Constrained Markov decision processes with compact state and action spaces are studied under long-run average reward or cost criteria. And introducing a corresponding Lagrange function, a saddle-point theorem is given, by which the existence of a constrained optimal pair of initial state distribution and policy is shown. Also, under the hypothesis of Doeblin, the functional characterization of a constrained optimal policy is obtained.
AbstTuct Coneerning with the topics of a fuzzy max order, a briefsurvey on orderi-g of fuzzy numbers is presented im this article, and we wil] consider an extensien to that of fuzzy sets. An extension of the fuzzy max order as a pseudo order is investigated and defined on a class of fuzzy sets on R" (n) 1). This order is developed by using a non-empLy closed convex cone and characterized by the projection into its dual cone. Especially a structure of the lattice can be illustrated with the class of rectang]e-type fuzzy sets.
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