EVOLUTIVE AND ACO STRATEGIES FOR SOLVING

THE MULTI-DEPOT VEHICLE ROUTING PROBLEM

H. I. Calvete

, C. Gal

and M. J. Oliveros

Dpto. de M

etodos Estad

ısticos, IUMA, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain

Dpto. de M

etodos Estad

ısticos, IUMA, Universidad de Zaragoza, Mar

ıa de Luna 3, 50018 Zaragoza, Spain

Dpto. de Ingenier

ıa de Dise

no y Fabricaci

on, Universidad de Zaragoza, Mar

ıa de Luna 3, 50018 Zaragoza, Spain

Keywords:

Multi-depot vehicle routing problem, Ant colony optimization, Genetic algorithm.

Abstract:

This paper addresses the multi-depot vehicle routing problem. This problem involves designing a set of routes

in order to deliver goods from several depots to a set of geographically dispersed customers. For solving this

problem, we propose two different approaches. Both have in common the use of an Ant Colony Optimization

algorithm to construct the routes from each depot. The approaches differ in the manner in which depots are

dealt with in terms of how customers are assigned to depots. In the ﬁrst method, called ACO-MDVRP, the

customer assignment process is controlled by the ant colony by adding a super-depot which is connected with

each depot by arcs with zero unit cost. The second method, called GA-MDVRP, is a hybrid algorithm in the

sense that an Ant Colony Optimization algorithm is embedded in a genetic algorithm. In order to construct a

feasible solution, the procedure uses a genetic algorithm to assign customers to depots. Then, under the given

data on each depot, the corresponding vehicle routing problems are solved by using Ant Colony Optimization.

1 INTRODUCTION

The vehicle routing problem (VRP) consists of de-

signing a set of routes for serving a number of geo-

graphically dispersed customers from a central depot.

The objective is to minimize total distance or total

travel time. In this problem customer demand is ﬁxed

and known in advance, vehicles are assumed to be

identical and cannot be overloaded, all routes start and

end at the depot and each customer is visited exactly

once by a single vehicle. Due to its theoretical and

practical importance, a lot of effort has been devoted

to solve this problem (Laporte, 2009). A lot of vari-

ants of the VRP have been proposed in the literature to

model more precisely real systems. The VRP and its

variants are NP hard and, usually, only small instances

can be solved to optimality within reasonable com-

putational time. Although some exact methods have

been proposed to solve the VRP, most techniques de-

scribed in the literature are heuristics or metaheuris-

tics which aim to provide cuasi-optimal solutions in

acceptable computational times.

A well known variant of the VRP is the multi-

depot vehicle routing problem (MDVRP). In this

problem a company owns several depots from which

it can serve the customers. There is a ﬂeet of vehicles

based at each depot. Each vehicle starts from one de-

pot, services a set of the customers assigned to that de-

pot and returns to the same depot. The objective of the

problem is to service all customers while minimizing

the total travel time. The MDVRP arises naturally in

a large variety of contexts and has considerable eco-

nomic importance. We refer the reader to (Renaud

et al., 1996) and (Cordeau et al., 1997) for more in-

depth discussion and details regarding the structure

and formulation of the MDVRP. Recent algorithms

for this problem or some variants have been proposed

in (Crevier et al., 2007; Giosa et al., 2002; Pisinger

and Ropke, 2007) and (Tansini and Viera, 2006).

When solving the MDVRP two decisions are

made: 1) how customers are assigned to depots and

2) which delivery routes are constructed. Therefore,

most methods developed in the literature to solve

the MDVRP propose a two-phase approach. In the

ﬁrst phase, customers are assigned to depots, usually

according to distance. In the second phase, routes are

constructed. Moreover, in order to improve the solu-

tion, different strategies of either relocating customers

in a route or transferring customers from a route to an-

other can be applied.

In this paper we propose two different meta-

heuristic procedures for solving the MDVRP: ACO-

I. Calvete H., Galé C. and J. Oliveros M..

EVOLUTIVE AND ACO STRATEGIES FOR SOLVING THE MULTI-DEPOT VEHICLE ROUTING PROBLEM.

DOI: 10.5220/0003673400730079

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 73-79

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

MDVRP and GA-MDVRP. Both have in common the

use of an Ant Colony Optimization (ACO) algorithm

to construct the routes from each depot. ACO is one

of the most powerful metaheuristics developed for

solving complex and large combinatorial problems

(Dorigo and St

utzle, 2004; Dorigo and St

utzle, 2010).

It is inspired by real behavior of ants when looking

for food. Ants are able to communicate information

about food sources by laying a chemical pheromone

trail on the ground which guides other ants. First

ACO algorithms were proposed in the early 1990s for

solving the traveling salesman problem. Since then,

several variants and extensions have been developed

and a variety of optimization models have been solved

by using ACO algorithms. Concerning the MDVRP,

(Calvete et al., 2011) study a complex hierarchical

production-distribution planning problem which has

embedded a MDVRP. They propose an ACO algo-

rithm for solving the MDVRP in which a super-depot

is added and connected with each depot by arcs with

zero unit cost. (Yu et al., 2011) introduce also a virtual

central depot and transform the MDVRP into a VRP

with this virtual depot as the origin. Then they solve

this VRP by using a parallel improved ACO.

The ACO-MDVRP method proposed in this pa-

per follows previous line of thinking and introduces a

super-depot node connected at zero cost with the de-

pots. In this method, there is no initial assignment

of customers to depots. Instead, every ant starts at

the super-depot and selects the real depot to visit ﬁrst.

From this depot, a route is built which serves a set of

customers. This set of customers is assigned to that

depot. The GA-MDVRP method is based on the two-

phase methodology. Hence, a genetic algorithm deals

with the problem of assigning customers to depots.

Having made this decision, as many VRP as depots

are solved by using an ACO algorithm. The paper is

organized as follows. Section 2 states the problem.

In Section 3 the algorithms are developed. Section 4

goes on to analyze the computational performance of

the procedures on a set of benchmark problems. Fi-

nally, conclusions are presented in Section 5.

2 THE MULTI-DEPOT VEHICLE

ROUTING PROBLEM

Let G = (N ,A) be a directed network where N =

,. . .,d

,1,2, ... ,n} is the set of nodes and A =

{(i, j) : i, j ∈ N } is the set of directed arcs. Node

represents the location of depot l, l = 1, ..., L.

Nodes 1 to n represent customer locations. Each arc

(i, j) ∈ A refers to a direct connection from i to j and

has an associated non-negative travel time t

i j

. We as-

sume that no routes connect depots to each other, so

there is no arcs (i, j) with i, j ∈ {d

,. . .,d

We assume that there is a ﬂeet of identical vehi-

cles with ﬁxed capacity U. Each vehicle is located

at the depot from which its route starts. The route of

each vehicle starts and ends at the same depot. Ve-

hicles are assumed not to be overloaded, hence to-

tal demand of all customers on one particular route

must not exceed the capacity of the vehicle assigned

to this route. Moreover, for every vehicle route, its to-

tal route length must not exceed a given bound usually

due to working time restrictions.

We assume that a ﬁxed quantity of goods q

has

to be delivered at customer i. This quantity is known

when delivery routes are established. Delivery at cus-

tomer i requires a service time s

. This delivery cannot

be divide up amongst vehicles, i.e. every customer

is served by a single vehicle and it is visited exactly

once. Therefore, it is assumed that q

6 U for all

i = 1,.. .,n. Otherwise, the corresponding customer

could not be served.

Let R be a feasible solution, i.e. R is a set

of routes verifying above constraints. Let R =

,. . .,i

} be a route in R which starts

at depot d

and sequentially visits customers i

, j =

1,. . .,k.

In order to be a feasible route, working hours reg-

ulations have to be met:

+ s

+ · ·· + s

≤ t

(1)

where t

refers to the maximum duration of a route.

Moreover, bearing in mind that vehicles cannot be

overloaded:

+ q

+ · ·· + q

≤ U (2)

The MDVRP consists of determining a set of

routes that minimizes the total travel time. A route

involves information about the depot in which the ve-

hicle is located, the customers which are visited and

the order in which they are visited.

3 ALGORITHMS

Both algorithms developed in this paper to solve

the MDVRP use artiﬁcial ants to build feasible so-

lutions. As indicated above, a feasible solution is a

set of routes which visit all customers and satisfy the

constraints (1) and (2). Each route must start and end

at one depot. Each ant constructs a feasible solution

and, when moving, lays a pheromone trail. This trail

and the heuristic information are used to guide next

movements of ants based on a stochastic rule.

At the end of every iteration, a number of feasible

solutions are available. The size of the ant colony in

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

each iteration is one of the algorithm parameters and

is ﬁxed a priori. Then, total travel time of these so-

lutions is compared with the best-so-far solution and

the best one is selected. This new best-so-far solu-

tion is used to guide other ants by depositing a quan-

tity of pheromone on its arcs. The customer selection

process and the guide process are main characteristics

of ACO algorithms. Following subsections explain in

more detail these processes for each of the algorithms

proposed.

3.1 The ACO-MDVRP Algorithm

The idea of this method is to add a super-depot node

which is connected with each real depot by arcs with

zero unit cost. As a consequence, the original MD-

VRP is transformed into a VRP with the super-depot

acting as the central depot. Hence, the ﬁrst ‘cus-

tomer’ of every route is selected from the set of de-

pots {d

,. . .,d

}. Afterwards, the route successively

selects the following customer from the set of acces-

sible customers not yet visited. Only when the route

leaves the real depot, real customers can be added to

the route. Summarizing, in this algorithm a depot is

assigned to each route in the same way as customers.

There is no a priori assignment of customers to de-

pots.

3.1.1 Initialization

In order to construct an initial feasible solution, in

the implementation of the algorithm we have used

a variant of a nearest neighbor heuristic. First, the

real depot which is reached from the super-depot is

randomly selected. Second, the ﬁrst customer of the

route is randomly selected amongst the reachable cus-

tomers from the depot. Then, while it is possible to

add a customer to the route, the nearest customer to

the incumbent customer is selected according to travel

time. When it is not possible to add a new customer

to the route, the artiﬁcial ant returns to the same depot

from which the route started and then goes back to

the super-depot. From the super-depot the ant starts a

new route.

This process continues until every customer has

been visited. It is worth mentioning that this vari-

ant of the nearest neighbor heuristic, which involves a

random selection in the ﬁrst steps, allows us to build

different initial solutions. This feature will be used to

improve the performance of the algorithm.

Once an initial solution has been computed, main

parameters of the algorithm are initialized. Let

T the

total travel time of the initial solution and τ = 1/n

T .

In the implementation of the algorithm, the initial

pheromone trail is set to τ

i j

= τ, (i, j) ∈ A; τ

= τ,

l = 1,... ,L where index 0 denotes the super-depot.

The heuristic information is set to η

i j

= 1/t

i j

, (i, j) ∈

A; η

= 1, l = 1,.. .,L. Let M denote the size of the

colony of ants in each iteration.

3.1.2 Main Iteration

In every iteration, each of M ants constructs a feasi-

ble solution of the MDVRP. For this purpose, the ant

selects the next node (a real depot at the beginning, a

real customer afterwards) to be visited by applying the

pseudo-random-proportional rule. Let i be the incum-

bent node and N

be the set of reachable customers

from i not yet visited. Let Q be generated from a uni-

form random variable in the interval (0, 1). Then, if

Q 6 Q

, j ∈ N

is selected as:

j = argmax

h∈N

{τ

(η

)

}

where β is a parameter which determine the relative

inﬂuence of the heuristic information.

If Q > Q

the ant chooses to go to customer j ∈ N

with probability:

i j

(η

i j

)

∑

h∈N

(η

)

Pheromone trails are updated locally and globally

in order to guide future ants to get improved solutions.

Immediately after the ant has crossed the arc (i, j), the

pheromone trail of this link is updated by using the

expression

i j

= (1 − ξ)τ

i j

+ ξτ, 0 < ξ < 1

where the parameter ξ reﬂects the tendency towards

exploration. The goal of this local pheromone trail

updating is to make this arc less desirable for the fol-

lowing ants in order to encourage the exploration of

arcs not yet visited.

Having built the M ants their feasible solution, the

best-so-far solution is selected. At this moment, it is

very useful to apply local search methods to improve

the quality of the solution selected. We have imple-

mented the 2-opt intra routes and the 2-opt inter route

procedures (Br

aysy and Gendreau, 2005). To end the

iteration, a global updating of the pheromone trail is

performed. Some pheromone is added to the arcs of

the best-so-far solution by using the expression:

i j

= (1 − ρ)τ

i j

+ ρ

τ, 0 < ρ < 1

where

τ is the inverse of the total travel time of the

best-so-far solution. The parameter ρ reﬂects the

speed of the pheromone evaporation.

EVOLUTIVE AND ACO STRATEGIES FOR SOLVING THE MULTI-DEPOT VEHICLE ROUTING PROBLEM

In order to improve the convergence of the algo-

rithm and avoid stalling problems, we also propose a

reinitialization of the algorithm after a number of it-

erations without decreasing the total travel time. Tak-

ing into account that the initial solution will be usu-

ally different due to the variant of the nearest neighbor

heuristic implemented, the reinitialization process im-

proves the exploration of different feasible solutions.

3.2 The GA-MDVRP Algorithm

In contrast to the previous algorithm, in the GA-

MDVRP algorithm customers are assigned to de-

pots before the construction of routes starts. More-

over, this process is controlled by a genetic algo-

rithm (GA). GA are stochastic search techniques in-

spired by natural biological evolution. They were in-

troduced by (Holland, 1975). When applying GA to

solving a problem, each solution is encoded as a string

of symbols which is called chromosome. Each posi-

tion in the string is a gene and its value is the allele

value. Each chromosome has a ﬁtness value associ-

ated which measures its quality, usually in terms of

the objective function.

To start the algorithm, an initial population of

chromosomes is generated and the ﬁtness of each

chromosome is computed. The size p of this popula-

tion is a parameter of the algorithm. In each iteration,

a new population is generated. New chromosomes are

formed by combining chromosomes from the current

population (parents) using a crossover operation or by

modifying a chromosome using a mutation operation.

After evaluating the ﬁtness of the resulting chromo-

somes, some of the chromosomes of the old popula-

tion and some new chromosomes (offspring) are se-

lected to form the new population. Some variants of

the selection procedure have been proposed in the lit-

erature. The most common is the elitist strategy in

which the best p chromosomes in terms of ﬁtness are

kept from one generation to the next. The algorithm

proceeds by building populations until the stopping

condition is met.

To solve the MDVRP, we have developed a GA

aiming to ﬁnd good customer assignments to depots

that lead to good solutions of the MDVRP in accept-

able computational times. The idea underlying the

proposed GA is to associate chromosomes with as-

signments. A gene corresponds to a customer and

the allele value indicates the depot which serves him.

The ﬁtness of a chromosome is the value of the to-

tal travel time of the solution of the MDVRP obtained

by putting together the best solutions provided by an

ACO algorithm applied to each of the L VRPs arisen

when considering each depot and its assigned set of

customers.

3.2.1 Initialization

The initial population is formed by p chromosomes.

Several routines for selecting the members of this

initial population can be envisaged. For instance, a

chromosome can be constructed by assigning each

customer to its nearest depot. Another possibility is

to assign the customer to a depot at random. This

possibility could also limit the number of customers

assigned to each depot or not. Initially, two rou-

tines were implemented in the algorithm: Nearest

depot and completely random assignment. The ﬁrst

one usually gives a good chromosome from the ﬁt-

ness point of view. However, random assignment

provides very bad chromosomes. Hence, in order

to have more promising chromosomes in the initial

population we implemented a routine in which ev-

ery customer has the possibility of being assigned

only to a set of depots which are close to him. In

order to deﬁne the closeness of a depot, let t

max

max{t

: (i,d

) ∈ A;i = 1,... ,n;l = 1,. ..,L} and

min

= min{t

: (i,d

) ∈ A; l = 1, ..., L}, i = 1,. ..,n.

The depot d

is close to the customer i if

−t

min

max

Then, the customer i is randomly assigned to one of

the depots close to him.

To evaluate the ﬁtness of each chromosome, we

compute a good solution of the MDVRP bearing

in mind the customer assignment determined by the

chromosome. For this purpose, we propose to solve

the VRP associated to each depot by using an ACO al-

gorithm. The description of this algorithm is similar

to the one given in Section 3.1. Instead of a super-

depot, L depots and n customers, there is one depot,

say d

, and n

customers (those previously assigned to

). Let T

be the total travel time of the best solu-

tion provided by the ACO algorithm after solving the

VRP corresponding to the depot d

. The ﬁtness of a

chromosome C is

f (C) =

∑

l=1

3.2.2 Main Iteration

The GA proceeds by performing crossover, mutation,

evaluation and selection, until the stopping condition

is met. In each generation a population formed by p

chromosomes is maintained. In the crossover process,

chromosomes are selected from the source population

and combined to form offspring which are potential

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

members of the successor population. A chromosome

of the current generation is selected for the crossover

operation with probability p

. The parents selected

are taken in pairs to provide offspring. Let us assume

that there are 4 depots and 10 customers. Let the two

parents selected be:

Parent 1:



1 2 1 1 3 4 1 2 4 3



Parent 2:



3 3 1 2 1 4 1 3 1 3



Then a location is randomly selected (for example the

seventh one) and the right-hand indices are exchanged

in pairs:

Parent 1:



1 2 1 1 3 4 1 3 1 3



Parent 2:



3 3 1 2 1 4 1 2 4 3



After crossover, the mutation operation is carried

out. A chromosome of the current population is se-

lected for the mutation operation with probability p

Once it has been selected, several customers are ran-

domly selected from the chromosome and a depot is

randomly assigned to each of them.

After mutation, the ﬁtness of the new chromo-

somes obtained from crossover and mutation is com-

puted. The offspring are added to the current popula-

tion and the whole set of chromosomes is passed on

to the selection step. We use the elitist strategy which

selects the best p chromosomes of the current set of

chromosomes available (current population plus off-

spring resulting from the crossover and mutation op-

erations). These chromosomes form the population of

the next iteration.

The population size, the probability of crossover,

the probability of mutation, the number of genes af-

fected by the mutation operation and the stopping

condition are control parameters of the algorithm

which have to be set in the beginning.

4 COMPUTATIONAL

EXPERIMENT

The performance of the algorithms has been

tested on the set of MDVRP benchmark instances

of (Cordeau et al., 1997), which can be found

in http://neumann.hec.ca/chairedistributique/data/.

The ten problems vary in size from 48 to 288 retailers

and have 4 or 6 depots. Table 1 displays the charac-

teristics of the problems and the total travel time of

the best known solution http://neo.lcc.uma.es/radi-

aeb/WebVRP.

The purpose of the experiment was twofold. On

the one hand, to check the algorithm efﬁciency in

terms of the closeness of the best solution provided

by each algorithm to the best known solution. On

the other hand, to compare both algorithms aiming

Table 1: Description of problems and total travel time of the

best known solution.

Problem # of customers # of depots f

1 48 4 861.32

2 96 4 1288.37

3 144 4 1782.58

4 192 4 2072.52

5 240 4 2343.66

6 288 4 2675.16

7 72 6 1085.61

8 144 6 1666.60

9 216 6 2153.10

10 288 6 2811.49

to conclude which one seems to have a better per-

formance. The experiments were performed on a PC

Core Quad 6600 at 2.4 GHz having 4 GB of RAM un-

der Windows 7. Both algorithms were coded in C++

using Borland Builder. For each problem ﬁve runs

with each algorithm were done.

Based on pilot testing, the general parameters of

the ACO-MDVRP algorithm have been set to ξ = ρ =

0.1, β = 2 and Q

= 0.5. Taking into account that

better solutions are obtained when combining ACO

algorithms with local search algorithms, we have im-

plemented the 2-opt intra routes and the 2-opt inter

routes local search procedures. These methods are ap-

plied to the best-so-far solution at the end of each iter-

ation. Reinitialization of the algorithm, as explained

in 3.1.2 is carried out after two iterations without ob-

jective function improvement. The number M of ants

in each iteration has been set to 50. The termina-

tion condition of this algorithm is established in terms

of computing time. In order to compare both algo-

rithms, the ACO-MDVRP algorithm is assigned for

each problem a computing time similar to that used

by the GA-MDVRP algorithm in the same problem.

Regarding the GA-MDVRP algorithm, each run

consists of eight iterations, the population size is p =

4, the probability of crossover is p

= 0.3, the proba-

bility of mutation is p

= 0.1 and the mutation oper-

ation is carried out on 5 genes (customers) randomly

selected. The general parameters of the ACO algo-

rithm applied to solve the VRP associated to each de-

pot are set to ξ = ρ = 0.1, β = 2 and Q

= 0.5 and lo-

cal search procedures are applied as explained above.

In this case, M = 4 ants are used in each iteration and

the ACO algorithm ends after 2 minutes of computing

time.

Tables 2 and 3 display the results of the experi-

ment. For every problem, the ﬁrst column indicates

the number of the problem. Columns two and three

refer to the objective function values, that is to say

EVOLUTIVE AND ACO STRATEGIES FOR SOLVING THE MULTI-DEPOT VEHICLE ROUTING PROBLEM

Table 2: Experimental results of the ACO-MDVRP algo-

rithm: Total travel time of the best solution f

, average of

the ﬁve runs

f and percentage of deviation %Dev.

Problem f

f %Dev

1 872.36 886.42 1.28

2 1366.34 1381.52 6.05

3 1926.05 1931.99 8.05

4 2208.90 2246.57 6.58

5 2543.51 2551.21 8.53

6 2934.56 2971.38 9.70

7 1126.78 1142.91 3.79

8 1760.57 1784.43 5.64

9 2334.86 2365.47 8.44

10 3088.71 3116.20 9.86

the total travel time, obtained in the ﬁve runs of each

instance. Column two shows the total travel time of

the best solution provided by the corresponding algo-

rithm and column three displays the average of the

ﬁve runs of the instance. In column four the best so-

lution obtained is compared with the best known so-

lution by using the expression

%Dev =

− f

× 100

where f

is the best objective function value provided

by the algorithm and f

is the objective function

value of the best known solution.

From the comparison of both tables, we can de-

rive that GA-MDVRP shows a better performance

than ACO-MDVRP. In fact, GA-MDVRP provides

a better solution than ACO-MDVRP for all prob-

lems. Moreover, GA-MDVRP provides good solu-

tions to all problems since only 3 out of 10 problems

(which are the larger ones) show a deviation higher

than 5.5%, but less than 9%, from the best known so-

lution. Figure 1 shows the percentage of deviation

for both algorithms. We can see that it is higher for

ACO-MDVRP than GA-MDVRP. Moreover, %Dev

increases in both algorithms with the size of the prob-

lem, specially with the number of customers.

Finally, it is worth making some comments on the

computing time invested by both algorithms. In or-

der to be able to compare both algorithms, we carried

out the experiment with the GA-MDVRP algorithm,

took the average time invested in the ﬁve runs and as-

signed the ACO-MDVRP algorithm a similar comput-

ing time for each run. However, we observed that the

best solution of each run of the ACO-MDVRP algo-

rithm was provided in much less time than assigned.

This suggests that we could have made fewer itera-

tions when applying this algorithm, thus saving the

time invested, without, in general, losing accuracy.

Table 3: Experimental results of the GA-MDVRP algo-

rithm: Total travel time of the best solution f

, average of

the ﬁve runs

f and percentage of deviation %Dev.

Problem f

f %Dev

1 861.32 880.82 0.00

2 1310.20 1339.54 1.69

3 1872.02 1884.23 5.02

4 2177.19 2203.61 5.05

5 2517.14 2534.88 7.40

6 2913.02 2945.42 8.89

7 1098.56 1110.16 1.19

8 1719.78 1745.84 3.19

9 2268.83 2280.82 5.37

10 3028.47 3064.58 7.72

Figure 1: %Dev versus Problem.

5 CONCLUSIONS

The MDVRP is a generalization of the standard VRP

in which more than one depot is available to deliver

goods to a set of geographically dispersed customers.

The objective of the problem is to design a set of

routes for serving the customers while minimizing the

total travel time. Each vehicle starts from one depot,

services a set of customers and returns to the same

depot. In this paper we have develop two algorithms

for solving this problem: ACO-MDVRP and GA-

MDVRP. They differ essentially in the way in which

customers are assigned to depots.

The ACO-MDVRP algorithm does not execute an

a priori assignment of customers to depots. By in-

troducing a super-depot, the ACO-MDVRP algorithm

uses an ACO approach in which the assignment of

customers to depots is controlled by the colony of ants

and is made as the route is being built.

On the contrary, the GA-MDVRP algorithm is a

hybrid algorithm which uses GA to assign a priori

customers to depots. A chromosome is an assignment

of customers to depots. The ﬁtness of a chromosome

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

is deﬁned as the total travel time of the routes which

serve the customers assigned to each depot. For the

purpose of computing the ﬁtness, an ACO algorithm

is applied to design the routes.

The results of the computational experiments car-

ried out using a set of MDVRP benchmark instances

indicate a better performance of the GA-MDVRP al-

gorithm. Moreover, this algorithm provides good re-

sults in terms of accuracy of the solution.

ACKNOWLEDGEMENTS

This research work has been supported by the Span-

ish Ministry of Education and Science under grant

MTM2010-17559

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EVOLUTIVE AND ACO STRATEGIES FOR SOLVING THE MULTI-DEPOT VEHICLE ROUTING PROBLEM