Milan Mares

Weak arithmetics of fuzzy numbers

Abstract Computation with fuzzy numbers appears to be a perspective branch of fuzzy set theory namely regarding the data processing applications. A very compact survey of the actual state of art is given in Dubois and Prade (1988). This paper aims to offer, in a brief and concentrated form, a survey of some results completing the classical ones of Dubois and Prade. Most of its content summarizes the approach submitted and discussed by Mares (1994).

Linear coalitional games and their fuzzy extensions

The paper deals with fuzzifications of the coalitional game models in which the expected pay-offs of players and coalitions are known only vaguely. There exists a class of games, here we call them linear coalitional games, which can be modelled as games with, as well as without, side payments. In the deterministic case both types of models of a linear coalitional game lead to the same results. As the fuzzification of games with side-payments is based on other principles than the fuzzification of more general games without side-payments, the linear coalitional games offer situations in which both approaches to the fuzzification of coalitional game models can be compared and their differences can be clearly illustrated. In this paper the attention is focused to the analysis of concepts of superadditivity and core of coalitional games.

T -partitions of the real line generated by idempotent shapes

Abstract The idea of generating fuzzy numbers as equivalence classes of particular T -equivalences on the real line R is fully exploited. Scales (or generators) are used to define certain (pseudo-)metrics on R . By means of a shape (function), these (pseudo-)metrics are then transformed into binary fuzzy relations on R . Shapes leading to T -equivalences, and hence to a class of fuzzy numbers forming a T -partition of R , are completely characterized in the case of a continuous generator. This characterization problem is shown to be closely related to determining the idempotents w.r.t. the T -addition of fuzzy numbers.

Verbally generated fuzzy quantities and their aggregation

The processing of vague data recently becomes one of attractive topics in the fuzzy set theory and its applications. As the vagueness is usually represented by some verbal expressions, this branch of the fuzzy sets is frequently called “computing with words”. Seemingly, but only seemingly, it could be understood as computational processing of fuzzy numbers or fuzzy quantities in the already classical sense. Other authors understand the computing with words rather as a fuzzy logical discipline being near to fuzzy reasoning methods and other related branches. Both approaches are rational and fully acceptable but, in the matter of facts, none of them appears to be complete. Their parallel existence offers a conclusion that the fair approach to computing with words can consist in some kind of their combination. Computing with words has two faces — quantitative and qualitative one — and each of them would be somehow reflected. The fuzzy set theoretical model of verbal variables, their generating and processing suggested in this contribution and in some of the referred papers is intended to offer such combined view on the quantitative — qualitative dualism existing in the “computing with words” and to develop at least elementary methods for manipulation with such dualistic verbal data.

Calculation over Verbal Quantities

Some verbal expressions represent quantitative data of essentially vague character and manipulation with them is frequently of computative type. This computation with words can be represented by fuzzy quantities. Nevertheless, each verbal quantitative expression is a combination of two components — of a quantitative one which connects it with some numerical (or quantitative) value, and of a semantic one which reflects its logical (or qualitative) features. The paper continues authors’ previous works in mathematical modelling of these two components and discussing their properties and interpretations.

Generated Fuzzy Quantities and Their Orderings

This paper is a substantial extension of [8]. In the following paragraphs the concepts and results presented in [8] are developed and commented in more details.

Deterministic Coalition Games

The theory of coalition games represents a standard mathematical model of cooperation among agents whose aim is to increase their individual profits. The theory is deeply developed and includes numerous concepts and results. Here, we recall its elementary parts that will be fuzzified in the following chapters.

Fuzzy Core and Effectivity in Games Without Side-Payments

The core C of the deterministic coalition game without side-payments (I, W) is, due to (3.21), a set of imputations which are accessible for some coalition structure (in the superadditive case for the coalition of all players), and which cannot be protested by any coalition.

Fuzzy Core and Effective Coalitions

The concept of a core in a fuzzy extension of a coalition game with side-payments will be naturally based on fuzzy analogues of (3.5) and (3.6). A heuristic analysis of the investigated model leads to the conclusion that a core of a fuzzified game is to be fuzzified as well. The vagueness entering the game by means of uncertain expectations of coalitional profits is also reflected by the vagueness connected with outputs of the game including the concept of a core. In the deterministic game, the core C is a subset of R I. Hence, in the fuzzy extension it should also be a fuzzy subset of the same set, as shown, e.g., in [33].

Fuzzy Additivity and Related Topics

Simple concepts of superadditivity and subadditivity belong to the basic ones in the theory of coalition games. They evidently influence the readiness of players to cluster into effectively cooperating coalitions. Results regarding this topic are briefly presented in some of the referred papers as, e.g., in [32], where the most detailed analysis of the problem is introduced, and where the formal proofs of statements that are rather less formally presented below, are given.