Document Type : Original Article


1 M.Sc., Bu-Ali Sina University.

2 Associate Professor, Bu-Ali Sina University.



A flowshop problem with objective functions of minimizing makespan and energy cost has been investigated. Reducing production costs is one of the goals that industries always have in mind. Increasing public awareness about the energy issues creates a new attitude toward minimizing energy costs. In order to make the problem more compatible with the real-world conditions, the problem is considered under uncertainty. An existing research gap inspired this study. It is assumed that machines can use the three slow, normal and fast speeds to process jobs. At high speeds, consumption rate increases and completion time decreases, and vice versa. The difference in machine processing speeds yields different and contradictory values in the objective functions. Therefore, a method should be proposed in which, in addition to the order of jobs, the speed of machines could be determined. A mathematical model is presented, and then a simulation-based genetic algorithm is used to solve the problem on a large scale. Simulation is used for each evaluation of the objective function in the genetic algorithm to consider the uncertainty of processing times. Due to the stochastic processing time, the expected value model is used to deal with uncertainty. The computational results indicate that the algorithm and approach show a good performance.


  1. Alam Tabriz, A. Roghanian, E. & Hoseinzadeh, M. (2012). Design and optimization of reverse logistics network under uncertainty using genetic algorithm. Journal of Industrial Management Perspective, 1, 61-89. (In Persian).
  2. Baker, K.R., Altheimer, D. (2012). Heuristic solution methods for the stochastic flow shop problem. European Journal of Operational Research, 216(1), 172-177.
  3. Chang, P.C., Chen, S.H., & Lin, K.L. (2005). Two-phase sub population genetic algorithm for parallel machine-scheduling problem. Expert Systems with Applications, 29, 705-712.
  4. Dantzig, G.B. (1955). Linear Programming under Uncertainty. Management Science, 1, 197-206.
  5. Ding, J.Y., Song, S., & Wu, C. (2016).Carbon-efficient scheduling of flow shops by multi-objective optimization. European Journal of Operational Research, 248, 758-771.
  6. Fatahi, P. Mohamadi, E. & Daneshamoz, F. (2019). Solve Multi-Objective Jobshop Scheduling Problem with One Step of Assembly and Considering the Flow of Cargo. Journal of Industrial Management Perspective, 9(33), 61-86. (In Persian).
  7. Framinan, J.M., P-Gonzalez, P. (2015). On heuristic solutions for the stochastic flowshop scheduling problem. European Journal of Operational Research, 246(2), 413-420.
  8. Gahm, C., Denz, F., Dirr, M., & Tuma, A. (2016). Energy-efficient scheduling inmanufacturing companies: A reviewand research framework. European Journal of Operational Research, 248, 744–757.
  9. Gonzalez-Neira, E. M., Ferone, D., Hatami, S. & Juan, A. (2017). A biased-randomized simheuristic for the distributed assembly permutation flowshop problem with stochastic processing times. Simulation Modelling Practice and Theory, 79, 23–36.
  10. Gourgand, M., Grangeon, N., & Norre, S. (2005). Markovian analysis for performance evaluation and scheduling in m machine stochastic flow-shop with buffers of any capacity. European Journal of Operational Research, 161(1), 126-147.
  11. Juan, A.A., Barrios, B.B., Vallada, E., Riera, D., & Jorba, J. (2014).Asimheuristic algorithm for solving the permutation flow shop problem with stochastic processing times. Simulation Modeling Practice and Theory, 46, 101-117.
  12. Liefooghe, A., Basseur, M., Jourdan, L., & Talbi, E. (2007). Combinatorial Optimization of Stochastic Multi-Objective Problems: An Application to the Flow-Shop Scheduling Problem. EMO 2007, LNCS 4403, 457-471.
  13. Luo, H., Du, B., Huang, G.Q., Chen, H., Li, X. (2013). Hybrid flow shop scheduling considering machine electricity consumption cost. International Journal of Production Economics, 146, 423-439.
  14. Mansouri, S.A., Aktas, E., & Besikci, U. (2016). Green scheduling of a two- machine flowshop: Trade-off between makespan and energy consumption. European Journal of Operational Research, 248, 772-788.
  15. Masmoudi, O., Yalaoui, A., Ouazene, Y., & Chehade, H. (2016). Solving a capacitated flow-shop problem with minimizing total energy costs. The International Journal of Advanced Manufacturing Technology volume 90, pages2655–2667.
  16. Schulz, S. (2018). A genetic algorithm to solve the hybrid flow shop scheduling problem with subcontracting options and energy cost consideration. SAT 2018, AISC 854, 263–273.
  17. Tanaev, V., Sotskov, Yuri, N., & Strusevich, V.A. (2012). Scheduling Theory: Multi-Stage Systems. Springer Science & Business Media.
  18. Tang, D., Dai, M., Salido, M.A., & Giret, A. (2016)Energy-efficient dynamic scheduling for a flexible flow shop using an improved particle swarm optimization. Computers in Industry, 81, 82–95.
  19. Wang, K., Choi, S.H., & Lu, H. (2015). A hybrid estimation of distribution algorithm for simulation-based scheduling in a stochastic permutation flowshop. Computers and Industrial Engineering, 90, 186-196.
  20. Zhai, Y., Biel K, Zhao, F., &Sutherland, J. (2017). Dynamic scheduling of a flow shop with on-site wind generation for energy cost reduction under real time electricity pricing. CIRP Annals - Volume 66, Issue 1, Pages 41-44.
  21. Zandieh, M. & Fotovat, A. (2015). Flowshop scheduling system with limitation of machine access and learning effect based on a hybrid model. Journal of Industrial Management Perspective, 1, 41-58. (In Persian).
  22. Zhang, H., Zhao, F., Fang, K., & Sutherland, J.W. (2014). Energy-conscious flow shop scheduling under time-of-use electricity tariffs. CIRP Annals - Manufacturing Technology, 63, 37-40.
  23., Available at February 2017.