On uncertainty problems in decision-making
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ON UNCERTAINTY PROBLEMS IN DECISION-MAKING V. I. Ivanenkoa and V. M. Mikhalevichb
UDC 519.71:330.42
The paper continues studying the uncertainty problem based on [2]. A decision-maker is assumed to have a prevalence relation on the consequences of his actions. The necessary and sufficient existence conditions for uncertainty in a decision-making problem for a nonparametric situation are given. Keywords: uncertainty, decision-making, matrix scheme, lottery scheme.
Decision-making problems (DMPs) are mainly considered as optimization ones, i.e., as problems of choosing optimal decisions. The set of such problems can be divided into two subclasses: DMPs without uncertainty (so-called deterministic problems) and DMPs with uncertainty. For such a classification, a criterion of the existence of uncertainty in a DMP is necessary. Let us specify some concepts relative to such a criterion. We say that a DMP arises in a decision-making system (DMS), which is formed always when a decision-maker (DM) appears in a decision situation (DS). In other words, a DM is in a situation that requires him to choose one decision (an alternative) d from a set of possible decisions D. A decision situation may belong to one of two classes: parametric and nonparametric situations. In the former case, there is a parameter w from a set W (in what follows, we will call it a space of unknown parameter, or briefly, a parametric space). If such a parameter is absent, the DS is called nonparametric. Let us assume that, in the general case, any decision d Î D in a DMS causes only one c ÎC d from the set of possible consequences C, where C d is a subset of the set of all consequences possible in this DMS, i.e., C = U C d . d ÎD
We also assume that the decision-maker has a personal preference relation (PR) { ³ } on the set of consequences C, and he has to construct a PR {Í} on the set of decisions D, which he will use to try to determine the best (optimal) decision. For simplicity, we will consider only the class of linear orderings (the set of consequences C is factored relative to equivalence, i.e., c1 = c 2 Û c1 coincides with c 2 ). Definition 1. We say that a decision d1 dominates over d 2 relative to (C, ³ ) if C d 1 > C d 2 , i.e., c1 ³ c 2 "c1 ÎC d 1 , "c 2 ÎC d 2 , Card (C d 1 Ç C d 2 ) £ 1, C d 1 ¹ C d 2 . Then we will name the procedure of forming a preference relation on a given set of decisions D under the conditions (i) C d 1 > C d 2 Þ d1 f d 2 "d1 , d 2 Î D; (ii) C d 1 = C d 2 , Card C d 1 = 1 Þ d1 » d 2 "d1 , d 2 Î D projecting. Definition 2. By a decision-making problem (briefly, decision problem (DP)) we will mean projecting a preference relation (C, ³ ) onto decisions D when the same preference on consequences can be projected into several preference relations on decisions, i.e., when the projection is ambiguous. Let us specify these concepts by using a DS model for a so-called complete uncertainty. Let us call such a model a DS scheme [2].
a
National Technical University “Kyiv Polytechnical Institute,” Kyiv, Ukraine, vivan@zeos
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