Minimization of Total Weighted Tardiness and Makespan for SDST Flow Shop Scheduling using Genetic Algorithm Kumar A. 1 *, Dhingra A. K. 1 1Department of Mechanical Engineering, University Institute of Engineering & Technology, Maharshi Dayanand University, Rohtak 124 001 Haryana INDIA * Corresponding author. Tel: + (91)01262-393291, E-mail: aman.k.ghangas@gmail.com ABSTRACT In the present work, SDST flow shop scheduling with minimizing the weighted sum of total weighted tardiness and makespan have been considered simultaneously. Four modified heuristic have been proposed for preliminary viable sequence. Sequence obtained from the modified heuristics is combined with the initial seed sequence of genetic algorithm and called as Genetic Algorithm (GA). Hence four different GAs named as GA 1, GA 2, GA 3 and GA 4 with further classical GA called as GA 5 have been presented for bi-criteria SDST flow shop scheduling problem for comparative analysis amongst them. Bi-criteria fitness function proposed for scheduling the jobs in SDST flow shop proves to be efficient and flexible for minimizing the due date and completion time related performance measures. From the comparative analysis among different GAs considered, it has been found that the performance of algorithm varies with the size of job and machines. It has also been concluded that GA 3 shows better results over other for all the problems considered upto 200 jobs and 20 machines including sequence dependent set up time. Keywords: Scheduling, Genetic Algorithm, Sequence Dependent Set up Time, Total weighted tardiness, makespan 1. Introduction A schedule is a systematic plan that generally tells when the things are made-up to happen; it shows the plan for the timing of the certain actions. The sequence is adapted which gives optimal results concerned with allocating limited resources to tasks to optimize certain objective functions. For standardization of performance, the level of performance should be a non-decreasing function of job completion times. The scheduling objective is to minimize the performance measures. To quantify the objectives in scheduling, it is very complicated as they are numerous, multifaceted and conflicting. Higher precision and accuracy of a particular work also contributes towards more complexity. Minimizing makespan, total flow time, total tardiness, maximum tardiness and number of tardy jobs are among the most generally measured performance measures. Makespan and total flow time deals with maximizing system deployment and inventory/stock, whereas the remaining measures are associated with the job due dates. Enhanced productivity and highest deployment of resources are directly related to scheduling with makespan criteria. If the industry is not competent in meeting the due dates on times, it will definitely results in the loss of customer and market competitiveness. As today s world, 483
deliverance of the goods is an important aspect for customer satisfaction. In modern days, cost of production of a product must be economical so that producer can survive. An optimal solution for tardiness based objectives is also very complex with the introduction of the just in time (JIT). Very efficient heuristics have been developed for the minimum makespan problem (Błazewicz et al., 1996). Many algorithms benefit from a schedule representation known as the disjunctive graph formulation (Adams et al., 1988). Particularly local-search algorithms like Simulated Annealing and Tabu Search have been applied with great success. It turned out that general purpose methods like GAs are only second best choice among the modern heuristics, particularly if applied to large makespan problems. Rajendran (1995) implemented the heuristic for flow shop scheduling for minimizing the multiple objectives of makespan, total flow time and idle time for machines. Sayin and Karabati (1999) considered the scheduling problem in a two machine flow shop environment by minimizing makespan and sum of completion times simultaneously. Gupta et al. (2001) minimized the total flow time and makespan in a two machines flow shop scheduling problems for optimal sequence. Ravindran et al. (2005) optimized the multiple objectives of makespan and total flow time together in a flow shop scheduling. They proposed three heuristics HAMC1, HAMC2 and HAMC3 and concluded that the proposed heuristics yields good results than the Rajendran heuristic CR (1995). Gowrishankar et al. (2001) considered two types of problems, firstly m-machine flow shop scheduling with minimizing variance of completion times of jobs and another with minimizing the sum of squares of deviations of the job completion times from a common due date. Ponnambalam et al. (2004) minimized the weighted sum of multiple objectives (i.e. minimizing makespan, mean flow time and machine idle time) with the proposed TSPGA multi-objective algorithm for flow shop scheduling. Blazewicz et al. (2005) examined different solution procedures for the two machine flow shop scheduling problem with a common due date and weighted latework criterion. Noorul Haq and Radha Ramanan (2006) used Artificial neural network (ANN) for minimizing bicriteria of makespan and total flow time in flow shop scheduling environment They showed that performance of ANN approach is better than constructive or improvement heuristics.rahimi-vahed and Mirghorbani (2007) developed multi objective particle swarm optimization for flow shop scheduling problem for minimizing the weighted mean completion time and weighted mean tardiness simultaneously. They also concluded that the proposed algorithm is effective from Genetic Algorithm for large sized problem. Chakraborty and Laha (2007) modified the NEH algorithm and achieved significant improvement in the quality of the solution while maintaining the same algorithmic complexity. They also showed that the original NEH and proposed NEH's outperform the best-known competitor to date.tamer Eren and Ertan Güner (2009) considered bi-criteria of minimizing a weighted sum of total completion time and total tardiness for m-identical parallel machine scheduling problem with a learning effect. Typical flow shop scheduling problems intends to reduce production time and enhance productivity and resource deployment as primarily focused towards the completion time associated objectives. As customer requirements are to be satisfied in time and simultaneously industry has to present a wide range of different and individual products for the assurance that they have made before. So, Bi-criteria SDST flow shop scheduling with minimization of makespan and total weighted tardiness has been considered in the present work. 484
2. Problem Statement Some of the assumptions made in the present work are as follows: a) Machines should never be break downed and should be accessible all through the scheduling period. b) Once a process is ongoing on a machine it must be completed otherwise another operation cannot be start on that machine. c) Each and every machine is frequently accessible for processing, without considerable distribution of the scale into shifts or days and without consideration of momentary unavailability such as breakdown or maintenance. d) Each and every processing time of each job on every machine must be known in advance, whatever may be the sequence of the jobs to be processed (deterministic, finite or independent). e) A job can be processed through each one of the m machines just the once. What's more, a job cannot be available to the next machine if processing on the current machine is going on. f) The first machine is assumed to be ready always; any job is to be processed on it first. g) Machines may be idle or at rest. h) If the next machine on the sequence needed by a job is not available, the job can wait and joins the queue at that machine i.e. in-process inventory is allowed. A number of researchers ignore the bi-criteria nature of SDST flow shop scheduling problem as more than one decision maker is involved in the decision making resulting into conflicting objectives. The accessibility of set of feasible solutions characterizing tradeoff between the different objective functions can be quite valuable, for adjustment of this situation. In many realistic problems, setup times depend on the type of job just completed as well as on the type about to be processed. In those situations, it is not valid to absorb the setup time for a job in its processing time, and explicit modifications must be made. Simultaneously, because of too much unnecessary intricacy in Bi-criteria SDST flow shop scheduling problems, a reduced amount of awareness has been provided to this dilemma. Bi-criteria fitness function considered here is the minimization of sum of makespan and total weighted tardiness which has been described below: The first performance measure for scheduling is makespan (C max ) which has been used for maximum utilization of resources to increase productivity and stated as maximum completion time of last job to exit from the system. C max = Max (C 1,., C n ) The second criterion for scheduling is to minimization of total weighted tardiness. Associated with each job j is a due date d j > 0. Let U j = 1 if due date for job j is smaller than the completion time C j of job j, otherwise U j = 0. The total weighted tardiness (t a ) is defined as Therefore Bi-criteria fitness function is obtained by combining both objectives into single scalar function and has been framed as: Where α and β are the weight values for the considered objective functions having constraints: And and 485
3. Proposed Genetic Algorithm Maintaining the seed sequence obtained from the developed heuristic together with a set of (Ps) randomly generated initial population according to population size Ps. As selection of a good set of chromosomes in the initial population results in improvement in the performance of a genetic algorithm. The proposed GA is described in the following: Step 1. Generate the seed sequence. Step 2. The algorithm then creates a sequence of new populations. At each step, the algorithm uses the individuals in the current generation to create the next population. To create the new population, the algorithm performs the following steps: a. Scores each member of the current population by computing fitness i.e. minimizing weighted sum of total weighted tardiness and makespan, simultaneously b. Selects members, called parents, based on their fitness. c. Some of the individuals in the current population that have lesser fitness are chosen as elite. These elite individuals are conceded to the next population. d. Produces offspring from the parents. Offspring s are produced either by combining the vector entries of a pair of parents crossover or by making random changes to a single parent mutation. e. Replaces the current population with the children to form the next generation. Step 3. The algorithm stops when computational time reaches n m 0.5 Seconds. Generation of seed sequence as in step 1 of GA has been obtained from the proposed modified heuristics. Therefore four modified heuristic based GA called as GA 1, GA 2, GA 3, GA 4 and one classical Genetic algorithm in which initial sequences have been generated randomly i.e. GA 5 have been developed for comparative analysis amongst them. 4. Proposed modified heuristic A NEH based heuristic of Nawaz et. al. (1983) for makespan minimization has been modified for Bi-criteria fitness function with minimizing the weighted sum of makespan and total weighted tardiness for SDST flow shop scheduling. The explanation of proposed modified heuristic is as follows: Step 1: Generate the initial sequence. Step 2: Fix k = 2. Select the earliest two jobs from the rearranged jobs list and schedule them in order to minimizing weighted sum of total weighted tardiness and makespan as if there are only two jobs. Set the better one as the current solution. Step 3: Increase k by 1. Create k candidate sequences by putting the first job in the left over job list into each slot of the current solution. Among these candidates, select the best one with the least partial minimization of weighted sum of total weighted tardiness and makespan. Update the selected partial solution as the new current solution. Step 4: If k = n, a schedule (the current solution) has been found and stops. If not, move to step 3. Initial sequence in step 1 of proposed modified heuristic is being generated by the following rules as described below:- 486
1. Schedule the jobs initially in descending order of m P i j as in original NEH (Nawaz et. i = 1 al., 1983). 2. Organize the jobs according to the earliest due date of jobs i.e. EDD (Kim, 1993). 3. Initially jobs are arranged according to least value of d j /w j (Parathasarthy and Rajendran, 1997). 4. Initially arrange the jobs according to earliest weighted due date i.e. w j d j. Thus the final sequence has been obtained by considering the initial sequence as above in step 1 of proposed modified heuristic and obtained four different NEH denoted as NEH 1, NEH 2, NEH 3, NEH 4. Thus four different modified heuristics have been developed and final sequence from these heuristic is taken as initial sequence in the GA procedure resulting into four modified heuristics based genetic algorithms (GAs). Table 1.1 Different heuristic for comparative analysis of proposed GA. S. No. Heuristic Description 1 NEH 1 Initial sequence is developed as proposed by Nawaz et al. (1983). 2 NEH 2 Initially jobs are arranged according to earliest due date and then final sequence of jobs is obtained by NEH procedure. 3 NEH 3 Initially jobs are arranged according to least value of d j /w j (Parthasarthi and Rajendran, 1997) and then final sequence is obtained by NEH procedure. 4 NEH 4 Initially jobs are ordered according to least value of w j d j and then final sequence is obtained by NEH procedure. Thus, the parameters defined in the proposed GA as follows:- Table 1.2: Parameters of GA. Parameter Value Weighted value for C max, α 0.5 Weighted value for Ta, β 0.5 Elite count 2 Generations 2000 Population size 50 Stopping criteria (Time limit) n m 0.5 seconds Selection function Roulette wheel Crossover function Order Mutation function Reciprocal Exchange Mutation fraction 0.2 Crossover fraction 0.8 Pareto fraction 0.35 Migration interval 20 487
5. Results & Discussions In the present work, Genetic A lgorithms (GAs) have been developed for Bi-criteria fitness function including weighted sum of total weighted tardiness & makespan for SDST flow shop scheduling. The analysis of all the algorithms implemented has been carried out using the instances (DD_SDST_125) developed by Taillard (1993) upto 200 jobs and 20 machines under SDST environment with due dates and weights allotted to each job. In the considered instances, SDST varies from 0 to 125. The instances considered has been divided into three categories as 5, 10 and 20 machine problems for analyzing the results with deviation of machines together with increase o f jobs. All the experimental tests are conducted on system configuration of THINKPAD R61i- INTEL CORE 2 DUO CPU T5750 @2.00GHz, RAM 1.99GHz, 3GB. Stopping limit of different GAs has been fixed to computational time limit i.e n m 0.5 seconds. Relative percentage deviation (RPD) (Naderi et. al., 2009) has been used for analyzing the performance which has been calculated as:- Where Mean solution is the solution obtained by Algorithm and Best solution is the best (minimum) solution obtained among all the GAs for particular problem. Best solution can be found out between the results obtained by running a particular algorithm five times for a particular problem and mean solution is the final average solution given by the algorithm for all the five runs. RPD near to zero gives the best results. 110 different Taillard (1993) benchmark problems have been solved for analyzing the results, which comprises different combinations of 20, 50, 100 and 200 jobs being processed on 5, 10 and 20 machines. Each combination of different machines and jobs has 10 different set of problems. After finding the RPD for 10 different set of problems of a particular combination their mean is also calculated called Average Relative Percentage Deviation (). In the present work both the performance measures have been given equal importance in the proposed fitness function. JOB SIZE Figure 5.1 for different GAs for 5 machine problems 488
JOB SIZE Figure 5.2 for different GAs for 10 machine problems JOB SIZE Figure 5.3 for different GAs for 20 machine problems 489
MACHINE SIZE Figure 5.4 for different GAs for 20 job problems MACHINE SIZE Figure 5.5 for different GAs for 50 job problems 490
MACHINE SIZE Figure 5.6 for different GAs for 100 job problems MACHINE SIZE Figure 5.7 for different GAs for 200 job problems Fig. 5.1 to 5.3 shows for 5, 10 and 20 machine problems and it is clear that GA 3 produce better results for different problem size. It can be seen from table 5.1 that for 5 machines, GA 3 presents improved results over other GAs for 20, 50 and 100 job problems with of 15.16%, 5.32% and 5.28% respectively. For 10 machines, GA 2 with of 15.16% shows better results for 20 job problems and GA 4 with of 22.67% shows improved results for 50 job problems. Whether for 100 491
and 200 job problems, GA 3 with of 2.22% and 10.67% shows improved results over other GAs. Also for 20 machines, GA 3 with of 14.52%, 10.98% and 2.78% shows improved results over others for 20, 50 and 200 job problems respectively. While for 100 job problem GA 1 with of 8.60% shows improved results. Table 1.3: for different GAs for different machine problems. Machines Jobs GA 1 GA 2 GA 3 GA 4 GA 5 20 22.61 16.76 15.16 17.69 23.81 5 50 16.09 10.18 5.32 14.16 21.52 100 22.01 15.85 5.28 20.45 30.13 20 17.43 15.63 20.95 19.48 23.39 10 50 33.03 25.86 24.42 22.67 31.72 100 12.14 9.22 2.22 13.44 17.90 200 14.61 10.70 10.67 17.05 24.82 20 20.11 17.10 14.52 21.55 20.00 20 50 18.24 13.78 10.98 18.50 23.48 100 8.60 9.689 9.19 10.53 10.97 200 14.54 10.67 2.78 15.88 20.43 Hence, from the analysis it has been concluded that performance of proposed GAs vary with the size of machines and jobs as GA 3 shows better results over other proposed GAs for all the three machine problems viz. 5,10 and 20 machines. Table 1.4: for different GAs for different job problems. Jobs Machines GA 1 GA 2 GA 3 GA 4 GA 5 20 50 100 5 22.61 16.76 15.16 17.69 23.81 10 17.43 15.63 20.95 19.48 23.39 20 20.11 17.10 14.52 21.55 20.00 5 16.08 10.18 5.32 14.16 21.52 10 33.03 25.86 24.42 22.67 31.72 20 18.24 13.78 10.98 18.50 23.48 5 22.00 15.85 5.28 20.45 30.13 10 12.14 9.22 2.22 13.44 17.90 20 8.60 9.68 9.19 10.53 10.97 10 14.61 10.70 10.67 17.05 24.82 200 20 14.54 10.67 2.78 15.88 20.43 Fig. 5.4 to 5.7 shows for 20, 50, 100 and 200 job problems and it is clear that GA 3 produce better results over others for different problem size. 492
It can be seen from table 5.2 that for 20 jobs, GA 3 presents improved results over other GAs for 5 and 20 machine problems with of 15.16%, and 14.52% respectively. While for 10 machine problem GA 2 with of 15.63% shows better results. For 50 jobs, GA 3 with of 5.32% and 10.98% shows improved results for 5 and 20 machine problems. While for 10 machine problem GA 4 with of 22.67% shows improved results. For 100 jobs, GA 3 with of 5.28% and 2.22% for 5 and 10 machine problems shows improved results over other GAs. As the machine size increased to 20 machines, GA 1 with of 8.60% shows better result. Also for 200 jobs, GA 3 with of 10.67% and 2.78% shows improved results over others for 10 and 20 machine problems respectively. Hence, from the analysis it has been concluded that performance of proposed GAs vary with the size of jobs and machines as GA 3 shows better results over other proposed GAs for all the four job problems viz. 20, 50,100 and 200 jobs. 6. Conclusion & Future Scope of work Bi-criteria fitness function proposed for scheduling the jobs in SDST flow shop proves to be efficient and flexible for minimizing the due date and completion time related performance measures. From the comparative analysis among different GAs considered, it has been found that the performance of algorithm varies with the size of job and machines. Amongst the five different GAs, it has been concluded that GA 3 shows better results over other for all the problems considered upto 200 jobs and 20 machines including sequence dependent set up time. The work has been restricted to Bi-criteria SDST flow shop scheduling only and several other performance measures apart from stated one s also exists, so presented work can be extended by focusing other performance measures also. Also, work has been restricted to flow shop scheduling problems only and the research can also be done with other scheduling environments. Work has been restricted to equal weight values for both the objective function and it may be extended for other weighted values. REFERENCES 1. Blazewicz, J., Lenstra, J.K., and Rinnooy Kan, A.H.G., 1983. Scheduling subject to resource constraints: classification and complexity. Discrete Applied Mathematics 5, pp.11-24. 2. Gowrisankar K., Chandrasekharan, R., and Srinivasan, G., 2001. Flow shop scheduling algorithm for minimizing the completion time variance and the sum of squares of completion time deviations from a common due date. Europeon Journal of Operation Research 132, pp.643 665. 3. Gupta, J.N.D., Venkata, G., Neppalli, R., and Werner, F., 2001. Minimizing total flow time in a two-machine flowshop problem with minimum makespan. International Journal of Production Economics 69, pp.323 338. 4. Kim, Y.D., 1993. A new branch and bound algorithm for minimizing mean tardiness in 2- machine flow shops. Computers and Operations Research 20, pp. 391-401. 493
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