Multiperson decision making with different preference representation structures: A direct consensus framework and its properties
Introduction
In multiperson decision making (MPDM) problems, it is quite natural that different decision makers may have different experience, cultures and educational backgrounds. As a result, these decision makers may use different preference representation structures to express their individual preference information.
Chiclana et al. [6] initiated a notable MPDM model based on fuzzy preference relations, where the preference information can be represented by means of preference orderings, utility functions and fuzzy preference relations. Chiclana et al. [7] further incorporated multiplicative preference relations in the MPDM model. Herrera et al. [22] proposed a multiplicative MPDM model involving three kinds of preference representation structures (preference orderings, utility functions, multiplicative preference relations), assuming the multiplicative preference relations as the uniform element of the preference representation structures. Dong et al. [16] presented a linguistic MPDM model based on linguistic preference relations, integrating fuzzy preference relations, different types of multiplicative preference relations and multigranular linguistic preference relations. Moreover, Herrera et al. [23], Herrera and Martínez [26], Herrera-Viedma et al. [31], Mata et al. [35], Chen and Ben-Arieh [5] and Jiang et al. [32] introduced several methods to solve the MPDM problems with multi-granularity linguistic evaluation information. In [30], Herrera-Viedma et al. proposed a consensus model for the MPDM problem with different preference representation structures. Palomares et al. [39] proposed an attitude-driven web consensus support system for heterogeneous group decision making.
Several desirable properties have been proposed in the MPDM model with different preference representation structures. Chiclana et al. [9] and Dong et al. [13] discussed the conditions under which the reciprocity property is maintained in the aggregation of fuzzy preference relations using the ordered weighted average (OWA) operator [46] guided by a relative linguistic quantifier [47]. Meanwhile, in the above MPDM models, the internal consistency is a key issue. The internal consistency refers to the ranking among alternatives derived from the transformed preference representation structure is the same one from the original preference representation structure. Chiclana et al. [7], [8] and Dong et al. [16] studied the conditions under which the internal consistency is maintained.
Inspired by the MPDM model initiated by Chiclana et al. [6] and the corresponding consensus model presented in Herrera-Viedma et al. [30], and also inspired by the direct approach presented in Herrera et al. [25], this study proposes a direct consensus framework for MPDM problems with different preference representation structures (preference orderings, utility functions, multiplicative preference relations and fuzzy preference relations). In the direct consensus framework, the individual selection methods, associated with different preference representation structures, are used to obtain individual preference vectors of alternatives. Then, the standardized individual preference vectors are aggregated into a collective preference vector. Finally, based on the collective preference vector, the feedback adjustment rules, associated with different preference representation structures, are presented to help the decision makers reach consensus. The results in this study are helpful to complete Chiclana et al.’s MPDM with different preference representation structures, based on the following reasons:
- (i)
The proposed framework can avoid internal inconsistency issue when using the transformation functions among different preference representation structures.
- (ii)
It satisfies the Pareto principle of social choice theory.
The rest of this paper is organized as follows. Section 2 introduces the preliminary knowledge regarding four kinds of preference representation structures (preference orderings, utility functions, multiplicative preference relations and fuzzy preference relations) and the OWA operator. A direct consensus framework for MPDM problems with different preference representation structures is proposed, and the differences between our proposal and models presented in [6], [7], [30] are analyzed in Section 3. Following this, the selection process is designed in Section 4. Subsequently, Section 5 proposes the consensus processes to help the decision makers reach consensus. Two desirable properties are presented in Section 6, and an illustrative example is provided in Section 7. Finally, concluding remarks are included in Section 8.
Section snippets
Preliminaries: Four kinds of preference representation structures and OWA operator
This section introduces four kinds of preference representation structures and the OWA operator.
Proposed framework
This section proposes a direct consensus framework for the MPDM with different preference representation structures. And the differences between our proposal and the models presented in Chiclana et al. [6], [7] and Herrera-Viedma et al. [30] are also analyzed.
Selection process
This section proposed the selection process in the direct consensus framework.
Consensus process
Classically, consensus is defined as the full and unanimous agreement of all the decision makers regarding all the possible alternatives. However, the complete agreement is not always necessary in real life. This has led to the use of “soft” consensus degree [33], [34]. A lot of researchers [2], [4], [15], [17], [24], [27], [30], [38], [40], [45], [49] have investigated the consensus process for MPDM problems, based on the uses of “soft” consensus. Cabrerizo et al. [3] and Herrera-Viedma et al.
Desirable properties
In this section, we introduce two desirable properties of the proposed direct consensus framework.
Illustrative example
In order to demonstrate the proposed direct consensus framework, let us consider the example presented in Herrera-Viedma et al. [30]. In this example, there are a set of eight decision makers E = {e1, e2, …, e8} and a set of six alternatives X = {x1, …, x6}. The decision makers e1 and e2 provide their opinions using the utility functions U(1) and U(2). The decision makers e3 and e4 provide their opinions using the preference orderings O(3) and O(4). The decision makers e5 and e6 provide their
Conclusion
In this study, we investigate the MPDM problem with different preference representation structures (preference orderings, utility functions, multiplicative preference relations and fuzzy preference relations). The main points presented are as follows:
- (1)
Inspired by the MPDM model initiated by Chiclana et al. [6] and the corresponding consensus model presented in Herrera-Viedma et al. [30], and also inspired by the direct approach presented in Herrera et al. [25], we propose a direct consensus
Acknowledgements
We would like to acknowledge the financial support of a grant (No. 71171160) from NSF of China, and a grant (No. skqx201308) from Sichuan University.
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