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Kansou, Ali; Yassine, AdnanSplitting algorithms for the multiple depot arc routing problem: application by ant colony optimization

International Journal of Combinatorial Optimization Problems and Informatics, vol. 3, nm. 3, septiembre-

diciembre, 2012, pp. 20-34

International Journal of Combinatorial Optimization Problems and Informatics

Morelos, Mxico

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International Journal of Combinatorial

Optimization Problems and Informatics,

ISSN (Electronic Version): 2007-1558

[email protected]

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Splitting algorithms for the multiple depot arc routing problem:

application by ant colony optimization

Ali Kansou, Adnan Yassine

Laboratoire de Mathmatiques Appliques du Havre, France.

Institut Suprieur dEtudes Logistiques (ISEL), France.

[email protected] and [email protected]

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 = (Vn Vd, E, A), where Vnis a no-depot nodes set, Vdis a depot nodes set,E is anedge set of G andAis an arc set of G. We call by (i, j)a link inEA. The traversal cost cijand the demand qijof each link (i, j)EA 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; qij>0}. We assume that mris the cardinal ofR (|R|=mr) and ndis 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 servedonce 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, 2011

Editorial Acadmica Dragn Azteca (EDITADA.ORG)


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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 Brando 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 W hlk [30] on the mixed CARP. Recently, the algorithms based on ant colony optimization are applied effectively

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) constructsa 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. Whlk,

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 todevelop 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

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2 An MD-CARP example solution

For what follows, we transform G into a graph = (V, W, B), where V is a new vertex set (the set of tasks). Each taskpVrepresents a required link arc or an one direction of a required edge of G. Ifpis a direction of edge, the other direction for the

same task is noted by p V . Ifpis an arclink, we take p=0 . Each taskpV has a routing cost cpand a demand qp> 0. Wisa 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 VW. Then, for each link (p,q)B we associate a costD(p,q)and the MD-CARP problem willbe 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,D2andD3), 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 tis 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 eis closest to one depotDi, and the other end of the same required edge eis closest to another depotDj, e is

randomly assigned to depotDi orDj. In our example, tasks from 14 are closest to depotD1, so they are assigned to a depot,

tasks from 56 are assigned to depotD2, becauseD2is their closest depot, and finally tasks from 7-11 are assigned to depotD3.

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 depotD1which are:D112 D1andD14 3D1; we have one route

associated to depotD2 which is:D25 6 D2and we have two routes associated to depot D3 which are: D378 9 D3

andD311 10D3.

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:

xijk=1 if the nodejis visited after iby the vehicle k(or route k) and equal 0 otherwise. lijk=1 if the vehicle kserves the link (i, j)

from i to jand equal 0 otherwise. Suppose that wijis the cost to serve a task (i, j) Ar E1r E

2r , whereE

1r andE

2rare

respectively the set of all the first and the second directions of required edges (Er=E1rE

2randE1E2=E). Then, we takeErand

Arrespectively the sets of all required arcs or edges, called the tasks. We denote by Kthe set of all vehicles.Lis the new set of




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links which contains all the arcs in GandLris the new set of tasks (new required arcs to be served). Then we will take L=E1 E2AandLr=E

1rE

2r Ar.

The objective function (1)minimizes the sum of the service cost and visiting cost without service cost. Constraints (2)assure

that each link edge is assigned to a single route and by unique direction. Constraints (3)require that each link arc must be

assigned to a single route. (4)are the flow conservation constraints. Constraints (5)ensure that each vehicle can be used at most

once from a single depot. Limits on vehicle capacity are imposed by constraints (6). Constraints (7)ensure that each required

link can not be served unless it is visited. Constraints (8)are the standard sub-tours elimination constraints, whereLr(S)={(i, j)Lr; i S and j S}. Experiments with a commercial linear solver (Cplex) showed that this linear programming model can besolved to optimality only for very small-sized instances. These disappointing results are not surprising since the MD-CARP is

a generalization of the CARP, which is already NP-hard. Due to the computational complexity of the MD-CARP, recent

research has been focused on developing heuristic algorithms. The proposed mathematical model is the following model:

Min kKi , jLr w ij l ijkK i , jL c ij x ijkl ijk (1)

s.t.

kKl ijkl jik=1

i , jEr (2)

kK lijk=1 i , j Ar (3)

i , j L xijk= j ,i L x jik

kK , i Vn

Vd (4)

wVd

jVnxwjk1

k K (5)

i , j Lr qij lijkQ k K (6)

xijkl ijk kK , i , jLr (7)

i , j LSx ijkl rsk kK , r , sLrS, SVn ;S2 (8)xijk, l ijk{0,1} kK , wVd, i , jVnVd (9)

4 An hybrid Ant colony SystemHACOfor the MD-CARP

The meta heuristic Ant Colony Optimization (ACO) was first proposed by Colorni et al. [8], Dorigo et al. [11] to solve thetraveling salesman problem (TSP). It is inspired by the behavior of real ant colonies. Real ants searching for food, are capable to

find the shortest path between a food source and their nest by exchanging information via a quantity of a chemical substance,

called pheromone. While walking on the roads between the nest and the food source, the pheromones are deposed by ants. As

other ants observe the pheromones trail and are attracted to it, the road is marked again and will attract even more ants to follow

the trail. In the first approachHACOto solve the MD-CARP problem, we propose an hybrid approach based on the ant colony

optimization and an insertion method. Kansou and Yassine [19] are developed the HACOmethod to resolve the MD-CARP

problem but in the case of an indirected graph. In this section, we generalize the method to a mixed graph. It is based on two

phases: sequencing followed by insertion of tasks. The insertion heuristic is then used to insert every task into the current

solution and the ACO is responsible for sequencing the tasks (or for find the order to insert the tasks). HACOstarts with an

initial solution obtained by an insertion heuristic (see Section 4.1.2). The two phases ofHACOwill be tackled repeatedly until

the algorithm termination condition is done.

4.1 Initialization

The phase Initialization is devoted to construct an initial solution of HACO. The method is called BDBP(Best Depot Best

Position). Initializationconsists of three followed stages:assigning,insertion,regularization . assigning assigns tasks to their

nearest depot. Next, for each depot independently, insertionwill sequence the tasks (on their best positions) assigned to it into a

single trip (a giant trip), starting and ending with the depot in question. Then, the regularizationstage will firstly, using a

procedure called split, divide up the giant trip associated with each depot into individual routes that do not violate vehicle

capacity. Finally the total cost for the solution will then by computed by the same procedure (see Section 4.1.3).




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4.1.1 Theassigningstage

In this stage, each task is assigned to the nearest depot (one depot) such that we have ndsetsNYd (d=1, , nd), where each task is

in the setNYd if it is the closest to depot damong all depots. Note:NYstands for not yet inserted, because members of the set

NYhave not yet been inserted into a sequence for a trip or a route. Each task-edge{p , p} is assigned to one setNYd , where dis

the closest depot to p or p . Then the end of this stage, we haveV= dW NYd , dW NYd= and

p NYdp NY

d dW .

4.1.2 The insertionstage

Secondly, following the assigningstage, comes the insertionstage. The tasks assigned to a given depot will all be sequenced

into a single giant trip, which starts and ends at the depot in question, and visits all the tasks-edges in one direction and all

tasks-arcs. A giant trip means a trip starts and ends at the same depot and not satisfied the constraint capacity. At the next stage

(regularizationstage), individual routes will be constructed, furthermore called by routes and they not violated the constraint

capacity. Each individual route of a giant trip starts and ends at the same depot allocated by the assigningstage. Then the term

giant trip is used to define a candidate solution before applying the regularizationstage. Each task appears in exactly one giant

trip associated with the depot to which the task is assigned. Thus, the giant solution is made up of a number of giant trips, one

giant trip for each depot.

A giant solution of MD-CARP problem is represented by S= S1, ..., Snd . Each Sd is a sub-solution of Swhich represents a

giant solution associated to depot d. Then, Sd contains the node depot dat the beginning and the ending, and between these, the

tasks sequence already inserted into the solution. For example, in Figure 1 we have nd=3, three sub-solutions

S1 =D11 24 3D1 , S2=D256 D2 and S3=D37891 1 1 0 D3 and 5 routes:

R1=D112D1 , R2=D143D1 , R3=D256D2 , R4=D3789D3 and

R5=D31110D3 . We denote by NYd the set of tasks that will be inserted in a solution Sd (not yet inserted in a

solution). If i indicates the position of a given taskSd , i in a sub-solution Sd then the best insertion cost Ipd of a taskp in aroute d is given by the formula

Ipd=Mini{ ,1 ...,Sd1}[D Sd ,i1 , pD p , Sd ,i w pD Sd ,i 1 , Sd , i] 1

Using formula (1), we can find the task that minimizes the cost of integration between two other tasks that are already in giant

trip. For each depotd* , we will insert all tasks ofNY

d*

into the initial sub-solution Sd

*

0one by one, with one direction

from each {p , p}NYd*

. However, to choose a taskp for insertion, we scan each task in NYd

*

starting by the closest one to

d* and we find the best positioni

* in Sd

*

0and direction p

* that gives Ip*d*=min{Ip d*, Ipd*} . Then we insert p* at the

positioni* in the route S

d*

0and we remove p

* and p*

from NYd

*

. To obtain S0 an initial MD-CARP giant solution, we

repeat choosing and insertion tasks for each d V d, until NYd is empty.

4.1.3 Theregularizationstage

Finally, we deal with the regularizationstage (realization and evaluation), in which the giant solution produced from the

previous stage, is transformed into a feasible solution of the MD-CARP by splitting all of the giant trips, associated with the

individual depots, into vehicle routes obeying the vehicle capacity constraint using the splitprocedure.

Thesplitalgorithm:The objective of the split procedure of Lacomme et al. [20] is to subdivide a permutation of tasks,

called a giant trip (consisting of all the tasks assigned to a particular depot), into a set of routes satisfying the vehicle capacity

constraint. split requires a giant trip which is a permutation of |R|tasks. We will explain in details in Sections 5.1, 5.2 and 5.3

because we will also generalize the split on a simple depot to the case of multiple depots (md-split). For each giant trip, we




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obtain a feasible CARP solution after applying split. The role of splitis also to calculate the cost of the feasible solutions CARP

which consist of the MD-CARP solution.

The initial MD-CARP solution ofHACO

C S0 is the cost of the initial solution S

0 , calculated after applying the split procedure to each sub-solution Sd0

. Thanks to

split, a CARP solution S not satisfying the constraints capacity is transformed into an optimal feasible CARP solution subject to

the order of tasks in S. At the start, the current best solution Sbest (at iteration 0 ofHACO algorithm) is initialized as S0 , and its

best cost isCbest=CS0 . The basic form of this heuristic is used by Lacomme et al. [20] on the undirected periodic arc

routing problem.

4.2 Construction of a giant solution for the MD-CARP Problem

We propose a colony of mants where each antf represents the order of tasks, called sequencef , which will be included in the

associated solutionSf (the size of sequencef is mr, the total number of tasks). We note by ij the amount of pheromone between

two tasks i andj, and firstly, we have ij=1/ CS0 for all (i,j )V V. The measure of desirability, denoted by ij takes into

account the distance between two tasks cij(the inverse of the travel cost between i andj ).

To build a giant solutionSf

of the MD-CARP problem at iteration t, by an antf, we apply two steps. The first step is toconstruct a sequencef of tasks to be inserted in the solution Sd

f. For this, we use the ACO method: the first task of sequencef is

chosen randomly byf and it constructs sequencef by successively choosing a task, continuing until each task of V has been in

sequencef . A task j is selected to be in sequencef after task i from a set NYf which contains the tasks that are not yet in

sequencef , according to the probability distribution:

P ijf t=[

ij t]

[ij]

/hNYf [iht]

[ih]

if j NY f

0 otherwise2

This probability distribution depends on the and parameters that determine the relative influence of the pheromone trails

and the desirability. The second step is to insert the tasks of sequencef in the solution Sf associated with antf using the order of

sequencef . Then for a task s, we find the best direction sd and the best positioni

d for each d V dand we insert sd

*

in the

positionid*

of the sub-solution Sd

*

fwhered

*=argmindVd

{Isd} .

4.2 The general algorithm of HACO

TheHACO methodstarts with an initial solution S0 (see Section 4.1). For each antf in the colony, we associate a solution S

f

and a costCf . During each iteration t of the general algorithm ofHACO, each antf constructs a sequencef , and there are m

giant solutions. With a probabilitypr, we improve each Sf using four procedures: (1) change the direction of each task by its

inverse, (2) change the current depot for each task by another depot, (3) exchange positions of tasks pand qwhich are in the

same Sdf

for each d, and finally(4) remove a taskp from its position and put it elsewhere in the same Sdf

and for every depot d.

The improvement phases will be done on the giant solutions where we do not take the capacity of the vehicle into consideration.

Thanks to split procedure we will obtain the feasible solutions and the values of objective functions. Depending on the value ofthe objective function of the solution, the pheromone trails are updated. In early ant system approaches, all the ants contributed

to the trail update (Dorigo et al. [11]). In our procedure, we update the pheromone (Bullnheimer et al. [7]) according to the

following expression:

pqt1 pq tf=1

f epq

f

(3)

withfpq =1 if (p, q) is insequencef and equal to zero otherwise.feis a number of elitist ants, is the trail persistence (0

1), thus the evaporation term is given by (1). TheHACO stops after a fixed number of iterations itermax is done. The pseudocode of the generalHACO algorithm is presented by the above Algorithm 1.




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Algorithm 1.The generalHACOalgorithm.

Initialize: pq(0)=1/Cbestfor allp and q V, Sbest=S0, iter=1

While iter maxiter repeat:

forallf=1, , nf do

NYf V , take an empty solution Sf

choose the first taskpto serve randomly byf

NY f NY f { p , p} whileNYf is non empty, do

choose a task qusing the formula (2)

NY f NY f {q , q}

endwhile

insert tasks by the order of sequencefin Sf (insertion phase, Section 4.1.2)

apply improvement stage onSf

applysplit procedureon eachSf and calculateC(Sf)

if C(Sf)


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The depot that minimize the cost to go from it, to serve the task t1=1 and to return to the same depot, is the depot D2. More

precisely, the required cost to go fromD2, to serve the task number 1 in Sand to return to the same depot is: 7 + 7 + 12 = 26.

ThenD2 1 D2represents a trip satisfied all constraints of the MD-CARP problem. Respecting the order of served tasks in S,

md-split constructs the auxiliary graphHwith mr + 1 nodes indexed from 0 to mr =6 (see Figure 4 for this example).

Each sub-sequence of tasks (tr, , ts) feasible for one feasible trip (the constraint capacity is satisfied and the vehicle associated

to this trip starts and ends at the same depot) gives an arc (r-1 , s) in the graphH. This arc(r-1 , s) is evaluated by the cost of theassociated feasible trip, for example the feasible tripD2 1 D2corresponds to the arc (0, 1) in the graphHand it has the cost

19 + 7=26. In Figure 4, we have next each arc the couplex(y)wherexrepresents the cost (without adding the served costs of the

tasks served by this trip) of the associated trip and y represents the depot assigned to this trip. For example, next the arc (0, 2)

we have 100(3) where 100 is the cost of the trip D3 1 2 D3 without adding the sum of service costs:

w t1w t2=711=26 . The complete cost of this trip is 100+26=126 and the value 3 between parentheses represents the

index of depot (the depot D3 in the example) which minimizes the cost to go from a depot, serves the task 1 and the task 3 and

finally to return to the same depot (which isD3). More generally, we will construct a matrixXof size (2mr+1; 2mr+1), where mris

the number of tasks that must be served and eachXpqcontains the value of the depot such that:

Xpq=argmin dVd {D d , pD q , d}

wherepand qare two tasks in V. In conclusion, the Figure 2 represents a giant MD-CARP solution, we construct an auxiliarygraphHlike in Figure 4 and finally we obtain the feasible MD-CARP solution Swhich is optimal respecting the tasks order in

the giant solution like in Figure 3 . The obtained MD-CARP solution for our example contains three feasible trips, each trip

starts and ends at the same depot, the capacity constraints are respected. The first trip is:D2 1 2 3 D2which the complete

cost1=7+(7)+38+(11)+44+(11)+14=132, it is associated to arc (0, 3) in H. The second trip is:D1 4 D1with the complete

cost2=15+(9)+21=45, it is associated to arc (3, 4) in H. The third trip is: D3 5 6 D3 with the complete

cost3=17+(12)+21+(8)+24=82, it is associated to arc (4, 6) in H. All trips respect the capacity constraint and form the optimal

feasible MD-CARP solution with the total cost= cost1+cost2+cost3 =259. Each trip of these three trips corresponds to an arc in

the shortest path inH, which contains three arcs.

Figure 2.giant solution S with 6 tasks-edges. Figure 3. obtained feasible trips (MD-CARP feasible solution).

Figure 4. auxiliary graph and optimal path.




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5.3 An efficient algorithm ofmd-split

We can calculate C P o1o2=C P0R in O(|R|2) without generate explicitly H. In our example, the path

P 0 6 o1 = ,0 o2=6 =R to go from 0 to node 6 is the best (with minimized cost) between all possible paths in H. For eachtask trin a given giant solution S, we takeZ[r] the cost of the optimal path from o1to trand P[r] its predecessor task. Each trip

tr,..., ts= nrscorresponds to an arc (r-1, s)inH. If there is an amelioration, we update the label of sinstead of storing the arc. Foreach r, we examine each possible trip nrs(s r)and we calculate its cost and load until load exceeds Qor tsreached the end of

S.The algorithm is flexible to support various additional constraints like a maximum cost do not to be extended by each trip.

We can use also the same Algorithm2 if the vehicles are not identical and have a maximal number. While the number of trips

cannot exceed the number of vehicles (called, nv), the procedure md-splitruns in O(mr nv)space and the complexity is O(mr2nv).

Algorithm 2.The general md-splitalgorithm for the MD-CARP problem.

Initialize: P[0]=0,Z[0]=0,Z[r]=1 for all r = 1, , mrFor all r=1, , mrdo

s=r, load=0

whiles mrand load Qrepeat:

load = load +q ts

ifr=sthen

cost=D Xtr tr, trwtrD tr, Xtr tr

else

cost= costD Xtrts1,trDts1, Xtr ts 1D Xtrts, tr

Dts, Xtr tsD ts1, tsw ts endif

if loadQthen

if(Z[r-1] + cost


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29

Algorithm 3.The generalDACOSalgorithm.

Initialize: pq(0)=1/Cbestfor allp and q V,fbest=S0, iter=1

While iter maxiterrepeat:

forallf=1, , nf do

NYf

R

choose the first taskpto serve randomly byf

NY f NY f { p , p}

whileNYf is non empty, do

choose a task qusing the formula (2) to serve byf

NY f NY f {q , q}

endwhile

apply improvement stage onf

applymd-split on eachf and calculateC(f)

if C(f)


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Concerning the Table 1, theDACOS generates only two better final solutions thanHACO, more precisely, in instances (mdgdb2

and mdgdb5). TheHACO method found better solutions thanDACOSin six instances: mdgdb3, mdgdb4, mdgdb8, mdgdb12,

mdgdb20and mdgdb23. Then theHACOmethod is better thanDACOSmethod on the Golden instances with multiple depots. In

all the remainder of instances, the two methods obtain the same values. We note also that the execution time is very small for

both our methods on the two files of instances: Golden and Benavent (Tables 1 and 3). Concerning the Table 3, the HACO

generates four better final solutions thanDACOS.

Table 1. Results on Golden instances with (multiple depots).

Instance BDBP HACO SEC1 DACOS SEC2

mdgdb1 351 300


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From Table 2, we can draw the following conclusions:DACOS obtains three better results thanHACO as an example, 1gdb9,

1gdb11 and 1gdb23.HACO is better thanDACOS at 1gdb8 only. The two methods provide rapid solutions. Comparing HACO

andDACOSto set of best methods of 1-CARP literature, we obtain almost same values (optimal results) except at following

instances: 1gdb8, 1gdb13, 1gdb22 and 1gdb23.Even we did not obtain same optimal values, we approaches them as at instances

1gdb22 and 1gdb23.As a result, the performance of HACOandDACOSis significantly better than the famous Hertz algorithm

(carpet)which is based on the tabu method.

From Table 4, we conclude that HACO algorithm is competitive with the best methods applied to 1-CARP problem but

DACOSbecomes less accurate if the size of instances augmented as we applied to Benavent instances which holds larger

instances than Golden (the size of instance presents the number of nodes and arcs).

Table 3. Results on Benavent instances with multiple depot.

Instance BDBP HACO SEC1 DACOS SEC2

mdb1a 449 415 1,3 415 1,2

mdb2a 471 368 0,8 368 1,1

mdb3a 204 167 1 167 1,5

mdb4a 843 710 3,1 748 3,2

mdb5a 801 702 3 737 2,5

mdb6a 632 530 3 544 2,4

mdb7a 754 634 3 5 666 3 1

mdb8a 719 634 3,3 663 2,9

mdb9a 887 772 7,5 824 6,3

mdb10a 918 821 8,7 883 7,4

mdb1b 449 415 1,2 421 1,5

mdb2b 471 368 1 380 1 7

mdb3b 204 167 1 167 2

mdb4b 843 717 3,5 764 3,2

mdb5b 801 697 3,2 749 3

mdb6b 632 530 2,2 547 2,7

mdb7b 754 634 3,7 670 3,6

mdb8b 719 642 3,1 671 3,3

mdb9b 887 773 7 859 6,2

mdb10b 918 826 7,6 896 6,5

mdb1c 449 415 1 7 435 1 5

mdb2c 505 410 1 422 1,2

mdb3c 204 167 1,4 174 1,5

mdb4c 843 710 3 710 3,3

mdb5c 801 716 3 756 2,9

mdb6c 684 595 1,8 612 2,5

mdb7c 778 640 2,9 666 3,5

mdb8c 739 695 2,5 722 3,2

mdb9c 887 776 6 825 6,6

mdb10c 918 836 7,3 903 7,2

mdb4d 857 754 3 1 759 3 4

mdb5d 841 749 2,4 795 3,3

mdb9d 911 796 6 846 6,7

mdb10d 964 887 6,5 931 7,2

Table 4. Results on Benavent instances with simple depot.

Best BDBP HACO SEC1 DACOS SEC2

173 173 173 1,7 173 1,6

227 250 227 1,4 227 1,4

81 85 81 1,9 81 2

400 436 400 37,6 404 25,2

423 453 423 40,2 423 38,3

223 243 223 3,6 223 3,2

279 296 279 1 8 279


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6.3 Experimental results of our approaches for solving instances with known optima

In this section, we examine the effectiveness of our methods in solving instances with known optima. In the literature of MD-

CARP problem with a mixed graph, the only work applied on instances with optimal values is given in [31] by Xing et al. The

authors proposed three different versions of evolutionary algorithm and the best one is HHAE3. They prove the superiority of

HHAE3 with others versions that are evaluated on 107 instances with up to 140 nodes and 380 arcs. For further evaluation

HHAE3with optimality, 20 MD-CARP instances with up to 100 nodes and 360 arcs with known optima were designed by thesame authors Xing et al. and they can be downloaded from http://xinglining.googlepages.com/home. The 20 new instances are

shown in columnNameof Table 5. Now, we change the parameters because the size of the instances is bigger, and the new

parameters used to obtain the results by the two methodsHACOandDACOSin Table 5 and Table 6 are the following: maxiter =

320, nf= 45,fe = 20, = 1, = 1, = 0.91 andpr = 0.4.

HACOet DACOSare applied on the CASEinstances and the experimental results are summarized in Table 5. The summary of

abbreviations used is as follows:Name: name of example,Np: number of nodes,NA: number of required arcs (each required edge

gives two required arcs),Lp: lower bound, F*: known optimal result. The computational time in second corresponds toHHAE3,

HACOandDACOSare given respectively in columns TA(S), TH(S)and TD(S).

Table 5. Results ofHHAE3,HACOandDACOSon CASE instances (over 50 trials) and the computational times.

Name NV NA Lp F* HHAE3 HACO DACOS TA(S) T H(S) T D(S)

CASE-C101 40 132 164 180 184 180 180 275.00 182.69 163.75

CASE-C102 40 132 218 262 262 262 262 178.84 180.26 172.11

CASE-C103 40 132 174 210 226 222 230 268.45 198.01 187.43

CASE-C104 40 132 218 262 266 262 262 182.48 179.19 162.84

CASE-C201 49 168 224 240 260 248 254 219.34 196.00 192.65

CASE-C202 49 168 296 324 342 332 332 153.57 213.76 188.20

CASE-C203 49 168 224 272 280 272 284 173.17 168.45 156.19

CASE-C204 49 168 296 324 340 326 340 154.81 134.00 115.96

CASE-C301 70 246 328 360 398 374 376 347.01 311.15 315.48

CASE-C302 70 246 424 492 520 492 518 254.19 253.61 248.87

CASE-C303 70 246 348 412 440 440 450 347.01 333.62 317.23

CASE-C304 70 246 424 492 520 498 516 254.90 246.05 245.29

CASE-C401 91 324 432 480 524 492 514 596.56 419.41 422.66

CASE-C402 91 324 556 668 702 676 702 416.59 398.00 387.24

CASE-C403 91 324 464 532 568 532 564 621.10 597.65 618.13

CASE-C404 91 324 556 668 700 678 700 418.05 402.53 412.09

CASE-C501 100 360 484 540 562 552 576 882.18 692.71 670.01

CASE-C502 100 360 630 754 786 764 782 616.84 578.55 562.43

CASE-C503 100 360 514 550 582 556 596 845.31 805.27 796.08

CASE-C504 100 360 630 754 788 762 790 616.14 577.18 592.34

Table 6. Computational errors betweenHHAE3,HACOandDACOS and the lower bounds and know optimal results.

101 102 103 104 201 202 203 204 301 302 303 304 401 402 403 404 501 502 503 504 M

FAE 12.2 20.2 29.9 22.0 16.1 15.5 25.0 14.9 21.3 22.6 26.4 22.6 21.3 26.3 22.4 25.9 16.1 24.8 13.2 25.1 21.2

FHE 9.8 20.2 27.6 20.2 10.7 12.3 21.4 10.1 14.0 16.0 26.4 17.5 13.9 21.6 14.7 22.0 14.0 21.3 8.2 21.0 17.1

FDE 9.8 20.2 32.2 20.2 13.4 12.2 26.9 14.9 14.6 22.2 29.3 21.7 20.0 26.3 21.6 25.9 19.0 24.1 16.0 25.4 20.8

FA* 2.2 0.0 7.6 1.5 8.3 5.6 2.9 4.9 10.6 5.7 6.8 5.7 9.2 5.1 6.8 4.8 4.1 4.2 5.8 4.5 5.3

FH* 0.0 0.0 5.7 0.0 3.3 2.5 0.0 0.6 3.9 0.0 6.8 1.2 2.5 1.2 0.0 1.5 2.2 1.3 1.1 1.2 1.8

FD* 0.0 0.0 9.5 0.0 5.8 2.5 4.4 4.9 4.4 5.2 9.2 4.9 7.1 5.1 6.0 4.8 6.7 3.7 8.4 4.8 4.9


http://xinglining.googlepages.com/homehttp://xinglining.googlepages.com/home


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According to the final results listed in Table 5, the performance of our methods HACOandDACOSis significantly better than

that ofHHEA3. In fact,HACOgives best results at all instances expect CASE-C102and CASE-C303.DACOSperforms good

results in comparison toHHAE3, but less competitive withHACO. DACOSbecomes less accurate as the size increases. For

example, DACOS (same as HACO) finds three optimal solutions from 4 instances (CASE-C101 to CASE-C104) where the

number of nodes is 40 and that of arcs is 132.In contrary, it gives weak solutions at CASE-C503and CASE-C504instanceswhere number of nodes is 100 and that of arcs is 360.According to our opinion, this occurs because we calculate giant solutions

before splitting them optimally utilizing md-splitprocedure.

HACOprovides six optimal solutions whileDACOSprovides three andHAAE3 founds only one optimal solution which are

founded by our methods (for the instance CASE-C102). Concerning the execution time, we note that our methods are very quick

compared toHAAE3because great number of crossover are used inHHAE3algorithm.

For more details about the experimental results, we calculated the following errors: FA*(resp. FA

E), FH*(resp. FH

E) and FD*(resp.

FDE) respectively between FAand F

*(resp. FAand Lp), between FHand F

*(resp. FHand Lp) and between FDand F

*(resp. FDand

Lp). As displayed in Table 6, there exists a large gap between FH*(resp. FH

E) and FA*(resp. FA

E). In Table 6,Mrepresents the

average of the corresponding error on all instances, frugally we can remark that we haveM(FH*)


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