On distribution function description of probabilistic sets and its application in decision making (Q791241)

According to \textit{K. Hirota} [ibid. 5, 31-36 (1981; Zbl 0442.60008)], the value of the membership function of a fuzzy set A, \(\mu_ A(x,\omega)\), is regarded as a random variable depending on the parameter x, i.e. it can be treated as a random process. So, the distribution function description of probabilistic sets is natural. The max and min functions and their distribution functions are presented as follows: If \(x_ 1,x_ 2,...,x_ n\) are probabilistic sets given by their distribution functions, then the distribution function of \(\max(x_ 1,x_ 2,...,x_ n)\) is given by \[ F_{\mu_{\max(x_ 1,...,x_ n)}(x^ 1\!_{i_ 1},x^ 2\!_{i_ 2},...,x^ n\!_{i_ n})}(\omega)= \] \[ F_{\mu_{x_ 1}(x^ 1\!_{i_ 1})\mu_{x_ 2}(x^ 2\!_{i_ 2})...\mu_{x_ n}(x^ n\!_{i_ n})}(\omega,...,\omega),\quad \omega \in \Omega_ c, \] and the distribution function of \(\min(x_ 1,x_ 2,...,x_ n)\) is given by \[ F_{\mu_{\min(x_ 1,...,x_ n)}(x^ 1\!_{i_ 1},...,x^ n\!_{i_ n})}(\omega)= \] \[ \sum^{n}_{j=1}F_{\mu_{x_ j}(x^ j\!_{i_ j})}(\omega)-\sum_{i\leq j<k\leq n}F_{\mu_{x_ j}(x^ j\!_{i_ j})\mu_{x_ k}(x^ k\!_{i_ k})}(\omega,\omega)+... \] \[ +(-1)^{n+1}F_{\mu_{x_ 1}(x^ 1\!_{i_ 1})\mu_{x_ 2}(x^ 2\!_{i_ 2})...\mu_{x_ n}(x^ n\!_{i_ n})}(\omega,\omega,...,\omega),\quad \omega \in \Omega_ c. \] The decision making problem is formulated by using the concept of probabilistic set and its distribution function description. Numerical examples are given to illustrate the technique.