Interval-valued Pythagorean fuzzy multi-criteria decision-making method based on the set pair analysis theory and Choquet integral.

This paper proposes a novel fuzzy multi-criteria decision-making method based on an improved score function of connection numbers and Choquet integral under interval-valued Pythagorean fuzzy environment. To do so, we first introduce a method to convert interval-valued Pythagorean fuzzy numbers into connection numbers based on the set pair analysis theory. Then an improved score function of connection numbers is proposed to make the ranking order of connection numbers more in line with reality in multi-criteria decision-making process. In addition, some properties of the proposed score function of connection numbers and some examples have been given to illustrate the advantages of conversion method proposed in the paper. Then, considering interactions among different criteria, we propose a fuzzy multi-criteria decision-making approach based on set pair analysis and Choquet integral under interval-valued Pythagorean fuzzy environment. Finally, a case of online learning satisfaction survey and a brief comparative analysis with other existing approaches are studied to show that the proposed method is simple,convenient and easy to implement. Comparing with previous studies, the method in this paper, from a new perspective, effectively deals with multi-criteria decision-making problems that the alternatives cannot be reasonably ranked in the decision-making process under interval-valued Pythagorean fuzzy environment.

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 , 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 : 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.

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.

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;

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.

Preliminaries

In this section, we first review the basic concepts of PFS, IVPFS, the SPA theory, fuzzy measure and Choquet integral.

Interval-valued Pythagorean fuzzy set

Definition 1

[11]. Let X be a universe of discourse, then a Pythagorean fuzzy set (PFS) P in X can be denoted aswhere , , for all The t(x) is the membership degree and m(x) is the non-membership degree of x, respectively, and is said to be the hesitancy degree of x to P. For convenience, is called a Pythagorean fuzzy number (PFN) by Zhang and Xu [29] with and .

Definition 2

[15]. Let X be a universe of discourse. An interval-valued Pythagorean fuzzy set (IVPFS) P in X is defined aswhere and such that for all . For convenience, an IVPFS can be expressed by an interval-valued Pythagorean fuzzy number (IVPFN) with the constrains , and and is denoted the hesitation interval index of P. Specially, for any , if , an IVPFN can reduce to a PFN .

Definition 3

[15]. Let be an IVPFN, a score function T of P can be shown asand an accuracy function of IVPFN P can be shown asSuppose there are two IVPFNs and , then Especially, if , then and ; if , then and .

( ’’ refer ’preferred to’);

, and

;

( ’’ refer ’equivalent to’).

Definition 4

[28]. An improved score function R of an IVPFN is denoted byIt is clear that . If and are two IVPFNs, then an order between and can be denoted as follows:

Definition 5

[12]. Let be an IVPFN. An improved accuracy function C of P is denoted bywhere . Let and be two IVPFNs, then the following order holds:

Set pair analysis and connection numbers

Definition 6

[31]. Let is a set pair consisting of two sets and for a given problem G . Suppose that there are total N features, I identical features , C contrary features , and discrepancy features in the problem G, respectively. The connection number (CN) of the set pair for the problem G is defined aswhere and show ’identity’, ’discrepancy’, and ’contrary’ degree between and , respectively. It is clear that , , and . In general, is the discrepancy coefficient; is the contrary coefficient.

Definition 7

[37]. For any two CNs and defined by Eq. (5), the following order between and holds:

Definition 8

[42]. For two CNs and defined by Eq. (5), a score function of can be denoted bywhere . In addition, the following order between and holds:

Fuzzy measure and Choquet integral

In the MCDM process, the criteria usually have interactions with each other [43]. To illustrate these interactions, a fuzzy measure (or a non-additive measure) of the criteria is proposed by Sugeno [44].

Definition 9

[44]. Let X is a universe of discourse. A fuzzy measure on X is a set function , satisfying the following conditions:where is the power of universe X.

Definition 10

[45]. A -fuzzy measure on a finite set satisfies the following conditions:where but .

From Eq. (8), we can obtainTherefore, can be uniquely determined by all and the Eq. (9).

Definition 11

[46] The discrete Choquet integral of function with respect to a fuzzy measure on X can be defined bywhere the subscript indicates a permutation on X such that and , .

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: In view of the above three findings, we introduce the following definition of conversion from IVPFNs into CNs.

The membership degree and non-membership degree of IVPFN are very close to the identity degree and contrary degree of CN respectively;

The hesitation interval index of IVPFN is an important influence factor for MCDM problems with interval-valued Pythagorean fuzzy information.

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.

Definition 12

For an IVPFN with , and , the CN of P can be determined bywhereand

Theorem 1

Let be an IVPFN with , and . Then the CN of P given by Eqs. (12–14) in Definition 12 is a rational CN.

Proof For an IVPFN , to prove that the CN given by Eqs. (12–14) is a rational CN, we need to prove that and . Since , then . In view of , we just need to prove .

In fact, since , thenThe above result is obtained using Mean inequality i.e. . Since as proved above, then and . Recall that , we also obtain .

Obviously, for the largest IVPFN , the CN of is ; for the smallest IVPFN , the CN of is ; for , the CN of P is .

In the following, a simple example is given to show how to calculate the CN of an IVPFN.

Example 1

There are two IVPFNs: and . Then the CN of by Eqs. (12–14) are That is, . In the same way, we get the CN of is .

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.

Example 2

Let and are three CNs. Using Eq. (6) in Definition 8, we get and . Since , we obtain . Obviously, this result is not only inconsistent with the result using Definition 7 ( by which we can get ), but also not in line with reality.

Therefore, to get a reasonable ordering of CNs, an improved score function of CNs is presented as follows.

Definition 13

Let be a CN denoted by Eq. (5). An improved score function of is defined as

According to Definition 13, we can obtain the following properties of the proposed score function :Obviously, from above, we can obtain that the value of score function of is between -1 and 1 for any CN . In addition, the larger the value of , the more forward position of CNs will take in the ranking order. That is,

Definition 14

For two CNs and , the following order between and holds:

Example 3

If we apply the score function as given in Eq. (15) to the Example 2, we get , and . Since , we obtain that the ranking order is . It is clear that the result is consistent with the result by Definition 7.

In the following, a few examples are presented to illustrate the advantage of proposed approach in this section.

Example 4

Suppose there are four different IVPFNs:Acording to Defination 3, we get the value of score function for the four IVPFNs by Eq. (1), and the value of accuracy function for them by Eq. (2). At the same time, according to Eq. (3) in Definition 4, we obtain . Therefore, we can get . Obviously, .

In the following, we use the proposed approach in this paper to reorder the four IVPFNs. That is, Therefore, the order of CNs is . It is noticed that the order is also consistent with the result by Definition 7.

Convert the four IVPFNs into CNs by Eqs. (12–14), then get

Calculate the score value of CNs by Eq. (15), then get

Compare the above score values, then get

Hence, based on above steps, the ranking order of the four IVPFNs is .

Example 5

Let and be two IVPFNs. According to the improved score function of IVPFNs defined in Eq. (4), we get and . Therefore, we obtain . However, it is obvious that , because is the largest IVPFN as mentioned before. In the following, we use the proposed method in this section to reorder the two IVPFNs. Similar to the steps in Example 4, Therefore, we obtain the order . It is clear that it is a reasonable order more in line with reality.

Convert and into CNs of them by Eqs. 12–14 and get

According to Eq. (15), calculate the score value of CNs, then get

Compare the above score values and get

In fact, let be the set of all IVPFNs. Then is a special subset of . If satisfy , by those previous methods [12, 28, 30], we get even if for any . 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 and in Example 1 are and . Recall the score values of CNs obtained above in Example 45, we get , then we obtain the ranking order . 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.

Interval-valued Pythagorean fuzzy multi-criteria decision making method

In this section, considering interactions among different criteria, we introduce a fuzzy MCDM method under interval-valued Pythagorean fuzzy environment by taking the advantages of proposed ranking technique above and Choquet integral.

Assume that, there are m alternatives, denoted by , and n criteria, indicated as . Let be the estimation for alternative i on criteria j. Here shows the degree of satisfaction, while is the degree of dissatisfaction of alternative i with respect to criteria j, where . In the following, to find the most desirable alternative in interval-valued Pythagorean fuzzy multi-criteria decision process, the specific steps of the proposed fuzzy MCDM method are given as follows: The complete flow chart of the proposed method is given in Fig. 1.

Fig. 1

Flow chart of the proposed method

Construct and normalize the interval-valued Pythagorean fuzzy decision matrix : where of which, represents the benefit type criteria subset, and is the cost type criteria subset.

Convert the above matrix into a CNs matrix: where is the CN of . In addition, are calculated by Eqs. (12–14) in Definition 12, respectively, for .

Calculate the value of score function of using Eq. (15) to obtain the score function matrix as follows:

Aggregate the above score values on each criterion to each alternative by Choquet integral. The specific implementation process is as follows:

Calculate the -fuzzy measure of each criterion set. In general, to obtain the -fuzzy measure of each criterion set, we should first get the weight, or the importance, of each criterion [43]. At present, there are three typical methods to determine the weight of criteria, i.e., the subjective method, the objective method and hybrid method [47]. No matter which method is used, the weight is obtained by a numerical value from 0 to 1 [46]. In this paper,we denote the weight of criteria j by . According to the Eq. (9), we get Then we can obtain the parameter by resolving Eq. (17) if we have get all in some way. Based on the parameter , the fuzzy measure of each criteria set can be derived by Eq. (7).

Calculate the Choquet integral of the score values of CNs for each alternative. According to the above fuzzy measures, we can aggregate the score values of CNs for each alternative with respect to each criterion as follows: for any . Here the subscript is a permutation on X such that and .

According to the aggregated values above, obtain a reasonable ranking order of all alternatives to find the most desirable one.

Discussion and comparative study

To illustrate the convenience and feasibility of our proposed approach, an example about the online learning satisfaction survey is given at first. Then the relevant discussion and comparative analysis between the proposed approach and other existing approaches [12, 15, 28, 30, 43, 49] are provided to validate the performance of the proposed approach in the paper.

Illustrative example

During the COVID-19 pandemic, a large number of Chinese universities implement large-scale online learning, which is a new model that can break the constraints of time and space for students. In the post-epidemic era, how to improve the effect of online learning model has become a very important topic. The online learning satisfaction survey is an important way that can help educators understand the real needs of students. The data in this section come from an online learning satisfaction survey conducted during the COVID-19 pandemic on a questionnaire survey platform [48]. By survey, it is found that the involved students can be divided into four categories according to the nature of their schools: research-oriented university (), application-oriented university (), higher vocational college (), and other types (). In this paper, we study the ranking order of online learning satisfaction for the four types of schools based on SPA and Choquet integral. We select four criteria for research, that is, the service of learning resource (), the communication between teachers and students (), the design of the course () and the communication between students (). By analysis, it is found that the four criteria are not independent(or say there are interactions among them). For example, the more extensive the service of learning resource, and the more interesting the teacher’s design of course, the more active the communication between teachers and students, and vice versa. In addition, the communication between teachers and students can also improve the communication between students, and vice versa. Therefore, the decision-making steps are described as below: Since , the ranking order of alternatives is . That is, the students from (application-oriented university) have the highest degree of satisfaction towards online learning. This result is consistent with the result of questionnaire survey.

Through analysis, it is found that all four criteria are benefit-type. Each type of school is given four evaluation values by their students which represent the degree of satisfaction on each criterion . The values are provided in the form of IVPFNs. Thus, the normalized Pythagorean fuzzy decision matrix can be obtained by Eq. (16): where the interval [0.5103, 0.7001] means the certain degree that alternative (research-oriented universities) satisfies (the service of learning resource), and [0.3565, 0.4103] is the uncertain degree that the alternative dissatisfies . The other values in have similar meanings.

Convert the above matrix into the CNs matrix by Eqs. (12–14), that is

Convert the above matrix into the score function matrix by Eq. (15):

The fuzzy measures of all the four criteria can be calculated by the method in [47]. This is an objective method by training a lot of data from the questionnaire survey platform [48]. The fuzzy measure of each criterion is obtained as follows [47]: Then, by Eq. (17), we get the parameter . And by Eq. (7), the fuzzy measure of each criteria subset can be obtained in Table 1.

Table 1

The fuzzy measures of criteria

C	g_λ(C)	C	g_λ(C)
{C_1}	0.35	{C_2}	0.25
{C_3}	0.20	{C_4}	0.18
{C_1,C_2}	0.63	{C_1,C_3}	0.56
{C_1,C_4}	0.50	{C_2,C_3}	0.58
{C_2,C_4}	0.43	{C_3,C_4}	0.48
{C_1,C_2,C_3}	0.75	{C_1,C_2,C_4}	0.70
{C_1,C_3,C_4}	0.78	{C_2,C_3,C_4}	0.74
{C_1,C_2,C_3,C_4}	1.00	∅	0

Calculate the overall satisfaction degree of all types schools based on Choquet integral by Eq. (18) as follows:

Discussion and comparison analysis

To illustrate the advantages of proposed approach in this paper, we conduct a brief discussion and a comparative analysis with some of the existing approaches [12, 15, 28, 30, 43, 49] under interval-valued Pythagorean fuzzy environment. These approaches corresponding to them are performed on the considered data and the rankings of alternatives which are summarized in Table 2. As shown in Table 2, it is easily observed that:

Table 2

Comparative analysis for examples in “Illustrative example”

Methodology	Core idea of method	Ordering
Garg [12]	Improved accuracy function	A_2≻A_3≻A_1≻A_4
Zhang [15]	Closeness index	A_2≻A_3≻A_1≻A_4
Garg [28]	Improved score function	A_2≻A_3≻A_1≻A_4
Garg [30]	Novel accuracy function	A_2≻A_3≻A_1≻A_4
Khan [43]	TOPSIS	A_2≻A_3≻A_1≻A_4
Khan [49]	IVPFCIA	A_3≻A_2≻A_1≻A_4
Proposed	SPA	A_2≻A_3≻A_1≻A_4

The approach proposed in this paper, from a new perspective, that is, SPA, deals with fuzzy information in the interval-valued Pythagorean fuzzy multi-criteria decision-making problems. To our knowledge, Zhang [15] proposed a closeness index for IVPFNs and presented the closeness index-based ranking method (QUALIFLEX MCDM approach) for ranking IVPFNs. In addition, Garg [12, 28, 30] proposed a novel accuracy function or an improved score function for IVPFNs for solving MCDM problems under interval-valued Pythagorean fuzzy environment. Khan et al. [43, 49] introduced an extension of TOPSIS method and a fuzzy GRA method under interval-valued Pythagorean fuzzy environment. Essentially, they are still ranking IVPFNs based on score function and accuracy function of IVPFNs [15] before integration process by the Choquet integral. However, the main idea of the approach proposed in this paper is to convert IVPFNs into CNs using SPA theory before integrating the values of alternatives by Choquet integral. Comparing with the existing approaches, this idea of transformation is first given under the general interval-valued Pythagorean fuzzy environment in the literature. According to the decision-making processes and results in Table 2, the proposed approach in this paper not only can effectively solve the MCDM problems under interval-valued Pythagorean fuzzy environment, but also it is simple, convenient and easy to implement. Meanwhile, the result of our proposed approach is consistent with most existing approaches.

The approach proposed in this paper can deal with the interval-valued Pythagorean fuzzy multi-criteria decision-making problems not only under the general interval-valued Pythagorean fuzzy environment, but also in some special cases which cannot be solved by existing methods [15, 28, 43, 49]. For example, suppose it is still the online learning satisfaction survey question in the above section and the fuzzy measure of all four criteria is also given by Eq. 19. Suppose that the evaluation values of an alternative on four criteria are in Example 4, respectively. Since the values of score function for the four IVPFNs are and the accuracy functions of them are , and the closeness index (for details,see [15]), we can get . In this case, the Choquet integral cannot be performed, because IVPFNs cannot be ranked reasonably. Therefore, the approaches [15, 28, 43, 49] are invalid. However, according to the proposed approach in this paper, we get , then as mentioned in Example 4. By simple calculation, we get the Choquet integral of the alternative is Therefore, the proposed approach in this paper is robust and can obtain the overall satisfaction evaluation from a new perspective.

The approach proposed in this paper takes the interactions among criteria into account using Choquet integral in MCDM under interval-valued Pythagorean fuzzy environment. While the existing approaches [12, 28, 30] can only deal with the MCDM problem in which the relationships among criteria are assumed to be independent in advance. At the same time, from the result of comparison which is shown in Table 2, we find that the result of our proposed approach is consistent with the method made by Zhang [15] and Khan [43] which takes into account the interaction between the criteria in MCDM under interval-valued Pythagorean fuzzy environment.

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 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.

a novel technique of converting IVPFNs into CNs is proposed considering the uncertainty and Pythagorean property of IVPFNs;

an improved score function of CNs is developed to overcome the shortcomings of previous formulas;

an effective ranking method for IVPFNs is introduced from a new perspective for the first time;

an interval-valued pythagorean fuzzy multi-critria decision-making method based on SPA and Choquet intergral is proposed by considering the interactions among criteria.

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.