Introduction
An important issue in multi-criteria decision-making (MCDM) is to obtain a reasonable ranking order of all alternatives. Due to the complexity of reality, fuzzy and uncertain information is naturally involved in MCDM process. For this reason, the theory of fuzzy set [
1], intuitionistic fuzzy set (IFS) [
2], interval-valued intuitionistic fuzzy set (IVIFS) [
3] and their applications have been put forward one after another with the development of research [
4‐
10]. However, Yager [
11] proposed such an example in real life: a decision maker may express his satisfaction with an alternative on a criterion is 0.6, but his dissatisfaction is 0.5. Because
\( 0.6 + 0.5 > 1\), the above special case cannot be modeled by the theory of IFS or IVIFS , which requires the sum of membership degree and the non-membership degree less than or equal to one [
12]. Therefore, a concept of Pythagorean fuzzy set (PFS) is introduced by Yager, of which the square sum of membership degree and non-membership degree is less than or equal to one [
11]. As extensions of PFS, Smarandache [
13] introduced the refined Pythagorean fuzzy sets, Ünver [
14] defined Spherical Fuzzy Sets and Zhang [
15] proposed a concept of interval-valued Pythagorean fuzzy set (IVPFS). As powerful tools to deal with vagueness and uncertainty involved in MCDM problems, theories and applications of these sets have recently been extensively studied in the literature. For instance, Ejegwa [
16] solved career placement problems under the Pythagorean fuzzy environment. Saeed et al. [
17] showed the properties, set-theoretic operations and axiomatic results for the refined Pythagorean Fuzzy Sets. For more details, please refer to [
18‐
26].
It is noteworthy that the ranking technique of fuzzy numbers is crucial in the fuzzy MCDM process [
15]. That is, if the ranking technique is not appropriate then no matter what fuzzy MCDM method you use, the results are unreliable or even completely inconsistent with reality [
27]. On the other hand, the interval-valued Pythagorean fuzzy numbers (IVPFNs) involves more uncertainties than other fuzzy numbers (e.g., intuitionistic fuzzy numbers (IFNs), interval-valued intuitionistic fuzzy numbers (IVIFNs), Pythagorean fuzzy numbers (PFNs),etc) which are usually able to adapt to higher degrees of uncertainty [
28]. In order to make the solutions to MCDM problems more reliable, it is necessary to develop a ranking method which not only ranks IVPFNs intuitively but also loses useful information as little as possible [
27]. Ever since IVPFNs’ appearance, many studies have focused on the ranking problems under interval-valued Pythagorean fuzzy environment. For instance, Zhang [
15] proposed a ranking method based on the closeness index of PFNs and IVPFNs and presented a Pythagorean fuzzy hierarchical qualitative flexible multiple criteria approach (QUALIFLEX) to solve the fuzzy MCDM problems. Although the method is relatively simple, it relies too much on the definition of distance for PFNs and IVPFNs. That is, different distances will get different ranking results which will bring some inconveniences in fuzzy MCDM process. Moreover, it is noteworthy that the score function and the accuracy function are important tools for ranking PFNs and IVPFNs [
15]. Zhang [
15] introduced the score function and accuracy function of IVPFNs, which generalized the definition for PFNs in [
29]. However, these definitions will lead to a certain loss of information, because they failed to consider the influence of hesitation of IVPFNs in fuzzy MCDM process under the interval-valued Pythagorean fuzzy environment. Therefore, a novel accuracy function of IVPFNs [
30], an improved accuracy function [
12] and an improved score function of IVPFNs [
28] were proposed by Garg considering the effect of hesitation interval index of IVPFNs. These above ranking methods have been widely used in the field of interval-valued Pythagorean fuzzy MCDM.
However, by browsing the literature, we find that there is one type of IVPFNs, the elements in which cannot be reasonably ranked in MCDM process using existing methods. This type of IVPFNs satisfies the following two properties: first, for each IVPFN, the lower limit of its membership degree is equal to the lower limit of its non-membership degree, and the upper limit of its membership degree is equal to the upper limit of its non-membership degree; second, the square sum of lower limit and upper limit of the membership degree of the one set is equal to the square sum of lower limit and upper limit of the membership degree of the other one. According to the existing methods, it can be concluded that these sets are equivalent even if they are completely different ones (see
Example 4 for details). In addition, we find the value of improved accuracy function for some IVPFNs may exceed one even if it is not the largest IVPFN (see
Example 5 for details). Obviously, all these results are not in line with reality. In view of the above analysis, it is necessary to propose a new ranking approach for IVPFNs from a new perspective to obtain a reasonable order between them.
The set pair analysis (SPA) theory is a new framework combining the certainty and uncertainty into a unified way [
31]. The connection number (CN) is a principal mathematical tool of SPA [
32]. It uses the degree of ’identity’, ’discrepancy’, and ’contrary’ to indicate the certainty, hesitancy, and uncertainty of a system, respectively [
33]. Since the SPA theory was proposed, researchers have done a lot of in-depth studies on its theory and applications under IFS and IVIFS environment [
34‐
37]. For example, based on the SPA theory, Garg and Kumar proposed some similarity measures of IFSs [
38] and some series of distance measures for IFSs [
39]. In [
40], they introduced a TOPSIS IVIFS MADM method in decision-making process using the SPA theory. And to rank different IVIFNs, Garg and Kumar proposed a new possibility measure of IVIFS based on the CNs of SPA [
41]. Kumar and Chen [
42] proposed a multi-attribute decision making method based on SPA under the interval-valued intuitionistic fuzzy environment and introduced a score function of connection numbers, and so on. Since IVPFS is the generalization of IVIFS, it can be inferred that the SPA theory can also be a useful tool to deal with uncertainty in MCDM process under interval-valued Pythagorean fuzzy environment. Unfortunately, we have not yet found any application of the SPA theory under the interval-valued Pythagorean fuzzy environment, let alone the research on ranking techniques and multi-attribute decision making methods under the interval-valued Pythagorean fuzzy environment. Moreover, the proposed score function of CNs [
42] has some shortcomings which are unable to get the reasonable sorting of alternatives in some MCDM processes under the interval-valued Pythagorean fuzzy environment (for details, see
Example2).
Motivated by above analysis, this paper first develops a novel ranking method for IVPFNs based on the SPA theory. That is, in order to get a reasonable order of IVPFNs in MCDM process, a technique to convert IVPFNs into CNs based on SPA is introduced at first by taking the hesitation interval index and Pythagorean property of IVPFNs into consideration properly. Then we propose an improved score function of CNs which can make the ranking order of CNs more in line with reality. The properties of the proposed score function of CNs and some examples are also given to illustrate the advantage of our proposed ranking method. Next, considering interactions among different criteria in the decision-making process, we propose a fuzzy MCDM approach under interval-valued Pythagorean fuzzy environment based on SPA and Choquet integral which is used to aggregate the evaluation information of criteria for each alternative.
Therefore, the innovations of this paper are summarized as follows :
(i)
A novel technique of converting IVPFNs into CNs based on SPA is proposed to rank IVPFNs in MCDM process from a new perspective for the first time in the literature. In addition, the idea of transformation fully takes into account the influence of hesitation interval index and Pythagorean property on information uncertainty under the interval-valued Pythagorean fuzzy environment.
(ii)
An improved score function of CNs is presented which can make the ranking order of CNs more in line with reality, and overcome the shortcomings of existing formulas;
(iii)
The aggregation of alternative evaluation information using SPA and Choquet integral considers uncertainty and interactions among criteria in MCDM under interval-valued Pythagorean fuzzy environment simultaneously.
This paper is organized as follows. The basic concepts of PFS, IVPFS, the SPA theory, the score function and accuracy function of CNs as well as fuzzy measure and Choquet integral are reviewed in the next section. The technique to convert IVPFNs into CNs and an improved score function of CNs are proposed and some examples are given to verify the advantage of the proposed converting method from IVPFNS into CNs in the following section. A novel MCDM approach based on SPA and Choquet integral under interval-valued Pythagorean fuzzy environment is introduced in the next section. An example of online learning satisfaction survey and a brief discussion and a comparative analysis with other existing methods are studied to illustrate the simplicity and viability of the proposed fuzzy MCDM approach in the following section. Conclusion of this paper is given in the last part.
Ranking method for IVPFNs based on SPA
In this section, we deal with the issue of ranking IVPFNs from a new perspective. First, based on the SPA theory, we propose a method to convert IVPFNs into CNs by taking the hesitation interval index and Pythagorean property into consideration simultaneously. Then an improved score function of CNs is given in the following part to make the ranking order of IVPFNs more in line with reality in fuzzy MCDM process.
Conversion from IVPFNs into CNs
Through the literature, we have three findings as follows:
(i)
The membership degree and non-membership degree of IVPFN are very close to the identity degree and contrary degree of CN respectively;
(ii)
The hesitation interval index of IVPFN is an important influence factor for MCDM problems with interval-valued Pythagorean fuzzy information.
(iii)
Since IVPFNs are the generalization of IVIFNs, the Pythagorean property should be fully considered to avoid the lack of uncertain information under interval-valued Pythagorean fuzzy environment as much as possible.
In view of the above three findings, we introduce the following definition of conversion from IVPFNs into CNs.
Proof For an IVPFN
\(P = \left\langle [a,b],[c,d] \right\rangle \), to prove that the CN
\( \mu _p = a_p+b_pi+c_pj\) given by Eqs. (
12–
14) is a rational CN, we need to prove that
\(0\le a_p \le 1,0\le b_p \le 1\) and
\(0\le c_p \le 1\). Since
\(a_p \ge 0, c_p \ge 0\), then
\(a_p + c_p \ge 0\). In view of
\(b_p = 1-a_p-c_p\), we just need to prove
\( a_p+c_p \le 1\).
In fact, since
\(0\le a^2+c^2 \le b^2+d^2 \le 1\), then
$$\begin{aligned} a_p +c_p= & {} \frac{a^2\sqrt{2-b^2-d^2}+b^2\sqrt{2-a^2-c^2}}{2}\\&+ \frac{c^2\sqrt{2-b^2-d^2}+d^2\sqrt{2-a^2-c^2}}{2}\\= & {} \frac{(a^2+c^2)\sqrt{2-b^2-d^2}+(b^2+d^2)\sqrt{2-a^2-c^2}}{2}\\\le & {} \frac{(a^2+c^2+b^2+d^2)}{2}\sqrt{2-a^2-c^2} \\= & {} \left( \sqrt{\frac{(a^2+c^2+b^2+d^2)}{2}}\right) ^2\sqrt{2-a^2-c^2}\\\le & {} \left( \frac{\frac{(a^2+c^2+b^2+d^2)}{2}.2+2-a^2-c^2}{3}\right) ^\frac{3}{2}\\= & {} \left( \frac{2+b^2+d^2}{3}\right) ^\frac{3}{2} \le 1^\frac{3}{2} = 1 \end{aligned}$$
The above result is obtained using Mean inequality
\(\big (\)i.e.
\(\frac{{x_1}^2+{x_2}^2+{x_3}^2}{3}\ge \root 3 \of {{x_1}^2{x_2}^2{x_3}^2}, \forall x_1,x_2,x_3 \ge 0\) \(\big )\). Since
\(0\le a_p+c_p \le 1\) as proved above, then
\( 0 \le a_p\le 1 \) and
\(0 \le c_p \le 1\). Recall that
\(b_p = 1-a_p-c_p\) , we also obtain
\(0\le b_p \le 1\).
\(\square \)
Obviously, for the largest IVPFN \(P_{max} = \left\langle [1, 1], [0, 0]\right\rangle \), the CN of \(P_{max}\) is \(\mu _{p_{max}} = 1+0i+0j\); for the smallest IVPFN \(P_{min} = \left\langle [0, 0], [1, 1]\right\rangle \), the CN of \(P_{min}\) is \(\mu _{p_{min} } = 0+0i+1j\); for \(P = \left\langle [0,0],[0,0]\right\rangle \), the CN of P is \(\mu _p = 0+i+0j\).
In the following, a simple example is given to show how to calculate the CN of an IVPFN.
Improved score function of CNs
Generally speaking, the methods for ranking CNs can be divided into two categories. One is to compare the identity degree and the discrepancy degree between CNs as mentioned before in Definition
7. However, by this method, only a partial order of IVPFNs can be obtained rather than a total order of IVPFNs. The other is to compare the values of score function of CNs to determine the ranking order of CNs as mentioned in Definition
8. However, as shown below, the score function in Definition
8 is unable to rank CNs correctly in some cases.
According to Definition
13, we can obtain the following properties of the proposed score function
\(S_F(\mu )\):
$$\begin{aligned} \begin{aligned} (P1 ) \mu&= 1 +0i +0j \Rightarrow S_F(\mu ) =1.\\ (P2 ) \mu&= 0 +1i +0j \Rightarrow S_F(\mu ) =0.\\ (P3 ) \mu&= 0 +0i +1j\Rightarrow S_F(\mu ) =-1.\\ (P4 ) \forall \mu&= a +bi +cj ( 0\le a, b, c\le 1,a+b+c=1)\\&\Rightarrow S_F(\mu ) \in [-1,1].\\ \end{aligned} \end{aligned}$$
Obviously, from above, we can obtain that the value of score function
\(S_F(\mu )\) of
\(\mu \) is between -1 and 1 for any CN
\(\mu = a +bi +cj\). In addition, the larger the value of
\(S_F(\mu )\), the more forward position of CNs will take in the ranking order. That is,
In the following, a few examples are presented to illustrate the advantage of proposed approach in this section.
In fact, let
\({\mathscr {P}} =\{ P_i =\left\langle [a_i,b_i],[c_i,d_i] \right\rangle | 0\le a_i\le b_i\le 1,0\le c_i\le d_i\le 1,b_i^2+d_i^2 \le 1, i\in \{1,2,\cdots ,n\}\}\) be the set of all IVPFNs. Then
\({\mathbb {P}} =\{ P_i\in {\mathscr {P}} |a_i = c_i,b_i=d_i,i\in \{1,2,\cdots ,n\}\}\) is a special subset of
\({\mathscr {P}}\). If
\(P_i,P_j \in {\mathbb {P}} (i \ne j )\) satisfy
\(a_i^2+b_i^2 = a_j^2+b_j^2\), by those previous methods [
12,
28,
30], we get
\(P_i \sim P_j\) even if
\(P_i \ne P_j\) for any
\( i \ne j\). However, by the approach proposed in this section, a reasonable order between them can be obtained as shown in Example
4 above. In fact, the proposed approach in this section is suitable for all IVPFNs ranking problems in interval-valued Pythagorean fuzzy MCDM process. For example, by Eq. (
15), the score values of CNs
\(\mu _{p_1}\) and
\(\mu _{p_2}\) in Example
1 are
\(S_F(\mu _{p_1})=-0.1247\) and
\(S_F(\mu _{p_2})=-0.1712\). Recall the score values of CNs obtained above in Example
4\(--\)5, we get
\(S_F(\mu _{p_2})<S_F(\mu _{p_1})<S_F(\mu _{p_5})< S_F(\mu _{p_6})<S_F(\mu _{p_3})< S_F(\mu _{p_4})<S_F(\mu _{p_8}) < S_F(\mu _{p_7})\), then we obtain the ranking order
\(P_2 \prec P_1 \prec P_5 \prec P_6 \prec P_3\prec P_4 \prec P_8\prec P_7\). That is, this proposed ranking approach is robust and it is more suitable for the ranking problem of IVPFNs than those previous methods. At the same time, as a ranking technique from a new perspective, the proposed approach is simple, effective and easy to implement.
Conclusions
IVPFS is a useful tool for dealing with uncertain information in MCDM process, while the SPA theory has the advantage of combining the certainty and uncertainty into a unified way. In view of these advantages, we propose a method to convert IVPFNs into CNs and introduce an improved score function of CNs for the first time. Furthermore, taking into account the uncertainty in decision-making and interactions between criteria simultaneously, we propose a fuzzy MCDM approach based on SPA and Choquet integral to get the overall degree of satisfaction for each alternative. The key contribution of this study is
(1)
a novel technique of converting IVPFNs into CNs is proposed considering the uncertainty and Pythagorean property of IVPFNs;
(2)
an improved score function of CNs is developed to overcome the shortcomings of previous formulas;
(3)
an effective ranking method for IVPFNs is introduced from a new perspective for the first time;
(4)
an interval-valued pythagorean fuzzy multi-critria decision-making method based on SPA and Choquet intergral is proposed by considering the interactions among criteria.
From examples shown above, we can see that the proposed approach successfully overcomes the drawbacks presented in [
12,
15,
28,
30,
42,
43,
49]. The proposed approach provides us with a very useful way to deal with MCDM problems in the IVPFSs context. At the same time, examples show that the proposed approach is simple and easy to implement in interval-valued Pythagorean fuzzy MCDM process.
However, the tendency of decision-makers towards fuzzy indexes (the membership, non-membership or hesitation) of IVPFSs has a significant impact on the outcome of the decision-making in many fields. Here the tendency refers to the decision-makers’ attitudes, habits, knowledge or experience, etc. In this paper, we failed to take into account the tendency of decision makers towards different fuzzy indexes of IVPFSs in the process of converting IVPFNs into CNs.In the future, on one hand, we will consider the tendency of decision makers in the process of converting IVPFNs into CNs, and on the other hand, we will also apply the proposed approach to more fields, such as pattern recognition, the security of industrial control, the big data of education, etc.
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