ACO and CP Working Together to Build a Flexible Tool for the VRP

Negar ZakeriNejad

1

, Daniel Riera

1

and Daniel Guimarans

2,3

1

Computing, Multimedia and Telecommunication Department, Universitat Oberta de Catalunya (UOC), Barcelona, Spain

2

Optimisation Research Group, National ICT Australia (NICTA), Sydney, Australia

3

Aviation Academy, Amsterdam University of Applied Sciences, Amsterdam, The Netherlands

Keywords:

Ant Colony Optimisation, Constraint Programming, Vehicle Routing Problem, Optimisation.

Abstract:

In this paper a ﬂexible hybrid methodology, combining Ant Colony Optimisation (ACO) and Constraint Pro-

gramming (CP), is presented for solving Vehicle Routing Problems (VRP). The stress of this methodology is

on the word ‘ﬂexible’: It gives reasonably good results to changing problems without high solution redesign

efforts. Thus a different problem with a new set of constraints and objectives requires no changes to the search

algorithm. The search part (driven by ACO) and the model of the problem (included in the CP part) are sep-

arated to take advantage of their best attributes. This separation makes the application of the framework to

different problems much simpler. To assess the feasibility of our approach, we have used it to solve different

instances of the VRP family. These instances are built by combining different sets of constraints. The results

obtained are promising but show that the methodology needs deeper communication between ACO and CP to

improve its performance.

1 INTRODUCTION

Vehicle Routing Problems (VRPs) are an optimisa-

tion problem in the ﬁeld of transportation and logis-

tics. It involves routing a ﬂeet of vehicles to serve

several customers. This problem is difﬁcult not only

because it is NP-hard but also for having many diverse

forms (Laporte et al., 2000; Golden et al., 2008)

Exact methods exist to solve VRPs optimally, but

they become computationally prohibitive for large

problem sizes. On the other hand approximate meth-

ods trade off optimality for feasibility of solving large

problem instances.

Also, hybridised methods, combining different al-

gorithms, have been developed to tackle VRPs (La-

porte et al., 2013; C´aceres-Cruz et al., 2014). One

successful hybridised approach has been combining

Constraint Programming (CP) with metaheuristics

(Solnon, 2010; Talbi, 2013; Shaw, 2011; Berbeglia

et al., 2012; Blum et al., 2011; Khichane et al., 2008;

Meyer and Ernst, 2004). CP is an exact method that

models a problem as a Constraint Satisfaction Prob-

lem (CSP) which can be solved using general purpose

solvers (Rossi et al., 2006; Khichane et al., 2010).

Metaheuristic algorithms are approximate methods

that explore the most promising regions of the solu-

tion space to ﬁnd a sufﬁciently good solutions (La-

porte, 2009).

Unfortunately, most of the research efforts for de-

veloping effective hybrid approaches are focused on

speciﬁc problems and are not easily extensible to oth-

ers (Vidal et al., 2013; Pisinger and Ropke, 2007) .

Accordingly, a fruitful challenge for researchers is de-

veloping a framework hybridising optimisation algo-

rithms and methods to address a large class of VRPs.

This would be useful because real-life applications

vary greatly and such a framework reduces design

costs. To satisfy this challenge a methodology is re-

quired that allows easy modeling (and re-modeling)

of different problems and a fast and effective search

engine.

The aim of our methodology is to maximise ﬂex-

ibility in terms of quick adaptation to different vari-

ants of VRPs, while staying reasonably close to the

optimal solutions. Our design hypothesis is based on

the high modelling power of Constraint Programming

(CP) and the ease of metaheuristics to explore search

spaces and quickly generate fairly good solutions. In

this case, Ant Colony Optimisation (ACO) (Dorigo

and Birattari, 2010) has been chosen as a metaheuris-

tic because of the closeness of the VRP instance rep-

resentations to the representation of the problems that

ACO solves.

The proposed approach has the merit of not de-

122

ZakeriNejad, N., Riera, D. and Guimarans, D.

ACO and CP Working Together to Build a Flexible Tool for the VRP.

DOI: 10.5220/0005668101220129

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 122-129

ISBN: 978-989-758-171-7

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

pending on the form of the problem and thus allow-

ing extensibility to a wide variety of VRPs, includ-

ing the ones that capture the high complexities, large

data sizes, uncertainties, and dynamisms that exist in

real-life. These kind of VRPs are called Rich Vehicle

Routing Problems (RVRPs) and are even more dif-

ﬁcult to solve than simpler VRPs (Drexl, 2012). The

ﬁnal aim of this work is to deal with RVRPs, although

for the initial assessment of the methodology we have

chosen classical VRP variants and incrementally con-

structed more complex problems.

In order to assess these goals, ﬁrst the collabora-

tion of two main parts of the framework needs to be

explored, which is the objective of this paper. This

is carried out by generating solutions using ACO as

a search algorithm and verifying their feasibility us-

ing CP. We aim to reduce the development time by

using the CP to model the constraints and check their

feasibility, and thus guide the approximate algorithm

towards better results.

The paper is organised as follows. In Section 2,

we propose our general framework by describing its

main two parts and how they cooperate to build feasi-

ble solutions. The experimental setup and the results

of the experiments are shown in Sections 3 and 4. Fi-

nally, the last section states the conclusions and some

future research ideas.

2 PROPOSED FRAMEWORK

The proposed framework is made up of two main ele-

ments: the CP and theselected heuristic/metaheuristic

(in this speciﬁc case, a basic version of the ACO).

The cooperation between these two parts is depicted

in Fig.1.

The aim of the CP part of the framework is

twofold. First, it provides a clear and powerful for-

mulation to encode any constraints relevant to a given

routing problem. Thus, different VRPs can be quickly

modelled and re-modelled as CSPs without the need

of changing the architecture of the framework. Sec-

ond, it encompasses a CP solver, which provides eval-

uations of solutions’ feasibility and also global search

information given partial solutions. This informa-

tion helps the metaheuristic to avoid exploring search

spaces with no feasible solutions.

The metaheuristic part of the framework is where

the main search through the solution space takes

place. In the proposed framework the metaheuristic

part is implemented in two distinct loops as shown in

ﬁgure 1. While the outer loop is mostly concerned

with the logic of the metaheuristic, the inner one uses

the interaction with the CP solver to guide the con-

struction of solutions.

The inner loop of the metaheuristic is effectively a

random construction algorithm. That is, in each iter-

ation the solution is augmented with a new part. The

selection of the new part is guided according to the

mechanism of the metaheuristic.

Because the metaheuristic is oblivious to the na-

ture of the problem, it frequently asks CP to verify the

solution. This is done by sending a partial solution to

the CP and having CP check that no constraints are

broken given the current assignment. As we will later

explain in more details, the CP checking performed in

the internal loop is allowed to ignore serving all cus-

tomers in a way that makes construction of a feasible

solution always possible. Therefore, in the outer loop

each solution is checked one last time to compute the

ﬁtness of the solution according to the objective func-

tion and possible unsatisﬁed customers.

The outer loop starts by constructing a new solu-

tion using the inner loop iterations. Then, it evalu-

ates the new solution quality and updates the state of

the metaheuristic according to the quality of the solu-

tion. For example in the case of ACO, it updates the

pheromone matrix.

2.1 CP Model

As said before there are many different problem sub-

classes (deﬁned by the speciﬁc active constraints) in

VRP, most of which have already been studied. This

makes it possible to build a constraint and objective

library and activate only the needed parts for solving

the given problem. In our case, we are working with

a CP-VRP library (Riera et al., 2009), built for the

ECLiPSe

1

CP system. This library allows the use of

constraints for capacity, distance, heterogeneous ve-

hicles, asymmetric routes, time windows, and pick up

and delivery, among others.

2.2 Search Algorithm

ACO is a well-known nature inspired metaheuristic

algorithm used for solving combinatorial optimisa-

tion problems. In particular, ACO is a probabilistic

technique which builds a solution iteratively through

a stochastic construction procedure. This class of

optimisation algorithms follows simple rules based

on a collective behaviour of the self-organised sim-

ple agents with no centralised control structure dictat-

ing how individual agents should behave (Dorigo and

Birattari, 2010; Yu and Yang, 2011; Balseiro et al.,

2011).

1

http://eclipseclp.org/

ACO and CP Working Together to Build a Flexible Tool for the VRP

123

Metaheuris c algorithm

Figure 1: Main parts of the proposed framework and the relationship between them.

At the moment, we use a basic version of the ACO

algorithm: Each ant constructs a solution determining

the trips for all vehicles in an iterative fashion. To

do this, it selects a path through a graph of nodes.

Each node corresponds to one customer, except for

the initial-ﬁnal node which corresponds to the depot.

Choosing the depot signals the route completion for

the current vehicle and the start of the journey of the

next vehicle.

At each iteration, the ant chooses a random node,

according to the distance of the nodes to the current

one and the amount of pheromone on the edge lead-

ing to the neighbours. Then a partial solution corre-

sponding with the current partial path is constructed

and passed to the CP checker. If the CP checker ﬁnds

the solution infeasible, the ant backtracks one step and

returns to the depot, ending the planning for the cur-

rent vehicle. If the CP checker ﬁnds no problem with

the current partial path (i.e. there might be feasible

solutions containing this partial one), the construction

of the path continues normally. These iterations ter-

minate when all vehicles have been planned for. Note

that, after termination some costumers may not have

been serviced.

After an ant constructs one solution, the CP solver

is invoked one last time to check the feasibility of the

solution found by the ant. We calculate the total dis-

tance of the trip as an objective function and use it

along with the number of unserviced customers to as-

sign a quality value to the solution found. The amount

of pheromone for each section of the solution is then

updated according to this computed quality. In Fig.2,

the ﬂowchart of the search procedure for one ant is

presented.

2.2.1 Ant Route Selection

An ant constructs its route by selecting nodes one by

one, in a stochastic manner. The probability of a node

being selected is computed according to two factors.

The ﬁrst factor is the amount of pheromone deposited,

τ

ij

, for the transition between the current node i and

the prospective node j. The second factor is the vis-

ibility heuristic deﬁned as the inverse of the distance

between nodes i and j,

1

d

ij

, and denoted by η

ij

. The

probability of choosing next node, P

ij

, can be stated

as follows:

P

ij

=

[τ

ij

]

α

· [η

ij

]

β

∑

l∈N

[τ

il

]

α

· [η

il

]

β

(1)

where α and β are parameters to tune the relative in-

ﬂuence of pheromone trail and heuristic information,

and N is the set of all nodes yet to be visited.

2.2.2 Pheromone Update

The process of updating pheromone trail consists of

two mechanisms. The ﬁrst is pheromone evaporation

which decreases the amount of pheromone values de-

posited on each edge in predeﬁned intervals. By ap-

plying this mechanism, the algorithm can avoid con-

vergence to sub-optimal solutions rapidly.

The second mechanism is pheromone deposition

which increases the pheromone trails of solution

edges according to the quality of the solution. The

aim of this mechanism is to make better solutions

more attractive for other ants and guide them towards

more promising regions.

The pheromoneupdate for each edge is commonly

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

124

Figure 2: Flowchart of the search procedure for a single ant.

implemented as:

τ

ij

← (1− ρ)· τ

ij

+

∑

k

∆τ

k

ij

(2)

In this formula, ρ is the pheromone evaporation

rate, and ∆τ

ij

is the amount of pheromone deposited

by ant k ant, computed as:

∆τ

k

ij

=

Q

L

k

(3)

where Q is a constant (problem-speciﬁc parameter

to tune), and L

k

determines the quality of a solution

found by ant k. In this case, it is equal to the total

distance travelled by the ant.

According to the type of a given problem and its

complexity, the process of updating pheromone trail

can be applied in different ways. In this work, we ap-

ply both pheromone evaporation and pheromone de-

position mechanisms simultaneously in predeﬁned in-

tervals.

3 EXPERIMENTAL SETUP

3.1 Problem Instances

In this paper, we study the interactions between the

two main parts of the framework and also the exten-

sibility aspect of the approach to adapt to newer in-

cremental problems without extra development. For

this, we have selected four VRP variants (C´aceres-

Cruz et al., 2014):

• Capacitated VRP (CVRP). It is one of the ear-

lier and most studied VRPs, where all vehicles

have the same capacity which cannot be exceeded.

The data used for CVRP is taken from a stan-

dard benchmark set proposed by Augerat et al.

(Augerat et al., 1995), where instance sizes are

between 15 and 100 customers.

• Heterogeneous ﬂeet VRP (HVRP). This is an ex-

tension of the CVRP with the difference that the

capacity and cost of vehicles are not homoge-

neous. To test the HVRP, eight problem instances

developed by Taillard (1999) (Imran et al., 2009)

are used. In this dataset, the number of customers

varies from 50 to 100 and the number of vehicle

types varies from 3 to 6.

• Distance-Constrained VRP (DCVRP). It is an ex-

tension of the CVRP with the difference that both

vehicle capacity and a maximum distance (for

each vehicle) are imposed as constraints. For

preliminary experiments, six problem instances

of the CVRP dataset

2

have been adapted for the

DCVRP.

• Asymmetric cost matrix VRP (AVRP). In this

type of problems the distance between each pair

of locations in the two directions is not the

same in both ways. Fourteen problem instances

based on realistic scenarios are selected from

AVRP benchmark proposed by Rodriguez and

2

Problem instances P-n16-k8, P-n20-k2, P-n40-k5, P-

n50-k7, P-n55-k10, and P-n101-k4.

ACO and CP Working Together to Build a Flexible Tool for the VRP

125

Ruiz (Rodr´ıguez and Ruiz, 2012)

3

. Because this

dataset provides a diverse number of tests, in-

stances were chosen so their sizes were between

50 and 100 customers. We also limited the dataset

to those instances that have higher demand and

nodes’ locations were randomly chosen within an

intra-city area.

4

3.2 Computational Setting

The metaheuristics part of the framework has been

implemented in Java. For the CP part, we chose the

ECLiPSe platform. A CP-VRP library (Riera et al.,

2009) has been used as a constraints pool. All experi-

ments were ran on a system with an Intel core i5-3470

CPU and 4GB RAM.

The performance of the ACO-based algorithms re-

lies on a set of correlated parameters and the values

chosen for them. Tuning the parameters is a time-

consuming process and can lead to a diverse set of pa-

rameters depending on the given problem. As the aim

of this experiment is to show the ability of the frame-

work to adapt to different types of VRPs, devoting

time to adjust parameters is not indispensable. As a

matter of fact, the need of adjusting parameters when

adding new constraints would be considered a penalty

for the methodology. Table 1 summarises the values

chosen for the ACO parameters used indistinctly in

all the experiments.

Table 1: Parameters used for conducting the tests.

Parameters Value

No of ants 2500

Pheromone update interval 50

Initial pheromone 2

α 3

β 3

ρ 0.0001

Q 100

4 EXPERIMENTAL RESULTS

In order to assess the efﬁciency of the proposed

framework, two different kinds of experiments are

considered. In the ﬁrst one, a small-sized example

is used as a preliminary test instance to show how the

3

http://soa.iti.es/problem-instances

4

Problem instances G-A-CAA0501, G-A-CAA0502,

G-A-CAA0503, G-A-CAA0504, G-A-CAA0505, G-

A-CAA1003, G-A-CAA1004, G-C-CAA0501, G-

C-CAA0502, G-C-CAA0503, G-C-CAA0504, G-C-

CAA0505, G-C-CAA1001, G-C-CAA1002.

framework behaves when solving different types of

VRPs (by adding constraints one by one). In the sec-

ond experiment, the framework is applied over four

classical benchmark sets, each related to a different

type of VRP, to check the quality of the solutions

found by the algorithm compared to the best known

5

.

The experimental results are given in detail below.

4.1 Laboratory Incremental Example

The ﬂexibility of the proposed method has been as-

sessed through a small-sized experimental test de-

scribed as follows. Table 2 shows the coordinates

of the depot and the coordinates and demand of the

clients to be served. The corresponding map can be

seen in Fig.3.

Table 2: summary of the simple model used for the test.

Node [X,Y] Demand (units)

Depot [40.348, -3.851] 0

C

1

[40.372, -3.593] 59

C

2

[40.295, -3.645] 64

C

3

[40.448, -3.760] 84

C

4

[40.444, -3.879] 70

C

5

[40.427, -3.548] 80

C

6

[40.335, -3.866] 69

C

7

[40.453, -3.528] 75

Furthermore, we have pre-calculated the (asym-

metric) distance matrix for the nodes, containing the

depot and the clients:

0 16 20 33 9 30 13 28 28 28

15 0 23 33 22 25 26 25 25 25

26 23 0 14 22 20 24 18 18 18

37 33 16 0 35 15 36 14 14 14

15 21 21 35 0 34 5 32 32 32

33 24 22 15 34 0 36 3 3 3

19 26 23 36 6 36 0 34 34 34

31 25 19 15 32 4 33 0 0 0

We have got three vehicles to design the routes.

In the homogeneous cases their capacity is 195 units,

while for the heterogeneous instances their capacities

are 210, 160, and 145.

Figures 3-6 show how the methodology adapts

from one kind of problem to another by only enabling

or disabling constraints. Notice that, as said before,

no changes in the search algorithm parameters have

been made for any experiment.

4.2 VRP Benchmarks

The second set of tests were performed to assess

the quality of the solutions found by the proposed

5

http://neo.lcc.uma.es/vrp/

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

126

C1

C2

C3

C4

C5

C6

C7

Depot

Figure 3: Conﬁguration of the small-sized instance.

C1

C2

C3

C4

C5

C6

C7

Depot

CVRP

d

5

=80

d

2

=64

d

1

=59

d

6

=69

d

7

=75

d

3

=84

d

4

=70

Figure 4: Solution found for CVRP/AVRP.

C1

C2

C3

C4

C5

C6

C7

Depot

HVRP

d

2

= 64

d

1

= 59

d

3

= 84

d

4

= 70

d

5

= 80

d

7

= 75

d

6

= 69

Figure 5: Solution found for HVRP.

C1

C2

C3

C4

C5

C6

C7

Depot

DCVRP D t e t

Figure 6: Solution found for DCVRP.

methodology. Although this is not the main aim, we

consider it important to not be very far from the state-

of-the-art best-known solutions. Note that we use the

same framework for solving different types of prob-

lems while the best-known solutions are speciﬁc for

that kind of problem.

For this, for every instance of each classical

benchmark set mentioned in Section 3, 50 runs with

different random seeds have been carried out. The

results are summarised in Table 3, where the compar-

isons between our methodology and the best-known

solutions are shown. For each benchmark, the num-

ber of problem instances and their size, according to

the number of customers and vehicles, has been indi-

cated. In the column “CSTR”, the type of constraints

activated for that benchmark is mentioned.

In the last two columns, we compare the quality of

the solutions found by our methodology with the best-

known solutions. We calculate the difference between

the average values of 50 runs and the best-known for

each instance. Then, we present the percentage gap

calculated over all instances for the given benchmark

set. In the same way, the difference is calculated by

consideringthe best of our solutions for each instance.

In this work, the computational time is not compared

because the methods used to obtain the best-known

solutions are vastly different from each other and in

some cases the timing is unknown.

In the CVRP experiment, the ﬂeet size of each

problem instance was ﬁxed to the minimum feasible

value speciﬁed in that instance. However, ﬁve in-

stances were too complex to be solved in a limited

time. For these instances

6

, the computations were

based on solutions using an additional vehicle.

To evaluate the results for HVRP, we compared

our results with the best-known solutions published

in (C´aceres Cruz, 2013) which are obtained in the

same situation according to the same objective func-

tion. Except the problem instance 13, all instances

were tested with the same type of vehicles introduces

in the standard benchmark.

Although our results are far from the best-known

solutions, the ﬂexibility of the approach is promising.

We hope that using the CP solver to search the solu-

tion space as opposed to just verifying the feasibility

of the solutions, would improve the quality of the re-

sults.

The aim in the DCVRP experiment was to show

that the framework could work properly and ﬁnd an

acceptable solution by adding a new constraint to the

problem. We selected six problems from the same

data set used for the CVRP tests ranging from small

to large size instances and imposed capacity and dis-

tance limitations. The results are summarised in Table

4. The best values of 50 runs for each instances are

shown in the table for both DCVRP and CVRP. The

results show that the algorithm can adapt to the new

situation without any modiﬁcation in the ACO part.

5 CONCLUSIONS AND FUTURE

WORK

In this work we have presented a ﬂexible hybrid

methodology to solve VRPs. The methodology fol-

6

P-n23-k8, P-n50-k8, P-n51-k10, P-n55-k15, P-n60-

k15.

ACO and CP Working Together to Build a Flexible Tool for the VRP

127

Table 3: Comparison between the proposed framework and the best-known solutions for three types of VRPs.

Prob Bench #Inst #Cust #Veh CSTR

a

Avg (%)

b

Best (%)

CVRP Augerat et al. setP 23 15-100 2-15 C 12.37%

c

6.68%

HVRP Taillard 8 50-100 7-17 C+H 26.89%

d

18.82%

AVRP Rodrguez et al. 14 50-100 2-6 C+A 13.26%

d

8.36%

a

The abbreviations are as follows: C = Capacity constraint, H = Heterogeneous ﬂeet of vehicles, D = Distance

constraint, A = Asymmetric cost matrix.

b

Avg.(%), means the average of the solution deviations, each deﬁned as ((AverageValue

50

runs− BKS)/BKS) × 100.

c

The ﬂeet size was increased by one for ﬁve problem instances of the benchmark set.

d

The results were compared with the best solutions found in (C´aceres Cruz, 2013).

Table 4: Computational results for DCVRP.

Instances Distance limit DCVRP-Best CVRP-Best

P-n16-k8 < 70 451 452

P-n20-k2 < 125 217 219

P-n40-k5 < 135 538 497

P-n50-k7 < 130 618 600

P-n55-k10 < 120 734 632

P-n101-k4 < 330 804 760

lows a constructive strategy, building partial solutions

through the collaboration of a metaheuristic/heuristic

and a CP solver. This provides a straightforward way

to share the search space between the two approaches

and take advantage of the characteristics of both ap-

proaches.

The methodology has been tested on different

VRP benchmarks, and also with combinations built

by mixing a set of constraints. The results show the

ﬂexibility of the framework in satisfying new con-

straints.

Currently we are including new constraints related

to time windows and the order clients are visited.

Once these constraints are added, the question that

we aim to answer is how to best share the problem

space between the CP solver and the metaheuristic

framework. More speciﬁcally, we would like to get

more feedbacks from the CP solver to guide the meta-

heuristic algorithm towards better solutions.

Other future work to be considered is the union of

ACO and CP in a single platform to avoid commu-

nication leaks, the inclusion of clustering techniques

in the problem, and an automatic ACO parameters’

tuning in order to improve its performance without

requiring additional effort from the ﬁnal user.

ACKNOWLEDGEMENTS

The authors would like to thank Amir Hossein

Bakhtiary Davijani for his kind help and collabora-

tion during the experiments. This work has been

partially supported by the Spanish Ministry of Econ-

omy and Competitiveness (grant TRA2013-48180-

C3-P), FEDER, and the Department of Universities,

Research & Information Society of the Catalan Gov-

ernment (Grant 2014-CTP-00001).

NICTA is funded by the Australian Government

through the Department of Communications and the

Australian Research Council through the ICT Centre

of Excellence Program.

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