Tailoring a Red Deer Algorithm to Solve a Generalized Network

Design Problem

Imen Mejri

, Safa Bhar Layeb

and Emna Drira

LR-OASIS, National Engineering, School of Tunis, University of Tunis El Manar, Tunis, Tunisia

Keywords: Network Design Problems, Metaheuristics, Bio-inspired Algorithms, Red Dear Algorithm.

Abstract: This work investigates solving a challenging network design problem using the recently introduced

evolutionary metaheuristic, namely the Red Deer Algorithm (RDA), that mimics the Scottish Red Deer’s

behavior during their breeding season. The RDA is tailored to solve a generalized network design problem

that aims to design a network with minimal cost while satisfying several practical constraints. To assess the

performance of this new bio-inspired metaheuristic on solving such NP-hard problem, computational

experiments were conducted on Benchmark as well as real-world instances. Computational experimentation

illustrates the accuracy of the RDA that outperforms all of the existing recent metaheuristics.

https://orcid.org/0000-0002-2182-8558

https://orcid.org/0000-0003-2536-7872

1 INTRODUCTION

Metaheuristics in general and bio-inspired

metaheuristics in particular are still of interest to

researchers as well as practitioners (e.g. Almufti et

al., 2021; Ma et al., 2019) for solving real-world

engineering design problems. The recent surveys of

Swan et al. (2021) and Osaba et al. (2021) are

dedicated to these cutting edges evolutionary

computation techniques. Actually, there are several

lately released metaheuristics that are increasingly

applicable to solve NP-hard problems. Precisely, the

Red Dear Algorithm (RDA) is a new nature-

inspired metaheuristic which paradigm was first

introduced by Fathollahi-Fard and Hajiaghaei-

Keshteli(2016) and then well-engineered by

Fathollahi-Fard et al. (2020a). The RDA mimics the

mating behavior of Scottish Red Deer during their

breading season. This new evolutionary bio-inspired

metaheuristic has shown its performance for solving

the vehicle routing problem, the travelling salesman

problem and the single-machine problem, which are

known as NP-hard problems (Fathollahi-Fard et al.,

2020a).

Due to their high ability to deal with a wide

range of real-world problems, Network Design

Problems (NDPs) represent the largest category of

combinatorial optimization in the fields of

industries, logistics, telecommunications and energy

(Mejri et al., 2021). In terms of graph theory, a NDP

is modeled by a set of nodes representing for

example power plants in an energy system (e.g.

Singh et al.,2021) or warehouses in a distribution

system (e.g. Layeb et al., 2018). Eventually, those

nodes have to inter-communicate to exchange

numerous types of merchandises, data, power, etc.

In this purpose, edges link adequate pairs of nodes

chosen properly. Those links are known as arcs

generally characterized by capacities, variable costs,

fixed costs, facilities, etc. Depending on the

situation, an edge can model a road in transportation

problems, a high voltage in energy systems, and so

on (e.g. Mejri et al., 2019a; 2019b). To sum up, the

objective of the network design problem consists on

generating the optimal configuration that satisfies

the maximum demands partially/totally with the

minimum possible total system cost. According to

the recent survey of Salimifard and Bigharaz (2020)

about the classification of network design problems

and their resolution approaches during the last two

decades, exact methods are less and less used since

2004. However, the progress of metaheuristics is

noticed during this period and is expected to

continue to grow in the future as the size of the

studied problems increases steadily.

Mejri, I., Layeb, S. and Drira, E.

Tailoring a Red Deer Algorithm to Solve a Generalized Network Design Problem.

DOI: 10.5220/0010689000003062

In Proceedings of the 2nd International Conference on Innovative Intelligent Industrial Production and Logistics (IN4PL 2021), pages 32-39

ISBN: 978-989-758-535-7

 2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reser ved

This work investigates a challenging variant of

NDPs, namely the Generalized Discrete Cost Multi-

commodity Network Design Problem (GDCMNDP)

(Khatrouch et al., 2019; Mejri et al., 2019c). Each

edge is characterized by several bidirectional

facilities with discrete prefixed costs and capacities.

In telecommunication, facilities model a set of

available technologies that provides high bandwidth

connections, each transmitting services and/or

information at different rates. Given the

multicommodity flow demands to route partially or

totally, the GDCMNDP mandates the installation of

no more than one facility on each edge to minimize

the sum of the fixed facility installation costs and

the penalties of unrouted demands. Not surprisingly,

the GDCMNDP is an NP-hard combinatorial

problem. This highly challenging generalized

network design problem was recently addressed by

Khatrouch et al. (2019). The authors proposed three

basic greedy heuristics as well as three

metaheuristics, namely a Genetic Algorithm (GA), a

Biogeography-Based Optimization method (BBO),

and a hybrid Genetic Algorithm coupled with a

Variable Neighborhood Search procedure (GA-

VNS). Computational experiments highlights that

the GA-VNS leads to the best performance.

Besides, a stochastic version of the GDCMNDP,

with uncertain amounts of flow demands, was

effectively solved using a simulation-based

optimization framework (Mejri et al., 2019c).

To the best of our knowledge, this work is the

first to tailor the Red Deer Algorithm in order to

solve a generalized Network Design Problem, and

more precisely, the highly challenging GDCMNDP.

Computational experiments on Benchmark instances

and real-world instances from the literature reveal

that the RDA generates good solutions within very

fair computation times and outperforms the existing

meta-heuristics (Khatrouch et al., 2019).

The rest of the article is organized as follows. A

mathematical formulation of the Generalized

Discrete Cost Multi-commodity Network Design

Problem is detailed in Section II. Then, Section III

presents the tailoring of the Red Deer Algorithm for

solving the GDCMNDP. The computational results

are detailed in Section IV. Finally, Section V draws

conclusions and future research avenues for this

work.

2 MATHEMATICAL MODEL

To model NDPs, combinatorial optimization

formulations (e.g. Ennaifer et al., 2016) and in

particular linear programming based models such as

the path-based formulations (e.g. Mejri et al., 2019c;

2019d) and the flow-based formulations (e.g. Mejri

et al., 2018; Layeb et al., 2017) are commonly used.

More precisely, the GDCMNDP is defined on an

undirected connected graph G=(V,E) where V is the

set of nodes indicating all possible switching centers

or customers, and E is the set of edges,

corresponding to all possible optical fibers

connections between the centers, in the field of

telecommunication. To derive a valid mathematical

model for the GDCMNDP, let’s begin by

introducing the following notations:

Sets:

V Set of nodes, indexed by i

E Set of edges, indexed by e={i,j}, i, j

∈

A Set of arcs derived from E such that for

each edge e = {i,j}

∈

E, two directed arcs

are generated (i,j) and (j,i)

∈

K Set of distinct point-to-point multi-

commodities, indexed by k

Set of potential facilities representing

types of optical fibers that can be installed

on edge e

∈

E, indexed by l

Parameters:

Fixed cost of installing facility l on edge e

Bidirectional capacity of facility l on edge

Source node of commodity k

Sink node of commodity k

Flow demand of commodity k to route

from s

to t

Service cost, representing penalty to be

paid per unit of commodity demand not

routed k

Decision Variables:

Continuous non-negative variable that

reflects the amount of flow of commodity

k circulating through arc (i,j)

Continuous non-negative variable that

reflects the commodity demand not routed

Binary variable that equals to 1 if facility l

is installed on edge e, 0 otherwise

Tailoring a Red Deer Algorithm to Solve a Generalized Network Design Problem

Using these notations, a Mixed Integer Linear

Programming (MILP) formulation of the

GDCMNDP can be stated as follows:

EeLl

ZpYcMinimize







(1)

The objective function (1) minimizes the total

installation fixed costs and penalties involved in

unrouted multicommodity demands. It should be

minimized subject to the following constraints:

EeY







(2)

For each edge, Constraints (2) impose the

selection of a maximum of one facility.





















Ajij

kkk

Aijj

tiifZd

KktisiViif

siifZd

),(:),(:

)(

,,,0

(3)

For each node and for each commodity,

Constraints (3) compel the flow conservation

principle.



EjieYuXX

KkLl







(4)

For each edge, Constraints (4) force that the bi-

directional flow through the two arcs to not surpass

the installed capacity.

KkAjiX

 ,),(0

(5)



LlEeY  ,,1,0

(6)

KkZ





 0

(7)

Constraints (5)-(7) indicate the nature of the

considered decision variables.

This yields to Model (1)-(7) that is clearly a

valid mathematical formulation of the DCMNDP.

Besides, it is a so-called compact MILP formulation,

in terms of the combinatorial optimization. Thus, it

is possible to solve it directly by a commercial MIP

solver. It is worthy to mention that Model (1)-(7)

will unsurprisingly become intractable in practice

and could not solve the GDCMNDP to optimality,

once the size of the instances increases. Therefore,

we invoked the RDA metaheuristic to find good

feasible solutions for this NP-hard problem.

3 RED DEER ALGORITHM

Recently addressed and well-established by

Fathollahi-Fardet al. (2020a), the Red Deer

Algorithm is a new nature-inspired population-based

metaheuristic inspired from the Scottish Red Deer

mating behavior during the breeding season.

Precisely, from the end of September till the end of

November begins the Scottish Red Deer breading

season, called also the rut. During mid to late

autumn, stags (Red Deer males) come back to the

hinds (groups of female Red Deer) territory in order

to be engaged in disputes with other males by

starting to show their masculine power and display

of dominance including roaring, parallel walks and

fighting. The reproductive success in males is

directly attached to their behavior during the mating

season and also their physical strength citing the

body size, the strength, the roaring and the

development of antlers. At the end of a battle, some

males can be seriously injured or can even die. After

that the dominant male chases the week one and

starts mating with the hinds. This is how the natural

selection is generated.

As an evolutionary metaheuristic, the proposed

RDA starts with an initial population of Red Deer

candidates composed of the Red Deer males and

Red Deer females (Fathollahi-Fard et al., 2020a).

After the roaring and fighting phase between males

also called 'intensification phase', males are divided

into two groups: ‘male commanders’ (the strongest

males to win the fights) and ‘stags’ (the chased

males due to their weakness or injuries during

fights) according to their strength and at this end

‘harems’ are formed. A harem is a group of hinds

whose size is related to the commanders' abilities

and power in the process of roaring, fighting and

mating. Thus, the more the male is strong, the more

the harem is large. Besides, the male commander of

each harem is mating with α percent of hinds in his

own harem and also with β percent of females of the

closest harem to his territory which represents the

so-called 'diversification phase'. Across the mating

process, new candidates appear as offspring of the

current candidates, i.e. the next generation.

Based on the research of Fathollahi-Fard et al.

(2020a), we tailor the RDA to solve the

IN4PL 2021 - 2nd International Conference on Innovative Intelligent Industrial Production and Logistics

GDCMNDP. A pseudo-code of the proposed

metaheuristic is presented in Algorithm 1 below.

Algorithm 1: Red Deer Algorithm.

Set the parameters: MaxIt, nPop, nMale,

α, β, µ;

Initialize the Red Deer population;

Select nMale male Red Deer;

RDA

= the best solution;

It=0;

While (It<MaxIt)

For each Red Deer male

Select the best male;

endFor

Form male commanders and stags;

For each male commander

Fight male commanders and stags;

Update male commanders and stages'

positions;

endFor

For each male commander

Mate male commander with hinds of

his harem;

Mate male commander with hinds

of another harem;

endFor

For each stag

Mate stag with any hinds of any

harem;

endFor

Select the next population using

the elitism principle;

UpdateS

RDA

;

It=It+1;

endWhile

Return S

RDA

First, it is worthy to note that Algorithm 1 is

based on 6 input parameters to tune: MaxIt=the

maximum number of iteration, nPop=the Red Deer

population size, nMale=the number of Red Deer

males, α=the percentage of mating inside a harem,

β=the percentage of mating outside a harem, and

µ=the percentage of male commanders.

Red Deer Representation:

Each Red Deer is represented by binary array that

corresponds to the vector of the decision network

variables

LlEe

 ,

)(

in Model (1)-(7). Obviously,

each Red Deer should correspond to a feasible

solution for the GDCMNDP. This condition is

ensured while Constraints (2) are respected.

Initial Population Generation:

The initial Red Deer population is randomly

generated of feasible candidates. During the

initialization of the Red Deer population, it is

simultaneously sorted in a descending order with

respect to the fitness. Then, the first nMale

candidates are selected as Red Deer males and the

rest of the population is considered as Red Deer

females, i.e. hinds.

Fitness Determination:

Similar to the genetic algorithm, the fitness function

express the value of a Red Deer referencing to his

grace, roaring power and fighting strength. In our

case, it corresponds to the objective function (1) in

Model (1)-(7). More precisely, it is calculated as the

sum of the installation costs and the penalties of

unrouted demands. Besides, let’s precise that this

amount of penalties is calculated through solving a

routing problem represented as a linear

programming model derived from Model (1)-(7)

once the

LlEe

 ,

)(

vector is fixed.

Male Commanders’ Selection:

For the Scottish Red Deer, intense roaring attracts

hinds, so that some males are more successful than

other in constructing harems due to their strength

and capacity to win several fights. Thus, Red Deer

males are divided according to their power into

commanders which are the strongest males and the

stags. The number of commanders is correlated to

the µ parameter and expressed as

nMaleCom = Round (µ. nMale) (8)

Naturally, once the male commanders identified,

the rest of the males are considered as stags.

Males’ Fighting and Mating:

As in (Fathollahi-Fard et al., 2020a), the Red Deer

males fight between each other based on the fitness

function. As an issue of each fight, one of them can

win the match. Thus, we update the male position

according to the bubble sorting principle. The

strongest male will take command of the harm while

the looser will be chased away. The harems as the

hinds mating selection are performed as proposed by

Fathollahi-Fard et al. (2020a).

Next Population Selection:

As experienced by Yadav and Sohal (2017) as well

as Khatrouch et al. (2019), the tournament selection

appears to be better than the rank-based solution,

since the tournament selection repetition is faster

Tailoring a Red Deer Algorithm to Solve a Generalized Network Design Problem

than the ranking section in generating the population

of individuals. In the terms of convergence, the

tournament selection shows more efficient results

than the roulette wheel selection. Hence, the ability

of reaching the maximum/minimum fitness with the

lowest number of generations is the highest

following the elitism selection technique. Thus the

choice of the next generation, we use the Elitism

selection according to the best ranked fitness.

4 COMPUTATIONAL

EXPERIMENTATION

In order to evaluate the performance of the RDA in

solving the GDCMNDP, the metaheuristic was

implemented in C++ language on the Microsoft

Visual C++ 2010 Express in concert with the

commercial MILP solver, ILOG CPLEX 12.5. The

code was run on a personal computer with8Go of

RAM and a Core i7-7500U at 2.70 GHz.

The test-bed consists of two types of Benchmark

instances. Six instances (MH01-MH06) from (Mrad

and Haouari, 2008) and five real-world instances

from telecommunication field: one instance denoted

by (EON) as European Optical Network (Fumagalli

et al., 1999) and four instances (NSFNET-

NSFNET4) expressing the National Science

Foundation Networks (Miyao and Saito, 1998).

Table I displays the main characteristics of the

considered instances and their data files are freely

available at (Layeb, 2018).

Table 1: The Main Instances Characteristics.

Instance |K| |V| |E|

NSFNet1 21 14 42

NSFNet2 21 14 42

NSFNet3 21 14 42

NSFNet4 21 14 42

EON 36 19 72

MH01 45 10 30

MH02 105 15 45

MH03 105 15 50

MH04 105 15 60

MH05 435 30 120

MH06 595 35 140

It is well-known that fixing the appropriate

algorithmic parameters has a great influence on the

efficiency and effectiveness of the algorithm

performance. It is very often a trade-off between the

quality of the solution and the required computation

time. Therefore, an empirical experimentation has

been conducted to fix the parameters values as:

MaxIt=10, nPop=100, nMale=10, α=70% β=30%,

and µ=rand[1,10]%. These settings yield to the best

solutions within a CPU time compromise.

Table II reports the numerical results of the

MILP Model (1)-(7), the proposed Red Deer

Algorithm (RDA), and the three metaheuristics

developed by Khatrouch et al. (2019): the Genetic

Algorithm (GA), the Biogeography-Based

Optimization method (BBO), and the hybrid Genetic

Algorithm coupled with a Variable Neighborhood

Search procedure (GA-VNS). Let’s denote by Sol*:

the optimal solution of each instance found by the

state-of-the-art MILP solver, Time: the CPU time in

seconds needed to the convergence of each

approach, Gap: the Gap en percentage between the

found solution and the best solution divided by the

best solution. In Table II, “-“ indicates that the

solver fails to find the optimal solution. Besides, it is

noteworthy that the GA, the GA-VNS, and the BBO

of Khatrouch et al. (2019) were run a machine with

similar technique characteristics than this work.

Therefore, the computation times reported in Table

II could be comparable.

From Table II, we can derive several

observations. Despite the sophisticated

combinatorial optimization tools built into

commercial MILP solvers, Model (1)-(7) become

intractable as the instance size increases. It could not

provide optimal solutions for instances with more

than 30 nodes. However, all the meta-heuristics

could solve all the tested instances.

Moreover, the Biogeography-Based

Optimization method lags far behind all the other

metaheuristics. It reveals very poor performance

both in terms of the solution quality and the high

computation times. Actually, the BBO requires

significant computational time while showing weak

performance. In other words, the BBO requires

much more computational effort to find solutions for

the GDCMNDP than the other evolutionary

population-based metaheuristics.

The Genetic Algorithm performs reasonably well

with an average gap of 1.7%. Besides, Strengthening

the GA with the Variable Neighborhood Search

procedure has enhanced its performance only in

terms of quality solution. In fact, the average gap of

the GA-VNS is about 1.4% while its average CPU

time has increased by 167 % compared to the CPU

time of the basic GA.

IN4PL 2021 - 2nd International Conference on Innovative Intelligent Industrial Production and Logistics

Table 2: Numerical Results.

Instance

Model (1)-(7)

(Khatrouch et al., 2019)

GA-VNS

(Khatrouch et al., 2019)

BBO

(Khatrouch et al., 2019)

RDA

Sol

Time Gap Time Gap Time Gap Time Gap Time

NSFNet1

13 100.6 0.0% 465.9 0.0% 523.4 0.0% 3235.8 0.0% 255.6

NSFNet2

21 0.6 0.0% 321.6 0.0% 344.3 0.0% 244.7 0.0% 304.6

NSFNet3

15 10.7 6.6% 239.8 6.6% 331.5 0.0% 2303.5 6.1% 305.7

NSFNet4

18 6.8 0.0% 384.2 0.0% 487.4 0.0% 2354.0 0.0% 1976.1

EON

14 114.4 7.1% 984.2 7.1% 1113.7 21.4% 4563.7 7.6% 126.7

MH01

993 11.2 1.2% 989.3 1.2% 1564.0 1.2% 5277.8 0.6% 136.6

MH02

2444 182.3 0.8% 1795.9 0.8% 2224.4 7.4% 6084.2 0.0% 243.3

MH03

3167 117.4 0.0% 2404.9 0.0% 3328.4 21.1% 8635.2 0.0% 286.3

MH04

3481 29.2 0.0% 3482.3 0.0% 6930.0 43.7% 9301.1 0.0% 550.8

MH05

- - 0.3% 6782.5 0.0% 8170.4 100.0% 57857.4 0.0% 2177.3

MH06

- - 2.8% 8265.3 0.0 % 18679.9 114.4% 110910.5 0.2% 3026.2

Average

1.7% 2374.2 1.4% 3972.5 28.1% 19160.7 1.3% 853.6

Table II shows that the Red Deer algorithm finds

the best solutions for 9 out of the 11 tested instances.

The proposed Red Deer Algorithm seems capable of

exploring the research space effectively in order to

identify almost the optimal solution. Actually, the

RDA lightly outperforms the GA-VNS in term of

quality solution as the average gap of the RDA is

about 1.3%. However, although the RDA finds

solutions very similar to those of the GA-VNS, the

RDA converges in extremely short computation

times and mostly unrivaled by those of the GA-

VNS. Indeed, the average CPU time required by the

RDA is about 21% of that required by the GA-VNS.

Thus, the Red Deer algorithm shows very

promising performance that can be further extended

by extensive computational experiments on

additional benchmark instances of the GDCMNDP

as well as other variants of the NDPs.

5 CONCLUSIONS

The scope of this work is to tailor the recently

introduced evolutionary Red Deer Algorithm to

solve the challenging Generalized Discrete Cost

Multi-commodity Network Design Problem

effectively. This generalized network design

problem has a significant spectrum of applications,

especially in distribution, logistics and

telecommunications, with a deep business impact on

the network companies. Moreover, the GDCMNDP

is known to be an NP-hard combinatorial

optimization problem and its resolution still remains

a hard task for researchers as well as practitioners,

especially when the size of the instances increases.

We have investigated the Red Deer Algorithm and

implemented it adequately to handle the particular

features of network design problems. The numerical

Tailoring a Red Deer Algorithm to Solve a Generalized Network Design Problem

results of the computational experiments on

Benchmark instances illustrate the effectiveness of

the Red Deer Algorithm when compared with the

metaheuristics from the existing literature. Actually,

the Red Deer Algorithm generates high-quality

solutions to large-size instances in very reasonable

computation times.

These promising results encourage going further

in investigating the Red Deer Algorithm

characteristics in order to enhance its performance.

As research avenues for future work, we suggest

improving the proposed approach by decreasing its

input parameters as recently proposed in

(Fathollahi‐Fard et al., 2020b). Having fewer

parameters to control seems to lead to deeper phases

of intensification and research that allow the best

solution to be found more efficiently.

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