an application of genetic algorithm for non-restricted space and pre-determined length width ratio...

7/30/2019 An Application of Genetic Algorithm for Non-restricted Space and Pre-determined Length Width Ratio Facility Layout Problem

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*Corresponding author ( J.Teeravaraprug ). Tel/ Fax: +66-2-5643001 Ext .3083. E-mail addresses:t j [email protected] u.ac.th . 2011 Internat ional Transacti on Journal of Engineering,Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860.eISSN 1906-9642. Online Avail able at htt p:/ / TuEngr. com/ V02/ 385-404.pdf

385

International Transaction Journal of Engineering,Management , & Applied Sciences & Technologies

http://www.TuEngr.com , http://go.to/Research

An Application of Genetic Algorithm for Non-restricted Space andPre-determined Length Width Ratio Facility Layout Problem

Jirarat Teeravarapruga*

, Tarathorn Kullpataranirunb

, and Boonchai Chinpaditsuka

a Department of Industrial Engineering, Faculty of Engineering, Thammasat University, THAILANDb Department of Industrial Management, Faculty of Business, Mahanakorn University of Technology,THAILAND

A R T I C L E I N F O A B S T RA C T Article history :Received 02 June 2011Received in revised form20 August 2011Accepted 24 August 2011Available online01 September, 2011

Keywords :Genetic algorithm;Facility layout problem;Two leveled chromosome

The use of a genetic algorithm is presented to solve afacility layout problem in the situation where there isnon-restricted space but the ratio of plant length and width is

pre-determined. A two-leveled chromosome is constructed. Sixrules are established to translate the chromosome to facility design.An approach of solving a facility layout problem is proposed. Anumerical example is employed to illustrate the approach.

2011 International Transaction Journal of Engineering, Management, &Applied Sciences & Technologies. Some Rights Reserved.

1. Introduction

Facility layout is one of the main fields in industrial engineering where a number of researchers have given elevated attentions. Various models and solution approaches for

several circumstances of facility layout have been proposed during the past three decades

(Kusiak and Heragu, 1987). Kusiak and Heragu (1987), Meller and Gau (1996), Heragu

(1997), and Balakrihnan and Cheng (1998) presented surveys of the layout problem and various

mathematical models. Moreover, Tavakkoli-Moghaddam and Shayan (1996) did a

comparative survey of the recent and advanced approaches in order to evaluate and select the

most suitable one of the facilities design problems.

2011 Internat ional Transaction Journal of Engineering, Management, & Applied Sciences & Technologies.


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386 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk

The problem in facility layout is to assign facilities to locations such that a given

performance measure is optimized. The problem commonly found in industries is how to

allocate facilities to either maximize adjacency requirement (Seppanen and Moore, 1970), or

minimize the cost of transporting materials between them (Koopmans and Beckmann, 1957).The maximize adjacency objective uses a relationship chart that qualitatively specifies a

closeness rating for each facility pair. This is then used to determine an overall adjacency

measure for a given layout. The minimizing of transportation cost objective, which is

considered in this paper, uses a value that is calculated by multiplying together the flow,

distance, and unit transportation cost per distance for each facility pair. The resulting values

for all facility pairs are then added.

However, solving the facility layout problem is elaborate because the facility layout

problem belongs to the class of non-polynomial hard (NP-hard) problems which are unsolvable

in polynomial time. It suggests that the problems complexity increases exponentially with the

number of facility locations (Adel El-Baz, 2004). Heuristic techniques were introduced to

seek near-optimal solutions at reasonable computational time for large-scaled problems

covering several well known methods such as improvement, construction and hybrid methods,

and graph-theory methods (Kusiak and Heragu, 1987). One of the well-liked tools is genetic

algorithm (GA), which is successfully applied in various types of problems. Wu and Appleton

(2002) applied GA to block layout by considering aisle. Lee, et al. (2003) proposes an improved

GA to derive solutions for facility layouts that are to have inner walls and passages. The

proposed algorithm models the layout of facilities on gene structures. Improved solutions are

produced by employing genetic operations known as selection, crossover, inversion, mutation,

and refinement of these genes for successive generations. Recently, Wu et al. (2007)

introduced a genetic algorithm for cellular manufacturing design and layout.

Based on the review, most researches give attention in minimization of transportation

cost in various circumstances by assigning fixed overall area of facilities. This paper considers

in the case that all facilities have not yet constructed. The overall area of facilities can be

changed, however the range of the ratio of width and length is given. This paper is then to

minimize transportation cost and overall area by enhancing the concept of genetic algorithm.


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2. Genetic Algorithm

Genetic algorithm (GA) introduced by Holland (1975) has increasingly gained popularity

in optimization. The main concept of GA is taken from natural genetics and evolution theory

(Tavakkoli-Moghaddam and Shayan, 1997; Venugopal and Narendran, 1992; Zhang et al.,

1997). GA is a simple algorithm that encodes a potential solution to a specific problem on a

simple chromosome like data structure and applies recombination operators to these structures

so as to improve the solution while preserving all critical information (Chan et al., 1996).

GA starts with an initial set of random solutions for the problem under consideration. This

set of solutions is called population. The individuals of the population are called

chromosomes. The chromosomes of the population are evaluated according to a predefined

fitness function. The chromosomes evolve through successive iterations called generations.

During each generation, merging and modifying chromosomes of a given population create a

new set of population. Merging chromosomes is known as crossover while modifying an

existing one is known as mutation. Crossover is the process in which the chromosomes are

mixed and matched in a random fashion to produce a pair of new chromosomes (offspring).

Mutation operator is the process used to rearrange the structure of the chromosome to produce a

new one. The selection of chromosomes to crossover and mutate is based on their fitness

function. Once a new generation is created, deleting members of the present population to

make room for the new generation forms a new population. The process is iterative until a

specific stopping criterion is reached.

In short, the typical steps required to implement GA are: encoding of feasible solutions into

chromosomes using a representation method, evaluation of fitness, setting of GA parameters,

selection strategy, genetic operators, and criteria to terminate the process (Goldberg, 1989).

Standard GAs utilize a binary coding of individuals as fixed-length strings over the alphabet{0,1}, a reproduction method based on the roulette wheel selection, a standard crossover

operator to produce new children and a mutation operator altering a bit string from a selected

individual. Tavakkoli-Moghaddain and Shayan (1998) introduced an improved robust GA

using non binary coding as well as different selection schemes and genetic operators.

In recent years, GA has been successfully applied to a vast variety of problems. Some

examples include constrained optimization (Homaifar, et al., 1994), multiprocessor scheduling


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(Hov, et al., 1994), jobshop scheduling (Davis, 1985), computer aided molecular design

(Venkatasubramanian, et al., 1994), and quadratic assignment problem (Tate and Smith, 1995).

The application of GA to facility layout problem are shown in Al-Hakim (2000), Gau and

Meller (1999), Hamamota (1999), Islier (1998), and Rajasekharan et al. (1998). Even thoughGA is popular, efficiency of applying GA depends on the nature of the problem and the process

of trial and error. Some experiments are required to analyze the suitability of genetic operators

in GA (Tavakkoli-Moghaddam and Shayan, 1997).

3. Two -leveled Genetic Algorithm with Facility Layout

To solve the facility layout problem, this paper introduces an enhanced genetic algorithm

called two-leveled genetic algorithm. Chromosome design is the starting task to solve the

problem. It is required to encode the candidate solutions in the solution space in the form of

symbolic strings. Then findings an appropriate fitness function and penalty function are next.

The uses of GA procedures of selection, crossover, and mutation are to acquire possible

chromosomes.

B11 B12 B13 B14 Z

B21 B22 B23 B24

Figure 1: Two-leveled chromosome.

3.1 Chromosome Design

The chromosome is designed in two levels shown in Figure 1. The number of genes in

each level is equal to the number of facilities plus one. The first level is used to identify which

side of the given facility is employed in designing the layout. B 1m is 0 or 1 value, where B 1m = 0

means the width of the facility m is utilized in designing the layout and B 1m=1 means the length

of the facility m is utilized. Z stands for the ratio of the plant length and the plant width and

then Z 1. The second level is the priority of facility arrangement. B 2ms are positive

integers.

The relation of chromosome and plant layout is based on (X,Y) coordinates. The facility

that B 2m = 1 is arranged first on (0,0) coordinate by considering B 1m. Figure 2 shows how to

arrange the first facility on (X,Y) coordinate. For arranging the remaining facilities, six rules

are set. The first rule is the remaining facilities do not use (0,0) coordinate as a starting point.For examples, the next facility that B 2m = 2 is arranged on the coordinate of the first facility but


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*Corresponding author ( J.Teeravaraprug ). Tel/ Fax: +66-2-5643001 Ext .3083. E-mail addresses:t j [email protected] u.ac.th . 2011 Internat ional Transacti on Journal of Engineering,Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860.eISSN 1906-9642. Online Available at htt p:/ / TuEngr. com/ V02/ 385-404.pdf

389

(0,0). For the left-hand side of Figure 2, the possible coordinates are (0,L), (W,L), and (W,0)

and for the right-hand side, the possible coordinates are (0,W), (L,W), and (L,0).

Figure 2: Arrangement of facility m that B 2m = 1.

The second rule is repetition points are cut off the next possible starting points. Figure 3

shows the proof of the rule. Based on Figure 3 (A), (1,1) and (1,0) coordinates are out and the

possible starting points are then (0,1), (2,1) and (2,0). Figures 3 (B and C) show if one of the

duplicate points is selected as a starting point, the overall area is greater than that not using a

duplicate point. The areas of layout shows in Figures 3 (B and C) are 5 and 4 respectively.

The third rule is that select the coordinate which has the lowest X if Ys are equal or select the

coordinate which has the lowest Y if Xs are equal (Figure 4). Based on Figure 3(A), there are

three possible starting points: (0,1), (2,1) and (2,0). Comparing between (0,1) and (2,1), (0,1)

should be selected and comparing between (2,1) and (2,0), only (2,0) should be selected.

Therefore, (0,1) and (2,0) are the possible starting points.

Figure 3: Proof of the second rule.


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Figure 4 shows the proof of the fourth rule. It can be seen that Figure 4(B) uses (2,1) as the

starting point, and its results the largest area, which is 7. The fourth rule is, utilize the defined Z

in the arrangement. Based on Figure 4, Zs equal to 1, 1.75, and 3.5 respectively. For

example, if the pre-defined Z equals 1 to 2, the only possible starting point is (0,1) and if the pre-defined Z equals to 3 to 4, the only starting point is (2,0). The fifth rule is in the case that Z

is out of the desired range, continuing arrange the remaining facilities. The last rule is each

facility cannot be overlapped.

Figure 4: Proof of the third rule.

3.2 Fitness Function In the fitness function, transportation expense and penalty are considered. The

transportation expense of chromosome k (TC k ) is shown in Eq. (1)

1 1

n n

k ij ij kiji j i

TC f C D= = +

= (1)

where


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391

ij f is frequency of transportation between facility i and facility j

ijC is the transportation expense per distance unit between facility i and facility j

kij D is the distance between facility i and facility j of chromosome k

n is the number of facilities

A penalty value is incurred when Z is out of the desired range in order to reduce the chance

of choosing in the selection process. This paper assumes a constant value of penalty.

Considering both transportation expense and penalty value, this paper multiplies those

values and called EV k (Eq.(2)).

EVk = TC k * PV (2)

where

PV is a penalty value and equals either 1 or a large value. It is 1 when Z is in the desired

range and it is a large value when Z is out of the desired range. So, the EV k would be very

large when Z is out of the desired range and it is the transportation cost when Z is in the desired

range.

The fitness function of chromosome k (F k ) is a measure of a solution to the objective

function. Therefore, the fitness function should be an inverse correlation with the cost. This

paper is assumed the fitness function as shown in Eq. (3).

1 /EVk k F = . (3)

3.3 Selection

In the chromosome selection process, this paper uses enlarged sampling space and roulette

wheel selection. The selection probability of chromosome k equals to the fitness value of the

chromosome k over the fitness values of population when the fitness value of population is the

summation of the fitness values over the population.


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3.4 Crossover

Chan and Tansri (1994) compared three crossover methods: CX (Cycle Crossover), OX

(Order Crossover), and PMX (Partially Matched Crossover), and concluded that CX operator

converges very rapidly in just a small number of generations, OX operator is the most

insensitive to the initial population, and PMX operator is a steady performer. PMX consistently

shows a steady trend of improvement in every graduation in generation. PMX has a mild

increase in the average fitness value and most often it produces the fittest solutions among the

three operators. PMX is expected to operate well and perform consistently for suitable

generation and population combinations. Therefore, this paper applies a well-known PMX as a

crossover method. Due to the uniqueness of the chromosome, the applied PMX crossover step

procedures are 1) randomly select a group of the population and called parents and randomly

select two positions in each selected parent, and 2) construct children by exchanging the genes

between two positions of the parents. In the case that there are duplications of B 2m in a

chromosome, the cells of B 2m that staying out of the mapping range are required to be legalized.

The process of legalization starts by finding duplicated numbers. Surely, one of duplications

stays in the mapping range and the other one is out of the mapping range. Find the genes

carrying the duplications in the range. Map the duplicated gene with the same gene of the

original chromosome. Replace both B 1m and B 2m of out of range duplicated number with the

gene of the original chromosome. In the case of remaining having duplications, take the

number of that to the other chromosome. Then replace both B 1m and B 2m of out of range

duplicated number with the numbers getting above. Check if there is duplication. If

duplication appears, redo the process. If not, the chromosome is legal.

Examples of the crossover and legalization process are shown in Figures 5-7. Twochromosomes are shown in Figure 5 as parents. The cutting points are at the second and

seventh. Proto-child 1 shows the crossover result when parent 1 is the main chromosome where

as proto-child 2 shows the result when parent 2 is the main one. It can be seen that there are

duplicating and lacking numbers in the results. For proto-child 1, there are two 1, 2, and 9 and

no 3, 4, and 5 in the second row. Contrarily, for proto-child 2, there are two 3, 4, and 5 and no

1, 2, and 9. Legalization process is then required. The process starts with the mapping range.

Considering the mapping range of proto-child 1, B 26 = 1. B 26 of the main chromosome equals to6, but the proto-child 1 already exists 6. So, considering 6 in the proto-child 1, it is on B 23. B23


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of the main chromosome equals to 3 and there is no 3 in the original proto-child 1. Therefore,

the1

3

is copied to0

1

of the out of mapping range of proto-child 1. Another example of

legalization process is the 2 duplication of the proto-child 1. The considering 2 is in themapping range: B 25. B25 of parent 1 equals to 5 and there is no 5 in the proto-child 1.

Therefore, the1

5

is copied to1

2

of the out of mapping range of proto-child 1. The last

legalization process of the proto-child 1 is 9. Considering the mapping range of proto-child 1,

B24 = 9. B 24 of parent 1 equals to 4 and there is no 4 in the proto-child 1. Therefore, the14

is copied to 19

of the mapping range of proto-child 1. Similarly, proto-child 2 is required to

do the legalization process. The process of legalization of the proto-child 1 is shown in Figure

6 and the offsprings are then shown in Figure 7.

3.5 Mutation

Insertion mutation, which is utilized in this paper, is a well-known mutation. Its process

includes:

1) Randomly select a group of chromosome from the population.

2) Randomly select a gene in each chromosome.

3) Randomly select a position in each chromosome.

4) Inserting the selected gene in the selected position.

Select two positions

0 1 1 1 1 0 0 0 11 1 3 4 5 6 7 8 9

1 0 1 1 0 0 0 0 15 4 6 9 2 1 7 8 3

Parent 1

Parent 2


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Exchange the genes between two positions

0 1 1 1 0 0 0 0 11 2 6 9 2 1 7 8 9

1 0 1 1 1 0 0 0 15 4 3 4 5 6 7 8 3

Figure 5: Crossover step procedures.

0 0 11 6 3

0 12 5

1 19 4

Figure 6 : Chromosome legalized.

1 1 1 1 0 0 0 0 13 5 6 9 2 1 7 8 4

0 1 1 1 1 0 0 0 0

2 9 3 4 5 6 7 8 1

Figure 7: Offspring.

Figure 8 : Insertion mutation.

An example of insertion mutation is shown in Figure 8.

3.6 The Program

Microsoft Visual Basic 6 is utilized to aid in calculation based on the concept of

chromosome design discussed in section 3.1, fitness function discussed in section 3.2, selection

discussed in section 3.3, crossover discussed in section 3.4, and mutation discussed in section

Proto-child 1

Proto-child 2

Offspring 1

Proto-child 1B2m= 1

Proto-child 1B2m= 2

Proto-child 1B2m= 9

Offspring 2


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*Corresponding author ( J.Teeravaraprug ). Tel/ Fax: +66-2-5643001 Ext .3083. E-mail addresses:t j [email protected] u.ac.th . 2011 Internat ional Transacti on Journal of Engineering,Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860.eISSN 1906-9642. Online Available at htt p:/ / TuEngr. com/ V02/ 385-404.pdf

395

4. Experiments and Results

Three departments are used. Each departments area is defined as shown in Table 1.

Frequencies of transportation between departments are shown in Table 2. Table 3 shows

transportation expenses between departments. The predetermined ratio of the plant length and

the plant width is between 1 and 2. An optimization technique provides eight patterns of

layouts as shown in Figure 9. Each pattern corresponds to chromosomes as shown in Figure

10. This example uses population size as 10, generation size as 10, crossover probability as

0.95, mutation probability as 0.001, and run as 10 times. After running the program for 10

times, the results show that one of the optimal solutions can be obtained in every run (Table 4).

The total transportation costs are 11.35.

Table 1: Defined departments areas.Department 1 2 3Width 1 1 2Length 1 2 3

Table 2: Transportation frequencies.Department 1 2 3

1 - 2 12 - - 1

Table 3: Transportation expenses.Department 1 2 3

1 - 1 22 - - 3

Figure 9: Optimal facility layout of the example.


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Figure 10 : Chromosomes of the optimal facility layouts.

Table 4: The results of the example.

Run Chromosome Width Length Ratio Area Costs1 1 1 0 3 3 1 9 11.35

2 3 12 1 1 1 3 3 1 9 11.35

3 2 13 0 0 0 3 3 1 9 11.35

3 2 14 1 1 1 3 3 1 9 11.352 3 1

5 0 0 1 3 3 1 9 11.353 2 1

6 1 1 0 3 3 1 9 11.353 2 1

7 0 0 1 3 3 1 9 11.353 2 1

8 0 0 0 3 3 1 9 11.352 3 1

9 1 1 1 3 3 1 9 11.353 2 1

10 1 1 0 3 3 1 9 11.352 3 1

0 0 0

1 2 3

0 0 0

1 23

0 0

1 2 3

0 0

1 23

11

0 0 0

12 3

0 0 0

12 3

0 0

12 3

1 0 0

12 3

1

0 0 0

1 23

0 01

1 23

0 0 0

123

0 0

123

1

0

1 2 3

1 1 0

1 23

1 1

1 2 3

1 1

1 23

1 111

0

12 3

11 0

12 3

1 1

12 3

11

12 3

1 11 1

1 23

111 0

1 23

11

0

123

1

123

1 1 11

Pattern 1

Pattern 8

Pattern 2

Pattern 3

Pattern 4

Pattern 5

Pattern 6

Pattern 7


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397

Based on the previous example, it is shown that the proposed approach and program can be

utilized. Another example is taken from Chan and Tansri (1994) and Mak et al.(1998). The

following plant specifications are used in this experiment:

Plant size 9-location plant consisting of 3 rows and 3 columns

Distance measure Rectilinear between centroids of locations

Evaluation criterion Quantitative (minimize total materials handling cost)

Frequency chart As shown in Table 5

Cost chart As shown in Table 6

The optimal facility layouts of the example providing by Chan and Tansri (1994) areshown in Figure 11. Based on the example, there is non- restricted space and there is no

limitation of the ratio of the plant length and width. Therefore, to verify the proposed

approach, the ratio is not utilized.

Table 5: Frequency (from-to) chart (number of trips per month).

From\To 2 3 4 5 6 7 8 9

1 100 3 0 6 35 190 14 122 6 8 109 78 1 1 1043 0 0 17 100 1 314 100 1 247 178 15 1 10 1 796 0 1 07 0 08 12

Table 6: Cost chart ($ per trip).

From\To 2 3 4 5 6 7 8 9

1 1 2 3 3 4 2 6 72 12 4 7 5 8 6 53 5 9 1 1 1 14 1 1 1 4 65 1 1 1 16 1 4 67 7 18 1


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Table 7: Population and generation sizes and number of trials.

No. Population size Generation size No. of trials

1 20 10 2002 40 10 400

3 100 10 1000

4 200 10 2000

5 500 10 5000

6 20 20 400

7 40 20 800

8 100 20 2000

9 200 20 4000

10 20 40 800

11 40 40 1600

12 100 40 4000

13 200 40 8000

14 20 100 2000

15 40 100 4000

16 100 100 10000

17 20 200 4000

18 40 200 8000

Table 8: Probabilities of crossover and mutation.

No. Probability of crossover Probability of mutation

1 0.5 0.000

2 0.6 0.001

3 0.7 0.003

4 0.8 0.005

5 0.9 0.010

6 1.0 0.030

7 - 0.050


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Table 9: Average costs and number of best found result.

No Population size Generation size No. of trials Average costs # of best found

1 20 10 200 5341.53 6

2 20 20 400 5254.11 9

3 40 10 400 5173.22 10

4 20 40 800 5181.32 17

5 40 20 800 5109.66 22

6 100 10 1000 5041.11 24

7 40 40 1600 5044.64 42

8 20 100 2000 5127.03 41

9 100 20 2000 4976.64 57

10 200 10 2000 4968.92 68

11 20 200 4000 5080.05 65

12 40 100 4000 5005.18 76

13 100 40 4000 4919.95 106

14 200 20 4000 4906.59 114

15 500 10 5000 4888.8 135

16 40 200 8000 4971.27 110

17 200 40 8000 4865.42 187

18 100 100 10000 4882.94 183

Table 10: Results by probabilities of crossover and mutation.

Probability of mutation

0.000 0.001 0.003 0.005 0.010 0.030 0.050

P r o

b a

b i l i t y o

f c r o s s o v e r 0.5 4919.7,0 4895.5,1 4891.4,4 4915.8,4 4874.7,4 4852.9,4 4860.4,4

0.6 4894.2,3 4879.8,5 4882.4,2 4878.4,3 4862.4,6 4857.2,6 4852.6,5

0.7 4841.0,5 4847.4,7 4881.5,3 4898.0,1 4878.2,4 4843.1,6 4833.2,7

0.8 4871.1,3 4879.1,3 4910.9,5 4878.0,4 4851.9,4 4851.9,4 4848.6,6

0.9 4851.9,4 4844.2,6 4867.2,2 4846.5,5 4822.4,9 4838.6,6 4838.7,7

1.0 4867.0,5 4894.3,5 4841.0,5 4874.9,2 4850.8,6 4835.6,6 4843.1,6

Since this research found that the appropriate population size and generation size are 200

and 40 respectively. Those settings then are used to determine appropriate probabilities of

crossover and mutation and it is found that the appropriate crossover and mutation for the Mak

et al. (1998) approach are 0.6 and 0.001 respectively. Utilizing the settings, the results show

that the average of best material handling costs among the 10 runs was 4840 and the number of


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runs which yielded one of the eight optimal solutions was 5 (Table 12). The comparison table

is shown in Table 12. It can be seen that the result of this research is better than that of Mak et

al. (1998) and the number of best found of this research is higher than that of Mak et al. (1998).

Therefore, the proposed approach is one of the good means to solve the facility layout problem.

Table 11: Comparative results of Mak et al. (1998) approach, PMX, OX, and CX.

Crossover approach Mak et al. (1998) approach PMX OX CX

Population size 100 100 100 100

Generation size 20 20 20 20

Probability of crossover 0.6 0.8 1.0 0.9

Probability of mutation 0.001 0.001 0.001 0.030

Average costs 4856.0 4979.3 5014.8 4986.9# best found 4 2 1 3

Table 12: The comparative result.

Mak et al. (1998) approach Proposed model

Population size 200 200

Generation size 40 40

Probability of crossover 0.6 0.9

Probability of mutation 0.001 0.010

Average costs 4840.0 4822.4

# best found 5 9

5. Conclusion

This research provides an approach to solve facility layout problem via genetic algorithm.

The research considers the case that the plant area is non-restricted but the ratio of the plantlength and width is pre-determined. Two-leveled chromosome is constructed to aid in solving

the problem. To translate the chromosome to facility layout, six rules are set. The fitness

function is based on transportation expense and penalty. Enlarged sampling space and roulette

wheel selection are used. The process of crossover and mutation are also utilized. A numerical

example is provided to illustrate the proposed approach. Furthermore, a comparison of the

proposed approach to Mak et al. (1998) is presented. The result shows that the proposed

approach provides less average costs than Mak et al. (1998) approach and the number of runs


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which yielded one of the eight optimal solutions of the proposed approach is higher than Mak et

al. (1998) approach.

6. Acknowledgements A very special thank you is due to Professor Dr. Chieh-Yuan Tsai ( Yuan Ze University,

Taiwan ) and Dr. Natapat Areeratkulkarn ( Dhurakij Pundit University, Thailand ) for insightful

comments, helping clarify and improve the manuscript.

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Dr. J. Teeravaraprug is an Associate Professor of Department of Industrial Engineering at ThammasatUniversity, Thailand. She holds a B.Eng. in Industrial Engineering from Kasetsart University, Thailand, anM.S. from University of Pittsburgh, and PhD from Clemson University, USA. Her research includes design of experiments, quality engineering, and engineering optimization.

Dr. T. Kullpataranirun is a lecturer of Department of Industrial Management at Mahanakorn University,Thailand. He holds a B.Eng in Industrial Engineering from Kasetsart University, an M.Eng fromChulalongkorn University, and Ph.D. from Sirindhorn International Institute of Technology, ThammasatUniversity, Thailand. His research includes industrial management, quality engineering, and engineeringoptimization.

B.Chinpaditsuk is a master student in the department of industrial engineering at Thammasat University. Heholds a B.Eng degree in Electrical Engineering from Kasetsart University.

Peer Review: This article has been internationally peer-reviewed and accepted for publication

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an application of genetic algorithm for non-restricted space and pre-determined length width ratio...

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