Evaluations of Crossover and Mutation Probability of ...ieomsociety.org/ieom_2016/pdfs/315.pdf · On the other hand, Genetic Algorithm used to solve facility layout problem in equal

Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management

Kuala Lumpur, Malaysia, March 8-10, 2016

Evaluations of Crossover and Mutation Probability of Genetic

Algorithm in an Optimal Facility Layout Problem

Maricar M. Navarro

Industrial Engineering Department

Technological Institute of the Philippines

Quezon City, Philippines

[email protected]

Bryan B. Navarro

Electrical Engineering Department

Technological Institute of the Philippines

Quezon City, Philippines

[email protected]

Abstract—Facility Layout Problem (FLP) is logic based combinatorial optimization problem. It is a meta-heuristic solution

approach that gained significant attention to obtained optimal facility layout. This paper examines the convergence analysis by

changing the crossover and mutation probability in an optimal facility layout. This algorithm is based on appropriate techniques

that include multipoint swapped crossover and swap mutation operators. Two test cases were used for the implementations of the

said technique and evaluate the robustness of the proposed method compared to other approaches in the literature.

Keywords—facility layout problem, genetic algorithm, material handling cost, meta-heuristics

I. INTRODUCTION

In today’s new generation, the ultimate increase in global competition in the manufacturing sector should response effectively to adjust facility layout to meet the changes of the market. Facility layout problem (FLP) is a major setback in manufacturing and service sector particularly focus on the layout of facilities or department. It can be state as the best assignment to a facilities to b locations to minimize a definite objective function. Facility layout planning contains an important part in the process of manufacturing and has a major impact in the profit of the company. Total material handling costs (summation of the product of unit material flow, unit material handling cost, and the rectilinear distance between the locations between equipment) constitutes the major part of total operating cost. Material handling cost range to 20% - 50% of the total operating cost, total manufacturing cost range to 10% - 80% and a good facility layout can reduce 10% -30% of material handling cost [1]. Hence little enhancement in material handling cost can contribute to lower down total operating cost. A good facility layout is essential to promote safe and efficient operations, minimize travel time, decrease material handling costs and avoid hindrances in material and movement of facilities.

Manufacturing sector is experiencing high operating expenses due to poor layout design, low productivity, ineffective process flow, etc. Ineffective layout design that results to high total material handling cost.

This study is conducted in order to develop a new method on designing an effective layout using genetic algorithm in the manufacturing industry that will minimize the total material handling cost. This study also aims to identify and illustrate the step by step procedure of the propose method. Show the robustness of the proposed method with no codification difficulties. Evaluate and show the effectiveness of the propose method by comparing to other approaches using benchmark numerical example, and evaluate the impact of mutation and crossover probability in an optimal facility layout design

Current genetic algorithm methodologies have codification difficulties which indicate a very long set up of program and rarely results to near optimal solution. This study is suited and capable to solve facility layout problem that will produce most of the optimal value.

This study will also gain a higher profit to the manufacturing and service industry which leads to minimization of total material handling cost.

This study may give precise information about an improved overall process of genetic algorithm to all researchers who used artificial intelligence that could solve facility layout problem.

This study is restricted using identified variables in minimizing total material handling costs in an equal area facility. These are the material flow among equipment, the unit material handling cost and the rectilinear distance between equipment but with

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no restrictions for the number of facilities. Also, this study used to solve facility layout problem using genetic algorithm in MATLAB platform.

A. Facility Layout Problem Solution Approaches

In the literature, there are numerous heuristics approach presented to solve facility layout problem, based on the survey conducted by [2]. Meta heuristics approaches are Tabu Search, Simulated Annealing, Ant Colony, and Genetic Algorithm approach used to solve facility layout problem (FLP).

Among the research presented in the literature who used to solve facility layout problem using meta heuristics approach are in [3] , who developed a tabu search method in solving facility layout problem that uses a neighborhood based including long term memory structure, dynamic tabu list size and other strategies. Furthermore, [4] developed a simulated annealing algorithm to solve facility layout problem with aspect ratio. They used pairwise exchange between facilities and random moves on the planar site. Likewise, [5] used two simulated annealing approaches for a dynamic facility layout problem. They used pairwise exchange method and improve simulated annealing called “look-ahead and look-back strategy”. Another research presented by [6] developed an ant colony algorithm in a sequence-dependent single row machine layout problem. On the other hand, [7] used ant colony algorithm in solving constrained and unconstrained dynamic layout problems.

There are still areas for consideration in solving facility layout problem. Considering whether the facility layouts are equal or unequal. In fact, most of the facility layout problem used genetic algorithm to minimize total material handling cost. However there are different methodologies used by different authors to obtain an optimal result.

B. Genetic Algorithm Approaches

Genetic Algorithms (GA) is an optimization and search technique based on the principle of natural selection. It was developed by John Holland in 1970’s. These algorithms program is a possible solution to a specific problem on a simple chromosome like data structure and apply recombination operators to these structures as to preserve critical information.

Execution of Genetic Algorithm starts with a population, usually random chromosomes are chosen and evaluation comes for reproduction opportunities. The chromosomes which give a higher rating to solve the target problem will be chosen for reproduction.

Many genetic algorithm models have been introduced by researchers mostly used for experimental purposes. Majority of these researchers are application oriented and interested in using genetic algorithms as an optimization tools.

On the other hand, Genetic Algorithm used to solve facility layout problem in equal and unequal area facilities. For unequal area facilities, among researcher’s who developed a genetic algorithm are [8] who developed a genetic algorithm with penalty function to minimize “transportation distance” in the workshop layout. And [9] presented an improved adaptive genetic algorithm for solving workshop layout. The crossover and mutation possibility adjusts adaptively with the fitness value in accordance with sigmoid function curve. Furthermore, [10] proposed genetic algorithm to solve the closed-loop layout problem with unequal-sized facilities. They proposed a GA resulted near optimal and compared to the outcome generated in Lingo software package. Reference [11] used a genetic algorithm and utilizes new encoding representation for designing plant layouts with unequal area facilities. As well as [12] who proposed an adaptive genetic algorithm to optimize material handling cost and workshop utilization and illustrated a multi-objective model in facility layout of cylinder block line.

For equal area facilities, research presented by [13] studied different genetic crossover operators to solve facility layout problem. They compared the partially mapped crossover (PMX), the order crossover (OX), and the cycle crossover (CX). The result shows that PMX operator provided excellent results. Also, [14] proposed genetic algorithm to minimize material handling costs in manufacturing layout problem. Likewise, [15] proposed genetic algorithm to solve the problem of optimal facilities layout in manufacturing systems design. He considers various material flow patterns of manufacturing environments. Also, [16] developed a genetic algorithm to solve facility layout problems. In the same way, [17] developed a multi objective genetic algorithm to solve facility layout problem based on slicing structure encoding. They used four objective functions of the block layout problem but they did not incorporate it into single objective function. Furthermore, [18] adopted a genetic algorithm methodology to solve quadratic assignment problems in order to minimize material handling cost.

Another research used to solve facility layout problem by [19] studied genetic algorithm in an equal area facility. They used the methodology of single crossover point. They compared their best optimal solution using the same GA parameters in 19 different sets of population and generation. Results shows that most of the optimal solution arrived using the proposed method and compare to the research of other author.

Genetic algorithm has been widely used in optimization with binary and continuous variable and quite popular in solving facility layout problem [20]. It is a popular method avoiding local optima in enhancing search techniques because it tries to parallel the process of biological evolution to find better solutions. Evolution and computation come together in Genetic Algorithms (GAs). It is software measures formulated after genetics and evolution. It is intended to efficiently search for better solutions to large computer problems [21].




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This study used genetic algorithm (GA) as a tool to solve facility layout problem and thus minimizing the total material handling cost. The objective function is to minimize the total material handling cost. The distinction of the proposed method to other works is the conversion of the objective values to its relative fitness before the selection operator. In the presented literature that uses genetic algorithm approaches in solving facility layout problem, most of the methods have codification difficulties. To make the coding of the GA simpler, a multi point swapped crossover [18] and swap mutation [14] was adopted. This paper investigates the convergence analysis by changing the crossover and mutation probability of genetic algorithm (GA) in an optimal facility layout

II. PROBLEM FORMULATION

For the facility layout problem, the objective function TC is the total material handling cost of the system. The objective in this study is to minimize the total material handling costs. The latter is the measure of how the facilities are set in an optimal arrangement. The calculation of TC is formulated as:

(1)

Where Fij is the amount of material flow among equipment i and j, Cij is the unit material handling cost between locations

of equipment i and j, and Dij is the rectilinear distance between the centroids of locations between equipment i and j and TC is the total material handling cost of the system.

The objective function is used to provide a measure of how individuals have performed in the problem domain. In the case of a minimization problem, the most fit individuals will have the lowest numerical value of the associated objective function.

III. FACILITY LAYOUT PROBLEM USING GENETIC ALGORITHM

The starting operator of the GA is the generation of initial population, which was randomly generated. The representation of an individual is a single-level string. The length of chromosome string is equal to the number of position of the facilities. Fig. 1 shows an example of twelve facilities with corresponding equipment location. The yellow-colored number is the facility location while the blue-colored number is the equipment number.

Fig. 1. Arrangement of facilities and encoding of chromosomes

As an example, equipment number 10 is located at facility location number 1; equipment number 4 is located at facility location number 2, etc. Fig. 1 show how the initial parent is encoded.

After the generation of initial population, it will call the objective function, which is the total material handling cost, and

passes the generated population as an input. For each individual, the objective function is then calculated.

The fitness function in this paper is given by [22].

(2)

Where xi is the position in the ordered population of individual i, MAX is the selective pressure or bias towards the fittest individual, and Nind is the number of individuals.

Selection is the process of determining the number of times, or trial, a particular individual is chosen for reproduction and, thus, the number of offspring that an individual will produce [22].

The selection used in this paper is the roulette wheel selection. This selection methodology is used to probabilistically select individuals based on total material handling cost.

The basic operator for producing new chromosomes in the GA is that of crossover. Like its counterpart in nature, crossover produces new individuals that have some parts of both parent’s genetic material. The multi-point crossover was employed. The crossover used in this study is swapped crossover based on [18]. This method works as a single parent instead of taking two parents as in other crossover methods to generate only feasible solution. It only changes the string of the original chromosome

1 2 3 4

10 4 5 2

5 6 7 8

9 7 3 11

9 10 11 12

6 12 1 8




of one parent and swapped their chromosome strings in the crossover points. Fig. 2 show how the multi-point swapped crossover was employed using 20 populations. The crossover operation is not necessarily performed on all strings in the population. Instead, it is applied with a probability, Px, when the pairs are chosen for breeding. Crossover probability of 0.90 means 90% of the entire population undergo multi-point swapped crossover.

Fig. 2. Multi-point swapped crossover and swap mutation

The mutation technique employed is swap mutation based on [14]. It simply selects two random chromosome strings and swapped their contents. Fig. 2 show an example of swap mutation using 20 populations. As in crossover, the mutation operation is not necessarily performed on all strings in the population. Instead, it is applied with a probability, Pm, a percentage of the entire population is to be mutated. Mutation probability of 0.90 means 90% of the entire population undergo swap mutation “once”

After the offspring are mutated, the objective value will be calculated. Fitness-based reinsertion combined with elitism was used in this paper. The termination depends on the maximum number of generations.

IV. RESULTS AND DISCUSSION

The crossover and mutation probability evaluation analysis is illustrated using two numerical examples described in the literature. The entire simulation was coded in MATLAB platform.

A. Numerical Example 1: 9 Facilities

The crossover and mutation probability evaluation analysis is illustrated using two numerical examples described in the literature. The entire simulation was coded in MATLAB platform. A comparative evaluation of their method is made using benchmark numerical example. The numerical example is taken from [13] with nine facilities and compared with the works of [14], [15], and [16] that used the same example to evaluate their work. Their work conducted 19 sets of experiments to determine the appropriate combination of the population size P and generation size G. All of them show the resulting optimal facility layouts having a total material handling cost of 4818. The result of the simulation shows that the proposed method of [19] is much efficient than the four other approaches presented in the literature. The results show that the proposed method produces all of the optimal solution using 20 trials of any combination of population and generation. This study uses the method of [19] for the GA approach because it shows robustness compared to other approaches.

B. Numerical Example 2: 12 Facilities

Another numerical example is from [19] with twelve facilities and used for the sensitivity analysis. In this example, the number of individuals and generations increase simultaneously to examine the behavior of optimal solution. The starting parameter of GA is 50 individuals and 50 generations. The optimal solution is compared in 20 trials. The starting parameter of GA is gradually increased by 50 until it reaches 300 for both individuals and generations. The optimal solution is 2040.2 and the optimal facility location is shown in Fig. 3.

Fig. 3. Optimal solution




Based on the results of [19], an increase of generation results to an improved objective value than an increase of individual. This study used the result to select the suitable individual and generation size for the crossover and mutation probability evaluation.

C. Crossover and Mutation Probability: Evaluation of 9 and 12 Facilities

Both numerical examples were examined for all crossover and mutation probability combinations. The crossover and mutation probability starts at both 0.1 and gradually increase by 0.1 until it reaches 1.0 for both. Both numerical examples run in 20 trials using 300 individuals and 300 generations in all simulations to be conservative.

Using numerical example 1 (9 Facilities), the simulation results to optimal solution of 4818 in all combination of crossover and mutation probability. To see the effect of GA operator (mutation and crossover) probability, convergence analysis needs to be examined. Table I shows the convergence characteristics at different crossover and mutation probability. Fig. 4 show the graphical representation of Table I. The result of the simulation shows that fast convergence can be obtained at mutation probability of 0.9 or 1.0 in any crossover probability.

Using numerical example 2 (12 Facilities), the simulation results to global and local optimal solution in all combination of crossover and mutation probability. Using the method of GA, there are few global optimal and also local optimal solution appears. Global optimal value is 2040.2 while local optimal value is 2041.8. The most appearance of global and local optimal solution is at crossover probability of 0.1 up to 0.5 and mutation probability from 0.5 up to 1.0.

Fig. 4. Graphical representation of convergence (iteration number) at different crossover and mutation probability

TABLE I. CONVERGENCE (ITERATION NUMBER) AT DIFFERENT CROSSOVER AND MUTATION PROBABILITY

V. CONCLUSION AND FUTURE WORKS

In this paper, we developed a methodology that minimizes total material handling cost using genetic algorithm. The proposed method is much efficient than the four other methods in the literature as a comparison using benchmark numerical example. The solution shows that an increase in generation results to an improved objective value than in an increase of individual.

The results of simulation from benchmark numerical example 1 (9 Facilities) shows that fast convergence of all optimal solution can be obtained at mutation probability 0.9 or 1.0 in any crossover probability. However in benchmark numerical example 2 (12 Facilities) shows the relative significance of global and local optimal solution. In this case, a small number of optimal solutions appeared; due to decimal point of rectilinear distance that leads to difficulty in obtaining optimal solution. Global and local optimal solution illustrate the best cross over and mutation probability was from 0.1 – 0.5 crossover and 0.5 – 1.0 mutation probability. Although the coding is simple, it shows robustness and generates good solution.

Px/Pm Pm=0.1 Pm=0.2 Pm=0.3 Pm=0.4 Pm=0.5 Pm=0.6 Pm=0.7 Pm=0.8 Pm=0.9 Pm=1.0 Px=0.1 104 148 125 8 15 9 5 16 16 9 Px=0.2 33 126 17 57 11 6 12 10 10 10 Px=0.3 92 172 19 21 24 14 14 9 14 22 Px=0.4 71 41 41 44 14 32 237 9 3 15 Px=0.5 22 11 34 8 1 100 20 19 11 15 Px=0.6 19 28 12 31 292 32 18 8 13 23 Px=0.7 271 31 39 9 41 12 39 17 18 47 Px=0.8 27 167 22 19 12 20 26 17 37 98 Px=0.9 195 195 12 60 39 43 1 42 34 16 Px=1.0 64 227 28 48 151 4 144 161 13 11




Future work may use the methodology described in this paper to a large number of facilities and apply to real-world case

studies. Also, expand the method in multi-constraint and multi-objective optimization problem.

REFERENCES

[1] J. A. Tompkins and J. A. White, Facilities Planning, 2nd Ed., New York, John Wiley, 1996.

[2] A. Drira, H. Pierreval, and S. Hajri-Gabouj, “Facility layout problems: a survey,” Annual Reviews in Control 31, pp. 255-267, 2007.

[3] W. C. Chiang and P. Kouvelis, “An improved tabu search heuristic for solving facility layout design problems,” International Journalof Production Research, vol. 34, no. 9, pp. 2565-2585, 1996.

[4] L. Chwif, M. R. Pereira Barreto, and L. A. Moscato, “A solution to facility layout problem using simulated annealing,” Computers inIndustry, vol. 36, nos. 1-2, pp. 125-132, 1998.

[5] A. R. McKendall, J. Shang, and S. Kuppusamy, “Simulated annealing heuristics for the dynamic facility layout problem,” Computersand Operations Research, vol. 33, no. 8, pp. 2431-2444, 2006.

[6] M. Solimanpur, P. Vrat, and R. Shankar, “An ant colony algorithm for the single row layout problem in flexible manufacturingsystems,” Computers and Operations Research, vol. 33, no. 8, pp. 583-598, 2005.

[7] A. Baykasoglu, T. Dereli, and I. Sabuncu, “An ant colony algorithm for solving budget constrained and unconstrained dynamicfacility layout problems,” Omega, vol. 34, no. 4, pp. 385–396, 2006.

[8] L. Tong-tong, L. Chao, and Z. Hu, “Optimal design for facility workshop layout based on genetic algorithm,” IEEE, 2011.

[9] Z. Yi, Z. Hu, F. Zi-tian, and W. Qiang, “Study on the facility layout in workshop based on improved adaptive genetic algorithm,”IEEE, 2009.

[10] R. Tavakkoli-Moghaddam and H. Panahi, “Solving a new mathematical model of a closed-loop layout problem with unequal-sizedfacilities by a genetic algorithm,” Proceedings of the 2007 IEEE IEEM, pp. 327-331, 2007.

[11] L. Salas-Morera, L. Garcia-Hernandez, and A. Arauzo-Azofra, “An evolutionary algorithm for the unequal area facility layoutproblem,” 2011 11th International Conference on Intelligent Systems Design and Applications, pp. 414-419, 2011.

[12] L. Xu, S. Yang, A. Li, and A. Matta, “An adaptive genetic algorithm for facility layout problem in cylinder block line,” IEEE, pp.749-753, 2011.

[13] K. C. Chan and H. Tansri, “A study of genetic crossover operations on the facility layout problem,” Computers and IndustrialEngineering, vol. 26, no. 3, pp. 537-550, 1994.

[14] I. Mihajlovic, Z. Zivkovic, N. Strbac, D. Zivkovic, and A. Jovanovic, “Using genetic algorithms to resolve facility layout problem,”Serbian Journal of Management, vol. 2, no. 1, pp. 35-46, 2007.

[15] M. Adel El-Baz, "A genetic algorithm for facility layout problems of different manufacturing environments,” Computers andIndustrial Engineering, vol. 47, pp. 233-246, 2004.

[16] K. L. Mak, Y. S. Wong, and T. S. Chan, “A genetic algorithm for facility layout problems,” Journal of Computer IntegratedManufacturing Systems, vol. 1, nos. 1-2, pp. 113-123, 1998.

[17] G. Aiello, G. La Scalia, and M. Enea, “A multi objective genetic algorithm for the facility layout problem based upon slicing structureencoding,” Expert Systems with Applications, 2012.

[18] P. Kulkarni and K. Shanker, “A Genetic algorithm for layout problems in cellular manufacturing systems,” IEEE, pp. 694-698, 2007.

[19] M. G. Misola and B. B. Navarro, “Optimal facility layout problem solution using genetic algorithm,” International Journal ofMechanical, Industrial Science and Engineering, World Academy of Science, Engineering and Technology (WASET), vol. 7, no. 8,pp. 622-627, 2013.

[20] R. Haupt and S. Haupt, Practical Genetic Algorithms, Second Edition, John Wiley and Sons, Inc.

[21] A. S. Ramkumar and S. G. Ponnambalam, “Design of single-row layout for flexible manufacturing systems using genetic algorithmand simulated annealing algorithm,” IEEE Conference on Cybernetics and Intelligent Systems, Singapore, December 2004, pp. 1143-1147, 2004.

[22] A. Chipperfield, P. Fleming, H. Pohlheim, and C. Ponseca, Genetic algorithm toolbox for use with Matlab user’s guide version 1.2,Department of Automatic Control and Systems Engineering of the University of Sheffield.

BIOGRAPHY

Maricar M. Navarro is an Assistant Professor with the Department of Industrial Engineering at the Technological Institute of the

Philippines. She earned her B.S. in Industrial Engineering from Technological Institute of the Philippines, Quezon City and Master of

Engineering major in Industrial Engineeering from Mapua Institute of Technology, Manila, Philippines. She has published journal and

conference papers. She has done research projects that deals in optimization of production, warehouse operations, and service operations.

Her research interests include manufacturing, simulation, optimization, facility layout and design. She is an active member of the

Phulippine Institute of Industrial Engineers (PIIE).

.

Bryan B. Navarro is an Assistant Professor with the Department of Electrical Engineering at the Technological Institute of the

Philippines. He earned his B.S. in Electrical Engineering from Technological Institute of the Philippines, Quezon City and Master of

Science in Electrical Engineering major in Power System from the University of the Philippines, Diliman, Quezon City.


Documents

Evaluations of Crossover and Mutation Probability of ...ieomsociety.org/ieom_2016/pdfs/315.pdf · On the other hand, Genetic Algorithm used to solve facility layout problem in equal