXT 25 MHz, –, 30 min. for all 8 methods; (2) Sun Sparc 10, 5 runs, 1.2 min.; (3) Sun Sparc 2, –, 0.08 min.; (4) Sun Sparc 10, –; (5) PC 486DX2/66 MHz,

–, 2.4 min.; (6) HP 9000/720, 3 runs, 20 min.; (7) DEC Alpha, 3 runs, 48.3 min.; (8) Pentium II 366 MHz, –, 5–30 min.; (9) Sun Ultra Sparc 1 167 MHz,

–, –; (10) Pentium II 266 MHz, –, –; (11) Pentium 500 MHz, –, 20 hours; (12) Pentium II 330 MHz, –, 4.6 min.; (13) Pentium 200 MHz, 1 run, 82.5 min.;

(14) Pentium III 545 MHz, 3 runs, 30 min.; (15) Sun Ultra 10, –, –; (16) Sun Ultra 10, 10 runs, 183.3 min.; (17) 5 ×IBM RS/6000, –, –; (18) AMD 700 MHz,

30 runs, 2.7 min.; (19) Pentium III 1 GHz, 1 run, 250 min.

All the methods use memory structures to facilitate

the search, and nine of them use special strategies or

operators to reduce the number of routes. Two and

ﬁve of the approaches employ ideas based on tabu

search and evolutionary algorithms (genetic algo-

rithms or evolution strategies), respectively. One uses

ant colonies, and two use multirestart variable neigh-

borhood search. All, except Bräysy (2003) include

some randomness in the search, and three approaches

allow infeasibilities during the search. Finally, four

methods include different techniques to speed up

the search. The method of Bent and Van Hentenryck

seems to give the best output, and the method of

Bräysy et al. (2002b) appears to be the fastest. The dif-

ferences in CNV and CTD between the 10 methods

are quite small, 0.5% and 1.2%, respectively. Based on

the above discussion, it seems that one cannot iden-

tify any special metaheuristic technique that would

perform better than the others. Usage of memory and

different traditional route construction and improve-

ment techniques seem to work well. Keeping in mem-

ory a set of solutions found during the search and

special strategies for reducing the number of routes is

also important.

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 135

Table 6 Comparison of Results Obtained with Limited Computational Effort for Solomon’s Benchmark Problems

Authors R1 R2 C1 C2 RC1 RC2 CNV/CTD

(1) Kontoravdis et al. (1995) 1258 309 1000 300 1263 350 427

132544 116427 8273 58965 150094 141421 64196

(2) Rochat and Taillard (1995) 1258 309 1000 300 1238 362 427

119742 95436 82845 59032 136948 113979 57120

(3) Potvin and Bengio (1996) 1258 309 1000 300 1263 338 427

1294711859 8610 6025146501476164679

(4) Taillard et al. (1997) 1233 300 1000 300 1190 338 417

122035 101335 82845 59091 138131 119863 58614

(5) Homberger and Gehring (1999) 1192 273 1000 300 1163 325 406

122806 96995 82838 58986 139257 114443 57876

(6) Gehring and Homberger (1999) 1242 282 1000 300 1188 325 415

1198 947 829 590 1356 1144 56946

(7) Brandão (1999) 1258 318 1000 300 1213 350 425

1205 995 829 591 1371 1250 58562

(8) Schulze and Fahle (1999) 1250 309 1000 300 1225 338 423

126842 105590 82894 58993 139607 130831 60651

(9) Gambardella et al. (1999) 1238 300 1000 300 1192 333 418

121083 96031 82838 59185 138813 114928 57583

(10) Kilby et al. (1999) 1267 300 1000 300 1213 338 423

120033 96656 83075 59224 138815 113342 57423

(11) Liu and Shen (1999) 1217 282 1000 300 1188 325 412

124957 101658 83006 59103 141287 120487 59318

(12) Bräysy (2003) 1192 273 1000 300 1150 325 405

122212 97512 82838 58986 138958 112838 57710

(13) Gehring and Homberger (2001) 1200 273 1000 300 1150 325 406

121757 96129 82863 59033 139513 113937 57641

(14) Li et al. (2003) 1208 291 1000 300 1175 325 411

121514 95343 82838 58986 138547 114248 57467

(15) Bent and Van Hentenryck (2004) 1217 273 1000 300 1163 325 409

120384 98031 82838 58986 137903 115891 57707

(16) Berger et al. (2003) 1217 273 1000 300 1188 325 411

125140 105659 82850 59006 141486 125815 60200

(17) Rousseau et al. (2002) 1208 300 1000 300 1163 338 412

121021 94108 82838 58986 138278 110522 56953

(18) Homberger and Gehring (2005) 1208 282 1000 300 1150 325 408

121167 95072 82845 58996 139593 113509 57422

(19) Bräysy et al. (2004b) 1200 273 1000 300 1150 325 406

122020 97038 82838 58986 139876 113937 57796

(20) Ibaraki et al. (2002) 1192 273 1000 300 1163 325 406

122002 96164 82838 58986 137872 113217 57480

Notes. (1) Sun Sparc 10, 5 runs, 1.2 (6.0) min.; (2) Silicon Graphics 100 MHz, 1 run, 92.2 (138) min.; (3) Sun Sparc 10, 1 run, 9.8 (9.8) min.;

(4) Sun Sparc 10, 1 run, 248 (248) min.; (5) Pentium 200 MHz, 10 runs, 13 (312) min.; (6) 4 ×Pentium 200 MHz, 1 run, 5 (48) min.; (7) Pen-

tium 200, 4 runs, 38.9 (373) min.; (8) Motorola PowerPC 604, 5 runs, 8.3 (270) min.; (9) Sun Ultrasparc 1, 1 run 30 (210) min.; (10) DEC Alpha,

3 runs, 48.3 (362) min.; (11) HP 9000/720, 3 runs, 20 (102) min.; (12) Pentium 200 MHz, 1 run, 82.5 (198) min.; (13) 4 ×Pentium 400 MHz,

5 runs, 13.5 (1,458) min.; (14) Pentium III 545 MHz, 3 runs, 30 (594) min.; (15) Sun Ultra 10, 5 runs, 30 (1,095) min.; (16) Pentium 400 MHz,

1 run, 30 (162) min.; (17) Sun Ultra 10, 10 runs, 183.3 (13,381) min.; (18) Pentium 400 MHz, 5 runs, 17.5 (473) min.; (19) AMD 700 MHz,

3 runs, 2.6 (106) min.; (20) Pentium III 1 GHz, 1 run, 33 (559) min.

The efﬁciency of the described metaheuristic

approaches is illustrated in Figure 1. Because the

number of vehicles is often considered as the pri-

mary optimization criterion, we consider the cumu-

lative number of vehicles (CNV) as a good measure

of the solution quality and robustness of a given

approach. The closer the point is to the lower left

corner in Figure 1, the better the method is consid-

ered to be. That is, the objective is to achieve low

CNV using as little CPU time as possible. In Figure 1

we included only approaches that report results for

all six datasets of Solomon. Moreover, to be able to

analyze the computational effort, here we consider

only results for which the time consumption and the

number of runs are described. However, to clarify the

ﬁgure, methods by Gehring and Homberger (2001),

Bent and Van Hentenryck (2004), and Rousseau et al.

(2002) are not considered due to their large CPU times.

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

136 Transportation Science 39(1), pp. 119–139, © 2005 INFORMS

To facilitate the comparison, the effect of different

hardware is normalized to equal Sun Sparc 10 using

Dongarra’s (1998) factors, described in detail in the

appendix. In addition, if the reported results are the

best ones over multiple experiments, we multiplied

the computation times by this number to obtain the

real computational effort.

From Figure 1 one can clearly see the develop-

ment of various VRPTW algorithms over the past

few years. In 1995, the best performing approach was

that of Russell (1995). Correspondingly, in 1999 the

Pareto optimal front in terms of solution quality and

time consumption was formed by the methods of

Russell (1995), Gehring and Homberger (1999), Liu

and Shen (1999), and Homberger and Gehring (1999).

At the moment, it seems that the only Pareto opti-

mal approaches in terms of solution quality and time

consumption are the local searches by Russell (1995),

Bräysy (2002), and Bräysy et al. (2004b), and the VNS

metaheuristic by Bräysy (2003).

More detailed results are provided in Table 6. As

in Figure 1, we consider here only results for which

the time consumption and the number of runs are

described. The computer, number of independent

runs, and the CPU time used to obtain the reported

results are described in the lower part of Table 6.

The number of runs is greater than one, only if the

reported result is best over multiple executions of the

given algorithm and the CPU time is reported only for

a single run. Two CPU time values are described: the

one reported by the authors and the modiﬁed CPU

time in parentheses, computed in the same way as the

values for Figure 1.

According to Table 6, the algorithms by Homberger

and Gehring (1999), Gehring and Homberger (2001),

Bräysy (2003), Ibaraki et al. (2002), and Bräysy et al.

(2004b) show the best overall performance. Regard-

ing individual problem groups, it seems that practi-

cally all approaches yield excellent results for problem

groups C1 and C2, having customers located in geo-

graphical clusters. Ibaraki et al. (2002) provides the

best output for group R1; Gehring and Homberger

(2001) report the best results for group R2 and Bräysy

(2003) for RC1 and RC2. In general, the differences

between the recent results, reported in 2001 and 2002

are small, varying within 2%. Here, one must note

that only Kilby et al. (1999), Liu and Shen (1999),

and Bräysy (2003) describe deterministic methods.

All other approaches in Table 6 are nondeterminis-

tic, and the presented results are often the best ones

over several runs. Thus, there is no guarantee for

obtaining similar results every time as is the case

with deterministic methods. For detailed best-known

results to Solomon’s benchmarks, we refer the reader

to http://www.top.sintef.no/.

During the last few years, several authors have

also tested their algorithms with other benchmark

problems, such as two real-life problems of Russell

(1995) and the extended Solomon’s benchmark prob-

lems by Gehring and Homberger (1999). Comparisons

regarding Russell’s test cases can be found in Bräysy

et al. (2004b), and at the moment the best results are

reported in Gehring and Homberger (2001), Bräysy

(2003), and Bräysy et al. (2004b). We are aware of

nine papers tackling the extended Solomon’s bench-

mark problems: Gehring and Homberger (1999, 2001),

Bräysy (2003), Li and Lim (2001), Bent and Van Hen-

tenryck (2004), Mester (2002), Homberger and Gehring

(2005), Le Bouthillier and Crainic (2005), and Bräysy

et al. (2004b). Bräysy (2003) reports the best average

performance for 200- and 400-customer data sets, and

is also faster than the competing approaches. The dif-

ferences are however small. On the larger benchmarks

Gehring and Homberger (2001) appear to perform

best. On the other hand, most of the best-known solu-

tions are reported in Mester (2002) and Bräysy et al.

(2004b). For details, see Bräysy et al. (2004b) and

http://www.top.sintef.no/.

5. Conclusions

NP-hardness of the VRPTW requires heuristic solu-

tion strategies for most real-life instances. In the pre-

vious sections, we have comprehensively surveyed

the remarkable evolution of metaheuristic VRPTW

methodologies. Currently, algorithms by Bent and

Van Hentenryck (2004), Bräysy (2003), Berger et al.

(2003), Homberger and Gehring (2005), and Ibaraki

et al. (2002) appear to achieve the best robustness. The

quality of the solutions obtained with different meta-

heuristic techniques is often much better compared

to traditional construction heuristics and local search

algorithms. At the same time, metaheuristics require

more CPU time and are more complex to imple-

ment and calibrate. This might prove quite signiﬁcant

in real, practical settings. Another issue of concern

when considering the choice of a solution approach

for real-life applications is that of ﬂexibility, i.e., how

well various approaches can handle the notoriously

“dirty” additional constraints that almost always clut-

ter practical instances. In general, local search and

metaheuristic techniques perform well in this respect,

but some methods may prove more effective in their

treatment of complicating constraints. This is the case,

for instance, of the CP-based approach of Rousseau

et al. (2002) that was speciﬁcally developed with this

objective in mind. The ﬁnal choice of the methodology

to apply in any given setting thus requires a careful

analysis to properly balance the different criteria that

need to be considered.

We believe that in the future the research on faster

metaheuristics, incorporating various sophisticated

Bräysy and Gendreau: Vehicle Routing Problem with Time Windows, Part II: Metaheuristics

Transportation Science 39(1), pp. 119–139, © 2005 INFORMS 137

speed-up techniques will intensify. Parallel imple-

mentations, hybridizations of different heuristics and

with exact techniques (and also CP), very large scale

neighborhoods, and different adaptive memory-based

approaches also sound like promising concepts. Pos-

sible future trends may also include tailored solution

approaches based on careful analysis of the problem

at hand and learning during the search process. On

the other hand, as pointed out in Cordeau et al. (2002),

the research on simpler and more ﬂexible, yet effec-

tive, metaheuristics will also increase.

Acknowledgments

This work was partially supported by the Emil Aaltonen

Foundation, the Canadian Natural Science and Engineering

Research Council, Liikesivistysrahasto and Volvo Founda-

tions, and the TOP project funded by the Research Council

of Norway. This support is gratefully acknowledged.

Appendix

Dongarra’s (1998) factors for the approaches described in

Figure 1 and Table 6.

Computer Mﬂops/s

16 ×Meiko T-800 7

PC 486/66 MHz 148

Sun Sparc 10 10

Silicon Graphics 100 MHz 15

Sun Ultra EnterPrise 450 44

8×Motorola PowerPC 604 65

Sun Ultra Sparc 1 167 MHz 70

Sun Ultra 10 73

DEC Alpha 25

HP 9000/720 17

Pentium 200 MHz 24

Pentium 366 MHz 48

Pentium 400 MHz 54

Pentium III 545 MHz 66

AMD 700 MHz 136

Pentium III 1 GHz 168

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