New Advances on Fuzzy Linear Programming Problem by Semi-Infinite Programming Approach

Abstract

As we are faced with more uncertainty problems in the real world, it is necessary to ‎provide ‎models that can provide appropriate solutions for dealing with these issues. In this ‎study, we ‎ proposed a new approach to solving linear programming problem in the fuzzy ‎environment ‎based on solving ‎a related multi-objective model. This kind of problem can be ‎reduced to a ‎fuzzy linear semi-infinite programming problem. In this way, we present a new ‎mixed ‎Multi-Objective Linear Semi-Infinite Programming (MOLSIP) model to solve the ‎main ‎problem, furthermore, as a practical case, we consider a fuzzy Data Envelopment ‎Analysis ‎‎(DEA) model which is a concern to‎ an evaluation of the performance of Decision-‎Making ‎Units (DMUs) in uncertainty environment, The new models show the advantage of ‎our ‎method over the previous ones in terms of certainty. Finally, numerical examples ‎are ‎included to illustrate the suggested solution procedure.‎

Keywords

• Fuzzy linear ‎programming
• Multi-objective linear ‎programming‎ ‎
• Linear semi-infinite ‎programming
• Data envelopment ‎analysis‎

References

1. [1] Bellman, R.E., Zadeh, L. A., “Decision making in a fuzzy environment”, Management Science, 17: 141-164, (1970).
2. [2] Sharif Uddin, M., Miah, M., Al-Amin Khan, M., AlArjani, A., “A solving approach for ‎fuzzy multi-objective linear fractional programming and application to an agricultural ‎planting structure optimization problem”, Chaos, Solitons & Fractals, 141: 110352 (2020).
3. [3] Efe, B., Efe, Ö. F., Kurt, M., “An integrated intuttionistic fuzzy set and mathematical programming approach for an occupational health and safety policy”, Gazi University Journal of Science, 30(2): 73 – 95, (2017).
4. [4] Nafei, A., Yuan, W., Nasseri, H., “A New Method for Solving Interval Neutrosophic Linear Programming Problems”, Gazi University Journal of Science, 33 (4): 796 – 808, (2020).
5. [5] Yang, G., Li, X., Huo, L., Liu, Q., “Goal programming tactic for uncertain multi-objective ‎transportation problem using fuzzy linear membership function”, Alexandria Engineering ‎Journal, 60(2): 2525-2533, (2021).‎
6. [6] Zimmermann, H.J., “Fuzzy programming and linear programming with several objective ‎functions”, Fuzzy Sets and Systems, 1: 45-55, (1978).
7. [7] Gadhi, N.A., “Necessary optimality conditions for a nonsmooth semi-infinite programming problem”, Journal of Global Optimization, 74(1): 161-168, (2019).
8. [8] Goberna, M.A., Gómez, S., Guerra, F., Todorov, M.I., “Sensitivity analysis in linear semi-infinite ‎programming: perturbing cost and right-hand-side coefficients”, European Journal of Operational Research, 181: 1069-1085, (2007).

9. [9] Jian, P.P., Li, L., XuZe, H. X., “An infeasible bundle method for nonconvex constrained optimization with application to semi-infinite programming problems”, Numerical Algorithms, 80(2): 397-427, (2019).
10. [10] Joshi, B.C., Mishra, Sh.K., Kumar, P., “On semi-infinite mathematical programming problems with equilibrium constraints using generalized convexity”, Journal of the Operations Research Society of China, 29: 1-18, (2019).
11. [11] Papp, D., “Semi-Infinite Programming”, Wiley StatsRef: Statistics Reference Online, 1-14, (2019).
12. [12] Fang, S.C., Hu, C.F., Wang, H.F., Wu, S.Y., “Linear programming with fuzzy coefficients in constraints”, Computers and Mathematics with Applications, 37: 63-76, (1999).
13. [13] Wu, S.Y., Fang, S.Ch., ‎Lin, Ch.J., “Analytic center based cutting plane method for linear semi-infinite programming”, In López, M.A. Semi-Infinite Programming: Recent Advances, Kluwer, Dordrecht, 221–233, ‎(2001)‎.
14. [14] Nasseri, S.H., Behmanesh, E., Faraji, P., Fallahzadeh Shahabi N., “Semi-infinite programming to solve linear programming with triangular fuzzy coefficients”, Annals of Fuzzy Mathematics and Informatics, 1: 213-226, (2013).
15. [15] Nasseri, S.H., Zavieh, H., “A multi-objective method for solving fuzzy linear programming based on semi-infinite models”, Fuzzy Information and Engineering, 10: 95-102, (2018).
16. [16] Uciński, D., “Sensor Selection with Non-smooth Design Criteria Based on Semi-Infinite Programming”, IFAC-PapersOnLine, 53(2): 75-39-7544, (2020).
17. [17] Geng, H., Liu, J., Wen-Luo, P., Cheng, L., Grant, S., Shi, Y., “Selective body biasing for post-silicon tuning of sub-threshold designs: A semi-infinite programming approach with Incremental Hyper-cubic Sampling”, Integration, 55: 465-473, (2016).
18. [18] Hale, W. T., Wlihelm, M. E., Palmer, K. A., Stuber, M. D., Bollas, G. M., “ Semi-infinite programming for global guarantees of robust fault detection and isolation in safety-critical systems”, Computers & Chemical Engineering, 126: 218-230, (2019).
19. [19] He, L., Huang, G., Lu, H., “Bivariate interval semi-infinite programming with an application to environmental decision-making analysis”, European Journal of Operational Research, 211(3): 452-465, (2011).
20. [20] Simić, V., Dabić-Ostojić, S., Bojović, N., “ Interval-parameter semi-infinite programming model for used tire management and planning under uncertainty”, Computers & Industrial Engineering, 113: 487-501, (2017).
21. [21] Djelassi, H., Mitsos, A., Stein, O., “Recent advances in nonconvex semi-infinite programming: Applications and algorithms”, EURO Journal on Computational Optimization, 9: 100006, (2021).
22. [22] Agarwal, Sh., “Efficiency Measure by fuzzy Data Envelopment analysis Model”, Fuzzy Information and Engineering, 6: 59-70, (2014).
23. [23] Gölcükcü, A., “Fuzzy DEA, Banking, Customer, Optimist and Pessimist Approaches”, Gazi University Journal of Science, 28 (4): 561 – 569, (2015).
24. [24] Jess, A., Jongen, H.Th., Neralic, L. Stein, O., “A semi-infinite programming model in data ‎envelopment analysis”, Optimization, 49: 369-385, ‎(2001)‎.
25. [25] Zhanxin, M. Ma., Wei, C., “Generalized fuzzy data envelopment analysis methods”, Applied Soft Computing, 19: 215-225, (2014).
26. [26] Buckley, J.J., “A fast method of ranking alternatives using fuzzy numbers”, Fuzzy Sets and ‎Systems, 30:337-348, (1989).
27. [27] Cao, B.Y., “Optimal Models and Methods with Fuzzy Quantities”, Studies in Fuzziness and Soft Computing, volume 248, Springer-verlag, Berlin, Heidelberg, (2010).‎
28. [28] Charnes, A., Cooper, W. W., Rhodes, E., “Measuring the efficiency of decision-making units”, European Journal of Operational Research, 2: 29-44, (1978).
29. [29] Hettich, R., Kortanek, K.O., “Semi-infinite programming: Theory, methods and applications”, SIAM Review, 35: 380-429, (1993).