Averaging aggregation operators with pythagorean trapezoidal fuzzy numbers and their application to group decision making

1 Introduction

The idea of fuzzy set theory was developed by L.A. Zadeh [2] in 1965. In fuzzy set theory the degree of member ship function was discussed. Fuzzy set theory has been studied in many fields such as, medical diagnosis, computer science, fuzzy algebra and decision making problems. Later on the great idea of intuitionistic fuzzy set (IFS) theory was developed by Atanassov [1] in 1989, and discussed the degree of membership and non-membership function. IFS is the generalization of fuzzy set theory. There are many advantages of IFS theory such as, using in engineering, management science, computer science [3, 4]. IFS theory is investigated by many authors [3, 6] and are used for decision making [7, 15], medical diagnosis [14, 15], pattern recognition. Atanassov also presented several relation and different mathematically operations such as, algebraic product, sum, union, intersection and complement [13]. He also introduced the concept of pseudo fixed points of all operators defined over the IFSs. In 1986, many scholars have done works in the field of IFS and its applications. Particularly, information aggregation is a very crucial research area in IFS theory that has been receiving more and more focus. Ban et al. [41] developed the idea of trapezoidal fuzzy numbers and interval-valued trapezoidal umbers. Later on Wang and Zhang [9] developed the definition of intuitionistic trapezoidal fuzzy numbers (ITFNs) and interval-valued intuitionistic trapezoidal fuzzy numbers (IV ITFNs). As compared to intuitionistic fuzzy numbers, the intuitionistic trapezoidal fuzzy numbers make their membership and non-membership degrees no longer relative to a fuzzy concept. Excellent. or. Good., but relative to the trapezoidal fuzzy number; thus can express decision information in different dimensions and avoid losing decision preference information. Correspondingly on aggregation methods, Wang and Zhang [9] first proposed the definition and Hamming distance formula for intuitionistic trapezoidal fuzzy numbers. Moreover, they developed the intuitionistic trapezoidal fuzzy weighted averaging (ITFWA) operator and the multi-criteria decision making method with incomplete certain information. Wei [42] developed the intuitionistic trapezoidal fuzzy ordered weighted averaging (ITFOWA) operator and the intuitionistic trapezoidal fuzzy hybrid aggregation (ITFHA) operator. Wan and Dong [43] defined the expectation and expectant score of intuitionistic trapezoidal fuzzy numbers from the geometric angle. Wu [44] further developed the intuitionistic trapezoidal fuzzy weighted geometric (ITFWG) operator, the intuitionistic trapezoidal fuzzy ordered weighted geometric (ITFOWG) operator and the intuitionistic trapezoidal fuzzy hybrid geometric (ITFHG) operator.

Liu et al. [36] proposed the orness measure to reflect the or-like degree of the continuous quasi ordered weighted averaging (C-QOWA) operator is proposed. Moreover, some desirable properties of the C-QOWA operator associated with its orness measure are investigated. In addition, we apply the C-QOWA operator to the aggregation of multiple interval arguments and obtain the weighted C-QOWA operator, the ordered weighted C-QOWA (OWA-QOWA) operator, the combine C-QOWA (CC-QOWA) operator. Liu et al. [34], defined the different parameter of the generalized ordered weighted hybrid averaging (GOWHA) operator is employed, various conventional aggregation operators can be deduced, such as the ordered weighted averaging (OWA) operator and the ordered weighted geometric averaging (OWGA) operator. We further demonstrate that the OWHA operator and the GOWHA operator exhibit monotonicity with respect to the weighting vector. They introduced the generalized hybrid harmonic averaging (GHHA) operator, which can reflect the importance degrees of both the given arguments and the ordered position of the arguments. Liu et al. [35] defined the generalized linguistic ordered weighted hybrid logarithm averaging (GLOWHLA) operator, which extends the GLOWLA operator. We also construct a nonlinear goal programming model to determine GLOWHLA weights from observational linguistic variable values under partial weight information.

He et al. [37] defined the ith-order polymerization degree function and propose a new ranking method to further compare different hesitant fuzzy sets. In order to obtain much more information in the process of group decision making, we combine the power average operator with the Bonferroni mean in hesitant fuzzy environments and develop the hesitant fuzzy power Bonferroni mean (BM) and the hesitant fuzzy power geometric Bonferroni mean. We investigate the desirable properties of these new hesitant fuzzy aggregation operators and discuss some special cases. He et al. [40] defined the complements to the existing generalizations of BM under intuitionistic fuzzy environment, this paper also considers the interactions between the membership function and nonmember ship function of different IFS and develops the intuitionistic fuzzy interaction BM and the weighted intuitionistic fuzzy interaction BM. We investigate the properties of these new extensions of BM and discuss their special cases. Kumar et al. [24] proposed the major component of the set pair analysis (SPA) known as connection number has been constructed based on the set pairs between two preference values consists of every attribute and ideal pairs of it. Based on these connection numbers, an extension of technique for order of preference by similarity to ideal solution method is developed by combining the proposed connection number for IVIFSs and hence finding the best alternative (s) using relative degree of closeness coefficient. Liu et al. [33] introduced the generalized ordered modular averaging (GOMA) operator, which is a special case of the OQMA operator. Some special cases of the GOMA operator are discussed. An orness measure to reflect the or-like degree of the GOMA operator is proposed. We further extend the GOMA operator to the generalized ordered hybrid modular (GOHM) operator, which focuses not only on the degree of importance with respect to input arguments but also their serial positions. He et al. [38] extended the Atanassov’s intuitionistic fuzzy interaction Bonferroni mean (EIFIBM) and the extended weighted Atanassov’s intuitionistic fuzzy interaction Bonferroni mean, which can evolve into a series of BMs by taking different generator functions that reflect the different preference attitudes of the decision makers. In addition, some of the EIFIBMs are consistent with aggregation operations on the ordinary fuzzy sets, and some of the EIFIBMs consider the interactions between the membership and non membership functions of different Atanassov’s intuitionistic fuzzy sets; thus, they can be used in more decision situations.

Xu and Yager [17] defined some basic geometric aggregation operators, such as intuitionistic fuzzy weighted geometric (IFWG) operator, intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and intuitionistic fuzzy hybrid geometric (IFHG) operator, and applied them to multiple attribute decision making (MADM) based on intuitionistic fuzzy information. Xu and Yager [18] introduced the notion of dynamic IFWA operator and developed a methods to solve the dynamic intuitionistic fuzzy multi attribute decision making (MADM) problems. Wei [10] also worked in this field and introduced the notion of some induced geometric aggregation operators with intuitionistic fuzzy information, and also applied them to group decision making problems.

Like the other scholars, Wang [8] also worked in the field of intuitionistic fuzzy set and introduced the idea of intuitionistic trapezoidal fuzzy (ITFNs) numbers and interval-valued intuitionistic trapezoidal fuzzy (IVITFNs) numbers. Wang [9] not only developed the idea of these numbers, but also introduced the concept of Hamming distance for TIFNs as well as, introduced a series of averaging aggregation operators for ITFNs such as ITFHWA, ITFOWA and ITFWA aggregation operators. Garg [28] introduced along with their two aggregation operators, namely, interval-valued Pythagorean fuzzy weighted average and weighted geometric operators for different IVPFS. Secondly, an improved accuracy function under IVPFS environment has been developed by taking the account of the unknown hesitation degree. Garg [29] presented in the interval-valued intuitionistic fuzzy sets (IVIFSs) environment by incorporating the idea of weighted average of the degree of hesitation between their membership functions. Secondly, an IVIFSs based method for solving the multi-criteria decision making (MCDM) problem has been presented with completely unknown attribute weights. A ranking of the different attributes is based on the proposed generalized improved score functions and the sensitivity analysis on the ranking of the system has been done based on the decision-making parameters. Garg [30] Based on it, a new class of generalized geometric interaction averaging aggregation operators using Einstein norms and conorms are developed, which includes the weighted, ordered weighted and hybrid weighted averaging operators. Furthermore, desirable properties corresponding to proposed operators have been stated.

In 2013, Yager [11] also worked in the field of Pythagorean fuzzy (PFs) set and introduced the concept of PFs, which is a generalization of IFSs, in which the square of their sum less than or equal to 1. Like the other scholars Gou et al. [32] defined the Pythagorean fuzzy sets (PFs), as the generalization of the fuzzy sets, can be used to effectively deal with this issue. Therefore, to enrich the theory of PFs it is very necessary to investigate the fundamental properties of Pythagorean fuzzy information. In this paper, we first describe the change values of Pythagorean fuzzy numbers (PFNs), which are the basic components of PFSs, when considering them as variables. Garg [27] defined propose a novel correlation coefficient and weighted correlation coefficient formulation to measure the relationship between two PFSs. Pairs of membership, nonmember ship, and hesitation degree as a vector representation with the two elements have been considered during formulation. He et al. [39] developed the robust fuzzy programming (RFP) approach to solve the multiple responses optimization (MRO) problems. The key advantage of the presented method is that it takes account of the location effect, dispersion effect and model uncertainty of the multiple responses simultaneously and thus can ensure the robustness of the solution. Garg [31] defined shortcoming of some existing aggregation operators has been identified and then new operational laws have been proposed for overcoming these shortcoming. Based on these operations, weighted, ordered weighted and hybrid averaging aggregation operators have been proposed by using Einstein operational laws. Furthermore, some desirable properties such as idempotency, boundedness, homogeneity etc, are studied. Garg [26] proposed the Pythagorean fuzzy Einstein weighted geometric, Pythagorean fuzzy Einstein ordered weighted geometric, generalized Pythagorean fuzzy Einstein weighted geometric, and generalized Pythagorean fuzzy Einstein ordered weighted geometric operators, are proposed in this paper. Garg [22] proposed function, degree of hesitation between the element of IVPFS has been taken into account during the analysis. Based on it, multi-criteria decision-making method has been proposed for finding the desirable alternative (s). Garg [23] defined the Under these environment and by incorporating the idea of the confidence levels of each Pythagorean fuzzy number, the present study investigated a new averaging and geometric operators namely confidence Pythagorean fuzzy weighted and ordered weighted operators along with their some desired properties. Garg [25] defined the Based on these preferences and an improved score function, a score matrix has been formulated and then a linear programming based method has been proposed to solve MCDM problems with unknown attribute weights. Obviously, PFS is more capable than IFSs to model the vagueness in the practical problems.

Due to the motivation and inspiration of the above discussion, we generalized the concept of Pythagorean trapezoidal fuzzy set which give us more accurate and precious result as compare to the above mention operators. Thus keeping an advantages of the above mention aggregation operators. In this paper, we develop a series of Pythagorean trapezoidal fuzzy aggregation operators, including the Pythagorean trapezoidal fuzzy weighted averaging (PTFWA) operator, Pythagorean trapezoidal fuzzy ordered weighted averaging (PTFOWA) operator and Pythagorean trapezoidal fuzzy hybrid averaging (PTFHA) operator.

In this paper, we develop the following sections; In Section 2, we briefly reviews some basic concepts and operation laws related to trapezoidal fuzzy numbers and Pythagorean trapezoidal fuzzy numbers. In Section 3, we propose some Pythagorean trapezoidal fuzzy aggregation operators, such as Pythagorean trapezoidal fuzzy weighted averaging

(PTFWA) operator, Pythagorean trapezoidal fuzzy ordered weighted averaging (PTFOWA) operator and Pythagorean trapezoidal fuzzy hybrid averaging (PTFHA) operator and their properties. In Section 4, we develop a multiple attribute group decision making method based on the proposed operators under Pythagorean trapezoidal fuzzy environment. In Section 5, we give numerical example. In Section 6, we discuss and compare the proposed operator with Pythagorean fuzzy weighted averaging (PFWA) operator. Concluding remarks are made in Section 7.

2 Preliminaries

Definition 2.1. [1] Let L be a fixed set. An IFS, U in L is an object having the form: U = { 〈 l , Ψ u ( l ) , ϒ u ( l ) 〉 | l ∈ L } where Ψ _u : L → [0, 1] and ϒ _u : L → [0, 1] represent the degree of membership and the degree of non-membership of the element l ∈ L to U, respectively, and for every l ∈ L: 0 ≤ Ψ u ( l ) + ϒ u ( l ) ≤ 1

For each (IFS), U in L, π U ( l ) = 1 - Ψ U ( l ) - ϒ U ( l ) , for all l ∈ L

Where π _A (l) is called the degree of indeterminacy of l to U.

Definition 2.2. [9] Let P be intuitionistic trapezoidal fuzzy number, its membership function is Ψ p ( l ) = { l - p q - p Ψ p , p ≤ l ≤ q ; Ψ p , q ≤ l ≤ r ; s - l s - r Ψ p , r ≤ l ≤ s ; 0 , otherwise ,(1) and its non-membership function is ϒ p ( l ) = { s - l + ϒ p ( l - p 1 ) q - p , p 1 ≤ l ≤ q ; ϒ p , q ≤ l ≤ r ; l - r + ϒ p ( s 1 - l ) s 1 - r , r ≤ l ≤ s 1 ; 0 , otherwise ,(2) where 0 ≤ Ψ α ˜ ≤ 1 ; 0 ≤ ϒ α ˜ ≤ 1 ; 0 ≤ ( Ψ α ˜ ) + ( ϒ α ˜ ) ≤ 1 ; p, q, r, s ∈ R. Then p ˜ = 〈 ( [ p , q , r , s ] ; Ψ α ˜ ) , ( [ p 1 , q , r , s 1 ] ; ϒ α ˜ ) 〉 is called intuitionistic trapezoidal fuzzy number. For convenience, p ˜ = ( [ p , q , r , s ] ; Ψ α ˜ , ϒ α ˜ ) .

Definition 2.3. [9] Let p ˜ j = ( [ p j , q j , r j , s j ] ; Ψ α ˜ j , ϒ α ˜ j ) ( j = 1 , 2 ) , be any two intuitionistic trapezoidal fuzzy numbers, and δ ≥ 0. Then,

p ˜ 1 ⊕ p ˜ 2 = ( [ p 1 + p 2 , q 1 + q 2 , r 1 + r 2 , s 1 + s 2 ] ; ( Ψ α ˜ 1 ) + ( Ψ α ˜ 2 ) - ( Ψ α ˜ 1 Ψ α ˜ 2 ) , ϒ α ˜ 1 ϒ α ˜ 2 ) ,

Definition 2.4. [11] Let L be a fixed set. The PFs, U in L is an object having the form: A = { 〈 l , Ψ U ( l ) , ϒ U ( l ) 〉 | l ∈ L } where Ψ _A : L → [0, 1] and represent the degree of membership and the degree of non-membership of the element l ∈ L to A, respectively, and for every l ∈ L such that, 0 ≤ Ψ U ≤ 1 , 0 ≤ ϒ U ≤ 1 , 0 ≤ Ψ U 2 ( l ) + ϒ U 2 ( l ) ≤ 1.

For each PFs, U in L, π U ( l ) = 1 - Ψ U 2 ( l ) - ϒ U 2 ( l ) , for all l ∈ L , where π _U (l) is called the degree of indeterminacy of l to U.

Definition 2.5. [20] Let p ˜ = [ Ψ α ˜ , ϒ α ˜ ] be a Pythagorean fuzzy value then we can defined the score function of p ˜ as follows; s ( p ˜ ) = Ψ α ˜ 2 - ϒ α ˜ 2 ,(3) where s ( α ˜ ) ∈ [ - 1 , 1 ] .

Definition 2.6. [20] Let p ˜ = [ Ψ α ˜ , ϒ α ˜ ] be a Pythagorean fuzzy value then the accuracy degree of p ˜ can defined as follows; h ( p ˜ ) = Ψ α ˜ 2 + ϒ α ˜ 2 ,(4) where h ( p ˜ ) ∈ [ 0 , 1 ] .

Definition 2.7. [20] Let p ˜ = ( Ψ α ˜ , ϒ α ˜ ) , p ˜ 1 = ( Ψ α ˜ 1 , ϒ α ˜ 1 ) and p ˜ 2 = ( Ψ α ˜ 2 , ϒ α ˜ 2 ) be any three PFNs and δ > 0. Then,

p ˜ c = ( ϒ α , Ψ α ) ,

Definition 2.8. Let p ˜ 1 = ( [ p 1 , q 1 , r 1 , s 1 ] ; Ψ α ˜ 1 , ϒ α ˜ 1 ) , and p ˜ 2 = ( [ p 2 , q 2 , r 2 , s 2 ] ; Ψ α ˜ 2 , ϒ α ˜ 2 ) , be any two PTF, numbers and δ ≥ 0. Then,

p ˜ 1 ⊕ p ˜ 2 = ( [ p 1 + p 2 , q 1 + q 2 , r 1 + r 2 , s 1 + s 2 ] ; ( Ψ α ˜ 1 ) 2 + ( Ψ α ˜ 2 ) 2 - ( Ψ α ˜ 1 Ψ α ˜ 2 ) 2 , ϒ α ˜ 1 ϒ α ˜ 2 ) ,

Definition 2.9. [7] Let p ˜ = ( [ p , q , r , s ] ; [ Ψ α ˜ , ϒ α ˜ ] ) be Pythagorean trapezoidal fuzzy numbers, a score function s can be defined as follows; s ( p ˜ ) = ( p + q + r + s 4 · ( Ψ α ˜ 2 - ϒ α ˜ 2 ) ) s ( α ˜ ) ∈ [ - 1 , 1 ] .(5)

Definition 2.10. [7] Let p ˜ = ( [ p , q , r , s ] ; [ Ψ , ϒ ] ) be Pythagorean trapezoidal fuzzy numbers, an accuracy function h can be defined as follows; h ( p ˜ ) = ( p + q + r + s 4 · ( Ψ α ˜ 2 + ϒ α ˜ 2 ) ) H ( α ˜ ) ∈ [ 0 , 1 ] .(6)

To determine the degree of an accuracy of the Pythagorean trapezoidal fuzzy numbers p ˜ where h ( p ˜ ) ∈ [ 0 , 1 ] . The bigger the estimation of h ( p ˜ ) the further level of accuracy of the Pythagorean trapezoidal fuzzy numbers p ˜ .

Definition 2.11. Let p ˜ 1 = ( [ p 1 , q 1 , r 1 , s 1 ] ; Ψ α ˜ 1 , ϒ α ˜ 1 ) , and p ˜ 2 = ( [ p 2 , q 2 , r 2 , s 2 ] ; Ψ α ˜ 2 , ϒ α ˜ 2 ) , be any two PTF numbers, and δ ≥ 0, then the normalized Hamming distance between p ˜ 1 and p ˜ 2 is defined as follows: d ( p ˜ 1 , p ˜ 2 ) = 1 8 ( | ( 1 + ( Ψ α ˜ 1 ) 2 - ( ϒ α ˜ 1 ) 2 ) p 1 - 2 | + | ( 1 + ( Ψ α ˜ 1 ) 2 - ( ϒ α ˜ 1 ) 2 ) q 1 - 2 | + | ( 1 + ( Ψ α ˜ 1 ) 2 - ( ϒ α ˜ 1 ) 2 ) r 1 - 2 | + | ( 1 + ( Ψ α ˜ 1 ) 2 - ( ϒ α ˜ 1 ) 2 ) s 1 - 2 | )(7)

Definition 2.12. For a normalized PTF decision making matrix Z ˜ = ( z ˜ ij ) m × n = ( [ p ij , q ij , r ij , s ij ] ; Ψ ij , ϒ ij ) , where 0 ≤ p ij ≤ q ij ≤ r ij ≤ s ij 1 ; 0 ≤ Ψ ij , ϒ ij ≤ 1 , 0 ≤ ( Ψ ij ) 2 + ( ϒ ij ) 2 ≤ 1 , is the PTF positive ideal solution and PTF negative ideal solution are defined as follows: z ˜ + = ( [ p + , q + , r + , s + ] ; Ψ + , ϒ + ) = ( [ 1 , 1 , 1 , 1 ] ; 1 , 0 ) ,(8) z ˜ - = ( [ p - , q - , r - , s - ] ; Ψ - , ϒ - ) = ( [ 0 , 0 , 0 , 0 ] ; 0 , 1 )(9)

Definition 2.13. Let p ˜ i = ( [ p i , q i , r i , s i ] ; Ψ α ˜ i , ϒ α ˜ i ) , are PTF number and z ˜ + be PTF the positive ideal solution. Then, the distance between p ˜ i and denoted as d ( p ˜ i , z ˜ + ) if d ( p ˜ 1 , z ˜ + ) ≤ d ( p ˜ 2 , z ˜ + ) . Then, p ˜ 1 ≥ p ˜ 2 .

Definition 2.14. [12] The normal distribution based method can be defined as follows: ℏ £ = e - ( £ - Ψ Φ ) 2 2 σ Φ 2 ∑ i = 1 n e - ( i - Ψ Φ ) 2 2 σ Φ 2 , £ = 1 , 2 , . . . , Φ ,(10) where Ψ _Φ is the mean of the collection of 1, 2, …, Φ, and σ _Φ (σ _Φ > 0) is the standard deviation of the collection of 1, 2, …, Φ, such that, Ψ Φ = 1 Φ ( 1 + Φ ) 2 = ( 1 + Φ ) 2 σ Φ = 1 Φ ∑ i = 1 Φ ( i - Ψ Φ ) 2 .

Theorem 2.15. Let p ˜ 1 = ( [ p 1 , q 1 , r 1 , s 1 ] ; Ψ α ˜ 1 , ϒ α ˜ 1 ) , and p ˜ 2 = ( [ p 2 , q 2 , r 2 , s 2 ] ; Ψ α ˜ 2 , ϒ α ˜ 2 ) , be any two PTFNs, and consider p ˜ 1 = p 1 ⊗ p 2 , p ˜ 2 = p δ . Then, both p ˜ 1 and p ˜ 2 be PTF numbers.

Proof. Let p ˜ 1 = ( [ p 1 , q 1 , r 1 , s 1 ] ; Ψ α ˜ 1 , ϒ α ˜ 1 ) , and p ˜ 2 = ( [ p 2 , q 2 , r 2 , s 2 ] ; Ψ α ˜ 2 , ϒ α ˜ 2 ) be two PTF numbers. Then, 0 ≤ Ψ α ˜ ≤ 1 ; 0 ≤ ϒ α ˜ ≤ 1 ; 0 ≤ ( Ψ α ˜ ) 2 + ( ϒ α ˜ ) 2 ≤ 1 .

Thus we obtain 0 ≤ Ψ α ˜ 1 Ψ α ˜ 2 ≤ 1 , 0 ≤ ( ϒ α ˜ 1 ) 2 + ( ϒ α ˜ 2 ) 2 - ( ϒ α ˜ 1 ) 2 ( ϒ α ˜ 2 ) 2 = 1 , Ψ α ˜ 1 2 Ψ α ˜ 2 2 + ( ϒ α ˜ 1 ) 2 + ( ϒ α ˜ 2 ) 2 - ( ϒ α ˜ 1 ) 2 ( ϒ α ˜ 2 ) 2 ≤ ( 1 - ϒ α ˜ 1 2 ) ( 1 - ϒ α ˜ 2 2 ) + ( ϒ α ˜ 1 ) 2 + ( ϒ α ˜ 2 ) 2 - ( ϒ α ˜ 1 ) 2 ( ϒ α ˜ 2 ) 2 = 1 .

Thus p ˜ 1 is PTF number. Since p ˜ = ( [ p , q , r , s ] ; Ψ α ˜ , ϒ α ˜ ) be PTF numbers, So 0 ≤ Ψ α ˜ δ ≤ 1 , 0 ≤ 1 - ( 1 - ϒ α ˜ 2 ) δ ≤ 1 . Ψ α ˜ 2 δ + 1 - ( 1 - ϒ α ˜ 2 ) δ ≤ ( 1 - ϒ α ˜ 2 ) δ + 1 - ( 1 - ϒ α ˜ 2 ) δ = 1

Thus p ˜ 2 is PTF number.

Theorem 2.16. Let p ˜ = ( [ p , q , r , s ] ; Ψ α ˜ , ϒ α ˜ ) , p ˜ 1 = ( [ p 1 , q 1 , r 1 , s 1 ] ; Ψ α ˜ 1 , ϒ α ˜ 1 ) , and p ˜ 2 = ( [ p 2 , q 2 , r 2 , s 2 ] ; Ψ α ˜ 2 , ϒ α ˜ 2 ) be any three PTFNs and δ, δ₁, δ₂, be any scalar numbers. Then,

( p ˜ 1 ⊕ p ˜ 2 ) = ( p ˜ 2 ) ⊕ ( p ˜ 1 )

Proof. (1) Proof is obvious.

(2) Using Definition 2.8 and operational law (1), such that, p ˜ 1 ⊕ p ˜ 2 = ( [ p 1 + p 2 , q 1 + q 2 , r 1 + r 2 , s 1 + s 2 ] ; ( Ψ α ˜ 1 2 ) + ( Ψ α ˜ 2 2 ) - ( Ψ α ˜ 1 2 Ψ α ˜ 2 2 ) , ϒ α ˜ 1 ϒ α ˜ 2 ) .

Then, by operational law (3) in Definition 2.8, it follows that; δ ( p ˜ 1 ⊕ p ˜ 2 ) = ( [ δ ( p 1 + p 2 ) , δ ( q 1 + q 2 ) , δ ( r 1 + r 2 ) , δ ( s 1 + s 2 ) ] ; 1 - ( 1 - ( Ψ α ˜ 1 2 + Ψ α ˜ 2 2 - Ψ α ˜ 1 2 Ψ α ˜ 2 2 ) ) δ , ( ϒ α ˜ 1 ϒ α ˜ 2 ) δ ) .

Also since δ ( p ˜ 1 ) = ( [ δ p 1 , δ q 1 , δ r 1 , δ s 1 ] ; ( 1 - ( 1 - Ψ α ˜ 1 2 ) δ ) , ( ϒ α ˜ 1 ) δ ) , δ ( p ˜ 2 ) = ( [ δ p 2 , δ q 2 , δ r 2 , δ s 2 ] ; ( 1 - ( 1 - Ψ α ˜ 2 2 ) δ ) , ( ϒ α ˜ 2 ) δ ) .

Then, we have δ ( p ˜ 2 ) ⊕ δ ( p ˜ 1 ) = ( [ δ ( p 1 + p 2 ) , δ ( q 1 + q 2 ) , δ ( r 1 + r 2 ) , δ ( s 1 + s 2 ) ] ; ( 1 - ( 1 - Ψ p ˜ 1 2 ) δ ) + ( 1 - ( 1 - Ψ p ˜ 2 2 ) δ ) - ( 1 - ( 1 - Ψ p ˜ 1 2 ) δ ) ( 1 - ( 1 - Ψ p ˜ 2 2 ) δ ) , ( ϒ p ˜ 1 ϒ α ˜ 2 ) δ ) , = ( [ δ ( p 1 + p 2 ) , δ ( q 1 + q 2 ) , δ ( r 1 + r 2 ) , δ ( s 1 + s 2 ) ] ; 1 - ( 1 - ( Ψ α ˜ 1 2 + Ψ α ˜ 2 2 - Ψ α ˜ 1 2 Ψ α ˜ 2 2 ) ) δ , ( ϒ α ˜ 1 ϒ α ˜ 2 ) δ ) .

Hence, δ ( p ˜ 1 ⊕ p ˜ 2 ) = δ ( p ˜ 2 ) ⊕ δ ( p ˜ 1 ) .

(3) By the operational law (3) in Definition 2.8, we obtain δ 1 ( p ˜ ) = ( [ δ 1 p , δ 1 q , δ 1 r , δ 1 s ] ; ( 1 - ( 1 - Ψ p ˜ 2 ) δ 1 ) , ( ϒ p ˜ ) δ 1 ) δ 2 ( p ˜ ) = ( [ δ 2 p , δ 2 q , δ 2 r , δ 2 s ] ; ( 1 - ( 1 - Ψ p ˜ 2 ) δ 2 ) , ( ϒ p ˜ ) δ 2 ) .

Then, δ 1 ( p ˜ ) ⊕ δ 2 ( p ˜ ) = ( [ ( δ 1 p + δ 2 p ) , ( δ 1 q + δ 2 q ) , ( δ 1 r + δ 2 r ) r , ( δ 1 s + δ 2 s ) ] ; ( 1 - ( 1 - Ψ α ˜ 2 ) δ 1 ) + ( 1 - ( 1 - Ψ α ˜ 2 ) δ 2 ) - ( 1 - ( 1 - Ψ α ˜ 2 ) δ 1 ) ( 1 - ( 1 - Ψ α ˜ 2 ) δ 2 ) , ϒ p ˜ δ 1 ϒ p ˜ δ 2 ) = ( [ ( δ 1 p + δ 2 p ) , ( δ 1 q + δ 2 q ) , ( δ 1 r + δ 2 r ) r , ( δ 1 s + δ 2 s ) ] ; ( 1 - ( 1 - Ψ α ˜ 2 ) δ 1 ) ( ( 1 - Ψ α ˜ 2 ) δ 2 ) , ϒ α ˜ δ 1 ϒ α ˜ δ 2 ) = ( [ ( δ 1 + δ 2 ) p , ( δ 1 + δ 2 ) q , ( δ 1 + δ 2 ) r , ( δ 1 + δ 2 ) s ] ; ( 1 - ( 1 - Ψ α ˜ 2 ) δ 1 + δ 2 ) , ϒ α ˜ δ 1 + δ 2 ) = ( δ 1 ⊕ δ 2 ) ( p ˜ ) .

We shall define averaging operators for aggregating Pythagorean trapezoidal fuzzy (PTF) numbers and study their properties as follows:

Definition 3.1. Let p ˜ £ = ( [ p £ , q £ , r £ , s £ ] ; Ψ α ˜ £ , ϒ α ˜ £ ) (j = £ =1, 2, …, Φ) be a collection of Pythagorean trapezoidal fuzzy numbers, suppose Ω be the set of Pythagorean trapezoidal fuzzy numbers, and let PTFWA, Ω ^Φ → Ω, such that, PTFWA ( p ˜ 1 , p ˜ 2 , . . . , p ˜ Φ ) = ( ℏ 1 p ˜ 1 ⊕ ℏ 2 p ˜ 2 . . . ⊕ ℏ Φ p ˜ Φ ) .(11)

Then, PTFWA is called Pythagorean trapezoidal fuzzy weighted averaging operator of dimension Φ. Especially, if ℏ = (ℏ ₁, ℏ ₂, …, ℏ _Φ ) ^T be the weighted vector of p ˜ £ ( j = £ = 1 , 2 , . . . , Φ ) , with ℏ_£ ∈ [0, 1] and ∑ £ = 1 Φ ℏ £ = 1 , if ℏ = ( 1 Φ , 1 Φ , . . . , 1 Φ ) T . Then, PTFWA operator is reduced to Pythagorean trapezoidal fuzzy averaging (PTFA) operator of dimension Φ, which is defined as follow; PTFA w ( p ˜ 1 , p ˜ 2 , . . . , p ˜ Φ ) = ( p ˜ 1 ⊕ p ˜ 2 . . . ⊕ p ˜ Φ ) 1 Φ(12)

By Definition 3.1 and Theorem (2.16), we can obtain the following result. In order to proof, we use mathematical induction.

Theorem 3.2. Let p ˜ £ = ( [ p £ , q £ , r £ , s £ ] ; Ψ α ˜ £ , ϒ α ˜ £ ) (£ =1, 2, …, Φ) be a collection of Pythagorean trapezoidal fuzzy numbers. Then, their aggregated value by using the PTFWA operator is an also PTF number such that, PTFWA ( p ˜ 1 , p ˜ 2 , . . . , p ˜ Φ ) = ( [ ∑ £ = 1 Φ ℏ £ p £ , ∑ £ = 1 Φ ℏ £ q £ , ∑ £ = 1 Φ ℏ £ r £ , ∑ £ = 1 Φ ℏ £ s £ ] ; , 1 - ∏ £ = 1 Φ ( 1 - Ψ α ˜ £ 2 ) ℏ £ , ∏ £ = 1 Φ ϒ £ ℏ £ )(13) where ℏ = ( 1 Φ , 1 Φ , . . . , 1 Φ ) T be the weighted vector of p ˜ £ ( £ = 1 , 2 , . . . , Φ ) with ℏ_£ ∈ [0, 1] and ∑ £ = 1 Φ ℏ £ = 1 .

Proof. The result first is follows from Definition 3.1 and Theorem 2.16, by mathematical induction we prove the second result, we show that Equation 13 satisfy the condition when Φ = 2. ℏ 1 p ˜ 1 = ( [ ℏ 1 p 1 , ℏ 1 q 1 , ℏ 1 r 1 , ℏ 1 s 1 ] ; 1 - ∏ £ = 1 Φ ( 1 - Ψ α ˜ 1 2 ) ℏ 1 , ∏ £ = 1 Φ ϒ α ˜ 1 ℏ 1 ) ℏ 2 p ˜ 2 = ( [ ℏ 2 p 2 , ℏ 2 q 2 , ℏ 2 r 2 , ℏ 2 s 2 ] ; 1 - ∏ £ = 1 Φ ( 1 - Ψ α ˜ 2 2 ) ℏ 2 , ∏ £ = 1 Φ ϒ α ˜ 2 ℏ 2 ) .

Then, PTFWA ( p ˜ 1 , p ˜ 2 ) = ℏ 1 p ˜ 1 ⊕ ℏ 2 p ˜ 2 = ( [ ℏ 1 p 1 + ℏ 2 p 2 , ℏ 1 q 1 + ℏ 2 q 2 , ℏ 1 r 1 + ℏ 2 r 2 , ℏ 1 s 1 + ℏ 2 s 2 ] ; 1 - ( 1 - Ψ α ˜ 1 2 ) ℏ 1 + 1 - ( 1 - Ψ α ˜ 2 2 ) ℏ 2 - ( 1 - ( 1 - Ψ α ˜ 1 2 ) ℏ 1 ) ( 1 - ( 1 - Ψ α ˜ 2 2 ) ℏ 2 ) , ( ϒ α ˜ 1 ℏ 1 ) ( ϒ α ˜ 2 ℏ 2 ) ) = ( [ ℏ 1 p 1 + ℏ 2 p 2 , ℏ 1 q 1 + ℏ 2 q 2 , ℏ 1 r 1 + ℏ 2 r 2 , ℏ 1 s 1 + ℏ 2 s 2 ] ; 1 - ( 1 - Ψ α ˜ 1 2 ) ℏ 1 ( 1 - Ψ α ˜ 2 2 ) ℏ 2 ) , ( ϒ α ˜ 1 ℏ 1 ) ( ϒ α ˜ 2 ℏ 2 ) )

If Equation 13 holds for Φ = k, that is PTFWG w ( p ˜ 1 , p ˜ 2 , . . . , p ˜ k ) = ( [ ∑ £ = 1 k ℏ £ p £ , ∑ £ = 1 k ℏ £ q £ , ∑ £ = 1 k ℏ £ r £ , ∑ £ = 1 k ℏ £ s £ ] ; ∏ £ = 1 k Ψ α ˜ £ ℏ £ , 1 - ∏ £ = 1 k ( 1 - v α ˜ £ 2 ) ℏ £ )

Then, if Φ = k + 1, by operational laws in Definition 2.8, we have PTFWA w ( p ˜ 1 , p ˜ 2 , . . . , p ˜ k + 1 ) = ( [ ∑ £ = 1 k + 1 ℏ £ p £ , ∑ £ = 1 k + 1 ℏ £ q £ , ∑ £ = 1 k + 1 ℏ £ r £ , ∑ £ = 1 k + 1 ℏ £ s £ ] ; 1 - ∏ £ = 1 k ( 1 - Ψ α ˜ £ 2 ) ℏ £ + 1 - ( 1 - Ψ α ˜ k + 1 2 ) ℏ k + 1 - ( 1 - ∏ £ = 1 k ( 1 - Ψ α ˜ £ 2 ) ℏ j ( 1 - ( 1 - Ψ α ˜ k + 1 2 ) ℏ k + 1 ) , ∏ £ = 1 k + 1 ϒ α ˜ £ ℏ £ ) = ( [ ∑ £ = 1 k + 1 ℏ £ p £ , ∑ £ = 1 k + 1 ℏ £ q £ , ∑ £ = 1 k + 1 ℏ £ r £ , ∑ £ = 1 k + 1 ℏ £ s £ ] ; , 1 - ∏ £ = 1 k + 1 ( 1 - Ψ α ˜ £ 2 ) ℏ £ , ∏ £ = 1 k + 1 ϒ α ˜ £ ℏ £ )

Therefore Equation 13 hold for Φ = k + 1. Hence Equation, 13 hold ∀ Φ. To study some properties of PTFWA operator, we have following Theorem.

Theorem 3.3. (Boundedness) Let p ˜ £ = ( [ p £ , q £ , r £ , s £ ] ; Ψ £ , ϒ £ ) (£ =1, 2, …, Φ) be a collection of Pythagorean trapezoidal fuzzy numbers, and ℏ = (ℏ ₁, ℏ ₂, …, ℏ _Φ ) ^T be the weighted vector of p ˜ £ ( £ = 1 , 2 , . . . , Φ ) , with ℏ_£ ∈ [0, 1] and ∑ £ = 1 Φ ℏ £ = 1 . Consider that Ψ - = min 1 ≤ £ ≤ Φ { Ψ £ } , Ψ + = max 1 ≤ £ ≤ Φ { Ψ £ } , ϒ - = min 1 ≤ £ ≤ Φ { ϒ £ } , ϒ + = max 1 ≤ £ ≤ Φ { ϒ £ } , then we get ( Ψ - , ϒ + ) ≤ PTFWA ( p ˜ 1 , p ˜ 2 , . . . , p ˜ n ) ≤ ( Ψ + , ϒ - ) .(14)

Proof. For any p ˜ £ ( £ = 1 , 2 , . . . , Φ ) , we can get Ψ^- ≤ Ψ_£ ≤ Ψ⁺, ϒ^- ≤ ϒ_£ ≤ ϒ⁺. Suppose that p ˜ min = ( Ψ - , ϒ + ) , p ˜ max = ( Ψ + , ϒ - ) . ∑ £ = 1 Φ ℏ £ Ψ - ≤ ∑ £ = 1 Φ ℏ £ Ψ ≤ ∑ £ = 1 Φ ℏ £ Ψ + , ∑ £ = 1 Φ ℏ £ ϒ - ≤ ∑ £ = 1 Φ ℏ £ ϒ ≤ ∑ £ = 1 Φ ℏ £ ϒ + s ( p min ) = ( Ψ - ) 2 - ( ϒ + ) 2 = ( ∑ £ = 1 Φ ℏ £ Ψ - ) 2 - ( ∑ £ = 1 Φ ℏ £ ϒ + ) 2 , s ( p max ) = ( Ψ + ) 2 - ( ϒ - ) 2 = ( ∑ £ = 1 Φ ℏ £ Ψ + ) 2 - ( ∑ £ = 1 Φ ℏ £ ϒ - ) 2 , s ( PTFWA ( p ˜ 1 , p ˜ 2 , . . . , p ˜ n ) ) = ( ∑ £ = 1 Φ ℏ £ Ψ £ ) 2 - ( ∑ £ = 1 Φ ℏ £ ϒ £ ) 2