Connections between Campos-Bolanos and Murofushi–Sugeno Representations of a Fuzzy Measure

Abstract

Nonadditivity of a fuzzy measure, as an indicator of defectiveness, makes a fuzzy mea-sure less useful in applications compared to additive, probabilistic measures. In order to neutralize this indicator of defectiveness to some degree, it is important to study the representations of fuzzy measures, including, in particular, additive, probabilistic representations. In this paper, we discuss a couple of probability representations of a fuzzy measure: the Campos-Bolanos representation (CBR) and the Murofushi–Sugeno representation (MSR). The CBR is mainly represented by the Associated Probability Class (APC). The APC is well studied and the aspects of its use can be found in many interesting studies. This is especially true for the environment of interactive attributes in their identification and multi-attribute group decision-making (MAGDM) models, related to the attributes’ Shapley values and interaction indexes. The MSR is a less-used tool in practice today. The main motivation of the research presented here was to explore the connections between these two representations, which will help increase the usability of the MSR in practice in the future. In the MSR, we constructed the nonequivalent representation class (NERC) of a fuzzy measure. This probabilistic new representation is somewhat similar to the APC in the CBR environment. The proposition on the existence of the MSR induced by the CBR was proven. The presented formula of the APC by the NERC was obtained. The duality property of fuzzy measures for the CBR is well studied with respect to fuzzy measures—Choquet second-order dual capacities. Significant properties were proven for the representation of a monotone expectation (ME) under the NERC conditions: as is known, the necessary and sufficient conditions for the existence of the second-order Choquet dual capacities are proven in the terms of the APC of a CBR and ME. After establishing the links between the APC of a CBR and the NERC of a MSR, we proved the same in the case of the MSR. A recursive connection formula between the interaction indexes, Shapley values, and the probability distribution of the NERC of a two-order additive fuzzy measure was obtained in the environment of a general MAGDM. A new distance concept was introduced for all fuzzy measures’ classes defined in finite sets in terms of the NERC. The distance between two fuzzy measures was defined as the distance between their NERCs. This distance is equivalent to the distance defined on the same class under the conditions of the APC of a CBR. The correctness proposition on the extension of the distance between fuzzy measures for the NERC was preserved: distances between any two fuzzy measures and between their dual fuzzy measures also coincided in the CBR as the MSR. After parameterization, the calculation formula of the new distance was obtained. An illustrative example was considered in order to easily present the obtained results. The connection schemes between the CBR and MSR and the sequential scheme of key facts and results obtained are presented at the end of this work.

Associated Probabilities in Insufficient Expert Data Analysis

“…Therefore, the two-pole characteristics of the incompleteness of expert data, fuzzy uncertainty and imprecision, fall into a certain conflicting resistance [10,[13][14][15]. Using fuzzy modeling approaches, the authors in their research [4,[11][12][13][14][16][17][18][19][20][21][22][23][24][25][26] try to reduce the impact of the uncertainty index on decision-making, but instead, the imprecision characteristics-the levels of fuzzy assignment of the fuzzy set increase. Our research is concerned with the information analysis of this complex uncertainty of IED and its use in modeling more precise decisions with minimal decision risks from the point of view of systems research [2][3][4][10][11][12]16,[27][28][29].…”

“…Only the probabilistic-fuzzy analysis guarantees more or less adequate results. When we assume the expert's evaluation of some decision-making approaches, the problems of modeling uncertainty and imprecision must be solved under the data information of the MGDM or multi-attribute group decisionmaking (MAGDM) [3,11,12,16,17,22,23,[29][30][31]. One way to solve this problem is presented in this article.…”

“…MAGDM modeling, which takes into account the interaction or dependence of possible alternatives, becomes particularly relevant. At the same time, such studies are more and more in demand, which consider expert evaluations of a bipolar nature, in the form of fuzzy uncertainty and imprecision [11,12,22,23]. In such MAGDM modeling of the experts' data aggregations, the authors largely use non-additive aggregation tools [10,36,37].…”