American Journal of Industrial Engineering
Volume 6, 2019 - Issue 1
Website: http://www.sciepub.com/journal/ajie

ISSN(Print): 2377-4320
ISSN(Online): 2377-4339

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Research Article

Open Access Peer-reviewed

Jaber S. Alzahrani^{ }

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

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.

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}

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.

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.

M: Set of machines

J: Set of jobs

D_{j}: Due date of job j, (j = 1, …, n) of J

SEQ: Processing sequence array

P_{ji}: Processing time for job j on m/c i (i=1, …, m) of M

S_{ji}: Starting time of job j on machine i,

F_{ji}: Finishing time of job j on machine i,

C_{j}: Completion time of job j,

E_{j}: Earliness of job j = (D_{j} - C_{j}) if D_{j} > C_{j}, and 0 otherwise,

T_{j}: Tardiness of job j = (C_{j} - D_{j}) if C_{j} > D_{j}, and 0 otherwise,

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

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(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:

(8) |

(9) |

Conjunction constraints:

(10) |

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.

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****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

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.

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.

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.

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.

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

**Figure****21****.**Inputs ranked by the effect on the 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.

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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Published with license by Science and Education Publishing, Copyright © 2019 Jaber S. Alzahrani

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Jaber S. Alzahrani. Multi-objective Job Shop Scheduling under Risk Using GA. *American Journal of Industrial Engineering*. Vol. 6, No. 1, 2019, pp 1-12. http://pubs.sciepub.com/ajie/6/1/1

Alzahrani, Jaber S.. "Multi-objective Job Shop Scheduling under Risk Using GA." *American Journal of Industrial Engineering* 6.1 (2019): 1-12.

Alzahrani, J. S. (2019). Multi-objective Job Shop Scheduling under Risk Using GA. *American Journal of Industrial Engineering*, *6*(1), 1-12.

Alzahrani, Jaber S.. "Multi-objective Job Shop Scheduling under Risk Using GA." *American Journal of Industrial Engineering* 6, no. 1 (2019): 1-12.

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[1] | G. Zhang, X. Shao, P. Li, and L. Gao, “An effective hybrid particle swarm optimization algorithm for multi-objective flexible job-shop scheduling problem,” Computers & Industrial Engineering, vol. 56, pp. 1309-1318, 2009. | ||

In article | View Article | ||

[2] | K.-L. Huang and C.-J. Liao, “Ant colony optimization combined with taboo search for the job shop scheduling problem,” Computers & operations research, vol. 35, pp. 1030-1046, 2008. | ||

In article | View Article | ||

[3] | A. Udomsakdigool and V. Kachitvichyanukul, “Multiple colony ant algorithm for job-shop scheduling problem,” International Journal of Production Research, vol. 46, pp. 4155-4175, 2008. | ||

In article | View Article | ||

[4] | M. S. Al-Ashhab, Munshi, S., Oreijah, M., & Ghulman, H., “Job Shop Scheduling Using Mixed Integer Programming,” International Journal Of Modern Engineering Research, vol. 7, p. 7, 2017. | ||

In article | |||

[5] | K. R. Baker and B. Keller, “Solving the single-machine sequencing problem using integer programming,” Computers & Industrial Engineering, vol. 59, pp. 730-735, 2010. | ||

In article | View Article | ||

[6] | M. Dell'Amico and M. Trubian, “Applying tabu search to the job-shop scheduling problem,” Annals of Operations research, vol. 41, pp. 231-252, 1993. | ||

In article | View Article | ||

[7] | A. Ponsich and C. A. C. Coello, “A hybrid differential evolution—tabu search algorithm for the solution of job-shop scheduling problems,” Applied Soft Computing, vol. 13, pp. 462-474, 2013. | ||

In article | View Article | ||

[8] | H. R. Lourenco, “Job-shop scheduling: Computational study of local search and large-step optimization methods,” European Journal of Operational Research, vol. 83, pp. 347-364, 1995. | ||

In article | View Article | ||

[9] | M. Faccio, J. Ries, and N. Saggiorno, “Simulated annealing approach to solve dual resource constrained job shop scheduling problems: layout impact analysis on solution quality,” International Journal of Mathematics in Operational Research, vol. 7, pp. 609-629, 2015. | ||

In article | View Article | ||

[10] | D. Sha and C.-Y. Hsu, “A hybrid particle swarm optimization for job shop scheduling problem,” Computers & Industrial Engineering, vol. 51, pp. 791-808, 2006. | ||

In article | View Article | ||

[11] | T.-L. Lin, S.-J. Horng, T.-W. Kao, Y.-H. Chen, R.-S. Run, R.-J. Chen, et al., “An efficient job-shop scheduling algorithm based on particle swarm optimization,” Expert Systems with Applications, vol. 37, pp. 2629-2636, 2010. | ||

In article | View Article | ||

[12] | L. Gao, G. Zhang, L. Zhang, and X. Li, “An efficient memetic algorithm for solving the job shop scheduling problem,” Computers & Industrial Engineering, vol. 60, pp. 699-705, 2011. | ||

In article | View Article | ||

[13] | R. Cheng, M. Gen, and Y. Tsujimura, “A tutorial survey of job-shop scheduling problems using genetic algorithms—I. Representation,” Computers & industrial engineering, vol. 30, pp. 983-997, 1996. | ||

In article | View Article | ||

[14] | Y. Wang, “A new hybrid genetic algorithm for job shop scheduling problem,” Computers & Operations Research, vol. 39, pp. 2291-2299, 2012. | ||

In article | View Article | ||

[15] | M. Al-Ashhab, “Multi-Objective Job Shop Scheduling Using a Lexicographic Procedure.” | ||

In article | |||

[16] | P. Quanke, W. Ling, and G. Liang, “Differential evolution algorithm based on blocks on critical path for job shop scheduling problems,” Journal of Mechanical Engineering, vol. 46, pp. 182-188, 2010. | ||

In article | View Article | ||

[17] | H. M. Wagner, “An integer linear‐programming model for machine scheduling,” Naval Research Logistics (NRL), vol. 6, pp. 131-140, 1959. | ||

In article | View Article | ||

[18] | A. S. Manne, “On the job-shop scheduling problem,” Operations Research, vol. 8, pp. 219-223, 1960. | ||

In article | View Article | ||

[19] | A. Scaria, K. George, and J. Sebastian, “An Artificial Bee Colony Approach for Multi-objective Job Shop Scheduling,” Procedia Technology, vol. 25, pp. 1030-1037, 2016. | ||

In article | View Article | ||

[20] | K. P. Yoon and C.-L. Hwang, Multiple attribute decision making: an introduction vol. 104: Sage publications, 1995. | ||

In article | View Article | ||

[21] | R. T. Marler and J. S. Arora, “Survey of multi-objective optimization methods for engineering,” Structural and multidisciplinary optimization, vol. 26, pp. 369-395, 2004. | ||

In article | View Article | ||

[22] | R. L. Becerra and C. A. Coello Coello, “Epsilon-constraint with an efficient cultured differential evolution,” in Proceedings of the 9th annual conference companion on Genetic and evolutionary computation, 2007, pp. 2787-2794. | ||

In article | View Article | ||

[23] | M. Laumanns, L. Thiele, and E. Zitzler, “An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method,” European Journal of Operational Research, vol. 169, pp. 932-942, 2006. | ||

In article | View Article | ||

[24] | E. Zitzler, K. Deb, and L. Thiele, “Comparison of multiobjective evolutionary algorithms: Empirical results,” Evolutionary computation, vol. 8, pp. 173-195, 2000. | ||

In article | View Article PubMed | ||

[25] | K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE transactions on evolutionary computation, vol. 6, pp. 182-197, 2002. | ||

In article | View Article | ||

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