A Hybrid Clonal Selection for the Single Row Facility Layout Problem with Unequal Dimensions

doi:10.4236/ib.2012.43027

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iBusiness, 2012, 4, 216-221

http://dx.doi.org/10.4236/ib.2012.43027 Published Online September 2012 (http://www.SciRP.org/journal/ib)

A Hybrid Clonal Selection for the Single Row Facility

Layout Problem with Unequal Dimensions

Hasan Hosseini-Nasab*, Leila Emami

Department of Industrial Engineering, Yazd University, Yazd, Iran.

Email: *[email protected]

Received May 10th, 2012; revised May 29th, 2012; accepted July 23rd, 2012

ABSTRACT

The single row facility layout problem (SRFLP) is an important combinatorial optimization problem where a given set

of facilities have to be arranged in a single row to minimize the weighted sum of the distances between all pairs of fa-

cilities. In this paper, a hybrid method for single row facility layout problem is proposed in which, the simulated an-

nealing (SA) is embedded in the clonal selection algorithm (CSA). The performance of the proposed algorithm is tested

on benchmark problems. Computational results show the efficiency of the proposed algorithm compared to other heu-

ristics.

Keywords: Single Row Facility Layout; Clonal Selection Algorithm; Simulated Annealing

1. Introduction

The facility layout problem (FLP) is to arrange of a given

number of departments while minimizing the total cost

associated with the (known or projected) interactions

between them. Types of the facility layout problem occur

in many environments, such as hospital layout and ser-

vice center layout. Generally, these problems are hard

problems; most types of these problems are NP-hard.

Single row facility layout problem (SRFLP) is a special

case of the facility layout that it is to arrange n depart-

ments on a straight line to minimize the total weighted

distances between all pairs of facilities. If all the facilities

have the same length, the SRFLP becomes an instance of

the linear arrangement problem [1,2] which is itself a

special case of the quadratic assignment problem [3]. Va-

rious applications of the SRFLP have been identified in

the literature. One such application is in the flexible ma-

nufacturing systems, where machines within manufac-

turing cells are often placed along a straight path trav-

elled by an automated guided vehicle [4]. The minimum

linear arrangement problem was proved NP-hard [5],

therefore the SRFLP is NP-hard because this is a gener-

alization of it. Exact approaches have been proposed for

the problem. A branch-and-bound algorithm is presente-

din [6-8]. Picard and Queyranne [9] presented dynamic

programming algorithms for the SRFLP. Heragu and Ku-

siak [10] presented a nonlinear model. Love and Wong

[11] and Amaral [12-14] proposed linear mixed-integer

programs.

The paper has the following structure. Section 2 gives

a literature review on SRFLP. Section 3 is the problem

description. Section 4 the principles of the original CSA

and SA method are briefly introduced, the proposed hy-

brid algorithm in more details for SRFLP described in

Section 5. In Section 6 computational results are given.

Finally, Section 7 contains conclusions.

2. Literature Review

Many heuristics and metaheuristic algorithms have been

presented to solve the SRFLP because this problem is a

NP-complete problem [15]. Neghabat [16] proposed an

algorithmto reach a complete solution by adding one ma-

chine at a time to the end of the current solution. Drezner

[17] presented a heuristic method based on the eigenvec-

tors of a transformed flow matrix. Another heuristic has

been presented by Heragu and Kusiak [10], in which a

pair of facilities with the largest adjusted flow is initially

laid and then the partial order is gradually completed

through a loop adding new machines to the right and left

of the order obtained in the previous iteration. Heragu

and Alfa [18] proposed a simulated annealing algorithm

to solve SRFLP. A constructive greedy heuristic has been

developed by Kumar and Hadjinicola [19]. In this algo-

rithm assigns facilities with the largest number of moves

between them to adjacent locations on the line. Braglia

[20] introduced an algorithm that is combined of simu-

lated annealing and genetic algorithm to minimize the

total backtracking in the linear ordering of machines.

*Corresponding author.

A Hybrid Clonal Selection for the Single Row Facility Layout Problem with Unequal Dimensions 217

Solimanpur et al. [21] presented a non-linear 0 - 1 pro-

gramming model for the SRFLP then was solved by an

ant algorithm.

Anjos et al. [22], Anjos and Vannelli [23], Anjos and

Kong [24], Hungerlander and Rendl [25], had presented

semi-definite programming relaxation providing a lower

bound on the optimum value of the SRFLP. Recently,

many researchers proposed meta-heuristic methods, such

as: a scatter search algorithm by Kumar [26], a hybrid

algorithm based on ant colony optimization and PSO by

Teo and Ponnambalam, [27], a genetic algorithm by Lin

[28], a PSO algorithm by Samarghandi et al. [29], and

genetic algorithm by Datta et al. [30].

In this paper, a hybrid clonal selection algorithm is

presented for SRFLP. The computational experiments

show the efficient performance of the proposed algorithm

on different instances of various sizes available in the

literature.

3. Problem Formulation

Heragu and Kusiak [4] presented a model for the prob-

lem, which they called ABSMODEL. Let be i



the

distance between the centroid of department i and the

line origin and let Sij be the minimum separation between

departments i and j. Their model is given by,

min

ij ij

iji

cu u













 

(1)



.: 2

1, ,;1, ,.

ij ijij

tuul ls

injin



 





(2)

As the absolute value for the distance between the cen-

troids are used, it does not matter if department i is to the

left or to the right of department j. Note that the mini-

mum value that ij

can assume could be set greater

than







ll if we are given a value Sij.

If n the set of all permutations π of .

The SRFLP can be mathematically formulated in Amaral,

[14]:





1, 2,,N



1π

π11

min

ij ij

iji



 

 (3)

where dij is the distance between departments i and j with

respect to a permutation π. New modeling of the SRFLP

in (3) implies that the ABSMODEL searches the set of

all permutations of numbers 1, to find the per-

mutation which minimizes the objective function.

2,, n

4. Clonal Selection and Simulated Annealing

4.1. AIS, CSA

Artificial immune algorithm (AIA) is a recent branch of

stochastic search algorithms and classified as a popula-

tion-based metaheuristic method. These properties impart

a high degree of robustness and performance and at-

tracted interest of researches in implementing it to engi-

neering systems. This adopted engineering analogue,

called AIS, emerged in the 1990s as a new computational

research area [31].

Three commonly applied types of AIAs are clonal se-

lection algorithm (CSA), immune network algorithm

(INA) and negative selection algorithm (NSA). Among

these models, CSA is reported to be successful in com-

binatorial optimization problems [32]. Therefore, a CSA

is employed to solve SRFLP in this study.

CSA is treated as biological and random search based

general-purpose heuristic methods. The most important

component of a CSA is how to represent the solution.

The feasible solutions are coded as individuals. Each in-

dividual in the population is called an antibody. A CSA

starts with a set of random antibodies as an initial popu-

lation. Any population in turn is a subset of the solution

space. Appropriate solutions are searched within the po-

pulation. The new generations are formed through copy-

ing (cloning) in proportion to concept called as the affin-

ity value. The algorithm does not have a crossover op-

erator as in GA. The mutation operators are utilized to

search and evaluate new regions in the solution space in-

versely proportional to the affinity value of the solution.

The mutation rates in CSA are higher compared to Gas.

It can be described by the following steps [33]:

1) Initialize the antibody pool init P including the sub-

set of memory cells (M);

2) Evaluate the fitness of all the antibodies (affinity

with the antigen) in population P;

3) Select the best candidates (Pr) from population P,

according to their fitness;

4) Clone Pr into a temporary antibody pool (C);

5) Generate a mutated antibody pool (C1). The muta-

tion rate of each antibody is inversely proportional to its

fitness;

6) Evaluate all the antibodies in C1;

7) Eliminate those antibodies similar to the ones in C,

and update C1;

8) Re-select the antibodies with better fitness from C1

to construct memory set M. Other improved individuals

of C1 can replace certain members with poor fitness in P

to maintain the antibody diversity;

9) Return back to Step 2, if a pre-set termination crite-

rion is not met.

4.2. Simulated Annealing

Simulated annealing (SA) proposed by Kirkpatrick et al.

[34], belongs to the class of stochastic search algorithms,

known as meta-heuristics. This algorithm motivated from

A Hybrid Clonal Selection for the Single Row Facility Layout Problem with Unequal Dimensions

218

an analogy between the physical annealing of solid mate-

rials and optimization problems. SA has been widely

applied to solve combinatorial optimization problems. It

is inspired by the physical process of heating a substance

and then cooling it slowly, until a strong crystalline

structure is obtained. This process is simulated by low-

ering an initial temperature by slow stages until the sys-

tem reaches an equilibrium point and no more changes

occur.

5. Proposed Hybrid Algorithm

5.1. Antibody Representation and Initialization

There are various techniques for encoding solutions in an

individual. In this paper, we use permutation representa-

tion. In this approach, Integer-valued n elements are the

facilities to be considered. Each permutation indicates a

layout of facilities. Each permutation allocating n rec-

tangular facilities to a straight line is a feasible solution

for the SRFLP therefore the solution is always feasible.

Each layout (antibody) represents a potential solution and

has a cost value that refers to the affinity value of that

antibody.

We have used two techniques for initializing individu-

als. These techniques are described in the following:

 Random initializing

This approach, we generate random permutation for an

individual.

 Length-Based Permutation (LBP)

This method proposed in Samarghandi et al. [29]. It is

based on the assumption that if the flow between facili-

ties is equal then the optimal permutations of the facili-

ties can be given as follows:

Sort the facilities in non-descending order such that

the shortest facility is denoted by 1 and the longest facil-

ity by n. Then, Figure 1 shows the optimum solution

when n is an odd number, and Figure 2 shows the opti-

mum solution when n is an even number [29].

5.2. Cloning Selection Procedure

For the selection of antibodies to constitute the mutating

pool, first the best antibody is selected and to fulfill the

rest, the binary tournament selection operator is used. It

selects two random individuals from the population and

stores a copy of the best individual (based on objective

values) in the mating pool. The process is repeated (pop-

size-1) times.

 

312 1ii 

Figure 1. Optimal layout when n is odd.

 

213 1ii 

Figure 2. Optimal layout when n is even.

5.3. Affinity Maturation

After generating the clone population all of the antibod-

ies existing in the pool undergo an operator which makes

a random change in the clones. This operator is called

hyper-mutation. Each antibody undergoes different rate

of change based on the affinity value. The inferior anti-

bodies undergo high rate of hyper-mutation whereas bet-

ter antibodies suffer a slight change. In this study, two

mutation procedure are used as follows:

As a low rate hyper-mutation, we utilize swap muta-

tion. In swap mutation randomly chooses two facilities

and exchanges their locations. As high rate hyper-muta-

tion, we use an operator working as follows:

 One solution is randomly chosen from the 20% of the

antibodies with the smallest objective values.

 Each gene of the antibody is checked against its cor-

responding gene in selected antibody for equality. If

they contain identical values, the antibody maintains

the gene; otherwise, the gene becomes empty.

Next, the values missing in the antibody are identified

and inserted into the empty genes in a random manner.

After defining the mutation operators, we need to de-

termine the condition under which we use one of the op-

erators. We sort the antibodies in non-decreasing order of

their objective function values. Swap mutation is applied

for H% of the best antibodies and the remaining antibod-

ies





1H% are generated through another operator.

For accepting the off spring we use simple simulated

annealing acceptance criterion. Besides the acceptance of

better offspring, inferior offspring might be accepted by

the following random mechanism:













0,1exp offspring creatorrnd T

In this paper, we use exponential cooling schedule as

follows:









1, 1

,1,2,,.

TAiBATTL L

BT AiL

 

 

(5)

where T0, Tf and L are initial temperature, final (stopping)

temperature and desired number of temperature levels

between T0 and Tf, respectively.

5.4. Stepwise Procedure

The stepwise procedure of the proposed hybrid algorithm

is as follows:

Step 1: Set the values of control parameters: P (anti-

body population size), T0 (initial temperature), Tf (final

temperature), L (desired number of temperature levels

between T0 and Tf), B (parameter of elimination ratio of

antibodies at each iteration), ;

1K

Step 2: Create a population of P antibodies based on

the procedure described in Section 5.1;

Step 3: Evaluate each antibody in the population;

A Hybrid Clonal Selection for the Single Row Facility Layout Problem with Unequal Dimensions

219

into two major sets, those with a proven optimal solution

and those without a proven optimal solution.

Step 4: Select antibodies using the tournament tech-

nique for mating pool;

Step 5: Sort the antibodies in non-decreasing order of

their objective function values. Swap mutation is applied

for H% of the best antibodies and replace that with off-

spring if its function improved or

 



0,1exp offspring creatorrnd T



and the

remaining antibodies



are generated through

another operator;



1H

The first set of problems with an optimal solution in-

cludes 15 problems. Objective function value and com-

putation time obtained by different heuristics are shown

in Table 1. For each problem the algorithms perform 10

runs and the best result obtained by the 10 runs presented.

Moreover, the computational time which the best solu-

tion appears for the first time by HCSA is reported. In

Table 1 the obtained best results of proposed algorithm

are compared with the solutions of Solimanpur et al. [21]

and with the optimal objective values given by the exact

methods. As shown in Table 1, proposed algorithm

could obtain the optimal solution for each of the prob-

lems. As mentioned the computational time in the tables

are not comparable because the computers are different.

Still, the computational time of meta-heuristics is con-

sidered here also as it is available in the literature.

Step 6: If generated new antibodies are less than (B*P)

go to Step 5 otherwise proceed to Step 7, 1KK





and adopt temperature;

Step7: If k = N stop. Otherwise go to Step 4.

6. Computational Results

In this section, we present the computational results of

the proposed HCSA algorithm applied to the existent

problems in the literature. The proposed algorithm are

coded in C++ and executed on a PC with 2.2 GHz Intel

core 2 Due and 4 GB of RAM memory. The proposed

hybrid algorithm includes six parameters which affect the

algorithm’s performance. For tuning the algorithm, ex-

tensive sensitivity analysis were conducted with different

sets of parameters. Based on these observations, the fol-

lowing values were obtained for the parameters:

We have also tested the algorithm on large-size prob-

lems with no proven optimal solution reported in the lit-

erature. This set of problems consist of 20 instances con-

sidered in Table 2. The best found solutions are reported

in boldface. It is observed the proposed HCSA generates

better solutions for 3 problems.

7. Conclusion

70, 35, 12, 15, 0.35PT NnLnB ,

This paper considers the single row facility layout problem

All the problems from the literature can be divided

Table 1. The best solutions obtained by the CSA for 15 traditional instances.

Solimanpur et al. (2005) Proposed algorithm

Problem no. No. of facility Optimal

objective value Reference

OFV Time Solution Time

1 4 78.00 Beghin-Picavet and Hansen (1982)78.00 0.00 78.00 0.00

2 5 151.00 Love and Wong (1976) 151.00 0.00 151.00 0.00

3 5 1.100 Nugent

et al. (1968) 1.100 0.00 1.100 0.00

4 6 1.990 Nugent

et al. (1968) 1.990 0.00 1.990 0.00

5 7 4.73 Nugent

et al. (1968) 4.73 0.00 4.73 0.00

6 8 6.295 Nugent

et al. (1968) 6.295 0.00 6.295 0.00

7 8 2324.5 Simmons (1969) 2324.5 0.00 2324.5 0.00

8 10 2781.5 Simmons (1969) 2781.5 0.01 2781.5 0.01

9 11 6933.5 Simmons (1969) 6933.5 0.03 6933.5 0.012

10 12 23.365 Heragu and Kusiak (1991) 23.365 0.06 23.365 0.01

11 15 44.600 Heragu and Kusiak (1991) 44.600 0.18 44.600 0.022

12 20 119.710 Heragu and Kusiak (1991) 119.71 1.8 119.710 0.081

13 20 15549.0 Heragu and Kusiak (1991) 15549.0 2.30 15549.0 0.12

14 30 334.870 Heragu and Kusiak (1991) 334.870 37.30 334.870 1.11

15 30 44965.0 Heragu and Kusiak (1991) 44965.0 37.30 44965.0 0.81

A Hybrid Clonal Selection for the Single Row Facility Layout Problem with Unequal Dimensions

220

Table 2. The best solutions obtained by the HCSA for 20 large-size instances.

Anjos et al. (2005) Samarghandi and Eshghi

(2010) Datta et al. (2011) HCSA

Problem

no. No. of

facility Objective

function Time (h) OFV Time (sec)OFV Time (sec) OFV Time (sec)

1 60 1479294.00 5 1477834.0 0.82 1477834.0 19.54 1477834.0 10.90

2 60 829792.00 5 841792.0 0.98 841792.0 22.34 841792.0 9.66

3 60 650167.00 5 648337.5 0.90 648337.5 68.81 648337.5 23.12

4 60 402214.00 5 398511.0 0.913 398468.0 20.71 398406.0 19.91

5 60 318805.00 5 318805.0 0.762 318805.0 26.41 318805.0 17.6

6 70 1531212.00 7 1529197.0 1.499 1528621.0 64.83 1529197.0 29.11

7 70 1440901.00 7 1441028.0 1.940 1441028.0 77.49 1441028.0 48.1

8 70 1518993.50 7 1518993.0 1.761 1518993.0 68.26 1518993.0 42.1

9 70 971090.00 7 969130.0 1.233 968796.0 100.59 969130.0 50.83

10 70 4216349.00 7 4218230.0 1.570 4218017.5 60.48

4218002.0 52.61

11 75 2396213.00 10 2393483.0 2.010 2393456.0 125.26 2393490.0 88.66

12 75 4325142.00 10 4321190.0 2.198 4321190.0 128.95 4321190.0 90.36

13 75 1256199.00 10 1248551.0 2.912 1248537.0 157.95 1248551.0 103.1

14 75 3941713.00 10 3942013.0 2.516 3941981.0 119.92 3942013.0 93.3

15 75 1801040.00 10

1791408.0 2.098 1791408.0 101.67 1791408.0 90.21

16 80 2104771.00 10 2069097.5 3.975 2069097.5 75.41 2069097.5 110.1

17 80 1919288.00 10 1921177.0 5.641 1921177.0 68.75 1921177.0 126.8

18 80 3291413.00 10 3251413.0 4.797 3251368.0 85.9 3251413.0 133.9

19 80 3751331.00 10 3746515.0 3.453 3746515.0 77.81 3746515.0 141.6

20 80 1593108.0 10 1589061.0 3.769 1588901.0 196.51

1588862.0 166.3

in which the size of facilities are different. This problem

belongs to an NP-hard class and traditional approaches

cannot reach to an optimal solution in a reasonable time.

Therefore, in this paper a hybrid clonal selection was

proposed to find optimal and near-optimal solutions for

SRFLP. To evaluate the efficiency the proposed hybrid

algorithm the known SRFLP test problems from the lit-

erature are used. The proposed HCSA is first tested on

set of problems with a proven optimal solution in which

it could obtain the optimal solution for all of them. Then

the algorithm is tested for large-size instances and the

results verify the efficiency of algorithm in finding good

quality solutions compared other heuristics and it suc-

cessfully could improve 3 of 20 solutions.

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