On stability in fuzzy linear programming problems
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.
scientific article; zbMATH DE number 4156236