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Applied Mathematical Modelling 40 (2016) 4 94 8–4 969

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Applied Mathematical Modelling

journal homepage: www.elsevier.com/locate/apm

Multi-objective multi-layer congested facility

location-allocation problem optimization with Pareto-based

meta-heuristics

Vahid Hajipour a , Parviz Fattahi a , Madjid Tavana

b , c , ∗, Debora Di Caprio

d , e

a Industrial Engineering Department, Faculty of Engineering, Bu-Ali Sina University, Hamedan, Iran b Business Systems and Analytics Department, Distinguished Chair of Business Analytics, La Salle University, Philadelphia, PA 19141, USA c Business Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, D-33098

Paderborn, Germany d Department of Mathematics and Statistics, York University, Toronto M3J 1P3, Canada e Polo Tecnologico IISS G. Galilei, Via Cadorna 14, 39100 Bolzano, Italy

a r t i c l e i n f o

Article history:

Received 22 January 2015

Revised 30 November 2015

Accepted 11 December 2015

Available online 18 December 2015

Keywords:

Location-allocation problem

Congested system

Multi-objective optimizations

MOVDO

MOHSA

a b s t r a c t

Facility location-allocation problems arise in many practical settings from emergency ser-

vices to telecommunication networks. We propose a multi-objective multi-layer facility

location-allocation (MLFLA) model with congested facilities using classical queuing sys-

tems. The goal is to determine the optimal number of facilities and the service allocation

at each layer. We consider three objective functions aiming at: (1) minimizing the sum of

aggregate travel and waiting times; (2) minimizing the cost of establishing the facilities;

and (3) minimizing the maximum idle probability of the facilities. The problem is for-

mulated as a multi-objective non-linear integer mathematical programming model. To find

and analyze the Pareto optimal solutions, we propose a Pareto-based multi-objective meta-

heuristic approach based on the multi-objective vibration damping optimization (MOVDO)

and the multi-objective harmony search algorithm (MOHSA). We demonstrate the effec-

tiveness of the proposed model and exhibit the efficacy of the procedures and algorithms

by comparing MOVDO and MOHSA with two well-known evolutionary algorithms, namely,

the non-dominated sorting genetic algorithm (NSGA-II) and multi-objective simulated an-

nealing (MOSA).

© 2015 Elsevier Inc. All rights reserved.

1. Introduction

Facility location-allocation (FLA) problems, originally introduced by Cooper [1] , are concerned with locating multi-

ple facilities in a Euclidian plane and allocating customers to a facility while minimizing the total costs, mainly com-

posed of transportation costs. In a typical FLA problem, suppliers, warehouses, and producers are considered as fa-

cilities, and retailers, purchasers, and service users are treated as customers. Researchers have studied multi-objective

∗ Corresponding author at: Distinguished Chair of Business Systems and Analytics, La Salle University, Philadelphia, PA 19141, United States. Tel.: + 1 215

951 1129; fax: + 1 267 295 2854.

E-mail addresses: [email protected] (V. Hajipour), [email protected] (P. Fattahi), [email protected] (M. Tavana), [email protected] ,

[email protected] (D. Di Caprio).

URL: http://tavana.us/ (D. Di Caprio)

http://dx.doi.org/10.1016/j.apm.2015.12.013

S0307-904X(15)00822-7/© 2015 Elsevier Inc. All rights reserved.


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V. Hajipour et al. / Applied Mathematical Modelling 40 (2016) 4948–4969 4 94 9

location-allocation problems extensively (see, among others, [2–8] ). In particular, FLA problems and the correspond-

ing solution methods play an important role in many and different areas such as: local clinics and hospitals centers;

emergency medical service (EMS) systems; rescue helicopter locations, relief distribution centers and reconstruction cen-

ter locations; education systems; police stations; truck terminals; hotels; city logistics terminals; press delivery net-

works, locating post boxes, post delivery services and fast delivery packing companies; solid waste disposal system;

air ports; telecommunication systems; fuel/gasoline service stations; blood banking centers; libraries; retail outlets; and

so forth.

The inclusion of multiple and conflicting objectives enhances the analysis of FLA problems by replacing the concept of

an optimal solution with an efficient (also referred to as non-dominated, non-inferior, or Pareto-optimal) solution [9] .

1.1. Congested facilities: a short review on queuing theory and evolutionary algorithms

The facilities are often used to meet average demands in a given production or service system. However, a facility may

not be able to fulfill all of the customers’ needs when the demand is at the maximum level. Facilities that cannot cope with

the additional demands are called congested facilities [10] .

Queuing theory is a useful method for studying congested facilities [11] . Berman and Larson [12] proposed a non-

linear congested facility location model with M/G/ 1 queues. Wang et al. [13] proposed a facility location model within

an M/M/1 queuing system that minimizes the aggregation of traveling and waiting times. Wang et al. [14] presented

a constrained location problem with the possibility for opening and closing of some facilities. They proposed three

heuristic algorithms including greedy interchange, Tabu search, and Lagrangian relaxation approximation to solve the

problem.

Silva and Figuera [15] considered demand as stochastic and formulated each of the facilities as an independent queue.

They combined stochastic models of manufacturing systems with deterministic location models to obtain a formula for

the backlogging probability at a potential facility location. A heuristic based on the greedy search procedure was proposed

to solve their model. Berman and Drezner [16] developed a single-objective facility location model within M/M/m queues.

Syam [17] proposed a nonlinear multiple server location-allocation model with relevant costs and other considerations.

Aboolian et al. [18] investigated a multiple-server center location problem with a nearest-facility constraint. Their goal was

to minimize the maximum time spent by any customer including travel time and waiting time. Raman et al. [19] hybridized

a queuing model with a genetic algorithm (GA) to solve the layout problem. This combination provides a unique opportunity

to consider the stochastic variations while achieving a good layout. Zarrinpoor and Seifbarghy [20] developed a competitive

location model with capacitated queues to minimize the total cost of the system and solve their model with GA and Tabu

search.

Zanjirani Farahani et al. [21] reviewed recent developments in multi-criteria location problems and classified them

into three categories: bi-objective, multi-objective, and multi-attribute problems. Pasandideh and Niaki [22] proposed a bi-

objective facility location problem within the M/M/1 queuing framework. Chambari et al. [23] presented two Pareto-based

algorithms including non-dominated sorting genetic algorithms (NSGA-II) and non-dominated ranking genetic algorithms

(NRGA) to solve a bi-objective facility location problem with M/M/1/k queues. Hajipour and Pasandideh [24] presented an

adaptive particle swarm optimization to optimize a bi-objective facility location model with batch arrivals. Pasandideh et al.

[7] developed a new multi-objective facility location problem with batch arrivals and utilized two meta-heuristics known as

simulated annealing (SA) and GA to solve the problem with integrated objectives using the LP-metric technique. LP-metric

method is a rigorous multi-objective technique used to navigate the search direction of GA.

A large number of closed form and approximation methods have been used in the literature to find Pareto fronts in

multi-criteria NP-hard location problems. Sherali and Nordai [25] showed that a multi-facility Weber problem (where the

facilities can be placed at any point in the Euclidian plane) with limited capacity and deterministic parameters is NP-hard

even if all the customers are located on a straight line. Therefore, the only reasonable approach for solving large-scale

problems is to use heuristics and meta-heuristics [7,13] . Farahani and Hekmatfar [26] , Farahani et al. [21] , Boloori et al. [27] ,

and Farahani et al. [28] present a comprehensive overview of these heuristics and meta-heuristics.

Several solution procedures involving simultaneous optimization of multiple objectives are proposed to find the Pareto

solution sets [29] . The multi-objective optimization algorithm based on GA and SA algorithms like NSGA-II [30] and the

multi-objective SA (MOSA) [31] are the meta-heuristics commonly used to find the Pareto front solutions in NP-hard multi-

objective problems [32] .

Two alternative approaches to solving FLA problems have been recently proven to be quite efficient, namely, the vibration

damping optimization (VDO) algorithm and the harmony search algorithm (HSA).

VDO is a meta-heuristic algorithm that uses the concept of vibration-damping in mechanical vibration improvisation of

musicians [33] . Mehdizadeh and Tavakkoli-Moghaddam [33] used the VDO algorithm to solve the parallel machine schedul-

ing problems. Mehdizadeh et al. (2011) proposed a hybrid VDO algorithm to solve multi-facility stochastic-fuzzy capacitated

location allocation problems. Mousavi et al. [34] developed a special type of VDO algorithm to solve capacitated multi-facility

location-allocation problems with probabilistic customers’ locations and demands.

Harmony search algorithm (HSA) is a music-inspired algorithm, conceptually simple and with very few parameters. HSA

is easy to implement and has been successfully applied to different problems including the Sudoku puzzle [35] , mechanical

structure design [36] , pipe network optimization [37] , inventory models [38] , and facility location [39,40] .

4950 V. Hajipour et al. / Applied Mathematical Modelling 40 (2016) 4948–4969

Table 1

A comparison of recent congested facility location models.

Problem characteristics Hajipour and

Pasandideh [24]

Pasandideh

et al. [7]

Rahmati

et al. [40]

Hajipour

et al. [41]

Hajipour

et al. [39]

This study

Congested/

queuing

system

M/M/1 √

M

[x] /M/1 √ √

M/M/m

√ √

Service

layers

Single √ √ √ √ √

Multiple √

Multi-

objective

optimizer

Integrated-based approach √ √ √

Pareto-based approach √ √ √

1.2. Contribution

In this study, we propose a multi-objective multi-layer facility location-allocation (multi-objective MLFLA) model with

random customer arrivals and congested facilities using classical queuing systems. More precisely, the model determines the

optimal number of facilities and the service allocation at each layer, when each selected facility acts like an M/M/1 queuing

system.

To the best of our knowledge, there is no research in the literature that uses queuing theory to study multi-objective

MLFLA problems with random demands. Despite being a standard approach in queuing theory, the identification of facilities

with M/M/1 queues is herein integrated into a highly complex environment where:

• a multi-objective non-linear integer mathematical programming problem is formulated to deal with the problem of si-

multaneously minimizing both traveling and waiting time of the customers forming the queues of a service system

consisting of several facilities in multiple layers and, at the same time, optimally allocating the facilities of the system

so as to minimize their cost; • a Pareto-based multi-objective meta-heuristic approach is proposed to solve this mathematical programming problem

where simultaneous and conflicting objectives must be optimized.

The novelty of our approach to multi-objective FLA can be summarized as follows:

• congested facilities are modeled as M/M/1 queues despite the complexity of the service system under consideration; • facilities are distributed in multiple service layers which provides an important improvement over the existing congested

facility location models dealing with single service layers; • two novel meta-heuristic algorithms are developed to find the Pareto-optimal solutions to the proposed multi-objective

MLFLA: a multi-objective version of the VDO algorithm (namely, MOVDO) and a multi-objective version of the HSA

algorithm (namely, MOHSA).

Table 1 summarizes the main features of the FLA models offered by the most recent literature compared to those of

our model. In particular, Hajipour et al. [41] proposed a reliable facility location-allocation problem and introduced a multi-

objective version of VDO to solve the model. However, to the best of our knowledge, this multi-objective version of VDO

has not been employed to solve NP-hard facility location problems.

Therefore, when compared with the already existing FLA models, our model shows three unique features including a con-

gested system, multiple service layers, and multi-objective optimizers. Indeed, there is no research in the literature on FLA

dealing with congestion in multiple service layers. At the same time, our model builds on a quite uncommon formulation

for a FLA problem, that is, as a Pareto-based multi-objective optimization problem.

Finally, to demonstrate the efficiency and robustness of the proposed solution methods, we compare the performance

of the newly introduced evolutionary heuristic algorithms (MOVDO and MOHSA) with that of two well-known evolution-

ary algorithms, that is, the non-dominated sorting genetic algorithm (NSGA-II) and the multi-objective simulated annealing

(MOSA).

Recall that “efficiency” refers to the simplicity and the quickness with which the solution can be obtained and it

is represented by the computational and time complexities, while “robustness” accounts for the ability of the model

to find the solution under different circumstances. In order to compare the four algorithms, both graphically and sta-

tistically, we used four standard performance metrics including the spacing (SP) metric, the number of Pareto so-

lutions (NPS) metric, the computational (CPU) time metric, and the multi-objective coefficient of variation (MOCV)

metric.

The results obtained by implementing the algorithms for solving 12 test problems show that, in general, MOVDO and

MOHSA have good performance with respect to NSGA-II and MOSA in terms of all the metrics considered except the NPS

metric according to which MOVDO and MOHSA perform worst and better than NSGA-II and MOSA, respectively. In particular,

MOVDO is shown to be the best method if CPU time is the most important metric for the decision maker, while both

V. Hajipour et al. / Applied Mathematical Modelling 40 (2016) 4948–4969 4951

Fig. 1. Graphical depiction of a multi-layer congested facility location problem.

MOVDO and MOHSA turn out to be more effective than NSGA-II when solving large problems. It follows that the choice of

the algorithm to use depends on the decision maker’s preferences in terms of performance metrics.

The remainder of this paper is organized as follows: in Section 2 , we formulate the problem through a mathematical

model. In Section 3 , we provide some basic background on multi-objective optimization problems and Pareto-based algo-

rithms. In Section 4 , we present the multi-objective optimization algorithms proposed in this study. Section 5 is dedicated to

testing the performance of the proposed methodology with several test problems. Finally, Section 6 presents our conclusions

and futures research directions.

2. The mathematical model

In this section, we describe the proposed problem, that is, a multi-layer congested facility location allocation problem

within a queuing framework. After defining the problem, we state the assumptions and introduce the notations, parameters

and decision variables that allow for its formalization. Finally, we formalize the problem defining the objective functions

and the constraints of the model.

2.1. Problem definition

Consider an immobile manufacturing or service system consisting of several facilities in multiple layers as shown in

Fig. 1 . A customer with a random demand that arrives to this system requires the service of one facility in each layer. Based

on a time-table for the traveling customer i (where i = 1 , 2 , . . . , M) to facility j (where j = 1 , 2 , . . . , N) in layer l (where

l = 1 , 2 , . . . , L ), the shortest routes should be selected. The first objective minimizes the traveling time and the customer

waiting time in the queue. The second objective finds the optimal allocation to the facilities based upon the minimization of

the maximum idle probability of the facilities. In other words, the facility with the highest idle probability has the highest

chance of being selected to serve a customer at a given time. Note that each customer gets different level of service in each

layer. Also, all the customers must be served in all the layers and cannot leave the system in the middle layers. Simultaneous

consideration of these two conflicting objectives makes the problem more realistic [29] . The goal is to determine the optimal

number of facilities at each layer as well as the allocation process of customers to the facilities in different layers.

The proposed problem has several applications in real-life situations. Consider, for instance, health care systems, reg-

istration systems, lending systems, and so forth. In particular, the current model could be extended into the facility loca-

tion branch of the geographic information systems (GIS) literature. This would provide a potentially valid complementary


alternative to the existing literature. See, among others, Vakalis et al. [42,43] Saadatseresht et al. [44] , Chang et al. [45] and

Lei et al. [46] .

2.2. Assumptions

The assumptions of the model are as follows:

• To receive service, customers move toward the facilities (immobile servers). • Each customer is assigned to only one facility in a layer. • Each facility behaves like an M/M/1 queue with an exponential service time. In addition, customers arrive independently

to the facilities based on a Poisson process with known arrival rates. • Customers cannot leave the system before going through all the layers of the system.

2.3. Indices and parameters

The indices and the parameters required for the formulation of the model are the following:

i : Index for a customer node; i = 1 , ..., M

l: Index for a service layer; l = 1 , ..., L

j: Index for an open facility node in the first layer l = 1 ; j = 1 , ..., N

s : Index for an open facility node in layer l > 1 ; s = 1 , 2 , . . . , N

m l : Maximum number of open facilities in layer l ; ∀ l = 1 , . . . , N, m l ≤ N

t i j : Traveling time of customer i to facility j in layer l = 1

t ′ i jsl

: Traveling time of customer i from facility j to the open facility s in layer l > 1

γ j : Demand rate for the facility j in the first layer l = 1

γ ′ sl

: Demand rate for the open facility s in layer l > 1

Z 1 : Sum of the traveling and waiting time in all layers

Z 2 : Total cost of establishing all the facilities in all layers

Z 3 : Maximum of the idle probability of the opened facilities in all layers

λi : Demand rate of customer i

ψ : A large number

μ j : Service rate of facility j in the first layer l = 1

μ′ sl

: Service rate of the open facility s in layer l > 1

2.4. Decision variables

The decision variables of the mathematical formulation of the problem are as follows:

x i j =

{1 if customer i is assigned to facility j 0 otherwise

y i jsl =

{1 if customer i of facility j is assigned to facility s in layer l ( l > 1 ) 0 otherwise

h j =

{1 if facility j in the first layer l = 1 is open

0 otherwise

h

′ sl =

{1 if facility s in layer l > 1 is open

0 otherwise

2.5. The mathematical formulation

Since each selected facility acts like an M/M/1 queuing system, the average waiting time at each facility is determined by

Eq. (1) . See Gross and Harris [47] .

w j =

1

μ j − γ j

; j = 1 , 2 , . . . , N

w sl =

1

μ′ sl

− γ ′ sl

; s = 1 , 2 , . . . , N, l = 2 , . . . , L, (1)


where γ j and γ ′ sl

are the demand rates at open facility j in the first layer and at the open facility s in layer l > 1 , respectively.

They are shown in Eq. (2) .

γ j =

M ∑

i =1

λi x i j ; j = 1 , 2 , . . . , N

γ ′ sl

=

M ∑

i =1

N ∑

j=1

λi y i jsl ; s = 1 , 2 , . . . , N, l = 2 , . . . , L.

(2)

Moreover, the idle probability of the open facility jin the first layer and the open facility s inlayer l > 1 are obtained by

Eq. (3) .

π0 , j = 1 − γ j

μ j

; j = 1 , 2 , . . . , N

π0 ,sl = 1 − γ ′ sl

μ′ sl

; s = 1 , 2 , . . . , N, l = 2 , . . . , L. (3)

Based on Eqs. (1) –( 3 ) and the above objectives and the assumptions, the multi-objective nonlinear integer programming

model can be described as follows:

Min Z 1 =

M ∑

i =1

N ∑

j=1

λi t i j x i j +

M ∑

i =1

N ∑

j=1

N ∑

s =1

L ∑

l=2

λi t ′ i jsl y i jsl +

N ∑

j=1

γ j

μ j − γ j

+

N ∑

s =1

L ∑

l=2

γ ′ sl

μ′ sl − γ ′

sl

, (4)

Min Z 2 =

N ∑

j=1

h j +

N ∑

s =1

L ∑

l=2

h

′ sl , (5)

M in Z 3 = M ax j=1 , 2 , ... , N

{(1 − γ j

μ j

)h j

}+ M ax

s = 1 , 2 , . . . , N

l = 2 , . . . , L

{(1 − γ ′

sl

μ′ sl

)h

′ sl

}. (6)

Subject to:

1 ≤N ∑

j=1

h j ≤ m l ; l = 1 . (7)

1 ≤N ∑

j=1

h

′ sl ≤ m l ; l = 2 , 3 , . . . , L. (8)

N ∑

j=1

x i j = 1 ; i = 1 , ..., M. (9)

N ∑

s =1

y i jsl = x i j ; i = 1 , ..., M , j = 1 , ..., N, l = 2 . (10)

N ∑

s ′ =1

y i jss ′ ,l+1 = y i jsl ; i = 1 , 2 , . . . , M, j = 1 , 2 , . . . , N, s = 1 , 2 , . . . , N, l = 2 , . . . , L − 1 . (11)

h j ≤M ∑

i =1

x i j ≤ M × h j ; j = 1 , 2 , . . . , N. (12)

h

′ sl ≤

M ∑

i =1

y i jsl ≤ M × h

′ sl ; j = 1 , ..., N, s = 1 , ..., N, l = 2 , ..., L. (13)

M ∑

i =1

λi x i j ≤ μ j ; j = 1 , ..., N. (14)

M ∑

i =1

N ∑

j=1

λi y i jsl ≤ μ′ sl ; s = 1 , ..., N, l = 2 , ..., L. (15)


N ∑

j=1

t i j x i j ≤ ( t i j − ψ) h j + ψ ; i = 1 , 2 , . . . , M, j = 1 , 2 , . . . , N. (16)

N ∑

j=1

t ′ i jsl y i jsl ≤ (t ′ i jsl − ψ) h

′ sl + ψ ; i = 1 , 2 , . . . , M, j = 1 , 2 , . . . , N, s = 1 , 2 , . . . , N, l = 2 , . . . , L. (17)

h j ∈ { 0 , 1 } ; j = 1 , 2 , . . . , N

h

′ sl

∈ { 0 , 1 } ; s = 1 , 2 , . . . , N, l = 2 , . . . , L

x i j ∈ { 0 , 1 } ; i = 1 , 2 , . . . , M, j = 1 , 2 , . . . , N

y i jsl ∈ { 0 , 1 } ; i = 1 , 2 , . . . , M, j = 1 , 2 , . . . , N, s = 1 , 2 , . . . , N, l = 2 , . . . , L.

(18)

The objective ( 4 ) minimizes the sum of the travelling time and the waiting time at each layer. The objective ( 5 ) minimizes

the costs of establishing all the facilities at each layer. The objective ( 6 ) minimizes the maximum of the idle probability

pertinent to each facility. Constraints ( 7 ) and ( 8 ) assure an upper bound for the on-duty facilities at each layer. Constraints

( 9 ) ensure that each customer must be assigned only to one facility. Constraints ( 10 ) and ( 11 ) ensure that all the customers

must be served in all the layers and cannot leave the system in the middle layers. Constraints ( 12 ) and ( 13 ) mean that

costumers can be assigned only to open facilities. Constraints ( 14 ) and ( 15 ) define the service capacity for each facility at

each layer. Constraints ( 16 ) and ( 17 ) force each costumer to be assigned to the nearest by open facility (nearest-facility

assignment). Finally, constraints ( 18 ) specify the range of the decision variables.

3. Background: MOPs and Pareto-based algorithms

In this section, we review some basic background information on multi-objective optimization problems.

A multi-objective optimization problem (MOP) consists of an objective vector function O ( � x ) = ( o 1 ( � x ) , . . . , o K ( � x )) , whose

coordinates are K conflicting objective functions. The values of O ( � x ) must be usually minimized subject to E constraints,

g ε ( � x ) ≤ 0 , ε = 1 , 2 , . . . , E, where � x denotes an n -dimensional vector that can be assumed to be a real, integer, or even

Boolean value. The set X of all � x satisfying the imposed constraints is called the feasible region (or feasible space).

Given a minimization model and two of its feasible solutions � p and

� q ( � p , � q ∈ X), we say that � p dominates � q , and write

� p ≺ �

q , if:

( d1 ) ∀ k ∈ { 1 , 2 , . . . , K} , o k ( � p ) ≤ o k ( � q )

and

( d2 ) ∃ k ∈ { 1 , 2 , . . . , K} , o k ( � p ) < o k ( � q ) .

A feasible solution

� x ∗ ∈ X is called a Pareto-optimal solution if it cannot be dominated by any other feasible solution:

∃

� x ∈ X : ∀ k ∈ { 1 , 2 , . . . , K} , o k ( � x ) < o k ( � x ∗) .

The set of all Pareto-optimal solutions is called the Pareto-optimal set, while the set of objective vectors O ( � x ∗) =( o 1 ( � x ∗) , . . . , o K ( � x ∗)) , obtained by evaluating a Pareto-optimal solution, is referred to as Pareto-optimal front.

The presence of conflicting objectives implies that at each improvement of one objective may correspond a worsening of

a different objective. Thus, the task is to look for those solutions that do not allow to improve any of the objectives without

worsening other ones. These solutions are the Pareto-optimal solutions described above. Hence, to solve a multi-objective

optimization problem means to find the Pareto-optimal set, and this can be quite expensive from a computational viewpoint.

It follows the need for algorithms allowing to generate a set of solutions whose objective values converge to the optimal

ones. In the following section, we propose two novel approaches to the proposed problem building on the search strategy

typical of evolutionary algorithms (EAs).

As it is the case for all evolutionary algorithms, an individual (or a chromosome) represents a feasible solution of our

problem, while a population represents a set of feasible solutions. During the optimization process, a given population is

changed through loops comprising evaluation/selection/recombination-mutation. The loop iterations stop on the basis of

a predefined criterion and the best individuals of the final population are taken as the optimal feasible solutions. Fig. 2

describes the pseudo code of a standard generic EA.

4. The novel Pareto-based meta-heuristics

In this section, two Pareto-based meta-heuristic algorithms, called MOVDO and MOHSA, are proposed to solve the multi-

objective optimization problem formulated in Section 2 . These two novel meta-heuristics methods are then compared with

two popular algorithms, NSGA-II and MOSA.

In Pareto-based approaches, the goal is to achieve the most convergence while keeping the highest diversity during the

evolution process [30] .


Fig. 2. Generic EA Pseudo-code.

4.1. The MOVDO algorithm

VDO is a meta-heuristic algorithm that uses the concept of vibration-damping in mechanical vibration improvisation of

musicians [33] . In this paper, we develop a multi-objective version of the VDO algorithm for the study of facility location

problems with congested systems. The details of this algorithm are explained in the following sections.

4.1.1. Solution representation

Following Pasandideh et al. [7] , we represent the solutions � x using the multi-strand type, in which the length of first

strand is equal to the total number of customers, the second one is related to the facilities, and the third one defines the

layers.

Some constraints are penalized using the method given in Yeniay and Ankare [48] . That is, infeasible solutions are fined

using Eq. (19) .

P ( � x ) = ξ × Max

{(g( � x )

b − 1

), 0

}, (19)

where ξ , g( � x ) and P ( � x ) represent a large number, the constraint under consideration, and the penalty value, respectively.

This equation is designed for a constraint of the form g( � x ) ≤ b and the penalty is added to the objective function value as

shown in Eq. (20) .

J( � x ) =

{O ( � x ) if � x ∈ X

O ( � x ) + P ( � x ) if � x / ∈ X

, (20)

where O ( � x ) is the objective function value of chromosome � x .

4.1.2. The VDO algorithm

In vibration theory, the concept of vibration can be identified with that of oscillation. If the damping is small, it has very

little influence on the natural frequencies of the system, and hence the calculations for the natural frequencies are almost

the same as those made in the no damping case. In the VDO algorithm, the higher is the amplitude (hence, the wider is the

scope of a solution) the larger is the probability of obtaining of a new solution. Therefore, when the amplitude is reduced,

the probability of obtaining a new solution decreases, and, consequently, the system stops because the amplitude state is

reached [33, 34] .

In the analogy between an optimization problem and the vibration-damping process, the states of the oscillation system

represent feasible solutions of the optimization problem, the energies associated with the states correspond to the objective

function values computed at those solutions, the minimum energy states correspond to the optimal solutions to the problem,

and rapid quenching can be viewed as local optimization.

The VDO algorithm starts by generating random solutions in the search space. Then, the algorithm parameters including

the initial amplitude ( A 0 ), the max of the iterations at each amplitude ( T ), the damping coefficient ( γ ) and the standard

deviation ( σ ) are initialized. Then, the solutions are evaluated by means of the objective function value (OFV). The algorithm

contains two main loops. The first loop generates a solution randomly. Successively, using a neighborhood structure, new


Fig. 3. MOVDO Pseudo-code.

solutions are obtained and the best one is chosen. However, similar to the SA algorithm, the solution with a lower OFV can

be selected with respect to the Rayleigh distribution function. In fact, the new solution is accepted if:

= OF V (NewSolution ) − OF V (CurrentSolution ) ≤ 0 . (21)

If, instead, > 0 , then a random number r is generated between (0, 1). The current solution is selected with respect to

the following criteria:

r < 1 − exp

(− A

2

2 σ 2

), (22)

where A is either the initial or the updated amplitude.

The second loop adjusts the amplitude, that is, is used to reduce the amplitude at each iteration t .

A t = A 0 exp

(−γ t

2

). (23)

The algorithm stops when the stopping criterion is met.

The pseudo-code of the standard VDO algorithm is reported in the non-boxed part of Fig. 3 . More precisely, given any

population P j (the initial population is randomly generated in VDO), the first and second loops described above correspond

to the loops composing the inner for-cycle relative to P j . The if-cycle incorporates Eqs. (21) and ( 22 ) and, hence, provides

the pseudo-code of the first loop. The update-cycle is based on Eq. (23) and, hence, gives the pseudo-code of the second

loop.


4.1.3. MOVDO main loop

We will now introduce a multi-objective version of the VDO algorithm, which we will refer to as MOVDO, and illustrate

the idea of applying it to solve and manage Pareto-optimal solutions. To compare solutions, we will apply two main concepts

of multi-objective meta-heuristics, namely, fast non-dominated sorting (FNDS) and the crowding distance (CD).

In FNDS, nPop initial populations (sets of feasible solutions) are compared and sorted. In order to do this, all chromo-

somes (solutions) in the first non-dominated front are first found. Since all the objective functions in the mathematical

model are to be minimized, the chromosomes are chosen using the concept of domination. Then, in order to find the chro-

mosomes in the next non-dominated front, the solutions of the previous fronts are disregarded temporarily. This procedure

is repeated until all the solutions are set into fronts.

After sorting the populations in different fronts, a CD measure is defined to evaluate solution fronts of populations in

terms of the relative density of individual solutions [30] . To do this, let S F and O ( � x ) = ( o 1 ( � x ) , . . . , o K ( � x )) be the set of non-

dominated solutions in a particular front F and the objective vector function, respectively. In addition, let δ be the cardinality

of S F and d ′ s denote the value of the CD associated with the solution

� s ∈ S F . Then, the CD is obtained using the following

steps:

(I) Initialize distance: ∀

� s ∈ S F , d ′ s = 0

(II) Sort the set S F as follows:

∀ k = 1 , 2 , . . . , K, sort in increasing order the set of objective values { o k ( � s ) : � s ∈ S F } obtained using the k th objective

function; obtain the ordered set { o k 1 , o k 2 , . . . , o k δ } ; ∀ k = 1 , 2 , . . . , K, sort the set S F using the ordered set { o k 1 , o k 2 , . . . , o k δ } ; obtain

S F = { � s 1 ,k , � s 2 ,k , ..., � s δ,k } . (III) For every k = 1 , 2 , . . . , K, assign infinite CD value to the boundary solutions � s 1 ,k and

� s δ,k : d

′ s 1 ,k

= ∞ and d ′ s δ,k = ∞

(IV) For every k = 1 , 2 , . . . , K, assign the CD to non-boundary solutions:

∀ i = 2 , . . . , δ − 1 , d ′ s i,k = d ′ s i,k + ( o k ( � s i +1 ,k ) − o k ( � s i −1 ,k )) .

The CD allows to define the so-called crowded tournament selection operator “�” as follows:

r x < r y or ( r x = r y and d ′ x > d ′ y ) ⇒

� x ≺ �

y [i . e . dominates � y ] ,

where r x and r y are the ranks and d ′ x and d ′ y are the crowding distances associated to the solutions � x and

� y . The crowded

tournament selection operator clearly defines a comparison criterion for the solutions to the MOVDO algorithm and can be

applied to select the individuals of the next generation [29] . Note that the individuals/solutions in the boundary are always

selected according to this comparison criterion since they are assigned infinite distance.

The crowded tournament selection process consists of the following steps.

Crowded tournament selection process

Step-1: choose n individuals in the population randomly.

Step-2: the non-dominated rank of each individual should be obtained and the CD of the solutions having equal non-

dominated rank is calculated.

Step-3: the solutions with the least rank are the selected ones. Moreover, if two or more individuals share the least rank,

the individual with the highest CD should be selected.

After operating the aforementioned concepts and operators, the parents and the offspring population should be com-

bined to ensure their elitism. Since the combined population size is naturally greater than the original population size | P | ,once more, non-domination sorting is performed. In fact, chromosomes with better ranks are selected and added to the

populations until the population size becomes | P | . The last front also consists of the population based on the crowding

distance. The algorithm stops when a predetermined number of iterations (or any stopping criteria) is reached.

4.1.4. Evolution process of MOVDO

The process is started by initializing the initial population of solution vectors P j . Hence, the VDO operators are imple-

mented on P j to create a new population Q j . The combination of P j and Q j creates R j to keep elitism. In this step, the

solution vectors in R j are sorted in several fronts using FNDS and CD. Using this selection method, a population of the next

iteration P j+1 is chosen to have a predetermined size.

Fig. 3 illustrates the pseudo-code of the MOVDO algorithm including the basic operators of a VDO algorithm and the

multi-objective operators described above. The main multi-objective parts of the pseudo-code have been boxed up. The

framework where to implement the evolution process of MOVDO is schematically represented in Fig. 4.

It is important to mention that the proposed MOVDO algorithm is a computationally efficient algorithm which uses a

selection method based on classes of dominant solutions.

4.2. The MOHSA

In this subsection, we propose another multi-objective evolutionary algorithm called MOHSA. The main differences be-

tween MOHSA and MOVDO are to be found in the evolution process that allows the algorithms to move from P j to Q j .


Fig. 4. Proposed MOVDO framework.

Fig. 5. Relationship among different HAS probabilities.

Fig. 6. A pitch-adjusting operator example.


Fig. 7. MOHSA Pseudo-code.

While the evolution process of the MOVDO is based on the VDO algorithm, the evolution process of the MOHSA builds on

a HSA. More precisely, while the framework behind both the MOVDO and the MOHSA is the same (see Fig. 3 ), the inner

loop interactions allowing for the mutation of P j into Q j follow a different structure. Accordingly, after generating or modi-

fying populations by means of single-objective operators of the basic algorithms (HSA or VDO), a multi-objective approach

is therein implemented in a similar fashion.

In the HSA, the objective function is interpreted as harmony and the aesthetic estimation of the player helps him/her to

find a good state of the harmony. Indeed, in this algorithm the qualitative improvisation process is modeled as a quantitative

optimization process. When a musician improvises with an instrument, she or he is faced with three possibilities:


Fig. 8. NSGA-II flowchart.

(

(i

(i) playing from her/his memory (with probability P HMCR ),

ii) adjusting the pitches slightly (with probability P pa ),

ii) composing randomly (with probability P rand ).

These options are formalized into three quantitative operators of HSA known as harmony memory (HM), pitch adjusting,

and randomization [49] , respectively. Therefore, the improvising process of HSA is the combination of these three operators.

A scheme of how these operators are applied, with the corresponding probabilities, is depicted in Fig. 5.

For a detailed description of the HSA operators, one can refer to Geem et al. [37] , Geem [35] , and Rahmati et al. [40] . In

particular, the design of the pitch adjusting operator of the HSA is similar to that of the neighborhood operator of the VDO

[49, 40] . Fig. 6 shows an example of how the operator works.

The main steps of the HSA can be outlined as follows. At each iteration, a random solution is first selected. Hence,

one/two operators of the HSA (based on their probabilities) is/are used to improvise the selected solution. After improvising,


Fig. 9. MOSA Pseudo-code.

if the obtained solution is better than the worst solution in the memory, the HSA is updated by replacing the poorest

solution with the new solution.

The multi-objective version of HSA is then obtained by adding the same main loop as in the MOVDO algorithm. Fig. 7

illustrates the pseudo code of the MOHSA. The boxed up lines describe the multi-objective parts of the algorithm, while

those in the middle (outside the boxes) correspond to the basic operators and the improvising process of the HSA.

To demonstrate the performance of the proposed MOVDO and MOHSA, two well-developed Pareto-based multi-objective

evolutionary algorithms, called NSGA-II and MOSA, are also applied. For the sake of completeness, we describe them quickly

in the following subsections.


Table 2

Algorithm parameter values.

Multi-objective algorithms Algorithm parameters Parameter descriptions Optimum values

MOVDO nPop Number of population 5

A 0 Initial amplitude 6

T Max of iteration at each amplitude 40

σ Standard deviation 1 .5

γ Damping coefficient 0 .05

MOHSA P HMCR Harmony memory considering rate 0 .75

P pa Pitch adjusting probability 0 .3

U Outer loop 50

T Inner loop 100

nPop Number of population 50

MOSA T 0 Initial temperature 500

nPop Number of population 5

nGen Maximum number of generation 500

β Temperature reduction rate 0 .99

NSGA-II nPop Number of population 25

P c Crossover probability 0 .6

P m Mutation probability 0 .01

nGen Maximum number of generation 100

Table 3

Test problem input parameters.

Test

problem

number M N L

1 10 7 2

2 20 10 2

3 35 15 2

4 50 27 3

5 85 46 3

6 110 76 3

7 135 97 5

8 165 110 5

9 220 145 5

10 310 175 10

11 450 220 10

12 800 550 10

Table 4

Computational results based on the MOCV metric for all the algo-

rithms.

Problem no . MOCV

MOVDO MOHSA NSGA-II MOSA

1 2 .26E −03 3 .13E −04 3 .30E −03 6 .62E −02

2 7 .68E −04 4 .26E −02 2 .22E −02 6 .29E −01

3 4 .32E −01 6 .22E −01 4 .68E −03 2 .83E −02

4 4 .33E −02 5 .24E −02 4 .56E −01 1 .25E −02

5 3 .37E −01 3 .88E −02 8 .75E −02 6 .52E −03

6 2 .77E −02 2 .81E −02 3 .36E −01 6 .90E −03

7 3 .12E −02 8 .94E −02 2 .13E −01 5 .64E −02

8 4 .13E −02 5 .67E −01 6 .21E −01 9 .07E −01

9 3 .31E −02 6 .79E −02 2 .60E −02 8 .09E −02

10 1 .01E + 00 7 .29E −01 7 .13E −01 2 .73E −02

11 1 .23E −02 4 .75E −02 8 .01E −02 3 .04E −01

12 1 .03E −01 4 .73E −02 7 .02E −01 4 .46E −01

4.3. Comparison with the NSGA-II

The main differences between NSGA-II and both MOVDO and MOHSA are again in the inner loop interaction structures

that allow for the mutation of P j into Q j and hence to define the evolution process of the algorithms. The main loop of

NSGA-II is still the one of an evolutionary algorithm using an elitist approach. However, while, in MOVDO and MOHSA,

we implemented a roulette wheel selection operator, in NSGA-II, the binary tournament selection strategy is applied. The


Table 5

Computational results based on the NPS metric for all the

algorithms.

Problem no. NPS


1 10 8 18 3

2 7 9 17 5

3 11 10 17 3

4 5 7 11 7

5 4 4 9 3

6 7 3 8 5

7 10 5 9 4

8 7 8 9 2

9 6 9 7 1

10 6 11 8 2

11 9 8 11 4

12 5 3 8 1

Table 6

Computational results based on the Spacing metric for all the algo-

rithms.

Problem no. Spacing


1 2 .12E + 08 2 .88E + 09 4 .85E + 08 3 .13E + 08

2 3 .54E + 08 9 .13E + 08 3 .54E + 09 3 .14E + 09

3 3 .12E + 09 1 .71E + 08 5 .42E + 08 1 .41E + 11

4 3 .15E + 08 9 .22E + 08 9 .54E + 09 9 .95E + 08

5 9 .93E + 08 8 .17E + 08 8 .18E + 08 9 .49E + 09

6 2 .37E + 08 1 .02E + 09 1 .93E + 09 5 .23E + 08

7 5 .15E + 09 9 .20E + 10 9 .18E + 09 9 .91E + 10

8 1 .91E + 10 2 .31E + 08 8 .19E + 10 9 .55E + 09

9 1 .10E + 09 1 .82E + 11 8 .12E + 10 8 .13E + 09

10 9 .98E + 10 1 .52E + 09 2 .23E + 11 5 .79E + 10

11 1 .23E + 10 8 .31E + 10 1 .23E + 10 8 .10E + 09

12 1 .22E + 11 3 .14E + 10 1 .50E + 11 4 .24E + 11

Table 7

Computational results based on the CPU time metric for all the al-

gorithms.

Problem no. CPU time


1 12 .1242 14 .8713 23 .6813 13 .8181

2 25 .2654 27 .2808 48 .9372 26 .9391

3 35 .8641 39 .9831 68 .9831 33 .8185

4 37 .8331 37 .8311 91 .8318 39 .7453

5 52 .8713 56 .9831 110 .9318 52 .8662

6 66 .8274 65 .9832 138 .8781 64 .9282

7 89 .3141 93 .9248 190 .9831 99 .2873

8 91 .7361 108 .7344 239 .7484 94 .8621

9 119 .4131 148 .8474 287 .6482 121 .7171

10 149 .6716 191 .8731 341 .6391 155 .7471

11 298 .8713 339 .7461 439 .8711 311 .7563

12 459 .8499 430 .9131 673 .718 469 .5711

mutation operator is based on a swap strategy and its design is similar to the neighborhood structure of MOHSA and

MOVDO. The NSGA-II framework is represented schematically by the flowchart in Fig. 8.

4.4. Comparison with the MOSA

Simulated annealing (SA) introduced by Kirkpatrick et al. [50] is another popular search algorithm. It exploits the ap-

plication of the principles of statistical mechanics to the behavior of a large number of atoms at low temperature, to find

minimal cost solutions to large optimization problems by minimizing the associated energy. For more details about MOSA,

refer to Ulungu et al. [31] and Bandyopadhyay et al. (2008).The neighborhood structure of MOSA is designed in a similar


Table 8

Analysis of variance results for the MOCV metric.

Source DF SS MS F P

Alg. 3 0 .0653 0 .0218 0 .26 0 .852

Error 44 3 .6454 0 .0828

Total 47 3 .7107

Table 9

Analysis of variance results for the NPS metric.

Source DF SS MS F P

Alg. 3 352 .83 117 .61 14 .88 0 .0 0 0

Error 44 347 .83 7 .91

Total 47 700 .67

Table 10

Analysis of variance results for the Spacing metric.

Source DF SS MS F P

Alg. 3 1 .16994E + 22 3 .89980E + 21 0 .62 0 .608

Error 44 2 .78535E + 23 6 .33034E + 21

Total 47 2 .90235E + 23

way as that of MOVDO and MOHSA. We provide a schematic representation of the MOSA framework by its pseudo-code in

Fig. 9.

In order to evaluate and demonstrate the applicability and performance of the proposed model and algorithms, in the

next section, we generate some test problems. The results will be analyzed using the standard metrics of multi objective

optimization and the comparisons made using statistical graphs.

5. Experiments and simulation output analysis

In this section, we compare the four multi-objective Pareto-based meta-heuristic algorithms described above when ap-

plied to solve the MLFLA problem for the congested system introduced in Section 2 in different sizes.

The algorithms parameters were tuned using the Taguchi method [39] and are reported in Table 2 .

The algorithms were coded in the MATLAB software (Version 7.10.0.499, R2010a) environment and the experiments are

performed on a two GHz laptop with eight GB RAM.

In order to evaluate the performances of the proposed MOHSA and MOVDO, we have applied four standard performance

metrics including the spacing metric, the number of Pareto solutions metric, the computational time metric, and the multi-

objective coefficient of variation metric.

• Spacing (SP): the spacing metric measures the standard deviation of the distances among the solutions belonging to the

Pareto front. That is,

SP =

√

1

| P F | ∑

� x ∈ PF

( ̄D − D x ) 2

where P F denotes the set of non-dominated solutions, | P F | the cardinality of PF, D x = min � y ∈ PF

∑ K k =1 ( o k ( � x ) − o k ( � y )) is the

minimal distance of the solution

� x from the set of remaining solutions and D̄ is the mean of all D x . See, among others,

Zitzler and Thiele [51] .

• Number of Pareto solutions (NPS): the NPS metric measures the size of the Pareto front, that is, the cardinality of the set

of all non-dominated solutions. Hence,

NP S = | P F | . Again, see, Zitzler and Thiele [51] .

• Computational (CPU) time: measures the algorithm running time to reach near to the optimum solutions. • MOCV: the MOCV metric is defined as the ratio between the mean ideal distance (MID) metric and the diversity metric,

that is,

M OCV =

M ID

di v ersity


Table 11

Analysis of variance results for the CPU time metric.

Source DF SS MS F P

Alg. 3 2327 776 0 .04 0 .988

Error 44 806,123 18,321

Total 47 808,451

Fig. 10. Individual-plot of the MOCV metric for algorithm comparison.

Fig. 11. Individual-plot of the NPS metric for algorithm comparison.

See, Rahmati et al. [40] . Recall that the MID metric measures the convergence rate of Pareto fronts towards a certain

point (0,0), while the diversity metric measures the extension of the Pareto front. Thus, the MOCV metric accounts for both

the convergence and the diversity of the Pareto solutions, simultaneously, two main goals of Pareto-based algorithms.

To evaluate and compare the performances of the solution methodologies under different environments, the experiments

have been implemented on 12 test problems whose input parameters M (number of costumers), N (number of facilities per

layers) and L (number of layers) are reported in Table 3.


Fig. 12. Individual-plot of the Spacing metric for algorithm comparison.

Fig. 13. Individual-plot of the CPU time metric for algorithm comparison.

The 12 instance problems have been solved by the four algorithms described in Section 4 . The results obtained from

using the four performance metrics described above for each of the algorithms analyzed in this study are reported in

Tables 4–7

The algorithms have also been compared form the statistical viewpoint using the analysis of variance method (ANOVA).

The results of these tests on each metric are represented in Tables 8–11

To allow for a better visual comparison of the results of the tests, the box-plots of the results obtained using the perfor-

mance metrics are shown in Figs. 10–13 . All performance metric results have also been plotted and are graphically compared

in Fig. 14.

Note that larger values are desired for the NPS metric while smaller values are desired for the spacing, MOCV and CPU

time metrics. Thus, in general, the proposed MOVDO and MOHSA could compete with NSGA-II and MOSA in terms of all

metrics considered except for the NPS metric. Indeed, the statistical comparisons of the metrics ( Figs. 11–14 ) show that in

terms of MOCV , Spacing , and CPU time the algorithms do not have significant differences; while, in terms of NPS, NSGA-II

and MOSA work better and worst, respectively. This conclusion is also confirmed at the 95% confidence level. To see this,

compare the P-values in the last columns of Tables 8 –11 : the P -values obtained by using ANOVA for the MOCV, Spacing,

and CPU time metrics are 0.852, 0.608 and 0.988, respectively, while the P -value assigned to the NPS metric is 0.0 0 0. Thus,

a significant difference in performance is present only when the algorithms are evaluated by the NPS criterion.


Fig. 14. A graphical comparison of the four algorithms (according to various metrics).

6. Conclusion and future research directions

In this study, we have considered a facility location-allocation problem with random customer arrivals and proposed a

multi-objective multi-layer facility location-allocation (multi-objective MLFLA) model with congested facilities to determine

the optimal number of facilities and the service allocation at each layer, when each selected facility acts like an M/M/1

queuing system.

A multi-objective non-linear integer mathematical programming problem has been formulated to deal with the problem

of simultaneously minimizing both traveling and waiting time of the customers forming the queues of the service system

under analysis and, at the same time, optimally allocating the facilities of the system so as to minimize their cost.

More precisely, we have formalized the MOFLA problem through a multi-objective non-linear integer mathematical

model with a three-fold objective: (I) minimizing the sum of the aggregate travel and waiting times, (II) minimizing the

cost of establishing the facilities and (III) minimizing the maximum idle probability of all the facilities at each layer.

In order to solve the mathematical programming problem we have employed a Pareto-based multi-objective meta-

heuristic approach and developed two novel meta-heuristic algorithms to find the Pareto-optimal solutions: ( 1 ) a multi-

objective version of the VDO algorithm (MOVDO) and ( 2 ) a multi-objective version of the HSA algorithm (MOHSA).

To demonstrate the efficiency and robustness of the proposed solution methods, we have compared the main perfor-

mance metrics of the newly introduced evolutionary heuristic algorithms (MOVDO and MOHSA) with those of two well-

known evolutionary algorithms, that is, the non-dominated sorting genetic algorithm (NSGA-II) and the multi-objective sim-

ulated annealing (MOSA) algorithm.

In order to compare the four algorithms, both graphically and statistically, we have used four standard performance

metrics: the spacing (SP) metric, the number of Pareto solutions (NPS) metric, the computational (CPU) time metric, and

the multi-objective coefficient of variation (MOCV) metric. The results obtained by implementing the algorithms for solving

12 test problems have shown that, in general, MOVDO and MOHSA have good performance with respect to NSGA-II and

MOSA in terms of all metrics except the NPS metric. In fact, with respect to the NPS metric, MOVDO and MOHSA proved to

perform worst and better than NSGA-II and MOSA, respectively. At the same time, MOVDO proved to be the best method

in terms of CPU time metrics, while both MOVDO and MOHSA turned out to be more effective than NSGA-II when solving

large problems.

Therefore, the choice of the algorithm to use depends on the decision maker’s preferences in terms of performance

metrics.

4 96 8 V. Hajipour et al. / Applied Mathematical Modelling 40 (2016) 4948–4969

Together with the identification of congested facilities with M/M/1 queues, the development of novel Pareto-based multi-

objective meta-heuristics constitutes the main contribution of the proposed model to the literature on FLA. Future research

can be developed in this direction by assuming the problem to use some other queue rules such as batch arrival and multi-

server facilities.

Finally, given the central role played by MLFLA problems and the relative solution methods in many and different real-

world applications, such as, health care systems, registration systems, lending systems, and so forth, the current model could

be extended into the facility location branch of the geographic information systems (GIS) literature so providing a potentially

valid complementary alternative to the existing literature.

Acknowledgment

The authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions.

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