A web based consensus support system for group decision making problems and incomplete preferences
Introduction
In group decision making (GDM) problems there are two processes to carry out before obtaining a final solution [21], [24], [31], [35]: the selection process and the consensus process. The former [26], [48] refers to how to obtain a solution set of alternatives from the opinions on the alternatives given by the experts, while the latter deals with the achievement of the maximum degree of consensus or agreement between the set of experts on the solution set of alternatives. Usually, this process is guided by the figure of a moderator [17], [24], [34], [35] and it is carried out before the selection process. Clearly, the consensus process is an important step in solving GDM problems because it aids to obtain solutions with high level of consensus among experts, which is usually a desirable property.
Good GDM processes require, before their application, the specification of several aspects about the problem to solve as well as the methodology to follow. This includes, but is not restricted to, the definition of the representation format(s) [45] available for the experts to express their preferences about the possible alternatives in the problem: linguistic [2], [19], [53] or fuzzy [32], [33], [51], [62], [63] formats. Many GDM approaches assume an homogeneous representation of the preference information provided by experts and therefore are based on the availability of one single representation format. However, it may well be the case that experts might feel more comfortable if different representation formats are available to express their preferences [8], [31]. Thus, the use of multiple representation formats has become a major area of research in GDM.
In [8] an approach with three different preference representation formats was proposed. In that approach, preference orderings and utility values are transformed into fuzzy preference relations in order to be able to operate with them. In [9], another preference representation format (multiplicative preference relations) was incorporated to the previous model to enhance it. Additionally, in [31] a consensus model for GDM problems with these four representation formats was presented. Fan et al. [18] proposed a goal programming approach where the preference information on alternatives provided by decision makers is represented in two different formats, multiplicative preference relations and fuzzy preference relations. In [42] an approach that deals with preference information represented in four different formats was also presented. These two approaches differ from the previous ones in that the ranking of alternatives or selection of the most desirable alternative(s) is obtained in a direct way, i.e. no unify process or aggregation of individual preferences are required. In [28] a model that tackles GDM situations with information expressed using 2-tuple linguistic values and interval valued preferences is presented. In [58] an interactive method for multiple attribute GDM dealing with exact numerical values and triangular fuzzy numbers was developed. Finally, in [64] some experimental results that validate the necessity of using multiple preference formats in decision making were presented.
A different issue that needs attention when dealing with real GDM problems is the lack of complete information [44], [55]. There might be situations where some of the experts might not not be able to efficiently express any kind of preference degree between two or more of the available options. Indeed, this may be due to an expert not possessing a precise or sufficient level of knowledge of part of the problem to be solved, or because that expert is unable to discriminate the degree to which some options are better than others. In such situations experts may prefer not to guess some values and thus, not to give part of the required information, that is, they would provide incomplete information [3], [16], [30], [37], [38], [56], [57], [59], [54].
Current decision support models and systems incorporate mechanisms to maximize the consensus in the decision process [5], [6], [4], [17], [25], [24], [34], [35], [36], [65]. For example, in [14] a consensus driven model is used in the selection of advanced technology field; in [22] some consensus reaching ideas are included as part of a decision support system for water resource management; while in [29] the authors developed a consensus model for GDM and incomplete information.
The aim of this paper is to present a new web based consensus support system (WBCSS) to deal with GDM problems under incomplete information situations and with experts’ preferences represented with different representation formats: fuzzy preference relations, linguistic preference relations and multi-granular linguistic preference relations. This consensus support system is based on the use of several consensus and consistency measures which are interactively computed when the experts provide their preferences. One of the main novelties in this contribution is the use of uninorms [40] to define the consistency measures as well as to tackle missing information. The consensus support system uses both kinds of measures to offer advice to the experts by means of easy to follow rules, thus providing a feedback mechanism to help experts to change their preferences in order to obtain solutions with a high level of consensus. The system also aims to help experts to maintain a high consistency level in their preferences to avoid self contradiction and, when that is the case, to reduce as much as possible incomplete information situations. This system is designed to help the moderator to carry out his duties during the different steps of the consensus process, but it could takeover the role of the moderator once the initialization steps have been completed. The system has been fully implemented and the experts can use it via a web interface which allows to carry out consensus processes in distributed environments. This means that the usual imperative condition of experts to physically meet together is eliminated and therefore the decision making process for group of experts, living for example in different countries, is facilitated.
The rest of this paper is organized as follows. Section 2 presents the theoretical model on which the WBCSS is based. In Section 3 the WBCSS for GDM problems with different kinds of preference relations is presented, and some of the technical details regarding its implementation and use are discussed. In Section 4 a toy application example of the system is used to illustrate the solving a simple GDM problem. Finally, in Section 5 we draw our conclusions and future improvements to the system are highlighted.
Section snippets
Theoretical model for the consensus support system
In [29] a consistency and consensus measures based theoretical model to guide the GDM consensus process with incomplete fuzzy preference relations was developed. This section briefly describes the extension of that model and the necessary improvements needed to allow the use of multi-granular linguistic preference relations, the use of uninorms as a new characterization of consistency and their use to deal with incomplete information. In order for this paper to be as self-contained as possible,
Web based consensus support system
Nowadays, many decision and consensus support systems are being implemented in order to aid experts to solve decision problems efficiently [22], [49]. Web-based applications are increasingly being used for GDM and Decision Support environments [65] because they offer many advantages. An example of these advantages is the possibility of accessing them from all over the world and thus, the possibility of carrying out distributed decision making processes where experts cannot meet physically
Example of application
In this section we present an example of application of the consensus support system to solve a simple GDM problem. The problem is that of selecting the best car from a set of four different alternatives:
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Black, economic and slow car: Black Car.
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Red, very small, fast and comfortable car: Red Car.
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White and very fast car. It consumes little but it is very expensive: White Car.
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Blue, very small and very cheap car: Blue Car.
Four experts (e1, e2, e3, e4) will give their preferences using different kinds
Conclusions and future works
We have presented a web consensus support system to deal with GDM problems with different kinds of incomplete preference relations (fuzzy, linguistic and multi-granular linguistic preference relations). The consensus reaching process is guided by both consistency and consensus measures. Consistency has been modelled via the multiplicative consistency property also known as the Cross Ratio uninorm, and it has been used to estimate unknown values of incomplete preference relations as well as to
Acknowledgments
This paper has been developed with the Financing of FEDER funds in FUZZYLING project (TIN2007-61079), PETRI project (PET2007-0460), Andalucian Excellence project (TIC-5299) and project of Ministry of Public Works (90/07).
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