Improved possibility degree method for ranking intuitionistic fuzzy numbers and their application in multiattribute decision-making

Original Paper
  • 54 Downloads

Abstract

In the present paper, we define an improved possibility degree method to rank the different intuitionistic fuzzy numbers (IFNs). To achieve it, we first present some shortcomings of the existing possibility degree method and score function of IFNs. The existing shortcomings are overcome by proposing a new possibility degree measure for IFNs. The desirable properties of it are analyzed in details. Afterward, based on proposed possibility degree measure, a decision-making approach presents to solve the multiattribute decision-making (MADM) problem under the intuitionistic fuzzy set environment. Finally, a real-life case is studied to manifest the practicability and feasibility of the proposed decision-making method.

Keywords

Multiattribute decision-making Possibility degree measure Intuitionistic fuzzy set Informative measures 

1 Introduction

Decision-making (DM) is the process to choose or to select the best alternative based on their attributes. Many decision-making processes are based on a single attribute but some are based on more than one attribute known as multiattribute decision-making (MADM). During this process, the preference values provided by the decision maker imbued with uncertainty. To handle it, Zadeh (1965) introduced the concept of fuzzy set (FS) theory to deal with uncertain or ambiguous data by assigning a membership value corresponding to each element whose range is in between 0 and 1. After their successful study, researchers are engaged in their extensions and out of that, intuitionistic fuzzy set (IFS) and interval-valued intuitionistic fuzzy set (IVIFS) theories as proposed by Atanassov (1986) and Atanassov and Gargov (1989), respectively, which takes the degree of membership as well as non-membership simultaneously are widely used in the decision-making fields. During the last decades, the researchers are paying more attention to these theories and successfully applied it to the various situations in the decision-making process. The two important aspects of solving the MADM problem are: first, to design an appropriate function which aggregates the different preference of the decision makers into the collective ones. Second, to design appropriate measures to rank the alternatives. For the former part, an aggregation operator is an important part of the decision-making which usually takes the form of mathematical function to aggregate all the individual input data into a single one.

In the aspect of aggregation operators, Xu and Yager (2006) and Xu (2007a) developed some weighted geometric and averaging aggregation operators to aggregate the different preferences of the decision makers in the form of the intuitionistic fuzzy numbers (IFNs). Wang and Liu (2012) extended these operators using Einstein norm operations. Garg (2016a, c) presented some improved interactive weighted aggregation operators by considering the degree of the hesitancy during the analysis using the Einstein norm operations. Zhao et al (2010) presented some generalized aggregation operators for the intuitionistic fuzzy numbers. Wang and Liu (2013) developed some new intuitionistic fuzzy hybrid weighted averaging aggregation operators under the interval numbers. Wan et al (2015) developed new method for interval-valued IFS to solve the decision making problems with incomplete attribute weight information. Chen et al (2016a) developed fuzzy multiattribute group decision-making method based on intuitionistic fuzzy sets and the evidential reasoning methodology. He et al (2014) developed an intuitionistic fuzzy geometric interaction averaging operator. Jamkhaneh and Garg (2017) presented some new operations over the generalized intuitionistic fuzzy sets. However, apart from these, some other kinds of the information aggregation (Arora and Garg 2018; Kaur and Garg 2018b, a; Garg 2018) and granular computing (Pedrycz and Chen 2011, 2015) got many attentions in the applications of our daily life.

In the aspect of similarity (or distance) measures, Chen and Randyanto (2013) presented a similarity measures between the intuitionistic fuzzy sets. Chen and Chang (2015) developed some similarity measures between the IFSs based on the transformation techniques. Garg et al (2017) presented a generalized entropy measure of order \(\alpha\) and degree \(\beta\) under the IFS environment and applied to solve the decision-making problems. Singh and Garg (2017) developed the distance measures between the type-2 intuitionistic fuzzy sets. Xu (2007b) defined similarity measures of IFSs based on the geometric distance model, the set-theoretic approach and the matching function, respectively, and applied them to solve the decision-making problems. Joshi and Kumar (2014) developed a technique of ordered preferences (TOPSIS) method based on intuitionistic fuzzy entropy and distance measure for solving the multiattribute decision-making problems. Kumar and Garg (2016, 2017) developed a connection number, of the set pair analysis theory, based TOPSIS approach to solving the decision-making problems under the IFS and interval-valued IFS environment. Garg (2017a) presented the distance and similarity measures for the intuitionistic multiplicative preference relation and apply it to solve the decision-making problems. Chen et al (2016b) utilized the concept of the centroid points of the intuitionistic fuzzy numbers and based on it they presented a similarity measures between them. Rani and Garg (2017) presented some distance measures for complex intuitionistic fuzzy sets.

In the aspect of ranking the numbers, Xu and Da (2003) defined the possibility degree method for ranking the interval numbers. Garg (2016b) presented a generalized improved score function to rank the different interval-valued IFNs. Zhang et al (2009) presented possibility degree method to rank the different interval-valued IFNs. Wan and Dong (2014) proposes a new ranking method of interval-valued IFNs based on the possibility degree from the probability viewpoint and their corresponding decision-making method. Wei and Tang (2010) presented the possibility degree measure for IFNs. An overview of the possibility measures of interval-valued IFSs and their applications to multicriteria decision-making (Gao 2013; Dammak et al 2016).

It is observed from the above studies that the various measures such as score, accuracy or the possibility measures are used by the researchers to rank the different IFNs. However, in some certain cases, it is found that they are some sort of deficiencies in ranking the alternatives. For instance, if we considered two different IFNs with same score values then the existing degree of the possibility measure remains same, and hence it is unable to distinguish between them. An altar to these, possibility theory is one of the mathematical theories for dealing with the certain types of uncertainty and is an alternative to probability theory. Under it, possibility degree measure is used to compare the two different objects, which reflects the probability of one object to another object.

Since the ranking of intuitionistic fuzzy sets is very important for the intuitionistic fuzzy decision-making. So, by motivating from the above studies under the IFS environment, the present study enhances the possibility measures for intuitionistic fuzzy number (IFNs) by overcoming the shortcomings of the existing measures. For it, from the probability viewpoint, an improved possibility degree measure of comparison between two intuitionistic fuzzy numbers is defined using the notion of the two-dimensional random vector, and a new method is then developed to rank IFNs. The properties of the proposed improved measures are also investigated. Furthermore, based on the proposed measures, a decision-making approach is presented to solve the multiattribute decision-making problems under the IFS environment. Therefore, under the IFS environment, the objective of this paper is divided into the three parts: (1) to introduce the ranking methods named as possibility degree measure for comparing the different IFNs, (2) to establish a group decision-making approach based on the proposed ranking method, and (3) to illustrate the developed approach with a numerical example and feasibility, superiority, as well as the validate of it, is demonstrated through a numerical example.

The rest of this article is classified as follows: Sect. 2 describes briefly the concepts of IFS and the existing ranking methods for the intuitionistic fuzzy numbers. In Sect. 3, we present a new possibility degree method for comparing IFNs that overcomes the certain drawbacks of existing possibility degree method under some certain cases. The advantages, as well as certain properties of the proposed measures, are also investigated. In Sect. 4, a decision-making approach to solve the MADM problems is presented based on the proposed possibility degree measure under the IFS environment. Section 5 deals with an illustrative example to show the effectiveness, feasibility, and superiority of the approach. Finally, Sect. 6 concludes the paper.

2 Preliminaries

In this section, we define some basic concepts of IFS theory on a universal set X as follows:

Definition 1

(Atanassov 1986) An IFS A is defined as an ordered pair given by
$$\begin{aligned} A=\Big \{\langle x, \tau (x) , \theta (x) \rangle \mid x \in X \Big \}, \end{aligned}$$
where \(\tau (x), \theta (x) \in [0,1]\), represents the degrees of membership and non-membership of the element x to A, respectively, such that \(0 \le \tau (x)+\theta (x) \le 1\) holds for all x. The hesitation degree towards the element x to A is given by \(\pi (x)= 1-\tau (x)-\theta (x)\). Usually, this pair is denoted as \(\gamma = \langle \tau , \theta \rangle\) and called as an intuitionistic fuzzy number (IFN) with the condition that \(\tau ,\theta \in [0,1]\) and \(\tau + \theta \le 1\).

Definition 2

(Xu and Yager 2006) The score function of an IFN \(\gamma =\langle {\tau },{\theta } \rangle\) is defined as
$$\begin{aligned} S(\gamma )= \tau -\theta ; \quad S(\gamma )\in [-1,1] \end{aligned}$$
(1)
and an accuracy function is
$$\begin{aligned} H(\gamma )=\tau +\theta ; \quad H(\gamma )\in [0,1]. \end{aligned}$$
(2)

Definition 3

(Xu and Yager 2006) For two IFNs \(\gamma _{1}=\langle \tau _1, \theta _1 \rangle\) and \(\gamma _2=\langle \tau _2, \theta _2\rangle\), we have
  1. (i)

    \(\gamma _1 = \gamma _2\) if and only if \(\tau _1 = \tau _2\) and \(\theta _1 = \theta _2\);

     
  2. (ii)

    \(\gamma _{1}\succ \gamma _{2}\) if \(\tau _{1}>\tau _{2}\) and \(\theta _{1}<\theta _{2}\).

     

Definition 4

(Xu and Yager 2006) For a collection of two IFNs \(\gamma _t = \langle {\tau _t}, {\theta _t}\rangle (t=1,2)\), \(S(\gamma _t)\) and \(H(\gamma _t)\) be the scores values and accuracy degrees of \(\gamma _t (t=1,2)\), respectively. Then, the comparison law to compare them is given as
  1. (i)

    If \(S(\gamma _{1})> S(\gamma _{2})\) then \(\gamma _1\) is greater than \(\gamma _2\), denoted by \(\gamma _{1} \succ \gamma _{2}\);

     
  2. (ii)
    If \(S(\gamma _{1}) = S(\gamma _{2})\) then
    1. (a)

      If \(H(\gamma _{1})> H(\gamma _{2})\) then \(\gamma _{1} \succ \gamma _{2}\);

       
    2. (b)

      If \(H(\gamma _{1})=H(\gamma _{2})\) then \(\gamma _1\) is equivalent to \(\gamma _2\), denoted by \(\gamma _{1}\sim \gamma _{2}\).

       
     

Definition 5

(Xu and Yager 2006) For a collection of IFNs \(\gamma _{k} = \langle \tau _k, \theta _k\rangle (k=1,2,\ldots ,n)\), an intuitionistic fuzzy weighted geometric (IFWG) aggregation operator is defined as
$$\begin{aligned} \text {IFWG}(\gamma _{1},\gamma _{2},\ldots ,\gamma _{n})=\left\langle \prod \limits _{k=1}^{n}\tau _{k}^{\omega _{k}}, 1-\prod \limits _{k=1}^{n}(1-\theta _{k})^{\omega _{k}} \right\rangle , \end{aligned}$$
(3)
where \(\omega _{k}\) is the weight of \(\gamma _{k}\) such that \(\omega _{k}> 0\) and \(\sum \nolimits _{k=1}^{n}\omega _{k}=1.\)
Later on, Garg (2016a) proposed an improved geometric aggregation named as intuitionistic fuzzy Einstein weighted geometric interactive averaging (IFEWGIA) operator by considering the degree of the interaction between the pairs of the membership degrees which is defined as
$$\begin{aligned} \text {IFEWGIA}(\gamma _{1},\gamma _{2},\ldots ,\gamma _{n})= & {} \left\langle \frac{2\left\{ \prod \limits _{k=1}^{n}\left( 1-\theta _{k}\right) ^{\omega _{k}} - \prod \limits_{k=1}^{n}\left( 1-\tau _k-\theta _k\right) ^{\omega _k}\right\} }{\prod \limits_{k=1}^{n}\left( 1+\theta _k\right) ^{\omega _k} + \prod \limits_{k=1}^{n}\left( 1-\theta _k\right) ^{\omega _k}}, \right. \nonumber \\&\left. \frac{\prod \limits_{k=1}^{n}\left( 1+\theta _k\right) ^{\omega _k} - \prod \limits_{k=1}^{n}\left( 1-\theta _k\right) ^{\omega _k}}{\prod \limits_{k=1}^{n}\left( 1+\theta _k\right) ^{\omega _k} + \prod \limits_{k=1}^{n}\left( 1-\theta _k\right) ^{\omega _k}}\right\rangle . \end{aligned}$$
(4)

Definition 6

(Wei and Tang 2010) Let \(\gamma _{1}=\langle \tau _{1},\theta _{1} \rangle\) and \(\gamma _{2}=\langle \tau _{2},\theta _{2}\rangle\) be two IFNs then the possibility degree, denoted by \(p^\prime (\gamma _{1} \succeq \gamma _{2})\), of \(\gamma _{1} \succeq \gamma _{2}\) is defined as
$$\begin{aligned} p^{\prime }(\gamma _{1} \succeq \gamma _{2})=\frac{\max \{0,(\tau _{1}+\pi _{1})-\tau _{2}\}- \max \{0,\tau _{1}-(\tau _{2}+\pi _{2})\}}{\pi _{1}+\pi _{2}}, \end{aligned}$$
(5)
where either \(\pi _{1}\ne 0\) or \(\pi _{2}\ne 0\).

From this measure, it is observed that under some certain cases it is unable to distinguish the different IFNs which are illustrated with a numerical example as follows.

Example 1

Consider \(\gamma _{1}=\langle 0.5, 0.3 \rangle\) and \(\gamma _{2}=\langle 0.4 ,0.2\rangle\) be two IFNs whose score values are equal. If we apply the existing possibility degree method on these two numbers then we get
$$\begin{aligned} p^{\prime }(\gamma _{1} \succeq \gamma _{2}) =\,& {} \frac{\max \{0,(0.5+0.2)-0.4\}- \max \{0,0.5-(0.4+0.4)\}}{0.2+0.4}\\=\,& \frac{0.3}{0.6}\\=\,& 0.5. \end{aligned}$$
Similarly, we can get \(p^{\prime }(\gamma _{2}\succeq \gamma _{1})=0.5\).

Therefore, according to this, neither the score function (except the accuracy function) nor possibility degree measure distinguish the given IFNs. Hence, there is a need to develop some new ranking methods which can work efficiently under those cases where the existing score as well as the possibility degree measure fails to rank. For it, an improved possibility degree measure is introduced in the next section.

3 New possibility degree measure for IFNs

In this section, we present a new possibility degree measure to rank the different IFNs \(\gamma _k=\langle \gamma _k, \theta _k\rangle (k=1,2,\ldots ,n)\).

Definition 7

Let \(\gamma _{1}=\langle {\tau _1}, {\theta _1} \rangle\) and \(\gamma _{2}=\langle {\tau _2}, {\theta _2} \rangle\) be two IFNs, then the possibility degree measure, \(p(\gamma _{1} \succeq \gamma _{2})\), of \(\gamma _{1} \succeq \gamma _{2}\) is proposed as
$$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})= \min \left( \max \left( \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}},0\right) ,1\right) , \end{aligned}$$
(6)
where either \(\pi _{1}\ne 0\) or \(\pi _{2}\ne 0\).
On the other hand, if \(\pi _1 = \pi _2 = 0\), then we define
$$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})= {\left\{ \begin{array}{ll} 1; \quad &{} \tau _{1}>\tau _{2}\\ 0; \quad &{} \tau _{1}< \tau _{2}\\ 0.5; \quad &{} \tau _{1}= \tau _{2}. \end{array}\right. } \end{aligned}$$
(7)

Theorem 1

Let \(\gamma _{1}\) and \(\gamma _{2}\) be two IFNs, then
  1. (i)

    \(0 \le p(\gamma _{1} \succeq \gamma _{2}) \le 1\);

     
  2. (ii)

    \(p(\gamma _{1} \succeq \gamma _{2})= 0.5\) if \(\gamma _{1}=\gamma _{2}\);

     
  3. (iii)

    \(p(\gamma _{1} \succeq \gamma _{2}) + p(\gamma _{2}\succeq \gamma _{1})=1\).

     

Proof

  1. (i)
    \(p(\gamma _{1}\succeq \gamma _{2}) \ge 0\) is obvious, so we need to prove only \(p(\gamma _{1}\succeq \gamma _{2})\le 1\). For it, we take
    $$\begin{aligned} x= \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}} \end{aligned}$$
    Then, the following cases are arising:
    1. (a)
      If \(x \ge 1\), then
      $$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})= & {} \min \left( \max \left( x,0\right) ,1\right) \\= & {} \min (x,1)=1. \end{aligned}$$
       
    2. (b)
      If \(0< x < 1\), then
      $$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})= & {} \min \left( \max \left( x,0\right) ,1\right) \\= & {} \min (x,1)=x. \end{aligned}$$
       
    3. (c)
      If \(x \le 0\), then
      $$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})= & {} \min \left( \max \left( x,0\right) ,1\right) \\= & {} \min (0,1)=0. \end{aligned}$$
       
    Hence, in all cases, we have \(0 \le p(\gamma _{1} \succeq \gamma _{2}) \le 1\).
     
  2. (ii)
    Let \(\gamma _{1}=\langle {\tau _1},{\theta _1} \rangle\), \(\gamma _{2}=\langle {\tau _2},{\theta _2}\rangle\) be two IFNs. If \(\gamma _{1}=\gamma _{2}\), which implies that \(\tau _{1}=\tau _{2}\) and \(\theta _{1}=\theta _{2}\). Then, Eq. (6) becomes
    $$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})= & {} \min \left( \max \left( \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}},0\right) ,1\right) \\= & {} \min \left( \max \left( \frac{1+\tau _{1}-2\tau _{1}-\theta _{1}}{\pi _{1}+\pi _{1}},0\right) ,1\right) \\= & {} \min \left( \max \left( \frac{\pi _{1}}{2\pi _{1}},0\right) ,1\right) \\= & {} \min (\max (0.5,0),1)\\=\,& {} 0.5. \end{aligned}$$
     
  3. (iii)
    For two IFNs \(\gamma _1 = \langle \theta _1, \tau _1\rangle\) and \(\gamma _2 = \langle \theta _2, \tau _2\rangle\), we take
    $$\begin{aligned} x= & {} \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}},\\ y= & {} \frac{1+\tau _{2}-2\tau _{1}-\theta _{1}}{\pi _{1}+\pi _{2}}, \end{aligned}$$
    such that
    $$\begin{aligned} x+y= & {} \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}+1+\tau _{2}-2\tau _{1}-\theta _{1}}{\pi _{1}+\pi _{2}}\\= & {} \frac{1-\tau _{1}-\theta _{1} +1-\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}}\\= & {} \frac{\pi _{1}+\pi _{2}}{\pi _{1}+\pi _{2}}=1. \end{aligned}$$
    Then, the following three cases are arising:
    1. (a)
      If \(x \le 0\) and \(y \ge 1\) then
      $$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})+ p(\gamma _{2} \succeq \gamma _{1})= & {} \min \left( \max \left( x,0\right) ,1\right) +\min \left( \max \left( y,0\right) ,1\right) \\= & {} \min (0,1)+\min (y,1)\\= & {} 0+1=1. \end{aligned}$$
       
    2. (b)
      If \(0< x,y < 1\) then
      $$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})+ p(\gamma _{2} \succeq \gamma _{1})= & {} \min \left( \max \left( x,0\right) ,1\right) +\min \left( \max \left( y,0\right) ,1\right) \\= & {} \min (x,1)+\min (y,1)\\= & {} x+y=1. \end{aligned}$$
       
    3. (c)
      If \(x \ge 1\) and \(y \le 0\) then
      $$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})+ p(\gamma _{2} \succeq \gamma _{1})= & {} \min \left( \max \left( x,0\right) ,1\right) +\min \left( \max \left( y,0\right) ,1\right) \\= & {} \min (x,1)+\min (0,1)\\= & {} 1+0=1. \end{aligned}$$
       
    Hence, in all cases, we have \(p(\gamma _{1} \succeq \gamma _{2}) + p(\gamma _{2} \succeq \gamma _{1}) = 1\).
     
\(\square\)

Theorem 2

For two IFNs \(\gamma _{1}=\langle {\tau _1}, {\theta _1} \rangle\) and \(\gamma _{2}=\langle {\tau _2}, {\theta _2} \rangle\), the proposed possibility degree measure \(p(\gamma _{1} \succeq \gamma _{2})\) satisfies the following characteristics:
  1. (i)

    \(p(\gamma _{1} \succeq \gamma _{2})= 1\) if \(\tau _{1}-\pi _{1}\ge \tau _{2}\);

     
  2. (ii)

    \(p(\gamma _{1} \succeq \gamma _{2})=0\) if \(\tau _{2}-\pi _{2}\ge \tau _{1}\).

     

Proof

For two IFNs \(\gamma _{1}=\langle {\tau _1}, {\theta _1} \rangle\) and \(\gamma _{2}=\langle {\tau _2}, {\theta _2} \rangle\), we have
  1. (i)
    Let \(\tau _{1}-\pi _{1}\ge \tau _{2}\) then we have
    $$\begin{aligned} \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}}= & {} \frac{\tau _{1}-\tau _{2}+1-\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}}\\= & {} \frac{\tau _{1}-\tau _{2}+\pi _{2}}{\pi _{1}+\pi _{2}}\\\ge & {} \frac{\tau _{2}+\pi _{1}-\tau _{2}+\pi _{2}}{\pi _{1}+\pi _{2}}\\= & {} 1. \end{aligned}$$
    Therefore, \(\min \left( \max \left( \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}},0\right) ,1\right) =1\). Hence, \(p(\gamma _{1} \succeq \gamma _{2})=1\).
     
  2. (ii)
    Let \(\tau _{2}-\pi _{2}\ge \tau _{1}\) then we have
    $$\begin{aligned} \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}}= & {} \frac{\tau _{1}-\tau _{2}+1-\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}}\\= & {} \frac{\tau _{1}-\tau _{2}+\pi _{2}}{\pi _{1}+\pi _{2}}\\\le & {} \frac{\tau _{1}-\tau _{1}}{\pi _{1}+\pi _{2}} = 0. \end{aligned}$$
    Therefore, \(\min \left( \max \left( \frac{1+\tau _{1}-2\tau _{2}-\theta _{2}}{\pi _{1}+\pi _{2}},0\right) ,1\right) =0\). Hence, \(p(\gamma _{1} \succeq \gamma _{2})=0\).
     
\(\square\)

Example 2

Let \(\gamma _{1}=\langle 0.4, 0.2\rangle\) and \(\gamma _{2}=\langle 0.3,0.2\rangle\) be two IFNs then
$$\begin{aligned} p(\gamma _{1} \succeq \gamma _{2})= & {} \min \left( \max \left( \frac{1+0.4-2\times 0.3-0.2}{0.4+0.5},0\right) ,1\right) \\= & {} \min \left( \max \left( 0.6667,0\right) ,1\right) \\= & {} 0.6667. \end{aligned}$$
Further, to rank the different IFNs, the inclusion-comparison probability of IFNs \(\gamma _k \succeq \gamma _j\); \(k,j\in \{1,2,\ldots ,n\}\) is denoted by \(p(\gamma _k \succeq \gamma _j)\) and their corresponding likelihood possibility degree measure matrix is denoted by \(P=(p_{kj})_{n\times n}\) where \(p_{kj}=p(\gamma _k\succeq \gamma _j); (k,j = 1,2,\ldots ,n)\) given byNow, the ranking value which represents the optimal degrees of the membership for the numbers \(\gamma _k(k=1,2,\ldots ,n)\) is given as follows:
$$\begin{aligned} r_k= \frac{1}{n(n-1)}\left( \sum \limits _{j=1}^{n}p_{kj}+\frac{n}{2}-1\right) . \end{aligned}$$
(8)
Then, the ranking order of all alternatives \(\gamma _k,(k=1,2,\ldots ,n)\) is found according to decreasing order of the values of \(r_k\)’s, and hence choose the best alternative.

Example 3

If we apply the proposed possibility degree measure on the dataset as considered in the Example 1 for comparing \(\gamma _{1}\) and \(\gamma _{2}\), then we have
$$\begin{aligned} p(\gamma _{1}\succeq \gamma _{2})= & {} \min \left( \max \left( \frac{1+0.5-2\times 0.4-0.2}{0.2+0.4},0\right) ,1\right) \\= & {} \min \left( \max \left( 0.8333,0\right) ,1\right) \\= & {} 0.8333. \end{aligned}$$
and \(p(\gamma _{2}\succeq \gamma _{1})=0.1667\). Therefore, the possibility degree matrix is formulated using Eq. (6) as
$$\begin{aligned} P = \begin{pmatrix} 0.5000 &{} 0.8333 \\ 0.1667 &{} 0.5000 \\ \end{pmatrix}. \end{aligned}$$
Thus, the optimal value of the membership for the IFNs \(\gamma _k(k=1,2)\) is computed using Eq. (8) and get \(r_1 = 0.66665\) and \(r_2 =0.33335\). Since \(r_1>r_2\), thus we get \(\gamma _1 \succ \gamma _2\).

Example 4

Let \(\gamma _{1}=\langle 0.5,0.3 \rangle\), \(\gamma _{2}=\langle 0.4,0.2 \rangle\), \(\gamma _{3}=\langle 0.3,0.2 \rangle\) and \(\gamma _{4}=\langle 0.2,0.4\rangle\) are four IFNs. Clearly, it is seen from the definition of score and accuracy functions that ranking order of these numbers is \(\gamma _1\succ \gamma _2\succ \gamma _3\succ \gamma _4\). However, to validate the stability of the ranking order by our proposed possibility degree measure also, we construct the possibility degree matrix \(P=(p_{kt})_{4\times 4}\) using Eq. (6) asBased on this matrix, the optimal membership degrees of the numbers are computed using Eq. (8) and get \(r_{1}= 0.3611\), \(r_{2}= 0.2569\), \(r_{3}= 0.1991\), and \(r_{4} = 0.1829\). From these values, it is seen that \(r_1> r_2>r_3>r_4\) and thus the ranking order of the given IFNs is \(\gamma _1 \succ \gamma _2\succ \gamma _3\succ \gamma _4\), which is same as that of original ones. Hence, the proposed possibility degree measure is a valid.

4 DM approach based on proposed possibility degree measure method for solving MADM problems

In this section, we present a decision-making method based on proposed possibility degree measure for solving the MADM problems under the IFS environment.

For this, consider a decision-making problem which consists of ‘m’ different alternatives \(H_1,H_2,\ldots ,H_m\) which are evaluated under the set of ‘n’ different attributes \(G_1,G_2,\ldots ,G_n\) by a decision maker. Assume that a decision maker evaluates the given alternatives under the environment of IFS and characterized in terms of IFNs \(\widetilde{\gamma }_{kt}=\langle \widetilde{\tau }_{kt},\widetilde{\theta }_{kt}\rangle\). Where \(\widetilde{\tau }_{kt},\widetilde{\theta }_{kt} \in [0,1]\) and \(\widetilde{\tau }_{kt} + \widetilde{\theta }_{kt}\le 1\) for \(k=1,2,\ldots ,m; t=1,2,\ldots ,n\). Further, considering the importance of the given attributes with weight vector \(\omega = (\omega _1, \omega _2, \ldots , \omega _n)^T\) such that \(\omega _t>0\) and \(\sum \nolimits _{t=1}^n \omega _t=1\). Then, in the following, we develop a method based on the proposed possibility degree measure to solve the decision-making problems with IFS information, which involves the following steps.

Step 1 Arrange the collective information of the alternatives as given by the decision maker in the form of the decision matrix \(\widetilde{R}\) as

Step 2 Normalize the decision matrix, if required by converting the cost type attribute into the benefit type using Eq. (9) as
$$\begin{aligned} \gamma _{kt}= {\left\{ \begin{array}{ll} \langle \widetilde{\tau }_{kt},\widetilde{\theta }_{kt}\rangle ; ~ &{} \text { for benefit type attribute} \\ \langle \widetilde{\theta }_{kt},\widetilde{\tau }_{kt}\rangle ; ~ &{} \text { for cost type attribute}\\ \end{array}\right. } \end{aligned}$$
(9)
and get the normalized decision matrix \(R=(\gamma _{kt})_{m\times n}\).
Step 3 Utilize IFEWGIA operator to aggregate the different preferences \(\gamma _{kt}(t=1,2,\ldots ,n)\) of the alternatives into the collective one \(\gamma _k (k=1,2,\ldots ,m)\) as
$$\begin{aligned} \gamma _k=\,& {} \langle \tau _k, \theta _k \rangle \nonumber \\=\,& {} \text {IFEWGIA}(\widetilde{\gamma }_{k1}, \widetilde{\gamma }_{k2}, \ldots ,\widetilde{\gamma }_{kn}) \nonumber \\=\,& {} \left\langle \frac{2\left\{ \prod \limits _{t=1}^n \left( 1-\widetilde{\theta }_{kt}\right) ^{\omega _t}-\prod \limits _{t=1}^n \left( 1-\widetilde{\theta }_{kt}-\widetilde{\tau }_{kt}\right) ^{\omega _t}\right\} }{\prod \limits _{t=1}^n \left( 1+\widetilde{\theta }_{kt}\right) ^{\omega _t} + \prod \limits_{t=1}^n \left( 1-\widetilde{\theta }_{kt}\right) ^{\omega _t}}, \right. \nonumber \\& \left. \frac{\prod \limits_{t=1}^n \left( 1+\widetilde{\theta }_{kt}\right) ^{\omega _t} - \prod \limits _{t=1}^n \left( 1-\widetilde{\theta }_{kt}\right) ^{\omega _t}}{\prod \limits _{t=1}^n \left( 1+\widetilde{\theta }_{kt}\right) ^{\omega _t} + \prod \limits _{t=1}^n \left( 1-\widetilde{\theta }_{kt}\right) ^{\omega _t}} \right\rangle . \end{aligned}$$
(10)
Step 4 Compute the possibility degree matrix \(P=(p_{kj})_{m\times m}\) as where \(p_{kj}= p(\gamma _{k}\succeq \gamma _{j}) (k,j=1,2,\ldots ,m)\) is defined as
  1. (i)
    If either \(\pi _{k}\ne 0\) or \(\pi _{j}\ne 0\) then
    $$\begin{aligned} p(\gamma _{k}\succeq \gamma _{j})= & {} \min \left( \max \left( \frac{1+\tau _{k}-2\tau _{j}-\theta _{j}}{\pi _{k}+\pi _{j}},0\right) ,1\right) . \end{aligned}$$
     
  2. (ii)
    If \(\pi _{k}=\pi _{j}=0\) then
    $$\begin{aligned} p(\gamma _{k}\succeq \gamma _{j})= {\left\{ \begin{array}{ll} 1: \quad &{} \tau _{k}>\tau _{j}\\ 0: \quad &{} \tau _{k}< \tau _{j}\\ 0.5: \quad &{} \tau _{k}= \tau _{j} \end{array}\right. }. \end{aligned}$$
     
Further, it is clearly observed that \(0 \le p_{kj}\le 1\) and \(p_{kj} + p_{jk}= 1; \,(k, j = 1, 2, \ldots ,m)\) which implies that P is the fuzzy complementary judgement matrix.
Step 5 The ranking value which represents the optimal degree of the membership for alternative \(H_k(k=1,2,\ldots ,m)\) is computed using
$$\begin{aligned} r_k= \frac{1}{m(m-1)}\left( \sum \limits _{j=1}^{m}p_{kj}+\frac{m}{2}-1\right) . \end{aligned}$$
(11)
Thus, based on these membership values, the ranking order of all alternatives is found according to decreasing order of the values of \(r_k (k=1,2,\ldots ,m)\)’s, and hence choose the best alternative.

5 Illustrative example

Due to the increase in the population and the infrastructure, the major cities of the country are facing sensitive traffic problem these days. Traffic jam is one the most crucial problem in front of the development authority of the region during the peak hours. To save the time, money and fuel of the people as well as the society, Government wants to take necessary steps against it. Here we are considering the problem of the traffic jam in the state New Delhi, Capital of India. To reduce this problem, New Delhi development authority (NDDA) wants to build the flyover on the heavy-duty intersection. In this context, NDDA and the government taken a considerable number of the flyover project. For this, the government issued the global tender to select the contractor for these projects in the newspaper and considered the five major attributes required for its, namely “project cost” \((G_{1})\), “completion time” \((G_{2})\), “technical capability” \((G_{3})\), “financial status”’ \((G_{4})\) and “company background” \((G_{5})\). The weightage factor assigned to these attribute is \(\omega = (0.3, 0.25, 0.10, 0.15, 0.20)^T\). The four companies taken as in the form of the alternatives, namely “PNC Infratech Ltd.” \((H_{1})\),“ Hindustan construction company” \((H_{2})\), “J.P. Construction” \((H_{3})\), “Gammon India Ltd.” \((H_{4})\) bid for these projects. For it, an expert is invited to evaluate these considered alternatives with respect to the attributes under the IFS environment. Then, the aim of the NDDA and the Government is to recognize the best company among for the required task.

5.1 By proposed approach

To achieve the required target, the following steps of the proposed method are executed as below:

Step 1 The rating values of the expert toward the given alternatives \(H_k(k=1,2,3,4)\) are summarized in Eq. (12) using IFNs.
Step 2 Since \(G_1\) and \(G_2\) are the cost type attributes, so their rating values are normalized using Eq. (9) and their results are summarized in Eq. (13).

Step 3 Aggregate the given preferences into the collective ones by utilizing Eq. (10) and hence we get \(\gamma _{1}= \langle 0.5708 , 0.2922\rangle,\) \(\gamma _{2}= \langle 0.3681 , 0.4361\rangle,\) \(\gamma _{3}= \langle 0.4196, 0.4494\rangle\) and \(\gamma _{4}= \langle 0.4507 , 0.5493\rangle\).

Step 4 Utilize Eq. () to construct the possibility degree matrix as

Step 5 The optimal degrees of the membership for the alternatives \(H_k(k=1,2,3,4)\) can be computed by Eq. (11) and get \(r_1= 0.3647\), \(r_2 = 0.1453\), \(r_3 = 0.1880\) and \(r_{4}= 0.3019\). Since, \(r_{1}> r_{4}> r_{3}> r_{2}\), therefore, ranking order of alternatives is \(H_{1} \succ H_{4} \succ H_{3} \succ H_{2}\). Thus, the best alternative for the required task is \(H_{1}\).

5.2 Validity test

Wang and Triantaphyllou (2008) established the following testing criteria to evaluate the validity of MADM methods.

Test criterion 1 “An effective MADM method does not change the index of the best alternative by replacing a non-optimal alternative with a worse alternative without shifting the corresponding importance of every decision attribute”.

Test criterion 2 “To an effective MADM method must be satisfied transitive property”.

Test criterion 3 “If we decomposed a MADM problem into the sub DM problems and same MADM method is utilized on subproblems to rank alternatives, the collective ranking of alternatives must be identical to ranking of un-decomposed DM problem”.

Validity of the proposed MADM approach is tested using these criteria as follows.

5.2.1 Validity test by criterion 1

Under this test, if we replace the rating values of any non-optimal alternative by its arbitrary worse rating values, then the best alternative remains the best after applying the proposed MADM method. To validate it, we arbitrary choose an non-optimal alternative (\(H_{4}\)) and replace their rating value with any worse rating values which are denoted by \(H_4^\prime\) define as \(H_4^\prime = \{\langle 0.5 ,0.3 \rangle\), \(\langle 0.2,0.4 \rangle\), \(\langle 0.5, 0.4 \rangle\), \(\langle 0.3, 0.5 \rangle\), \(\langle 0.1,0.8\rangle \}\). Therefore, the complete rating value of the alternatives is summarized as follows:Now, by applying the proposed decision-making method to this transformed data then the optimal degree of the membership for the alternatives \(H_k(k=1,2,3,4)\) can be obtained using Eq. (11), and hence get \(r_1 = 0.3750\), \(r_2 = 0.2032\), \(r_3= 0.2703\) and \(r_4^\prime = 0.1514\), respectively. Therefore, the ranking order of the alternatives is \(H_{1}\succ H_{3}\succ H_{2}\succ H_{4}^{\prime }\) with the ranking order of original problem and it validate the test criterion 1.

5.2.2 Validity test by criteria 2 and 3

Under this test, we decomposed the original decision-making problem into three subproblems which contain the alternatives \(\{H_{1},H_{2},H_{4}\}\), \(\{H_{1},H_{3},H_{4}\}\) and \(\{H_{2},H_{3},H_{4}\}\). If we apply the proposed MADM approach on these sub decision-making problems then we get ranking order of alternatives as \(H_{1}\succ H_{4} \succ H_{2}\), \(H_{1}\succ H_{4} \succ H_{3}\) and \(H_{4}\succ H_{3} \succ H_{2}\), respectively. After combining together the ranking of the alternatives of these smaller problems, we get the final ranking order as \(H_{1}\succ H_{4}\succ H_{3} \succ H_{2}\) which is same as un-decomposed DM problem and show transitive property. Hence, the proposed DM approach is valid under the test criteria 2 and 3.

5.3 Superiority of the proposed approach over the existing approaches

Here, we present some examples which shows that under some certain cases, the existing approaches are unable to classify the alternatives while the proposed one does.

Example 5

Assume that there are three alternatives \(H_1, H_2\) and \(H_3\) which are assessed under the set of four attributes \(G_1,G_2\), \(G_3\) and \(G_4\) with their weight vectors 0.2, 0.3, 0.3 and 0.2, respectively. An expert evaluates these alternatives and represents their information in the form of an intuitionistic fuzzy decision matrix \(D_1\) as follows:If we followed an aggregation operator IFWG (Xu and Yager 2006) to aggregate these values then we get the aggregated value of each alternative as \(\gamma _1 = \langle 0.3704, 0.2878\rangle\), \(\gamma _2= \langle 0.3704, 0.2878\rangle\), \(\gamma _3= \langle 0.3704, 0.2878\rangle\) and \(\gamma _4= \langle 0.3704, 0.2878\rangle\), respectively. Thus, using the existing possibility degree measure as mentioned in Wei (2010), we construct their possibility degree matrix as
$$\begin{aligned} P_1^{\prime }= \begin{pmatrix} 0.5000 &{} 0.5000 &{} 0.5000\\ 0.5000 &{} 0.5000 &{} 0.5000\\ 0.5000 &{} 0.5000 &{} 0.5000 \end{pmatrix}. \end{aligned}$$
Therefore, the optimal degree of the alternatives is computed using Eq. (11), and hence get \(r_1 = 0.5\), \(r_2=0.5\) and \(r_5=0.5\) which implies that \(H_1 \sim H_2\sim H_3\). However, clearly seen from the input argument that \(H_1\not \sim H_2\not \sim H_3\), and therefore, Wei (2010) approach is unable to classify these alternatives.
On the other hand, if we apply the proposed decision-making approach to the above considered data, then by following the step 3 of the proposed approach, we get the aggregated values of each alternative as \(\gamma _1 = \langle 0.4118, 0.2780\rangle\), \(\gamma _2 = \langle 0.4036, 0.2780\rangle\) and \(\gamma _3 = \langle 0.4214, 0.2780\rangle\). The possibility degree matrix corresponding to these alternatives is computed using step 4 of the proposed approach and get
$$\begin{aligned} P_1 = \begin{pmatrix} 0.5000 &{} 0.5194 &{} 0.4763 \\ 0.4806 &{} 0.5000 &{} 0.4569 \\ 0.5237 &{} 0.5431 &{} 0.5000 \end{pmatrix}. \end{aligned}$$
Thus, the optimal degrees of each alternative are obtained using Eq. (11), and hence we get \(r_{1}= 0.3326\), \(r_{2} = 0.3229\) , \(r_{3}= 0.3445\). Since, \(r_{3}>r_{1}>r_{2}\) and thus the ranking order of the alternatives is \(H_{3}\succ H_{1} \succ H_{2}\). Therefore, \(H_{3}\) is the best alternative.

Example 6

Consider two alternatives \(H_1\) and \(H_2\) which are evaluated by an expert under the set of the attributes \(G_1\) and \(G_2\) with their weight vectors 0.5 and 0.5, respectively. The rating values of these alternatives given by an expert are summarized in the form of the decision matrix \(D_2 = (\gamma _{kt} )_{2\times 2}\), as follows:Using this information, the collective values of the alternative \(H_k(k=1,2)\) are computed using IFWG operator and get \(\gamma _{1}=\langle 0.24,0.52\rangle\) , \(\gamma _{2}=\langle 0.00,0.28\rangle\). Thus, the possibility degree matrix is constructed using Wei (2010) measure and we get
$$\begin{aligned} P_2^{\prime }= \begin{pmatrix} 0.5000 &{} 0.5000 \\ 0.5000 &{} 0.5000 \\ \end{pmatrix}. \end{aligned}$$
From this matrix, the optimal values of the membership degree of the alternative are obtained as \(r_1=r_2=0.5\). Thus, the possibility degree measure as proposed by Wei (2010) gives us that \(H_1\sim H_2\). However, clearly seen from the input argument that \(H_1\not \sim H_2\), and therefore, Wei (2010) approach is unable to distinguish between the alternatives.
On the other hand, if we apply the steps of the proposed decision making approach to this considered information then we get the possibility degree matrix as
$$\begin{aligned} P_2= \begin{pmatrix} 0.5000 &{} 0.8248 \\ 0.1752 &{} 0.5000 \\ \end{pmatrix}. \end{aligned}$$
Thus, the optimal degree of the membership for the alternatives \(H_k(k=1,2)\) using Eq. (11) is obtained as \(r_{1}= 0.6624 ,r_{2} = 0.3376\). Since \(r_{1}> r_{2}\) therefore ranking order of alternatives \(H_{1}\succ H_{2}\). Thus, we get \(H_{1}\) is the best alternative.

Example 7

Consider a set of two alternatives \(\{H_1,H_2\}\) which is evaluated under the set of three attributes \(\{G_1,G_2,G_3\}\) by taking an equal weightage to these attributes. For this, an external expert evaluate these under the IFS environment, and hence their rating values are summarized in the form of decision matrix \(D_3\) asBy applying the existing possibility degree measure Wei (2010) to this information, we construct the possibility degree matrix corresponding to the two alternatives \(H_k(k=1,2)\) as
$$\begin{aligned} P_3^{\prime }= \begin{pmatrix} 0.5000 &{} 0.5000 \\ 0.5000 &{} 0.5000 \\ \end{pmatrix}. \end{aligned}$$
Therefore, the optimal measurement values for the alternatives \(H_1\) and \(H_2\) are obtained as 0.5 and 0.5. Clearly, from it we are unable to select the best one. On the other hand, by proposed approach with equal weightage to each attribute we get the possibility degree matrix as
$$\begin{aligned} P_3 = \begin{pmatrix} 0.500 &{} 0.000 \\ 1.000 &{} 0.500 \\ \end{pmatrix}. \end{aligned}$$
Thus, the optimal value for the alternatives is obtained using Eq. (11) and get \(r_1=0.25\) and \(r_2=0.75\) which implies \(H_2\succ H_1\). Therefore. \(H_2\) is preferred over \(H_1\).

5.4 Comparative studies

To compare the efficiency of the proposed method with respect to some state of art, we implemented the various existing measures and the different aggregation operators on to the considered data. The analysis corresponding to these is summarized as below.
  1. 1.
    If we utilize the existing possibility degree measure, as proposed by Wei and Tang (2010) given in Eq. (5), to rank the alternatives then we construct the possibility degree matrix \(P^{\prime }= (p_{kk}^\prime )_{4\times 4}\) in the step 4 of the proposed approach using Eq. (5). The results corresponding to this measure is constructed as Thus, the optimal degree of the membership for the alternatives \(H_k(k=1,2,3,4)\) obtained using \(r_{k}=\frac{1}{4}\sum \nolimits _{j=1}^{4}p^{\prime }_{kj} (k=1,2,3,4)\) is \(r_{1}= 0.8723\), \(r_{2} = 0.4192\), \(r_{3} = 0.4832\), and \(r_{4} = 0.2253\). Since, \(r_{1}> r_{3}> r_{2}> r_{4}\), therefore, ranking order of alternatives is \(H_{1} \succ H_{3} \succ H_{2} \succ H_{4}\) and coincides with the proposed ones.
     
  2. 2.

    However, apart from this, if we compare the proposed approach results with some of the existing approaches such as aggregation operator (Xu 2007a; Garg 2016c; He et al 2014; Garg 2017b), TOPSIS method (Kumar and Garg 2017), distance measure (Joshi and Kumar 2014), possibility degree (Wei and Tang 2010), then their corresponding results along with their ranking order are summarized in Table 1. It is clearly seen from this table that the results obtained from the proposed approach coincide with these existing approaches, and hence conclude that the proposed measures and the approach can equivalency solve the decision-making problem. Furthermore, it is noted from this table that the final ranking order of the proposed approach is different from these approaches which are quite obvious. This is mainly due to the changeable decision environments. For instance, in the approaches (Xu 2007a; Garg 2016c; He et al 2014; Garg 2017b), weighted averaging and geometric aggregation operators were introduced and do not take into account the information related to the dependency factor between the pairs of the IFNs. Also, during the ranking, score function is used. On the other hand, the proposed method aggregates the different preferences using weighted geometric interactive aggregation operator and an improved possibility degree measure is used to rank them. In (Kumar and Garg 2017; Joshi and Kumar 2014; Wei and Tang 2010), authors used the distance or possibility measures to solve the decision-making problem. In these, measures are computed from their ideal alternatives without considering the interaction between them, and hence results may sometimes give an inconsistency. In addition, based on the comparison in Examples 5-7, we conclude that our proposed method have benefits over the existing methods.

     
Table 1

Comparison with existing approaches

Author

Rating values of alternatives

Ranking

 

\(H_{1}\)

\(H_{2}\)

\(H_{3}\)

\(H_{4}\)

 

Xu (2007a)

0.2522

− 0.1083

− 0.0638

− 0.2827

\(H_{1}\succ H_{3}\succ H_{2}\succ H_{4}\)

Wei and Tang (2010)

0.8723

0.4192

0.4832

0.2253

\(H_{1}\succ H_{3}\succ H_{2}\succ H_{4}\)

Kumar and Garg (2017)

0.9919

0.2532

0.5160

0.1158

\(H_{1}\succ H_{3}\succ H_{2}\succ H_{4}\)

Joshi and Kumar (2014)

0.8697

0.4242

0.5680

0.2083

\(H_{1}\succ H_{3}\succ H_{2}\succ H_{4}\)

Garg (2016c)

0.2854

− 0.0736

0.0455

− 0.3757

\(H_{1}\succ H_{3}\succ H_{2}\succ H_{4}\)

He et al (2014)

0.2639

− 0.0803

− 0.0562

− 0.1221

\(H_{1}\succ H_{3}\succ H_{2}\succ H_{4}\)

Garg (2017b)

0.2888

− 0.0688

0.0571

− 0.3647

\(H_{1}\succ H_{3}\succ H_{2}\succ H_{4}\)

Proposed method

0.3647

0.1453

0.1880

0.3019

\(H_{1}\succ H_{4}\succ H_{3}\succ H_{2}\)

6 Conclusion

This paper develops an improved possibility degree measure for ranking the different IFNs. In this proposed degree measure, we first overview some of the existing methods to rank the different alternatives and then we present some of the drawbacks of these under some certain cases during ranking the numbers. To improve it, in the present paper, a new possibility degree measure by incorporating the factors of the degree of hesitancy into the analysis are brought which successfully overcomes the drawbacks of the existing measures. The prominent features of the measure are studied. Furthermore, we utilized it to develop some approaches to solve the multiattribute decision-making problems with IFS environment. The effectiveness, feasibility, and superiority of the approach are evidenced by a numerical example and some comparison is given. A validity test is conducted to show the superiority of the proposed approach. In the future, the result of this paper can be extended to some other uncertain and fuzzy environments.

References

  1. Arora R, Garg H (2018) Prioritized averaging/geometric aggregation operators under the intuitionistic fuzzy soft set environment. Scientia Iranica 25(1):466–482.  https://doi.org/10.24200/SCI.2017.4410
  2. Atanassov K, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349MathSciNetCrossRefMATHGoogle Scholar
  3. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96CrossRefMATHGoogle Scholar
  4. Chen SM, Chang CH (2015) A novel similarity measure between atanassov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Inf Sci 291:96–114CrossRefGoogle Scholar
  5. Chen SM, Randyanto Y (2013) A novel similarity measure between intuitionistic fuzzy sets and its applications. Int J Pattern Recognit Artif Intell 27(7):1350021(34pages)Google Scholar
  6. Chen SM, Cheng SH, Chiou CH (2016a) Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology. Inform Fusion 27:215–227CrossRefGoogle Scholar
  7. Chen SM, Cheng SH, Lan TC (2016b) A novel similarity measure between intuitionistic fuzzy sets based on the centroid points of transformed fuzzy numbers with applications to pattern recognition. Inf Sci 343–344:15–40MathSciNetCrossRefGoogle Scholar
  8. Dammak F, Baccour L, Alimi AM (2016) An exhaustive study of possibility measures of interval-valued intuitionistic fuzzy sets and application to multicriteria decision making. Adv Fuzzy Syst 2016:10 (Article ID 9185,706)Google Scholar
  9. Gao F (2013) Possibility degree and comprehensive priority of interval numbers. Syst Eng Theory Pract 33(8):2033–2040Google Scholar
  10. Garg H (2016a) Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making. Comput Ind Eng 101:53–69CrossRefGoogle Scholar
  11. Garg H (2016b) A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems. Appl Soft Comput 38:988–999.  https://doi.org/10.1016/j.asoc.2015.10.040 CrossRefGoogle Scholar
  12. Garg H (2016c) Some series of intuitionistic fuzzy interactive averaging aggregation operators. SpringerPlus 5(1):999.  https://doi.org/10.1186/s40064-016-2591-9
  13. Garg H (2017a) Distance and similarity measure for intuitionistic multiplicative preference relation and its application. Int J Uncertain Quantif 7(2):117–133CrossRefGoogle Scholar
  14. Garg H (2017b) Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application. Eng Appl Artif Intell 60:164–174CrossRefGoogle Scholar
  15. Garg H (2018) Some arithmetic operations on the generalized sigmoidal fuzzy numbers and its application. Granul Comput 3(1):9–25.  https://doi.org/10.1007/s41066-017-0052-7 CrossRefGoogle Scholar
  16. Garg H, Agarwal N, Tripathi A (2017) Generalized intuitionistic fuzzy entropy measure of order \(\alpha\) and degree \(\beta\) and its applications to multi-criteria decision making problem. Int J Fuzzy Syst Appl 6(1):86–107CrossRefGoogle Scholar
  17. He Y, Chen H, Zhau L, Liu J, Tao Z (2014) Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making. Inf Sci 259:142–159MathSciNetCrossRefMATHGoogle Scholar
  18. Jamkhaneh EB, Garg H (2017) Some new operations over the generalized intuitionistic fuzzy sets and their application to decision-making process. Granul Comput.  https://doi.org/10.1007/s41066-017-0059-0
  19. Joshi D, Kumar S (2014) Intuitionistic fuzzy entropy and distance measure based topsis method for multi-criteria decision making. Egypt Inform J 15(2):97–104CrossRefGoogle Scholar
  20. Kaur G, Garg H (2018a) Cubic intuitionistic fuzzy aggregation operators. Int J Uncertain Quantif.  https://doi.org/10.1615/Int.J.UncertaintyQuantification.2018020471
  21. Kaur G, Garg H (2018b) Multi attribute decision-making based on bonferroni mean operators under cubic intuitionistic fuzzy set environment. Entropy 20(1):65.  https://doi.org/10.3390/e20010065 CrossRefGoogle Scholar
  22. Kumar K, Garg H (2016) TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment. Comput Appl Math.  https://doi.org/10.1007/s40314-016-0402-0
  23. Kumar K, Garg H (2017) Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making. Appl Intell.  https://doi.org/10.1007/s10489-017-1067-0
  24. Pedrycz W, Chen SM (2011) Granular computing and intelligent systems: design with information granules of higher order and higher type. Springer, HeidelbergCrossRefGoogle Scholar
  25. Pedrycz W, Chen SM (2015) Granular computing and decision-making: interactive and iterative approaches. Springer, Heidelberg, GermanyCrossRefGoogle Scholar
  26. Rani D, Garg H (2017) Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision-making process. Int J Uncertain Quantif 7(5):423–439MathSciNetCrossRefGoogle Scholar
  27. Singh S, Garg H (2017) Distance measures between type-2 intuitionistic fuzzy sets and their application to multicriteria decision-making process. Appl Intell 46(4):788–799CrossRefGoogle Scholar
  28. Wan S, Dong J (2014) A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making. J Comput Syst Sci 80(1):237–256MathSciNetCrossRefMATHGoogle Scholar
  29. Wan SP, Xu GL, Wang F, Dong JY (2015) A new method for atanassov’s interval-valued intuitionistic fuzzy magdm with incomplete attribute weight information. Inf Sci 316:329–347CrossRefGoogle Scholar
  30. Wang W, Liu X (2012) Intuitionistic fuzzy information aggregation using einstein operations. IEEE Trans Fuzzy Syst 20(5):923–938CrossRefGoogle Scholar
  31. Wang W, Liu X (2013) Interval-valued intuitionistic fuzzy hybrid weighted averaging operator based on einstein operation and its application to decision making. J Intell Fuzzy Syst 25(2):279–290MathSciNetMATHGoogle Scholar
  32. Wang X, Triantaphyllou E (2008) Ranking irregularities when evaluating alternatives by using some electre methods. Omega Int J Manag Sci 36:45–63CrossRefGoogle Scholar
  33. Wei CP, Tang X (2010) Possibility degree method for ranking intuitionistic fuzzy numbers. In: 3rd IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology (WI-IAT ’10), pp 142–145Google Scholar
  34. Wei G (2010) Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Appl Soft Comput 10:423–431CrossRefGoogle Scholar
  35. Xu ZS (2007a) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15:1179–1187CrossRefGoogle Scholar
  36. Xu ZS (2007b) Some similarity meeasures of intuitionistic fuzzy sets and their applications to multiple attribute decision making. Fuzzy Optim Decis Mak 6:109–121MathSciNetCrossRefMATHGoogle Scholar
  37. Xu ZS, Da QL (2003) Possibility degree method for ranking interval numbers and its application. J Syst Eng 18:67–70Google Scholar
  38. Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433MathSciNetCrossRefMATHGoogle Scholar
  39. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefMATHGoogle Scholar
  40. Zhang X, Yue G, Teng Z (2009) Possibility degree of interval-valued intuitionistic fuzzy numbers and its application. In: Proceedings of the international symposium on information processing, pp 33–36Google Scholar
  41. Zhao H, Xu Z, Ni M, Liu S (2010) Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 25(1):1–30CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsThapar Institute of Engineering and Technology (Deemed University)PatialaIndia

Personalised recommendations