Future Makespan Heuristic for Job Shop Scheculing Problem

Anan Mungwattana Department of Industrial Engineering, Faculty of

Engineering, Kasetsart Universityline,

Bangkok, Thailand fenganm@ku.ac.th

Kanate Ploydanai Department of Industrial Engineering, Faculty of

Engineering, Kasetsart Universityline,

Bangkok, Thailand Kanate_engineer@hotmail.com

Abstract—Job shop scheduling problem is well known NP-hard problem, this problem is difficult to find the good solution but industrials require the solution to competition and survive in the market. This paper focuses on developing algorithm to solve job shop scheduling problem. The new algorithm is called Future Makespan Heuristic. The concept of new algorithm is jobs are evaluated the makespan before selection one job to the schedule. Each iteration of selection, the new makespan is less than or equal to the makespan of the previous iteration. Iteration makes decision tree by using the candidate jobs (available jobs). Each branch of tree, the candidate jobs are evaluated the makespan by using basic heuristic such as dispatching rules. By this technique, the makespan is improved for all time of scheduling. The experiment reveals the solution is highly improved when comparing with original dispatching rules. Moreover, the computation time of new algorithm is short. The mathematic proof shows that the new algorithm can find the high quality solution in polynomial time algorithm if the evaluating procedure is polynomial time algorithm.

Keywords-job shop scheduling; heuristic; algorithm; optimization

I. INTRODUCTION The scheduling is an important problem because the

industrial requires an affective production planning for high productivity under limited constraint such as machines and routing of jobs. This problem is well known NP-Hard problem [1]. In this point, many researchers are interested in this problem. In the set of NP-Hard problem, job shop scheduling problem (JSP) is a popular problem to develop algorithm because the problem is too hard.

Nowadays, there is none algorithm that can find the optimal solution in polynomial time. That means algorithm can find optimal solution in the small size problem. When the size of problem is growing, to fast obtain the optimal solution is extremely difficult. When the optimal solution requires long computation time, industrial cannot wait the optimal solution. In addition, we present heuristic that used for solving job shop scheduling problem. Heuristic is effective for the specific problem. For example Rajendran studied performance of 13 dispatching rules for analysis the flow shop and job shop scheduling problems [2]. For this study, he used many

objective functions; for example, minimizing mean flow time, maximum flow time, proportional of tardy job, mean tardiness, maximum tardiness, and variance of tardiness. He suggested that the performance of dispatching rules is being influenced by the routing of job and shop floor configurations. Jansen presented an approximation algorithm for solving non-preemptive job shop scheduling problem [3]. The problem is a generation of the classical job shop scheduling problem. He presented that the problem can solve in a linear computation time. Lengyel solved the job shop scheduling problem based on high variety of product and tight due date [4]. The objective function is to minimize number of tardy and early jobs by using a function namely feasibility function. He developed a real-time algorithm to solve the problem and applied to simulation model. The results showed that the feasibility function is more effective in job shop scheduling problem if on-time completion is importance. Monch presented distributed shifting bottleneck heuristic for solving complex job shop scheduling problem with total weight tardiness objective [5]. The problem includes two special properties. The one is sequence-dependent setup time and the rest is reentrant process flows. He used two-layer algorithm. The upper layer works on aggregated model and the lower layer use shifting bottleneck heuristic. The result is similar with original shift bottleneck heuristic. Al-Anzi [6] modified the job shop scheduling problem with objective function total completion time to the mixed graph coloring problem with the criterion of minimizing the sum of maximal colors. After that, he used branch and bound algorithm to solve the new problem. The result showed the methodology can be used to minimize total completion time in the job shop scheduling problem. Wenqi solved job shop scheduling problem with the minimize makespan objective function. He presented a new algorithm that was developed from shifting bottleneck procedure namely improved shifting bottleneck procedure. The study suggests that the new procedure is more efficient and effective than the original shifting bottleneck procedure [7]. Wang presented algorithm to solve dynamic job shop scheduling problem [8]. Tardiness is the objective function of the problem. He used the algorithm namely filtered beam search-based algorithm. The result is very well for dynamic rescheduling. Baptiste solved job shop scheduling problem by Lagrangian relaxation [9]. First, the precedence constraints are relaxed. Second, machine

constraints are relaxed. In addition, dynamic programming and branch and bound algorithm are used to solve the problem. The solution is a new lower bound. Finally the solution is improved again by using the new lower bound and local search algorithm.

This paper aims to develop an algorithm namely future makespan heuristic. All jobs are considered to schedule. Candidate jobs are evaluated by dispatching rule. The solution is not similar with dispatching rule. The quality of solution is highly improved. The computation time of algorithm is still short.

II. PROBLEM FORMULATION Job shop scheduling problem is the problem to sequence N

jobs to M machines. The routing of jobs is different form the routing of the other jobs. In general, the processing time of job operation in each machine is different.

There are !N for sequencing job in each machine. In job shop scheduling problem, there are M machines. Each machine might use different sequence. So, there are mN )!( ways to find optimal solution. Because of this, a mathematician said the problem is strongly NP-hard. The following symbols will be used in the problem formulation. Let

jit , Start time of operation of job j that perform on machine i

jip , Processing time of operation of job j that perform on machine i

maxC Makespan or the last completing time of schedule

A Set of precedent constraint

N Set of jobs to schedule

jr Available time of job j

jd Due date of job j

Minimize maxC (1)

( ) NjiptC jiji ∈∀≥− ,;,,max (2)

( ) ( ) Ajhjiforptt jijijh ∈→≥− ,,,,, (3)

jijikikikiji pttORptt ,,,,,, ≥−≥− ( ) ( )kiandji ,,;∀ (4)

( ) Njit ji ∈∀≥ ,0, (5)

rt ji ≥, (6)

jjiji dpt ≤+ ,, (7)

There are many type of objective function in job shop scheduling problem. The popular objective is to minimize makespan. Equation (1) shows makespan of the problem. The makespan is the last finish time of the last finish operation in the system. That is shown in Equation (2). There are many constraints. That depends on researchers who setup the problem. In general, the operation of job j must process depend on steps. Equation (3) is the precedence constraint. Equation (4) expresses that each machine can process only one operation at a time. Equation (5) implies that process start times are non negative value. Some paper added the other constraint depend on the problem in the industrial. For example, the equation (6) is use when the release times of jobs are constraint. When job j cannot finish after its due date, Equation (7) is added to the mathematic model.

III. METHODOLOGY In this paper, we use evaluation function to choose job in

each step of sequence. If evaluation function is effective, the solution is close to the optimal solution. In this paper, dispatching rule is used to be the evaluation function. For iteration, algorithm selects one candidate job to schedule. Before selection of job, the candidate jobs are evaluated the makespan. All times to operate, the makespan is less than or equal to the makespan of previous iterations. The future makespan heuristic is shown in Fig. 1.

The algorithm is shown. First, each candidate jobs are evaluated according to the evaluation function. In evaluating

step, each candidate job is fixed in the sequence number thg (first sequence 1=g ). other job are sequenced after the candidate job by using dispatching rule. After that, the evaluation value of candidate job is set by equal to makespan of the sequence. The candidate job that has best evaluated value is chosen. Algorithm sets only the best evaluated value candidate job to the real sequence. In addition, new job or new process of job might be added to candidate job list. Each iteration runs until all process of job are in the sequence.

The algorithm used in this paper includes two phases. The former phase is selection procedure and the rest is evaluation procedure. First the selection procedure is described in Fig. 2. In step W there are four candidate jobs. Candidate jobs are job that are not finish to process and require perform on machines. Each candidate job has precedent constraints. When a candidate job is chosen in first time, the first operation of this job is performed. Whenever the job is chosen again, the next operation of this job is performed. For instant, job 2 is chosen. That means the first operation of job 2 is chosen. When job 2 is chosen again, the next operation of job 2 is chosen. If all operation complete, the job is not the candidate job. For each step, one job is selected and selected job are perform only one operation.

Figure 1. Future makespan heuristic.

For evaluation procedure, candidate jobs are chosen depends on rules. There are three example rules in the procedure. The first one is to select job that has earliest start time. The next criterion is to select job that can process and finish earliest. The last one is to choose the job by shortest processing time.

In addition, when most of operations of jobs complete, the makespan is determine. The makespan is used to become the evaluating value. In other word, the evaluating procedure is dispatching rule procedure. The main concept of future makespan heuristic is selecting procedure that is described hereafter.

Figure 2. Selection procedure.

In addition, the selection procedure is presented in Fig3. Fig. 3 shows step W and step W+1. Each step of figure includes candidate jobs evaluating procedure and evaluating value. The selection procedure is choosing a job for each step like as the previous procedure. On the contrary, the main concept is each candidate jobs are determined makespan before the job are selected. In each step, candidate jobs are evaluated by assume that if this job is in the sequence in this step. After that algorithm sequences the next job until all operation of all jobs are sequence and calculating makespan. The sequence is

rough planning in order to find rough makespan. In the rough planning, evaluating procedure is employed. The rough makespan use to be evaluating value. When all evaluating values of candidate jobs are computed, algorithm chooses only one job to the sequence and only one operation is performed. After that the evaluating value of the candidate jobs are calculated again and choosing job process is used again. The iteration is recalled until all of operations complete.

Figure 3. Evaluation procedure.

The first advantage of this algorithm is the result of algorithm is the feasible solution. That is different from some JSP algorithms that might have the infeasible solution because of interlock event of sequence in machines. The next advantage is each step of sequence brings to the better solution, for example, if the makespan of the before step is 3000, then the makespan of the after step might be 2500. The solution is developed when the step near the last step.

In brief, the algorithm includes two main loops. The first is outer loop. That uses for selecting jobs to the schedule. The last is the inner loop. This loop use to sequence the job to calculate makespan of each candidate job by the sequence of inner loop depends on the criteria.

IV. COMPUTATIONAL RESULTS To demonstrate the performance of Future makespan

heuristic, this paper use problem set from internet to be the case study. The case studies include the problem with 20 to 100 jobs. Each job performs on machines. There are 15 to 20 machines in each case study. In this paper there are 240 experiments. There are 40 test problems. Each problem is solved by six criteria. There are 15 to 50 jobs in the test problem. There are 15 to 20 machines to perform these jobs.

First, the three criteria are used to the 40 problems. Those include the criteria to choose the job that earliest start time to

the sequence. The second criterion is to choose the earliest finish time to the sequence and the last criteria is to choose the shortest processing time to the sequence.

After that, the three criteria are applied with future makespan heuristic. There are new three algorithms from the three elder criteria techniques. Table1 illustrate the performance of the three new algorithms by comparing with three old criteria technique. The result presented in Table1 conclusively demonstrates that the new algorithms are better than the original algorithm.

The first column is the number of problem in this experiment. The second and third columns are detail of problems. The second column is quantity of jobs and the third column is quantity of machines. The forth and the fifth column is makespan by using earliest start time criteria respectively. The fourth is original earliest start time and the fifth is EST that applies by the future makespan heuristic. The sixth and seventh column is makespan of earliest finish time. The sixth one is the results of the original and the seven one is the future makespan heuristic, respectively. Final, the eighth and ninth columns present the result of original shortest processing time criteria and better to better shortest processing time criteria algorithms, respectively.

TABLE I. RESULT OF ALGORITHMS

The data form the experiment reveal that the good results are from the earliest start time criteria is high perform on the other criteria in the experiment. The earliest finish time criteria techniques are better than the shortest processing time.

TABLE II. RESULT OF ALGORITHMS

The different value between maximum and minimum are shown in Table2. The first column is minimum makespan of each algorithm. The second one is maximum makespan of each algorithm. The different value of them is shown in third column. The forth column is the average of improve value between the original version and future makespan heuristic. The last column is standard deviation of the improved value. The first row presents values of original earliest start time criteria. The second row is the values of original earliest finish time criteria and the third row is shortest processing time criteria. The forth is value when applying future makespan heuristic to earliest start time criteria. Then fifth one is value from future makespan heuristic that applies to earliest finish time criteria. And the last row value for future makespan heuristic that applies to shortest processing time criteria.

The table illustrates that the new algorithm bring to the better result of makespan. Moreover, the result of new algorithm narrow swings (between the short ranges) by comparison with the three original three criteria. This can be seen that the new algorithm improve the results of original dispatching rules. The best solution is from the hybrid between new algorithm and earliest start time criteria. The next good solution from the hybrid algorithm from the new algorithm and earliest finish time criteria. In this experiment, the hybrid of new algorithm and shortest processing time criteria gives the poor solution than the others.

The experiment could reveal that the new algorithm that mixes with the dispatching rule give the higher quality of solution. Some priority rule might not be performance on minimum makespan objective function, however; the solution is highly improved when this priority rule is mixed together with the new algorithm. If the priority is optimal for the objective function that solutions form the hybrid algorithm is improve to close the optimal solution.

Not only does the new algorithm give high performance to find the better solution, but it also spends a short computation time. Next this part shows how fast to find the solution of new algorithm.

There are N jobs to perform on M machines. For calculating each sequence, algorithm requires T1=N×M time to calculate. For the three priorities dispatching rules, each time to choose the candidate jobs there are N possible alternate ways. Consequently, the time to use for dispatching rule is followed that:

MNTdis ×= 2 (8)

For new algorithm, similarly, the number of job in the sequence is MN × and there are N candidate jobs in each time to sequence. Before selecting candidate jobs, their evaluating values are calculated. Let fT be time to calculate the value. The computation time of the new algorithm is followed that;

fnew TMNT ××= 2 (9)

In the fact that, the computation time is polynomial time if

fT is polynomial time. In this case, the dispatching rules are use to evaluated the evaluating value. So, the computation time is followed that;

24MNTnew = (10)

All of evidence presented in this section emphasized that there are two advantage of the new algorithm. The former is the improvement of the quality of solution and the rest is how fast to discover the solution.

V. CONCLUSION AND FUTURE WORK Much of research focuses on development algorithms to

solve job shop scheduling problem, this paper, however, aim to develop a new algorithm to mix with other algorithm to solve the problem. The new algorithm could apply to other algorithms such as dispatching rule heuristic to the new (hybrid) algorithm. The study could be demonstrated that the result of future makespan heuristic is higher quality that the original result. At this point, the computation time is polynomial when the original algorithm that uses to construct is polynomial algorithm.

In the future, we present the new algorithm that mixes with many dispatching rules to construct one solution. That different from this paper because this paper uses only one dispatching to calculate evaluated value. In future study, we calculate the evaluated value by many dispatching rules. By the effect, the selection of candidate jobs is more affective and the solution of algorithm is the hybrid solution of many dispatching rule.

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