By Dr. J. Wilhelm (auth.)
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Extra resources for Objectives and Multi-Objective Decision Making Under Uncertainty
B< with ICP1 and xe:I. +< and hence (P 1 'P 2 ) cannot be a cut. of genetopothere we have 2) Suppose X to have no Dedekind-cuts. +<. If there is no smallest element in I. I) is a cut contrary to the assumption. Hence there is a smallest element in I and I is closed. a<. Hence all open unbounded intervals are closed sets. Accordingly. all sets consisting of one element are at the same time open and closed which proves the assertion. 32 By 1) and 2) the proposition is completely proved. 5. Let X be preference ordered and P be a subset of X.
Def. 1. 7 •• E 43 Usually this formulation is felt to be to pessimistic because it implies that for many problems there are no feasible actions at all. y)eA partially violating condition (1') or (2') (for some wen) makes the statement of the problem incomplete: there is no well -defined objective; for. ) bankruptcy has taken place by assumption (because of illiquidity; by assumption there are no additional activities available in t 1 ). e 2 will be no longer defined. a 2 has immediately to be paid.
Two facts show that nothing fundamental has been changed compared with the initial statement of the problem: on the one hand, the same problems arise when Zv=0 holds, problems of evaluating the case of illiquidity; on the other hand, this evaluation must probably be carried out in t2 too: it may be thought the decision maker is not sufficiently informed of final states in t2 to determine the associated objective-achievement; here also bottlenecks of liquidity may occur, the decision maker cannot say a priori whether or not they can be compensated.