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

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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 …

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.

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  • 1. 2011 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies http://www.TuEngr.com, http://go.to/ResearchAn Application of Genetic Algorithm for Non-restricted Space andPre-determined Length Width Ratio Facility Layout Problem a* b aJirarat Teeravaraprug , Tarathorn Kullpataranirun , and Boonchai Chinpaditsuka Department of Industrial Engineering, Faculty of Engineering, Thammasat University, THAILANDb Department of Industrial Management, Faculty of Business, Mahanakorn University of Technology,THAILANDARTICLEINFO A B S T RA C TArticle history: The use of a genetic algorithm is presented to solve aReceived 02 June 2011Received in revised form facility layout problem in the situation where there is20 August 2011 non-restricted space but the ratio of plant length and width isAccepted 24 August 2011 pre-determined. A two-leveled chromosome is constructed. SixAvailable online01 September, 2011 rules are established to translate the chromosome to facility design. An approach of solving a facility layout problem is proposed. AKeywords: numerical example is employed to illustrate the approach.Genetic algorithm;Facility layout problem;Two leveled chromosome 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 ofresearchers have given elevated attentions. Various models and solution approaches forseveral 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 variousmathematical models. Moreover, Tavakkoli-Moghaddam and Shayan (1996) did acomparative survey of the recent and advanced approaches in order to evaluate and select themost suitable one of the facilities design problems.*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. 385eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 2. The problem in facility layout is to assign facilities to locations such that a givenperformance measure is optimized. The problem commonly found in industries is how toallocate facilities to either maximize adjacency requirement (Seppanen and Moore, 1970), orminimize the cost of transporting materials between them (Koopmans and Beckmann, 1957).The maximize adjacency objective uses a relationship chart that qualitatively specifies acloseness rating for each facility pair. This is then used to determine an overall adjacencymeasure for a given layout. The minimizing of transportation cost objective, which isconsidered 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 valuesfor all facility pairs are then added. However, solving the facility layout problem is elaborate because the facility layoutproblem belongs to the class of non-polynomial hard (NP-hard) problems which are unsolvablein polynomial time. It suggests that the problem’s complexity increases exponentially with thenumber of facility locations (Adel El-Baz, 2004). Heuristic techniques were introduced toseek near-optimal solutions at reasonable computational time for large-scaled problemscovering 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 geneticalgorithm (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 improvedGA to derive solutions for facility layouts that are to have inner walls and passages. Theproposed algorithm models the layout of facilities on gene structures. Improved solutions areproduced 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 transportationcost in various circumstances by assigning fixed overall area of facilities. This paper considersin the case that all facilities have not yet constructed. The overall area of facilities can bechanged, however the range of the ratio of width and length is given. This paper is then tominimize transportation cost and overall area by enhancing the concept of genetic algorithm. 386 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
  • 3. 2. Genetic Algorithm  Genetic algorithm (GA) introduced by Holland (1975) has increasingly gained popularityin 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 asimple chromosome like data structure and applies recombination operators to these structuresso 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. Thisset of solutions is called ‘population’. The individuals of the population are called‘chromosomes’. The chromosomes of the population are evaluated according to a predefinedfitness function. The chromosomes evolve through successive iterations called ‘generations’.During each generation, merging and modifying chromosomes of a given population create anew set of population. Merging chromosomes is known as ‘crossover’ while modifying anexisting one is known as ‘mutation’. Crossover is the process in which the chromosomes aremixed 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 anew one. The selection of chromosomes to crossover and mutate is based on their fitnessfunction. Once a new generation is created, deleting members of the present population tomake room for the new generation forms a new population. The process is iterative until aspecific stopping criterion is reached. In short, the typical steps required to implement GA are: encoding of feasible solutions intochromosomes 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 crossoveroperator to produce new children and a mutation operator altering a bit string from a selectedindividual. Tavakkoli-Moghaddain and Shayan (1998) introduced an improved robust GAusing 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. Someexamples include constrained optimization (Homaifar, et al., 1994), multiprocessor scheduling*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. 387eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 4. (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 andMeller (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 processof trial and error. Some experiments are required to analyze the suitability of genetic operatorsin 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 algorithmcalled two-leveled genetic algorithm. Chromosome design is the starting task to solve theproblem. It is required to encode the candidate solutions in the solution space in the form ofsymbolic 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 possiblechromosomes. 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 ineach level is equal to the number of facilities plus one. The first level is used to identify whichside of the given facility is employed in designing the layout. B1m is 0 or 1 value, where B1m = 0means the width of the facility m is utilized in designing the layout and B1m=1 means the lengthof the facility m is utilized. Z stands for the ratio of the plant length and the plant width andthen Z ≥ 1. The second level is the priority of facility arrangement. B2ms are positiveintegers. The relation of chromosome and plant layout is based on (X,Y) coordinates. The facilitythat B2m = 1 is arranged first on (0,0) coordinate by considering B1m. Figure 2 shows how toarrange the first facility on (X,Y) coordinate. For arranging the remaining facilities, six rulesare 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 388 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
  • 5. (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 3shows the proof of the rule. Based on Figure 3 (A), (1,1) and (1,0) coordinates are out and thepossible starting points are then (0,1), (2,1) and (2,0). Figures 3 (B and C) show if one of theduplicate points is selected as a starting point, the overall area is greater than that not using aduplicate 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 thecoordinate which has the lowest Y if Xs are equal (Figure 4). Based on Figure 3(A), there arethree 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.*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. 389eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 6. Figure 4 shows the proof of the fourth rule. It can be seen that Figure 4(B) uses (2,1) as thestarting point, and its results the largest area, which is 7. The fourth rule is, utilize the defined Zin the arrangement. Based on Figure 4, Zs equal to 1, 1.75, and 3.5 respectively. Forexample, if the pre-defined Z equals 1 to 2, the only possible starting point is (0,1) and if thepre-defined Z equals to 3 to 4, the only starting point is (2,0). The fifth rule is in the case that Zis out of the desired range, continuing arrange the remaining facilities. The last rule is eachfacility cannot be overlapped. Figure 4: Proof of the third rule.3.2 Fitness Function  In the fitness function, transportation expense and penalty are considered. Thetransportation expense of chromosome k (TCk) is shown in Eq. (1) n n TCk = ∑ ∑ f ij Cij Dkij (1) i =1 j =i +1 where 390 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
  • 7. fij is frequency of transportation between facility i and facility j Cij is the transportation expense per distance unit between facility i and facility j Dkij 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 chanceof choosing in the selection process. This paper assumes a constant value of penalty. Considering both transportation expense and penalty value, this paper multiplies thosevalues 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 desiredrange and it is a large value when Z is out of the desired range. So, the EVk would be verylarge when Z is out of the desired range and it is the transportation cost when Z is in the desiredrange. The fitness function of chromosome k (Fk) is a measure of a solution to the objectivefunction. Therefore, the fitness function should be an inverse correlation with the cost. Thispaper is assumed the fitness function as shown in Eq. (3). Fk = 1/ EVk . (3)3.3 Selection  In the chromosome selection process, this paper uses enlarged sampling space and roulettewheel selection. The selection probability of chromosome k equals to the fitness value of thechromosome k over the fitness values of population when the fitness value of population is thesummation of the fitness values over the population.*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. 391eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 8. 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 operatorconverges very rapidly in just a small number of generations, OX operator is the mostinsensitive to the initial population, and PMX operator is a steady performer. PMX consistentlyshows a steady trend of improvement in every graduation in generation. PMX has a mildincrease in the average fitness value and most often it produces the fittest solutions among thethree operators. PMX is expected to operate well and perform consistently for suitablegeneration and population combinations. Therefore, this paper applies a well-known PMX as acrossover method. Due to the uniqueness of the chromosome, the applied PMX crossover stepprocedures are 1) randomly select a group of the population and called parents and randomlyselect two positions in each selected parent, and 2) construct children by exchanging the genesbetween two positions of the parents. In the case that there are duplications of B2m in achromosome, 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 duplicationsstays in the mapping range and the other one is out of the mapping range. Find the genescarrying the duplications in the range. Map the duplicated gene with the same gene of theoriginal chromosome. Replace both B1m and B2m of out of range duplicated number with thegene of the original chromosome. In the case of remaining having duplications, take thenumber of that to the other chromosome. Then replace both B1m and B2m of out of rangeduplicated number with the numbers getting above. Check if there is duplication. Ifduplication 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 andseventh. Proto-child 1 shows the crossover result when parent 1 is the main chromosome whereas proto-child 2 shows the result when parent 2 is the main one. It can be seen that there areduplicating and lacking numbers in the results. For proto-child 1, there are two 1, 2, and 9 andno 3, 4, and 5 in the second row. Contrarily, for proto-child 2, there are two 3, 4, and 5 and no1, 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 to6, but the proto-child 1 already exists 6. So, considering 6 in the proto-child 1, it is on B23. B23 392 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
  • 9. of the main chromosome equals to 3 and there is no 3 in the original proto-child 1. Therefore, ⎡1 ⎤ ⎡0 ⎤the ⎢ ⎥ is copied to ⎢1 ⎥ of the out of mapping range of proto-child 1. Another example of ⎣3⎦ ⎣ ⎦legalization process is the 2 duplication of the proto-child 1. The considering 2 is in themapping range: B25. B25 of parent 1 equals to 5 and there is no 5 in the proto-child 1. ⎡1 ⎤ ⎡1 ⎤Therefore, the ⎢ ⎥ is copied to ⎢ 2 ⎥ of the out of mapping range of proto-child 1. The last ⎣5 ⎦ ⎣ ⎦legalization process of the proto-child 1 is 9. Considering the mapping range of proto-child 1, ⎡1 ⎤B24 = 9. B24 of parent 1 equals to 4 and there is no 4 in the proto-child 1. Therefore, the ⎢ ⎥ ⎣ 4⎦ ⎡1 ⎤is copied to ⎢ ⎥ of the mapping range of proto-child 1. Similarly, proto-child 2 is required to ⎣9 ⎦do the legalization process. The process of legalization of the proto-child 1 is shown in Figure6 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 processincludes: 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 Parent 1 1 1 3 4 5 6 7 8 9 1 0 1 1 0 0 0 0 1 Parent 2 5 4 6 9 2 1 7 8 3*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. 393eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 10. Exchange the genes between two positions 0 1 1 1 0 0 0 0 1 Proto-child 1 1 2 6 9 2 1 7 8 9 1 0 1 1 1 0 0 0 1 Proto-child 2 5 4 3 4 5 6 7 8 3 Figure 5: Crossover step procedures. Proto-child 1 0 0 1 B2m= 1 1 6 3 Proto-child 1 0 1 B2m= 2 2 5 Proto-child 1 1 1 B2m= 9 9 4 Figure 6: Chromosome legalized. 1 1 1 1 0 0 0 0 1 Offspring 1 3 5 6 9 2 1 7 8 4 0 1 1 1 1 0 0 0 0 Offspring 2 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 ofchromosome design discussed in section 3.1, fitness function discussed in section 3.2, selectiondiscussed in section 3.3, crossover discussed in section 3.4, and mutation discussed in section 394 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
  • 11. 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 showstransportation expenses between departments. The predetermined ratio of the plant length andthe plant width is between 1 and 2. An optimization technique provides eight patterns oflayouts as shown in Figure 9. Each pattern corresponds to chromosomes as shown in Figure10. This example uses population size as 10, generation size as 10, crossover probability as0.95, mutation probability as 0.001, and run as 10 times. After running the program for 10times, 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.*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. 395eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 12. 0 0 0 0 0 0 1 0 0 1 0 0 Pattern 1 1 2 3 1 3 2 1 2 3 1 3 2 0 0 0 0 0 0 0 0 1 0 1 0 Pattern 2 2 1 3 2 3 1 2 3 1 2 1 3 0 0 0 0 1 0 Pattern 3 3 1 2 3 1 2 Pattern 4 0 0 0 0 0 1 3 2 1 3 2 1 Pattern 5 0 1 1 0 1 1 1 1 1 1 1 1 1 2 3 1 3 2 1 2 3 1 3 2 1 0 1 1 1 0 1 1 1 1 1 1 Pattern 6 2 1 3 2 3 1 2 1 3 2 3 1 Pattern 7 1 0 1 1 1 1 3 1 2 3 1 2 1 1 0 1 1 1 Pattern 8 3 2 1 3 2 1 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 1396 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
  • 13. Based on the previous example, it is shown that the proposed approach and program can beutilized. Another example is taken from Chan and Tansri (1994) and Mak et al.(1998). Thefollowing 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 nolimitation of the ratio of the plant length and width. Therefore, to verify the proposedapproach, the ratio is not utilized. Table 5: Frequency (from-to) chart (number of trips per month). FromTo 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). FromTo 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*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. 397eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 14. Figure 11: Optimal facility layouts for the Chan and Tansri (1994) example. Normally, a large number of numerical experiments provides a better solution but risestime consuming. Chan and Tansri (1994) concluded that the number of numericalexperiments should not exceed 3% of all possible solutions. The possible solution of theexample given above is 362,880 (9!) solutions. Therefore, the number of experiment shouldnot exceed 10,886. Five levels of population sizes and generation sizes are given. Excludingthat 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 numberof experiments turns to be 756 (6*7*18) experiments. Each experiment has been done in 10runs. The result shows in Table 5. It can seen that the larger experimental numbers, the bettersolutions. Based on Table 9, it can be seen that the appropriate population size and generationsize are 200 and 40 respectively. Table 10 shows the results when changing the probabilitiesof crossover and mutation based on the appropriate population size and generation size. It canbe seen that the appropriate probabilities of crossover and mutation are 0.9 and 0.01respectively and the number of runs which yielded one of the eight optimal solutions is 9 out of10 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 ofcrossover as 0.6, and the probability of mutation as 0.001. The average of best materialhandling costs among the 10 runs was 4856 and the number of runs which yielded one of theeight optimal solutions was 4 (Table 11). 398 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
  • 15. 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*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. 399eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 16. 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 0.5 4919.7,0 4895.5,1 4891.4,4 4915.8,4 4874.7,4 4852.9,4 4860.4,4 Probability of crossover 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 200and 40 respectively. Those settings then are used to determine appropriate probabilities ofcrossover and mutation and it is found that the appropriate crossover and mutation for the Maket al. (1998) approach are 0.6 and 0.001 respectively. Utilizing the settings, the results showthat the average of best material handling costs among the 10 runs was 4840 and the number of 400 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
  • 17. runs which yielded one of the eight optimal solutions was 5 (Table 12). The comparison tableis shown in Table 12. It can be seen that the result of this research is better than that of Mak etal. (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 95. 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 solvingthe problem. To translate the chromosome to facility layout, six rules are set. The fitnessfunction is based on transportation expense and penalty. Enlarged sampling space and roulettewheel selection are used. The process of crossover and mutation are also utilized. A numericalexample is provided to illustrate the proposed approach. Furthermore, a comparison of theproposed approach to Mak et al. (1998) is presented. The result shows that the proposedapproach provides less average costs than Mak et al. (1998) approach and the number of runs*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. 401eISSN 1906-9642. Online Available at http://TuEngr.com/V02/385-404.pdf
  • 18. which yielded one of the eight optimal solutions of the proposed approach is higher than Mak etal. (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 insightfulcomments, helping clarify and improve the manuscript.7. References   Adel El-Baz, M. (2004). A genetic algorithm for facility layout problems of different manufacturing environments. Computer & Industrial Engineering, 47, 233-246.Al-Hakim, L. (2000). On solving facility layout problems using genetic algorithms. International Journal of Production Research, 38(11), 2573-2582.Balakrihnan, J. and Cheng, C.H. (1998). Dynamic layout algorithms: A state-of-the-art survey. Omega, 26(4), 507-521.Chan, K.C. and Tansri, H. (1994). A study of genetic crossover operations on the facility layout problem. Computers & Industrial Engineering, 26(3), 537-550.Chan, W.T., Chaua, D.K. and Kannan, G. (1996). Construction resource scheduling with genetic algorithms. Journal of Construction Engineering Management, ASCE, 122, 125-132.Davis, L. (1985). Job shop scheduling with genetic algorithms, International Journal of Grefenstette, editor, Proceedings of an International Conference on Genetic Algorithms and their Applications, Hillsdale, 136-140.Gau, K.Y. and Meller, R.D. (1999). An iterative facility layout algorithm. International Journal of Production Research, 37(16), 3739-3758.Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Addison Wesley.Hamamota, S. (1999). Development and validation of genetic algorithm-based facility layout a case study in the pharmaceutical industry. International Journal of Production Research, 37(4), 749-768.Heragu, S. (1997). Facilities Design. PWS Publishing Company, Boston, MA.Holland, J.H. (1975). Adaptation in Natural and Artificial Systems. The University of Michigan Press. Addison-Wesley, Reading. 402 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk
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  • 20. algorithms. Computer Industrial Engineering, 35(3-4), 527.530.Venkatasubramanian, V., Chan, K. and Caruthers, J.M. (1994). Computer-aided molecular design using genetic algorithms. Computers & Chemical Engineering, 18(9), 833-844.Venugopal, V. and Narendran, T.T. (1992). A genetic algorithm approach to the machine-component grouping problem with multiple objectives. Computers in Industrial Engineering, 22(4), 469-480.Wu, X., Chu, C.H. Wang, Y. and Yan, W. (2007). A genetic algorithm for cellular manufacturing design and layout. European Journal of Operational Research, 181, 156-167.Wu, Y. and Appleton, E. (2002). The optimization of block layout and aisle structure by a genetic algorithm. Computer & Industrial Engineering, 41(4), 371-387.Zhang, Y., Zhu, X. and Lou, Y. (1997). Applying genetic algorithms to task planning of multi-agent systems. Proceeding of 22nd International Conference on Computer and Industrial Engineering, 411-414. 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.Peer Review: This article has been internationally peer-reviewed and accepted for publication according to the guidelines given at the journal’s website. 404 Jirarat Teeravaraprug, Tarathorn Kullpataranirun, and Boonchai Chinpaditsuk

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