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Vehicle Routing Problem With Time Windows, Part II: Metaheuristics

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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.
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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 Ns. 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-optoperator. 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-optNo —
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 nmcustomers
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,
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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
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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
=CSCS. 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-optwith 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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Ant colony optimization (ACO) has been found to be useful on several vehicle routing problem variations. In this work, ACO is applied to the electric vehicle routing problem with time windows (E-VRPTW). The E-VRPTW has a hierarchical multiple objective function, which is to minimize the number of electric vehicles and the total distance traveled. A multiple ACO is applied to E-VRPTW in which two colonies cooperate to minimize the objectives in parallel. A local search is embedded in ACO to improve the quality of the output. The experimental results on a set of benchmark instances show that the multiple ACO is competitive with existing methods.
... For an overview of exact methods for the VRPTW we refer to [1]. Heuristic methods are reviewed in [3] and [4], and a compact review of exact and heuristic solving approaches for the VRPTW can be found in [6]. ...
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Attended home delivery services like online grocery shopping services require the attendance of the customers during the delivery. Therefore, the Vehicle Routing Problem with Time Windows occurs, which aims to find an optimal schedule for a fleet of vehicles to deliver goods to customers. In this work, we propose three sweep algorithms, which account for the family of cluster-first, route-second methods, to solve the Vehicle Routing Problem with Time Windows. In the first step, the customers are split into subsets such that each set contains as many as possible customers that can be served within one tour, e.g., supplied with one vehicle. The second step computes optimal tours for all assigned clusters. In our application, the time windows follow no special structure, and hence, may overlap or include each other. Further, time windows of different lengths occur. This gives additional freedom to the company during the planning process, and hence, allows to offer discounted delivery rates to customers who tolerate longer delivery time windows. Our sweep algorithms differ in the clustering step. We suggest a variant based on the standard sweep algorithm and two variants focusing on time window length and capacity of vehicles. In the routing step, a Mixed-Integer Linear Program is utilized to obtain the optimal solution for each cluster. The paper is concluded by a computational study that compares the performance of the three variants. It shows that our approach can handle 1000 customers within a reasonable amount of time.KeywordsVehicle routingTime windowsSweep algorithmAttended home deliveryTransportationLogistics
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A route-directed hybrid genetic approach to address the Vehicle Routing Problem with Time Windows is presented. The proposed scheme relies on the concept of simultaneous evolution of two populations pursuing different objectives subject to partial constraint relaxation. The first population evolves individuals to minimize total traveled distance while the second focuses on minimizing temporal constraint violation to generate a feasible solution, both subject to a fixed number of tours. Genetic operators have been designed to incorporate key concepts emerging from recent promising techniques such as insertion heuristics and large neighborhood search to further explore the solution space. Results from a computational experiment over common benchmark problems show that the proposed technique matches or outperforms some of the best heuristic routing procedures, providing six new best-known solutions. In comparison, the method proved to be fast, cost-effective and highly competitive.RésuméOn présente une approche génétique hybride guidée par routes pour le problème de tournées de véhicules avec fenêtres de temps. Le schéma proposé est basé sur l’évolution simultanée de deux populations visant des objectifs différents avec relaxation partielle de contraintes. Une première population de solutions cherche à minimiser la distance totale parcourue tandis que l’autre s’efforce de minimiser la violation de contraintes temporelles en tentant de générer une solution réalisable, les deux populations étant assujetties à un nombre fixe de routes. Des opérateurs génétiques ont été élaborés afin d’inclure des concepts issus de certaines techniques récentes de routage telles que des heuristiques d’insertion et de recherche. Les résultats de simulations menées sur des exemplaires de problème bien connus indiquent que la technique proposée se compare avantageusement aux meilleures procédures heuristiques, obtenant six nouvelles meilleures solutions. La méthode est rapide, efficace et très concurrentielle
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