Fuzzy aggregation and averaging for group decision making: A generalization and survey

Abstract

This article presents some systematic sorting and ordering of approaches dealing with fuzzy aggregation and fuzzy averaging from different authors. The aggregation of fuzzy information from a group of experts for developing collective opinion or verdict is the important question in the expert systems theory and practice. This is to obtain a more comprehensive and realistic solution to the given decision problem. This note tries to outline an overall formal umbrella to various methods to aggregate several fuzzy sets, which describe the individual points of view of experts, or results of judgements from the various characteristics.

Introduction

Many authors use different approaches when dealing with fuzzy aggregation and fuzzy averaging. This letter tries to bring some systematic sorting and ordering of approaches from different authors.

When the given decision making process is ill-structured, vague, and exhibits multiple decision aspects or views, the need for human expertise is one effective solution to such kind of problems. Ill-structuredness and vagueness require the reliance on human experts’ thinking, intuition and control, and the multi-aspect decision making, in this situation, needs multiple corresponding expertise in order to obtain a comprehensive or complete solution to the decision problem. Studies conducted in [29] suggest that it may be preferable to use the information provided by several experts as a more robust classification scheme. An appealing reason for using the opinions of several experts when solving a problem is that a group approach may produce better solutions to complex problems. There are many reported evidences about the importance of integrating several experts or expert systems. Also, the literature showed that a group problem solving (GPS) approach, where a problem is decomposed into smaller sub-problems, improves a decision maker’s performance in certain problem domains [18]. This article deals with the group decision making (GDM) situations in which several experts or decision makers express their opinions or preferences in linguistic terms or fuzzy sets. Since the aggregation operators are a necessary mechanism in order to pool the opinions of several experts, the main focus of this article is on the generalization of the methods of how to aggregate several fuzzy sets, describing the individual judgements of experts.

Fuzzy-set theory applied to decision making allows a more flexible framework, where it is possible to stimulate human ability to deal with the fuzziness of human judgments quantitatively and therefore to incorporate more human consistency or “Human Intelligence” in decision models. Different fuzzy decision making models have been proposed. A classification for all of them, depending on the number of stages before the decision is reached, can be found in [12]. The interest of this article is about the fuzzy multi-person decision making model applied in group decision theory, which is one fuzzy model of a one-stage decision making.

In group preference aggregation a synthesizing mechanism is used to derive a collective decision, representative in some way to the individual opinions. Because the aggregation of individual judgments or preference constitutes a major success factor in integrating the knowledge and expertise of multiple decision makers or experts, the aggregation of preferences has been widely studied by researchers. The aggregation of preference rankings has wide applications in group decision making, social choice, committee election and voting systems. How to aggregate individual preferences or a set of ordinal rankings into a group preference or consensus ranking is a typical group decision making problem. A large amount of research has already been conducted in this area [1], [2], [3], [17], [28], [29].

In a fuzzy environment a group decision problem is taken out as follows. It is assumed that there exists a finite number of alternatives, as well as a finite set of experts. Each expert may have a vague knowledge about the performance of each alternative, and cannot estimate his/her preferences with an exact numerical value. Hence, a more realistic approach is to use linguistic assessments and natural language of the human expert, instead of using exact numerical values. Thus, each variable involved in the decision problem will be assessed by means of linguistic terms [26], [31], [7]. The aggregation and synthesizing methods are the key mechanisms to realize the comprehensive feature of GDM.

Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined in some way to produce a single representative either fuzzy or crisp set. Several related research attempts have considered this issue. Czogala and Zimmermann [5] dealt with some important classes of aggregation operations on various kinds of sets applied to decision making problems. These operations are mainly based on general concepts such as triangular norms (t- and s-norms). Cholewa [6] considered the problem of aggregation of fuzzy opinions obtained from a group of experts. In 1985, Dubois and Prade [9] presented an extensive survey on fuzzy sets theoretic aggregation operations useful in decision analysis, quantitative psychology and information processing, and emphasized the relevance of the theory of functional equations in the axiomatic construction of classes of such operations and the derivation of functional representations.

A landmark research was conducted in 1988, in which Yager [30] presented a new type of operator for aggregation of multi-criteria, called an ordered weighted average (OWA). He investigated the properties of this operator, and pointed out that this operator has the property of lying between the “and”, requiring satisfaction of all criteria, and the “or”, requiring satisfaction of at least one of the criteria. In 1991, Dubois and Koning [10] viewed the problem of aggregating n fuzzy sets F₁, F₂, … , F_n on a set ω as one of merging the opinions of n individuals (e.g. experts) that rate objects belonging to ω. They expressed various natural properties of a voting procedure. Leclerc [15] dealt with the problem of aggregation of fuzzy preferences.

Krishnapuram and Lee [14] introduced a new methodology based on fuzzy-set-theoretic connectives to achieve information fusion in computer vision systems, and utilizing the neural network scheme. Ralescu and Ralescu [23] discussed the equivalence between aggregation of fuzzy sets and integration with respect to a special class of non-additive set functions. Yager and Rybalov [32] introduced a number of properties associated with the aggregation of scores. Notable among these is the property of self-identity. They showed how the requirement of self-identity imposes a useful restriction on the weights associated with this aggregation. Moon and Kang [19] proposed a technique which utilizes fuzzy-set theory in the aggregation of expert judgments. They employed two main key concepts: linguistic variables and fuzzy numbers.

In 2000, Beliakov [4] considered the issue of giving the set of aggregation operators, the extension operators to the original max and min of fuzzy set, a meaningful and simple interpretation. Sasikala and Petrou [25] utilized the non-conventional aggregation operators of the fuzzy sets theory for conjunctive and disjunctive reasoning in dealing with the problem of assessing the risk of desertification after a forest fire. Ribeiro and Pereira [24], in the context of multiple attributes decision making, presented an aggregation scheme based on generalized mixture operators using weighting functions, and compared it with two standard aggregation methods: weighted averaging and ordered weighted averaging.

Extending the Yager’s research work, Llamazares [16] used the ordered weighted averaging OWA operators in order to aggregate individual preferences represented by fuzzy preferences through values located between 0 and 1. They generalized simple and absolute special majorities by means of OWA operators. Dubois and Prade in 2004 [11] discussed the role of the existing body of fuzzy sets aggregation operations in various kinds of problems where the process of fusion of items coming from several sources is central. Pasi and Yager [22] studied the issue of defining a decision strategy which takes into account the individual opinions of the decision makers in group decision making. They expressed the concept of majority using fuzzy-set theory by a linguistic quantifier (such as most), which is formally defined as a fuzzy subset. Dimova et al. [8] analyzed the problems of ranked local criteria aggregation in MCDM and presented some new theoretical results which can be useful for a proper choice of aggregation method. They proposed a new method for generalization of aggregation schemes based on level-2 fuzzy sets mathematical tools. In 2006, Ölçer et al. [21] employed an attribute based aggregation technique for homogeneous and heterogeneous groups of experts, and used it for dealing with fuzzy opinions aggregation. In 2006, Tsabadze [27] proposed a method for fuzzy aggregation based on group expert evaluations. The approach was based on the coordination index and similarity of the finite collections of fuzzy sets.

The above review of past research attempts reveals the wide spread of the situations in which the needs for generalized aggregation operators occurs frequently, in order to cope with and fuse the multi-aspect, multi-source information. Most researchers have considered developing different new aggregation operators. One well-known and widely utilized aggregation operator is the OWA. Some of these researchers have concentrated on generalizing binary operators utilized in the theory of fuzzy sets, like t-norms and t-co-norms operators, into continuous aggregation operators on the interval 〈0,1〉. Little researches, and may be no researches, have considered the generalization of such aggregation operators and methods. This article is concerned with generalizing the fuzzy sets aggregation operators and their properties. This is in order to allow for wider application of such operators in experts group decision making situations.

The next section presents a generalization of aggregation operators of linguistic terms or fuzzy sets. It follows the consideration in [13], [20].