Towards a Better Understanding of Genetic Operators for Ordering

Optimization: Application to the Capacitated Vehicle Routing Problem

S. Ben Hamida

, R. Gorsane

and K. Gorsan Mestiri

LAMSADE CNRS UMR 7243, Paris Dauphine University, PSL Research University, France

InstaDeep, Tunisia

Keywords:

Genetic Algorithms, Ordering Optimization, CVRP, Hybridization, Exploitation/Exploration.

Abstract:

Genetic Algorithms (GA) have long been used for ordering optimization problems with some considerable

efforts to improve their exploration and exploitation abilities. A great number of GA implementations have

been proposed varying from GAs applying simple or advanced variation operators to hybrid GAs combined

with different heuristics. In this work, we propose a short review of genetic operators for ordering optimization

with a classiﬁcation according to the information used in the reproduction step. Crossover operators could be

position (”blind”) operators or heuristic operators. Mutation operators could be applied randomly or using

local optimization. After studying the contribution of each class on solving two benchmark instances of the

Capacitated Vehicle Routing Problem (CVRP), we explain how to combine the variation operators to allow

simultaneously a better exploration of the search space with higher exploitation. We then propose the random

and the balanced hybridization of the operators’ classes. The hybridization strategies are applied to solve

24 CVRP benchmark instances. Results are analyzed and compared to demonstrate the role of each class of

operators in the evolution process.

1 INTRODUCTION

Ordering Optimization is a sub-ﬁeld of combinatorial

optimization consisting of searching for an arrange-

ment solution of an ordering problem such as order-

ing tasks in a scheduling problem or ordering cities

for the traveling salesman problem. Solutions are of-

ten encoded with an order based scheme using a rela-

tive order of the symbols in the genotype. A solution

is then a sequence of n ordered objects. A genotype

might be a string of numbers that represent a position

(number) in the sequence. Given n unique objects,

n! permutations of the objects exist, which make the

problem NP-Complete and hard to solve. The interest

in using GA for ordering application has been going

on since the 1990s. Throughout these decades, much

effort has been invested to propose powerful variation

operators as well as hybrid methods replacing opera-

tors with heuristic or meta-heuristic techniques. The

present study focus on genetic operators for ordering

optimization.

We propose in this paper a short review of vari-

ation operators for ordering optimization with GA.

We then propose to classify these operators accord-

ing to the information from the parent chromosomes

implied in the reproduction step. Thus, for crossover

operators, we distinguish between position operators,

using only the position of the genes (symbols) in the

chromosomes, and advanced or heuristic operators

using additional information such as the neighbor-

hood and the performance of the parents. Similarly,

we classify mutation operators into two categories:

Position mutation applying a random variation, and

local mutation applying local optimization on the mu-

tated solution.

Each class of operators is applied to solve some

selected benchmark instances of the Capacitated Ve-

hicle Routing Problem (CVRP). The performance of

each class is analyzed according to the distance of the

best solution to the optimum and the population con-

vergence speed. We then study the combination of

the different operators’ classes according to the explo-

ration/exploitation ability of each class. Otherwise, to

handle constraints of the CVRP application, we pro-

pose a simple procedure based on a segregational rank

that is introduced in the selection operators. This pro-

cedure gives advantage randomly to feasible or unfea-

sible solutions to preserve diversity in the population.

The paper is organized as follows. Section 2 re-

views the different categories of variation operators

Ben Hamida, S., Gorsane, R. and Mestiri, K.

Towards a Better Understanding of Genetic Operators for Ordering Optimization: Application to the Capacitated Vehicle Routing Problem.

DOI: 10.5220/0009832704610469

In Proceedings of the 15th International Conference on Software Technologies (ICSOFT 2020), pages 461-469

ISBN: 978-989-758-443-5

461

for order based optimization and explains the oper-

ational mode of the operators implemented for the

experimental study. Section 3 formulate the CVRP

problem, introduce the segregational selection for

handling constraints, and summarize the general loop

of the implemented GA and its parameters. The dif-

ferent classes of operators are studied in the section 4

through the ﬁrst series of tests. Then, the hybridiza-

tion strategies and their corresponding experimental

study are detailed in section 5.

2 REVIEW AND

CLASSIFICATION OF THE

VARIATION OPERATORS

Standard order-based variation operators for GAs are

essentially ”blind” operators acting regardless of the

ﬁtness of the solutions to cross or mutate. For order-

ing optimization, ”blind” crossover operators are in-

spired from standard binary crossovers, such as the

uniform crossover (Syswerda, 1989) and the order-

based crossover (OX) (Davis, 1985; M. et al., 1987;

Deep and Adane, 2011) described below. We design

this class of operators as ”Position Crossovers” since

they need only the sequences of symbols and their

positions to be applied. Similarly, several mutation

operators for ordering optimization change randomly

one or several symbols in different positions in the

chromosome. They are designed as ”Position Muta-

tion” operators. Four operators in this category are

described and illustrated in section 2.3.

To increase the GA exploitation ability, a great

number of advanced and heuristic crossover operators

have been proposed. An advanced crossover operator

considers an individual as the origin of its search and

tries to produce a better solution. They are designed in

this paper as ”Heuristic Crossovers”. They use some

heuristic information from the parents to hopefully

produce better offspring. The heuristic information

could be the adjacency describing gene neighbors in

the parents, the objective function, the neighborhood

map, distance between genes, etc. Three crossovers

are selected from this class for the experimental study:

the Distance Preserving crossover (DPX) (Freisleben

and Merz, 1996), the Alternating Edges Crossover

(AEX) (Grefenstette et al., 1985) and the Edge Re-

combination crossover (ERX) (Whitley et al., 1989).

As for Heuristic crossover, advanced mutation op-

erators use a local search algorithm that starts from a

solution and ends up in a local minimum where no

further improvement is possible. It aims at intensify-

ing the search by exploiting search paths determined

by the neighborhood of the corresponding solution

(Neri et al., 2012). The classical local search λ-opt

is studied in this work with λ=2 or 3. For further

intensiﬁcation of the search around the best area, a

stochastic Hill-climbing technique is implemented as

a local search operator.

The following subsection describes the variation

operators implemented for the experimental study.

The set of these operators and their corresponding

classes are summarized in table 1.

Table 1: Variation operators’ classes.

Operator Class Selected Operators

Position Crossover (PosCross): OX1, OX4, PMX

Heuristic Crossover (HeurCross): DPX, AEX, ERX

Position Mutation (PosMut): ISM, SWM, SHM, DM

Local Mutation (LocalMut): λ-Opt(λ=2/3), Hill-Climbing

2.1 Position or ”Blind” Crossovers

Position or ”blind” crossovers recombine the genetic

material of the parents into a new conﬁguration with-

out considering their initial performance in the pur-

pose to explore new research directions. Thus, they

can be seen as explorative more than exploitative op-

erators. The oldest operator in this category is the

uniform order-based crossover (Syswerda, 1989) that

generates a random binary template to decide from

which parent the gene is selected at each position.

Several operators were proposed in the same cate-

gory which main purpose is to produce feasible solu-

tions without considering their performance, such as

the order based crossover and the partially matched

crossover implemented for this work.

The Order-based Crossover (OX): OX, proposed

by Davis (Davis, 1985), is a variation of the uniform

crossover introduced in the purpose of preserving the

relative order of symbols in the sequences to be com-

bined. The ﬁrst implementation of the OX operator

(OX

) generates two cut points for both parents. The

symbols between the two cut points are copied to the

children. Then, starting from the second cut point

of one parent, the symbols from the other parent are

copied to one offspring in the same order, omitting

those which already exist.

Deep and Mebrahtu (Deep and Adane, 2011) pro-

posed three other variations of OX. In the ﬁrst variant,

the cut points in the two parents are at different posi-

tions, but the size of the substring between the cut

points is the same for both parents. The symbols are

copied to the offspring respecting the same rules de-

ﬁned for the original OX. This variant might also be

applied with different substring sizes (called OX

Partially Matched (or Mapped) Crossover (PMX):

The PMX crossover is a well-known operator pro-

posed by Goldberg and Lingle (Goldberg and Lin-

ICSOFT 2020 - 15th International Conference on Software Technologies

462

gle, 1985). It has two tasks: cross two parents and

prevent the repetition of symbols in the offspring. It

operates in two main steps: swap and map. The

ﬁrst step is quite similar to the OX. Two crossover

points are selected randomly. The sub-strings be-

tween these two points are exchanged to create two

incomplete offspring. In the second step, each off-

spring is completed using the remaining elements in

the corresponding parent. If the element is already

present, then a correction is performed according to

the mapping deﬁned with the matching section.

Other Position Crossovers: Several other crossover

operators for ordering optimization could be clas-

siﬁed as Position crossovers, such as the Max-

imal Preservative operator (PMX) (M

uhlenbein

et al., 1988), the voting recombination crossover

uhlenbein, 1989) and the Cycle Crossover (M.

et al., 1987). A description of these operators could

be found in (P

etrowski and Ben-Hamida, 2017).

Otherwise, some variants for most of the posi-

tion operators were proposed either for comparative

purposes or to be adjusted to the handled ordering

problem. For example, the Non-Wrapping Order

Crossover (Cicirello, 2006) is a variant of OX and

the Extend PMX (Tao, 2008) is a variant of PMX. A

short review with a comparative sutdy could be found

in (Karakati

c and Podgorelec, 2015).

2.2 Heuristic Crossovers

Distance Preserving Crossover (DPX): DPX

(Freisleben and Merz, 1996) generates an offspring

close to its parents by preserving some common

sub-strings. It starts by detecting common sub-paths

of two parents using the Hamming distance. Then, it

uses a greedy method to reconnect them and produce

a child.

Alternating Edges Crossover (AEX): AEX (Grefen-

stette et al., 1985) considers a genotype as a directed

cycle of arcs, where an arc is deﬁned by two succes-

sive symbols. To create a child from two parents, arcs

are chosen in alternation from the two parents, with

some additional random choices in case of unfeasibil-

ity. AEX starts by choosing the arc (g

→ g

) from

the ﬁrst parent. Next, the arc having as ﬁrst gene g

the second parent is selected which the second gene

is added to the child. AEX proceeds in the same way

until the child is complete.

Genetic Edge Recombination Crossover (ERX):

ERX (Whitley et al., 1989) considers a genotype as an

edge and it is based on the gene adjacency. The aim is

to keep as many neighbor as possible in the offspring.

ERX starts by building the ”edge-map” that gives for

each vertex the set of edges that start or ﬁnish on it.

To produce an offspring, ERX starts from a ran-

dom symbol designed as current symbol x

. It ﬁnds

the element in the neighbor set of x

having the lower

neighbor set cardinality. This element is added to the

child, removed from the ”edge-map” and becomes the

new current element. The process continues until all

entries are visited.

Other Heuristic Crossovers: Grefenstette proposed

in (Grefenstette, 1987) a set of Heuristic crossover for

the TSP: HGreX, HRndX, HpPoX. As AEX, HGreX

considers the genotype as a directed cycle of arcs. The

Inver-Over operator (Tao and Michalewicz, 1998) can

be classiﬁed in the same category as it considers a

solution as a directed graph. It invert some substring

from a given parent in the hope to improve the ﬁtness.

2.3 Position or ”Blind” Mutation

Random or ”blind” mutation is widely used for com-

binatorial order-based optimization for explorative

purpose and to maintain the population diversity

etrowski and Ben-Hamida, 2017). Four operators

are selected and implemented for this work: exchange

mutation or swap mutation (SWM) (M. et al., 1987),

displacement mutation (Michalewicz, 1992) (DM),

shifting mutation (SHM) (Fogel, 1988) and insertion

mutation (ISM). SWM selects two random positions

along the parent string and exchange the correspond-

ing symbols. SHM chooses randomly two positions

on the parent genotype. Then, the symbol in the ﬁrst

position is shifted from its current position to the sec-

ond position. DM displaces a randomly selected sub-

sequence in the parent genotype.

Note that several other mutation operators having

the same goal have been proposed for ordering op-

timization such as the scramble operator (Syswerda,

1991) that selects a random sub-sequence in the par-

ent genotype and scrambles its symbols, or the in-

version mutation operator (Fogel, 1988) that inverts a

randomly selected sub-string in the parent genotype.

2.4 Local Mutation

The idea of introducing a local search technique in the

GA engine for combinatorial problems is supported

by several researchers since many decades ago. Sev-

eral works in the literature demonstrated that such hy-

bridization improves the exploitation of best solutions

and might speed convergence to the optimal or to a

near-optimal solution (Neri et al., 2012). We present

below two simple techniques to implement a local

search in the mutation step, the λ-Opt local search and

the stochastic Hill-climbing.

Towards a Better Understanding of Genetic Operators for Ordering Optimization: Application to the Capacitated Vehicle Routing Problem

463

λ-Opt Mutation: λ-Opt is a simple local search al-

gorithm. The main idea behind it is to take a route

that crosses over itself and reorder it so that it does

not. It works as follows. λ edges are removed from

the tour of the corresponding solution. The remain-

ing segments are reconnected in all possible ways. If

any improvement is identiﬁed, then the corresponding

movements are applied to the solution.

Stochastic Hill-Climbing: The simplest way to per-

form a local search in a combinatorial space is to per-

turb a trial solution S and to replace it by the new solu-

tion if it performs better. This procedure is known as

Hill-Climbing (HC). It tries to explore the neighbor-

hood of a given solution and to move it to the eventu-

ally best direction. HC can be applied using the ran-

dom genetic mutation operators to generate solutions

in the neighborhood of the trial solution. However, for

an ordering problem, the neighborhood is very large

and a full exploration is impossible. The stochastic

Hill-Climbing is widely applied in this case. It picks

a single random neighbor and accepts it if it is better

than S. The algorithm terminates upon a computa-

tional limit.

3 APPLICATION TO THE

CAPACITATED VEHICLE

ROUTING PROBLEM

Ordering optimization problems in the real world are

generally submitted to a set of constraints. For this

reason, we chose for our study the Capacitated Vehi-

cle Routing Problem (CVRP) described in the follow-

ing subsection. The rest of the section describes the

GA implemented for the experimental study.

3.1 CVRP Formulation

The vehicle routing problem (VRP) is an important

problem class in the ﬁeld of operations research and

transport logistics optimization. Its original formula-

tion has been deﬁned over 60 years ago by Dantzig

and Ramser(Dantzig and Ramser, 1959) and consists

of a ﬂeet of identical vehicles serving a set of cus-

tomers with a certain demand from a single depot

and having a certain capacity. This formulation is re-

ferred to as the Capacitated Vehicle Routing Problem

(CVRP).

The mathematical formulation of the CVRP can

be deﬁned as follows: let G = (N,A) be an undirected

graph, where N is the set of nodes N =

{

0,1,··· ,n

}

and E =

{

(i, j) : i, j ∈ N,i 6= j

}

is the set of edges

joining the nodes. Node 0 refers to the depot from

which the vehicles start from and comeback to. The

other nodes represent the customers having each a

known non-negative demand q

for customer i. A set

of K vehicles N =

{

,··· ,V

}

having each a max-

imum capacity Q is provided. The travel distance be-

tween node i and j is deﬁned by d

i j

> 0.

The CVRP can be stated as follows:

min

∑

i=0

∑

j=0

∑

k=1

i j

i, j

(1)

Where

i, j



1 if V

travels from node i to node j

0 otherwise

(2)

Subject to:

∑

k=1

∑

i=0

i, j

= 1, j ∈

{

1···N

}

,i 6= j (3)

∑

k=1

∑

j=0

i, j

= 1,i ∈

{

1···N

}

,i 6= j (4)

∑

i=0

∑

j=0

i, j

≤ Q

,k ∈

{

1···K

}

(5)

∑

j=1

0, j

−

∑

j=1

j,0

= 0,k ∈

{

1···K

}

(6)

∑

k=1

0, j

≤ K (7)

The objective function deﬁned in (1) minimizes the

total travelled distances by all the vehicles. Constraint

sets (3) and (4) guarantee that each customer is visited

only once. Constraint set (5) ensures that the total re-

quest of the customers being served by a vehicle does

not exceed the vehicle capacity. Constraint set (6)

guarantees that all routes start and ﬁnish at the depot,

and constraint (7) requires that there are a maximum

of K routes for serving the customers.

To encode a VRP solution, several representations

are possible: vector, matrix, graph, etc (P

etrowski

and Ben-Hamida, 2017). The most common encoding

technique is the intuitive path-encoding in which each

vehicle route is designed by a sequence of numbers

corresponding to the requests indexes in the request

vector R. In the present work, to encode a CVRP

solution S, two strings of positive numbers are used:

S = (P,L), where P designs the global path of the so-

lution deﬁned by the connection of the different ve-

hicle routes without including the depot, and L is a

string vector designing the routes lengths of all the

vehicles.

ICSOFT 2020 - 15th International Conference on Software Technologies

464

3.2 Handling Constraints

To handle constraints without implementing costly

tools or adding new parameters, we propose the segre-

gational selection for the parental selection. The main

idea of the segregational selection is to sort the pop-

ulation (or a set of individuals for tournament selec-

tion) either according ﬁrst to ﬁtness (total distance for

CVRP) and then violation rate (capacity overﬂow for

CVRP), or according ﬁrst to violation rate and then

total distance. Segregational selection aims to bal-

ance between best feasible solution and unfeasible so-

lutions having the lowest total distance.

3.3 The Algorithm

For the experimental study, the implemented gener-

ational genetic loop respects the Steady State GA

(SSGA). Indeed, with the SSGA, at each iteration,

only two parents are selected to generate a new off-

spring that replaces the worst offspring in the popula-

tion. In our case, the new offspring is generated with

a reproduction procedure using the different variation

operators described in section 2.

Algorithm 1: SSGA.

1: Initialize population with µ feasible solutions

2: Evaluate initial population

3: while non STOP condition do

4: select two parents (P

, P

) using the segrega-

tional tournament selection

5: C ← Reproduction(P

, P

)

6: set W the worst solution according to either ﬁt-

ness or violation rate

7: Replace W with C in the parent population

8: end while

3.4 Benchmark Data and Parameters

Setting

SSGA with the variation operators is implemented in

Python language. Its performance is then validated

on some selected benchmark instances from the set of

CVRPs proposed by (Augerat et al., 1995) available

on the ”VRP Web” site

. The set A contains 27 in-

stances with a number of requests n varying from 32

to 80 and a number of vehicles k varying from 5 to

10. The vehicle capacity is set to 100 for all the prob-

lems. For a comparison purpose, the optimal or the

best-known solution (BKS) is given with the SSGA

results. A set of preliminary tests have been applied

to tune the SSGA parameters and the retained values

http://www.bernabe.dorronsoro.es/vrp/

are summarized in table 2. As shown in the table,

some parameters may have different values from one

experimental study to another.

Table 2: Parameters used for SSGA.

Hyperparameter Value

Crossover probability 0.8

Mutation probability 0.4

Population size 200

Tournament size 8

Maximum nb of it 50000 (1

study)

300000 (2

study)

It without improvement 10000

Runs per Instance 20 (1

study), 30 (2

study)

4 COMPARING OPERATORS

CLASSES

To study the SSGA behavior when applying one of the

operator classes introduced above, two benchmarks

instances are selected: the ﬁrst has 34 requests and 5

vehicles (n34-K5) and the second has 39 requests and

5 vehicles (n39-K5). The corresponding BKS are 778

and 822 respectively. The GA parameters applied for

this study are set as given in table 2. A ﬁrst series of

tests are run using a single operator class at a time. In

the second series of tests, each crossover class is ap-

plied with one operator selected randomly from the

two mutation classes. Four combinations are then

studied: (PosCross+PosMut), (PosCross+LocalMut),

(HeurCross+PosMut), (HeurCross+LocalMut).

To compare the different combinations, we ex-

tract some synthetic information from the obtained

results. First, we compute the minimum and aver-

age relative distance (loss) to the optimum: (%loss =

100 ∗ (Best − optimum)/optimum). Results are illus-

trated in ﬁgure 1. Second, we record the relative con-

vergence speed according to the maximum number of

iterations deﬁned in the parameters’ settings. The aim

is to visualize if the evolution has stopped due to a

stagnation problem or because of the maximum num-

ber of iterations is reached. A high value generally

reﬂects rapid, often premature convergence. Results

are illustrated in the ﬁgure 2. Apart from that, to visu-

alize the exploration/exploitation ability of the opera-

tors classes along the evolution, the best ﬁtness spot

of the best run of each test case is illustrated with a

curve in ﬁgure 3.

Figures 1 (a) and (b) show clearly that the per-

formance of each class of operators depends on the

problem to solve. For the n34-K5 case, mutation op-

erators, either in PosMut or LocalMut class, were able

to bring the population near the optimal solution. It

wasn’t the case with n39-k5, which is more difﬁcult

Towards a Better Understanding of Genetic Operators for Ordering Optimization: Application to the Capacitated Vehicle Routing Problem

465

0,0%

10,0%

20,0%

30,0%

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

PosCross HeurCross PosMut LocalMut

1,8%

6,0%

4,2%

7,9%

0,9%

5,5%

0,9%

5,5%

(a) n34-K5 instance

0,0%

10,0%

20,0%

30,0%

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

PosCross HeurCross PosMut LocalMut

9,2%

14,2%

4,5%

26,0%

4,7%

27,4%

6,9%

27,3%

(b) n39-K5 instance

-1,0%

4,0%

9,0%

14,0%

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

PosC+PosM PosC+LocalMHeurC+PosMHeurC+LocalM

0,9%

5,2%

1,2%

4,6%

0,0%

3,4%

1,8%

4,7%

n34-K5 instance

0,0%

2,0%

4,0%

6,0%

8,0%

10,0%

12,0%

14,0%

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

Min

Loss

Avg

Loss

PosC+PosMPosC+LocalMHeurC+PosMHeurC+LocalM

1,0%

7,6%

0,5%

9,2%

1,8%

12,1%

0,5%

9,2%

n39-K5 instance

Figure 1: Minimum and average loss according to BKS ob-

tained by each class or combination of classes of operators

for the CVRP instances n34-K5 and n39-K5.

to solve than n34-k5. For the difﬁcult cases, the use

of both crossover and mutation operators is needed

in evolutionary search. Indeed, Figures 1(c) and (d)

show that the deviation to the optimum is greatly

improved with the combined classes. For example,

with the combination (PosCross+LocalMut), the de-

viation of best solution is about only 0.5% to the op-

timum (Fig 1(d)), while it is about 9.2% for PosCross

and 6.2% for LocalMut (Fig 1(b)) when applied with

combination. Similarly, combining operators classes

for the n34-K5 case allows a better exploration of

the search space, especially for the HeurCross class

which its combination with PosMut class has allowed

it to easily ﬁnd the optimum.

0,0% 20,0% 40,0% 60,0% 80,0%

PosCross

HeurCross

PosMut

LocalMut

PosCross

HeurCross

PosMut

LocalMut

K34-K5 K39-K5

Average convergence speed of the different operator

classes

0% 10% 20% 30% 40% 50% 60% 70%

PosC+PosM

PosC+LocalM

HeurC+PosM

HeurC+LocalM

PosC+PosM

PosC+LocalM

HeurC+PosM

HeurC+LocalM

K34-K5 K39-K5

Average convergence speed of different operator

classes' combinations

Figure 2: Convergence Speed according to the maximum

number of iterations for each class or combination of classes

of operators for the CVRP instance n34-K5 and n39-K5.

Otherwise, it can be noted through the ﬁgures 1

and2 that some classes of operators have a great abil-

ity to perform local search around the best solutions.

This great exploitation capacity, applied with a high

probability since the ﬁrst steps of the evolution, bring

the population around local optima. The SSGA diver-

siﬁcation process helps in some cases to explore other

regions but ends up falling back into the trap of pre-

mature convergence. This is the case for HeurCross

class operators.

Other classes of operators have the opposite prob-

lem. Their exploration capacity allows the population

to continuously explore new regions in the research

space and avoid local optima. It is the case of the Pos-

Mut class. However, the convergence is too slow and

does not necessarily lead to the optimum region. This

is due to the lack of effective exploitation of the best

solutions. The question now is how to take advantage

of the strengths of each class and create a perfect bal-

ance between exploitation and exploration along the

evolution.

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(b) n39-K5: Evolu�on of the best ﬁtness

with one operator class

Pos Mut

Local mut

Heur Cross

Pos Cross

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HeurC+LocalM

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(d) n39-K5: Evolu�on of the best ﬁtness

with operator classes combina�on

PosC+LocaM

PosC+PosM

HeurC+PosM

HeurC+LocalM

Figure 3: Evolution of the best ﬁtness of the best run in the

series of tests for solving the cases n34-k5 and n39-k5.

Combining Genetic Algorithms with local search

techniques in the mutation step is widely used and

proved to be efﬁcient in speeding up the evolution

process for ordering optimization. However, when

applied with heuristic crossover, this advantage can

become a difﬁculty. Indeed, if the two steps of re-

production have as common objective to exploit the

best solutions to produce more competitive ones, the

exploration of the research space is reduced and the

population is attracted to local optima. This is the fa-

mous dilemma between exploitation and exploration

for genetic algorithms. We believe that it is important,

for ordering optimization, to control the simultaneous

use of different classes of operators. Thus, we pro-

pose and test in the following section two strategies

to allow a simultaneous use of all classes of opera-

tors. The second strategy introduces in addition a re-

strictive combination between classes to control the

population convergence.

ICSOFT 2020 - 15th International Conference on Software Technologies

466

5 COMBINING VARIATION

OPERATORS

5.1 Hybridization Strategies

As concluded in the ﬁrst comparative study, the ex-

ploitation/exploration abilities of the variation oper-

ators differ from a class to another. Each opera-

tor can help in the evolution of the population to-

wards better areas in the search space but in a dif-

ferent way. We then make the ﬁrst hypothesis: ”Us-

ing variation operators from different classes improve

the GA efﬁciency for ordering optimization.” How-

ever, heuristic crossover when applied with local mu-

tation can lead to a premature convergence. To pre-

vent this phenomenon, we propose to combine heuris-

tic crossover with random mutation, and similarly, po-

sition crossover with local mutation. The objective is

that random operators control the convergence speed

of heuristic/local operators and vise versa . Thus, we

make the second hypothesis: ”An operator class with

high exploitation ability could be controlled with an

operator class with a high exploration ability”.

To test the two hypotheses, we propose below two

hybridization strategies. The ﬁrst one is the random

hybridization where the crossover and mutation oper-

ators are chosen randomly from the set of all the op-

erators’ classes. The second strategy is the balancing

hybridization which is a conditional hybridization. If

one class of operators is used for crossover, it will not

be used again for mutation. Note that the operator

within the given class is chosen randomly.

5.1.1 Random Hybridization (RH)

Random hybridization is quite equivalent to the multi-

operator mechanism proposed in (Pulji

c and Manger,

2013) with an additional condition on the set of vari-

ation operators which must includes some operators

from each class. To produce an offspring with the RH

strategy, a crossover operator is selected randomly

from the PosCross or HeurCross classes. The pro-

duced offspring is then mutated using a random op-

erator from PosMut or LocalMut classes (Algo. 2).

There is no additional restriction on the choice of op-

erators either for crossover or mutation.

Algorithm 2: RH Reproduction(P

, P

1: select randomly opX in PosCross ∪ HeurCross

2: With a probability p

, apply opX to (P

, P

) and

get the new child C

3: SELECT randomly opM in LocalMut ∪ PosMut

4: With a probability p

, apply opM to C

5.1.2 Balancing Hybridization (BH)

Balancing hybridization aims to balance the GA ex-

ploration and exploitation abilities by mixing heuris-

tic operators with position operators. Thus, if an

exploitative offspring is generated with an heuristic

crossover, it can be mutated only with a position mu-

tation operator to add an exploratory dimension. Sim-

ilarly, if an exploratory child is generated with a po-

sition crossover, it can be mutated only with a lo-

cal search based mutation for an exploitative purpose

(Algo. 3).

Algorithm 3: BH Reproduction(P

, P

1: select randomly opX in PosCross ∪ HeurCross

2: With a probability p

, apply opX to (P

, P

) and

get the new child C

3: if opX ∈ PosCross then

4: SELECT opM from LocalMut class

5: else

6: SELECT opM from PosMut class

7: end if

8: With a probability p

, apply opM to C

5.2 Results and Discussion

For each series of 30 runs, we record: the cost of the

best obtained solution (Best), the average cost over

the thirty runs (Average) and the percentage of the

relative deviation (loss) from the best-known solution

(BKS): %Loss = 100 ∗ (Best − BKS)/BKS (%Devia-

tion). Results for the second experimental study are

given in table 3. The relative deviation values (loss)

are also illustrated in ﬁgure4 to make easier the com-

parison between the RH and BH procedures.

The different values recorded for each bench-

mark instance demonstrate that the SSGA efﬁciency

is greatly improved with the hybridization strategies,

either random or balanced hybridization. Indeed, the

deviation or loss to the optimum is less than 1% for

11 instances with the ”RH reproduction” and for 10

instances with the ”BH Reproduction” as illustrated

in ﬁgure 4. These results conﬁrm our ﬁrst hypothesis

that the application of the multi-operator mechanism

including operators from the four classes enhances the

efﬁciency of the Genetic Algorithm. However, except

for some cases, ”BH Reproduction” was not able to

outperform the ”RH Reproduction” as we supposed in

the second hypothesis. As illustrated in ﬁgure 4, with

the RH Reproduction”, the GA is able to get closer

to the optimum than with ”BH Reproduction”, espe-

cially for the cases with a number of requests greater

than 42 (except for the n65-k5 case). Through these

Towards a Better Understanding of Genetic Operators for Ordering Optimization: Application to the Capacitated Vehicle Routing Problem

467

Table 3: Results with CVRP A.

Random Hybridization Balancing Hybridization

Instance BKS Best Average % Deviation Best Average % Deviation

A-n32-k5.vrp 784 784 806 0 784 803 0

A-n33-k5.vrp 661 661 689 0 661 694 0

A-n33-k6.vrp 742 742 757 0 743 761 0.13

A-n34-k5.vrp 778 778 797 0 778 800 0

A-n36-k5.vrp 799 805 831 0.75 814 840 1.88

A-n37-k5.vrp 669 670 706 0.15 670 703 0.15

A-n37-k6.vrp 949 953 1009 0.42 953 1036 0.42

A-n38-k5.vrp 730 731 763 0.14 733 759 0.41

A-n39-k5.vrp 822 834 863 1.46 825 864 0.36

A-n39-k6.vrp 831 839 866 0.96 839 875 0.96

A-n44-k7.vrp 937 947 984 1.07 957 981 2.13

A-n45-k7.vrp 1146 1166 1203 1.75 1176 1206 2.62

A-n46-k7.vrp 914 953 986 4.27 923 978 0.98

A-n48-k7.vrp 1073 1115 1164 3.91 1107 1150 3.17

A-n53-k7.vrp 1010 1031 1100 2.08 1038 1189 2.77

A-n54-k7.vrp 1167 1193 1362 2.23 1222 1548 4.71

A-n55-k9.vrp 1073 1075 1168 0.19 1104 1507 2.89

A-n60-k9.vrp 1408 1414 1596 0.43 1405 1619 3.77

A-n63-k10.vrp 1315 1378 1846 4.79 1349 1456 4.57

A-n63-k9.vrp 1634 1685 2441 3.12 1717 2960 5.08

A-n64-k9.vrp 1402 1472 1903 4.99 1459 2136 4.07

A-n65-k9.vrp 1177 1289 2698 9.52 1219 1754 3.37

A-n69-k9.vrp 1168 1186 1491 1.54 1219 1549 4.37

results, we can understand that there is a synergy be-

tween the different classes of operators. So it is use-

ful to combine the different classes without restric-

tion. However, HeurCross and LocMut classes are

helpful to reﬁne the best solutions in the second step

of the evolution but could penalize the search process

at the beginning of the evolution. Thus, we think that

is advantageous to control the application of the dif-

ferent combination between operators classes with a

dynamic probability.

A-n32-k5

A-n33-k5

A-n33-k6

A-n34-k5

A-n36-k5

A-n37-k5

A-n37-k6

A-n38-k5

A-n39-k5

A-n39-k6

A-n44-k7

A-n45-k7

A-n46-k7

A-n48-k7

A-n53-k7

A-n54-k7

A-n55-k9

A-n60-k9

A-n63-k10

A-n63-k9

A-n64-k9

A-n65-k9

A-n69-k9

% devia�on to the BKS

% devia�on of best obtained solu�on for each

benchmark case

RH BH

Figure 4: Relative Deviation to the optimum given by the

best solution obtained with RH and BH reproduction.

6 CONCLUSION

This paper proposes a classiﬁcation of the variation

operators used for evolutionary ordering optimiza-

tion into four classes: Position crossover, Heuris-

tic crossover, Position mutation and Local mutation.

After studying the exploitation/exploration ability of

each class on solving two benchmark instances of

the Capacitated Vehicle Routing Problem, we pro-

posed two hybridization strategies between operators

classes: Random Hybridization and Balancing Hy-

bridization. Results obtained for 24 CVRP bench-

mark instances showed clearly that combining vari-

ation operators from different classes increases the

GA robustness when solving constrained ordering op-

timization problems. However, the balancing hy-

bridization needs further control on the application of

the different classes combinations. Future works aim

to introduce this control with a dynamic or adaptive

probability and apply the implemented GA for solv-

ing other CVRP benchmark series.

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