Keywords

1 Introduction

Recent developments in information processing have heightened the need for a highly efficient information aggregation operators. The Choquet integral is one of the aggregation operators in which it is used to aggregate information and calculate the global score. It was introduced by Choquet [10] as an aggregation operator to solve the interrelationship among criteria of decision problems where ordering of the individual criteria is the ultimate result. Ordering of criteria is more challenging particularly when there exists some uncertainty regarding the criteria. In light of this difficulty, the concept of fuzzy measure is used in Choquet integral where interaction phenomena among the decision criteria can be modeled [40]. The Choquet integral uses the concept of fuzzy measure to indicate the weights or the importance of multiple interdependent criteria in decision making [12]. The main idea of Choquet integral is that the interrelationship between criteria can be modeled through a fuzzy measure where assigning a weight is not only to each criterion but also to each subset of criteria [11, 36]. In other words, information about criteria is expressed via a fuzzy measure. It is a non-additive fuzzy integral or a numerical-based approach where interactivity of subjective judgment experts can be eliminated. The Choquet integral encompasses the property of non-additive capacity and corresponds to a large class of aggregation functions [8, 34]. Sub additive or super additive operators are used to integrate functions with respect to the fuzzy measures where many extensions and generalizations of fuzzy sets could be inserted into fuzzy measures.

One of the generalizations of fuzzy sets is intuitionistic fuzzy sets (IFS). Atanassov [5] generalized fuzzy sets to IFS. Element in IFS is expressed by an ordered pair, and each ordered pair is characterized by a membership degree and non-membership degree. The sum of the two degrees must be less than or equal to one [5]. The IFS has been received much attention since its inception and broadly applied as an aggregation operator. In the context of fuzzy measures, instead of using fuzzy sets, Tan and Chen [42] used IFS to propose IFS Choquet integral based on t-norms and t-conorms. Murofushi and Sugeno [33] used the Choquet integral to propose the interval-valued IFS correlated averaging operator and interval-valued IFS correlated geometric operator to aggregate interval-valued IFS information and applied them to a practical decision making problem. Wang et al. [49] proposed a method based on intuitionistic fuzzy dependent aggregation operators and applied them to supplier selection. Recently, Liang et al. [31] proposed arithmetical average operator and geometric average operator based on IFS to aggregate the intuitionistic fuzzy information. Abdullah et al. [4] propose a combination of interval-valued IFS and Choquet integral to allow a strong interrelationship between criteria of a decision making model. In a case of personnel matching management, very recently Yu and Xu [51] proposed novel IFS Choquet integral aggregation. This aggregation operator is integrated with multi-objective decision-making model to find optimal personnel matching results.

It can be seen that these aggregation operators are built using the dual memberships of IFS where the sum of these two memberships is less than or equal to one. However, in some circumstances, the sum of membership and non-membership degree of criteria could be greater than one. This situation could not be described by the IFS. In order to address this problem, Yager [50] introduced another generalization of fuzzy set which is called as Pythagorean fuzzy set (PFS). Unlike IFS, the sum squares of PFS memberships is less than or equal to one. The PFS has emerged as an effective tool compared to IFS in depicting uncertainty of criteria in decision problems. For example, Chen [9] proposed Chebyshev distance measures in the ELimination Et Choice Translating REality (ELECTRE) method for addressing multiple criteria decision-making problems under uncertainty of PFS. Very recently, the author developed operational laws and their corresponding weighted aggregation operators based on PFS then proposed an algorithm to solve the multiple attribute group decision making problems [18–23]. Some other publications that related to PFS and its applications can be retrieved from Ejegwa [14, 16, 17], Wan Mohd et al. [46], Abdullah and Mohd [3], Wan Mohd and Abdullah [45] and Ejegwa and Awolola [15].

In relation to application of Choquet integral under PFS environment, Khan [30] used the Choquet integral to develop a very comprehensive Pythagorean fuzzy aggregation operator. With no specific application, the author developed a specific aggregation operator and illustrated the proposed method with a group decision making problem. Wang et al. [48] developed the concept of interval-valued hesitant Pythagorean fuzzy sets. To ease the application in selecting project private partner, the authors supported the concept with technique for order preference by similarity to ideal solution and Choquet integral. Khan et al. [29] proposed several aggregation operators based on Pythagorean hesitant fuzzy Choquet integral and applied the developed operators to multi-attribute decision making problem. It can be seen that these literature do not provide any specific real-life applications. It is also noticed that these literature tend to focus on hesitant PFS and various types of aggregator operators instead of real decision making applications. Far too little attention has been paid to introduce PFS in the aggregation operator of Choquet integral. In other words, the two memberships of PFS are not fully utilized in aggregating information through the Choquet integral operator. To bridge this knowledge gap, this chapter proposes PFS-Choquet integral where the concept of fuzzy measures is extended to PFS. The proposed PFS-Choquet is expected to provide better tool in handling uncertain and incomplete information of a case study of sustainable solid waste management (SWM). The rest of this paper is organized as follows. In Sect. 2, we describe the concepts of PFS and Pythagorean fuzzy measures, and some knowledge about the Choquet integral. The proposed PFS-Choquet integral is presented in Sect. 3. In Sect. 4, a case study of SWM is presented to illustrate the proposed method. Finally, conclusion is made in Sect. 5.

2 Preliminaries

This section recalls the definitions of fuzzy measures, PFS, and Choquet integral. These definitions are fully utilized in proposing the PFS-Choquet integral.

Definition 1 (Yager [50])

A Pythagorean Fuzzy Sets P in a finite universe of discourse is

$$P = \left\{ {\left. { < x,\mu_{P} (x),v_{P} (x) > } \right|x \in X} \right\} >$$

where \(\mu_{p} ,v_{P} :X \to [0,1][0,1]\) with the condition that the square sum of its membership degree and non-membership degree is less than or equal to 1.

$$\left( {\mu_{P} \left( x \right)} \right)^{2} + \left( {v_{P} \left( x \right)} \right)^{2} \le 1$$

It is clearly seen that sum of squares of its memberships is less than or equal to one.

Definition 2 (Grabisch [24]; Sugeno [41])

A fuzzy measure on X is a set function \(\mu :2^{x} \to [0,1]\) satisfying the following axioms.

  1. (1)

    \(\mu (\phi ) = 0,\mu (X) = 1\) (boundary conditions)

  2. (2)

    \(A \subseteq B \Rightarrow \mu (A) \le \mu (B)\) (monotonicity)

Definition 3 (Murofushi and Sugeno [33])

A fuzzy density function of a fuzzy measure \(\mu\) on a finite set X is a function \(s:X \to [0,1]\) satisfying,

$$s(x) = \mu (\left\{ x \right\}),x \in X$$

\(s(x)\) is called the fuzzy density of singleton x.

Definition 4 (Tan and Chen [42])

Let f be a real-valued function on X, and \(\mu\) be a non-additive measure (fuzzy measure) on X. Then, the Choquet integral of f with respect to non-additive \(\mu\) is represented as,

$$\left. {\left. {C_{\mu } (f)} \right) = \sum\limits_{i = 1}^{n} {\mu \left( {x_{(i)} } \right) - \mu (x_{(i + 1)} )} } \right]f(i)$$
(1)

where \(\left( . \right)\) is finite order of permutation \(f(1) \le \cdots \le f(n),A(i) = \left\{ {i, \ldots ,n} \right\}\), and \(A_{(n + 1)} = \phi\).

It shows that Choquet integral is also called Lebesgue integral up to reordering of the indices. In other words, when the fuzzy measure is additive, it shows that the Choquet integral reduces to Lebesgue integral.

Conditions of fuzzy measures are given in Definition 5.

Definition 5 (Sugeno [40])

A fuzzy measure on X is a set function \(\mu :P(X) \to [0,1]\), satisfying:

  1. i.

    Boundary condition: \(\mu (\phi ) = 0\) and \(\mu (X) = 1\);

  2. ii.

    Monotonicity: If \(A,B \in P(X)\) and \(A \subseteq B\), then \(\mu (A) \le \mu (B)\).

For \(A,B \in P(X)\) with \(A \cap B \in \varphi\), the fuzzy measure is said to be:

  1. i.

    an additive measure, if \(\mu (A \cup B) = \mu (A) + \mu (B)\);

  2. ii.

    a super additive measure, if \(\mu (A \cup B) > \mu (A) + \mu (B)\);

  3. iii.

    a sub additive measure if \(\mu (A \cup B) < \mu (A) + \mu (B)\).

and,

$$\mu \left( {A \cup B} \right) = \mu (A) + \mu (B) + \lambda \mu (A)\mu (B),\;\lambda \in \left[ { - 1,\infty } \right),\;\forall A,B \in P\left( X \right)\;{\text{and}}\;A \cap B = \varphi$$
(2)

This equation is required to find fuzzy measure.

If X is finite, then the parameter \(\lambda\) of a fuzzy measure satisfies

$$\mu (X) = \frac{1}{\lambda }\left( {\prod\limits_{i = 1}^{n} {\left( {1 + \lambda \mu \left( {x_{i} } \right)} \right)} - 1} \right),\lambda \ne 0$$
(3)

The parameter \(\lambda\) can be determined with the boundary condition \(\mu (x) = 1\), i.e.,

$$\lambda + 1 = \prod\limits_{i = 1}^{n} {\left( {1 + \lambda \mu \left( {x_{i} } \right)} \right)}$$
(4)

Definition 6 (Choquet [10])

Let \(\mu\) be a fuzzy measure on N. The Choquet integral of \(x = \left( {x_{1} , \ldots ,x_{n} } \right) \in [0,1]^{n}\) with respect to \(\mu\) is defined as

$$C_{\mu } \left( {x_{1} , \ldots ,x_{n} } \right) = \sum\limits_{i = 1}^{n} {\left[ {\mu \left( {A_{(i)} } \right) - \mu \left( {A_{{\left( {i + 1} \right)}} } \right)} \right]} \,x_{i}$$
(5)

where (.) indicates a finite order of permutation on N such that \(x_{(1)} \le x_{(2)} \le \ldots \le x_{n}\), and \(A_{(i)} = \left\{ {(i), \ldots (n)} \right\}\), \(A_{(n + 1)} = \phi\)

Definition 7 (Tan and Chen [42])

Let xi = (txi, fxi) (i = 1,2. …..n) be a collection of intuitionistic fuzzy values on X, and μ be a fuzzy measure on X. The (discrete) intuitionistic fuzzy Choquet integral of xi with respect to μ is defined by

$$IF\,C_{\mu } \left( {x_{1} , \ldots ,x_{n} } \right) = \sum\nolimits_{i = 1}^{n} {\left[ {\mu \left( {A_{(i)} } \right) - \mu \left( {A_{(i + 1)} } \right)} \right]x_{i} }$$

where (·) indicates a permutation on X such that \(x_{(1)} \le x_{(2)} \le \ldots \le x_{n}\) and \(A_{(i)} = \left\{ {(i), \ldots (n)} \right\}\), \(A_{(n + 1)} = \phi\)

These definitions are generally used in the proposed method in which the ultimate decision of Choquet value could be made based on Definition 6. Detailed explanation of the proposed method is presented in the following section.

3 Pythagorean Fuzzy Choquet Integral

It is known that the characteristics of PFSs are closely related to IFSs despite its differences in the condition of dual memberships. Similar to IFSs where arithmetic operations, aggregation operators have been widely discussed in literature, several attempts have been made to understand the algebraic operations of PFS. Peng and Yang [37] for example, proposed division and subtraction operations of PFS and discuss their properties. The properties of aggregation operators (see Definition 5) such as boundedness, idempotency, and monotonicity were also investigated. They developed a Pythagorean fuzzy superiority and inferiority ranking method to solve multi-criteria decision making (MCDM) problem instead of a special aggregation operator. In this paper, we used the knowledge of aggregations operator properties and extended it to propose PFS-Choquet integral. In this paper, the PFS-Choquet integral is defined as follows.

Let xi = (txi, fxi) (i = 1,2. …..n) be a collection of Pythagorean intuitionistic fuzzy values on X, and μ be a fuzzy measure on X. The (discrete) Pythagorean fuzzy Choquet integral of xi with respect to μ is defined by

$${PICI}_{\mu }\left({x}_{1},...,{x}_{n}\right)={\sum }_{i=1}^{n}\left[\mu \left({A}_{(i)}\right)-\mu \left({A}_{\left(i+1\right)}\right)\right]\hspace{0.33em}{x}_{i}$$

where (·) indicates a permutation on X such that \(x_{(1)} \le x_{(2)} \le \ldots \le x_{n}\) and \(A_{(i)} = \left\{ {(i), \ldots (n)} \right\}\), \(A_{(n + 1)} = \phi\).

Differently from the method of Peng and Yang [37] where membership and non-membership are calculated separately at the end of the computational procedures, this proposed method substitutes the separation method with the score functions. This proposed method is not fitted with the separation of membership and non-membership as it is undermined the concept of interrelationship of dual memberships of PFS. In response to this limitation, the score function proposed by Zhang and Xu [53] is substituted to the newly PFS-Choquet integral. In this proposed method, the membership and non-membership are combined to get the score of all criteria of MCDM problems. In addition, linguistic terms used in evaluation are defined in PFS in which the sum of squares of two memberships for linguistic terms is less than or equal to one. The computational procedure of the proposed PFS-Choquet integral method is given as follows.

Step 1: Construct a function Xm where X = {x1, x2, ⋯, xn}. Identify input n value (number of evaluation items) and m value (number of inputs).

Step 2: Construct decision matrix using the linguistic variable defined in PFS.

Step 3: Calculated weighted Pythagorean fuzzy set using Eq. (6).

$${\uplambda }_{m} P_{i,m} = \left\langle {\sqrt {1 - \left( {1 - \mu_{i,m}^{2} } \right)^{\lambda } } , \left( {v_{i,m} } \right)^{\lambda } } \right\rangle$$
(6)

Step 4: Calculate the aggregation for each input using Eq. 7

$${\uplambda }_{{m_{1} }} P_{{i,m_{1} }} \oplus {\uplambda }_{{m_{2} }} P_{{i,m_{2} }} = \left\langle {\sqrt {\mu_{{{\uplambda }_{{m_{1} }} P_{{i,m_{1} }} }}^{2} + \mu_{{{\uplambda }_{{m_{2} }} P_{{i,m_{2} }} }}^{2} - \mu_{{{\uplambda }_{{m_{1} }} Z_{{m_{{1P_{{i,m_{1} }} }} }} }}^{2} \mu_{{{\uplambda }_{{m_{2} }} P_{{i,m_{2} }} }}^{2} } , v_{{{\uplambda }_{{m_{1} }} P_{{i,m_{1} }} }} v_{{{\uplambda }_{{m_{2} }} P_{{i,m_{2} }} }} } \right\rangle$$
(7)

Step 5: Calculate the score function using Eq. (8)

$$s_{i} = \mu_{{P_{i} }}^{2} - v_{{P_{i} }}^{2}$$
(8)

Step 6: Obtain relative weight for each criteria using Eq. (9)

$$\frac{{\lambda_{m} P_{i,m} }}{{\sum\limits_{i = 1}^{n} {\lambda_{m} P_{i,m} } }}$$
(9)

The relative weight of criteria is obtained using Eq. (9) in which the defuzzification is already implemented beforehand using the score function equation.

Step 7: Obtain \(v(i)\) using Eq. (10)

$$v(i,i + 1) = w_{{C_{i} }} + w_{{C_{i + 1} }}$$
(10)

Step 8: Calculate value of the Pythagorean Choquet integral.

$${PICI}_{\mu }\left({x}_{1},...,{x}_{n}\right)={\sum }_{i=1}^{n}\left[\mu \left({A}_{(i)}\right)-\mu \left({A}_{\left(i+1\right)}\right)\right]\hspace{0.33em}{x}_{i}$$
(11)

where t is a permutation satisfying \(x_{t\left( 1 \right)} \le \ldots \le x_{t\left( n \right)}\).

The ultimate output of the proposed method is the Choquet value in which the strength of aggregated criteria is measured.

4 Application to Sustainable Solid Waste Management

The Choquet integral is one of the aggregation operators that has been used to aggregate information. It has been applied with great success to many different information aggregation cases where vast majority of information in real-world applications is characterized by high level of uncertainties. The success of Choquet integral in information aggregation can be witnessed in economics, insurance, finance, quality of life, and social welfare [27]. It is mainly owing to the non-additive characteristics of Choquet integral in which it can play a key role or provide a capacity for recent advances in decision theory [25]. Moreover, the role of fuzzy measures in dealing uncertainties, and the role of operator integrals in computational aspects are the vital component in Choquet integral that guarantee success in real applications. Several real application research using Choquet integral are supply chain management strategy measurements [47], job-shop scheduling problem [38], selection of optimal supplier in supply chain management [44]. Zhang [52], used Choquet integral for screening geological CO2 storage sites. Very recently, Olawumi and Chan [35] used generalized Choquet fuzzy integral method to determine the importance weights of the sustainability assessment criteria. The sustainable solid waste management problem and methods used in solving the problems are also discussed by many authors. Very recently, Sharma, et al. [39] applied the clustering method k-means algorithms framework for municipal solid waste management. In another recently published work, Tascione et al. [43], used a linear programming model to identify the best scenario of managing solid waste. With about a similar model, Batur et al. [7] used a mixed integer linear programming model for solid waste management by developing a long term system. This paper adds another application of the proposed Choquet integral to the case of sustainable SWM. Specifically, this chapter presents an evaluation of sustainable SWM of two major cities in Malaysia using the proposed method. The two cities are Kuala Lumpur and Johor Bahru. Kuala Lumpur is the capital of Malaysia where it is located at the west coast of Peninsular Malaysia. Johor Bahru is located at the southern part of Peninsular Malaysia and separated by a causeway with Singapore. The evaluation includes the selection of the best city in Malaysia in the context of managing solid waste. The ultimate goal of this application is to obtain the optimized value of Choquet integral in which the better city in managing solid waste could be suggested. Detailed descriptions of the evaluation are given as follows.

4.1 Experts and Criteria

The evaluation begins with the identification of experts and criteria. The evaluation criteria that influence sustainable SWM are retrieved from literature while the weight and priority of the criteria are provided by a group of experts in the field that related to sustainable SWM. This group of four experts comprises several important key personnel at number of sustainable SWM companies and some are academicians that are attached to environmental studies academic program at a public university in Malaysia. Personal communications with experts were conducted to collect linguistic evaluation. These verbal communications mainly aimed at obtaining the weight of importance of criteria of sustainable SWM. Table 1 shows the linguistic terms and rating scale used in this study.

Table 1 Importance of criteria and PFS rating scale
Full size table

The criteria for this study are retrieved from several literature in sustainable SWM (see Herva and Roca [1, 28] In this study, the selected criteria are Relative Cost (C1), Environmental Health (C2), Socio-culture (C3), Public Awareness(C4), Institutional(C5), Technical (C6), Operation & Maintenance Challenges (C7), Population Size (C8), Human Health (C9), and Consumption Habits (C10). Detailed description of the criteria can be retrieved from Abdullah and Goh [2]. The collected linguistic evaluations with respect to criteria are then computed using the proposed method. Detailed computations are described as follows.

4.2 Computation

Step 1: Construct evaluation matrix.

Based on the defined linguistic in Table 1, the evaluation matrix of Expert 1, Expert 2, Expert 3, Expert 4 are summarized in Table 2.

Table 2 Rating of importance of criteria from experts
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Step 2: Calculate weighted PFS matrix.

Prior to calculating weighted PFS, the weight of each expert is obtained based on their working experience, knowledge, and also seniority in their company. Table 3 presents the weight and relative weight of experts.

Table 3 Weight score and relative weight of experts
Full size table

The next step is to calculate the weighted rating. It is calculated using Eq. (6).

The weighted rating of Expert 1, Expert 2, Expert 3, and Expert 4 are summarized in Table 4.

Table 4 Weighted rating for criteria
Full size table

Step 3: Calculate the aggregated matrix.

The aggregated matrix for Kuala Lumpur, for example, is calculated using Eq. (7).

$$\begin{aligned} {\uplambda }_{1} P_{1} \oplus {\uplambda }_{2} P_{1} & = \left\langle {\sqrt {0.3965^{2} + 0.5715^{2} - 0.3965^{2} \times 0.5715^{2} } , 0.6645 \times 0.5780} \right\rangle \\ & = \left\langle {0.6576,0.3841} \right\rangle \\ {\uplambda }_{1} P_{2} \oplus {\uplambda }_{2} P_{2} & = \left\langle {\sqrt {0.5866^{2} + 0.5715^{2} - 0.5866^{2} \times 0.5715^{2} } , 0.5572 \times 0.5780} \right\rangle \\ & = \left\langle {0.7472,0.3221} \right\rangle \\ \end{aligned}$$

The remaining calculation for aggregated rating is computed similarly. Table 5 presents aggregated rating for Kuala Lumpur and Johor Bahru.

Table 5 Aggregated weighted rating of cities
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Table 6 Aggregated weighted rating of Kuala Lumpur and Johor Bahru.

Table 6 Score function of criteria for cities
Full size table

Step 4: Calculate the score function.

Equation (8) is used to calculate score function for the cities.

For example, the score function for Kuala Lumpur is calculated as,

$$s_{1,1} = 0.6576^{2} - 0.3841^{2} = 0.2850\quad s_{2,1} = 0.7472^{2} - 0.3221^{2} = 0.4546$$

The remaining calculation for score functions are executed similarly. Summarily, the score functions of Kuala Lumpur and Johor Bahru are presented in Table 6.

Step 5: Obtain the relative weight of criteria using Eq. (9) and total weight of criteria using Eq. (10). The relative weights are presented in Table 7.

Table 7 Relative weight and total weight of criteria
Full size table

Step 6: Obtain the total weight of criteria \(v(i)\) using Eq. (10)

For example,

$$\begin{aligned} v(1,2,3,4,5,6,7,8,9,10) & = 0.0007 + 0.1492 + 0.0064 + 0.0495 + 0.0859 \\ & \quad { +} 0.1151 + 0.1683 + 0.1332 + 0.1485 + 0.1431 \\ & = {1}{\text{.0000}} \\ v(2,3,4,5,6,7,8,9,10) & = 0.1492 + 0.0064 + 0.0495 + 0.0859 \\ & \quad { +} 0.1151 + 0.1683 + 0.1332 + 0.1485 + 0.1431 \\ & = {0}{\text{.9993}} \\ \end{aligned}$$

The total weight \(v(i)\) of all criteria is presented in Table 7.

Step 7: Calculate the value of Choquet integral using Eq. (11).

For example,

$$\begin{aligned} & x_{1} \left[ {v\left( {1, \ldots ,n} \right) - v\left( {2, \ldots ,n} \right)} \right] = 0.2850\left[ {1.0000 - 0.9993} \right] = 0.0002 \\ & x_{2} \left[ {v\left( {2, \ldots ,n} \right) - v\left( {3, \ldots ,n} \right)} \right] = 0.4546\left[ {0.9993 - 0.8501} \right] = - {0}{\text{.0044}} \\ \end{aligned}$$

The remaining calculation for \(x_{i} \left[ {v\left( {i, \ldots ,n} \right) - v\left( {i + 1, \ldots ,n} \right)} \right]\) are implemented similarly.

From the calculation, it is found that the Choquet integral for Kuala Lumpur and Johor Bahru are \(C_{v(1, \ldots ,10),1} \left( x \right) = - 0.3715\) and \(C_{v(1, \ldots ,10),2} \left( x \right) = - 0.3867\), respectively.

The values of Choquet integral for Kuala Lumpur and Johor Bahru are very close to each other which indicates that the two major cities in Malaysia are not much different in managing solid waste. However, on close inspection, the values show that Kuala Lumpur provides is slightly better than Johor Bharu in managing solid waste.

5 Conclusions

In recent years, there has been an increasing interest in developing various types of aggregation operators based on many sets including PFS. These developments have led to a renewed interest in developing Choquet integral operations under PFS environment. In this chapter, we developed Choquet integral based on PFS by combining relative weight of criteria and a score function. In this aggregation, the Choquet integral has been considered to model the interaction between criteria. This new amalgamation has successfully overcome the issues of independence among criteria of decision problems under PFS environment. The proposed method has been applied to a case of sustainable SWM where establishing total weights of criteria and Choquet integral value are the ultimate decision. The proposed eight-step aggregation operator was implemented in the evaluation of two big cities in Malaysia pertaining to criteria of sustainable SWM. Kuala Lumpur is suggested as the better city in managing solid waste based on the values of Choquet integrals. This study has proved that the Choquet integral under PFS environment is one of the potent tools in aggregating incomplete and vague information. Nevertheless, this study has some recommendations for future research. The proposed aggregation operators can be extended by considering the other methods that directly deal with interrelationships among criteria. The Bonferroni mean [26] and the Shapley value [6, 32] are two aggregators that possibly could be utilized to deal with independence and interrelationship among criteria of decision problems. Another possible future research is developing linguistic interval-valued under Pythagorean fuzzy environment [20, 21].