Distance-based consensus reaching process for group decision making with intuitionistic multiplicative preference relations

Highlights

• We propose new intuitionistic multiplicative distance measures and their weighted forms.

• A new intuitionistic multiplicative consensus measure is defined.

• A new intuitionistic multiplicative group decision-making method is proposed.

• A numerical example is given to illustrate the effectiveness and practicability of this new method.

Abstract

The intuitionistic multiplicative preference relation uses Saaty’s $1∕9$-$9$ scale to depict people’s opinions from the prior and not prior aspects. Due to its objectivity in representing people’s cognition, the intuitionistic multiplicative preference relation has attracted extensive attentions of scholars. In this paper, we study the distance-based consensus measures in the context of group decision-making with intuitionistic multiplicative preference relations. First of all, some new distance measures between intuitionistic multiplicative numbers/sets are proposed, which contains the improved Hamming distance, the improved Euclidean distance, and their weighted forms. Then, we investigate their desirable properties. To aid the group decision-making process, we further develop a new consensus measures regarding intuitionistic multiplicative preference relations based on the proposed distance measures. Afterwards, a new group decision-making method is proposed to solve the complex group decision-making problems with intuitionistic multiplicative preference relations. Finally, an example concerning the project investment selection is given to demonstrate the proposed group decision-making method, and then we compare the proposed method with other existing group decision-making methods with intuitionistic multiplicative preference relations in detail.

Introduction

Due to time pressure and lacking of relevant information or knowledge on decision-making problems, it is difficult for experts to make accurate decisions. Meanwhile, there are lots of imprecise information and unbalanced or non-symmetrical cases in our life such as the law of diminishing marginal utility in economics [1]. For example, when the same investments are added, companies with poor economic performance can yield more profits than those with poor economic performance. In this sense, to describe experts’ opinions more effectively and intuitively, Xia et al. [1] proposed a new kind of information representation tool, namely, intuitionistic multiplicative preference relation (IMPR). The IMPR can depict the degree that one object is prior to another and the degree that the object is not prior to another, simultaneously. It uses the asymmetric scale, i.e., Saaty’s 1/9-9 [2], instead of the symmetric scale, i.e., 0–1 scale, used in the intuitionistic fuzzy preference relation (IFPR), to express the preferences of experts. The 1/9-9 scale is suitable for the description of non-symmetric or unbalance preference information. In recent years, the IMPR has been studied by many scholars since it was proposed. These studies can be divided into four categories: (1) Interval-value intuitionistic multiplicative set [3]; (2) intuitionistic multiplicative aggregation operators [1], [4], [5], [6]; (3) intuitionistic multiplicative distance measures [7], [8], [9], [10], [11]; and (4) decision-making methods with IMPRs [12], [13], [14], [15]. In this paper, we mainly discuss the distance measures and group decision-making (GDM) methods with IMPRs.

Distance measure is used to describe the difference between two numbers or sets. It has been applied to many fields such as pattern recognition, medical diagnosis, engineering, and artificial intelligence [4]. Additionally, it has also been widely researched in many multi-criteria decision-making settings, such as fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and hesitant fuzzy linguistic term sets. Analogously, distance measures also play an important role in the researches related to IMPRs. Based on the quantitative relationship between intuitionistic multiplicative number (IMN) and intuitionistic fuzzy number (IFN), Jiang et al. [7] proposed the Manhattan distance measure, Euclidean distance measure, and generalized Minkowski distance measure between IMNs (or intuitionistic multiplicative sets, IMSs). Later, Garg [11] further developed the normalized weighted Euclidean distance measure, normalized weighted Manhattan distance measure, as well as Hausdorff distance measure, and then applied them to deal with the decision-making problems concerning pattern recognition and medical diagnose. Liao et al. [9] put forward a series of distance measures between IMNs/IMSs from the discrete and continuous forms, respectively, and then extended the classical TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method and VIKOR (Vlsekriterijumska Optimizacija I Kompromisno Resenje in Serbian) method to intuitionistic multiplicative environment. In addition, Liao et al. [10] proposed two distance measures between IMNs/IMSs, namely, the projection-based distance measure and psychological distance measure. The former represents the deviation in terms of both distance and angle, while the later depicts the preferences of experts more precisely and objectively.

There are many GDM problems in which several individuals or experts with different opinions are invited to discuss the decision problem to obtain a common solution [16]. As for the decision-making problems with IMPRs, many methods have been proposed [9], [10], [12], [13], [14], [15]. For example, Ren et al. [12] introduced the intuitionistic multiplicative analytic hierarchy process (IMAHP), and then applied it to address the GDM problems with IMPRs. However, limitations existed in the IMAHP method, including the unreasonable concept of IMPR and the questionable priority-generation method. To overcome these limitations, Zhang and Pedrycz [13] introduced a new definition of IMPR, and then developed a new GDM method [14], that is, the intuitionistic multiplicative group analytic hierarchy process (IMGAHP). Additionally, Mou et al. [15] put forward the intuitionistic multiplicative best-worst method (IMBWM). The above-mentioned works show that decision-making with IMPRs has turned out to be a hot research topic in recent years, especially for the GDM problems with IMPRs. Due to the differences between experts’ personal experiences and knowledge levels, the evaluation information given by experts inevitably has some deviations or even contradictions in GDM process. Generally, the procedure to solve the GDM problem consists of three steps: the consistency checking and repairing process of each preference relation, the consensus checking and reaching process of the group, and the selection process [17]. To solve GDM problems effectively, the three sub-problems mentioned above must be considered, simultaneously. However, only the first and third sub-problems were considered in [12], [14] without considering the second problem. In addition, only the first sub-problem was discussed in [15]. In fact, the consensus checking and reaching process is particularly important because only this process can ensure that the decision results are accepted by all group members despite their different opinions [18]. Therefore, in this paper, we investigate the consensus checking and reaching process of the GDM problem with IMPRs.

Consensus measure is considered as an important component of the consensus checking and reaching process, and the basic tool to deal with GDM problems (see Section 2.3 for details). As we know, there is little research on the intuitionistic multiplicative consensus measure. Also, there are some limitations in the existing intuitionistic multiplicative distance measures: the existing intuitionistic multiplicative distance measures are based on the transformation mechanism function between IMPR and IFPR, so there is a loss of information in the computing process and the calculation is complex due to the transformation in computing process. Motivated by this, in this paper, we propose some new distance measures between IMNs/IMSs. After that, based on the intuitionistic fuzzy PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) method [19], [20] and the consensus-based GDM method [20], [21], new distance-based consensus measure is introduced under the IMS circumstance. Finally, a novel intuitionistic multiplicative GDM method is developed, and it is further demonstrated by a practical GDM problem about project investment. Furthermore, detailed comparative analyses and discussions are given to illustrate the effectiveness of our proposed intuitionistic multiplicative (IM) distance measures and GDM method.

In summary, both the scientific and practical contributions of this paper can be highlighted as follows:

(1) New IM distance measures including the improved IM Hamming distance, improved IM Euclidean distance and improved generalized IM distance are investigated with weighted forms and non-weighted forms, respectively. Unlike the existing distance measures [7], [8], [9], [10], [11], the distance measures proposed in this paper are not based on the transformation function between IMPR and IFPR. Thus, the difference between IMNs or IMSs are directly derived based on the membership degrees, non-membership degrees and hesitancy degrees of IMNs, but not the transformation function.

(2) Based on the defined IM outranking flow and the proposed distance measures, an IM consensus measure is defined to measure the consensus degree of experts in GDM. After that, an IM GDM method is introduced, which takes into account the consensus checking and reaching processes as well as the alternative selection processes. The proposed GDM method can be used to deal with the GDM problems with IMPRs. Since both the consensus checking and reaching processes are considered in the proposed IM GDM method, the obtained result is acceptable by experts in real case.

(3) The proposed IM GDM method is applied to address the problem of project investment decision-making. It cannot only aid the managers to make scientific decisions in GDM but also enrich the decision-making method system of the project investment.

The rest of this paper is organized as follows: The basic concepts and operations of IMPRs are reviewed in Section 2. Then, some new distance measures between IMNs/IMSs are defined and their properties are discussed in Section 3. Based on the proposed IM distance measures and the outranking flow of IMPR, new IM consensus measure is defined, based on which, a novel GDM method is developed in Section 4. In Section 5, a numerical example is provided to illustrate the applicability of the IM consensus measure and the GDM method, based on which, detailed discussions are further made in this part. Finally, the paper ends with some concluding remarks in Section 6.