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*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://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 and Pre-determined Length Width Ratio Facility Layout Problem Jirarat Teeravarapruga*, Tarathorn Kullpataranirunb, and Boonchai Chinpaditsuka
a Department of Industrial Engineering, Faculty of Engineering, Thammasat University, THAILAND b 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 2011 Received in revised form 20 August 2011 Accepted 24 August 2011 Available online 01 September, 2011 Keywords: Genetic algorithm; Facility layout problem; Two leveled chromosome
The use of a genetic algorithm is presented to solve a facility layout problem in the situation where there is non-restricted space but the ratio of plant length and width is pre-determined. A two-leveled chromosome is constructed. Six rules are established to translate the chromosome to facility design. An approach of solving a facility layout problem is proposed. A numerical 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 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies.
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 problem’s 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.
*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
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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
388 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
(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 though
GA 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. Twoleveled 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. B1m is 0 or 1 value, where B1m = 0
means the width of the facility m is utilized in designing the layout and B1m=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. B2ms are positive
integers.
The relation of chromosome and plant layout is based on (X,Y) coordinates. The facility
that B2m = 1 is arranged first on (0,0) coordinate by considering B1m. 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 B2m = 2 is arranged on the coordinate of the first facility but
*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://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 B2m = 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.
390 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
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 (TCk) is shown in Eq. (1)
1 1
n n
k ij ij kiji j i
TC f C D= = +
=∑∑ (1)
where
*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
391
ijf is frequency of transportation between facility i and facility j
ijC is the transportation expense per distance unit between facility i and facility j
kijD 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 EVk (Eq.(2)).
EVk = TCk * 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 EVk 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 (Fk) 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 kF = . (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.
392 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
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 B2m in a
chromosome, the cells of B2m 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 B1m and B2m 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 B1m and B2m 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. Two
chromosomes 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, B26 = 1. B26 of the main chromosome equals to
6, but the proto-child 1 already exists 6. So, considering 6 in the proto-child 1, it is on B23. B23
*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
393
of the main chromosome equals to 3 and there is no 3 in the original proto-child 1. Therefore,
the 13⎡ ⎤⎢ ⎥⎣ ⎦
is copied to 01⎡ ⎤⎢ ⎥⎣ ⎦
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 the
mapping range: B25. B25 of parent 1 equals to 5 and there is no 5 in the proto-child 1.
Therefore, the 15⎡ ⎤⎢ ⎥⎣ ⎦
is copied to 12⎡ ⎤⎢ ⎥⎣ ⎦
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. B24 of parent 1 equals to 4 and there is no 4 in the proto-child 1. Therefore, the 14⎡ ⎤⎢ ⎥⎣ ⎦
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 offspring’s 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 1 1 1 3 4 5 6 7 8 9
1 0 1 1 0 0 0 0 1 5 4 6 9 2 1 7 8 3
Parent 1
Parent 2
394 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
Exchange the genes between two positions 0 1 1 1 0 0 0 0 1 1 2 6 9 2 1 7 8 9
1 0 1 1 1 0 0 0 1 5 4 3 4 5 6 7 8 3
Figure 5: Crossover step procedures.
0 0 1 1 6 3
0 1 2 5
1 1 9 4
Figure 6: Chromosome legalized.
1 1 1 1 0 0 0 0 1 3 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 1 B2m= 1
Proto-child 1 B2m= 2
Proto-child 1 B2m= 9
Offspring 2
*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
395
4. Experiments and Results Three departments are used. Each department’s 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 department’s areas.
Department 1 2 3 Width 1 1 2 Length 1 2 3
Table 2: Transportation frequencies. Department 1 2 3
1 - 2 1 2 - - 1
Table 3: Transportation expenses.
Department 1 2 3 1 - 1 2 2 - - 3
Figure 9: Optimal facility layout of the example.
396 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
Figure 10: Chromosomes of the optimal facility layouts.
Table 4: The results of the example. Run Chromosome Width Length Ratio Area Costs
1 1 1 0 3 3 1 9 11.35 2 3 1
2 1 1 1 3 3 1 9 11.35 3 2 1
3 0 0 0 3 3 1 9 11.35 3 2 1
4 1 1 1 3 3 1 9 11.35 2 3 1
5 0 0 1 3 3 1 9 11.35 3 2 1
6 1 1 0 3 3 1 9 11.35 3 2 1
7 0 0 1 3 3 1 9 11.35 3 2 1
8 0 0 0 3 3 1 9 11.35 2 3 1
9 1 1 1 3 3 1 9 11.35 3 2 1
10 1 1 0 3 3 1 9 11.35 2 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
*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
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) are
shown 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 12 2 6 8 109 78 1 1 104 3 0 0 17 100 1 31 4 100 1 247 178 1 5 1 10 1 79 6 0 1 0 7 0 0 8 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 7 2 12 4 7 5 8 6 5 3 5 9 1 1 1 1 4 1 1 1 4 6 5 1 1 1 1 6 1 4 6 7 7 1 8 1
398 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
Figure 11: Optimal facility layouts for the Chan and Tansri (1994) example.
Normally, a large number of numerical experiments provides a better solution but rises
time consuming. Chan and Tansri (1994) concluded that the number of numerical
experiments should not exceed 3% of all possible solutions. The possible solution of the
example given above is 362,880 (9!) solutions. Therefore, the number of experiment should
not exceed 10,886. Five levels of population sizes and generation sizes are given. Excluding
that all numerical experiments exceeds 10,886, 18 sets of experiments are shown in Table 7.
Probabilities of crossover and mutation are given as shown in Table 8. Therefore, the number
of experiments turns to be 756 (6*7*18) experiments. Each experiment has been done in 10
runs. The result shows in Table 5. It can seen that the larger experimental numbers, the better
solutions. Based on Table 9, it can be seen that the appropriate population size and generation
size are 200 and 40 respectively. Table 10 shows the results when changing the probabilities
of crossover and mutation based on the appropriate population size and generation size. It can
be seen that the appropriate probabilities of crossover and mutation are 0.9 and 0.01
respectively and the number of runs which yielded one of the eight optimal solutions is 9 out of
10 times.
Then this research compares the result to that of Mak et al. (1998). Mak et al. (1998)
showed that their methodology is more efficient than PMX, OX, CX. Mak et al. (1998)
concluded the appropriate population size as 100, generation size as 20, the probability of
crossover as 0.6, and the probability of mutation as 0.001. The average of best material
handling costs among the 10 runs was 4856 and the number of runs which yielded one of the
eight optimal solutions was 4 (Table 11).
*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
399
Table 7: Population and generation sizes and number of trials. No. Population size Generation size No. of trials
1 20 10 200
2 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
400 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
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
Prob
abili
ty o
f cro
ssov
er 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
*Corresponding author (J.Teeravaraprug). Tel/Fax: +66-2-5643001 Ext.3083. E-mail addresses: tjirarat@engr.tu.ac.th. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. . Volume 2 No.4. ISSN 2228-9860. eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
401
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 plant
length 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
402 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
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 Thammasat University, Thailand. She holds a B.Eng. in Industrial Engineering from Kasetsart University, Thailand, an M.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 from Chulalongkorn University, and Ph.D. from Sirindhorn International Institute of Technology, Thammasat University, Thailand. His research includes industrial management, quality engineering, and engineering optimization.
B.Chinpaditsuk is a master student in the department of industrial engineering at Thammasat University. He holds a B.Eng degree in Electrical Engineering from Kasetsart University.
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