Fuzzy and stochastic programming
A comparison of fuzzy and stochastic programming is performed in a framework of fuzzy variables. A fuzzy variable X is defined as a function \(X: \Gamma\to R\) with \(\Gamma\) equipped with a weight \(\sigma\) such that \(\sigma (\emptyset)=0\), \(\sigma (\Gamma)=1\), \(\sigma\) (\(\cup_{\alpha}A_{\alpha})=\sup_{\alpha} \sigma (A_{\alpha})\) for any \(A_{\alpha}\) being an element of P(\(\Gamma)\). Then a membership function being a basic notion in fuzzy sets theory results from \(\sigma\) as follows: \[ \mu_ X(x)=\sigma (X^{-1}(\{x\}),\quad x\in R \] Discussed are problems of fuzzy mathematical programming of the following form, \[ \max \sigma \{f_ 0(x,\gamma)=0\},\quad s.t.\quad \sigma \{f_ i(x,\gamma)=0\}\geq \alpha_ i,\quad \alpha_ i\in [0,1],\quad i=1,2,...,m,\quad x\in R_{n+},\quad \gamma \in \Gamma \] with different forms of functions \(f_ i's\). These tasks are compared with an analogous stochastic programming problem, namely \[ \max P\{f_ 0(x,\omega)\leq 0\},\quad s.t.\quad P\{f_ i(x,\omega)\leq 0\}\geq p_ i,\quad i=1,2,...,m,\quad x\in R_{n+}. \] Their solutions are discussed in detail. Moreover, nonfuzzy analogies of the above fuzzy mathematical programs are derived.
scientific article; zbMATH DE number 4012319
scientific article