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Multi-objective Job Shop Scheduling under Risk Using GA

Jaber S. Alzahrani

Department of Industrial Engineering, Engineering College at Alqunfudah, Umm Al-Qura University, Saudi Arabia

Abstract: In this study, a multi-objective job-shop scheduling model is developed to optimize makespan, maximum job tardiness and maximum and idle time of machines under risk. The model considers multi-jobs and multi-machines. Each task has a specific due date and random processing times of specific probability distribution. The model is solved using @RiskOptimizer.

Keywords: job-shop scheduling, optimization, uncertainty, @RiskOptimizer, genetic algorithm

Received August 12, 2019; Revised September 16, 2019; Accepted October 07, 2019

Citation: Jaber S. Alzahrani, Multi-objective Job Shop Scheduling under Risk Using GA. American Journal of Industrial Engineering, 6 (1). 1-12. January 2019. doi:10.12691/ajie-6-1-1

1. Introduction

The aim of Job Shop Scheduling (JSS) problem may be defined as a main of the planning functions of production that comprises a set of hard optimization problems [1].

Ant Colony Optimization [2,3], Mathematical Programming [4,5], Tabu Search [6,7], Simulated Annealing [8,9], Particle Swarm Optimization [10,11], Memetic Algorithm [12], Genetic Algorithms [13,14], Goal Programming [15], and differential evolution algorithm [16] have been used in optimization of JSSP.

Most researches tackled the scheduling problems as a single objective optimization problem [5,13,14,17,18,19] like tardiness [5] and makespan [17,18] optimization. The multi-objective treatment is required in scheduling process to consider conflicting objectives [19]. So, researchers often deal with problems that involve multiple usually conflicting criteria [20].

Regarding Multi-Objective Job Shop Scheduling (MOJSS) optimization, R. T. Marler and J. S. Arora [21] presented a comprehensive study about the methods of multi-objective optimization. Some researches have been performed using different methods like Pareto front [22,23,24,25], the ϵ-constraint method [26,27,28], and the preemptive objectives procedure M. S. Al-Ashhab [29]. Genetic Algorithms (GA) [30,31,32,33], Ant Colony System (ACS) [34,35,36,37], Particle Swarm Optimization (PSO) [38,39,40,41,42], A Non-Dominated Algorithm [43] and Distribution Estimation Algorithm [44,45,46,47].

In realistic job shops, the working environment bears some uncertainties, such as job release time delay that may stem from late deliveries of subcontractors [48], or late arrivals of raw materials from a supplier [49]. When facing uncertainties, the optimum schedule performance may deteriorate.

In the existing literature, although Multi-Objective Scheduling has been used to solve flow shop scheduling [50,51], it has not yet been investigated to make robust schedules in job shops with uncertainties, where the environment is much more complex than the deterministic permutation flow shop.

In this paper, an MOJSS model using Pre-emptive goal programming procedure has been formulated to minimize the makespan, maximum job tardiness and maximum and idle time of machines under risk. The model has been formulated using Excel spreadsheets and solved using @RiskOptimizer to consider the uncertain processing time. This work is an extension of the work done by Jaber S. Alzahrani [52]

2. Problem Description and Assumptions

The processing sequences of three jobs on four machines are shown in Table 1. The durations and the due date of each job are shown in Table 2.

Table 1. Job Sequences

Table 2. Durations and the due dates

The optimization process starts with accurate problem modelling. For any given set of decision values, called adjustable cell values, the model evaluates an objective function, which required to be optimized. @RiskOptimizer searches for the solution, the objective function provides feedback about the solution quality. @RiskOptimizer continues to search for better solutions until no considerable improvements can be obtained in a predefined number of trials.

In the beginning, the problem is solved considering deterministic processing times after that the durations are assumed to follow different well-known distributions. The following assumptions have been considered in the model:

Ÿ Jobs are mutually independent,

Ÿ Jobs have equal processing priorities,

Ÿ The due date of each job is known and constant,

Ÿ The processing times of some tasks are uncertain,

Ÿ Each job will be processed by the same machine only one time,

Ÿ Any operation is not allowed to be processed until its preceding operations are completed,

Ÿ All jobs and machines are ready at time zero,

Ÿ Only one job can be processed on each machine at a time,

Ÿ No circulation is allowed.

3. Model Formulation

3.1. Sets

M: Set of machines

J: Set of jobs

3.2. Parameters

Dj: Due date of job j, (j = 1, …, n) of J

SEQ: Processing sequence array

Pji: Processing time for job j on m/c i (i=1, …, m) of M

3.3. Decision Variables

Sji: Starting time of job j on machine i,

Fji: Finishing time of job j on machine i,

Cj: Completion time of job j,

Ej: Earliness of job j = (Dj - Cj) if Dj > Cj, and 0 otherwise,

Tj: Tardiness of job j = (Cj - Dj) if Cj > Dj, and 0 otherwise,

The formula of each objective is given in Equations (1- 3).

[picture] | (1)

[picture] | (2)

[picture] | (3)

[picture] | (4)

[picture] | (5)

[picture] | (6)

[picture] | (7)

The make-span measure in (1) is calculated as the difference between the maximum completion time and the minimal starting time of all jobs. The maximum tardiness in (2) is defined as the maximum of differences between the due date of each job and completion time in which its due date is larger than the completion time. The maximum machine idle time in (3) is defined as the maximum difference between the maximum completion time on a machine and summation of all processing times on the same machine.

Subject to conjunctive (Equation 8) and disjunction (Equation 9, 10) constraints

Disjunction constraints:

[picture] | (8)

[picture] | (9)

Conjunction constraints:

[picture] | (10)

4. Computational Results and Analysis

In this section, the results of applying the proposed model are introduced and analyzed. The model has been solved using @RiskOptimizer solver. In consequence, the model has been solved using @RiskOptimizer which operated in an Intel® Core™ i3-2310M CPU @2.10 GHz.

The accuracy of the model is verified through solving and analysing a 5J*4M problem ot ten precent variation for all tasks, which was solved seven times to optimise the following objectives:

1) makespan

2) Maximum job tardiness

3) Maximum machine idle time

4) Multi-objectives (Equal weights)

5) Multi-objectives (double weighted makespan)

6) Multi-objectives (double weighted maximum job tardiness)

7) Multi-objectives (double weighted maximum machine idle time)

The Total or maximum earliness will not be optimized since its optimization gives no practical schedules [52].

The model has been solved using Evolver assuming constant processing times and the results are presented in Table 3. These results verified the accuracy of the model since the optimized objective has gotten its best value all over the seven cases. The Gantt chart of minimizing the makespan is depicted in Figure 1.

Table 3. Optimization results of the seven cases

Figure 1. Gantt chart of makespan objective [[52]]

4.1. Makespan

In this case, the problem has been solved to optimize the makespan. The resultant optimal distribution of the makespan is presented in Figure 2 and processing times of the tasks have been ranked by the effect on the output mean of the makespan as shown in Figure 3 from which it is noticed that the duration of the second task of job 5 on machine 3 has the greatest effect on the resultant optimal mean of the makespan which is logic according to the Gantt chart shown in Figure 1.

The resultant distributions of the other objectives while optimizing the makespan are presented in Figure 4 to Figure 9.

Figure 2. Optimal makespan distribution

Figure 3. Inputs ranked by the effect on makespan mean

Figure 4. The resulted in maximum tardiness while optimizing the makespan

Figure 5. The resulted in maximum machine idle time while optimizing the makespan

Figure 6. The resulted in multi-objectives (Equal weights) while optimizing the makespan

Figure 7. The resulted in Multi-objectives (double weighted makespan) while optimizing the makespan

Figure 8. The resulted in Multi-objectives (double weighted maximum job tardiness) while optimizing the makespan

Figure 9. The resulted in Multi-objectives (double weighted maximum machine idle time) while optimizing the makespan

4.2. Maximum Job Tardiness

In the second case, the maximum job tardiness has been minimized. The resultant optimal distribution of the maximum job tardiness is presented in Figure 10 and processing times of the tasks have been ranked by the effect on the output mean of the makespan as shown in Figure 11 from which it is also noticed that the duration of the second task of job 5 on machine 3 has the greatest effect on the resultant optimal mean of the maximum job tardiness which is logic according to the Gantt chart shown in Figure 1.

Figure 10. Optimal maximum job tardiness distribution

Figure 11. Inputs ranked by the effect on maximum job tardiness

4.3. Maximum Machine Idle Time

In the third case, the problem is solved to minimize the maximum machine idle time. The resultant optimal distribution of the maximum job tardiness is presented in Figure 12 and processing times of the tasks have been ranked by the effect on the output mean of the makespan as shown in Figure 13 from which it is noticed that the duration of the fourth task of job 4 on machine 1 has the greatest effect on the resultant optimal mean of the maximum machine idle time.

Figure 12. Optimal maximum machine idle time distribution

Figure 13. Inputs ranked by the effect on the maximum machine idle time

Figure 14. Optimal multi-objectives (Equal weights) distribution

Figure 15. Inputs ranked by the effect on the multi-objectives (Equal weights)

4.4. Multi-objectives (Equal Weights)

In the fourth case, the problem has been solved to minimize the fourth objective that has been presented in Equation 4. The resultant optimal distribution of the fourth objective is presented in Figure 14 and processing times of the tasks have been ranked by the effect on the output mean of the makespan as shown in Figure 15 from which it is noticed that the duration of the last task of job 5 on machine 3 has the greatest effect on it.

4.5. Multi-objectives (Double Weighted Makespan)

In the fifth case, the problem has been solved to minimize the fourth objective that has been presented in Equation 5. The resultant optimal distribution of the fourth objective is presented in Figure 16 and processing times of the tasks have been ranked by the effect on the output mean of the makespan as shown in Figure 17 from which it is noticed that the duration of the last task of job 5 on machine 3 has the greatest effect on it.

4.6. Multi-objectives (Double Weighted Maximum Job Tardiness)

In the sixth case, the problem has been solved to minimize the fourth objective, which has been presented in Equation 6. The resultant optimal distribution of the fourth objective is presented in Figure 18 and processing times of the tasks have been ranked by the effect on the output mean of the makespan as shown in Figure 19 from which it is noticed that the duration of the last task of job 5 on machine 3 has the greatest effect on it.

Figure 16. Optimal multi-objective (double weighted makespan) distribution

Figure 17. Inputs ranked by the effect on the multi-objectives (double weighted makespan)

Figure 18. Optimal multi-objective (double weighted maximum job tardiness) distribution

Figure 19. Inputs ranked by the effect on the multi-objectives (double weighted maximum job tardiness)

Figure 20. Optimal Multi-objective (double weighted maximum machine idle time) distribution

Figure 21. Inputs ranked by the effect on the multi-objectives (double weighted maximum machine idle time)

4.7. Multi-objectives (Double Weighted Maximum Machine Idle Time)

In the seventh case, the problem has been solved to minimize the fourth objective, which has been presented in Equation 7. The resultant optimal distribution of the fourth objective is presented in Figure 20 and processing times of the tasks have been ranked by the effect on the output mean of the makespan as shown in Figure 21 from which it is noticed that the duration of the last task of job 5 on machine 3 has the greatest effect on it.

5. Conclusion

In this study, a multi-objective job-shop scheduling model under uncertainty is developed to optimize makespan, maximum job tardiness and the maximum idle time of machines under risk. The model considers multi-jobs and multi-machines. Each task has a specific due date and random processing times of specific probability distribution. The model is solved using both Evolver and @RiskOptimizer.

The accuracy of the model has been verified through logical analysis of its results. Throughout the gotten results, it may be concluded that it is necessary to reduce the variation of the task’s processing times by applying more training of the personnel and removing or at least reducing the resources of variation to get more stable schedules.

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