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TRANSPORTATION SCIENCE
Vol. 39, No. 1, February 2005, pp. 104–118
issn 0041-1655eissn 1526-54470539010104
informs®
doi 10.1287/trsc.1030.0056
© 2005 INFORMS
Vehicle Routing Problem with Time Windows,
Part I: Route Construction and Local
Search Algorithms
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 presents a survey of the research on 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. Both traditional heuristic
route construction methods and recent local search algorithms are examined. The basic features of each method
are described, and experimental results for Solomon’s benchmark test problems are presented and analyzed.
Moreover, we discuss how heuristic methods should be evaluated and propose using the concept of Pareto
optimality in the comparison of different heuristic approaches. The metaheuristic methods are described in the
second part of this article.
Key words: vehicle routing; time windows; heuristics; local search metaheuristics; tabu search; genetic
algorithms
History : Received: December 2001; revision received: December 2002; accepted: December 2002.
Transportation is an important domain of human
activity. It supports and makes possible most other
social and economic activities. Whenever we use a
telephone, shop at a food store, read our mail, or
fly for business or pleasure, we are the beneficia-
ries of some system that has routed messages, goods,
or people from one place to another. Freight trans-
portation, in particular, is one of today’s most impor-
tant activities. Let us mention that the annual cost of
excess travel in the United States has been estimated
at some $45 billion (King and Mast 1997), and the
turnover of goods transportation in Europe is some
$168 billion per year. In the United Kingdom, France,
and Denmark, for example, transportation represents
some 15%, 9%, and 15% of national expenditures,
respectively (Crainic and Laporte 1997, Larsen 1999).
It is estimated that distribution costs account for
almost half of the total logistics costs and in some
industries, such as in the food and drink business,
distribution costs can account for up to 70% of the
value added costs of goods (De Backer et al. 1997,
Golden and Wasil 1987). Halse (1992) reports that in
1989, 76.5% of all the transportation of goods was
done by vehicles, which also underlines the impor-
tance of routing and scheduling problems.
The vehicle routing problem with time windows
(VRPTW) is an important problem occurring in many
distribution systems. The VRPTW can be defined as
follows. Let G=V E be a connected digraph con-
sisting of a set of n+1 nodes, each of which can be
serviced only within a specified time interval or time
window and a set Eof arcs with nonnegative weights,
dij , and with associated travel times, tij . The travel
time, tij , includes a service time at node i, and a vehicle
is permitted to arrive before the opening of the time
window, and wait at no cost until service becomes pos-
sible, but it is not permitted to arrive after the latest
time window. Node 0 represents the depot. Each node
i, apart from the depot, imposes a service requirement,
qi, that can be a delivery from or a pickup for the
depot. In most of the surveyed papers the objective is
to find the minimum number of tours, K∗, for a set
of identical vehicles such that each node is reached
within its time window and the accumulated service
up to any node does not exceed a positive number Q
(vehicle capacity). A secondary objective is often either
to minimize the total distance traveled or the duration
of the routes. All problem parameters, such as cus-
tomer demands and time windows, are assumed to be
104
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
Transportation Science 39(1), pp. 104–118, © 2005 INFORMS 105
known with certainty. Moreover, each customer must
be served by exactly one vehicle, thus prohibiting split
service and multiple visits. The tours correspond to
feasible routes starting and ending at the depot. Some
of the most useful applications of the VRPTW include
bank deliveries, postal deliveries, industrial refuse col-
lection, national franchise restaurant services, school
bus routing, security patrol services, and JIT (just in
time) manufacturing.
The VRPTW has been the subject of intensive
research efforts for both heuristic and exact optimiza-
tion approaches. Early surveys of solution techniques
for the VRPTW can be found in Golden and Assad
(1986, 1988), Desrochers et al. (1988), and Solomon and
Desrosiers (1988). Desrosiers et al. (1995) and Cordeau
et al. (2001) focus mainly on exact techniques. Fur-
ther details on these exact methods can be found
in Larsen (1999) and Cook and Rich (1999). Because
of the high complexity level of the VRPTW and its
wide applicability to real-life situations, solution tech-
niques capable of producing high-quality solutions
in limited time, i.e., heuristics, are of prime impor-
tance. Over the last few years, many authors have
proposed new heuristic approaches, primarily meta-
heuristics, for tackling the VRPTW. To our knowl-
edge, these have not been comprehensively surveyed
and compared. The purpose of this two-part survey
is to fill this gap. In the first part, we examine tradi-
tional heuristic approaches, that is, route construction
and route improvement (local search) methods. These
are of interest by themselves because they can pro-
vide good solutions with a low computational effort,
but also because they are a major component of all
metaheuristics for the VRPTW. Metaheuristics are dis-
cussed in the second part of this survey.
The remainder of this paper is organized as follows.
Section 1 is devoted to a discussion of how heuristics
are to be evaluated. Route construction techniques are
reviewed in §2 and route improvement (local search)
methods in §3. Finally, §4 concludes the paper.
1. Evaluation of Heuristics
Evaluation of any heuristic method is subject to the
comparison of a number of criteria that relate to
various aspects of algorithm performance. Examples
of such criteria are running time, quality of solu-
tion, ease of implementation, robustness, and flexi-
bility (Barr et al. 1995, Cordeau et al. 2002). Because
heuristic methods are ultimately designed to solve
real-world problems, flexibility is an important con-
sideration. An algorithm should be able to easily han-
dle changes in the model, the constraints, and the
objective function. As for robustness, an algorithm
should not be overly sensitive to differences in prob-
lem characteristics: A robust heuristic should not per-
form poorly on any instance. Moreover, an algorithm
should be able to produce good solutions every time
it is applied to a given instance. This is to be high-
lighted because many heuristics are nondeterminis-
tic, and contain some random components such as
randomly chosen parameter values. The output of
separate executions of these nondeterministic meth-
ods on the same problem is, in practice, never the
same. This makes it difficult to analyze and compare
results. Using only the best results of a nondetermin-
istic heuristic, as is often done in the literature, may
create a false picture of its real performance. Thus,
we consider average results based on multiple execu-
tions on each problem an important basis for the com-
parison of nondeterministic methods. Furthermore, it
would also be important to report the worst-case per-
formance. Extensive discussions on these subjects can
be found in Cordeau et al. (2002).
The time a heuristic takes to produce good qual-
ity solutions can be crucial when choosing between
different techniques. Similarly, the quality of the final
solution, as measured by the objective function, is
important. How close the solution is to the optimal
solution is a standard measure of quality, or if the
heuristic is designed to simply produce feasible solu-
tions, then the ability of the heuristic to provide such
solutions is important.
There is generally a trade-off between run time and
solution quality—the longer a heuristic is run, the bet-
ter the quality of the final solution. A compromise is
needed so that good quality solutions are produced in
a reasonable amount of time. Basically, this trade-off
between run time and solution quality can be viewed
in terms of a multiobjective optimization in which
the two objectives are balanced. Performance mea-
sures for heuristics can be plotted in two-dimensional
space, with the first dimension corresponding to run
time and the second to solution quality. In that space,
points such that there exist no other points with bet-
ter values on both dimensions are said to be Pareto
optimal; they define effective compromises between
the objectives. This is illustrated in Figure 10 of §3,
where points Antes and Derigs (1995), Russell (1995),
and Bräysy (2003) are the Pareto optimal ones. The
choice between different heuristic approaches yield-
ing Pareto optimal results depends on the preferences
of the decision maker and the situation at hand.
By far, the most common method of evaluating the
solution quality of a heuristic algorithm is empiri-
cal analysis. In general, empirical analysis involves
testing the heuristic across a wide range of problem
instances to get an idea of overall performance. To
arrive at conclusions that have meaning in a statistical
sense, experimental design should ideally be used on
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
106 Transportation Science 39(1), pp. 104–118, © 2005 INFORMS
different levels of the various algorithm parameters
and the results compared by appropriate techniques.
In the actual comparison of heuristics, one often
faces a number of difficulties. The most obvious dif-
ficulty is making the competition fair. Differences
between machines first come to mind. In this paper,
we address this issue by adjusting reported running
times by the factors given by Dongarra (1998). Even
more difficult issues to address are differences in cod-
ing skill, tuning, and effort invested (Hooker 1995).
Another difficulty faced, especially in the VRPTW
context, is that often only the best results obtained
during the whole computational study are reported.
Moreover, in some cases the authors do not report
the number of runs or CPU time required to get the
reported results. In these cases it is impossible to con-
clude anything about the efficiency of the methods, or
compare these methods with other approaches. The
only adequate basis for comparison of these meth-
ods would be optimal solutions, because if enough
time is available, it is always preferable to solve the
problems to optimality using exact methods. To be
able to reach proper conclusions, in addition to the
number of runs and time consumption, one should
answer questions such as what are the limits of the
given algorithm, i.e., how good are the best results
that can be obtained using the particular approach,
and how good a solution can be obtained in a given
amount of time. One should, in other words, report
results obtained using different computation times,
and observe how much time is needed to obtain
results of a given quality. Moreover, in our opinion,
figures describing the relationship between solution
quality and computation time would greatly facili-
tate the analysis. Taillard (2001) extensively discusses
this issue and proposes reporting an absolute com-
putational effort, such as the number of iterations
instead of computating times, and using probability
diagrams based on repeated Mann-Whitney statisti-
cal tests. Obviously, such an approach is not possible
when one relies on previously published results as we
do here.
In the VRPTW context, the most common way
to compare heuristics is the results obtained for
Solomon’s (1987) 56 benchmark problems. These
problems have 100 customers, a central depot, capac-
ity constraints, time windows on the time of deliv-
ery, and a total route time constraint. The C1 and
C2 classes, have customers located in clusters, and in
the R1 and R2 classes the customers are at random
positions. The RC1 and RC2 classes contain a mix of
both random and clustered customers. Each class con-
tains between 8 and 12 individual problem instances,
and all problems in any one class have the same cus-
tomer locations and the same vehicle capacities; only
the time windows differ. In terms of time window
density (the percentage of customers with time win-
dows), the problems have 25%, 50%, 75%, and 100%
time windows. The C1, R1, and RC1 problems have a
short scheduling horizon and require 9 to 19 vehicles.
Short horizon problems have vehicles that have small
capacities and short route times, and cannot service
many customers at one time. Classes C2, R2, and RC2
are more representative of “long-haul” delivery with
longer scheduling horizons and fewer (two to four)
vehicles. Both travel time and distance are given by
the Euclidean distance between points.
The results are usually ranked according to a hier-
archical objective function, where the number of vehi-
cles is considered as the primary objective, and for
the same number of vehicles, the secondary objective
is often either total traveled distance or total dura-
tion of routes. Therefore, a solution requiring fewer
routes is always considered better than a solution
with more routes, regardless of the total traveled dis-
tance. According to Bräysy (2001) these two objectives
are very often conflicting, meaning that the reduction
in number of vehicles often causes increase in total
traveled distance. Thus, a better solution in terms of
total distance can be obtained by increasing the num-
ber of routes. Some other papers report similar find-
ings; see, for example, Caseau and Laburthe (1999).
2. Route Construction Heuristics
Route construction heuristics select nodes (or arcs)
sequentially until a feasible solution has been created.
Nodes are chosen based on some cost minimization
criterion, often subject to the restriction that the selec-
tion does not create a violation of vehicle capacity
or time window constraints. Sequential methods con-
struct one route at a time, while parallel methods
build several routes simultaneously.
Solomon (1986) proposes a so-called route-first
cluster-second scheme using a giant-tour heuristic.
First, the customers are scheduled into one giant tour,
which is then divided into a number of smaller routes.
The initial giant tour could, for example, be gener-
ated as a traveling salesman tour without considering
the capacity and time constraints. No computational
results are given in the paper for the heuristic.
Solomon (1987) describes several heuristics for the
VRPTW. One of the methods is an extension to
the savings heuristic of Clarke and Wright (1964). The
savings method, originally developed for the classical
VRP, is probably the best-known route construction
heuristic. It begins with a solution in which every
customer is supplied individually by a separate
route. Combining the two routes serving, customers
iand j, respectively, results in a cost savings of
Sij =di0+d0j−dij . Clarke and Wright (1964) select the
arc (ij) linking customers iand jwith maximum Sij ,
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
Transportation Science 39(1), pp. 104–118, © 2005 INFORMS 107
subject to the requirement that the combined route
is feasible. With this convention, the route combi-
nation operation is applied iteratively. In combining
routes, one can simultaneously form partial routes for
all vehicles or sequentially add customers to a given
route until the vehicle is fully loaded. To account for
both the spatial and temporal closeness of customers,
Solomon sets a limit to the waiting time of the route.
The savings method is illustrated in Figure 1.
The second heuristic, a time-oriented nearest neigh-
bor, starts every route by finding an unrouted cus-
tomer closest to the depot. At every subsequent
iteration, the heuristic searches for the customer clos-
est to the last customer added into the route and adds
it at the end of the route. A new route is started any
time the search fails to find a feasible insertion place,
unless there are no more unrouted customers left. The
metric used to measure the closeness of any pair of
customers attempts to account for both geographical
and temporal closeness of customers.
The most successful of the three proposed sequen-
tial insertion heuristics is called I1. A route is first
initialized with a “seed” customer and the remain-
ing unrouted customers are added into this route
until it is full, with respect to the scheduling hori-
zon and/or capacity constraint. If unrouted customers
remain, the initializations and insertion procedures
are then repeated until all customers are serviced. The
seed customers are selected by finding either the geo-
graphically farthest unrouted customer in relation to
the depot or the unrouted customer with the low-
est allowed starting time for service. After initializing
the current route with a seed customer, the method
uses two subsequently defined criteria, c1iuj and
c2iuj, to select customer ufor insertion between
adjacent customers iand jin the current partial route.
Let (i0i
1i
2i
mbe the current route, with i0and
imrepresenting the depot. For each unrouted cus-
tomer u, we first compute its best feasible insertion
cost on the route as
c1iu u j u =min
=1 m c1i−1ui
(1)
ijij
Figure 1 The Savings Heuristic
Note. In the left part, customers iand jare served by separate routes; in the
right part, the routes are combined by inserting customer jafter i.
Next, the best customer u∗to be inserted in the route
is the one for which
c2iu∗ u∗ju
∗
=max
uc2iu u j u uis unrouted and
the route is feasible(2)
Client u∗is then inserted into the route between iu∗
and ju∗. When no more customers with feasible
insertions can be found, the method starts a new
route, unless it has already routed all customers. More
precisely c1iuj is calculated as
c1iuj =1c11 iuj+2c12iuj (3)
where
1+2=1
1≥0
2≥0
c11iuj =diu +duj −dij ≥0(4)
c12iuj =bju −bj(5)
and diu,duj , and dij are distances between customers i
and u,uand j, and iand j, respectively. Parameter
controls the savings in distance, and bju denotes the
new time for service to begin at customer j, given
that uis inserted on the route and bjis the beginning
of service before insertion. The criterion c2iuj is
calculated as follows
c2iuj=d0u−c1iuj ≥0(6)
Parameter is used to define how much the best
insertion place for an unrouted customer depends
on its distance from the depot, and on the other
hand how much the best place depends on the extra
distance and extra time required to visit the cus-
tomer by the current vehicle. The second type of pro-
posed insertion heuristics (I2) aims to select customers
whose insertion costs minimize a measure of total
route distance and time, and the third approach (I3)
accounts for the urgency of servicing a customer.
Dullaert (2000) and Dullaert and Bräysy (2003) ar-
gue that Solomon’s time insertion criterion c12iuj
underestimates the additional time needed to insert
a new customer ubetween the depot and the first
customer in the partially constructed route. This can
cause the insertion criterion to select suboptimal
insertion places for unrouted customers. Thus, a route
with a relatively small number of customers can have
a larger schedule time than necessary. The author
introduces new time insertion criteria to solve the
problem and concludes that the new criteria offer sig-
nificant cost savings starting from more than 50%.
These cost savings, however, decrease as the number
of customers per route increases.
Solomon’s (1987) time-oriented sweep heuristic is
based on the idea of decomposing the problem into
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
108 Transportation Science 39(1), pp. 104–118, © 2005 INFORMS
a clustering stage and a scheduling stage. In the
first phase, customers are assigned to vehicles as in
the original sweep heuristic (Gillett and Miller 1974).
Here, a “center of gravity” is computed and the cus-
tomers are partitioned according to their polar angle.
In the second phase, customers assigned to a vehicle
are scheduled using an insertion heuristic of type I1.
Potvin and Rousseau (1993) introduce a parallel
version of Solomon’s insertion heuristic I1, where the
set of mroutes is initialized at once. The authors
use Solomon’s sequential insertion heuristic to deter-
mine the initial number of routes and the set of seed
customers. The selection of the next customer to be
inserted is based on a generalized regret measure over
all routes. A large regret measure means that there is
a large gap between the best insertion place for a cus-
tomer and its best insertion places in the other routes.
Foisy and Potvin (1993) implemented the above-
described parallel version of Solomon’s insertion
heuristic on parallel hardware consisting of two to
six Sun 3 workstation transputers. The parallelism is
exploited in the calculation of insertion cost for each
customer. The selection of the best customer for inser-
tion is then run only on half of the available pro-
cessors. To reduce the unequal workload among the
processors, unrouted customers are reassigned among
the processors, so as to reduce the average processor’s
idle time. The authors conclude that the overall reduc-
tion in computation time is linear with the number
of processors for the distributed part of the heuristic
algorithm.
Ioannou et al. (2001) use the generic sequential
insertion framework proposed by Solomon (1987) to
solve a number of theoretical benchmark problems
and an industrial example from the food industry. The
proposed approach is based on new criteria for cus-
tomer selection and insertion that are motivated by
the minimization function of the greedy look-ahead
solution approach of Atkinson (1994). The basic idea
behind the criteria is that a customer uselected for
insertion into a route should minimize the impact of
the insertion on the customers already on the route
Table 1 Route Construction Heuristics
Author R1 R2 C1 C2 RC1 RC2 CNV/CTD
(1) Solomon (1987) 1358 327 1000 313 1350 388 453
1436714024 9519 6927159651682173004
(2) Potvin et al. (1993) 1333 309 1067 338 1338 363 453
150904 138667 134369 79759 172372 165105 78834
(3) Ioannou et al. (2001) 1267 309 1000 313 1250 350 429
1370 1310 865 662 1512 1483 67891
Note. For all algorithms, the average results for Solomon’s benchmarks are described. The notations CNV and CTD
in the rightmost column indicate the cumulative number of vehicles and cumulative total distance over all 56 test
problems.
(1) DEC 10, 1 run, 0.6 min.; (2) IBM PC, 1 run, 19.6 min.; (3) Intel Pentium 133 MHz, 1 run, 4.0 min.
under construction, on all nonrouted customers, and
on the time window of customer u, himself.
Balakrishnan (1993) describes three heuristics for
the vehicle routing problem with soft time windows
(VRPSTW). The heuristics are based on nearest neigh-
bor and Clarke-Wright savings rules, and they differ
only in the way used to determine the first customer
on a route and in the criteria used to identify the
next customer for insertion. The motivation behind
VRPSTW is that by allowing limited time window
violations for some customers, it may be possible
to obtain significant reductions in the number of
vehicles required and/or the total distance or time
of all routes. Among the soft time window problem
instances, dial-a-ride problems play a central role.
Bramel and Simchi-Levi (1996) propose an asymp-
totically optimal heuristic based on the idea of solving
the capacitated location problem with time windows
(CLPTW). In CLPTW, the objective is to select a sub-
set of possible sites, to locate one vehicle to each
site, and to assign customers to the vehicles. In the
VRPTW context, this selection of locations for vehi-
cles refers to selecting a set of seed customers that
initialize the routes. The authors use a Lagrangian
relaxation-based technique to solve the CLPTW and
the other customers are inserted in greedy order into
simple tours by favoring customers that least increase
the distance traveled. The authors conclude that their
heuristic provides a better solution than Solomon’s
heuristic for 25 of the 56 problems using reasonable
running times.
Table 1 compares some of the described route con-
struction algorithms. The first column to the left indi-
cates the authors. Columns R1, R2, C1, C2, RC1, and
RC2 present the average number of vehicles and aver-
age total distance with respect to the six problem
groups of Solomon (1987). Finally, the rightmost col-
umn indicates the cumulative number of vehicles and
cumulative total distance over all 56 test problems.
In the lower part of the table, we report information
regarding the computer used, number of runs, and
average time consumption in minutes as reported by
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
Transportation Science 39(1), pp. 104–118, © 2005 INFORMS 109
the authors. We could not include all the algorithms
described in the table due to lack of information (not
all authors report results properly or use Solomon’s
problem set). In Table 1, the number of vehicles is the
primary minimization objective, and the secondary
objective is the total duration of routes in Solomon
(1987) and Potvin and Rousseau (1993) and the total
distance in Ioannou et al. (2001). Thus, in some cases,
there may be a slight overestimation of the total
distance values of Solomon (1987) and Potvin and
Rousseau (1993). The methods by Solomon (1987) and
Potvin and Rousseau (1993) are coded in Fortran;
Ioannou et al. (2001) do not report the programming
language used. Finally, because we used rounded dis-
tance measures reported by other authors to calculate
the cumulative total distance (CTD), we rounded the
values to integers in Tables 1 and 2.
It seems that Ioannou et al. (2001) produce the
best results, though at the cost of higher computation
times. As for the other two methods, Solomon (1987)
seems to be better than Potvin and Rousseau (1993)
in clustered problem groups C1 and C2, while the
opposite is true for the other problem groups. These
heuristics are very fast, and there are no significant
differences in the computational burden if one takes
into account the differences in the hardware used.
Compared to local search approaches, these construc-
tion heuristics are considerably faster, as one can see
Table 2 Local Search Algorithms
Author R1 R2 C1 C2 RC1 RC2 CNV/CTD
(1) Thompson et al. (1993) 1300 318 1000 300 1300 371 438
135692 127600 91667 64463 151429 163443 68916
(2) Potvin et al. (1995) 1333 327 1000 313 1325 388 448
1381912934 9029 6532154531595169285
(3) Russell (1995) 1266 291 1000 300 1238 338 424
1317 1167 930 681 1523 1398 65827
(4) Antes et al. (1995) 1283 309 1000 300 1250 338 429
138646 136648 95539 71731 154592 159806 71158
(5) Prosser et al. (1996) 1350 409 1000 313 1350 513 471
124240 97712 84384 60758 140876 111137 58273
(6) Shaw (1997) 1231 — 1000 — 1200 — —
120506 82838 136040
(7) Shaw (1998) 1233 — 1000 — 1195 — —
120179 82838 136417
(8) Caseau et al. (1999) 1242 309 1000 300 1200 338 420
123334 99099 82838 59663 140374 122099 58927
(9) Schrimpf et al. (2000) 1208 282 1000 300 1188 338 412
121153 94927 82838 58986 136176 109763 56830
(10) Cordone et al. (2001) 1250 291 1000 300 1238 338 422
124189 99539 83405 59178 140887 113970 58481
(11) Bräysy (2003) 1217 282 1000 300 1188 325 412
125324 103956 83288 59349 140844 124496 59945
Note. For each method two average results for Solomon’s benchmarks are presented. The rightmost CNV and CTD indicate the
cumulative number of vehicles and cumulative total distance over all test problems.
(1) PC/AT 12 MHz, 4 runs, 1.8 min.; (2) Sparc workstation, −, 3.0 min.; (3) PC/486/DX2 66 MHz, 3 runs, 1.4 min.; (4) RS6000/530,
4 runs, 3.6 min.; (5) −−−; (6) DEC Alpha, 3 runs, 2 hours; (7) Sun Ultra Sparc 143 MHz, 6 runs, 1 hour; (8) Pentium 300 MHz,
1 run, 5 min.; (9) RS 6000, −, 30 min.; (10) Pentium 166 MHz, 1 run, 15.7 min.; (11) Pentium 200 MHz, 1 run, 4.6 min.
from Figure 10. However, these simple procedures
lack in solution quality compared to more sophisti-
cated approaches.
3. Solution Improvement Methods
Classical local search methods form a general class
of approximate heuristics based on the concept of
iteratively improving the solution to a problem by
exploring neighboring ones. To design a local search
algorithm, one typically needs to specify the follow-
ing choices: how an initial feasible solution is gen-
erated, what move-generation mechanism to use, the
acceptance criterion, and the stopping test. The move-
generation mechanism creates the neighboring solu-
tions by changing one attribute or a combination of
attributes of a given solution. Here attribute could
refer, for example, to arcs connecting a pair of cus-
tomers. Once a neighboring solution is identified,
it is compared against the current solution. If the
neighboring solution is better, it replaces the current
solution, and the search continues. Two acceptance
strategies are common in the VRPTW context, namely
first-accept (FA) and best-accept (BA). The first-accept
strategy selects the first neighbor that satisfies the pre-
defined acceptance criterion. The best-accept strategy
examines all neighbors satisfying the criterion and
selects the best among them.
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
110 Transportation Science 39(1), pp. 104–118, ©2005 INFORMS
The local optimum produced by any local search
procedure can be very far from the optimal solu-
tion. Local search methods perform a myopic search
because they only sequentially accept solutions
that produce reductions in the objective function
value. Thus, the outcome depends heavily on initial
solutions and the neighborhood generation mecha-
nism used. Most iterative improvement methods that
have been applied to vehicle routing and scheduling
problems are edge-exchange algorithms.
The edge-exchange neighborhoods for a single
route are the set of tours that can be obtained from
an initial tour by replacing a set of kof its edges by
another set of kedges. Such replacements are called
k-exchanges, and a tour that cannot be improved
by a k-exchange is said to be k-optimal. Verifying
k-optimality requires Onktime. Figure 2 illustrates
2-exchange or 2-opt. It tries to improve the tour by
replacing two of its edges by two other edges and
iterates until no further improvement is possible.
Russell (1977) reports early work on the VRPTW
for a k-optimal improvement heuristic. The so-called
M-tour approach was effective in solving an actual
problem with a few time-constrained customers. A
solution for a 163-customer problem with 15% time-
constrained customers was generated in less than
90 seconds of IBM 370/168 CPU time.
Efficient implementations for speeding up the
screening of infeasible solutions and the evaluation
of the objective function are reported in Savelsbergh
(1986), Solomon and Desrosiers (1988), Solomon et al.
(1988), Savelsbergh (1990), and Savelsbergh (1992).
The techniques used involve preprocessing, tai-
lored updating mechanisms, and lexicographic search
strategies.
Baker and Schaffer (1986) report on a computa-
tional study of route improvement procedures, which
are applied to heuristically generated initial solu-
tions. Time-oriented nearest neighbor and three dif-
ferent cheapest insertion algorithms with differing
cost functions are used for solution construction pur-
poses. The cost functions consider one or more of the
following components: distance, increase in arrival
j
i
j
+1
j
i
j
+1
Figure 2 2-Opt Exchange Operator
Note. The edges (i,i+1) and (j,j+1) are replaced by edges (i,j and
(i+1, j+1), thus reversing the direction of customers between i+1 and j.
time, and waiting time. The improvement meth-
ods considered are extensions to the VRPTW of the
2-opt and 3-opt edge exchange procedures of Lin
(1965). Both intraroute and interroute exchanges are
tested. The authors conclude that the best overall
solutions are usually obtained from the best start-
ing solutions, and that, generally, the cheapest inser-
tion procedures outperformed the nearest neighbor
ones. The authors also conclude that only less than
10% of the solution improvements involve the rever-
sal of the orientation of a sequence of two or more
customers.
Van Landeghem (1988) presents a bicriteria heuris-
tic based on the savings method of Clarke and Wright
(1964). More precisely, the author proposes combining
the original savings measure in terms of travel time
with so-called “loss of flexibility.” The flexibility is
defined as the difference between customer time win-
dow length and route time window length after com-
bining. Route time window refers to the difference
between time slots, inside which a vehicle can start
servicing the first and last customers on the route. In
the end, the results are improved using simple cus-
tomer reinsertions. A closely related operator is the
Or-opt introduced by Or (1976) for the traveling sales-
man problem. The basic idea is to relocate a chain of l
consecutive vertices (customers). This is achieved by
replacing three edges in the original tour by three new
ones without modifying the orientation of the route
as illustrated in Figure 3.
Koskosidis et al. (1992) describe an extension of
the cluster-first, route-second algorithm of Fisher and
Jaikumar (1981) for the variant of the VRP with soft
time window constraints, where the time windows
can be violated at a cost. The problem is heuristically
decomposed into a capacitated clustering problem
and a series of traveling salesman problems with soft
time windows. The clustering problem is solved with
a greedy heuristic procedure that assigns customers
to selected seeds according to a regret function repre-
senting the penalty of not assigning the customer to
its closest seed. Then, an attempt is made to find new
j
i
–1
i
j
i
j
+1
i
–1
i+
i+
1
1
Figure 3 The Or-Opt Operator
Note. Customers iand i+1 are relocated to be served between two cus-
tomers jand j+1, instead of customers i−1 and i+2. This is performed
by replacing three edges (i−1, i), (i+1, i+2), and (j,j+1) by the edges
(i−1, i+2), (j,i), and (i+1, j+1), preserving the orientation of the route.
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
Transportation Science 39(1), pp. 104–118, © 2005 INFORMS 111
seed customers with lower clustering cost, and local
exchanges between all customer pairs belonging to
different clusters are performed. For the routing part,
the authors propose a combination of a total enu-
meration algorithm, a reduced gradient scheduling
algorithm, and the branch exchange heuristic of Lin
and Kernighan (1973). The iterative master-slave solu-
tion procedure approximates linearly the clustering
costs and improves the approximation successively
through the actual routing costs obtained. Numerical
results based on randomly generated and benchmark
problem sets indicate that the algorithm outperforms
Solomon’s insertion heuristic and 2-opt and 3-opt
improvement heuristics of Baker and Schaffer (1986),
though at the cost of a clearly higher computational
effort.
Potvin and Rousseau (1995) compare different edge
exchange heuristics for VRPTW (2-opt, 3-opt, and Or-
opt) and introduce a new 2-opt∗exchange heuristic.
The basic idea in 2-opt∗is to combine two routes so
that the last customers of a given route are intro-
duced after the first customers of another route, thus
preserving the orientation of the routes. The opera-
tor is illustrated in Figure 4, where the edges i i +1
and j j +1are replaced by ij +1and j i +1,
i.e., the end portions of two routes are exchanged.
As a special case, it can combine two routes into
one if edge i i +1is the first one on its route
and edge j j +1the last one on its route or vice
versa. A hybrid approach based on Or-opt and 2-opt∗
exchanges is found to be particularly powerful. This
approach oscillates between the two neighborhoods
by changing the operator each time local minimum is
found. The authors also test an implementation where
the two operators are merged together. The initial
solutions are created with Solomon’s I1 heuristic.
Prosser and Shaw (1996) compare intraroute
2-opt by Lin (1965) and interroute operators relo-
cate, exchange, and cross, originally proposed by
Savelsbergh (1992) for the classical VRP. The 2-
opt works by reversing part of a single route (see
j
j
+1
j
ij
+1
i
Figure 4 2-Opt∗Operator
Note. The customers served after customer ion the upper route are rein-
serted to be served after customer jon the lower route, and customers after
jon the lower route are moved to be served on the upper route after cus-
tomer i. This is performed by replacing edges (i,i+1) and (j,j+1) with
edges (i,j+1) and (j,i+1).
j
i
j
+1
i
–1
j
i
j
+1
i
–1
Figure 5 Relocate Operator
Note. The edges (i−1, i), (i,i+1), and (j,j+1) are replaced by (i−1,
i+1), (j,i), and (i,j+1), i.e., customer ifrom the origin route is placed
into the destination route.
Figure 1). The relocate operator simply moves a cus-
tomer visit from one route to another. It is illustrated
in Figure 5.
The exchange heuristic swaps two visits in differ-
ent routes. This is pictured in Figure 6. Finally, cross
is similar to 2-opt∗proposed by Potvin and Rousseau
(1995) for VRPTW. Initially, a virtual vehicle, which
performs the visits not carried out by the real vehi-
cles, exists. This virtual vehicle is different from the
real ones in two respects. First, the virtual vehicle
can make an unlimited number of customer visits.
Second, the cost incurred by the virtual vehicle when
it performs a customer visit is typically higher than
that incurred by a real vehicle.
De Backer et al. (1997) report research similar to
Prosser and Shaw (1996) in the constraint program-
ming (CP) context. CP is a paradigm for representing
and solving a wide variety of problems. Tackling com-
binatorial problems generally involves manipulating
variables that can take a finite number of values.
In CP, a domain is associated with every variable.
The domain is created by using constraints on vari-
ables that restrict the possible combinations of values
for the variables. Looking locally at a particular con-
straint, the CP algorithm attempts to remove from the
domain of each variable involved in that constraint
values that cannot be part of any solution. This reduc-
tion of a variable’s domain triggers the examination of
all constraints involving this variable, which in turn
j
j
+1
j
+1
i
j
i
i
jj
–
–
–
–
Figure 6 The Exchange Operator
Note. The edges (i−1, i), (i,i+1), (j−1, j), and (j,j+1) are replaced
by (i−1, j), (j,i+1), (j−1, i), and (i,j+1), i.e., two customers from
different routes are simultaneously placed into the other routes.
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
112 Transportation Science 39(1), pp. 104–118, ©2005 INFORMS
may reduce other domains. This recursive process
stops when either no new domain reduction has taken
place or a domain becomes empty. In constraint pro-
gramming, the computation is driven by constraints,
thus giving them an active role. The problems are
then solved using often complete search techniques
such as depth-first search and branch and bound. For
more details on CP, see, for example, Jaffar and Lassez
(1986) and Jaffar and Maher (1994).
Other frequently applied neighborhood operators
are the -interchange of Osman (1993), the CROSS-
exchange of Taillard et al. (1997), the GENI-exchange
of Gendreau et al. (1992), and ejection chains (Glover
1991, 1992). The -exchange generation mechanism
can be described as follows: Given a solution for the
problem represented by a set of routes S=r1
rpr
qr
k ,a-interchange between a pair
of routes (rpr
q) is a replacement of a subset of
customers S1⊆rpof size S1≤by another
subset S2⊆rqof size S2≤to get two new
routes rp=rp−S1∪S2and rq=rq−S2∪
S1and a new neighboring solution S=r1
r
pr
qr
k . The neighborhood NS of a
given solution Sis the set of all neighbors Sgener-
ated in this way for a given value of .
In CROSS-exchanges, the basic idea is to first
remove two edges (i−1i) and (kk +1) from a first
route, while two edges (j−1j), and (l l +1) are
removed from a second route. Then the segments i−k
and j−l, which may contain an arbitrary number
of customers, are swapped by introducing the new
edges (i−1, j), (l,k+1), (j−1, i), and (kl +1) as
illustrated in Figure 7.
Ejection chains (Glover 1991, 1992) are based on
the notion of generating compound sequences of
moves, leading from one solution to another, by
linked steps in which changes in selected elements
cause other elements to be ejected from their current
state, position, or value assignment. In the VRP con-
text, moves refer to the removal of a customer from
its route and its reinsertion in another route. The goal
j
l
+1
i
i
–1
j
j
i
i
–1
k
l
l
+1
l
k
–1–
Figure 7 CROSS-Exchange
Note. Segments (i,k) on the upper route and (j,l) on the lower route are
simultaneously reinserted into the lower and upper routes, respectively. This
is performed by replacing edges (i−1, i), (k,k+1), (j−1, j), and (l,
l+1) by edges (i−1, j), (l,k+1), (j−1, i), and (k,l+1). Note that the
orientation of both routes is preserved.
j
j
+1
i
i
j
j
+1
i
i
k
k
+1
kkk
+1
k
–
–
–
–
Figure 8 The GENI-Exchange Operator
Note. Customer ion the upper route is inserted into the lower route between
the customers jand kclosest to it by adding the edges (j,i) and (i,k).
Because jand kare not consecutive, one has to reorder the lower route.
Here, the feasible tour is obtained by deleting edges (j,j+1) and (k−1, k)
and by relocating the path j +1k−1.
is to “make room” for a new customer in a route by
first removing another customer from the same route.
In each phase within the ejection chain, one customer
remains unrouted. The removal and insertion proce-
dures are repeated until one can insert a customer into
another route without the need to remove (eject) any
customer. The GENI operator is an extension of the
relocate neighborhood in which a customer can also
be inserted between the two customer nodes on the
destination route that are nearest to it, even if these
customer nodes are not consecutive. The operator is
illustrated in Figure 8.
Thompson and Psaraftis (1993) propose a method
based on the concept of cyclic k-transfers that involves
simultaneously transferring kdemands from route Ij
to route I#j for each jand fixed integer k. The set
of routes Ir ,r=1m constitutes a feasible solu-
tion, and #is a cyclic permutation of a subset of
1m . In particular, when #has fixed cardinal-
ity C, we obtain a C-cyclic k-transfer. By allowing
kdummy demands on each route, demand trans-
fers can be performed among permutations rather
than cyclic permutations of routes. Due to the com-
plexity of the cyclic transfer neighborhood search,
it is performed heuristically. A general methodology
developed by Thompson and Orlin (1989) is used
for searching cyclic transfer neighborhoods. They
transform the search for negative cost cyclic transfers
into a search for negative cost cycles in an auxiliary
graph. Savelsbergh’s (1986) 2-opt procedure is used
to maintain local optimality of the routes at all times,
and the initial solutions are constructed using the I1
heuristic of Solomon. The three-cyclic, two-transfer
operator is illustrated in Figure 9.
Antes and Derigs (1995) propose a parallel con-
struction approach that constructs and improves sev-
eral tours, simultaneously. The approach is based on
the concept of negotiation between customers and
tours. First, each unrouted customer requests a service
cost from every tour and sends a proposal to the tour
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
Transportation Science 39(1), pp. 104–118, © 2005 INFORMS 113
a
b
c
d
e
f
g
h
i
j
2
3
k
lm
n
o
p
4
1
l
m
k
n
o
p
e
d
c
a
b
g
h
i
j
f
1
2
3
4
Figure 9 The Cyclic Transfer Operator
Note. The basic idea is to simultaneously transfer the customers denoted by
white circles in a cyclical manner between the routes. More precisely, here
customers aand cin Route 1, fand jin Route 2, and oand pin Route 4 are
simultaneously transferred to Routes 2, 4, and 1, respectively, and Route 3
remains untouched.
that offered the lowest price, and second, each tour
selects the most efficient proposal. The prices are cal-
culated according to Solomon’s evaluation measures
for insertion (heuristic I1). Once a feasible solution is
constructed, the number of tours is reduced by one
and the problem is resolved. The authors propose
also a post-optimization approach, where some of the
most inefficient customers are first removed from the
tours and then reinserted using the negotiation pro-
cedure described above.
Russell (1995) embeds global tour improvement
procedures within the tour construction process. The
construction procedure used is similar to that in
Potvin and Rousseau (1993). Nseed points represent-
ing fictious customers are first selected using the seed
point generation procedure of Fisher and Jaikumar
(1981), originally proposed for the classical VRP. The
basic idea is to use vehicle capacity information to
create sectors and decide the distance of the seeds
from the depot within each sector. Three ordering
rules are used to select next customer for insertion,
namely earliest time window, farthest distance from
depot, and width of the time window augmented
by distance from the depot. The local search method
employs a scheme developed by Christofides and
Beasley (1984). In this scheme, a move is performed
by deleting and reinserting four customer points close
to each other. For each customer, the best two routes
are first determined according to the insertion cost
of Solomon (1987), because it would be computation-
ally intractable to evaluate all route assignments. This
interchange procedure is applied after every kcus-
tomer has been routed. This approach is compared to
the k-opt multiple tour branch exchange heuristic of
Russell (1977). The author concludes that the hybrid
approach of embedding improvement into the con-
struction procedure is superior compared to the tra-
ditional two-phase approach, i.e., route construction
followed by solution improvement.
Thangiah et al. (1995) examine the vehicle rout-
ing problems with time deadlines (VRPTD), i.e., with-
out earliest time window. They create two heuristics
based on principles of time-oriented sweep and
cheapest insertion procedures for solving the VRPTD,
followed by -interchanges of Osman (1993). The
authors conclude that the two proposed heuristics
perform well for problems in which the customers are
tightly clustered or have long deadlines.
Hamacher and Moll (1996) describe a heuristic for
real-life VRP’s with narrow time windows in the con-
text of delivery of groceries to restaurants. The algo-
rithm is divided into two parts. In the clustering step,
the customers are partitioned into regionally bounded
sets using the structure of the minimal spanning tree
(MST). The MST is divided into subtrees, where nodes
of each subtree represent the customers belonging to
one tour. Several weight functions based on the num-
ber of customers, distance, total demand, and time
window types are used to determine whether a sub-
tree leads to a cluster. Then, customers within these
sets are routed using a simple cheapest insertion algo-
rithm followed by a local improvement phase, which
cuts out pieces of the tour and inserts them back at
another feasible location within the same tour. If a
feasible solution is not found, the remaining unrouted
customers are scheduled manually.
Shaw (1997) describes a large neighborhood search
(LNS) based on rescheduling selected customer visits
using CP techniques. LNS is analogous to the shuf-
fling technique used in job-shop scheduling (see, for
example, Applegate and Cook 1991), which is, itself,
inspired from the shifting bottleneck procedure of
Adams et al. (1988). The search operates by choosing
in a randomized fashion a set of customer visits. The
selected customers are removed from the schedule,
and then reinserted at optimal cost. To create opportu-
nity for interchange of customer visits between routes,
the removed visits are chosen so that they are related.
Here, the term related refers to customers that are
geographically close to each other, served by the same
vehicle, have similar demand for goods and simi-
lar starting times for service. A branch-and-bound
method coupled with CP is then used to reschedule
removed visits. In the initial solution, each customer
is served by a separate vehicle. Due to high compu-
tational requirements, this approach can be applied
only to problems where the number of customers per
route is relatively low.
Shaw (1998) uses an LNS approach similar to
Shaw’s (1997) above approach for solving vehicle
routing problems. The basic difference is the usage of
constraint-based limited discrepancy search (LDS) in
the reinsertion of customers within the branch-and-
bound procedure. For more details about LDS, see
Harwey and Ginsberg (1995). The number of visits to
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
114 Transportation Science 39(1), pp. 104–118, ©2005 INFORMS
be removed is increased during the search each time
a number of consecutive attempted moves have not
resulted in an improvement of the cost. LDS is used to
explore the search tree in order of an increasing num-
ber of discrepancies, a discrepancy being a branch
against the best insertion places. For instance, a single
discrepancy would consist in inserting a customer at
its second cheapest position.
Schrimpf et al. (2000) introduce also a methodol-
ogy similar to LNS that the authors name “ruin and
recreate.” Three strategies are first used to remove a
set of customers from a solution. The removed cus-
tomers are then reinserted in random order using a
greedy cheapest insertion heuristic. The removal pro-
cedure removes customers randomly, based on the
distance to a randomly selected key customer and a
set of succeeding customers on the same route with
the key customer. To minimize the number of routes,
a fixed penalty is set for routes exceeding the desired
minimum number. During the search, solutions that
worsen the objective function value are also accepted
if the worsening is within a threshold.
Caseau and Laburthe (1999) describe a heuristic
specifically designed for large routing problems. The
authors introduce an LDS variation to the parallel
cheapest insertion heuristic that branches between the
best and second best alternative routes for each cus-
tomer if the differences in insertion costs are small.
During solution construction, three moves are consid-
ered after each insertion, namely 2-opt∗, reinsertion
of a chain of consecutive customers from a route r
into another route r, as well as a simple customer
transfer move. When no feasible insertion place can
be found, three different types of moves are consid-
ered to make room for the unrouted customer. The
first move, swap, removes a chain of consecutive cus-
tomers from rand inserts it into another route r. The
second move, relocate, removes a vertex from r, and
inserts it into another route r, which may recursively
require that another vertex is removed from r, etc.,
followed by reoptimization of each route concerned
by the move. The last move, flush and relocate, first
removes from rall nodes that can be directly relo-
cated into another route, before trying to insert cus-
tomer ci. In cases where the number of customers on a
route is less than 30, the order of the customers within
the route is optimized using the exact constraint-
based branch-and-bound algorithm by Caseau and
Laburthe (1997). Otherwise, in case of longer routes,
3-opt is used to modify routes after each insertion.
The authors also try to restrict the customers included
in each route to a particular geometric zone.
Hong and Park (1999) propose a two-phase heuris-
tic algorithm that consists of a parallel insertion
method for clustering and a sequential linear goal
programming procedure for routing. The primary cri-
terion for the algorithm is the minimization of the
total traveled distance instead of the number of vehi-
cles, and the second criterion is the minimization of
the total customer waiting time. The seed customers
are selected by identifying customers that cannot be
served on the same route due to time or vehicle con-
straints. The remaining customers are inserted into
these initialized tours so that the increase in route dis-
tance and waiting time is minimal. Similar to Potvin
and Rousseau (1993), customers with a small num-
ber of feasible insertion locations and a large differ-
ence between the best and next best insertion places
are considered for clustering first (regret measure). At
the end of the clustering stage, groups are reformed
using Or-opt and 2-opt improvement procedures. In
the routing stage, the goal programming model is
decomposed into two linear programming subprob-
lems, where either total distance or waiting time
is minimized first. The authors report slightly bet-
ter results than Potvin and Rousseau (1993), though
using longer computation time.
Cordone and Wolfer-Calvo (2001) propose a deter-
ministic heuristic based on classical k-opt exchanges
combined with a procedure to reduce the number of
routes. The special feature of the algorithm is that it
alternates between minimization of total distance and
of total route duration to escape from local minima.
The algorithm builds a set of initial solutions using
Solomon’s insertion heuristic I1, applies a local search
procedure (exchanges two or three arcs) to each of
them, and chooses the best one. The route reduc-
tion procedure tries to insert each customer of one
route at a time into another route. If simple insertion
fails, a simple ejection chain (Glover 1991, 1992) is
tried, where a customer, cj, is first removed from the
target route, rn, and inserted into some other route,
rm, before inserting the current customer ciinto rn.
The authors use special implementation techniques to
reduce the computation time. The first technique is
based on so-called macronodes. The macronode is a
collapse of whatever sequence of nodes into a sin-
gle one that is easier to handle (see Cordone and
Wolfler-Calvo 1997). The other techniques are explor-
ing the k-neighborhood in lexicographic order (for
details, see Savelsbergh 1986) and keeping in mind
the best exchange for each route, each pair, and each
triplet of routes.
Bräysy (2003) describes several local search heuris-
tics using a new three-phase approach for the
VRPTW. In the first phase, several initial solutions are
created using the route construction heuristics with
different combinations of parameter values. In the sec-
ond phase, an effort is put to reduce the number
of routes using a new ejection chain-based approach
(Glover 1991, 1992) that also considers reordering of
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
Transportation Science 39(1), pp. 104–118, © 2005 INFORMS 115
the routes. In the third phase, Or-opt exchanges are
used to minimize total traveled distance. The first
construction heuristics borrow their basic ideas from
the studies of Solomon (1987) and Russell (1995).
Routes are built, one at a time, in sequential fash-
ion, and after kcustomers have been inserted into the
route, the route is reordered using Or-opt exchanges.
In addition, new seed selection schemes are intro-
duced. The second heuristic draws its basic con-
cepts from the savings heuristic of Clarke and Wright
(1964). Here, a parallel version of the savings heuristic
is implemented, and the original measure of savings
is modified to also consider changes in waiting times.
Moreover, the customers in the combined route are
reordered before evaluating the savings incurred by
uniting the two routes.
Table 2 summarizes some of the results obtained
by described local search algorithms. We could not
include all the algorithms described in the table due
to lack of information (not all authors report results
properly or use Solomon’s problem set). In Table 2,
most of the algorithms are deterministic in nature.
The only stochastic approaches are that of Russell
(1995), Shaw (1997, 1998), and Schrimpf et al. (2000).
Russell (1995) and Cordone and Wolfler-Calvo (2001)
implemented their algorithm in Fortran, and Potvin
and Rousseau (1995), Antes and Derigs (1995), Shaw
(1998), and Caseau and Laburthe (1999) used C.
Thompson and Psaraftis (1993), Prosser and Shaw
(1996), Shaw (1997), and Schrimpf et al. (2000) do
not report the software used. The number of vehi-
cles is considered as a primary optimization criterion
by all authors except Prosser and Shaw (1996), where
only the total distance of the routes is minimized. The
secondary objective is total distance in most papers.
Thompson and Psaraftis (1993), Potvin and Rousseau
Solomon (1987) and
Potvin et al. (1993)
Thomps on et
al.(1993)
Russell (1995)
Antes et al. (1995)
Cordone et al. (2001)
Caseau et al. (1999)
Ioannou et al. (2001)
Bräysy (2003)
405
410
415
420
425
430
435
440
445
450
455
460
0 5 10 15 20 25 30
Time in minutes
CNV
Figure 10 The Efficiency of the Described Methods
Note. The notation CNV refers to the cumulative number of vehicles required to solve all 56 test problems. Note that the time consumption of each method is
normalized to Sun Sparc 10 using Dongarra’s (1998) factors to facilitate the analysis.
(1995), and Russell (1995) optimize the total duration
of routes; this may cause an overestimation of the
total distance values and should be taken into account
in the comparison.
According to Table 2, the methods of Schrimpf et al.
(2000) and Bräysy (2003) are the best ones with respect
to solution quality. The difference in the cumulative
number of vehicles is about 14%, compared to the
worst method by Prosser and Shaw (1996). The rea-
son for this can be found in the optimization criteria
used: In Prosser and Shaw (1996), only the total dis-
tance of the routes is considered. Schrimpf et al. (2000)
dominates all other methods for four problem groups.
For the easy clustered problem group C1, Shaw (1997,
1998), and Caseau and Laburthe (1999) yield equally
good output, and in RC2 Bräysy (2003) performs best.
It is difficult to conclude anything regarding the com-
putational effort, as many of the authors do not report
the CPU time or the number of runs required to get
the reported results. Given the information available,
the methods of Russell (1995), Caseau and Labur-
the (1999), and Bräysy (2003) appear to be the most
efficient ones. It should also be noted that, due to
poor performance, Shaw (1997, 1998) do not report
the results for the problem groups R2, C2, and RC2.
Thus, these two procedures are not comparable with
other approaches in terms of robustness.
The efficiency of the described methods is illus-
trated in Figure 10. In Figure 10 we included only
approaches where a sufficient amount of information
is provided by the authors. At least the computer,
number of computational runs, the time consump-
tion, and the number of vehicles must be reported.
From Figure 10, one can see that difference in time
consumption between Solomon (1987), Potvin and
Bräysy and Gendreau: Vehicle Routing Problem, Part I: Route Construction and Local Search Algorithms
116 Transportation Science 39(1), pp. 104–118, ©2005 INFORMS
Rousseau (1993), Thompson and Psaraftis (1993), and
Antes and Derigs (1995) is quite small. Therefore,
only Antes and Derigs (1995), Russell (1995), and
Bräysy (2003) can be considered as Pareto optimal
in terms of solution quality and time consumption.
There is no clear rule to determine which Pareto
optimal approach is the best. The choice depends on
the preferences of the decision maker. The methods
by Antes and Derigs (1995) and Russell (1995) are a
lot faster than the one in Bräysy (2003), but they fall
behind in solution quality.
4. Conclusions
The vehicle routing problem with time windows is
one of the classical research areas in operations re-
search with considerable economic significance. The
NP-hardness of the VRPTW requires heuristic solution
strategies for most real-life instances. The research on
approximation methods has, over the years, produced
a wide variety of heuristic approaches for the VRPTW.
In this paper, methods based on classical solution con-
struction and improvement techniques were compre-
hensively reviewed.
VRPTW heuristics are usually measured against
two criteria: solution quality in terms of objective
function value, and speed. In our opinion, simplic-
ity of implementation, flexibility, and robustness are
also essential attributes of good heuristics. By flexi-
bility, we mean the ability to accommodate the var-
ious side constraints encountered in a majority of
real-life applications. As for robustness, an algorithm
should still able to produce results under difficult
circumstances, such as when a problem instance is
pathological. These issues, as well as the question of
how to evaluate heuristics, are discussed in §1.
Recent composite heuristics were found to perform
best in terms of solution quality, the most efficient
being those of Russell (1995) and Bräysy (2003). These
methods provide better results than earlier simple
heuristics, while still being quite fast. As heuristics
need to be especially effective for very large-scale
problems, we expect work on these to intensify.
Acknowledgments
This work was partially supported by the Emil Aaltonen
Foundation, the Canadian Natural Science and Engineer-
ing Research Council, Liikesivistysrahasto and Volvo Foun-
dations, and TOP project funded by the Research Council
of Norway. This support is gratefully acknowledged. The
authors also thank Dr. Geir Hasle (SINTEF Applied Mathe-
matics, Norway) for his valuable comments.
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