A chi-square method for priority derivation in group decision making with incomplete reciprocal preference relations

Abstract

This paper proposes a chi-square method (CSM) to obtain a priority vector for group decision making (GDM) problems where decision-makers’ (DMs’) assessment on alternatives is furnished as incomplete reciprocal preference relations with missing values. Relevant theorems and an iterative algorithm about CSM are proposed. Saaty’s consistency ratio concept is adapted to judge whether an incomplete reciprocal preference relation provided by a DM is of acceptable consistency. If its consistency is unacceptable, an algorithm is proposed to repair it until its consistency ratio reaches a satisfactory threshold. The repairing algorithm aims to rectify an inconsistent incomplete reciprocal preference relation to one with acceptable consistency in addition to preserving the initial preference information as much as possible. Finally, four examples are examined to illustrate the applicability and validity of the proposed method, and comparative analyses are provided to show its advantages over existing approaches.

Introduction

Group decision making (GDM) [9], [13], [14], [18], [21], [24], [35] is a procedure of drawing on the combined wisdom and experience of experts from different domains to rank a finite number of alternatives. Reciprocal preference relations [21], [23], [27], [34], [39] are commonly used to represent decision-makers (DMs)’ preferences over a set of possible alternative solutions, and have received considerable research attention in the past decades. However, owing to time pressure, lack of knowledge, and the DM’s limited expertise in the specific problem domain [1], [4], [5], [33], [36], [37], [41], [42], [43], [44], sometimes a DM can at best furnish his/her judgment on alternatives as a reciprocal preference relation with missing or incomplete entries. Therefore, the method to derive priorities from incomplete reciprocal preference relations [3], [10], [11], [15], [41] has presented itself as an important and promising research topic, and attracted considerable research interest.

For example, Xu and Da [32] put forward a normalizing ranking aggregation method (NRAM) to derive priorities from an incomplete reciprocal preference relation. Xu and Wang [40] extended the well-known eigenvector method (EM) for priority derivation for an incomplete reciprocal preference relation, and the improvement method therein not only increases the consistency level but also preserves the initial preference information as much as possible. It is worth noting that the aforementioned NRAM and EM can only be applied to a single incomplete reciprocal preference relation. Xu [41] proposed two goal programming models (GPMs) to obtain a collective priority vector from several incomplete reciprocal preference relations. Gong [17] put forward a least-square method (LSM) to generate a collective priority vector from incomplete reciprocal preference relations furnished by multiple DMs. Gong’s approach results in a simple equation. But it cannot be applied to obtain a priority vector when the matrix Q is singular or Q⁻¹ does not exist. In contrast to LSM, which is only applicable to the case with at least one multiplicative inconsistent incomplete reciprocal preference relation, the logarithmic least squares method (LLSM) put forward by Xu et al. [38] can be used for all incomplete reciprocal preference relations regardless of their multiplicative consistency property. In real-world decision processes, different DMs often carry heterogeneous power in reaching the final recommendation. It is noted that the aforementioned methods did not take into account DMs’ weights in the decision process.

This paper extends a chi-square method (CSM) to prioritize alternatives in a GDM context when DMs furnish their judgment as incomplete reciprocal preference relations. The CSM was initially developed for priorities by Jensen [19], and was later cited by Blankmeyer [7]. The original approach is complicated and has rarely been used. Wang and Fu [28] developed a convergent and simple iterative algorithm to facilitate its application in practice. Due to its nonlinear property, this improved algorithm has many advantages such as ease in computer implementation. As such, the extended CSM has arisen as a simple but efficient approach to deal with incomplete reciprocal preference relations.

The key motivations to adopt the CSM can be summarized as follows: (1) The CSM can be used to obtain a collective priority vector from several incomplete reciprocal preference relations, while other methods such as EM and NRAM can only be applied to a single incomplete reciprocal preference relation. This advantage makes it a natural choice for handling GDM problems. (2) The CSM is convenient in considering different DMs’ weights in the decision process while this issue has been largely omitted by other methods. (3) By properly setting model parameters, the CSM can be flexibly employed to handle both complete and incomplete reciprocal preference relations. (4) Compared with other methods, the CSM is known for its better fitting performance, rank preservation capability and discrimination power. After Wang and Fu [28]’s extension, the improved CSM has become an efficient and convenient tool to handle incomplete reciprocal preference relations. By exploiting CSM to derive priority weights from incomplete reciprocal preference relations in a GDM context, this article further enhances its applications and enriches the theory and methodology of priority derivation.

An important issue in GDM with incomplete reciprocal preference relations is consistency test and inconsistency repairing because consistency of the judgment given by DMs has a direct impact on the final decision results [22]. Xu and Wang [40] adapted Saaty [26]’s consistency ratio (CR) to a fuzzy context and introduced a so-called fuzzy consistency ratio (FCR), which can be applied to incomplete reciprocal preference relations. By adopting Saaty’s suggested threshold, an incomplete reciprocal preference relation is deemed to be acceptably consistent if FCR < 0.1 [24]. If an incomplete reciprocal preference relation given by the DM does not possess acceptable consistency, it has to be repaired so that its consistency reaches the acceptable threshold. This paper will put forward a CSM-based algorithm to accomplish this task.

The remainder of the paper is organized as follows. Section 2 provides a review on basic concepts of reciprocal preference relations, incomplete reciprocal preference relations and an acceptable FCR. An associated theorem is also presented. In Section 3, the CSM is extended to obtain a priority vector from incomplete reciprocal preference relations based on the multiplicative transitivity property, resulting in an iterative algorithm. Section 4 puts forward an approach to repair an unacceptably inconsistent incomplete reciprocal preference relation to derive one with acceptable consistency. In Section 5, four examples are examined to show how to apply the proposed CSM and its effectiveness in handling GDM problems. Comparative analyses with existing methods demonstrate its validity and advantages. Concluding remarks are furnished in Section 6.