A new consistency definition of interval multiplicative preference relation

Abstract

This paper presents a new consistency definition of interval multiplicative preference relation (IMPR). Several desirable properties of this new consistency are fully discussed, and the sufficient and necessary conditions of the new consistency definition are also provided. It is proven that this new consistency of IMPR has robustness and invariance with respect to permutation of the compared objects' labels. Moreover, comparative analyses are conducted to reveal the relationships among five existing consistency definitions and this new consistency definition of IMPR. Based on the new consistency definition of IMPR, a new consistency index is introduced to measure the consistency degree of IMPR. An iterative algorithm is proposed to improve the consistency level of an inconsistent IMPR. Subsequently, a goal programming model is built to derive an interval priority vector from an acceptably consistent IMPR. Eventually, a new individual decision making method with an IMPR is put forward. Numerical examples and simulation analyses are conducted to illustrate the superiority and validity of the proposed individual decision making method.

Introduction

Analytic hierarchy process (AHP) [1], [2] is one of the most widely adopted multi-criteria decision making methods [3]. By means of pairwise comparison technology, pairwise judgments are provided by decision-makers (DMs) to construct preference relations (PRs) (or pairwise comparison matrices). However, off-diagonal elements of PRs are affected by DMs' subjective judgments. In many real-life problems, it is very difficult for DMs to express their feeling with exact positive real numbers [4] when comparing two objects. To better deal with the imprecise and uncertain information, fuzzy sets [5] were successfully employed to express DMs' evaluations. Up to now, based on different transitive formulations and fuzzy sets, different types of PRs have been developed to describe the uncertain information of decision-making problems, such as fuzzy preference relations [6], [7], [8], interval multiplicative preference relations (IMPRs) [9], [10], [11], [12], [13], [14], [15], [16], [17], interval fuzzy preference relations (IFPRs) [18], [19], [20], [21], [22], triangular multiplicative preference relations [23], [24], [25], [26], triangular fuzzy preference relations [27], [28], intuitionistic fuzzy preference relations [29], [30], hesitant fuzzy preference relations [31], [32] and linguistic preference relations [33]. For some complex systems with uncertain and ambiguous information, it is wiser for DMs to describe their opinions in terms of intervals [17], [34], [35], [36], [37] rather than exact numbers. In current study, more and more scholars have paid great concern on IMPR and obtained abundant theoretical results on IMPR.

Based on the real continuous Abelian linearly ordered (Alo) groups, Cavallo & Brunelli [36] studied a general unified approach to IMPRs and IFPRs and further discussed the reciprocity and consistency conditions of IMPRs whose elements are expressed as any positive real numbers satisfying the condition of multiplicative reciprocity. Cavallo & Brunelli [36] and Cavallo & D'Apozzo [38] pointed out that several different representations of PRs share the same algebraic structure. Furthermore, Cavallo & D'Apozzo [38] provided a general unified framework for interval PRs. In fact, the property of reciprocity is an intrinsic characteristic of PRs [13], [36]. Additionally, the consistency of a PR means that there are no contradictions among all the preference values in the PR. At present, the consistency definition and consistency measurement of IMPR are two hot topics [36], [39] in decision analysis. As is well known, the consistency definition of PR should preserve the property of invariance with regards to the permutation of the compared objects' labels [13], [40]. Usually, highly inconsistent IMPRs maybe result in the wrong conclusions or reverse the ranking order of the compared objects, especially for high-order IMPRs.

In current studies, interval PRs are generally divided two types: IMPRs and IFPRs with multiplicative or additive transitivity [22]. Some scholars [11], [13], [15], [17], [35] presented different consistency definitions and consistency indices of IMPR [14], [36]. Brunelli & Fedrizzi [41] characterized the property of invariance as an axiom to characterize the consistency indices of PR. Up to now, great achievements have been made on the consistency issue of IMPR [11], [13], [15], [17], [35]. On the basis of the feasible region idea, Wang et al. [17] presented the consistency definition of IMPR. Later, the consistency definition of IMPR was put forward in [35] by means of the lower and upper boundaries of interval preference values. From the perspective of the geometrical means of interval endpoints, Li et al. [11] gave the geometrical consistency concept. Krejčí [13] referred an IMPR as a consistent IMPR by adding some special constraints to the consistency definition of IMPR proposed in [17]. Based on the concepts of quasi-positive interval and quasi-consistent IFPR, a consistency definition of IFPR and a consistency definition of IMPR were proposed in [19] and [15], respectively. Combining the idea in [13] with the multiplicative consistency definition of fuzzy PRs, a new multiplicative consistency definition of IFPR is presented in [22]. Up to now, although some consistency definitions of IMPR have been put forward from different perspectives, there is not yet one definition that has gained common recognition.

Consistency test ensures that DMs are neither illogical nor arbitrary when giving their pairwise comparisons with the help of the consistency index of PR. A highly consistent PR is usually beneficial to improve the quality of subject judgments and derive the more credible ranking order of compared objects. At present, there is no a common consistency index computed directly from an IMPR [36]. Based on Alo-groups, Brunelli & Cavallo [9] generalized some consistency conditions for IMPRs and/or IFPRs. By means of suitable isomorphisms between Alo-groups, Brunelli & Cavallo [9] also presented transformation functions among the multiplicative Alo-group, the additive Alo-group and the fuzzy Alo-group. According to Cavallo & Brunelli [36], the consistency index and indeterminacy index can be regarded as the means of assessing the discrimination and ambiguity of DMs' pairwise judgments.

How to derive the more reliable and reasonable priority vector from an IMPR is a vital process of applying AHP with interval preference values. Sugihara et al. [42] and Entani & Tanaka [43] suggested that the priority vector should be estimated as an interval vector because of indeterminacy of DMs' subjective judgments. Liu [35] derived an interval priority vector from an acceptably consistent IMPR by extending the geometric mean method [44]. Based on the consistency index defined by Saaty [1], Wang et al. [17] proposed a nonlinear programming model to generate an interval priority vector from an IMPR. Wan et al. [20] designed a fuzzy logarithmic programming model to derive an interval priority vector from an IFPR. Sugihara et al. [42] and Entani & Tanaka [43] constructed an upper approximation model to obtain an interval priority vector from an traditionally multiplicative preference relation (MPR). Islam et al. [45] employed a lexicographic goal programming (LGP) to derive a crisp priority vector from an IMPR. Xu & Chen [46] constructed several linear programming models to derive an interval priority vector from a consistent IFPR. Guo & Tanaka [47] built linear programming and quadratic programming models to estimate interval probabilities of MPRs by minimizing the imprecision of DMs' judgments. Entani & Sugihara [48] presented three programming models to derive an interval priority vector from an MPR from three perspectives of interval entropy, width and ignorance, respectively. Meng et al. [49] presented two methods to determine an interval priority vector from an IMPR.

Literature review illustrates that great contributions have been made on IMPRs. Nevertheless, there still exist several limitations as follows:

(i) The consistency definition of IMPR should be independent on the labels of the compared objects. That is to say, the consistency definition of IMPR should possess the property of invariance with respect to permutation of the compared objects' labels. However, Liu's consistency [35] is dependent on the compared objects' labels.

(ii) It is impracticable and impossible to require DMs to provide perfectly consistent IMPRs in real-life decision-making problems. In fact, unacceptably consistent IMPRs are more common. Thus, it is necessary to discuss the derivation of the ranking order from an unacceptable IMPR. However, Li et al. [11] did not consider how to derive the ranking order from an unacceptably consistent IMPR.

(iii) To generate the compared objects' ranking order, the priority vector obtained from the upper triangular parts of an IMPR should be identical to that from the lower triangular parts of the same IMPR. However, when using LGP [45] to solve an decision-making problem with an IMPR, the ranking order derived from the upper part of an IMPR may be different from that from the lower triangular part of the same IMPR.

To make up the aforesaid limitations, it is necessary to further study the consistency issue of IMPR. This paper proposes a new consistency definition of IMPR and discusses several useful properties of the new consistency. It is proven that this new consistency definition of IMPR has robustness and invariance [13], [36], [41] with respect to permutation of the compared objects' labels. Comparative analyses are conducted to explore the relationships among five existing consistency definitions and this new consistency definition. Based on the new consistency definition of IMPR, a new consistency index of IMPR is designed to check whether an IMPR is acceptably consistent or not. Subsequently, an iterative algorithm is presented to improve the consistency level of an inconsistent IMPR. Moreover, a goal programming model is established to obtain an interval priority vector from an IMPR. Thereby, a new method for individual decision making with an IMPR is put forward. Eventually, simulation analyses illustrate the effectiveness and superiority of the proposed method.

The paper is organized as follows. Section 2 reviews some basic concepts and analyzes five existing consistency definitions of IMPRs. Section 3 provides a new consistency definition of IMPR and reveals several useful properties. Moreover, comparative analyses with five existing consistency definitions of IMPR are given in Section 3. Section 4 presents a new consistency measurement and an iterative algorithm of improving the consistency level of an inconsistent IMPR. Following this, Section 5 constructs a goal programming model to derive an interval priority vector from an IMPR. Illustrative examples and simulation analyses are given in Section 6. Finally, some constructive remarks are included in Section 7.