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Kansou, Ali; Yassine, Adnan
Splitting algorithms for the multiple depot arc routing problem: application by ant colony optimization
International Journal of Combinatorial Optimization Problems and Informatics, vol. 3, núm. 3, septiembrediciembre, 2012, pp. 20-34
International Journal of Combinatorial Optimization Problems and Informatics
Morelos, México
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Available in: http://www.redalyc.org/articulo.oa?id=265224452002
Red de Revistas Científicas de América Latina, el Caribe, España y Portugal
Sistema de Información Científica
Kansou, Ali; Yassine, AdnanSplitting algorithms for the multiple depot arc routing problem: application by ant colony optimizationInternational Journal of Combinatorial Optimization Problems and Informatics, vol. 3, núm. 3, septiembre-diciembre, 2012, pp. 20-34International Journal of Combinatorial Optimization Problems and InformaticsMorelos, México
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International Journal of Combinatorial Optimization Problems and Informatics,
ISSN (Electronic Version): 2007-1558editor@ijcopi.orgInternational Journal of CombinatorialOptimization Problems and InformaticsMéxico
www.redalyc.org
Non-Profit Academic Project, developed under the Open Acces Initiative
Splitting algorithms for the multiple depot arc routing problem: application by ant colony optimization
Ali Kansou, Adnan Yassine
Laboratoire de Math
é
matiques Appliqu
é
es du Havre, France. Institut Sup
é
rieur d’Etudes Logistiques (ISEL), France.ali.kansou@yahoo.fr and adnan.yassine@univ-lehavre.fr
Abstract.
This paper handles the Capacitated Arc Routing Problem with Multiple Depots (MD-CARP). The well known CARP problem consists of designing a set of vehicle trips, so that each vehicle starts and ends at the single depot. The MD-CARP on a mixed graph involves the assignment of tasks (arcs and edges), which have to be served, to depots and the determination of vehicle trips for each depot. The MD-CARP is NP-hard, to resolve it efficiently, two ant colony approaches are developed. The first proposed work is based on ant colony optimization (ACO) combined with an insertion heuristic: the ACO is used to optimize the order of insertion of the tasks, and the heuristic is devoted to inserting each task in the solution. A generalization for the splitting method of Ulusoy is incorporated with the ant colony optimization in the second approach. Computational results on benchmark instances show the good quality of the proposed methods and the superiority of the first algorithm compared to the second method on the large instances. Another experimental results of instances with known optima suggest the performance of the two methods is significantly better than the methods of literature.
Keywords:
Capacitated arc routing problem, MD-CARP problem, insertion heuristic, ant colony optimization.
1 Introduction
The purpose of this paper is to present two ant colony approaches for an important extension of the classical Capacitated Arc Routing Problem (CARP): the Multi-Depot Capacitated Arc Routing Problem (MD-CARP) on a mixed graph. The MD-CARP problem is defined on an undirected graph
G = (V
n
∪
V
d
, E, A)
, where V
n
is a no-depot nodes set,
V
d
is a depot nodes set,
E
is an edge set of
G
and
A
is an arc set of
G
. We call by
(i, j)
a link in
E
∪
A
. The traversal cost
c
ij
and the demand
q
ij
of each link
(i, j)
∈
E
∪
A
are known in advance. Each link which has a strictly positive demand is called a ‘task’ (required edge or required arc), and it is an element of a set
R
of required links,
R=
{
(i, j) E
∈ ∪
A; q
ij
>0
}
.
We assume that
m
r
is the cardinal of
R
(|
R|
=
m
r
) and
n
d
is the number of depots. A fleet of identical vehicles with limited capacity
Q
is located at the depots. All the depots of MD-CARP problem have unlimited capacity. MD-CARP consists of determining a set of vehicle routes of minimal cost satisfying the following conditions: 1. each vehicle route starts and ends at the same depot to which it is assigned, 2. each task is served once and exactly once by one vehicle, and finally 3. the total demand of the every route does not exceed the capacity
Q.
In this paper, we focus on the MD-CARP problem with the following three points: (1) G is a mixed graph, then we have two sets of links (a link is an arc or edge and a required link is called task); (2) a cost to visit a task and another cost for serving it; (3) windy edge: an edge have different cost by each direction. Tasks do not have to be visited at all and further, any link (required or non-required) can be visited any number of times, if it needs to be traversed more often to ensure that all required edges and arcs are served. The total cost of routing is minimized and the set of all vehicle routes forms a feasible MD-CARP solution. The MD-CARP problem is NP-hard and arises naturally in garbage disposal, snow removal and winter gritting, emergency, routing of street sweepers, school bus service, electric power lines and inspection of gas pipelines, etc.
© International Journal of Combinatorial Optimization Problems and Informatics, Vol. 3, No. 3,Sep-Dec 2012, pp. 20-34. ISSN: 2007-1558.
Received April 26, 2010 / Accepted Dec 31, 2011Editorial Académica Dragón Azteca (EDITADA.ORG)
colony optimization. IJCOPI Vol. 3, No. 3, Sep-Dec 2012, pp. 20-34. ISSN: 2007-1558.
Originally, the CARP was proposed by Golden and Wong [15] and we refer to the survey of Assad and Golden (1995) for more details about the CARP problem. Among the used meta heuristics, we cite the simulated annealing (Li, 1992 and Eglese, 1994), the tabu search (Belenguer [3] et al., Hertz et al. [17] and Brand
â
o et al. [6]), the scatter search of Greistorfer [16], the memetic algorithms by Prins and Lacomme [20]. Belengeur et al. [3] present a lower and upper bounds and a new lower bounds was proposed by Whlk [30] on the mixed CARP. Recently, the algorithms based on ant colony optimization are applied effectively
ϕ
on the CARP problem by Lacomme [21], and on the mixed CARP problem by Bautista [2]. To resolve the multi-depot vehicle routing problem (MD-VRP) problem, several algorithms are available in the literature. Laporte [9] studied a family of multi-depot asymmetrical problems and have developed exact branch-and-bound algorithms and tabu search algorithms. The early heuristics developed on the MD-VRP problem, based on simple construction and improvement procedures, have been developed by Tillman [27], Gillett and Johnson [12], Golden et al. [15]. A tabu search heuristic is developed by Renaud et al. [26] and Cordeau et al. [9] which is probably the best known algorithm. Two hybrid genetic algorithms (GA) are also developed by Ho et al. [18] where the MD-VRP problem is divided into three sub-problems: Assigning customers to depots (grouping problem), assigning customers in each depot to routes (routing problem) and sequencing each route in every depots (scheduling problem). There exist heuristics for the CARP with Intermediate Facilities CARPIF, we refer to the works of Ghiani et al. [13], Polacek et al. [25] based on the variable neighborhood search algorithm VNS and to the Ghiani et al. [14] for the arc routing problem with intermediate facilities under capacity and length restrictions CLARPIF. In 2006, Bouhafs et al. [5] proposed a combination of simulated annealing (SA) and ant colony system (ACS) for the capacitated location-routing problem CLRP, where the (SA) searching the good facility configuration and the (ACS) constructs a good routing that corresponds to this configuration.The MD-CARP problem has been studied by Amberg et al. [1], where a route first cluster-second algorithm is proposed. Wøhlk, in her dissertation [29] considered the undirected MD-CARP when the vehicles have various fixed costs. She considered the MD-CARP from a theoretical point of view and she gave a mathematical model with two lower bounds for the routing cost and for the fixed cost. Zhu et al. [33] have developed a hybrid genetic algorithm for the MD-CARP problem. Kansou and Yassine [19] have used two evolutionary algorithms to resolve the same problem, where the network is an indirected graph and using dead heading cost only. The first one is based on the ant colony optimization and an insertion heuristic, the second one is a genetic algorithm based on an specific cross over. An evolutionary algorithm approach to the MD-CARP with time-limited service was developed and evaluated on 20 instances, with known optima, with up 100 nodes and 360 arcs (Xing et al. [31]).Due to the complexity of the MD-CARP problem, the optimal resolution is extremely expensive. In this work, to treat the problem effectively, we propose two different methodologies based on the Ant Colony Optimization ACO. The first combines the ACO with an insertion heuristic, called hybrid ant colony optimization (
HACO
). The second one is based on the generalization of splitting method to the MD-CARP problem combined with the ACO, called directed ant colony optimization with splitting (
DACOS
). The splitting method of CARP problems (with one depot), called
split
, developed by Lacomme et al. [20], is generalized in this work to adapt the multi-depots cases (MD-CARP problems) by a procedure, called
md-split.
The good results of the ACO obtained on the problems: bin packing problem and PCARP problem, where the resolution by ACO is equivalent and appropriate to the MD-CARP problem on a mixed graph, motivated us to apply the same principle to MD-CARP problem. An ACO is used for the bin packing problem by Levine and Ducatelle [23] and Yalaoui and Chu [32], so that the order in which objects are placed into bins is optimized by the ACO and an insertion method is used to insert the objects into bins. Kansou and Yassine (2009) have developed an ACO method combined with an heuristic method for the periodic CARP (PCARP) problem and this hybrid method outperforms the genetic algorithm developed by Lacomme et al. [20]. The same hybrid method is applied on the indirected MD-CARP in [19] by the same authors. The main goal of this work is to develop the splitting method for the MD-CARP and to prove their performance when integrate it with the evolutionary algorithms. The novelty of the present paper is that we apply two approaches to the resolution of MD-CARP problem, hybrid ACO method and ACO method with splitting MD-CARP solutions that have not been tried before.The MD-CARP problem is presented in Section 2 by an example. The mathematical model of the problem is proposed in Section 3. Resolution methods are developed respectively in Section 4 and Section 5. Section 6 is devoted to computational results: the comparison between the two methods and the algorithms of literature is presented after a preliminary testing is done on multi-depot data that we adapted and on the instances with known optima. Finally, our conclusion is presented in Section 7.
Ali Kansou and Adnan Yassine / Splitting algorithms for the multiple depot arc routing problem: application by ant 21
22
2 An MD-CARP example solution
For what follows, we transform
G
into a graph
=
Γ
(V, W, B)
, where
V
is a new vertex set (the set of tasks). Each task
p
∈
V
represents a required link arc or an one direction of a required edge of
G.
If
p
is a direction of
edge
, the other direction for the same task is noted by
p
∈
V
. If
p
is an
arc
link, we take
p
=
0
. Each task
p
∈
V
has a routing cost
c
p
and a demand
q
p
> 0.
W
is a set of no depot nodes.
B
is a set of the fictitious links that are evaluated by the costs of shortest paths calculated by Dijkstra algorithm and linking nodes of
V
∪
W
. Then, for each link
(p,q)
∈
B
we associate a cost
D(p,q)
and the MD-CARP problem will be seen as a MD-VRP problem except that for the MD-CARP problem, we take into account, both directions of a task (see Belenguer et al. [3] on the periodic CARP). From each task pair,
{
p, p
}
, exactly one direction is selected to appear in a MD-CARP solution, and each task is chosen exactly once, with no task left out of a solution.In Figure 1, an MD-CARP example solution with eleven tasks (thick lines, the tasks are only edges in this example) identified by indexes from 1 to 11 and three depots (
D1
,
D2
and
D3
), is considered. (
x
,
y
) coordinates are given for the depots (in brackets under each depot) and for each task (at its extremities), and the demand associated with each task
t
is given in brackets. For example, (1, 7) and (1, 10) are the
(x, y) coordinates and (3) is the demand of task 3 (
Q
= 7). Thin lines represent the shortest paths linking tasks between them and the tasks with the depots.To construct initial solutions for our two methods, at the first level, each task is assigned to the nearest depot. If one end of a required edge
e
is closest to one depot
Di
, and the other end of the same required edge
e
is closest to another depot
Dj
,
e
is randomly assigned to depot
Di
or
Dj
. In our example, tasks from 1–4 are closest to depot
D
1, so they are assigned to a depot, tasks from 5–6 are assigned to depot
D2
, because
D2
is their closest depot, and finally tasks from 7-11 are assigned to depot
D3
. At the second level, the tasks in each group are divided into routes using an insertion algorithm to minimize the costs (see Section 4.2). We have two routes associated to depot
D1
which are:
D1
−
1
−
2
−
D1
and
D1
−
4
−
3
−
D1
; we have one route associated to depot
D
2 which is:
D2
−
5
−
6
−
D2
and we have two routes associated to depot D3 which are:
D3
−
7
−
8
−
9
−
D3
and
D3
−
11
−
10
−
D3
.
Figure 1.
An MD-CARP example solution with 11 tasks (
Q
=7).
3 Mathematical model of MD-CARP
In this section, we give a mathematical model for the mixed MD-CARP. Our model formulation uses two binary variables:
x
ijk
=1 if the node
j
is visited after
i
by the vehicle
k
(or route
k
) and equal 0 otherwise.
l
ijk
=1 if the vehicle
k
serves the link
(i, j)
from
i
to
j
and equal 0 otherwise. Suppose that
w
ij
is the cost to serve a task
(i, j)
∈
A
r
∪
E
1r
∪
E
2r
, where
E
1r
and
E
2r
are respectively the set of all the first and the second directions of required edges (
E
r
=E
1r
∪
E
2r
and
E
1
∪
E
2
=E
). Then, we take
E
r
and
A
r
respectively the sets of all required arcs or edges, called the “tasks”. We denote by
K
the set of all vehicles.
L
is the new set of
colony optimization. IJCOPI Vol. 3, No. 3, Sep-Dec 2012, pp. 20-34. ISSN: 2007-1558.Ali Kansou and Adnan Yassine / Splitting algorithms for the multiple depot arc routing problem: application by ant

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