# Decision Making Using Interval-Valued Aggregation

## Abstract

In this chapter we present the results connected with multicriteria (or similarly multiagent) decision making problems under uncertainty. The aspects of decision making form a wide branch of considerations both in fuzzy sets theory and its extensions (cf. [1, 2, 3]). We consider only some of the basic concepts for the case of interval-valued fuzzy relations. Here, there is examined preservation of transitivity properties by aggregation operators. Similar considerations may be performed for the remaining properties. We concentrate on transitivity as an exemplary property since this is one of the most important properties which may guarantee consistency of choices of decision makers. Namely, following crisp notion of transitivity we see that if *x* is preferred to *y* and *y* is preferred to *z*, mathematically *R*(*x*, *y*) and *R*(*y*, *z*), then *x* should be preferred to *z*, in mathematical notions it means that *R*(*x*, *z*) holds. This is intuitive and natural assumption. We provide the results connected with the notions of pos-*B*-transitivity, nec-*B*-transitivity and preservation of these properties by aggregation operators in decision making. We propose to apply the respective notion of transitivity and aggregation method, depending on the requirements of the given problem. Namely, if for a given problem we require that at least one element in the first interval is smaller or equal to at least one element in the second interval, then the notions related to \(\preceq _{\pi }\) would be suitable (pos-*B*-transitivity, pos-aggregation function). If for a given problem we require that each element in the first interval is smaller or equal to each element in the second interval, then the notions related to \(\preceq _{\nu }\) would be suitable (nec-*B*-transitivity, nec-aggregation function). We think that such approach may lead to the more meaningful results and better choice of the solution alternatives (however, please note that the mentioned classes of aggregation operators are not disjoint, cf. Corollary 2.2 and Theorem 2.9).

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