PROBABILISTIC DIVERSIFICATION AND
INTENSIFICATION IN LOCAL SEARCH FOR
VEHICLE ROUTING
Yves Rochat
Département de mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Suisse
e-mail : rochat@dma.epfl.ch WWW : http://dmawww.epfl.ch/rose.mosaic/rochat.html
Éric D. Taillard
Centre de recherche sur les transports, Université de Montréal, C. P. 6128, succursale Centre-Ville, Montréal, Canada H3C 3J7
e-mail : taillard@crt.umontreal.ca WWW : http://www.iro.umontreal.ca/people/taillard
Abstract :
This paper presents a probabilistic technique to diversify, intensify and parallelize a local search adapted for solving vehicle routing problems. This
technique may be applied to a very wide variety of vehicle routing problems and local searches. It is shown that efficient first level taboo searches
for vehicle routing problems may be significantly improved with this technique. Moreover, the solutions produced by this technique may often be
improved by a post-optimization technique presented in this paper too. The solutions of nearly 40 problem instances of the literature have been
improved.
Key words : Vehicle routing, local searches, parallel algorithms.
1. INTRODUCTION
More and more, local search methods are used to find
good solutions to combinatorial optimization problems.
Throughout the paper, we use the term local search as a
synonym of neighbourhood search. Local searches are
sometimes restricted to steepest descent algorithms but
we also include taboo search and simulated annealing in
local search methods. These techniques have two main
weaknesses : first they may sometimes be trapped in a
very poor local optimum and this difficulty may be hard
to overcome even by a fastidious tuning of the local
search parameters ; second, these methods require a
large computational effort. Nowadays, the most prom-
ising way of cutting down computational time is to use
parallel computers. Unfortunately, local search is a
process that is intrinsically sequential and therefore not
always easy to parallelize.
In this paper, we describe a technique that overcomes
both weaknesses. We illustrate this technique on two
taboo searches we have developed for vehicle routing
problems (VRPs) : first for the most elementary VRP
(one depot, identical vehicles) and second for the VRP
with time windows (VRPTW). This technique may also
be applied to other local searches or other VRPs. In
section 2, we briefly describe the problems treated and
the local search used to solve them. In section 3, we
present the new technique that allows us first to diversify
the search by exploring solutions that are very different
from each other, second to intensify the search in order
to identify better local optima in a promising region of
the set of feasible solutions, and third to parallelize the
process. Then, we present a post-optimization technique
that allows frequent improvement in the solutions
produced by the diversification and intensification
procedure. In section 4, we compare the new technique
to the original local searches on which they are based
and finally we conclude in section 5.
2. VRP AND LOCAL SEARCH
2.1. Presentation of the problems
The first problem we treat is the following : identical
vehicles with a fixed carrying capacity Q must deliver
order quantities q
i
(i =1,…,n) of goods to n customers
from a single depot (i = 0). Knowing the distance d
ij
between customers i and j (i, j =0,…,n), the problem is
to find tours for the vehicles in such a way that :
• The total distance travelled by the vehicles is
minimized.
• Only one vehicle handles the deliveries for a given
customer.
• The total quantity of goods that a single vehicle
delivers cannot be larger than Q.
The second problem we treat is similar to the
elementary VRP, but distances are interpreted as dura-
tions and each customer i requires a service time s
i
and
must be served during a time window [b
i
, e
i
]. When a
vehicle arrives at customer i before time b
i
, it must wait.
Solomon (1987) has proposed a set of 56 problems
where the first objective is to minimize the number of
vehicles needed to deliver all orders and the second
objective, subject to optimizing the first, is to minimize
the total distance travelled by the vehicles.
P
ublished in Journal of Heuristics, vol. 1, pp. 147-167,
which should be cited to refer to this work.
DOI: 10.1007/BF02430370
2 Yves Rochat, Éric Taillard
2.2. Local search for VRP
Very generally, a local search can be formulated as
follows :
(a) Choose an initial solution s
0
; set k := 0 ;
(b) While a stopping criterion is not satisfied, repeat :
(b1) Choose a solution s
k +1
∈ N(s
k
), the set of
neighbour solutions of s
k
.
(b2) Set k := k +1
The choice of a policy for points (a) to (b2) leads to
various iterative searches. Generally, the initial solution
is chosen in such a way that its generation is fast and
easy. The choice of the stopping criterion is often related
to the type of iterative search chosen : for example, a
descent method stops as soon as there is no better solu-
tion than s
k
in N(s
k
) ; a taboo search stops when the
number k of iterations is greater than a threshold K. In
fact, common terminology equates local search with a
descent method. We use a more general terminology
based on allowing the rules for choosing among its
elements to follow a broader design than customary.
From the perspective of the formulation adopted here,
the way of defining N(s) and the way of selecting a solu-
tion in N(s) are the most difficult parts of the design of a
local search. For the VRP, the reader may refer to the
works of Taillard (1993), Gendreau et al. (1994) or
Gendreau, Laporte and Potvin (1995) for more details
on these choices.
The method of Taillard (1993) is one of the most effi-
cient for the elementary VRP. Its main feature is to
partition large problems into independent subproblems
and to optimize each subproblem independently. A
partition generates sets (subproblems) involving four to
eight tours (or 30 to 60 customers) that are near one
another. Once the subproblems are optimized, all the
tours of the subproblems are grouped together to
construct a solution to the original problem and this
solution is again partitioned, and so on. There is a
random component in the partition process. Therefore,
the algorithm may produce very different solutions from
one run to the next. The taboo search we have used in
this paper has been slightly changed from the original
version of Taillard (1993) : the partition procedure has
been improved and the exact procedure used for the
optimization of the tours has been replaced by a
heuristic approach that is more reliable. Generally, these
modifications improve the behaviour of the method on
non-uniform problem instances.
For uniform problems (i. e., with customers
uniformly distributed in the plane around the depot and
ordering quantities much smaller than the vehicle
capacity), this algorithm finds good solutions in a small
amount of computation time. If the problems are non-
uniform, the search may be trapped in a poor solution
with a high probability. Since most real-life problems
are non-uniform (see e. g. the 120-city problem of
Christofides et al. (1979), Taillard (1993), Semet and
Taillard (1993), Fisher (1994), Rochat and Semet
(1994)), it is worth designing a method that works well
on this type of problems.
The taboo search we used for the VRPTW is derived
from the adaptation of Rochat and Semet (1994) for a
real-life problem that is more complex than the
VRPTW ; for example this real-life problem incorpo-
rates an heterogeneous fleet, driver’s breaks and
accessibility constraints (where each customer can only
be reached by a subset of the vehicles). So our taboo
search for the VRPTW is more nearly a simplification of
this complex taboo search than a new taboo search
specially designed for the Solomon’s VRPTW
instances. However, the resulting method is competi-
tive, since it succeeds in improving the quality of 16
previous best known solutions out of the 56 Solomon
instances (see table 6). There is a random component in
our taboo search for the VRPTW. This means that two
runs of the method will generally produce two different
solutions.
3. IMPROVING LOCAL SEARCHES
APPLIED TO THE VRP
3.1. Probabilistic diversification and
intensification technique
A fundamental principle of taboo search is to exploit
the interplay between diversification and intensification
where diversification drives the search to examine new
regions, and intensification focuses more intently on
regions previously found to be good. (Intensification
typically operates by re-starting from high quality solu-
tions, or by modifying choice rules to favour the
inclusion of attributes of these solutions).
Our approach to achieving such an exploitation is
based on two primary perspectives, which we describe
as follows. The first of these comes from probabilistic
taboo search, which is founded on the idea of translating
information generated by the search history, coupled
with current measures of attractiveness, into evaluations
that are monotonically mapped into probabilities of
selection. Operating in a neighbourhood framework, the
approach then successively selects among available
alternatives according to a probability assignment that is
strongly biased to favour the choice of higher evalua-
tions. (In contrast to some terminology, taboo search
refers to neighbourhoods that are constructive and
destructive as well as transitional, since it includes strat-
egies not only of restarting, but also of alternating
between constructive and destructive steps).
Probabilistic diversification and intensification in local search for vehicle routing 3
This approach is motivated by the following
premise : an intelligent use of randomization, which is
not blindly uniform but embedded in probabilities that
account for history and measures of attractiveness,
offers a useful type of diversification that can substitute
for more complex uses of memory. As noted in Glover
(1989), “
… the use of randomization, via assigned prob-
abilities, allows a gain in efficiency by obviating
extensive record keeping and evaluation operations that
a more systematic pursuit of diversity may require”. We
take advantage of this means of achieving diversity in a
special way, by rules, we indicate subsequently.
The second main perspective that underlies our
approach derives from one of the most basic (and
earliest) types of intensification strategies. The heart of
this approach lies in generating solutions by reference to
the notions of strongly determined and consistent varia-
bles. A strongly determined variable is one whose value
cannot be changed except by inducing a disruptive effect
on the objective function value or on the values of other
variables. The identification of strongly determined
variables is by reference to best solutions from previous
solution efforts, which motivates us to measure their
strength by the quality of the solutions in which varia-
bles lie (at particular values). This weighting by
objective function values corresponds to the practice of
emphasizing relative attractiveness in probabilistic
taboo search, allowing us to exploit both approaches
together.
A consistent variable is one that is frequently
strongly determined at a particular value (or in a narrow
range). Specifically, the more often a variable receives a
particular value in the best solutions (where we weight
these solutions by their objective value), the more highly
it qualifies as consistent.
The rationale for isolating such variables, and the
strategy for exploiting them, are embodied in the
following expectations. First, a variable that is highly
consistent is likely to receive its preferred value (or lie
in its preferred range) in optimal and near-optimal solu-
tions. Second, once some variables are assigned specific
values, other variables that previously did not seem
highly consistent will now become a good deal more so.
Third, imposing narrow restrictions on selected varia-
bles will yield increasingly reliable measures of the
relative consistency of remaining variables, given the
imposed restrictions.
The strategy to take advantage of these tendencies
may then be summarized as follows (Glover, 1977).
(a) Select one or more variables with greatest relative
consistencies and constrain these to their preferred
values.
(b) Determine new relative consistencies for the vari-
ables on the basis of the restriction of step (a).
(c) Repeat the process until all variables have been
constrained to specific values.
This process is then joined with a heuristic improve-
ment procedure to transform its value assignments into
a new solution, thus creating an iterative method for
obtaining progressively more refined outcomes.
In our approach for executing the preceding steps in
the vehicle routing setting we select variables in blocks,
composed of specific tours to which customers have
been assigned in the best previous solutions. The new
relative consistency measures produced for step (b) then
result by observing that once particular variables are
assigned values, we cannot reassign them within the
same solution — that is, we cannot select a new tour
containing customers already assigned on a previous
step. Our improvement heuristics for obtaining better
solutions, and which are the drivers of the method, are
those of Taillard (1993) and Rochat and Semet (1994).
We now sketch the details of our approach more
precisely. We emphasize that our method can be applied
to most forms of local search (as encompassed by our
rather general definition), and to most types of VRPs. To
play a diversification rôle, a local search must be able to
produce solutions that are very different from each
other, but not necessarily among the very best. This
condition is generally satisfied by iterated local searches
that either start with solutions randomly generated or
that apply a random component in their subsequent deci-
sions, thereby causing different runs generally to
produce different solutions. However, by incorporating
the principle of probabilistic taboo search to guide our
choices, we create a more strategic type of diversifica-
tion than that embodied in simple randomization. The
technique we employ may be described as follows:
In a first phase (initialization), the search is diversi-
fied by generating, with the local search, I solutions that
are different from one another. For our approach to the
elementary VRP, this generation is done by considering
various initial decompositions of the problem. For the
VRPTW — as well as for elementary VRPs of small
size that cannot be decomposed — the non-determin-
istic characteristics of our local search guarantee the
diversity of the solutions.
By generating several initial solutions with the local
search, one hopes that all the information necessary to
create solutions of very high quality exists in these solu-
tions, but in a non-apparent way. In the case of the VRP,
this information is included in the tours. So, one hopes
that the initialization creates a set of tours that includes
members not very different from the tours of a good
solution. It is not unrealistic to think that it is easy to
create good tours. The challenge is to find a set of tours
whose members are simultaneously good for all
customers.
4 Yves Rochat, Éric Taillard
The generation of I initial solutions creates a set T of
tours. In figure 1, we give a subset of tours of T after the
initialization phase for a problem of Christofides et al.
(1979) with 199 customers. This figure may be
compared with figure 2 where a very good solution to
the same problem is given. We see that several tours are
similar, two of them being identical. However, T
contains several other tours and, after the initialization
phase, the good tours cannot be directly identified. In
these figures, the area of the customers (empty circles) is
proportional to the quantity ordered ; the filled circle is
the depot and its area is proportional to the capacity of
one vehicle ; the first and last trips of each vehicle are
not drawn.
In a second phase, the goal is to extract these good
tours and improve them. We start from the principle that,
if a solution s includes tours that belong to a good solu-
tion, then the objective function value of s is probably
better than that of a solution that does not contain such
tours. The second phase thus favours the extraction of
tours that belong to the best solutions generated during
the initialization phase. This extraction must not totally
exclude tours belonging to bad solutions.
To implement this phase, each tour is labelled with
the value of the solution to which it belongs. The set T is
sorted by increasing values of the labels and the tours
with only one customer are removed from T, since they
do not contain interesting information. Then, we choose
tours of T probabilistically, by giving preference to tours
with low labels and by ignoring tours that include
customers belonging to tours already extracted. This
choice is repeated until it is not possible to extract new
tours from T.
Figure 1 : Example of tours belonging to T after the initializa
-
tion phase.
Let S be the set of tours thus extracted. Since S may
not contain all the customers of the problem, S is a
partial solution. In order to construct a feasible solution
from S, the set of customers not belonging to the tours of
S may be considered as an independent VRP (of small
size) that can be solved by the local search. The tours of
this independent VRP are added to S to create a feasible
solution to the initial VRP. This feasible solution is
considered as the initial solution of a local search that
will try to improve its quality. Another way to initialize
the local search is to modify the procedure, embedded in
the local search, that produces an initial solution from a
set of customers. This procedure is modified in such a
way that it also accepts the tours of the partial solution
S. In our implementations, we have chosen this way to
initialize the local searches.
Once a new solution is generated by the current
application of local search, the tours of this solution are
labelled with the value of the objective function and are
included in the set T. It is important to note that the same
tour may occur in more than one solution. In order to
represent this fact appropriately (in the sense of identi-
fying the relative consistency of consistent variables), a
tour is included in T multiple times, one for each solu-
tion in the set T. The extraction of tours of T, followed
by the optimization with the local search and the inser-
tion of the new tours in T is repeated until a stopping
criterion is met. More formally, this algorithm may be
formulated as follows :
Figure 2 : Best known solution for the problem of Christofides e
t
al. (1979) with 199 customers.
Probabilistic diversification and intensification in local search for vehicle routing 5
Diversification and intensification algorithm :
Initialization :
(a) Generate I different initial solutions with the local
search.
(b) Label each tour with the value of the solution to
which it belongs.
(c) Remove the tours having only one customer.
(d) Insert the remaining tours in a set T of tours.
(e) Sort T by increasing values of labels.
Diversification and intensification (to repeat until a
stopping criterion is satisfied) :
(1) Set T’:=T, S := ∅.
(2) While T’ ≠∅, repeat
(2a) Choose t ∈ T’, probabilistically, based on its
current relative evaluation.
(2b) Set S := S ∪ {t}.
(2c) Remove from T’ all the tours including one
or several customers belonging to t.
(3) If some customers are not covered by the tours of
S, construct a feasible solution S’, including them,
using the partial solution S.
(4) Improve with the local search the solution S’.
(5) Label the tours of the improved solutions, remove
tours with only one customer, insert the remaining
tours in T, sort T as in steps (b) to (e) of the
initialization.
To favour the generation of good solutions, the tours
are not uniformly chosen at step (2a), but the ith worst
tour of T’ has a probability of 2i/(|T’||T’ + 1|) of being
chosen. For practical reasons, it is necessary to limit the
size T to a value L. After each sorting of T, the last |T|–
L tours are removed from T if |T|>L.
We note this procedure also contains an implicit
aspect of “combining” solutions since we construct new
ones out of components of previous ones. Hence in this
sense our approach embodies some of the spirit of
genetic algorithms (see, e. g., Holland (1976) and Davis
(1987)). This “implicit combination by intensification”
(by exploiting strongly determined and consistent varia-
bles), whose origins are approximately
contemporaneously with those of genetic algorithms
provides a useful counterpart to combination by
“genetic” operators.
The working principles of this algorithm may be
explained as follows : If I and L are large enough, the
initialization phase guarantees that various regions of
the solution space will be explored. The creation of
partial solutions at the beginning of the second phase
(step (2)) allows the search to extend the diversity of the
solutions visited.
As far as the process goes, T grows and the partial
solutions are more and more complete. At the end, they
are often feasible or even, in few cases, better than the
best solution found so far. After having performed the
second phase several times, the process automatically
intensifies the search in promising regions, since the
tours are not uniformly chosen at step (2a) and the worst
solutions of T are removed. Moreover, identical tours
are not removed from T and, more and more often, there
exist tours that are not modified by the local search. So,
the best tours of T are more and more frequently
extracted during the construction of the partial solution
and the search progressively changes from a diversifica-
tion to an intensification process.
This process may easily be parallelized. The steps (a)
and (4) (applying the local search) are those consuming
the most computation time. A master-slave approach is
convenient : The master process executes steps (b) to (e)
of the initialization phase and steps (1), (2), (3) and (5)
of the diversification and intensification phase. In addi-
tion I slave processes independently perform one local
search of step (a), then transmit one initial solution to the
master process and enter a loop in which they wait for a
partial solution from the master process, improve this
solution with the local search (step (4)) and finally
transmit the improved solution to the master process.
3.2. A post-optimization technique
When examining the empirical behaviour of this
procedure, it turns out that during the diversification and
intensification phase, the process sometimes succeeds in
improving the best solution directly from the solution
constructed at step (2) before applying the local search
of step (4). This means that, with the tours contained in
the set T, it may be possible to build solutions better than
those already produced.
Let c
j
be the length of the j
th
tour of T (j = 1,…, |T|)
and, for i =1,…,n and j = 1,…, |T|:
Then, the best solution that can be built using tours
of T may be found by solving the following set parti-
tioning problem :
The assignment x
j
= 1 indicates that the j
th
tour of T
is chosen. By extension, the solution space may be
a
ij
1 if customer ij
th
tour of T∈
0Otherwise
=
min c
j
x
j
j 1=
T
∑
s. t. a
ij
x
j
j 1=
T
∑
1= i 1 … n,,=
x
j
01{,}∈ j 1 … T,,=
