Proposing a mathematical model of parallel machine scheduling with the aim of minimizing the task completion time and energy cost using a meta-heuristic algorithm.

Document Type : Research Paper


1 Department of Industrial Management, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

2 Department of Industrial Engineering, School of Management and Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran

3 Department of Industrial Engineering, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran


This research proposes and solves a mathematical problem of parallel machine scheduling to minimize the total completion time and energy cost. This research aims to design and optimize a multi-objective mathematical model by minimizing energy consumption and total completion time for the parallel machine scheduling problem in Semnan Polyethylene Factory. First, the mathematical model of the problem is provided, and then the solution method is investigated using the epsilon constraint method in the GAMS optimization software and the meta-heuristic imperialist competitive algorithm (ICA). The mathematical model is validated using GAMS software and the constraint epsilon method and a real problem is implemented in large dimensions regarding the case study of the polyethylene factory in Semnan province using the meta-heuristic ICA. Finally, the performance of the ICA is measured in terms of the RPI index for small dimensions and the MID index for examples with large dimensions. Numerical results show that the value of the index for distance from the ideal point in the ICA is lower than that of the index obtained from solving the problem in GAMS. With these interpretations, it can be concluded that the ICA has a better performance than GAMS for optimizing the parallel machine scheduling problem in this research. According to the obtained answers, it can be concluded that with the increase in the time to do a task, the time to complete all tasks also increases and the cost of energy remains constant. While the cost of doing the task and the price of the electricity signal increase, energy costs increase and the time to complete tasks remains constant.


Main Subjects

Abikarram, J. B., McConky, K., & Proano, R. (2019). Energy cost minimization for unrelated parallel machine scheduling under real time and demand charge pricing. Journal of cleaner production, 208, 232-242.
Abolghasemian, M., Pourghader Chobar, A., AliBakhshi, M., Fakhr, A., & Moradi Pirbalouti, S. (2021). Delay scheduling based on discrete-event simulation for construction projects. Iranian Journal of Operations Research, 12(1), 49-63.
Al-Harkan, I. M., & Qamhan, A. A. (2019). Optimize unrelated parallel machines scheduling problems with multiple limited additional resources, sequence-dependent setup times and release date constraints. IEEE Access, 7, 171533-171547.
Al-harkan, I. M., Qamhan, A. A., Badwelan, A., Alsamhan, A., & Hidri, L. (2021). Modified Harmony Search Algorithm for Resource-Constrained Parallel Machine Scheduling Problem with Release Dates and Sequence-Dependent Setup Times. Processes, 9(4), 654.
Atashpaz-Gargari, E., & Lucas, C. (2007, September). Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition. In 2007 IEEE congress on evolutionary computation (pp. 4661-4667). Ieee.
Babaeinesami, A., Ghasemi, P., Chobar, A. P., Sasouli, M. R., & Lajevardi, M. (2022). A New wooden supply chain model for inventory management considering environmental pollution: A genetic algorithm. Foundations of Computing and Decision Sciences, 47(4), 383-408.
Cataldo, A., Perizzato, A., & Scattolini, R. (2015). Production scheduling of parallel machines with model predictive control. Control Engineering Practice, 42, 28-40.
Chobar, A.P., Adibi, M.A. & Kazemi, A. Multi-objective hub-spoke network design of perishable tourism products using combination machine learning and meta-heuristic algorithms. Environ Dev Sustain (2022).
Daneshvar, A., Radfar, R., Ghasemi, P., Bayanati, M., & Pourghader Chobar, A. (2023). Design of an Optimal Robust Possibilistic Model in the Distribution Chain Network of Agricultural Products with High Perishability under Uncertainty. Sustainability, 15(15), 11669.
Ding, J. Y., Song, S., Zhang, R., Chiong, R., & Wu, C. (2015). Parallel machine scheduling under time-of-use electricity prices: New models and optimization approaches. IEEE Transactions on Automation Science and Engineering, 13(2), 1138-1154.
Kianpour, P., Gupta, D., Krishnan, K., & Gopalakrishnan, B. (2021). Optimising unrelated parallel machine scheduling in job shops with maximum allowable tardiness limit. International Journal of Industrial and Systems Engineering, 37(3), 359-381.
Kianpour, P., Gupta, D., Krishnan, K., & Gopalakrishnan, B. (2021). Optimising unrelated parallel machine scheduling in job shops with maximum allowable tardiness limit. International Journal of Industrial and Systems Engineering, 37(3), 359-381
Li, Z., Yang, H., Zhang, S., & Liu, G. (2016). Unrelated parallel machine scheduling problem with energy and tardiness cost. The International Journal of Advanced Manufacturing Technology, 84(1), 213-226.
Maadanpour Safari, F., Etebari, F., & Pourghader Chobar, A. (2021). Modelling and optimization of a tri-objective Transportation-Location-Routing Problem considering route reliability: using MOGWO, MOPSO, MOWCA and NSGA-II. Journal of optimization in industrial engineering, 14(2), 83-98.
Meng, L., Zhang, C., Shao, X., Ren, Y., & Ren, C. (2019). Mathematical modelling and optimisation of energy-conscious hybrid flow shop scheduling problem with unrelated parallel machines. International Journal of Production Research, 57(4), 1119-1145.
Moon, J. Y., Shin, K., & Park, J. (2013). Optimization of production scheduling with time-dependent and machine-dependent electricity cost for industrial energy efficiency. The International Journal of Advanced Manufacturing Technology, 68(1), 523-535.
Pan, Z., Lei, D., & Zhang, Q. (2018). A new imperialist competitive algorithm for multiobjective low carbon parallel machines scheduling. Mathematical problems in engineering, 2018.
Pourghader Chobar, A. (2022). Mathematical modeling and problem solving Integrated production planning and preventive maintenance with limited human resources. Journal of New Researches in Mathematics, 8(39), 5-24.
Rego, M. F., Pinto, J. C. E., Cota, L. P., & Souza, M. J. (2022). A mathematical formulation and an NSGA-II algorithm for minimizing the makespan and energy cost under time-of-use electricity price in an unrelated parallel machine scheduling. PeerJ Computer Science, 8, e844.
Rubaieea،Saeed، Yildirim،Mehmet Bayram،2019، An energy-aware multiobjective ant colony algorithm to minimize total completion time and energy cost on a single-machine preemptive scheduling، Computers & Industrial Engineering،Volume 127، January 2019، Pages 240-252.
Salahi, F., Daneshvar, A., Homayounfar, M., & Pourghader Chobar, A. (2023). Presenting an integrated model for production planning and preventive maintenance scheduling considering uncertainty of parameters and disruption of facilities. Journal of Industrial Management Perspective, 13(1, Spring 2023), 105-139.
Shao, W., Shao, Z., & Pi, D. (2020). Modeling and multi-neighborhood iterated greedy algorithm for distributed hybrid flow shop scheduling problem. Knowledge-Based Systems, 194, 105527.
Sohrabi, R., Pouri, K., Sabk Ara, M., Davoodi, S. M., Afzoon, E., & Pourghader Chobar, A. (2021). Applying sustainable development to economic challenges of small and medium enterprises after implementation of targeted subsidies in Iran. Mathematical Problems in Engineering, 2021, 1-9.
Zhou, S., Li, X., Du, N., Pang, Y., & Chen, H. (2018). A multi-objective differential evolution algorithm for parallel batch processing machine scheduling considering electricity consumption cost. Computers & Operations Research, 96, 55-68.