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Parametric orders on fuzzy numbers and their roles in fuzzy optimization problems. (English) Zbl 0898.90134
The paper introduces and studies two types of parametric order relations for the class of symmetric fuzzy numbers defined as $A\left(x\right)=max\left(0,L\left(\left(x-m\right)/\alpha \right)\right)$. Here $m$ being a modal value of $A$ and $\alpha$ forming a spread of this fuzzy number; furthermore (i) $L\left(x\right)=L\left(-x\right)$, (ii) $L\left(x\right)=1$ iff $x=0$, (iii) $L\left(x\right)$ is nonincreasing on $\left[0,\infty \right)$. It is shown that the proposed order relations introduce a parametric total order. A number of additional properties are also discussed. An application of these order relations is given to the fuzzy shortest route problem (giving rise to a generalized Dijkstra algorithm).
##### MSC:
 90C70 Fuzzy programming