Elsevier

Knowledge-Based Systems

Volume 144, 15 March 2018, Pages 122-143
Knowledge-Based Systems

A programming-based algorithm for interval-valued intuitionistic fuzzy group decision making

https://doi.org/10.1016/j.knosys.2017.12.033Get rights and content

Abstract

Interval-valued intuitionistic fuzzy preference relations (IVIFPRs) are powerful to express the uncertain preferred and non-preferred judgments of decision makers simultaneously. After reviewing previous researches about IVIFPRs, we find that several limitations exist. Especially, previous methods are insufficient to address inconsistent and incomplete cases. Considering these issues, this paper first analyzes the limitations of the previous consistency concepts for IVIFPRs and then introduces a new one that avoids the disadvantages of previous ones. 0–1 mixed programming models are built for judging the multiplicative consistency of IVIFPRs. Subsequently, several multiplicative consistency-based 0–1 mixed programming models are constructed for determining missing values that can address the situation where ignored objects exist. For group decision making, a distance measure on IVIFPRs is defined, by which the weights of the decision makers are derived. Meanwhile, a consensus index is offered. A multiplicative consistency and consensus based algorithm to group decision making with IVIFPRs is proposed. Finally, two practical decision-making problems are offered to show the application of the new algorithm, and theoretical and numerical analysis of several related methods is made.

Introduction

Preference relations are powerful tools to decision making that only need decision makers (DMs) to compare two objects at one time. Since Saaty [18] first introduced the concept of multiplicative preference relations, researches about decision making with preference relations have been largely developed. At present, it is still one of the most important decision-making methods. According to elements in preference relations, it can be roughly classified into two types: crisp preference relations [17], [18], [22] and fuzzy preference relations [3], [12], [19], [21], [27]. The former uses exact values to represent the comparisons between objects, while the latter adopts fuzzy numbers, such as interval values, triangular fuzzy numbers, hesitant fuzzy variables, and uncertain linguistic variables. Crisp preference relations are simpler than fuzzy preference relations to rank objects, while fuzzy preference relations are more powerful than crisp preference relations to express the judgements of the DMs.

All of the above-mentioned preference relations can only represent the preferred information of the DMs for the comparison objects. In some times, this might be insufficient, and the DMs may want to express their non-preferred judgements too. Considering this situation, intuitionistic fuzzy sets (IFSs) [1] are good choices that can express the preferred and non-preferred judgements simultaneously. Later, Xu [28] introduced IFSs for preference relations and proposed the concept of intuitionistic fuzzy preference relations (IFPRs). In past ten years, many researchers [5], [6], [9], [24], [26], [29] devoted themselves to researching the theory and application of IFPRs. Recently, Meng et al. [13] noted that IFPRs can only express the quantitative judgements of the DMs and put forward intuitionistic linguistic fuzzy preference relations (ILFPRs). This type of preference relations allows the DMs to apply linguistic variables and intuitionistic fuzzy variables to represent the qualitative and quantitative judgements.

Although IFPRs and ILFPRs have some advantages to express the opinions of the DMs, both of them only allow the DMs to use exact variables. This is still too restrictive to denote uncertain preferences. To address this issue, Xu [30] proposed interval-valued intuitionistic fuzzy preference relations (IVIFPRs) that permit the DMs to apply intervals in [0, 1] to denote the preferred and non-preferred judgments rather than real values. Such an extension further endows the DMs with more flexibility. After the original work of Xu [30], Wang et al. [23] introduced a consistency concept for IVIFPRs. However, one can easily check that several limitations of this concept exist. When an IVIFPR is additively consistent, there might be no normalized interval priority weight vector satisfying formula (16) [23]. Meanwhile, the offered programming model to calculate the interval priority weight vector cannot address inconsistent case. Additionally, the interval priority weight vector cannot reflect the preferred and non-preferred opinions of the DMs. Similar to reciprocal preference relations, there are two types of consistent IVIFPRs. Following Tanino's multiplicative consistency concept [20], there are three multiplicative consistency concepts for IVIFPRs. On the basis of the operational laws on interval-valued intuitionistic fuzzy values (IVIFVs) [31], Xu and Cai [32] gave a multiplicative consistency concept for IVIFPRs. Nevertheless, one can check that this multiplicative consistency concept holds if and only if all IVIFVs in an IVIFPR are identical. Furthermore, Liao et al. [10] and Xu and Cai [33] gave another multiplicative consistency concept, respectively. However, the main issue is that this concept does not satisfy robustness. This means that contradictory conclusions can be drawn for different comparison orders of objects. Following Liao and Xu's consistency concept for IFPRs [11], Wan et al. [25] presented a convex combination based multiplicative consistency concept for IVIFPRs. This concept considers an IVIFPR to be consistent when the associated induced IFPR is consistent [11]. Note that Wan et al.’s consistency concept is dependent of the comparison order too. Xu and Yager [34] defined a similarity measure on intuitionistic fuzzy sets and developed a group decision-making method with IVIFPRs based on consensus analysis. Nevertheless, it does not consider consistency.

After reviewing the literature, we find that several limitations of previous researches about IVIFPRs exist: (I) Previous concepts are insufficient to judge the consistency of IVIFPRs; (II) none of them considers incomplete IVIFPRs; (III) none of them discusses how to determine the weights of the DMs; (IV) most of them do not study consensus; (V) all of them cannot deal with inconsistent IVIFPRs completely; (VI) different rankings may be derived for different comparison orders. This paper continues to study IVIFPRs and develops a new method for group decision making with IVIFPRs that can address inconsistent and incomplete cases. The rest part is organized as follows:

Section 2 first recalls several basic concepts to help us understand the following. Then, it reviews four previous consistency concepts for IVIFPRs and analyzes their limitations from theoretical and numerical aspects. Section 3 introduces two new types of preference relations called 2-tuple preferred fuzzy interval preference relations (TPFIPRs) and quasi-TPFIPRs (QTPFIPRs). Then, following Meng et al.’s consistency concept for fuzzy interval preference relations [14], a new multiplicative consistency concept for IVIFPRs is proposed. Section 4 builds 0–1 mixed programming models to judge the consistency of IVIFPRs. Section 5 focuses on incomplete IVIFPRs and constructs several models to determine missing values. Meanwhile, a method for deriving completely multiplicative consistent IVIFPRs from inconsistent ones is introduced. Section 6 mainly discusses group decision making with IVIFPRs. A method for determining the weights of the DMs is provided, and a consensus index is defined. Then, a multiplicative consistency and consensus based algorithm to group decision making with IVIFPRs is developed. Section 7 offers two practical examples to show the application and feasibility of the new method and to compare it with several previous ones. Conclusions are made in the last section.

Section snippets

Preliminary

This section contains two parts. The first part reviews several basic concepts, including: (interval-valued) intuitionistic fuzzy sets, intuitionistic fuzzy preference relations (IFPRs), interval-valued intuitionistic fuzzy preference relations (IVIFPRs), and fuzzy interval preference relations (FIPRs). The second part recalls four previous consistency concepts for IVIFPRs [10], [23], [25], [33] and then analyzes their limitations, by which one can find that they are insufficient to judge the

A new consistency concept

To ensure the rational application of IVIFPRs, it is necessary to further study the consistency of IVIFPRs [10], [11], [12], [14], [25], [26]. This section introduces a new multiplicative consistency concept that avoids the limitations of previous concepts. To do this, we first review some operational laws on intervals.

Let a¯=[a,a+] and b¯=[b,b+] be any two positive intervals. Following Minkowski operations, we derive

  • (i)

    a¯b¯=[a+b,a++b+];

  • (ii)

    a¯b¯=[ab,a+b+];

  • (iii)

    a¯/b¯=[a/b+,a+/b];

  • (iv)

    a¯λ=[aλ,a+λ]λ0;

  • (v)

    λ

Programming models for judging consistency of IVIFPRs

For any given IVIFPR R˜=(r˜ij)n×n, Definition 11 shows that we can apply the consistency of the associated QFIPRs γ¯=(r¯ij)n×n and λ¯=(λ¯ij)n×n to judge its consistency. However, it is not an easy thing because many different QFIPRs exist. To address this problem, we construct programming models to judge the consistency of IVIFPRs. First, we introduce the concept of 0–1 indicator variables. For each pair of (i, j), let θij={1r¯ij=[μl,ij,1vl,ij]0r¯ij=[1vl,ij,μl,ij] and ϑij={1λ¯ij=[μu,ij,1vu,ij

Programming models for addressing incomplete IVIFPRs

As researchers noted that some judgments in preference relations might be missing for various kinds of reasons. In this case, we can only obtain incomplete preference relations [11], [12], [13], [14], [32], [33]. To rank objects, we first need to determine missing judgments. This section focuses on incomplete IVIFPRs. At present, we only find two references [32], [33] that considered incomplete IVIFPRs. However, they are based on Definitions 5 and 6, and some limitations exist. Following the

A method for group decision making with IVIFPRs

With the increasing complexity of decision-making problems, it becomes more and more impossible to require one DM to consider all aspects of a decision-making problem. Considering this situation, group decision making has become a hot researching topic. This section focuses on group decision making with IVIFPRs. In the setting of preference relations, the study of consensus is a key step [8], [12], [14], [25], [26], which reflects the degree of agreements between the DMs’ opinions.

Without loss

Case study and comparison analysis

This section provides two practical decision making problems to show the specific application of the new algorithm for group decision making with IVIFPRs. In addition, we compare the new algorithm with several previous methods.

Example 5

[25]. The AHEAD information technology company that is a famous software enterprise of China focuses on medical information integrating and service since its inception in 2003. Now, this company desires to develop a new-type rural cooperative medical care management

Conclusions

Considering previous researches about IVIFPRs, we find that several limitations exist, which are directly or indirectly caused by the adopted consistency concepts. Therefore, we wrote this paper. To address the issues of the previous consistency concepts for IVIFPRs, we introduced a new one by using the multiplicative consistency concept for FIPRs [14]. From discussion about the properties of the new concept, one can find that it is a natural extension for Tanino's multiplicative consistency

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 71571192, 71671188, and 71501189), the Innovation-Driven Planning Foundation of Central South University (No. 2016CXS027), the State Key Program of National Natural Science of China (No. 71431006), the Projects of Major International Cooperation NSFC (No. 71210003), the Hunan Province Foundation for Distinguished Young Scholars of China (No. 2016JJ1024), and the China Postdoctoral Science Foundation (No.

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