A new method for solving fuzzy multi-objective linear programming problems

Document Type: Research Paper

Authors

Jiangxi University of Finance and Economics

Abstract

The purpose of this paper is to develop a new two-stage method for fuzzy multi-objective linear program and apply to engineering project portfolio selection. In the fuzzy multi-objective linear program, all the objective coefficients, technological coefficients and resources are trapezoidal fuzzy numbers (TrFNs). An order relationship for TrFNs is introduced by using the interval expectation of TrFNs. In the first stage, the fuzzy multi-objective linear program with TrFNs is transformed into an interval multi-objective linear program according to the order relationship of TrFNs. Combining the ranking order relation between intervals with the satisfactory crisp equivalent forms of interval inequality relations, the interval multi-objective linear program is further transformed into a crisp multi-objective linear program. In the second stage, the positive and negative ideal solutions are calculated as well as the closeness degrees from the positive ideal solution to all objectives on the basis of the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution). Then, using the closeness degrees, we convert the crisp multi-objective linear program into mono-objective program to solve. The proposed method is not only mathematically rigorous, but also can adequately consider the acceptance degree of decision maker that the fuzzy constraints may be violated. The other possible cases of the fuzzy multi-objective linear program are also discussed. The proposed method is illustrated by means of a project portfolio selection problem.

Keywords


[1] M. H. Alavidoost, H. Babazadeh, S. T. Sayyari, An interactive fuzzy programming approach for bi-objective straight and
U-shaped assembly line balancing problem, Applied Soft Computing, 40 (2016), 221{235.
[2] M. Arenas, A. Bilbao, B. Prez, M. V. Rodrguez, Solving a multiobjective possibilistic problem through compromise program-
ming, European Journal of Operational Research, 164 (2005), 748{759.
[3] M. Arenas, A. Bilbao, M. V. Rodrguez, Solving the multiobjective possibilistic linear programming problem, European Journal
of Operational Research, 117 (1999), 175{182.
[4] M. Arenas, A. Bilbao, M. V. Rodrguez Ura, Solution of a possibilistic multiobjective linear programming problem, European
Journal of Operational Research, 119 (1999), 338{344.
[5] J. M. Cadenas, J. L. Verdegay, Using ranking functions in multiobjective fuzzy linear programming, Fuzzy Sets and Systems,
111 (2000), 47{53.
[6] N. B. Chang, Y. L. Chen, C.G. Wen, A fuzzy multi-objective programming approach for optimal management of reservoir
watershed, European Journal of Operational Research, 99 (1997), 289{302.
[7] V. Chankong, Y. Y. Haimes, Multiobjective decision making: theory and methodology, New York: North-Holland, 1983.
[8] S. Dhamar, J.R. Rao, R. N. Tiwari, Fuzzy goal programming-an additive model, Fuzzy Sets and Systems, 24 (1987), 27{34.
[9] S. Dhouib, A. Kharrat, H. Chabchoub, Goal programming using multiple objective hybrid metaheuristic algorithm, Journal of
the Operational Research Society, 62 (2011), 677{689.
[10] K. F. Doerner, M. Gendreau, P. Greistorfer, W. J. Gutjahr, R. F. Hartl, M. Reimann, Meta heuristics Progress in Complex
Systems Optimization, Springer Science, New York, 2007.
[11] M. Dorigo, T. Sttzle, Ant Colony Optimization, Mass MIT Press, Cambridge, 2004.
[12] D. Dubey, A. Mehra, A bipolar approach in fuzzy multi-objective linear programming, Fuzzy Sets and Systems, 246 (2014),
127{141.
[13] D. Dubois, H. Prade, Fuzzy sets and systems: Theory and applications. Academic Press, New York, 1980.
[14] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems, 24 (1987), 279{300.
[15] Y. Gao, G. Q. Zhang, J. Ma, J. Lu, A λ-cut and goal-programming-based algorithm for fuzzy-linear multiple-objective bilevel
optimization, IEEE Transaction on Fuzzy Systems, 18 (2010), 1{13.
[16] P. Gupta, M. K. Mehlawat, A new possibilistic programming approach for solving fuzzy multiobjective assignment problem,
IEEE Transactions on Fuzzy Systems, 22 (2014), 16{34.
[17] E. L. Hannan, Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems, 6 (1981), 235{248,.
[18] F. Hassanzadeh, M. Collan, M. Modarres, A practical approach to R&D portfolio selection using the fuzzy pay-off method,
IEEE Transaction on Fuzzy Systems, 20 (2012), 615{622.
[19] F. Hassanzadeh, H. Nemati, M. H. Sun, Robust optimization for interactive multiobjective programming with imprecise
information applied to R&D project portfolio selection, European Journal of Operational Research, 238 (2014), 41{53.
[20] C. F. Hu, S. C. Fang, Set covering-based topsis method for sloving sup-T equation constrained multi-objective optimization
problems, Journal of Systems Science and Systems Engineering, 24(3) (2015), 258{275.
[21] C. Y. Hu, R. B. Kearfott, A. D. Korvin, V. Kreinovich, Knowledge processing with interval and soft computing, Springer
Verlag, London, 2008, pp.168-172.
[22] C. L. Hwang, K. Yoon, Multiple attributes decision making methods and applications, Springer: Berlin Heidelberg, 1981.

[23] M. Inuiguchi, M. Sakawa, Possible and necessary efficiency in possibilistic multiobjective linear programming problems and
possible efficiency test, Fuzzy Sets and Systems, 78 (1996), 231{241.
[24] H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function, European Journal of
Operation Research, 48 (1990), 219{225.
[25] K. Y. Lee, M.A. El-Sharkawi, Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems, John
Wiley & Sons, Inc., New Jersey, 2008.
[26] S. Y. Li, C. F. Hu, An interactive satisfying method based on alternative tolerance for multiple objective optimization with
fuzzy parameters, IEEE Transaction on Fuzzy Systems, 16 (2008), 1151{1160.
[27] D. F. Li, J. X. Nan, M. J. Zhang, Interval programming models for matrix games with interval payoffs, Optimization Methods
and Software, 27(1) (2012), 1{16.
[28] D. F. Li, S. P Wan, A fuzzy inhomogenous multiattribute group decision making approach to solve outsourcing provider
selection problems, Knowledge-Based Systems 67 (2014), 71{89.
[29] G. P. Liu, J.B. Yang, J. F. Whidborne, Multiobjective Optimisation and Control, Research Studies Press, Philadelphia, PA,
2001.
[30] D. G. Luenberger, Y. Yu Ye, Linear and Nonlinear Programming, third ed., Springer Science, New York, 2008.
[31] M. K. Luhandjula, Multiobjective programming problems with possibilistic coefficients, Fuzzy Sets and Systems, 21 (1987),
135{145.
[32] M. K. Luhandjula, M. J. Rangoaga, An approach for solving a fuzzy multiobjective programming problem, European Journal
of Operational Research, 232 (2014), 249{255.
[33] S. Rivaz, M. A. Yaghoobi, Minimax regret solution to multiobjective linear programming problems with interval objective
functions coefficients, Central European Journal of Operations Research, 21 (2013), 625{649.
[34] S. Rivaz, M. A. Yaghoobi, Weighted sum of maximum regrets in an interval MOLP problem, International Transactions in
Operational Research, 25(2018): 1659-1676.
[35] F. Ruiz, M. Luque, J. M. Cabello, A classifi cation of the weighting schemes in reference point procedures for multiobjective
programming, Journal of the Operational Research Society, 60 (2009) 544-553.
[36] R. Tavakkoli-Moghaddam, B. Javadi, F. Jolai, A. Ghodratnama, The use of a fuzzy multi-objective linear programming for
solving a multi-objective single-machine scheduling problem, Applied Soft Computing, 10(3)( 2010) 919-925.
[37] C. S. Tu, C. T. Chang, Using binary fuzzy goal programming and linear programming to resolve airport logistics center
expansion plan problems, Applied Soft Computing, 44 (2016), 222{237.
[38] S. P. Wan, D. F. Li, Atanassovs intuitionistic fuzzy programming method for heterogeneous multiattribute group decision
making with Atanassovs intuitionistic fuzzy truth degrees, IEEE Transaction on Fuzzy Systems, 22 (2014), 300{312.
[39] H. C. Wu, The KarushCKuhnCTucker optimality conditions in multiobjective programming problems with interval-valued
objective functions, European Journal of Operational Research, 196 (2009), 49{60.
[40] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{356.
[41] M. Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New York, 1982.
[42] M. Zhalechian, R. Tavakkoli-Moghaddam, Y. Rahimi, F. Jolai, An interactive possibilistic programming approach for a
multi-objective hub location problem: Economic and environmental design, Applied Soft Computing, 52 (2017), 699{713.
[43] H. J. Zimmermann, Fuzzy programming and linear programming with several objectives functions, Fuzzy Sets and Systems,
2 (1978), 45{55.