Average-case consistency measurement and analysis of interval-valued reciprocal preference relations
Introduction
Reciprocal preference relations are based on the pairwise comparison method, and are widely used preference representation structures in decision-making problems. Various types of reciprocal preference relations have been proposed, such as additive preference relations (also called fuzzy preference relations) [2], [11], [13], [14], [24], and multiplicative preference relations [3], [21], [22], [23]. It is well known that quantifying consistency is a very important issue in decision-making with preference relations. The lack of consistency can lead to inconsistent conclusions. In the specialised literature, a number of consistency measurement methods of reciprocal preference relations have been proposed (see, among others, [1], [7], [15], [17], [18], [34], [38]).
However, due to the complexity and uncertainty involved in real-world decision problems, it is sometimes unrealistic to acquire exact judgments. Thus, reciprocal preference relations are extended to interval-valued reciprocal preference relations (see, among others, [27], [35]). In this paper, we focus on the consistency of interval-valued reciprocal preference relations. Existing studies regarding the measurement of consistency of interval-valued reciprocal preference relations can be broadly classified as implementing one of two different measures that we refer to as: the “classical” consistency measure [8], [12], [27], [35], and the “boundary” consistency measure [19], [20], which are described in Section 2.2. However, based on the definitions of the classical and boundary consistency measures (see Eqs. (3) and (4)), we can find that:
- (1)
The classical consistency degree of an interval-valued reciprocal preference relation is determined by its associated reciprocal preference relation with highest consistency degree, while
- (2)
The boundary consistency degree is determined by its two associated boundary reciprocal preference relations.
It is natural that the consistency index of an interval-valued reciprocal preference relation should be determined by taking into account all its associated reciprocal preference relations. Motivated by this, in this paper a new average-case consistency analysis of interval-valued reciprocal preference relations is suggested, defined and analyzed. Furthermore, this paper also proposes an average-case consistency improving method, based on the relationship among the average-case consistency measure, the classical consistency measure, and the worst consistency measure.
The rest of the paper is organised as follows. Section 2 introduces a basic description of the interval-valued reciprocal preference relation, the classical consistency measure and the boundary consistency measure. Section 3 presents the average-case consistency analysis of the interval-valued reciprocal preference relation (Section 3.1), as well as a numerical analysis (Section 3.2) and the different consistency measure internal mechanisms (Section 3.3). Section 4 is dedicated to the average-case consistency improving method. Finally, concluding remarks are included in Section 5.
Section snippets
Preliminaries
This section provides the basic knowledge regarding interval-valued reciprocal preference relations, as well as the classical consistency measure and the boundary consistency measure for interval-valued reciprocal preference relations.
Average-case consistency measure of IVRPRs
This section proposes the average consistency index (ACI) of IVRPRs, followed by numerical examples and a comparative study to justify the feasibility of the new ACI to measure consistency of IVRPRs.
Average-case consistency improving method
For RPRs of unacceptable consistency, consistency improving methods [8], [9], [10], [36], [37] have been developed. In this section, an average-case consistency improving method with the aim of obtaining a modified IVRPR with a required ACI is developed.
Conclusion
In this paper, the average-case consistency measure of IVRPRs has been proposed, analyzed and compared against the two existing consistency measures of IVRPR: the classical consistency measure and the boundary consistency measure. The underlying idea of the average consistency measure consist in measuring the consistency degree of the IVRPR using the average of the consistency degrees of all its associated RPRs. The internal mechanisms of the different consistency measures have been analyzed,
Acknowledgments
This work was supported by the grants (Nos. 71171160 and 71571124) from NSF of China, the grant (No. skqy201606) from Sichuan University, the grant (No. TIN2013-40658-P) from the FEDER funds, and the grant (No. TIC-5991) from the Andalusian Excellence Project.
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