Abstract
This paper introduces the concept of softarison. Softarisons merge soft set theory with the theory of binary relations. Their purpose is the comparison of alternatives in a parameterized environment. We develop the basic theory and interpretations of softarisons. Then, the normative idea of ‘optimal’ alternatives is discussed in this context. We argue that the meaning of ‘optimality’ can be adjusted to fit in with the structure of each problem. A sufficient condition for the existence of an optimal alternative for unrestricted sets of alternatives is proven. This result means a counterpart of Weierstrass extreme value theorem for softarisons; thus, it links soft topology with the act of choice. We also provide a decision-making procedure—the minimax algorithm—when the alternatives are compared through a softarison. A case-study in the context of group interviews illustrates both the application of softarisons as an evaluation tool, and the computation of minimax solutions.
Highlights
This article proposes a model that combines soft set theory with binary relations
Its argument has been exported to maximization of binary relations, the most basic result being the Bergstrom–Walker theorem [17, 40]: it asserts that any continuous acyclic relation defined on a compact topological space has maximal elements
We shall need the notion of soft compactness of a soft topology, that in turn uses the concept of cover of ðF; EÞ 2 SSEðXÞ: it is a collection W of soft sets over X such that ðF; EÞY t fðF0; EÞ : ðF0; EÞ 2 Wg
Summary
This article proposes a model that combines soft set theory with binary relations. The performance of binary relations is enhanced by the explanatory ability of soft set theory for vague intelligence. The parameters correspond to the relevant attributes of the situation This novel model appears to be natural and applicable since it uses the same primitive elements as standard soft set theory, namely, alternatives and their attributes, together with ordinary soft sets for this context. Its argument has been exported to maximization of binary relations, the most basic result being the Bergstrom–Walker theorem [17, 40]: it asserts that any continuous acyclic relation defined on a compact topological space has maximal elements. We adopt a procedural position in order to provide a decision-making algorithm for situations where a softarison captures the soft-theoretical comparisons between pairs of alternatives. This procedure relies on the idea of a minimax alternative. It points out at several lines for future research on this topic
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