Document Type : Original Article

Author

1 Department of Mathematics, College of Science and Arts, Qassim University, Al-Badaya 51951 Saudi Arabia.

2 Department of Operations and Management Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt.

10.22105/cand.2023.194229

Abstract

The current study investigates a two-machine Flow Shop Scheduling (FSS) problem with piecewise quadratic fuzzy processing time. It is illogical to consider that the processing time is exact but uncertain because it varies due to human factors. One of the most popular approximate intervals, namely, close interval approximation for the Piecewise Quadratic Fuzzy Number (PQFN), is introduced. A solution method with the help of Johnson's algorithm [1], the close interval approximation of PQFNs, and the modified McCahon and Lee's algorithm [2] is developed to determine the minimization of the expected makespan. Numerical experimentation is performed to demonstrate the effectiveness of the suggested methodology.

Keywords

[1]     Johnson, S. M. (1954). Optimal two‐ and three‐stage production schedules with setup times included. Naval research logistics quarterly, 1(1), 61–68. DOI:10.1002/nav.3800010110
[2]     McCahon, C. S., & Lee, E. S. (1990). Job sequencing with fuzzy processing times. Computers and mathematics with applications, 19(7), 31–41. DOI:10.1016/0898-1221(90)90191-L
[3]     Morton, T. E., & Pentico, D. W. (1993). Heuristic scheduling systems: with applications to production systems and project management (Vol. 3). John Wiley & Sons.
[4]     Agrawal, S., Gautam, A., Chauhan, D. K., Tiwari, L. M., & Kapoor, S. (2012). A flow shop scheduling problem with transportation time and separated setup time of jobs. Procedia engineering, 38, 1327–1332.
[5]     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. International journal of advanced manufacturing technology, 73(5/8), 601–611. DOI:10.1007/s00170-014-5841-4
[6]     Ren, T., Guo, M., Lin, L., & Miao, Y. (2015). A local search algorithm for the flow shop scheduling problem with release dates. Discrete dynamics in nature and society, 2015. DOI:10.1155/2015/320140
[7]     Laribi, I., Yalaoui, F., Belkaid, F., & Sari, Z. (2016). Heuristics for solving flow shop scheduling problem under resources constraints. IFAC-papersonline, 49(12), 1478–1483. DOI:10.1016/j.ifacol.2016.07.780
[8]     Yazdani, M., & Naderi, B. (2016). Modeling and scheduling no-idle hybrid flow shop problems. Journal of optimization in industrial engineering, 10(21), 59–66. http://www.qjie.ir/article_261.html
[9]     Qu, C., Fu, Y., Yi, Z., & Tan, J. (2018). Solutions to no-wait flow shop scheduling problem using the flower pollination algorithm based on the hormone modulation mechanism. Complexity, 2018. DOI:10.1155/2018/1973604
[10]   Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353.
[11]   Bellman, R., & Zadeh, L. (1970). Decision-making in a fuzzy environment. Management science, 17(4), B--141. DOI:10.1142/9789812819789_0004
[12]   Dubois, D. J., & Prade, H. (1980). Fuzzy sets and systems: theory and applications (Vol. 144). Academic Press.
[13]   Panda, A., & Pal, M. (2015). A study on pentagonal fuzzy number and its corresponding matrices. Pacific science review b: humanities and social sciences, 1(3), 131–139. DOI:10.1016/j.psrb.2016.08.001
[14]   Yager, R. R. (1981). A procedure for ordering fuzzy subsets of the unit interval. Information sciences, 24(2), 143–161. DOI:10.1016/0020-0255(81)90017-7
[15]   Litoiu, M., & Tadei, R. (2001). Fuzzy scheduling with application to real-time systems. Fuzzy sets and systems, 121(3), 523–535. DOI:10.1016/S0165-0114(99)00176-1
[16]   Yao, J. S., & Lin, F. T. (2002). Constructing a fuzzy flow-shop sequencing model based on statistical data. International journal of approximate reasoning, 29(3), 215–234. DOI:10.1016/S0888-613X(01)00064-0
[17]   Hejazi, S. R., Emami, S., & Arkan, A. (2009). A heuristic algorithm for minimizing the expected makespan in two-machine flow shops with fuzzy processing time. Journal of uncertain systems, 3(2), 114–122.
[18]   Ishibuchi, H., Murata, T., & Lee, K. H. (1996). Formulation of fuzzy flowshop scheduling problems with fuzzy processing time. IEEE international conference on fuzzy systems (Vol. 1, pp. 199–205). IEEE. DOI: 10.1109/fuzzy.1996.551742
[19]   Temiz, I., & Erol, S. (2004). Fuzzy branch-and-bound algorithm for flow shop scheduling. Journal of intelligent manufacturing, 15(4), 449–454. DOI:10.1023/B:JIMS.0000034107.72423.b6
[20]   Ignall, E., & Schrage, L. (1965). Application of the branch and bound technique to some flow-shop scheduling problems. Operations research, 13(3), 400–412. DOI:10.1287/opre.13.3.400
[21]   Gupta, D., Aggarwal, S., & Sharma, S. (2012). A Fuzzy logic based approach to minimize the rental cost of machines for specially structured three stages flow-shop scheduling. Advances in applied science research, 3(2), 1071–1076.
[22]   González-Neira, E. M., Montoya-Torres, J. R., & Barrera, D. (2017). Flow-shop scheduling problem under uncertainties: Review and trends. International journal of industrial engineering computations, 8(4), 399–426. DOI:10.5267/j.ijiec.2017.2.001
[23]   Komaki, G. M., Sheikh, S., & Malakooti, B. (2019). Flow shop scheduling problems with assembly operations: a review and new trends. International journal of production research, 57(10), 2926–2955. DOI:10.1080/00207543.2018.1550269
[24]   Janaki, E., & Mohamed Ismail, A. (2020). Flow shop scheduling in which processing time connected with probabilities and job delay due to maintenance for m * n machine [presentation]. Advances in intelligent systems and computing (Vol. 933, pp. 651–657). DOI: 10.1007/978-981-13-7166-0_65
[25]   Khalifa, H. A. (2020). On single machine scheduling problem with distinct due dates under fuzzy environment. International journal of supply and operations management, 7(3), 272–278. DOI:10.22034/IJSOM.2020.3.5
[26]   Abd El-Wahed Khalifa, H., Alodhaibi, S. S., & Kumar, P. (2021). Solving constrained flow-shop scheduling problem through multistage fuzzy binding approach with fuzzy due dates. Advances in fuzzy systems, 2021, 1–8. DOI:10.1155/2021/6697060
[27]   Alharbi, M. G., & El-Wahed Khalifa, H. A. (2021). On a flow-shop scheduling problem with fuzzy pentagonal processing time. Journal of mathematics, 2021, 1–7. DOI:10.1155/2021/6695174
[28]   Alburaikan, A., Garg, H., & Khalifa, H. A. E. W. (2023). A novel approach for minimizing processing times of three-stage flow shop scheduling problems under fuzziness. Symmetry, 15(1), 130. DOI:10.3390/sym15010130
[29]   Ren, J., Ye, C., & Yang, F. (2021). Solving flow-shop scheduling problem with a reinforcement learning algorithm that generalizes the value function with neural network. Alexandria engineering journal, 60(3), 2787–2800. DOI:10.1016/j.aej.2021.01.030
[30]   Wang, C. N., Hsu, H. P., Fu, H. P., Phan, N. K. P., & Nguyen, V. T. (2022). Scheduling flexible flow shop in labeling companies to minimize the makespan. Computer systems science and engineering, 40(1), 17–36. DOI:10.32604/CSSE.2022.016992
[31]   Wang, C. N., Porter, G. A., Huang, C. C., Nguyen, V. T., & Husain, S. T. (2022). Flow-shop scheduling with transportation capacity and time consideration. Computers, materials and continua, 70(2), 3031–3048. DOI:10.32604/cmc.2022.020222
[32]   Jemmali, M., & Hidri, L. (2022). Hybrid flow shop with setup times scheduling problem. Computer systems science and engineering, 44(1), 563–577. DOI:10.32604/csse.2023.022716
[33]   Koulamas, C., & Kyparisis, G. J. (2022). Flow shop scheduling with two distinct job due dates. Computers and industrial engineering, 163, 107835. DOI:10.1016/j.cie.2021.107835
[34]   Jain, S. (2010). Close Interval Approximation of Piecewise Quadratic Fuzzy Numbers for Fuzzy Fractional Program. Iranian journal of operations research, 2(1), 77–88.