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Fuzzy Sets and Systems
Article . 2000 . Peer-reviewed
License: Elsevier TDM
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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
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Using ranking functions in multiobjective fuzzy linear programming

Authors: José Manuel Cadenas; José L. Verdegay;

Using ranking functions in multiobjective fuzzy linear programming

Abstract

The following multiobjective linear programming problem with fuzzy coefficients in the objective functions is considered: \[ \min[c^f_1 x,c^f_2 x,\dots, c^f_k x],\quad\text{s.t. }Ax\leq b,\;x\geq 0,\tag{1} \] where \(A\) is an \(m\times n\)-matrix, \(b\) an \(m\)-vector, \(x\) an \(n\)-vector and each \(c^f_j\), \(j= 1,\dots, k\), an \(n\)-vector of fuzzy numbers. By applying the constraint approach the problem (1) is replaced with the following one: \[ \min c^f_1 x,\quad \text{s.t. }c^f_jx\leq_f \lambda^f_j,\quad j= 2,\dots, k,\quad Ax\leq b,\;x\geq 0,\tag{2} \] where \(\lambda^f_j\), \(j= 2,\dots, k\), are fuzzy numbers (fuzzy goals) assigned to the corresponding objective functions and \(\leq_f\) is a certain relation between fuzzy numbers which may be defined in different ways. The authors propose to use for comparison of fuzzy numbers so-called ranking functions. If \(I: FN(\mathbb{R})\to [0,1]\) is any ranking function \((FN(\mathbb{R}))\) is the set of all fuzzy numbers) then \[ \forall A^f, B^f\in FN(\mathbb{R}) A^f\leq_f B^f\overset{\text{def}}\Leftrightarrow I(A^f)\leq I(B^f). \] Using the above way of comparison of fuzzy numbers the authors transform the problem (2) to the following crisp mathematical programming problem: \[ \min I_1(c^f_1 x),\quad\text{s.t. }I_j(c^f_j x)\leq I_j(\lambda^f_j),\quad j= 1,2,\dots, k,\quad Ax\leq b,\;x\geq 0.\tag{3} \] Since, the authors assume that the objective functions in (1) may come from different decision makers, the ranking functions \(I_j\) applied in (3) may be different for different \(j, j= 1,\dots, k\). Then the authors derive the form of the problem (3) for two special situations when \(I_j\) is for each \(j\) equal to the comparison index of Yager or that proposed by Adamo. In each case, assuming that the all fuzzy parameters in (2) and (3) are triangular fuzzy numbers, the problem (3) is reduced to a linear programming. The full usefulness of the given approach is illustrated by a biobjective linear programming problem for land use, involving fuzzy coefficients in the objective functions.

Related Organizations
Keywords

ranking function, fuzzy coefficients, Linear programming, crisp mathematical programming problem, Fuzzy and other nonstochastic uncertainty mathematical programming, multiobjective linear programming, Multi-objective and goal programming

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
66
Top 10%
Top 10%
Average
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