Approximate resolution of an imprecise goal programming model with nonlinear membership functions

Jiménez, Mariano; Arenas, Mar; Bilbao, Amelia; Uría, María Victoria Rodríguez

doi:10.1016/j.fss.2004.05.001

Cited by 10 publications

(4 citation statements)

References 23 publications

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“…. .,N,+ I N+1 (z) = Yocj Iz-gjz+0.5 (5) j=0 Even in the case in which the membership function is linear (see Fig. 3), we propose to consider it as a piecewise linear function, then according to (5) we write: U(z) = -Iz-gl+ I z-(g+t) +0.5 (6) 2t 2t…”

Section: Resultsmentioning

confidence: 99%

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On unfeasible fuzzy goal programming problems

Jiménez

2007

2007 IEEE International Fuzzy Systems Conference

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In a Fuzzy goal programming problem (FGP), it is possible that the tolerance thresholds fixed by the Decision Maker (DM) for the different fuzzy goals were to be incompatible, therefore obtaining an unfeasible problem. In this case the DM doesn't know the amount in which the tolerance thresholds have been surpassed and, so, neither which changes should be introduced in his aspiration levels to obtain an acceptable solution. To avoid this drawback, we propose to consider the fuzzy goal membership functions as piecewise defined, including all the values that surpass the tolerance threshold, which logically have a satisfaction degree equal to 0. This way a problem that before was unfeasible becomes feasible, although, logically, the optimal solution will have an aggregate degree of satisfaction equal to 0. This solution can be used as a reference by the DM to decide how much the aspiration levels should be changed in order to obtain a satisfying solution.

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Section: Resultsmentioning

confidence: 99%

“…However the rate of increase (or decrease) of the satisfaction degree must not always be constant, and then fuzzy goal membership functions are nonlinear. Jimenez et al [5] showed how any nonlinear membership function can be approximated by a piecewise linear function. Inuiguchi Fig.…”

Section: Analytical Expression Of a Piecewise Linear Membership Fmentioning

confidence: 99%

On unfeasible fuzzy goal programming problems

Jiménez

2007

2007 IEEE International Fuzzy Systems Conference

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“…One of the most common assumptions in theory (Zimmermann, ; Narasimhan, ; Hannan, ; Aouni et al., ) and applications (Rommelfanger, ; Arenas‐Parra et al., ; Mekidiche et al., ; Aouni et al., ) is that this drop can be represented by a linear function. As Verdegay () states: “It was shown that possible further changes of those membership functions do not affect the former optimal solution, … This sensitivity analysis … shows the convenience of using linear functions instead of other more complicated ones.” Therefore, for the sake of simplicity and due to the relevance of the linear case in the literature (see Jiménez et al., ; Chang, ; Huang, , among others), we focus our approach on this type of membership functions. But, in any case, we want to point out that, as it is shown in Appendix B, the proposed method can be applied, in an easy way, to nonlinear membership functions.…”

Section: A Review Of Fuzzy Gpmentioning

confidence: 99%

“…shows the convenience of using linear functions instead of other more complicated ones." Therefore, for the sake of simplicity and due to the relevance of the linear case in the literature (see Jiménez et al, 2005;Chang, 2010;Huang, 2014, among others), we focus our approach on this type of membership functions. But, in any case, we want to point out that, as it is shown in Appendix B, the proposed method can be applied, in an easy way, to nonlinear membership functions.…”

Section: Membership Functionmentioning

confidence: 99%

A model for solving incompatible fuzzy goal programming: an application to portfolio selection

Jiménez

Bilbao-Terol

Arenas-Parra

2017

Int Trans Operational Res

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For many fuzzy goal programming (GP) approaches, in order to build the membership functions of fuzzy aspiration levels, a tolerance threshold for each one of them should be determined. In this paper, we address the case in which the decision maker proposes incompatible thresholds, which could lead to an infeasible problem. We propose an alternative algebraic formulation of the membership functions, which allows us to formulate models capable of providing solutions, although some tolerance thresholds are surpassed. The objective values that do not violate their corresponding threshold are evaluated positively according to the degree of achievement to their fuzzy target, and in turn those who violate the threshold are penalized according to their unwanted deviation with respect to the threshold. Thus, our model jointly uses the fuzzy GP approach and the standard GP approach, which also allows incorporating fuzzy and crisp targets into the same problem. The proposed procedure is applied to socially responsible portfolio selection problems.

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