Bi-level Multi-objective Programming Problems with Fuzzy Parameters: Modified TOPSIS Approach
International Journal of Management and Fuzzy Systems
Volume 2, Issue 5, October 2016, Pages: 38-50
Received: Oct. 5, 2016; Accepted: Nov. 30, 2016; Published: Dec. 27, 2016
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Authors
Ibrahim A. Baky, Department of Mathematics, Faculty of Science, Tabuk University, Tabuk, Saudi Arabia; Department of Basic Engineering Sciences, Faculty of Engineering, Benha University, El Qalyoubia, Egypt
M. A. El Sayed, Department of Basic Engineering Sciences, Faculty of Engineering, Benha University, El Qalyoubia, Egypt
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Abstract
In this paper, a modified TOPSIS (techniques for order preference by similarity to ideal solution) approach for solving bi-level multi-objective programming (BL-MOP) problems with fuzzy parameters is presented. These fuzzy parameters are assumed to be characterized by fuzzy numerical data, reflecting the experts' imprecise or fuzzy understanding of the nature of the parameters in the problem formulation process. Firstly, the corresponding non-fuzzy bi-level programming model is introduced based on the α-level set. Secondly, a modified TOPSIS approach is developed, in which the fuzzy goal programming (FGP) approach is used to solve the conflicting bi-objective distance functions instead of max-min operator. As the FGP approach utilized to achieve the highest degree of each membership goal by minimizing the sum of the unwanted deviational variables. Finally, an algorithm to clarify the modified TOPSIS approach, as well as Illustrative numerical example and comparison with the existing methods, are presented.
Keywords
Bi-level Programming, Fuzzy Sets, Fuzzy Parameters, TOPSIS, Fuzzy Goal Programming, Multi-objective Programming
To cite this article
Ibrahim A. Baky, M. A. El Sayed, Bi-level Multi-objective Programming Problems with Fuzzy Parameters: Modified TOPSIS Approach, International Journal of Management and Fuzzy Systems. Vol. 2, No. 5, 2016, pp. 38-50. doi: 10.11648/j.ijmfs.20160205.11
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Mahmoud. A. Abo-Sinna and A. H. Amer, Extensions of TOPSIS for multi-objective large-scale nonlinear programming problems, Appl. Math. Comput. 162 (2000) 243-256.
[2]
Mahmoud. A. Abo-Sinna, Extensions of the TOPSIS for multi-objective dynamic programming problems under fuzziness, Adv. Model. Anal. (AMSE Press, France) B 43 (4) (2000) 1-24.
[3]
Mahmoud. A. Abo-Sinna, A. H. Amer and A. S. Ibrahim, Extensions of TOPSIS for large-scale multi-objective nonlinear programming problems with block angular structure, Appl. Math. Model. 32 (2008) 292-302.
[4]
Ibrahim A. Baky and M. A. Abo-Sinna, TOPSIS for bi-level MODM problems, Appl. Math. Model. 37 (3) (2012) 1004-1015.
[5]
Ibrahim A. Baky, Fuzzy goal programming algorithm for solving decentralized bi-level multi-objective programming problems, Fuzzy Sets Systems, vol. 160, no. 18, pp. (2009) 2701-2713.
[6]
Ibrahim A. Baky, Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach, Applied Mathematical Modelling, 34 (9) (2010) 2377-2387.
[7]
Ibrahim A. Baky, M. H. Eid, M. A. Elsayed, Bi-level multi-objective programming problem with fuzzy demands: a fuzzy goal programming algorithm, Journal of Operational Research Society of India (OPSEARCH), 51 (2) (2014) 280-296.
[8]
Ibrahim A. Baky, M. H. Eid, M. A. Elsayed, Bi-level multi-objective programming problem with fuzzy parameters: TOPSIS approach, the 38th International Conference for statistics, computer science and its applications, Cairo, Egypt, 8-18 April, 13-33, 2013.
[9]
C. T. Chen, Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy Set Syst. 114 (2000) 1-9.
[10]
H. Deng, C. H. Yeh and R. J. Willis, Inter-company comparison using modified TOPSIS with objective weights, Comput. Oper. Res. 17 (2000) 963-973.
[11]
C. L. Hwang and K. Yoon, Multiple attribute decision making: methods and applications, Springer-Verlag, Heidelberg, 1981.
[12]
Y.-J. Lai, T.-J. Liu and C. L. Hwang, TOPSIS for MODM, Euro. J. Oper. Res. 76 (1994) 486-500.
[13]
E. S. Lee and R. J. Li, Fuzzy multiple objective programming with Pareto optimum, Fuzzy Sets and Systems 53, (1993) 275-288.
[14]
R. H. Mohamed, The relationship between goal programming and fuzzy programming, Fuzzy Sets and Systems, 89 (1997) 215-222.
[15]
S. Pramanik and T. K. Roy, Fuzzy goal programming approach to multi-level programming problems, European Journal of Operational Research, vol. 176, no. 2, pp (2007) 1151-1166.
[16]
S. Pramanik and P. P. Dey, Bi-level multi-objective programming problem with fuzzy parameters. International Journal of Computer Applications, 30(11)(2011) 13-20.
[17]
B. B. Pal, B. N. Moitra, and U. Maulik, A goal programming procedure for fuzzy multi-objective linear fractional programming problem, Fuzzy Sets and Systems, 139(2) (2003) 395-405.
[18]
S. Sinha, Fuzzy programming approach to multi-level programming problems, Fuzzy Sets and Systems, 136 (2003)189-202.
[19]
M. Sakawa and I. Nishizaki, Interactive fuzzy programming for decentralized two-level linear programming problems, Fuzzy Sets and Systems, 125(2002) 301-315.
[20]
G. Zhang, J. Lu and T. Dillon, Decentralized multi-objective bi-level decision making with fuzzy demands, Knowledge-Based System, 20 (5)(2007), 495-507.
[21]
L. A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965) 338-353.
[22]
B. N. Moitra and B. B. Pal, A fuzzy goal programming approach for solving bi-level programming problems, in Advances in Soft Computing – AFSS, N. R. Pal and M. Sugeno, Eds., vol. 2275 of Lecture Notes in Computer Science, PP. 91-98, Springer, Berlin, Germany, 2002.
[23]
M. Sakawa, Fuzzy sets and interactive multi-objective optimization, Plenum Press, New York, NY, USA, 1993.
[24]
M. Zeleny, Multiple criteria decision making, McGraw-Hill, New York, 1982.
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