Solving fuzzy multiobjective linear bilevel programming problems based on the extension principle

Document Type : Research Article

Author

• A. Abbasi Molai

School of Mathematics and Computer Sciences, Damghan University, Damghan, P.O.Box 36715-364, Iran.

10.22067/ijnao.2021.11304.0

Abstract

Fuzzy multiobjective linear bilevel programming (FMOLBP) problems are studied in this paper. The existing methods replace one or some deterministic model(s) instead of the problem and solve the model(s). Doing this work, we lose much information about the compromise decision, and it does not make sense for the uncertain conditions. To overcome the difficulties, Zadeh’s extension principle is applied to solve the FMOLBP problems. Two crisp multiobjective linear three-level programming problems are proposed to find the lower and upper bound of its objective values in different levels. The problems are reduced to some linear optimization problems using one of the scalarization approaches, called the weighting method, the dual theory, and the vertex enumeration method. The lower and upper bounds are estimated by the resolution of the corresponding linear optimization problems. Hence, the membership functions of compromise objective values are produced, which is the main contribution of this paper. This technique is applied for the problem for the first time. This method applies all information of a fuzzy number and does not estimate it by a crisp number. Hence, the compromise decision resulted from the proposed method is consistent with reality. This point can minimize the gap between theory and practice. The results are compared with the results of existing approaches. It shows the efficiency of the proposed approach.

Keywords

• Fuzzy mathematical programming
• Fuzzy number
• Fuzzy multi objective bilevel programming
• Extension principle
• Vertex points
• Weighting method

20.1001.1.24236977.2021.11.1.1.8

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References

1. Abdelaziz, F.B. and Mejri, S. Multiobjective bi-level programming for shared inventory with emergency and backorders, Ann. Oper. Res., 267, (2018) 47–63.
2. Abo-Sinna, M.A. and Baky, I.A. Interactive balance space approach for solving bilevel multiobjective programming problems, Adv. Model. Anal. B49 (3-4) (2006) 43–62.
3. Audet, C., Haddad, J. and Savard, G. A note on the definition of a linear bilevel programming solution, Appl. Math. Comput. 181 (2006) 351–355.
4. Baky, I.A. Fuzzy goal programming algorithm for solving decentralized bilevel multiobjective programming problems, Fuzzy Set. Syst. 160(18)(2009) 2701–2713.
5. Baky, I.A. Solving multilevel multiobjective linear programming problems through fuzzy goal programming approach, Appl. Math. Model. 34(9) (2010) 2377–2387.
6. Baky, I.A., Eid, M.H. and El Sayed, M.A. Bi-level multiobjective program ming problem with fuzzy demands: A fuzzy goal programming algorithm, OPSEARCH. 51, (2014) 280–296.
7. Bard, J. Practical bilevel optimization: Algorithms and applications, Amsterdam, Kluwer Academic Publishers, 1998.
8. Bard, J. and Falk, J. An explicit solution to the programming problem, Comput. Oper. Res. 9 (1982) 77–100.
9. Bard, J.F. and Moore, J.T. A branch and bound algorithm for the bilevel programming problem, SIAM J. Sci. Stat. Comput. 11 (1990) 281–292.
10. Bazaraa, M.S., Jarvis, J.J. and Sherali, H.D. Linear programming and network flows, Second edition, Wiley, New York, 1990.
11. Bellman, R.E. and Zadeh, L.A. Decision-making in a fuzzy environment, Manag. Sci. 17 (1970) 141–164.

12. Bialas, W.F. and Karwan, M.H. Two-level linear programming, Manag. Sci. 30(8) (1984) 1004–1020.
13. Budnitzkia, A. The solution approach to linear fuzzy bilevel optimization problems, Optimization 64(5) (2015) 1195–1209.
14. Candler, W. and Townsley, R. A linear two-level programming problem, Comput. Oper. Res. 9 (1982) 59–76.
15. Colson, B. and Marcotte, P. and Savard, G. An overview of bilevel optimization, Ann. Oper. Res. 153 (2007) 235–256.
16. Gang, J., Tu, Y., Lev, B., Xu, J., Shen, W. and Yao, L. A Multi-objective bi-level location planning problem for stone industrial parks, Comput.  Oper. Res. 56 (2015) 8–21.
17. Hansen, P., Jaumard, B. and Savard, G. New branch-and-bound rules for linear bilevel programming, SIAM J. Sci. Statist. Comput. 13 (1992)1194–1217.
18. Kamal, M., Gupta, S., Chatterjee, P., Pamucar, D. and Stevic, Z. Bi-level multi-objective production planning problem with multi-choice parameters: A fuzzy Goal programming algorithm, Algorithms 12(7) (2019) 143.
19. Liu, Q.-M. and Yang, Y.-M. Interactive programming approach for solving multilevel multiobjective linear programming problem, J. Intell. Fuzzy Syst. 35(1) (2018) 55–61.
20. Lu, J., Shi, C. and Zhang, G. An extended branch and bound algorithm for bilevel multifollower decision making in a referential-uncooperative situation, Int. J. Inf. Tech. Decis. Mak. 6(2) (2007) 371–388.
21. Mersha, A.G. and Dempe, S. Linear bilevel programming with upper level constraints depending on the lower level solution, Appl. Math. Comput. 180 (2006) 247–254.
22. Moitra, B.N. and Pal, B.B. A fuzzy goal programming approach for solving bilevel programming problems, Pal, N.R., Sugeno, M. (eds.) In: Advances in Soft Computing – AFSS. vol. 2275 of Lecture Notes in Computer Science, PP.91–98, Springer, Berlin, Germany, 2002.
23. Mohamed, R.H. The relationship between goal programming and fuzzy programming, Fuzzy Set. Syst. 89 (1997) 215–222.
24. Peric, T., Babic, Z. and Omerovi, M. A fuzzy goal programming approach to solving decentralized bilevel multiobjective linear fractional programming problems, Croat. Oper. Res. Rev. 10 (2019) 65–74.
25. Pramanik, S. and Dey, P.P. Bi-level multiobjective programming problem with fuzzy parameters, Int. J. Comput. Appl. 30(11) (2011) 13–20.

26. Pramanik, S. and Roy, T.K. Fuzzy goal programming approach to multilevel programming problems, Eur. J. Oper. Res. 176(2) (2007) 1151–1166.
27. Sakawa, M. Fuzzy sets and interactive multiobjective optimization, Plenum press, New York, 1993.
28. Sakawa, M., Nishizaki, I. and Uemura, Y. Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters, Fuzzy Set. Syst. 115 (2000) 93–103.
29. Sakawa, M., Nishizaki, I. and Uemura, Y. Interactive fuzzy programming for multilevel linear programming problems with fuzzy parameters, Fuzzy Set. Syst. 109 (2000) 3–19.
30. Shi, C., Lu, J. and Zhang, G. An extended Kth-best approach for linear bilevel programming, Appl. Math. Comput. 164 (2005) 843–855.
31. Shi, C., Lu, J., Zhang, G. and Zhou, H. An extended branch and bound algorithm for linear bilevel programming, Appl. Math. Comput. 180 (2006)529–537.
32. Shi, C., Zhang, G. and Lu, J. On the definition of linear bilevel program ming solution, Appl. Math. Comput. 160(1) (2005) 169–176.
33. Shi, C., Zhang, G. and Lu, J. An extended Kuhn-Tucker approach for linear bilevel programming, Appl. Math. Comput. 162(1) (2005) 51–63.
34. Shih, H.S., Lai, Y.J. and Lee, E.S. Fuzzy approach for multilevel programming problems, Comput. Oper. Res. 23 (1983) 73–91.
35. Shih, H.S. and Lee, E.S. Compensatory fuzzy multiple level decision making, Fuzzy Set. Syst. 14 (2000) 71–87.
36. Toksari, M.D. and Bilim, Y. Interactive fuzzy goal programming based on Jacobian matrix to solve decentralized bi-level multi-objective fractional programming problems, Int. J. Fuzzy Syst. 17 (2015) 499–508.
37. Von Stackelberg, H. The theory of the market economy, Oxford University Press, York, Oxford, 1952.
38. Wan, Zh.P. and Fei, P.Sh. The theory and algorithm of optimization, The Wuhan University Press, 2004.
39. Wang, G., Wang, X., Wan, Z. and Lv, Y. A globally convergent algorithm for a class of bilevel nonlinear programming problem, Appl. Math. Comput. 188 (2007) 166–172.
40. Zadeh, L.A. Fuzzy sets, Inform. Control, 8 (1965) 338–353.
41. Zadeh, L.A. Fuzzy sets as a basis for a theory of possibility, Fuzzy Set. Syst. 1 (1978) 3–28.

42. Zhang, G. and Lu, J. The definition of optimal solution and an extended Kuhn-Tucker approach for fuzzy linear bilevel programming, The IEEE Computational Intelligence Bulletin 5 (2005) 1–7.
43. Zhang, G. and Lu, J. Model and approach of fuzzy bilevel decision making for logistics planning problem, J. Enterp. Inf. Manag. 20 (2007) 178–197.
44. Zhang, G., Lu, J. and Dillon, T. A branch-and-bound algorithm for fuzzy bilevel decision making, in: D. Ruan, P. D’hondt, P. Fantoni, M. De Cock, M. Nachtegael, E. Kerre (Eds.), Applied Artificial Intelligence, World Scientific, Singapore, 2006, pp. 291–298.
45. Zhang, G., Lu, J. and Dillon, T. An approximation branch and bound approach for fuzzy linear bilevel decision making, The 1st International Symposium Advances in Artifical Intelligence and Applications (AAIA06) Wisla, Poland, November 6–10, 2006.
46. Zhang, G., Lu, J. and Dillon, T. Decentralized multiobjective bilevel decision making with fuzzy demands, Knowl-Based Syst. 20 (2007) 495–507.
47. Zheng, Y., Wan, Z. and Wang, G. A fuzzy interactive method for a class of bilevel multiobjective programming problem, Expert Syst. Appl. 38 (2011)10384–10388.