Compatibility measures and consensus models for group decision making with intuitionistic multiplicative preference relations
Graphical abstract
Highlights
► Develop compatibility measures for intuitionistic multiplicative values. ► Develop compatibility measures for intuitionistic multiplicative preference relations. ► Establish consensus models for checking, reaching and improving the group consensus level.
Introduction
Due to the high complexity of socioeconomic environments, it is difficult and impracticable for a single decision maker to consider all important aspects in practical decision making problems. Therefore, group decision making (GDM), especially with preference information, has caught attention widely in the decision making field and many desirable results have been derived over the last few decades [1], [2], [3], [4], [5]. The preference information is certain, if the decision maker clears about which one is preferred to another absolutely; otherwise, the preference information is uncertain or fuzzy. Usually, in actual GDM problems, due to time pressure and lack of information or knowledge, the decision makers may not express their preferences precisely and have to refer to fuzzy preferences. Therefore, the fuzzy preferences are used commonly in GDM problems of education, engineering, social sciences and economics. In general, the fuzzy preferences can be modeled qualitatively or quantitatively. Qualitatively, it is represented by linguistic labels, such as “good, fair and poor” [6], [7], [8]; while quantitatively it is represented by the numerical values indicating preference intensities [9], [10] or degrees of preference [11], [12], each of which can be interpreted as the grade or strength of one object is preferred to another.
From dozens of researches on GDM, we have found that fuzzy preference relations [11] are useful tools to express the preferences about a set of alternatives or criteria. The fuzzy preference relations given by decision makers can be diversiform including interval fuzzy preference relations [13], [14], triangular fuzzy preference relations [15], intuitionistic fuzzy preference relations [16], [17], incomplete interval fuzzy preferences relations [18], incomplete triangular fuzzy preferences relations [18], etc. It is noted that each element in a fuzzy preference relation is expressed by using 0.1–0.9 scale or 0–1 scale which assumes that the grades between “extremely not preferred” and “extremely preferred” are distributed uniformly and symmetrically. But in some practical problems, the decision makers need to assess their preferences with the unsymmetrically grades [19], [20], and unbalanced distribution may appear due to the characteristics of the corresponding problems. The law of diminishing marginal utility in economics is a good example. To increase the same consumption, a company with bad performance yields more utility than that with good performance, in other words, the gap between the grades expressing good information should be bigger than the one between the grades expressing bad information. Saaty's 1–9 scale (see Table 1) is a useful tool to deal with such a situation and the multiplicative preference relations [21], represented by Saaty's 1–9 scale, have attracted much attention. There are lots of literature studying on multiplicative preferences [2], [21], [22], such as incomplete multiplicative preferences [23], [24], interval multiplicative preferences [25], incomplete interval multiplicative preferences [18], triangular fuzzy multiplicative preferences [15], incomplete triangular fuzzy multiplicative preferences [18], etc. Recently based on the concept of interval multiplicative preference relation, Xia and Xu [26] proposed a new type of preference relation called intuitionistic multiplicative preference relation and introduced its desirable properties. In addition, several aggregation operators have been introduced to aggregate intuitionistic multiplicative preference information. More importantly, we can find that the intuitionistic multiplicative preference relation is based on the unbalanced scale, i.e., the Saaty's 1–9 scale, which is more reasonable, comprehensive and practical in some problems.
Due to the powerfulness of preference relations to rank alternatives or criteria, researches on them have been paid attention to in the last decades, such as the methods to determine the weight vectors based on preference relations [2], [27], the verification or modification methods of consistency [28], [29], [30], [31], [32], [33], [34], and the incomplete preferences evaluation methods [3], [4], [18], [35], and so on. For all kinds of preference relations, compatibility is a very efficient tool used for measuring the consensus of opinions and plays an important role in GDM. The lack of acceptable compatibility can lead to unsatisfied or even incorrect results because there are unavoidable differences and even contradictions among the preference relations provided by decision makers in GDM. Chen et al. [36] proposed the compatibility degree of uncertain additive linguistic preference relations, Xu [13] introduced some compatibility measures for the intuitionistic preference values, the interval-valued intuitionistic preference values, the intuitionistic preference relations and the interval-valued intuitionistic preference relations. Moreover, Saaty [37] put forward the compatibility to judge the difference between two multiplicative preference relations. Similarly, it is vital but still a blank to discuss the compatibility of intuitionistic multiplicative preference relations in GDM.
This paper defines the compatibility to measure the intuitionistic multiplicative preference information and develops two consensus models with respect to the intuitionistic multiplicative preference relations aiming at actual GDM problems. The rest of the paper is arranged as follows: Section 2 introduces the basic knowledge regarding the intuitionistic multiplicative preference relation. Section 3 develops the compatibility measure between intuitionistic multiplicative preference values and the method to derive compatibility index between intuitionistic multiplicative preference relations. In Section 4, we introduce two consensus models to help decision maker check, reach and improve the consensus level of his/her subjective preferences. An illustrative example is discussed in Section 5. Concluding remarks are included in Section 6.
Section snippets
Preliminaries
In this section, we review some basic definitions and concepts, so as to understand the proposal of the compatibility in intuitionistic multiplicative environments.
Let be a fixed set, then a fuzzy set defined on X can be given as , where the value μC(xi) (0 ≤ μC(xi) ≤ 1) is the membership degree of the element xi in X. It is noted that only the membership degrees are concerned in a fuzzy set. To describe the uncertainty and vagueness in more detail, Atanossov [28]
Compatibility degree between a pair of IMVs
Compatibility is an efficient and important tool which can be used to measure the consensus of opinions within a group of decision makers. In this subsection, we define the compatibility degree of IMVs and investigate its properties, and some detailed analysis will be presented sequentially. Definition 3.1 Let and be two IMVs, then we call
a compatibility degree between α1 and α2. Theorem 3.1 The compatibility degree C(α1, α2) derived from Eq. (8) satisfies the
Consensus models for intuitionistic multiplicative preference relations
We have known that the compatibility of preference relations is used to measure the consensus of different opinions in GDM in the above sections. When preference relations of a group of decision makers are “similar”, there exists consensus among the decision makers. This section proposes two consensus models in GDM in order to modify inadequate compatibility among intuitionistic multiplicative preference relations.
Illustrative example
In this section, we will offer an example (adopted from [41]) to illustrate our procedure.
Suppose that there is a GDM problem involving the evaluation of four schools in a university. Four experts (whose weight vector is w = (1/4, 1/4, 1/4, 1/4)T) compare the four schools by using Saaty's 1–9 scale. The experts provide their intuitionistic multiplicative preference relations respectively, as listed below:
Concluding remarks
In this paper, we have focused on the GDM problems where uncertain preference information given by the decision makers is expressed as intuitionistic multiplicative preference relations. We have defined some compatibility measures for intuitionistic multiplicative values and intuitionistic multiplicative preference relations respectively, and studied their desirable properties in detail. Based on these compatibility measures, we have developed two consensus models with respect to intuitionistic
Acknowledgements
The authors are very grateful to the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China (Nos. 71071161 and 61273209).
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