Mehar ranking method for comparing connection numbers and its application in decision making

Abstract

Kumar and Garg (Applied Intelligence, 2017, 10.1007/s10489-017-1067-0) pointed out the limitations of some existing methods for solving intuitionistic fuzzy multi-attribute decision-making (MADM) problems. Also, to overcome the limitations, Kumar and Garg proposed a connection number (CN) based method for solving intuitionistic fuzzy MADM problems. In this paper, it is shown that the ranking method, used in Step 5 of Kumar and Garg’s method for comparing connection numbers (CNs), fails to compare two distinct CNs. Hence, Kumar and Garg’s method fails to rank the alternatives of intuitionistic fuzzy MADM problems. Furthermore, to overcome the limitation of Kumar and Garg’s method, a new ranking method (named as Mehar ranking method) is proposed for comparing CNs.

Keywords

Set pair analysis connection number (CN)intuitionistic fuzzy set decision-making problems

1 Introduction

In several daily life problems, there is need to follow a process for selecting the best alternatives from all the available alternatives having same conflicting attributes or for the ranking of all the available alternatives having same conflicting attributes. This process is called MADM and such problems are called MADM problems [5]. For example, to find the best student of a class the arithmetic mean of the marks, obtained by each student in different subjects, is calculated and then the obtained average marks is used to find the best student of the class.

One of the important steps of the MADM is to collect the information/data regarding the problem. It is pertinent to mention that it is not always possible to represent the collected data/information as a real number. For example, the rating of a movie review cannot be represented by a real number. It can be expressed in linguistic terms such as poor, average, good, excellent etc. In the literature, different ways have been introduced to represent linguistic terms. CN is one of the way to represent the linguistic terms [19].

Kumar and Garg [11] pointed out that several researchers [3 , 13–15] have used the CN for solving MADM problems under crisp environment as well as MADM problems under fuzzy environment [16]. However, till now no one has used the CN for solving MADM problems under intuitionistic fuzzy environment [1]. Kumar and Garg [11] also considered some intuitionistic fuzzy MADM problems to show that the existing intuitionistic fuzzy MADM method [2] fails to rank the alternatives of the considered intuitionistic fuzzy MADM problem.

Kumar and Garg [11] pointed out that this limitation of the existing intuitionistic fuzzy MADM method [2] can be resolved with the help of a CN. Since, to do the same there was need to transform each intuitionistic fuzzy number (IFN) [1] of the intuitionistic fuzzy decision matrix into a CN. But, there was no method in the literature to transform an IFN into a CN. Therefore, Kumar and Garg [11] firstly proposed a method to transform an IFN into a CN. Then using this method, Kumar and Garg [11] proposed a method to solve intuitionistic fuzzy MADM problems.

In this paper, it is shown that the ranking method, used in Step 5 of Kumar and Garg’s method for comparing connection numbers (CNs), fails to compare two distinct CNs. Hence, Kumar and Garg’s method fails to rank the alternatives of intuitionistic fuzzy MADM problems. Furthermore, to overcome the limitation of Kumar and Garg’s method, a new ranking method (named as Mehar ranking method) is proposed for comparing CNs.

This paper is organized as follows. In Section 2, some basic definitions are presented. In Section 3, Kumar and Garg’s intuitionistic fuzzy MADM method [11] is presented. In Section 4, limitations of Kumar and Garg’s intuitionistic fuzzy MADM method [11] are pointed out. In Section 5, reasons for the occurrence of the limitations are pointed out. In Section 6, a new ranking method (named as Mehar ranking method) is proposed for comparing CNs as well as its validity is discussed. In Section 7, the exact preference order of the alternatives of an intuitionistic fuzzy MADM problem is obtained by the modified Kumar and Garg’s intuitionistic fuzzy MADM method. Section 8, concludes the paper.

2 Preliminaries

In this section, some basic definitions are presented.

Definition 2.1. [16] A set $\tilde{A} = {〈 x, μ_{\tilde{A}} (x) 〉$ $| x \in X, 0 \leq μ_{\tilde{A}} (x) \leq 1},$ defined on the universal set X, is said to be a fuzzy set (FS), where $μ_{\tilde{A}} (x)$ represents the degree of membership of the element x in $\tilde{A}$ .

Definition 2.2. [1] A set $\tilde{A} = {\begin{matrix} 〈 x, μ_{\tilde{A}} (x), ν_{\tilde{A}} (x) 〉 | x \in X, 0 \leq μ_{\tilde{A}} (x) \leq 1, \\ 0 \leq ν_{\tilde{A}} (x) \leq 1, μ_{\tilde{A}} (x) + ν_{\tilde{A}} (x) \leq 1 \end{matrix}},$ defined on the universal set X, is said to be an intuitionistic fuzzy set (IFS), where, $μ_{\tilde{A}} (x)$ and $ν_{\tilde{A}} (x)$ represents the degree of membership and degree of non-membership respectively of the element x in $\tilde{A}$ .

Definition 2.3. [19] If two sets A and B, having same number of characteristics (say N), are put together to form a pair H with respect to the problem W then the pair H is called a set pair and the number $μ (H, W) = (\frac{S}{N}) + (\frac{P}{N}) j$ is called a connection number of the set pair H, where, S represents the number of identity characteristics and P represents the number of contrary characteristics. Furthermore, $\frac{S}{N}$ and $\frac{P}{N}$ are called identical degree and the contrary degree of these two sets respectively. Assuming $\frac{S}{N} = a$ and $\frac{P}{N} = c,$ the CN $μ (H, W) = (\frac{S}{N}) + (\frac{P}{N}) j$ can also be written as μ (H, W) = a + cj .

3 Kumar and Garg’s intuitionistic fuzzy MADM method

Kumar and Garg [11] proposed the following intuitionistic fuzzy MADM method for solving intuitionistic fuzzy MADM problems.

Step 1. Check that all the attributes of the intuitionistic fuzzy decision matrix D are of same type or not.

Case (i) If all the attributes are of same type then go to Step 2.

Case (ii) If some attributes are benefit attributes and others are cost attributes then normalize the intuitionistic fuzzy decision matrix by transforming the cost attributes into the benefit attributes, by the following way:

If the p ^th attribute is a cost attribute then replace all the elements 〈μ _ip, ν _ip〉 of the p ^th column of the decision matrix D with 〈ν _ip, μ _ip〉.

Step 2. Find the positive ideal scheme $〈 μ_{t}^{+}, ν_{t}^{+} 〉 = 〈 max_{1 \leq i \leq m} {μ_{it}}, min_{1 \leq i \leq m} {ν_{it}} 〉,$ t = 1 to n as well as the negative ideal scheme $〈 μ_{t}^{-}, ν_{t}^{-} 〉 = 〈 min_{1 \leq i \leq m} {μ_{it}}, max_{1 \leq i \leq m} {ν_{it}} 〉,$ t = 1 to n .

Step 3. Find the CN, a _it + c _it j, i = 1 to m, t = 1 to n where $a_{it} + c_{it} j = (a_{it}^{+} \times c_{it}^{-}) + (c_{it}^{+} \times a_{it}^{-}) j$ $\begin{matrix} = [(\frac{μ_{it}}{μ_{t}^{+}} \times \frac{ν_{t}^{+}}{ν_{it}}) \times (\frac{μ_{it} - μ_{t}^{-}}{μ_{it}} \times \frac{ν_{t}^{-} - ν_{it}}{ν_{t}^{-}})] \\ + [(\frac{μ_{t}^{+} - μ_{it}}{μ_{t}^{+}} \times \frac{ν_{it} - ν_{t}^{+}}{ν_{it}}) \times (\frac{μ_{t}^{-}}{μ_{it}} \times \frac{ν_{it}}{ν_{t}^{-}})] j . \end{matrix}$

Step 4. Find the CN, $μ_{i} = a_{i} + c_{i} j = \sum_{t = 1}^{n} (w_{t} a_{it}) + \sum_{t = 1}^{n} (w_{t} c_{it}) j, i = 1$ to m .

Step 5. Find, $T (μ_{i}) = \frac{a_{i}}{a_{i} + c_{i}}, i = 1$ to m and check that T (μ _p) > T (μ _q) or T (μ _p) < T (μ _q) or T (μ _p) = T (μ _q) .

Case (i) If T (μ _p) > T (μ _q) , then the preference order for the alternatives A _p and A _q is A _p > A _q .

Case (ii) If T (μ _p) < T (μ _q) , then the preference order for the alternatives A _p and A _q is A _p < A _q .

Case (iii) If T (μ _p) = T (μ _q) , then the preference order for the alternatives A _p and A _q is A _p = A _q.

4 Limitations of Kumar and Garg’s intuitionistic fuzzy MADM method

In this section, an intuitionistic fuzzy MADM problem, having four alternatives and two benefit attributes, is solved by Kumar and Garg’s intuitionistic fuzzy MADM method [11] to show that,

Kumar and Garg’s intuitionistic fuzzy MADM method [11] fails to rank two alternatives of the considered intuitionistic fuzzy MADM problem.

The obtained preference order of the remaining two alternatives is not correct.

Let D = 〈 α _ij 〉 = (〈 μ _ij, ν _ij 〉) _4×2 $\begin{matrix} G_{1} G_{2} \\ = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \end{matrix} [\begin{matrix} 〈 0.1, 0.2 〉 \\ 〈 0.2, 0.4 〉 \\ 〈 0.3, 0.6 〉 \\ 〈 0.4, 0.8 〉 \end{matrix} \begin{matrix} 〈 0.3, 0.4 〉 \\ 〈 0.3, 0.4 〉 \\ 〈 0.3, 0.4 〉 \\ 0.3, 0.4 \end{matrix}] \end{matrix}$ represents the intuitionistic fuzzy decision matrix of an intuitionistic fuzzy MADM problem, where A ₁, A ₂, A ₃ and A ₄ are the alternatives, G ₁ and G ₂ are the benefit attributes and α _ij =〈 μ _ij, ν _ij 〉 is the rating value of the ith alternative over the jth attribute.

Since, the rating value of all the alternatives over the second attribute G ₂ are same. Therefore, the preference order of the alternatives will depend only upon the rating value of all the alternatives over the attribute G ₁. Furthermore, as the rating value of all the alternatives over the attribute G ₁ are different therefore the preference order of any of these alternatives cannot be same i.e., A _i ≠ A _j for any value of i and j .

While, in this section, it is shown that if Kumar and Garg’s intuitionistic fuzzy MADM method [11] is applied to find the preference order of the alternatives of the considered intuitionistic fuzzy MADM problem then,

This method fails to evaluate the ranking of the alternatives A ₁ and A ₄.

The obtained ranking of the alternatives A ₂ and A ₃ are same, which is mathematically incorrect.

Using Kumar and Garg’s intuitionistic fuzzy MADM method [11], discussed in Section 3, the ranking of the alternatives for the considered intuitionistic fuzzy MADM problem can be obtained as follows:

Step 1. Since both the attributes are benefit attributes therefore according to Case (i) of Step 1 of Kumar and Garg’s intuitionistic fuzzy MADM method [11], there is no need to normalize the given intuitionistic fuzzy decision matrix.

Step 2. Using Step 2 of Kumar and Garg’s intuitionistic fuzzy MADM method [11] $(i) 〈 μ_{1}^{+}, ν_{1}^{+} 〉 = 〈 \begin{matrix} max {0.1, 0.2, 0.3, 0.4}, \\ min {0.2, 0.4, 0.6, 0.8} \end{matrix} 〉 = 〈 0.4, 0.2 〉 .$ $(ii) 〈 μ_{2}^{+}, ν_{2}^{+} 〉 = 〈 \begin{matrix} max {0.3, 0.3, 0.3, 0.3} \\ min {0.4, 0.4, 0.4, 0.4} \end{matrix}, 〉 = 〈 0.3, 0.4 〉 .$ $(iii) 〈 μ_{1}^{-}, ν_{1}^{-} 〉 = 〈 \begin{matrix} min {0.1, 0.2, 0.3, 0.4}, \\ max {0.2, 0.4, 0.6, 0.8} \end{matrix} 〉 = 〈 0.1, 0.8 〉 .$ $(iv) 〈 μ_{2}^{-}, ν_{2}^{-} 〉 = 〈 \begin{matrix} min {0.3, 0.3, 0.3, 0.3}, \\ max {0.4, 0.4, 0.4, 0.4} \end{matrix} 〉 = 〈 0.3, 0.4 〉 .$

Step 3. Using Step 3 of Kumar and Garg’s intuitionistic fuzzy MADM method [11],

The CN corresponding to the intuitionistic fuzzy number〈0.1, 0.2〉 is 0 + 0j .

The CN corresponding to the intuitionistic fuzzy number 〈0.2, 0.4〉 is $\frac{1}{16} + \frac{1}{16} j .$

The CN corresponding to the intuitionistic fuzzy number 〈0.3, 0.6〉 is $\frac{1}{24} + \frac{1}{24} j .$

The CN corresponding to the intuitionistic fuzzy number 〈0.4, 0.8〉 is 0 + 0j .

The CN corresponding to the intuitionistic fuzzy number 〈0.3, 0.4〉 is 0 + 0j .

Replacing the elements 〈μ _ij, ν _ij〉 of the intuitionistic fuzzy decision matrix D with the obtained corresponding CNs, the following decision matrix of CNs is obtained. $\begin{matrix} D = {(〈 μ_{ij}, ν_{ij} 〉)}_{4 \times 2} \\ G_{1} G_{2} \\ = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \end{matrix} [\begin{matrix} 0 + 0 j \\ \frac{1}{16} + \frac{1}{16} j \\ \frac{1}{24} + \frac{1}{24} j \\ 0 + 0 j \end{matrix} \begin{matrix} 0 + 0 j \\ 0 + 0 j \\ 0 + 0 j \\ 0 + 0 j \end{matrix}] \end{matrix}$

Step 4. Assuming the weights of the attributes G ₁ and G ₂ as w ₁ and w ₂ where w ₁, w ₂ ≥ 0 and w ₁ + w ₂ = 1 and using Step 4 of Kumar and Garg’s intuitionistic fuzzy MADM method [11], $\begin{matrix} μ_{1} = a_{1} + c_{1} j = w_{1} (0 + 0) \\ + w_{2} (0 + 0) j = 0 + 0 j, \\ μ_{2} = a_{2} + c_{2} j = w_{1} (\frac{1}{16} + 0) \\ + w_{2} (\frac{1}{16} + 0) j = \frac{w_{1}}{16} + \frac{w_{2}}{16} j, \\ μ_{3} = a_{3} + c_{3} j = w_{1} (\frac{1}{24} + 0) \\ + w_{2} (\frac{1}{24} + 0) j = \frac{w_{1}}{24} + \frac{w_{2}}{24} j, \\ μ_{4} = a_{4} + c_{4} j = w_{1} (0 + 0) \\ + w_{2} (0 + 0) j = 0 + 0 j . \end{matrix}$

Step 5. Using Step 5 of Kumar and Garg’s intuitionistic fuzzy MADM method [11], $(i) T (μ_{1}) = \frac{a_{1}}{a_{1} + c_{1}} = \frac{0}{0 + 0} = \frac{0}{0} .$ $(ii) T (μ_{2}) = \frac{a_{2}}{a_{2} + c_{2}} = \frac{\frac{1}{16}}{\frac{1}{16} + \frac{1}{16}} = \frac{1}{2} .$ $(iii) T (μ_{3}) = \frac{a_{3}}{a_{3} + c_{3}} = \frac{\frac{1}{24}}{\frac{1}{24} + \frac{1}{24}} = \frac{1}{2} .$ $(iv) T (μ_{4}) = \frac{a_{4}}{a_{4} + c_{4}} = \frac{0}{0 + 0} = \frac{0}{0} .$

Since, T (μ ₂) = T (μ ₃), so according to Step 5 of Kumar and Garg’s intuitionistic fuzzy MADM method [11], A ₂ = A ₃, which is mathematically incorrect as rating value of A ₂ over the attributes G ₁ is not equal to the rating value of alternative A ₃ over G ₁ Furthermore, the values of T (μ ₁) and T (μ ₄) are indeterminate values. Therefore, it is not possible to conclude any result about the ordering of A ₁ and A ₄.

5 Reasons for the occurrence of the limitations

Kumar and Garg’s intuitionistic fuzzy MADM method [11] fails to find the preference order of the alternatives for the intuitionistic fuzzy MADM problem, considered in Section 4, due to the following reasons.

It is obvious from Step 5 of Kumar and Garg’s intuitionistic fuzzy MADM method [11] that Kumar and Garg [11] have used the following method for comparing two CNs, μ ₁ = a ₁ + c ₁ j and μ ₂ = a ₂ + c ₂ j .

Find $T (μ_{1}) = \frac{a_{1}}{a_{1} + c_{1}}$ and $T (μ_{2}) = \frac{a_{2}}{a_{2} + c_{2}}$ and check that T (μ ₁) > T (μ ₂) or T (μ ₁) < T (μ ₂) or T (μ ₁) = T (μ ₂) .

Case (i) If T (μ ₁) > T (μ ₂) then μ ₁ > μ ₂ .

Case (ii) If T (μ ₁) < T (μ ₂) then μ ₁ < μ ₂.

Case (iii) If T (μ ₁) = T (μ ₂) then μ ₁ = μ ₂.

However, the following clearly indicates that it is not appropriate to use this method.

If μ ₁ = 0 +0j and $μ_{2} = \frac{1}{16} + \frac{1}{16} j$ then $T (μ_{1}) = \frac{0}{0}$ and $T (μ_{2}) = \frac{\frac{1}{16}}{\frac{1}{16} + \frac{1}{16}} = \frac{1}{2} .$ Since T (μ ₁) is an indeterminate value so none of the conditions T (μ ₁) > T (μ ₂) , T (μ ₁) < T (μ ₂) and T (μ ₁) = T (μ ₂) is satisfying. Hence, it is not possible to compare μ ₁ = 0 +0j and $μ_{2} = \frac{1}{16} + \frac{1}{16} j .$

If $μ_{1} = \frac{1}{16} + \frac{1}{16} j$ and $μ_{2} = \frac{1}{24} + \frac{1}{24} j$ then $T (μ_{1}) = T (μ_{2}) = \frac{1}{2}$ i.e., μ ₁ = μ ₂ . While, it is obvious that μ ₁ ≠ μ ₂ .

6 Proposed Mehar ranking method for comparing CNs and its validity

It is obvious from Section 5 that to overcome the limitations of Kumar and Garg’s intuitionistic fuzzy MADM method [11], there is need to propose a ranking method for comparing CNs. Keepincg the same in mind in this section, a new ranking method (named as Mehar ranking method) is proposed for comparing CNs, μ ₁ = a ₁ + c ₁ j and μ ₂ = a ₂ + c ₂ j .

The steps of the proposed Mehar ranking method are as follows:

Step 1. Check that a ₁ - c ₁ > a ₂ - c ₂ or a ₁ - c ₁ < a ₂ - c ₂ or a ₁ - c ₁ = a ₂ - c ₂

Case (i) If a ₁ - c ₁ > a ₂ - c ₂ then μ ₁ > μ ₂.

Case (ii) If a ₁ - c ₁ < a ₂ - c ₂ then μ ₁ < μ ₂.

Case (iii) If a ₁ - c ₁ = a ₂ - c ₂ then go to Step 2.

Step 2. Check that a ₁ + c ₁ > a ₂ + c ₂ or a ₁ + c ₁ < a ₂ + c ₂ or a ₁ + c ₁ = a ₂ + c ₂

Case (i) If a ₁ + c ₁ > a ₂ + c ₂ then μ ₁ > μ ₂.

Case (ii) If a ₁ + c ₁ < a ₂ + c ₂ then μ ₁ < μ ₂.

Case (iii) If a ₁ + c ₁ = a ₂ + c ₂ then μ ₁ = μ ₂.

To prove that the proposed Mehar ranking method is valid, it is sufficient to show that on comparing two CNs, μ ₁ = a ₁ + c ₁ j and μ ₂ = a ₂ + c ₂ j, with the help of proposed Mehar ranking method, the relation μ ₁ = μ ₂ will hold only if a ₁ = a ₂ and c ₁ = c ₂.

According to proposed Mehar ranking method, μ ₁ = μ ₂ ⇒ a ₁ - c ₁ = a ₂ - c ₂ as well as a ₁ + c ₁ = a ₂ + c ₂. It is obvious that both equations will be satisfied only if a ₁ = a ₂ and c ₁ = c ₂ i.e. μ ₁ = μ ₂ ⇒ a ₁ = a ₂ and c ₁ = c ₂.

7 Exact preference order of the alternatives for the considered intuitionistic fuzzy MADM problem

It is obvious from the results, discussed in Section 5, that Kumar and Garg’s intuitionistic fuzzy MADM method [11] fails to rank the alternatives due to using the inappropriate ranking method for comparing CNs. If in Step 5 of Kumar and Garg’s intuitionistic fuzzy MADM method [11], Mehar ranking method, proposed in Section 6, will be used instead of the existing ranking method then neither Kumar and Garg’s intuitionistic fuzzy MADM method [11] will fail to rank the alternatives nor on using Kumar and Garg’s intuitionistic fuzzy MADM method [11], incorrect ranking of the alternatives will be obtained.

To validate this claim the intuitionistic fuzzy MADM problem, considered in Section 4, is solved by Kumar and Garg’s intuitionistic fuzzy MADM method [11].

Using modified Kumar and Garg’s intuitionistic fuzzy MADM method [11], the exact preference order of the alternatives for the considered intuitionistic fuzzy MADM problem can be obtained as follows:

Step 1. Use Step 1 to Step 4 of Section 3, μ ₁ = a ₁ + c ₁ j = w ₁ (0) + w ₂ (0) j = 0 +0j, $μ_{2} = a_{2} + c_{2} j = w_{1} (\frac{1}{16}) + w_{2} (\frac{1}{16}) j,$ $μ_{3} = a_{3} + c_{3} j = w_{1} (\frac{1}{24}) + w_{2} (\frac{1}{24}) j$ and μ ₄ = a ₄ + c ₄ j = w ₁ (0) + w ₂ (0) j = 0 +0j .

Now according to Step 1 of the proposed Mehar ranking method, a ₁ - c ₁ = 0, $a_{2} - c_{2} = (w_{1} - w_{2}) \frac{1}{16},$ $a_{3} - c_{3} = (w_{1} - w_{2}) \frac{1}{24}$ and a ₄ - c ₄ = 0 .

Case (i) If w ₁ = w ₂, then a ₁ - c ₁ = a ₂ - c ₂ = a ₃ - c ₃ = a ₄ - c ₄ = 0 and hence according to Case (ii) of Step 1 proposed Mehar ranking method there is need to go to Step 2.

Case (ii) If w ₁ > w ₂ then a ₁ - c ₁ = a ₄ - c ₄ = 0 but a ₂ - c ₂ > a ₃ - c ₃ . Therefore according to Case (ii) of Step 2 of proposed Mehar ranking method A ₂ > A ₃ but to find the relation between A ₁ and A ₄ there is need to go to Step 2.

Case (iii) If w ₂ > w ₁ then a ₁ - c ₁ = a ₄ - c ₄ = 0 but a ₃ - c ₃ > a ₂ - c ₂ . Therefore according to Case (ii) of Step 2 of proposed Mehar ranking method A ₃ > A ₂ but to find the relation between A ₁ and A ₄ there is need to go to Step 2.

Step 2. a ₁ + c ₁ = 0, $a_{2} + c_{2} = \frac{1}{16},$ $a_{3} + c_{3} = \frac{1}{24}$ and a ₄ + c ₄ = 0 .

Since a ₁ + c ₁ = a ₄ + c ₄ and a ₂ + c ₂ > a ₃ + c ₃ for all values of w ₁ and w ₂. Therefore,

Case (i) If w ₁ = w ₂ then (A ₁ = A ₄) < A ₃ < A ₂ .

Case (ii) If w ₁ > w ₂ then A ₁ = A ₄ .

Case (iii) If w ₁ < w ₂ then A ₁ = A ₄ .

8 Conclusion

The limitations of Kumar and Garg’s intuitionistic fuzzy MADM method [11] are pointed out. Also, it is shown that these limitations are occurring due to using an inappropriate ranking method for comparing CNs. Furthermore, to overcome the limitations of Kumar and Garg’s intuitionistic fuzzy MADM method [11], a new ranking method (named as Mehar ranking method) is proposed for comparing CNs.

In future, the existing methods [4 , 18] for solving crisp multi-attribute group decision-making problems may be generalized for solving intuitionistic fuzzy multi-attribute group decision-making problems.

Footnotes

Acknowledgments

The authors would like to thank Associate Editor “Professor Yucheng Dong” and the anonymous Reviewers for their constructive suggestions which have led to an improvement in both the quality and clarity of the paper.

References

K.T.

Atanassov , Intuitionistic fuzzy sets, Fuzzy Sets Syst 20 (1986), 87–96.

F.E.

Boran ,

Genċ and

Akay , Personnel selection based on intuitionistic fuzzy sets, Hum Factors Ergon Manuf 21(5) (2011), 493–503.

Chang Jian , Application of the set pair analysis theory in multiple attribute decision-making, J Mech Strength 029 (2007), 1009–1012.

Chen ,

Zhang and

Dong , The fusion process with heterogeneous preference structures in group decision making: A survey, Inf Fus 24 (2015), 72–83.

Dong ,

Liu ,

Liang ,

Chiclana and Herrera-Viedma

, Strategic weight manipulation in multiple attribute decision making, Omega 75 (2018), 154–164.

Dong ,

Zha ,

Zhang ,

Kou ,

Fujita ,

Chiclana and

Herrera-Viedma , Consensus reaching in social network group decision making: Research paradigms and challenges, Knowl-Based Syst (2018) 10.1016/j.knosys.2018.06.036.

Dong ,

Zhan ,

Kou ,

Ding and

Liang , A survey on the fusion process in opinion dynamics, Inf Fus 43 (2018), 57–65.

Dong ,

Zhang and

Herrera-Viedma , Consensus reaching model in the complex and dynamic MAGDM problem, Knowl-Based Syst 106 (2016), 206–219.

Fu and

Zhou , Triangular fuzzy number multi-attribute decision-making method based on set-pair analysis, J Softw Eng 11(1) (2017), 116–122.

10.

Hu and

Yang , Dynamic stochastic multi-criteria decision making method based on cumulative prospect theory and set pair analysis, Syst Eng Procedia 1 (2011), 432–439.

11.

Kumar and

Garg , Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making, Appl Intell 48(8) (2018), 2112–2119.

12.

I.J.

Perez ,

F.J.

Cabrerizo ,

Alonso ,

Y.C.

Dong ,

Chiclana and

Herrera-Viedma , On dynamic consenseus processes in group decision-making problems, Inf Sci 459 (2018), 20–35.

13.

Rui ,

Zhongbin and

Anhua , Multi-attribute group decision making based on set pair analysis, Int J Advancements in Computing Technology 4(10) (2012), 205–213.

14.

J.Q.

Wang and

Gong , Interval probability stochastic multi-criteria decision-making approach based on set pair analysis, Control and Decision 24 (2009), 1877–1880.

15.

Xie ,

Zhang ,

Cheng and

Li , Fuzzy multi-attribute decision making methods based on improved set pair analysis, In: Sixth International Symposium on Computational Intelligence and Design 2, 2013, pp. 386–389

16.

L.A.

Zadeh , Fuzzy sets, Inf Control 8 (1965), 338–353.

17.

Zhang ,

Dong and

Chen , The 2-rank consensus reaching model in the multi-granular linguistic multiple-attribute group decision making, IEEE Trans Syst Man Cybern: Syst (2017). 101109/TSMC20172694429.

18.

Zhang ,

Palomares ,

Dong and

Wang , Managing non-cooperative behaviors in consensus-based multiple attribute group decision making: An approach based on social network analysis, Knowl.-Based Syst (2018). 10.1016/j.knosys.2018.06.008

19.

Zhao , Set pair and set pair analysis-a new concept and systematic analysis method, Proceedings of the National Conference on System Theory and Regional Planning, 1989, pp. 87–91.