A Split based Approach for the Vehicle Routing Problem

with Route Balancing

Philippe Lacomme

, Caroline Prodhon

, Christian Prins

Xavier Gandibleux

, Boris Beillevaire

and Libo Ren

Université Blaise Pascal (LIMOS UMR 6158),Campus des Cézeaux, 63177 Clermont-Ferrand, France

ICD-LOSI (UMR-STMR CNRS 6279), Université de Technologie de Troyes,

12 rue Marie Curie, CS 42060, 10004 Troyes Cedex, France

Université de Nantes (LINA UMR 6241), UFR Science et Techniques,

2 rue de la Houssinières, BP 92208, 44322 Nantes Cedex 3, France

Keywords: Vehicle Routing, Multi-objective Optimization, Split Algorithm, Meta-heuristic.

Abstract: The vehicle routing problem with route balancing is a bi-objective routing problem, in which the total route

length and the balance of routes (i.e. the difference between the maximal and minimal route length) have to

be minimized. In this paper, we propose an approach based on two solution representations: a giant tour

representing a sequence of customers (indirect representation) and a complete solution with a decomposition

of the giant tour, combined with a split algorithm to alternate between them. This approach offers a mainly

efficient way to explore the solution space. Our motivation is based on the possibility to generate efficiently

several solutions a time using an indirect representation which has been already proved to be efficient in

numerous routing problems resolution. The originality here is to tune the split algorithm considering two

objectives. An evolutionary path relinking algorithm is embedded to improve the obtained solutions. The

proposed approach is evaluated on classical vehicle routing problem instances and the results push us into

accepting that the method is competitive with the best published mono-objective methods (on criteria one :

the total route length). On a bi-objective point of view, our method is competitive with the lexicographic

solutions reported in the literature in the sense that it provides similar or better results in comparable

computational time.

1 INTRODUCTION

This paper addresses the vehicle routing problem

with route balancing (VRPRB), which is a variant of

the bi-objective vehicle routing problem (bi-

objective VRP). The VRP is a class of routing

problems that consist in visiting a set of customers

using a homogeneous fleet of capacitated vehicles

with the objective of minimizing the total route

length. The bi-objective versions of VRP consist, for

the second objective, in maximization of a profit or

equity between routes by minimizing the route

balancing. The VRPRB holds on the second.

1.1 Vehicle Routing Problems

The basic version of the VRP is the capacited

vehicle routing problem (CVRP). It can be defined

on a complete and undirected graph ,,

where  0,…, is the set of nodes and 

, |∀,  ∈ ,    is the set of edges. The depot

is represented by node 0, where an unlimited fleet of

identical vehicles with a capacity  is available to

serve the demand 



of each customer 

corresponding to nodes from 1 to . Each edge , 

is the shortest path from node  to node  and

associated with a non-negative cost 



. The

objective is to find the set of routes of minimal cost

to serve all customers with the following constraints:

 demands cannot be split (each customer must be

served by a single visit);

 each route starts and ends at the depot;

 the total demand of the customers served by one

vehicle must fit its capacity.

159

Lacomme P., Prodhon C., Prins C., Gandibleux X., Beillevaire B. and Ren L..

A Split based Approach for the Vehicle Routing Problem with Route Balancing.

DOI: 10.5220/0004764801590166

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 159-166

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

In several publications, the number of vehicles is

limited and/or a time limit is given to perform a trip.

In the latter case, a service time is added to each

customer. The CVRP is NP-hard since the mono-

vehicle case, corresponding to the traveling

salesman problem (TSP) is known to be NP-hard.

When multiple objectives are identified, they are

frequently in conflict. For this reason, adopting a

multi-objective point of view can be interesting.

1.2 VRP with Route Balancing

In the VRPRB is an extension of the CVRP in which

two objectives have to be optimized:

 Minimization of the distance traveled by the

vehicles.

 Minimization of the difference between the

longest and the shortest route length.

Even if very efficient methods exist to solve the

CVRP, they manage only the first objective.

Lacomme et al. (Lacomme et al., 2006) concerns the

resolution of an arc routing problem using an

NSGA-II approach. To the best of our knowledge,

the last publication on VRPRB is the one proposed

by Jozefowiez et al. (Jozefowiez et al., 2009).

Among the proposed approaches in the literature for

multi-objective (MO) problems, NSGAII (Deb,

2001) is intensively used. However, to provide

quality results on the CVRP, its general structure

requires efficient specific developments. More

generally, taking advantages of ranking schemes

seems to be a good approach in routing problem as

stressed by Coello Coello (2000) in a survey. For a

complete introduction on MO optimization, it is

possible to refer to the annotated bibliography from

Ehrgott and Gandibleux (2002) which provides a

suitable entry point for general definitions and

pertinent references.

In this paper, a new approach is proposed to

obtain a set of efficient solutions through a

technique that is based on an indirect representation

of solutions for routing problems: the mapping

function denoted split in the majority of publications

(Prins, 2004). The original version is here adapted to

tackle the multi-objective feature of the problem and

a Path Relinking (PR) algorithm is embedded to

explore the solution space.

The remainder of this paper is organized as

follows: section 2 presents the proposed approach;

computational results are introduced on Section 3

and the paper concludes with section 4.

2 PROPOSED APPROACH

The proposed algorithm is based on a Split

algorithm, a procedure that has proven its efficiency

on routing problems and that is here adapted to

handle multi-objective functions.

2.1 Split based Approaches for Routing

Problems

The split algorithm was proposed by Beasley as the

second phase in a “route-first, cluster-second”

heuristic for the CVRP (Beasley, 1983). The first

phase consists in creating a giant tour by relaxing

both vehicle capacity and maximum tour length, and

the second phase constructs a cost network and then

applies a shortest path algorithm to find least cost

feasible trips. However, the real rise of the approach

appears in 2001 when it has been implemented

within more general frameworks for routing

problems providing methods competitive with the

best published ones from 2001 to 2008 on the

Capacitated Arc Routing Problem - CARP

(Lacomme et al., 2001) (Lacomme et al., 2004) and

the VRP (Prins, 2004). In this context, the number of

split applications in routing increases strongly as

pointed by Duhamel et al. (2011) and covers now

CARP, VRP, Location routing and numerous

extensions which represent a set of more than 40

publications. Moreover, Duhamel et al. (2011) gives

a fully generic description of split functions and

proves that some ones require shortest path with

resource constraints and several labels on nodes.

The split algorithm is a function which ensures a

mapping from one indirect representation of solution

(denoted QDRS in the Figure 1) and a solution of

Figure 1: Efficient routing framework outlines according

to (Duhamel et al., 2011).

Determine a

QDRS

A quasi-direct

representation of

solution (QDRS)

A solution S.

Improved solution S’

A quasi-direct

representation of

solution (QDRS)

Heuristics

dedicated to the

problem

A solution S.

A quasi-direct

representation of

solution (QDRS)

Initial set of

QDRS

Initialization of the framework

Diversification

Process

Local Search

(LS)

Improvement of solution

Framework

iterations

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the problem. The oscillation between the set of

QDRS (giant tours in routing problem) and the set of

solutions (solutions of the VRP for example) has

been proved to be a strongly efficient way for space

exploration.

The mapping function between one QDRS and a

solution could be classified into several categories

(Figure 2) as stressed for years by Cheng et al.

(1996).

Figure 2: The mapping from coding

to solution (Cheng et al., 1996).

2.2 Search Space Exploration Strategy:

SPR

The search strategy is made by a path relinking

approach enforced by a multi-start scheme to bring

some diversity, and by an alternation between

solution spaces thanks to a new adaptation of the

classical split procedure for VRP. The resulting

method is called SPR (Split based Path Relinking

approach) and provides an approximation of the

Pareto front – referred to as GPOP - updated all

along the search process, by keeping non-dominated

solutions. Three main components characterize the

method (Figure 3):



Generation of giant tours, either heuristically

(through a randomized nearest neighbor

algorithm or a random sequencing) at the

beginning of each restart, or with PR between

existing giant tours;

 Evaluation of the giant tours by transformation

into potentially non-dominated solutions through

the proposed split procedure followed by local

searches;

 Inclusion of the obtained non-dominated

solutions within the current population POP, and

GPOP which model the Pareto front. The

insertion may result in some solution deletions in

the populations due to dominance rules.

The originality of the proposed approach remains

on the alternation between two search spaces taking

advantages of the split procedure. Several non-

dominated solutions can be derived from a single

giant tour. Using such an approach, a strongly

limited number of giant tours permits to obtain a

larger population with a time efficient split

algorithm. However, the population is limited to





solutions sorted by increasing solution cost.

Thus, the generation of the initial giant tours at

the beginning of each global iteration or restart (line

12 in procedure 1) aims at creating 



giant tours

either with a randomized nearest neighbor algorithm

or a random sequencing. A set of solutions is

Figure 3: SPR strategy principle.

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161

Procedure 1: Multi_Start_Split_based_Path_Relinking_Approach

1 global parameters

2 iter_max: maximal number of iterations

3. Nr : number of replications

4. input/output parameters

5. GPOP : population

6. local variables

7. POP: a population of solutions

8. begin

9. GPOP := 

10. for j:=1 to iter_max do

11. POP = 

12. call Generate_new_sol_in_Population (POP)

13. call Path_relinking (iter_max ,POP,POP.n)

14 GPOP := GPOP + POP

15 end for

16. call Post-optimization(iter_max ,GPOP,GPOP.n)

17. return GPOP

obtained from the evaluation of each giant tour by

the proposed split procedure. Then, these solutions

are improved using local search procedures, and

inserted into both populations POP (the current

population made of non-dominated solutions

encounter since the beginning of the related global

iteration) and GPOP (the global population, made of

the pool of non-dominated solutions encounter since

the beginning of the whole algorithm). Once

inserted, some solution deletions in the populations

may result of the dominance rules.

The second part of the algorithm (line 13)

explores the solution space around solutions in POP

through a path relinking between them. Two inner

loops are successively called. The first one performs

a path relinking between the best solution of POP

according to the cost criteria and 



randomly

selected other solutions of the population. The

second loop does the same process but with the best

solution according to the balance criteria. The

encountered solutions on the paths are tested to enter

in both POP and .

The third part (line 16) is a post optimization

that also performs a path relinking, but this time, it is

made between the subset of  solutions of the

approximate efficient front contained in GPOP.

More precisely, a loop with  from 1 to /2 create a

path between solutions  et .

2.3 An Adaptation of Split Procedure

for Multi-objective VRP

The split procedure allows to obtain the lowest cost

feasible trips from a given giant tour . To do so, an

auxiliary acyclic graph  based on a sequence of

tasks (giant tour of customers) is first constructed.

The graph contains 1 nodes numbered from 0

to , 0 being an artificial initial node. An arc , 

corresponds to a subsequence of consecutive

customer from position 1 to  in tour  and

visited in a single trip starting and ending at the

depot. Splitting  corresponds to the computation of

a min-cost path from node 0 to node  in . On

VRP, using Bellman algorithm for acyclic graph, the

splitting of the giant tour is optimal. On more

complex VRP versions, it might be useful to

compute a resource-constrained shortest path

(Desrochers, 1988) that is typically done by a label-

correcting algorithm, involving to manage several

labels per node.

In VRPRB, there exist also several labels per

node since a label  can be defined as a structure

with its cost .  and its balance. . This definition

does not allow for a simple comparison to claim that

one dominates or is at least equal to another, as this

is the case when only a cost is used to compare

labels. Unfortunately, the second criteria, the

balance of the routes, is not regular and the only

dominance rule which could guarantee the split

optimality is weak and does not permit to prune

enough labels to obtain strongly time efficient split

algorithm.

Thus, we introduce approximate dominance rules

which cannot guarantee the split optimality but

which are consistent with objectives and should

offer a compromise between split quality and

computation time related to the number of labels

kept on nodes.

Label 



approximately dominates label 



and only if

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



. 



. and 



. 



.





. 



. and 



. 



.

The dominance rule reduces the number of labels

stored at each node to a small subset. However, a

large number of labels could still be generated.

Thence, other time-saving approaches can be

proposed, such as limiting the number of labels per

node or the total number of lables generated during

the split process. Here, the number is limited only on

each label to n



. This principle, added into the

approximate dominance rules, results in some labels

pruning whereas they should not. Such restrictions

may allow to strongly reducing the CPU time, but

they are also raisons explaining why the proposed

algorithm does not guarantee to generate optimal

splitting.

For a detailed presentation of a shortest path

algorithm with resource constraints including a

specific algorithm for label comparison, it’s possible

to refer to (Duhamel et al., 2011) where a generic

algorithm dedicated to split with resource

constrained is introduced. We introduced hereafter,

basic split example decomposition from one giant

tour into a set of non dominated solutions.

2.4 Local Searches

The local search procedures implemented in the

framework rely on the first improvement selection

strategy. The local search is composed of 3 parts.

• Improvement of each trip by using classical VRP

neighborhoods such as 2-OPT and insertion

technique. It is limited to 



iterations per call. It

focuses only on cost reduction.

• Closure of the shortest trip with the objective to

minimize the solution trips balance (second criterion

to minimize).

• Reduction of the longest trip by using a careful

nodes transfer technique in existing trips. During

this part of the algorithm, worsening of the criterion

1 (the cost) is acceptable in the limit of 1.1 time the

initial cost. The objective is to avoid excessive waste

of time in exploring non-promising solutions for the

criterion 1.

2.5 Path Relinking

Numerous distance measures could be investigated

as stressed in the overview of (Sörensen and

Schittekat, 2013). In this paper, the one proposed by

(Zhang 2005) is used. They design an efficient

algorithm to compute the distance relates to the

minimum numbers of permutations required to

transform a sequence  into a sequence. The size

of the two sequences has to be the same and that is

totally compliant with the giant tour definition. The

proposed path relinking relies on giant tour and also

introduces progressively attributes from a guiding

solution into an initial solution to reduce the distance

defined by (Zhang 2005). Let us note  as a solution

and  the corresponding giant tour obtained by split

(the inverse function of the split algorithm, consiste

of concatenating the trips of a solution into a geant

tour). Given that a small change on  may produce

very distant solutions from , the path relinking

works only on promising solutions (and not

necessarily distant as preconized in most of the

papers). Thus, two solutions, selected in a given

population (POP or GPOP), are transformed into

tours and then linked through a path in this reduced

solution space. Each giant tour on the path, with a

given probability 



undergoes an evaluation

by split (producing potentially several solutions) and

local searches (also generating potentially several

non-dominated solutions). This strategy offers a

strong exploration.

3 NUMERICAL EXPERIMENTS

Numerical experiments were achieved on 14

classical problems first introduced by (Christofides

and Eilon, 1969) and (Christofides and al.,1979).

They are made of two groups, problems 1-5 and 11-

12 having a maximum vehicle range (travel time)

and problems 6-10 and 13-14 having not. In these

instances, the number of customers varies from 50 to

199.

3.1 Parameters

Results presented in this section have been achieved

by setting the parameters of the method (called SPR

in the sequel) as follow. Parameters were defining

using a short time tuning empirical process.

 n



/5

 n



25

 n



 100

 n



5

 n



 /2

 n



2∗

 prob











3.2 Comparative Study

To provide a fair comparative study, the computational

ASplitbasedApproachfortheVehicleRoutingProblemwithRouteBalancing

163

time of each method has been scaled by the

performance factor presented in Table 2. This

coefficient takes into account the MIPS performance

of each processor. A special attention must be

directed on the RISC 6000 computer which used 8

processors and author take advantages of the 8

processors. Since the 1.1 Ghz proc. is ranked about

125 MFlops, the whole computer provides a global

performances about 1Gflops (Table 1).

Table 1: Relative performances of computers.



(Jozefowiezet

al.,2009)

Ourproposal

(SPR)

Approachof

resolution

bi‐objective bi‐objective

Computer

RISC 6000

1.1Ghs

8processors

IntelXeon

2.40Ghz

OS Unix

Language C C

MFlops 1000Mflops 4850MFlops

Speedfactor 0.2 1

3.3 Analysis of Solutions

The numerical experiments encompasses the 14

well-known instances but the comparative study

with (Jozefowiez et al., 2009) is limited to instance

1-5 and 11-12 since their method (MOEA) is only

dedicated to this subset of instances, without vehicle

range.

3.3.1 Best Solution Cost

Table 2 exposes the results on the cost criteria.

Column  indicates the number of customers in the

related instance. BKS provides the best-known

solution cost.

Columns 3 (9) and 4 (10) indicate the best solution

cost obtained during the 5 runs of MOEA (SPR) and

the corresponding gap to BKS. Column 5 (11) gives

the balance associated to the solution cost. 



is the

CPU time reported in (Jozefowiez et al., 2009) in

seconds. 



Norm is the scaled time in seconds

according to Table 2. These times are the average

ones over the 5 replications and are representative of

the time efficiency of the method. Boldface

solutions represent dominance over the other

method.

For the first set of instances (1-5 and 11-12),

MOEA provides a gap of less than 1% for an

average computational time of 500 seconds and SPR

provides a gap of 2.24% with a computation time

350 seconds. It is possible to state that SPR

competes with the MOEA in terms of computational

time but provides a deviation greater than 2%.

However, this result is mainly due to instance 11 for

which SPR achieves very poor solution cost with a

gap around 10% from BKS. A comparison removing

this instance would lead to gaps equal to 1.12 and

0.96 for MOEA and SPR respectively, giving the

advantage to SPR. In fact, MOEA seems more stable

but it does not tackle instances with limitation on the

service provided by vehicles (problems 6-10 and 13-

14). SPR does and provides a good global

performance with a gap of 1.92% to BKS and by

retrieving 7 best known solutions. To conclude for

the solution cost, it is possible to state that SPR as a

range of application greater than MOEA. Although a

fair comparative study is difficult to manage, one

can note SPR and MOEA have similar computation

time and MOEA is quite better for a subset of

Table 2: Results for the extreme solutions on the cost criteria.

MOEA (Jozefowiez et al. 2009) SPR(ourproposal)

Instance n BKS

Cost Gap(cost) Balance









Norm

Cost Gap(cost) Balance





Norm

1 50 524.61 524.6 0.00 20.07 613.20 122.64 524.6 0.00 20.07 30.40

2 75 835.26

835.3 0.01 78.1 1522.80 304.56

843.7 1.01 93.08 142.40

3 100 826.14 827.4 0.15 67.55 2113.20 422.64 827.4 0.15 67.55 219.20

4 150 1028.14

1047.35 1.84 74.78 3936.00 787.20

1038.8 1.01 94.92 496.80

5 199 1291.45 1352.46 4.72 76.6 4968.00 993.60 1337.6 3.57 90.04 902.80

6 50 555.43 - - - - - 555.4 0.00 116.78 62.60

7 75 909.68 - - - - - 909.7 0.00 32.75 152.20

8 100 865.94 - - - - - 865.9 0.00 48.57 245.60

9 150 1162.55 - - - - - 1175.4 1.11 29.02 565.40

10 199 1395.85 - - - - - 1434.7 2.78 36.45 1363.40

11 120 1042.11 1042.11 0.00 146.67 2418.00 483.60 1145.7 9.94 135.89 419.00

12 100 819.56 819.6 0.00 93.43 2125.80 425.16 819.6 0.00 93.43 235.80

13 120 1541.14 - - - - - 1655.0 7.39 49.18 629.80

14 100 866.37 - - - - - 866.4 0.00 532.56 353.00

Avg. scale time (s) (instance

1-5 + 11-12)

505.63

349.49

Gap %

(instance 1-5 + 11-12)

0.96 2.24

Avg. scale time (s) (instance

1-14)

342.69

Gap %

(instance 1-14)

1.92

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instances. A second part of the analysis can focus on

the best solutions in terms of balance.

3.3.2 Best Solution Balance

In this section, the other extreme of the Pareto front

is analyzed, i.e. the best solutions according to the

balance criteria. In Table 3, the two first columns are

related to the instance. Cost and Balance represents

respectively the cost and the balance of the best

solution on the balance criteria found over the five

replications for both MOAE and SPR. Cost' and

Balance' are respectively the cost and the balance of

the solution found by SPR over the five replications

that have the closest (but smaller) balance to the

solution reported with MOEA. Boldface solutions

represent dominance over the other method.

Firstly, let us note that on a great majority of

instances, the right end solution of the front has a

lower balance with SPR than with MOEA. For

example, with the instance 1, the right hand solution

has a balance of 0.03 with SPR and 0.24 with

MOEA. This remark holds for all the instances. This

remark pushes us into accepting than the Pareto

front is well spread in the balance with SPR.

Table 3: Results for the right hand solutions on the balance

criteria.

MOEA SPR

n Cost Balance Cost Balance Cost' Balance'

1 50 618.22 0.24 783.39 0.03

611.50 0.20

2 75 1203.98 0.59 2153.38 0.31

1194.34 0.57

3 100 1871.06 0.29

1296.07 0.11 994.85 0.28

4 150 1484.48 0.80 1704.33 0.18

1338.50 0.71

5 199 1902.64 1.38 2571.97 0.30

1767.56 1.25

6 50 690.89 1.40

7 75 1141.87 3.58

8 100 1052.83 1.71

9 150 1662.29 4.13

199

1943.96 5.32

120 2388.30 0.10

1920.21 0.03 1485.09 0.10

100 1429.90 1.15

1272.55 0.10 1203.57 0.85

120

2502.85 0.64

100

1383.91 0.17

Avg. 1556.94 0.65 1671.70 0.15 1233.51 0.57

Secondly, when scanning the front obtained by SPR

to identify the closest solution balance to the best

solution balance found with MOEA, it appears that

the MOEA solution is always dominated. For

example, the right hand solution for instance 1 with

MOEA is (618.22; 0.24). The closest solution in the

SPR front (with respect to the balance criteria) is the

solution (615.58; 0.22) which has a lower cost. This

analysis suggests that the Pareto front with SPR

could compete with the MOEA front.

3.3.3 Front Analysis

Our research has been directed first on the cost

minimization and on the balance but not especially

on the quality of the front. This quality varies from

replications but on a wide majority of instances and

replications, the solutions minimizing the balance

are better that solutions reported by (Jozefowiez et

al., 2009). This comment must be moderated since

(Jozefowiez et al., 2009) does not provide any

evaluation of the obtained fronts and only solutions

for instances 1-5 and 9-10 are graphically presented.

The solution (524.61;20.06) at the left is the best

known solution of the problem considering the cost

criteria, and it is retrieved by both methods. The

solution (618.22; 0.24) is the best solution found by

MOEA on the balance criteria. Table 4 gives the

details of the right part of the front with solutions

closed to the MOEA solution. On this particular run,

the best balance achieved by SPR is related to the

solution (1239.62;0.05) and the closest but better

balance that MOEA brings is the solution

(615.58;0.22) which dominates the MOEA solution

(618.22;0.24).

Table 4: Details of the front.

Cost Balance

1 524.611 20.06

2 531.643 17.09

… … …

20 610.04 0.32

21 615.58 0.22

22 666.20 0.20

23 690.79 0.17

24 785.00 0.16

25 791.00 0.15

26 935.97 0.09

27 1239.62 0.05

4 CONCLUDING REMARKS

In this paper we have proposed a split based

algorithm for a bi-objective VRP, i.e. VRP with

route balancing in which both the total length and

the balance of the routes have to be minimized

(VRPRB). The proposition encompasses all the

well-known bi-objective VRP instances including

instances with range vehicle constraints, which have

not been addressed by previous papers. Preliminary

experiments show that the proposition permits to

obtain high quality solutions for the set of 14

instances and competes for the subset of 7 instances

with previous published works. Our research is now

directed to the Pareto front and to definition of an

ASplitbasedApproachfortheVehicleRoutingProblemwithRouteBalancing

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approach which could be validated on several bi-

objective routing problems.

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