On stability in fuzzy linear programming problems
type: Item
instance of: scholarly article
title: On stability in fuzzy linear programming problems
zbMATH Open document ID: 0704.90101
DOI: 10.1016/0165-0114(89)90026-2
author: Elsie Sterbin Gottlieb
published in: Fuzzy Sets and Systems
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://citeseerx.ist.psu.edu/viewdoc/summary
review text: This paper proves a certain stability property of the optimal value of a fuzzy linear programming problem (FLPP) whose coefficients are symmetrical triangular fuzzy numbers. An FLPP is formulated as: \((a_{i1},\alpha)x_ 1+...+(a_{in},\alpha)x_ n\lesssim (b_ i,\beta_ i)\), \(i=1,...,m\), where \((a_{ij},\alpha)\) denotes a symmetrical triangular fuzzy number with centers \(a_{ij}\in R\) and width \(\alpha\). The operations and relations \(\lesssim\) have been defined on a fuzzy set. The degree of satisfaction of the i-th restriction at \(x\in R^ n\) is defined and it is denoted by \(\mu_ i(x)\). The solution of FLPP is a fuzzy set with membership function \(\mu (x)=\min \mu_ i(x)\). An optimal solution \(x'\) satisfies the equation \[ \mu (x')=\mu '=\max_{x\in R^ n}\mu (x). \] The author defines a problem P(\(\delta\)) in which only approximations \(a_{ij}(\delta)\), \(b_ i(\delta)\) of the exact centers are known. P(\(\delta\)), \(\mu^{\delta}\) and \(\mu^{'\delta}\) are defined. The main result of the paper is that \[ \| \mu -\mu \delta \| =\sup | \mu (x)-\mu^{\delta}(x)| \leq \delta [\frac{1}{\alpha}+\frac{1}{\min_{0\leq i\leq m}\beta_ i}]. \] It is recommended that the solution of FLPP with \(\alpha >0\) and \(\beta_ i>0\) may be instable with respect to changes of \(a_{ij}\) and \(b_ i\). A concluding remark states that the stability or the instability of the maximizing solution is an open problem.
Mathematics Subject Classification ID: 90C70
Mathematics Subject Classification ID: 90C05
Mathematics Subject Classification ID: 90C31
zbMATH DE Number: 4156236
zbMATH Keywords: stability
zbMATH Keywords: optimal value
zbMATH Keywords: fuzzy linear programming
zbMATH Keywords: symmetrical triangular fuzzy number
reviewed by: S. M. Allende-Alonso
MaRDI profile type: Publication
OpenAlex ID: W2071297453
cites work: Systems of linear fuzzy constraints
cites work: Sensitivity analysis in fuzzy linear programming
cites work: A value of information in FLP problems via sensitivity analysis
cites work: DESCRIPTION AND OPTIMIZATION OF FUZZY SYSTEMS
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description: scientific article; zbMATH DE number 4156236