Flow Shop Scheduling Problem with Machine Availability Constraints and Learning Effect based on a Hybrid Model

Document Type : Original Article


1 Associate Professor, Shahid Beheshti University.

2 M.Sc., Raja Institute of Higher Education.


During the recent decades, flow shop scheduling problem has investigated with different assumptions. One of the most important one that has attracted many researchers, is consideration the concept of learning effect. In real situation of work environment, learning effect is not limited to job position, in other word, workers experiments also should be considered during the process of operations. In one hand, there are factors that can cause machines and equipment unavailability in the planning horizon. This study investigates flow shop scheduling problem with machine availability constraints and learning effect based on a hybrid model. This learning model includes not only job positions but also total logarithmic processing time of jobs. First, a mixed integer liner programming model has been proposed to formulate the problem. Because of high complexity of this model, two meta-heuristic algorithms, simulated annealing (SA) and cloud theory-based simulated annealing (CSA) have been used to find nearly optimal solutions. Finally it is cleared that, CSA could be more successful in generating nearly optimal solutions than SA.


1. Adeli, M., Zandieh, M. (2013). Multiobjective simulation-optimization approach for intrgrated sourcing and inventory decisions. Journal of Industrial Management Perspective3(11), 89-110.
2. Aggoune, R. (2004). Minimizing the makespan for the flow shop scheduling problem with availability constraints. European Journal of Operational Research153(3), 534–543.
3. Aggoune, R., & Portmann, M.-C. (2006). Flow shop scheduling problem with limited machine availability: A heuristic approach. International Journal of Production Economics99(1-2), 4–15.
4. Allaoui, A., & Artiba, A. (2006). Scheduling two-stage hybrid flow shop with availability constraints. Computers & Industrial Engineering33(5), 1399–1419.
5. Biskap, D. (1999). Single-machine scheduling with learning considerations. Euroupean journal of operational research115(1), 173-178.
6. Biskup, D. (2008). A state-of-the-art review on scheduling with learning effects. European Journal of Operational Research188(2), 315-329.
7. Breit, J. (2004). An improved approximation algorithm for two-machine flow shop scheduling with an availability constraint. Information Processing Letters90, 273–278.
8. Cheng, T., Kou, W.-H., & Yang, D.-L. (2013). Scheduling with a position-weighted learning effect based on sum-of-logarithm-processing-times and job position. Information Sciences221(1), 490-500.
9. Cheng, T., Lai, P., Wu, C., & Lee, W. (2009). Single-machine scheduling with sum-of logarithm-processing-times-based learning considration. Information Sciences179(18), 3127–3135.
10. Cheng, T., Wu, C., & Lee, W. (2008). Some scheduling problems with sum-of-processing-times-based and job-position-based learning effects. Information Sciences178(11), 2476–2487.
11. Deyi, L., & Yi, D. (2005). Artificial intelligence with uncertainty. Chapman & Hall.
12. Deyi, L., Haijun, M., & Xuemei, S. (1995). Membership clouds and membership cloud generators. Journal of Computer Research and Development32(6), 15–20.
13. Eren, T., & Güner, E. (2008). A bicriterion flowshop scheduling with a learning effect. Applied Mathematical Modelling32(9), 1719–1733.
14. Ghodratnama, A., Rabbani, M., Tavakkoli-Moghaddam, R., & Baboli, A. (2010). Solving a single-machine scheduling problem with maintenance, job deterioration and learning effect by simulated annealing. Journal of Manufacturing Systems29(1), 1-9.
15. Janiak, A., & Rudek, R. (2008). Viewpoint on: complexity results for single-machine scheduling with positional learning effects. Journal of the Operational Research Society, 59(10), 1430.
16. Lee, C. (1999). Two-machine flowshop scheduling with availability constraints. European Journal of Operational Research114(2), 420–429.
17. Lee, W.-C., & Wu, C.-C. (2004). Minimizing total completion time in a two-machine flowshop with a learning effect. International Journal of Production Economics88(1), 85–93.
18. Liao, L., & Tsai, C. (2009). Heuristic algorithms for two-machine flowshop with availability constraints. Computers & Industrial Engineering56(1), 306–311.
19. Lv, P., Yuan, L., & Zhang, J. (2009). Cloud theory-based simulated annealing algorithm and application. Engineering Applications of Artificial Intelligence, 22(4-5), 742–749.
20. Ma, Y., Chu, C., & Zou, C. (2010). A survey of scheduling with deterministic machine availability constraints. Computers & Industrial Engineering58(2), 199-211.
21. Rahimi, H., Azar, A., Rezaei Pandari, A. (2015). Designing a multi objective job shop scheduling model and solving it by simulated annealing. Journal of Industrial Management Perspective5(19), 39-64.
22. Vahedi-Nouri, B., Fattahi, P., & Ramezanian, R. (2013b). Minimizing total flow time for the non-permutation flow shop scheduling problem with learning effects and availability constraints. Journal of Manufacturing Systems, 32(1), 167-173.
23. Vahedi-Nouri, B., Fattahi, P., Rohaninejad, M., & Tavakkoli-Moghaddam, R. (2013a). Minimizing the total completion time on a single machine with the learning effect and multiple availability constraints. Applied Mathematical Modelling37(5), 3126-3137.
24. Vahedi-Nouri, B., Fattahi, P., Tavakkoli-Moghaddam, R., & Ramezanian, R. (2014). A general flow shop scheduling problem with consideration of position-based learning effect and multiple availability constraints. The International Journal of Advanced Manufacturing Technology73(5-8), 601-611.
25. Wu, C., & Lee, W. (2009). Single-machine and flowshop scheduling with a general learning effect model. Computers & Industrial Engineering56(4), 1553-1558.
26. Yang, S.-J. (2010). Single-machine scheduling problems with both start-time dependent learning and position dependent aging effects under deteriorating maintenance consideration. Applied Mathematics and Computation, 217(7), 3321–3329.
27. Zhang, X., Yan, G., Huang, W., & Tang, G. (2012). A note on machine scheduling with sum-of-logarithm-processing-timebased and position-based learning effects. Information Sciences187, 298-304.