An effective new hybrid optimization algorithm for solving flow shop scheduling problems

Transcription

1 Malaya Journal of Matematik, Vol. S, No., 07-3, An effective new hybrid optimization algorithm for solving flow shop scheduling problems Harendra Kumar, Pankaj Kumar * and Manisha Sharma2 Abstract A flow shop is a production system in which machines are arranged in the order in which operations are performed on jobs. The flow shop is characterized by a flow of work that is unidirectional. In this paper, a new hybrid optimization (NHO) algorithm combining branch and bound (B&B) technique with genetic algorithm (GA) is proposed for solving flow shop scheduling problem (FSSP). Triangular and trapezoidal fuzzy numbers are used to represent processing times of jobs on each machines which are more realistic and general in nature. The present algorithm is divided into two phases. In the first phase, an initial schedule is constructed by using branch and bound technique. The processing times have been defuzzified into crisp one. The second phase finds the best schedule of the jobs by genetic algorithm for fuzzy processing times. The performance of a genetic algorithm depends very much on the selection of the proper genetic operators. In this paper, partially matched crossover operator for crossover and shift mutation operator for mutation are used. Numerous examples are illustrated to explain the proposed approach. Finally, the experimental results show the suitability and efficiency of the present NHO algorithm for optimal flow shop scheduling problem. Keywords Flow shop scheduling, branch and bound technique, genetic algorithm, fuzzy numbers, fuzzy processing times. Department of Mathematics and Statistics, Gurukula Kangri University, Hardwar , Uttarakhand, India. of Mathematics, Panjab University, Chandigarh-004, Punjab, India. *Corresponding author: pmittalvce@gmail.com Article History: Received 24 December 207; Accepted 2 January Department Contents Introduction Background information Some basic fuzzy set theory: Defuzzification: Genetic Algorithm: Problem description Notations Description of the proposed new hybrid optimization (NHO) algorithm Results and Discussion Conclusion and Future Scope References Introduction In the past several decades, flow shop scheduling (FSS) problems are widely researched. It consists in scheduling of c 208 MJM. n jobs on m machines with given processing times to meet a desired objective or measure of performance. In FSS, the sequence of processing a job on all machines is identical and unidirectional for each job. Flow shop scheduling is a special case of job shop scheduling where there is strict order of all operations to be performed on all jobs. In most of the studies concerned for FSSP, processing times were taken as certain and fixed value. But in the real world application, information is often ambiguous, vague and imprecise. Several techniques are proposed for managing uncertainty. To solve vague situations in real problems, the first systematic approach related to fuzzy set theory was successfully applied in many areas such as in scheduling problems. In recent studies, scheduling problems were fuzzified by using the concept of fuzzy due date and fuzzy processing times. Branch and bound technique is an integer programming solving technique and it was first applied to scheduling problems by Ignall and Schrage []. McMahon and Burton [2] applied the branch and bound technique to the three machine flow shop problem where the objective is to minimize makespan. McCahon and Lee [3] used triangular and trapezoidal fuzzy

2 An effective new hybrid optimization algorithm for solving flow shop scheduling problems 08/3 numbers to represent those vague job processing times in job shop production systems and modified Johnson and Ignall Schrage job sequencing algorithms and calculated fuzzy makespans and fuzzy mean flow times. McCahon and Lee [4] modified CDS (Campbell, Dudek and Smith) algorithm for job sequencing to accept trapezoidal fuzzy processing times. Tsujimura et al. [] illustrated a methodology for solving job sequencing problem and used triangular fuzzy numbers to represent the processing times of jobs on each machine. Cheng et al. [] considered a m-machine permutation flow shop scheduling problem to minimize the makespan by using branch and bound technique. Temiz and Erol [7] applied the fuzzy concept to the FSSP and modified the branch and bound algorithm of Ignall Schrage and then rewritten for three-machine flow shop problems with fuzzy processing time to determine the minimum completion time. Xu and Gu [8] presented a fuzzy scheduling model for FSSP with uncertain processing time based on fuzzy programming theory and proposed a hybrid algorithm by combining the immune algorithm with the branch and bound method which can avoid the blind search of the immune algorithm. Ashour [9] developed a branch and bound algorithm for finding the sequence of jobs on machines which minimizes the schedule time. Parveen and Ullah [0] discussed the more recent literature on scheduling using multi criteria decision making and addressed both job shop and flow shop scheduling problem. Jannatipour et al. [] addressed the problem of FSS with linear job deterioration to minimize the sum of fuzzy earliness and tardiness penalties. This problem is investigated in an uncertain environment, and fuzzy theory is applied to describe this situation. Chaudhry and Khan [2] proposed a genetic algorithm for the no-wait flow shop problem. Sunitha et al. [3] have extended the results of single transport facility on two machines scheduling problem under fuzzy processing time, into a double transport facility on three machines scheduling problem with fuzzy processing time. Ambika and Uthra [4] presented a branch and bound technique in FSSP with imprecise processing to minimize the total elapsed time by taking processing times by triangular membership functions. Talapatra et al. [] dealt with the branch and bound technique for solving m machines and n jobs in FSSP and obtained the optimal sequence of jobs through minimizing the total elapsed time by a lower bounding method based on the branch and bound algorithm. Pour et al. [] considered the processing time as a trapezoidal fuzzy numbers and find an optimum sequence in such a way that the completing time of jobs to be minimized. Kumar et al. [20] have developed a tasks allocation model considering cost for each task as a fuzzy number which is more realistic and general in nature. Vinoj and Tijo [7] used genetic algorithm for the solution of the flow shop scheduling problem with the objective of minimizing mean flow time and P. Kumar et al. [8] proposed a promising genetic algorithm with a random population generation in terms of makespan. Sathish and Ganesan [9] proposed a method to minimize the total makespan without converting the fuzzy processing 08 time to classical numbers by using a new type of fuzzy arithmetic and a fuzzy ranking method. Kumar [22] has discussed some resent defuzzification methods used for solving in real world problems. Recently, Selvamalar and Vinoba [2] used octagonal fuzzy numbers to represent those vague job processing times in flow shop production systems and modified the branch and bound algorithm of Ignall Schrage to solve the m jobs and n machines problem. In the present work, an attempt is made to develop a new hybrid optimization (NHO) algorithm for solving a FSSP having processing times as a fuzzy number by combining B&B technique with GA to minimize the total elapsed time of machines. 2. Background information 2. Some basic fuzzy set theory: Fuzzy set theory was introduced by Zadeh (9) in order to provide a scheme for handling a variety of problems in which a fundamental role is played by an indefiniteness arising more from a sort of intrinsic ambiguity than from a statistical variation. Fuzzy sets are sets whose elements have degrees of membership. Fuzziness has so far not been defined uniquely semantically, and probably never will be. It will mean different things, depending on the application area and the way it is measured. Here, in the present paper, triangular A = (a, a2, a3 ) and trapezoidal A = (a, a2, a3, a4 ) fuzzy membership functions are used to represent the uncertainty involved in processing times of jobs. 2.2 Defuzzification: In many situations, for a system whose output is fuzzy, it is easier to take a crisp decision if the output is represented as a single scalar quantity. This conversion of a fuzzy set to single crisp value is called defuzzification and is the reverse process of fuzzification. In the present paper, we are using the following methods for defuzzification: a) Centre of maximum b) Robust s ranking method c) Mean of maximum d) Centre of area 2.3 Genetic Algorithm: The most popular technique in evolutionary computation research has been the genetic algorithm. In 97, Holland developed this idea in his work Adaptation in natural and artificial systems. He described how to apply the principles of natural evolution to optimization problems and built the first genetic algorithms. Holland s theory has been further developed and now genetic algorithms stand up as a powerful tool for solving search and optimization problems. Genetic algorithms are based on the principle of genetics and evolution. The power of mathematics lies in technology transfer: there exist certain models and methods, which describe many different phenomena and solve wide variety of problems. Today,

3 An effective new hybrid optimization algorithm for solving flow shop scheduling problems 09/3 GAs is used to resolve complicated optimization problems, like, timetabling, flow shop scheduling, games playing. among jobs in σ σ : Partial sequence δ : The set of jobs that are not contained in the partial sequence σ Jr : Partial schedule of r scheduled jobs 0 Jr : The set of remaining (n r) free jobs Ch(i) : Chromosome number in the population FCh(i) : Fitness values of each chromosome. 3. Problem description The following assumptions are made before proceeding with the mathematical formulation in developing the hybrid algorithm for flow shop scheduling problems which has fuzzy processing time: a) All the n jobs are available for processing at time zero. b) Pre-emption is not allowed. Once an operation is started on the machine it must be completed before another operation can begin on that machine. c) Machines never breakdown and are available throughout the scheduling period. d) Machines may be idle. e) There is no travel time between stages; jobs are available for processing at a stage immediately after completing processing at the previous stage. f) Set up times are known and are included in processing times. Now, let us consider n jobs, say i =, 2, 3,, n are processed on three machines X, Y and Z in the order XYZ. A job (i =, 2, 3,, n) has fuzzy processing times in the form of triangular fuzzy numbers (ai, bi, ci ) and trapezoidal fuzzy numbers (ai, bi, ci, di ). The given flow shop problem can be written in the matrix form as: Table. Flow Shop Problem in Matrix Form Jobs i Machine X Machine Y Machine Z 2 3 n P P 2 P 3 P n P 2 P 22 P 23 P 2n P 3 P 32 P 33 P 3n. Description of the proposed new hybrid optimization (NHO) algorithm In this section, we propose a hybrid approach for solving flow shop scheduling problems based on B&B technique and GA to minimize the total elapsed time. The present NHO algorithm for finding an optimal solution of FSSP consists of two phases: Phase-I: To find an initial scheduling by using branch and bound technique. Phase-II: To find the best scheduling from the initial scheduling of jobs by using GA. Phase I: Method for finding the initial scheduling by using branch and bound technique :- The various steps involved in this phase for finding the initial schedule by using B&B technique are listed in the form of algorithm which is given below: Algorithm-I: Step : Defuzzify the fuzzy processing times P ji into crisp one Pji on machines X, Y and Z. Step 2: Evaluate the lower bounds of the makespan on machines, 2 and 3 respectively by using the following equations: l b = LC (σ ) + P i + min P 2i + P 3i (.) l b 2 = LC 2 (σ ) + P 2i + min P 3i Our objective is to find the optimal schedule of all jobs which minimize the total elapsed time by developing a new hybrid optimization algorithm combining B&B technique with GA. (.2) l b 3 = LC 3 (σ ) + P 3i (.3) Step 3: Calculate the lower bound LB(σ ) on makespan as: 4. Notations Following notations are used throughout the paper: n : Number of jobs to be scheduled M j : Machine j, j =, 2, 3 S : Sequence of jobs i =, 2, 3,, n Sk : Initial sequence of jobs obtained by applying branch and bound technique Sl : Best sequence of jobs obtained by applying genetic algorithm approach P ji : Fuzzy processing time of job i on machine j Pji : Processing time of job i on machine j in crisp one LC j (σ ) : Fuzzy completion time of the last job on machine j 09 LB(σ ) = max{l b, l b 2, l b 3 } (.4) We calculate LB(σ ) first for the n classes of permutations, i.e. for these starting with, 2, 3,, n respectively having labeled the appropriate vertices of the scheduling tree by these values. Step 4: Now, explore the vertex with lowest label. Evaluate LB(σ ) for the (n - ) subclasses starting with this vertex. If LB(σ ) is equal for more than one job then we take that job which has minimum total processing time and again concentrate on the lowest label vertex. Continuing this way, until we reach at the end of the tree represented by two single permutations for which we calculate the total work duration. Thus,

4 An effective new hybrid optimization algorithm for solving flow shop scheduling problems 0/3 Table 4. Lower Bounds for Each Partial Schedule we get the initial schedule of the jobs. Step : Prepare In-Out table for the initial sequence obtained in Step 4 for the original fuzzy processing times and get the total elapsed time. Example: To demonstrate how the above algorithm is employed, an example of jobs and 3 machines problem from the data set is considered to construct an initial sequence from a set of jobs to minimize the total elapsed time. The processing times for each job on each machine are taken in fuzzy environment (in triangular form) and these are shown in table 2. Table 2. Fuzzy Processing Times for the Jobs Jobs Machine M Machine M2 Machine M3 J J (2, 3, 4) (8, 9, 0) (0,, 2) (8, 0, 2) (, 7, 8) (,, 7) (4,, ) (,, 7) (,, 7) (3, 4, ) (4,, ) (, 7, 8) (2, 3, 4) (8, 9, 0) The fuzzy processing times are defuzzified into crisp one by using the centre of maximum method which are shown in table 3. Table 3. Jobs with Crisp Processing Times Jobs Machine M Machine M2 Machine M3 J J The lower bounds of the makespan on machines, 2 and 3 respectively by using equations ()-(3) are: 0 For J = (), J () = {2, 3, 4, } l b =, l b 2 = 43, l b 3 = 3 T hen, LB(J ) = max{, 43, 3} = Similarly, we can calculate LB( ) =, LB( ) =, Proceeding in this way, we find lower bound values LB(Jr ) for each partial schedule Jr which are shown in table 4. Thus, the initial schedule Sk for the above FSSP is The In-Out flow for the initial sequence of the jobs is shown in table. Phase II: Method for finding the best schedule of jobs by using GA :- In the present section of the paper, we shall discuss the GA to find the best schedule of jobs by using initial schedule obtained by algorithm-i. A population consists Partial schedule Jr Lower bounds LB(Jr ) of number of individuals being tested and some information about search space. A chromosome in a population is a sequence of gene. Therefore, in order to apply GA to the above described flow shop problem, the structure of a chromosome is expressed as a sequence of the jobs. The length of the chromosomes is equal to the number of jobs. The various steps involved in this phase for finding the best schedule of the jobs to minimize the total elapsed time of machines are listed in the form of algorithm which is given below: Algorithm-II: Step : Initialization: The sequence obtained by using the above algorithm-i is taken as one of the initial sequence to the GA for the further improvement. Since GA deals with a population of solutions and not with a single solution, therefore population is initialized with one sequence obtained by algorithm-i as a one chromosome and other chromosomes of the population are generated by taking reverse of the initial sequence. For the above example, the sequence is generated by algorithm-i. Therefore, the sequence is considered here as one chromosome for the initial population. And other chromosomes are generated by taking reverse of the initial sequence. Using the above encoding method, two different chromosomes for the initial population can be: Population Ch() = Population Ch(2) = Step 2: Calculation of fitness values of each chromosome: The following fitness function is used for evaluating the fitness value FCh(i), for the ith chromosome: 0 FCh(i) = Ci, i =, 2, 3,, n (.) where Ci is the total elapsed time of the ith job in fuzzy environment. Step 3: Reproduction of new population: In reproduction process, chromosomes are copied into next generation mating

5 An effective new hybrid optimization algorithm for solving flow shop scheduling problems /3 Machines M2 In Out (3,, 7) (, 7, 9) (9, 22, 2) (23, 27, 3) (28, 33, 38) (33, 38, 43) (38, 44, 0) (4,, 7) (0, 7, 4) Parent If for the above example, two crossover points are selected at random at positions 2 and 4 then the genes 3 is exchanged by gene 4 and gene 4 is exchanged by gene 3. So, child looks like: Child Out (, 7, 9) (23, 27, 3) (33, 38, 43) (4,, 7) Child 2 also is formed as by exchanging the same genes in parent 2: Child Therefore, each offspring contains ordering information partially determined by each of its parents. Step : Mutation operator: After crossover, the strings are subjected to mutation. Mutation helps to maintain genetic diversity in the population. It introduces new genetic structures in the population and prevents the algorithm to be trapped in a local minimum. There are many different forms of mutation for the different kinds of representation. In a problem where a chromosome represents a sequence, mutation needs to be defined differently. In the present work, shift mutation method for mutation is used. In this a single job is selected randomly and inserted in a random position. For example, Before Mutation J J In (0, 0, 0) (, 7, 9) (23, 27, 3) (33, 38, 43) M Jobs Table. In-Out Flow Table of Jobs M3 In Out (3,, 7) (, 9, 22) (9, 22, 2) (2, 29, 33) (28, 33, 38) (3, 42, 48) (38, 44, 0) (0, 7, 4) (0, 7, 4) (,, 70) pool with a probability associated with their fitness value. In this phase, steady state approach is used for reproduction. In this approach, after the application of crossover, only one offspring (child) is produced as compared to other approaches where, two offspring are produced. The resulting offspring solution is put back into the population. If the fitness of the offspring is better than the other members of the population then least fit member of the population is discarded, otherwise the offspring solution is discarded. Step 4: Crossover operator: The crossover operator is the primary operator in GA. There are a lot of crossover operators in the literature. But in the present research, partially matched crossover is used, which is one of the better performers among the others. In the present paper, the crossover probability is assumed to be.0, i.e. all offspring are made by crossover. In partially matched crossover, two chromosomes are aligned, and two crossover points are selected uniformly at random along the length of the chromosomes. The two crossover points give a matching selection. The crossover is performed as position-by-position exchange operations. For example if following are the two parents: Parent After Mutation Step : Termination criteria: If there is no improvement in the best solution found for a pre-specified number of generations or if the algorithm reaches maximum number of generations then the algorithm terminates. Step 7: End On applying the algorithm-ii, the best sequence Sl, is obtained and total elapsed time for the above schedule is {,, 70}.

6 An effective new hybrid optimization algorithm for solving flow shop scheduling problems 2/3. Results and Discussion Table 8. Jobs with Crisp Processing Times To evaluate the effectiveness of the proposed NHO algorithm, the computational experiments were conducted. The test for FSS problems of different sizes were randomly generated with the following information: Example: To illustrate Jobs Machine M Machine M2 Machine M3 Table. Information for the Data Generation Parameter Values Number of Jobs Number of Machines Processing Time of the Jobs Reproduction Scheme Crossover Operator Crossover Probability (Pc ) Mutation Operator Mutation Probability (Pm ) Termination Criteria 4,,, 8, 0, 20, 30 2, 3, 4,, 0 Fuzzy Numbers Steady State Approach Partially Matched Crossover.0 Shift Mutation 0.09 Number of Iterations (40) J of 0.24 at 9% confidence level; hence the mean values of total elapsed time of the present NHO algorithm and reference methods do not differ significantly at 9% level of significant. 7. Conclusion and Future Scope In this paper, we investigate a new hybrid optimization (NHO) algorithm combining branch & bound (B&B) technique with genetic algorithm (GA) in a FSSP. The proposed NHO algorithm consists of two phases: Initial schedule by using B&B technique is computed in phase I. Phase II of the NHO algorithm finds the best scheduling from the initial scheduling of jobs by using GA. In this phase, population is the procedure outline of the present NHO algorithm, 4 jobs initialized with one sequence obtained by algorithm-i as a one and 3 machines FSSP with trapezoidal fuzzy processing times chromosome and other chromosomes of the population are given in table 7 is considered. The fuzzy processing times are generated by taking reverse of the initial sequence. The performance of the NHO algorithm has been measured for various Table 7. Fuzzy Processing Times for the 4 Jobs problems. The results are very encouraging as it generates Jobs Machine M Machine M2 Machine M3 better solutions in comparison to other well known algorithms in the literature. J (2, 3, 4, ) (,, 7, 8) (0,,, ) This work may be extended in many directions. The algorithm (3, 4, 8, 9) (0,,., 2) (, 2, 3, ) may be extended to solve the scheduling problems with multi (9, 0,, ) (2, 3, 4, ) (, 7, 8, 20) ple objectives and related to various performance measures. (4,, 8, 9) (2,,, 8) (4, 4.,, ) The present hybrid approach is general enough to be applied to other optimization problems also. Different proportions can defuzzified into crisp one by using Robust s ranking method also be tried to improve the initial solution of B&B technique. which are shown in table 8. New genetic operators can be developed to increase the evoluthe best sequence obtained by the present NHO algorithm is tion and convergence speed. Further studies can focus on the with minimum total elapsed time {, 0, 70, 8}. sensitivity and the parameter and their possible relationships In order to demonstrate the general propose and customiza- with the convergence rate of the algorithm. tion feature of the proposed NHO algorithm, five example problems from the literature were taken. A summary of the References results is given in table 9. It is worth mentioning here that [] E. J. Ignall and L. E. Scharge, Applications of the branch and bound the problems were formulated within the same model as was technique for flow shop scheduling problems, Operational Research, built for finding the minimum total elapsed time. The results 3(9), [2] G. B. McMohan and P. G. Burton, Flow shop scheduling with the branch show that the present hybrid approach for solving flow shop and bound method, Institute of Operations Research and Management scheduling problems is much better than the other well known Sciences, (3)(97), heuristics. [3] C. S. McMohan and E. S. Lee, Job sequencing with fuzzy processing Analysis of variance: To evaluate the performance, the protimes, Computers Math. Applications, 9(7)(990), 3-4. [4] C. S. McMohan and E. S. Lee, Fuzzy job sequencing for a flow shop, posed algorithms are coded in C++ and tested for a large European Journal of Operational Research, (992), number of randomly generated flow shop problems. For sim[] Y. Tsujimura, S. H. Park, I. S. Chang and M. Gen, An effective method ulation study, we have taken 30 samples. The numbers of for solving flow shop scheduling problems with fuzzy processing times, machines are taken from 2 to 0 and numbers of jobs are Computers and Industrial Engineering, 2(-4)(993), [] J. Cheng, H. Kise, and H. Matsumoto, A branch and bound algorithm taken from 4 to 20. One way ANOVA was carried out in with fuzzy inference for a permutation flow shop scheduling problem, MINTAB8 between the results (total elapsed time in crisp European Journal of Operational Research, Elsevier, 9(997), 78one) obtained by present NHO algorithm and reference meth90. [7] I. Temiz, and S. Erol, Fuzzy branch and bound algorithm for flow shop ods which is shown in table 0. The results of the ANOVA show that the P-value is 0.3 which is greater than F-value scheduling, Journal of Intelligent Manufacturing, (2004),

7 (, 0, 70, 8) (34, 0, 0) (2,, ) (,, 70) (2,, ) Table 0. ANOVA Analyze Source Factor Error Total [8] [9] (, 0, 70, 8) (3,, 3) (49,, 4) (,, 70) (2,, ) [0] [] [2] McCahon and Lee (990) Branch and Bound Technique Tsujimura et al. (993) Fuzzified Branch and Bound Technique Sunitha et al. (203) Heuristic Approach Ambika and Uthra (204) Branch and Bound Technique Sathish and Ganesan (20) Fuzzy Ranking Method [3] [4] [] [] [7] [8] [9] [20] [2] [22] By Present NHO Algorithm Sequence Total Elapsed Time By Reference Method Sequence Total Elapsed Time Approach Reference Problem Table 9. Performance Comparison of the Results Obtained by the Proposed NHO Algorithm with Existing Methods An effective new hybrid optimization algorithm for solving flow shop scheduling problems 3/3 3 DF 8 9 Analysis of Variance Adj SS Adj MS F-value P-value 0.3 Z. Xu, and X. Gu, A hybrid algorithm for scheduling problems of flow shop with uncertain processing time, Proceedings of the th World Congress on Intelligent Control and Automation, June 2-23, Dallan, China, 200. S. Ashour, A branch and bound algorithm for flow shop scheduling problems, AIIE Transactions, Taylor Francis, 2(2)(2007), S. Parveen and H. Ullah, Review on job shop and flow shop scheduling using multi criteria decision making, Journal of Mechanical Engineering, 4(2)(200), M. Jannatipour, B. Shirazi and I. Mahdavi, Fuzzy simulation based genetic algorithm for just-in-time flow shop scheduling with linear deterioration function, International Journal of Computer Applications, 7()(202), 7-4. I. A. Chaudhry, and A. M. Khan, Minimizing makespan for a no-wait flow shop using genetic algorithm, Sidhana, 37()(202), B. Sunitha, T. P. Singh and P. Allawalia, Optimal / Heuristic approach in solving certain production scheduling problems in fuzzy and non - fuzzy environment, Ph.D. Thesis, Shodhganga, 203. G. Ambika, and G. Uthra, Branch and bound technique in flow shop scheduling using fuzzy processing times, Annals of Pure and Applied Mathematics, 8(2)(204), S. Talapatra, S. Rahman, C. Das and U. K. Dey, Application of branch and bound algorithm for solving flow shop scheduling problem comparing it with tebu search algorithm International Conference on Mechanical, Industrial and Energy Engineering, 2-2 December, Khulna, Bangladesh, 204. N. S. Pour, M. H. Abolhasani Ashkezari, H. Sheikhi, H. M. Andargoli and H. A. Ashkezari, A novel genetic algorithm for a flow shop scheduling problem with fuzzy processing time, Int. J. of Research in Industrial Engineering, 3(4)(204), -2. K. Vinoj, and J. Tijo, Flow shop scheduling using genetic algorithm, International Journal of Latest Trends in Engineering and Technology, 7()(20), P. Kumar, Pawan Kumar, R. Bhool and V. Upneja, An efficient genetic algorithm approach for minimising the makespan of job shop scheduling problems, International Journal of Science, Engineering and Technology Research, ()(20), S. Sathish and K. Ganesan, Scheduling of flow shop problems on 3 machines in fuzzy environment with double transport facility, American Institute of Physics, 20, doi: 0.03/.49230, -8. H. Kumar, N. K. Chauhan and P. K. Yadav, Dynamic tasks scheduling algorithm for distributed computing systems under fuzzy environment, International Journal of Fuzzy System Applications, (4)(20), N. Selvamalar and V. Vinoba, Fuzzy Branch and Bound Solution for a Flow Shop Scheduling Problem, Journal of Computer and Mathematical Sciences, 8(3)(207), 8-9. H. Kumar, Some Recent Defuzzification Methods, Theoretical and Practical Advancements of Fuzzy System Integration, Hershey, PA: IGIGlobal, 3-48, 207.