Elsevier

Information Fusion

Volume 17, May 2014, Pages 83-92
Information Fusion

Using consensus and distances between generalized multi-attribute linguistic assessments for group decision-making

https://doi.org/10.1016/j.inffus.2011.09.001Get rights and content

Abstract

This paper proposes a mathematical framework and methodology for group decision-making under multi-granular and multi-attribute linguistic assessments. It is based on distances between linguistic assessments and a degree of consensus. Distances in the space of qualitative assessments are defined from the geodesic distance in graph theory and the Minkowski distance. The degree of consensus is defined through the concept of entropy of a qualitatively-described system. Optimal assessments in terms of both proximity to all the expert opinions in the group and the degree of consensus are used to compare opinions and define a methodology to rank multi-attribute alternatives.

Introduction

This paper proposes a mathematical framework and a methodology for multi-attribute group decision-making under multi-granular linguistic assessments. Multiple attributive group decision-making seeks to apply methods to find a collective solution to a decision problem in situations where a group of experts express their preferences on multiple attributes [12], [13]. Finding the collective solution usually requires compiling, representing and fusing expert opinions.

In this context, there are several methods for compiling, representing, and fusing expert opinions on the significant features of a specific real-world unstructured problem [7], [22], [30], [35], and this is an essential issue in group decision and negotiation. These methods usually involve a consensus process or consensus measures to ensure the quality of the result, namely the convergence of expert opinions [1], [21], [22], [26], [36], [37]. A wide range of applications from managerial to medical and engineering areas has proven the suitability of these methods [3], [5], [8], [17], [27]. These processes are often guided by a moderator and involve several rounds. The moderator’s role is to ensure impartiality and avoid group polarization (i.e. the tendency of the group to converge on extreme solutions and inappropriate biases due to social pressure or unfounded trends).

Regarding the representation of the information supplied by the experts, expert assessments are frequently expressed using ordinal scales corresponding to linguistic labels [19], [20]. The importance of offering experts the choice of different scales of linguistic labels for different features, and different levels of precision within each feature [20], [22] has been demonstrated. Such an approach enables the experts to give their opinions in a flexible way, without being forced to deal with strict scales. In this paper, linguistic modelling is tackled by means of qualitative order-of-magnitude models. The introduction of qualitative reasoning, in particular order-of-magnitude models, makes it possible not only to deal with incomplete and ordinal judgments with different levels of precision, but also to handle the different degrees of strictness of the expert assessments without previous normalization.

With respect to the processes of information fusion, there are several methods for combining individual expert opinions into a global solution [1], [21], [24], [26], [9], [31], [35], [33]. Some of these methods use a weighted aggregation operator, other methods involve consensus processes to obtain a collective solution, and others are based on reference point methods.

In this paper we introduce a mathematical framework and a methodology into group decision-making using distances and consensus within linguistic information. The proposed distances, between multi-dimensional linguistic labels, are defined from the geodesic distance in graph theory and the Minkowski distance. A generalization of the degree of consensus introduced in [28] is presented.

The methodology presented provides a convenient tool for helping multi-attribute decision-making in management and engineering environments where decisions involve qualitative or linguistic values due to the lack of precise information or the specific nature of the problem.

Two characteristics of the presented methodology, which are the main advances of this paper with respect to previous works, can be highlighted.

Firstly, the proposed approach can be applied to multi-dimensional multi-granular linguistic information, i.e. linguistic labels belonging to different ordinal scales where all levels of precision are allowed, and without the need for a unification of this linguistic information through a linguistic hierarchy [23]. Some features can be described in a more refined granularity than others. The new methodology accommodates these differences.

Secondly, the definition of distances among experts (considering experts to be defined by their judgments) and a degree of consensus are presented and used to define optimal representative solutions. The distances, defined in multi-dimensional qualitative spaces, enable us to find optimal representative solutions in terms of proximity to all the expert opinions in the group, as well as estimating the difficulty of achieving consensus and measuring the differences between each expert opinion and the optimal representative solutions. Optimal representative solutions that give most information are used as the final consensus solutions to the decision-making problem under consideration.

This paper is structured as follows. In Section 2, multi-dimensional qualitative assessments involving different order-of-magnitude spaces are introduced and a definition of entropy in these spaces, which provides a measure for the information regarding multi-dimensional qualitative assessments, is presented. In Section 3, the extension of a degree of consensus in these spaces is provided. Section 4 introduces a distance into the set of qualitative assessments based on the geodesic distance from graph theory and the Minkowski distance. In Section 5, the concept of optimal representative assessment is defined. In Section 6, the results of the previous sections are used to propose a ranking method in group decision-making. A comparison with other methods is given in Section 7. Finally, in Section 8, the conclusions and some future lines of research are drawn.

Section snippets

Multi-dimensional qualitative assessments involving different order-of-magnitude spaces

This section extends the theory of consensus based on qualitative reasoning developed in [28]. The extension is made by allowing qualitative assessments over different order-of-magnitude qualitative spaces with different granularity Sn [32], and different measures μ over the basic labels of each space Sn. Recall that the set of basic labels is denoted by Sn={B1,,Bn}, which is totally ordered as a chain: B1 <  < Bn, and the complete description universe for the order-of-magnitude space Sn with

A degree of consensus for generalized qualitative assessments

This section introduces a degree of consensus for group decision-making under multi-granular and multi-attribute linguistic assessments. The proposed degree of consensus is based on the concept of the entropy of generalized qualitative assessments.

Distance between generalized qualitative assessments

This section is devoted to defining a distance in a space Λ,Sni,{μ}, similar to the family of distances defined over a set of basic labels Sni presented in [11].

Earlier work on distance measures in linguistic information was done in [34], which introduces a distance between linguistic labels based on deviation degree and similarity degree.

Our generalization presents three main advantages. Firstly, it enables experts to judge different alternatives over different order-of-magnitude spaces.

Optimal representative generalized qualitative assessments

The introduced distance between generalized qualitative assessments, defined without the necessity of consensus, enables us to measure how far an expert is from another – or from a group of experts. In this section, we present a mathematical framework and a methodology to find optimal generalized qualitative assessments that minimize the distance to a given set of expert generalized qualitative assessments.

Let (Q,dp) be the metric space of the generalized qualitative assessments of Λ,Sni,{μ},

Ranking multi-attribute alternatives

This section introduces a method for ranking a set of alternatives described by several features, such as the three firms in Example 4.

Let E={α1,,αM} be a committee or group of experts who must rank a set of alternatives A={c1,,cN} through multidimensional linguistic labels using a generalized description space Λ,Sni,{μ}. Let us denote by QEci the evaluations (generalized qualitative assessments) provided by the group for the alternative ciA,i=1,,N.

Definition 15

Let Λ,Sni,{μ} be a generalized

Comparison with other methods

This section outlines an analysis and comparison of the main features of other existing multi-attribute group decision-making approaches associated with consensus and the method presented.

There are numerous methods associated with consensus in the group decision-making literature. Group decision-making problems of ranking or choice involve a set of alternatives and a group of experts who provide their assessments of these alternatives. Experts assess alternatives either considering them

Conclusion and future research

This work presents a mathematical framework and a methodology to be considered in multi-attribute group decision-making problems under multi-granular linguistic assessments. The results obtained can be applied to tackle evaluation and ranking problems which require different sets of ordinal labels to qualify features.

Directions for future work include applying the methodology studied using real data and assessing the suitability of the entire process. The methodology is currently being

Acknowledgments

The Authors wish to express their gratitude to anonymous referees for their valuable comments and suggestions. This research has been partially supported by the SENSORIAL Research Project (TIN2010-20966-C02-01 and TIN2010-20966-C02-02), funded by the Spanish Ministry of Science and Information Technology, and the Barcelona Tech (UPC) CERMET Project.

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