Method of MAGDM based on pythagorean trapezoidal uncertain linguistic hesitant fuzzy aggregation operator with Einstein operations

Abstract

In this paper, we define some Einstein operations on Pythagorean trapezoidal uncertain linguistic hesitant fuzzy (PTULHF) set and developed Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein weighted geometric (PTULHFEWG) operator, Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein ordered weighted geometric (PTULHFEOWG) operator and Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein hybrid weighted geometric (PTULHFEHG) operator. Furthermore, we origin ate the relationship between the current aggregation operators and suggested operators and establish many properties of these operators. We apply the proposed aggregation operators to deal with multi-attribute group decision making (MAGDM) problems. Finally, an illustrative example is given to illustrate the decision-making steps and to demonstrate its practicality and effectiveness. We also compare the result of proposed method with existing methods.

Keywords

Trapezoidal fuzzy numbers pythagorean trapezoidal fuzzy numbers einstein operation aggregation operation multiple attribute group decision-making

1 Introduction

The complexities of decision making problems and decision making situations, it is frequently intricate to communicate attribute values of alternatives by present values. Multi-attribute group decision making (MAGDM) problems widely survive in human social life. In the development of decision-making (DM), decision makers are usually requested to present their evaluation information against a set of alternatives over multiple attributes and then the best one (s) or ranking order result will be obtained by means of some MAGDM methods. Since the classical fuzzy set was anticipated by L.A. Zadeh [2] to express the uncertain and imprecise decision information, a lot of fruitful research achievements have been achieved. Meanwhile, many different forms of extensions of fuzzy sets have been put forward to depict decision-making objects. Intuitionistic fuzzy set (IFS), is generalization of fuzzy sets (FS) developed by Atanassov [1] and discussed the degree of membership degree (MD) and non-membership degree (N-MD). Atanassov [1] also presented several relation and different mathematically operations such as, algebraic product, sum, union, intersection and complement. With the help of membership work, we get data which makes feasible for us to fulfill the conclusion. The requirement of production with uncertainty in existent humanity troubles has been a continuing research experiment that has originated different methodologies and theories. In the study of fuzzy decision making along with their extensions have been discussed the different types of the problems. Fuzzy decision methods have become more and more accepted in decision making for personnel collection. Different decision making approaches have been planned to explain the difficulty. According to Liang [5] partial estimations are then concluded with facts inserted from trusted proficient’s. In accumulation regarding Liang [5] such opinions are supplementary modified by simulating their evaluation with respect to collective influence. If suppositions are not assembling, ordinary aggregation operators are used to choose the paramount alternative. Though, in existent assessment creation, there is a massive covenant of qualitative information that preserve be articulated by linguistic variables (LV), such as “good” and “bad” and so on, and associated to fuzzy numbers, the linguistic variables explain that its membership degree is 1 and its non membership degree is 0. On the establishment of the IFS and the LV set. Wang and Li [9] projected the notion of intuitionistic linguistic sets (ILS) whose fundamentals are intuitionistic linguistic (IL) numbers. As its emergence, the MAGDM method with ILS, particularly the analysis of the IL aggregation technique have obtained progressively consideration. Some intuitionistic linguistic power aggregation operators is proposed by [Liu 13]. Su [10] extensive the idea of order weighted averaging distance operator to intuitionistic linguistic operator.

Zhang et al. [30] proposed an approach to deriving a priority weight vector from an incomplete hesitant fuzzy preference relations (HFPR) using the logarithmic least squares method. Based on the priority weight vector, the consistency index of an incomplete HFPR is defined, which calculates the average deviation between the priority weight vector and all elements of the incomplete HFPR. Zhang et al. [31] also defined group decision making with intuitionistic multiplicative preference relation (IMPR). On this basis, a linear programming-based algorithm is developed to check and improve the consistency of an IMPR. Secondly, discussed the relationships between an IMPR and a normalized intuitionistic multiplicative weight vector. Zhang et al. [32] proposed a novel computational model based on the use of extended linguistic hierarchies, which not only can be used to operate with multi granular linguistic distribution assessments but also can provide interpretable linguistic results to decision makers. Yu Wen et al. [33] proposed a new method to deal with multi-criteria group decision making problems with unbalanced hesitant fuzzy linguistic term sets. The uncertain linguistic hesitant fuzzy set theory is a useful tool to deal with uncertainty in decision-making problems, which is proved by many scientific papers. This approach represents the situation in which different membership functions are considered possible in respect of decision situation. The elements in HFLSs are called hesitant fuzzy linguistic numbers (HFLNs). That is to say, for one object, an HFLS is reduced to an HFLN, which can be considered as a special case of HFLSs. For example < s₂, (0.3,0.4,0.5)>is an HFLNs and 0.3, 0.4, and 0.5 are the possible membership degrees to the linguistic terms. HFLSs have enabled great progress in describing linguistic information and to some extent may be considered an innovative construct. The main advantage of HFLSs is that they can describe two fuzzy attributes of an object: a linguistic term and a hesitant fuzzy element. The former provides an evaluation value, such as “excellent’‘ or “good.” The latter describes the hesitancy for the given evaluation value and denotes the membership degrees associated with the specific linguistic term.

In some practical multiattribute group decision making problems, the related criteria may be at different priority levels. For instance, a young couple wants to choose a toy for their child, the criteria of the toy they will consider are safety and price; obviously, the criteria safety has a higher priority than price. Aggregation operator plays a crucial role in the process of information fusion. In real life, the criteria sometimes have different priority levels. For example, safety has a higher priority than price when a couple chooses a toy for their child. Xiao [11] discussed the intuitionist linguistic suggest order weighted averaging operator, and pre-meditated its function in financial decision making problems. Liu and Jin [12] further planned the intuitionistic uncertain linguistic variables (IULVs) by extending LV and defined some basic operational laws of intuitionistic uncertain linguistic variables. Since linguistic variables are the unique cases of the uncertain linguistic variables, the IULVs are more general than the intuitionistic linguistic variables (ILS). P Liu et al. [14] developed the idea of linguistic fuzzy and discussed many operators in this field. In 2013, Yager [18, 19] constructed the idea of Pythagorean fuzzy set (PFS) differentiated by a MD and N-MD degree, which satisfies the provision that the square sum of its MD and N-MD degree is less than or equivalent to 1. Yager [19] gave an example to state of this condition, a DM gives his maintain for MD of an alternative as $(\frac{\sqrt 3}{2})$ and his maintain beside MD is $\frac{1}{2}$ . Due to the addition of two values is larger than 1, they are not accessible for IFS, but they are obtainable for PFS since ${(\frac{\sqrt 3}{2})}^{2} + {(\frac{1}{2})}^{2} \leq 1$ . Peng and Y. Yang [20] introduced the notion of the Pythagorean fuzzy weighted averaging (PFWA) operator, Pythagorean fuzzy weighted power averaging (PFWPA) operator, and the Pythagorean fuzzy weighted power geometric (PFWPG). In [3] H. Garg utilized the Einstein sum and introduced the view of the Pythagorean fuzzy aggregating operators such as, Pythagorean fuzzy Einstein weighted averaging (PFEWA) operator Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA) operator. Shakeel et al. [4] developed some aggregation operators to use the decision information represented by Pythagorean trapezoidal fuzzy numbers, 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. Shakeel et al. [21] extended the work of aggregation operators into interval-valued Pythagorean trapezoidal fuzzy weighted averaging (IVPTFWA) operator.Shakeel et al. [22] also introduced the idea of induced interval-valued Pythagorean trapezoidal fuzzy Einstein ordered weighted geometric (I-IVPTFEOWG) operator and induced interval-valued Pythagorean trapezoidal fuzzy Einstein hybrid geometric (I-IVPTFEHG) operator. Shakeel et al. [24] extended the idea of interval-valued Pythagorean trapezoidal fuzzy aggregation operators into interval-valued Pythagorean trapezoidal fuzzy Einstein weighted geometric, ordered weighted, and hybrid geometric operator.

The proposed operators are the advanced version of the already existing operators, treating the uncertainty problems in more accurate way. A variety of the problems can be addressed using these operators., which may not be covered by the existing literature. Based on the above analysis in this paper we consider Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein aggregation operators. From the above literature review the geometric Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein aggregation operator is not define. Therefore we develop a series of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein aggregation operators, including the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein weighted geometric (PTULHFEWG) operator, Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein ordered weighted geometric (PTULHFEOWG) operator and Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein hybrid geometric(PTULHFEHG) operator. We also develop the score and accuracy functions for these aggregation operators. Further, ranking order with the alternative highly depends on the weights of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein numbers, so choosing the weight vector is an important task. In this paper, the unknown weight vector of the Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein numbers is determined by using the aggregation operators. The main contributions of this work are summarized as below:

To overcome the shortcomings, some new operation laws for Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein numbers and based on it, some series of weighted, ordered weighted and hybrid weighted geometric aggregation operators have been proposed. The various desirable relations between them have been investigated in details.

The proposed operators introduce a new representation by considering the dependency between the degrees of the membership and non-membership functions during the aggregation process. Furthermore, based on it, a multi-criteria decision-making approach presented for solving the decision-making problems.

In this paper we develop the following sections; In Section 2, we briefly assess various necessary notions of intuitionistic fuzzy numbers, Pythagorean fuzzy numbers, trapezoidal fuzzy numbers, and Einstein operations on Pythagorean trapezoidal uncertain linguistic fuzzy numbers. In Section 3, we suggest some Pythagorean trapezoidal uncertain linguistic hesitant fuzzy aggregation operators based on Einstein operations, such as Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein weighted geometric (PTULHFEWG) operator, Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein ordered weighted geometric (PTULHFEOWG) operator and Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein hybrid geometric (PTULHFEHG) operator. In Section 4, we develop a MAGD method based on the proposed operators under Pythagorean trapezoidal uncertain linguistic hesitant fuzzy environment. In Section 5, we give numerical example. In Section 6, we compare the result with existing methods. Concluding remarks are completed in Section 7.

2 Preliminaries

In this section, we provide some basic definitions and results of intuitionistic uncertain linguistic set, intuitionistic trapezoidal fuzzy numbers, Pythagorean fuzzy numbers and Pythagorean trapezoidal fuzzy numbers.

Definition 1. [12] Let $[s_{θ (p)}, s_{t (p)}] \in \bar{s}$ , and P is a given domain as follows; $U = {[s_{θ (p)}, s_{t (p)}], (Ψ_{u} (p), ϒ_{u} (p))] : p \in P},$ (1) is called an intuitionistic uncertain linguistic set (IULS), where $s_{θ (p)}, s_{t (p)} \in \bar{s}$ . Where Ψ_u (p) and ϒ_u (p) are represent the MD and N-MD respectively, with the condition 0≤ Ψ_u (p) + ϒ_u (p) ≤1, ∀ p ∈ P. The MD and N-MD of the element p to the ULV [s_θ(p), s_t(p)]. For each ILS, U in p, if π_U (p) =1 - Ψ_U (p) - ϒ_U (p), for all p ∈ P, then π_u (p) is called the degree of indeterminacy degree of p to the ULV [s_θ(p), s_t(p)]. It is obvious that 0≤ π_u (p) ≤1, ∀ p ∈ P.

Definition 2. [12] Let U = {[s_θ(p), s_t(p)] , (Ψ_u (p) , ϒ_u (p))]} : p ∈ P}, be IULS, the quaternion {[s_θ(p), s_t(p)] , (γ_u (p) , ς_u (p))} is called an IULV, and U can also be viewed as a collection of the IULV. So, it can also be expressed as follows; $U = {[s_{θ (p)}, s_{t (p)}], (Ψ_{u} (p), ϒ_{u} (p))]} : p \in P} .$ (2)

Definition 3. [28] Let ${\tilde{α}}_{1} = ([p_{1}, q_{1}, r_{1}, s_{1}]; Ψ_{{\tilde{α}}_{1}},$ $ϒ_{{\tilde{α}}_{1}})$ , and ${\tilde{α}}_{2} = ([p_{2}, q_{2}, r_{2}, s_{2}]; Ψ_{{\tilde{α}}_{2}}, ϒ_{{\tilde{α}}_{2}})$ are two ITF numbers and δ ≥ 0. Then,

${\tilde{α}}_{1} \oplus {\tilde{α}}_{2} = ([p_{1} + p_{1}, q_{1} + q_{1}, r_{1} + r_{1}, s_{1} + s_{1}]; (Ψ_{{\tilde{α}}_{1}}) + (Ψ_{{\tilde{α}}_{2}}) - (Ψ_{{\tilde{α}}_{1}} Ψ_{{\tilde{α}}_{2}}), ϒ_{{\tilde{α}}_{1}} ϒ_{{\tilde{α}}_{2}})$ ,

${\tilde{α}}_{1} \otimes {\tilde{α}}_{2} = ([p_{1} p_{2}, q_{1} q_{2}, r_{1} r_{2}, s_{1} s_{2}]; Ψ_{{\tilde{α}}_{1}}$ $Ψ_{{\tilde{α}}_{2}}, (ϒ_{{\tilde{α}}_{1}}) + (ϒ_{{\tilde{α}}_{2}}) - (ϒ_{{\tilde{α}}_{1}} ϒ_{{\tilde{α}}_{2}})$ ,

$δ \tilde{α} = ([δ p, δ q, δ r, δ s]; 1 - (1 - Ψ_{\tilde{α}})^{δ}; (ϒ_{\tilde{α}})^{δ})$ ,

${\tilde{α}}^{δ} = ([p^{δ}, q^{δ}, r^{δ}, s^{δ}]; Ψ_{\tilde{α}}^{δ}, 1 - (1 - ϒ_{\tilde{α}})^{δ})$ .

Now we define the definition of Yager concept of Pythagorean fuzzy sets and some basic properties

Definition 4. [18] Let L be a fixed set. The PFS, U in L is an object having the form;

$A = {(l, Ψ_{U} (l), ϒ_{U} (l)) | l \in L},$ (3) where Ψ_A : L → [0, 1] and ϒ_A : L → [0, 1] represent the ND and N-MD of the element l ∈ L to A, respectively, and for every l ∈ L; $0 \leq Ψ_{U} \leq 1, 0 \leq ϒ_{U} \leq 1, 0 \leq Ψ_{U}^{2} (l) + ϒ_{U}^{2} (l) \leq 1 .$

For each PFS, U in L, $π_{U} (l) = \sqrt{1 - Ψ_{U}^{2} (l) - ϒ_{U}^{2} (l)}, for all l \in L,$ where π_U (l) is called the degree of indeterminacy of l to U.

Definition 5. [20] Let $\tilde{α} = (Ψ_{\tilde{α}}, ϒ_{\tilde{α}})$ , ${\tilde{α}}_{1} = (Ψ_{{\tilde{α}}_{1}},$ $ϒ_{{\tilde{α}}_{1}})$ and ${\tilde{α}}_{2} = (Ψ_{{\tilde{α}}_{2}}, ϒ_{{\tilde{α}}_{2}})$ be three PFNs and δ > 0. Then,

${\tilde{α}}_{1} \oplus {\tilde{α}}_{2} = (\sqrt{(Ψ_{{\tilde{α}}_{1}})^{2} + (Ψ_{{\tilde{α}}_{2}})^{2} - (Ψ_{{\tilde{α}}_{1}}^{2} Ψ_{{\tilde{α}}_{2}}^{2})},$ $ϒ_{{\tilde{α}}_{1}} ϒ_{{\tilde{α}}_{2}})$ ,

${\tilde{α}}_{1} \otimes {\tilde{α}}_{2} = (Ψ_{{\tilde{α}}_{1}} Ψ_{{\tilde{α}}_{2}},$ $\sqrt{(ϒ_{{\tilde{α}}_{1}})^{2} + (ϒ_{{\tilde{α}}_{2}})^{2} - (ϒ_{{\tilde{α}}_{1}}^{2} ϒ_{{\tilde{α}}_{2}}^{2})})$ ,

$δ \tilde{α} = \sqrt{1 - {(1 - Ψ_{\tilde{α}}^{2})}^{δ}}; (ϒ_{\tilde{α}})^{δ})$ ,

${\tilde{α}}^{δ} = (Ψ_{\tilde{α}}^{δ}, \sqrt{1 - {(1 - ϒ_{\tilde{α}}^{2})}^{δ}})$ .

Now we define the definition of Shakeel et al. concept of Pythagorean trapezoidal fuzzy numbers and some basic properties

Definition 6. [34] Let $\tilde{α} = {[p, q, r, s]; (Ψ_{\tilde{α}}, ϒ_{\tilde{α}})}$ , ${\tilde{α}}_{1} = {[p_{1}, q_{1}, r_{1}, s_{1}]; (Ψ_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{1}})}$ , and ${\tilde{α}}_{2} = {[p_{2},$ $q_{2}, r_{2}, s_{2}]; (Ψ_{{\tilde{α}}_{2}}, ϒ_{{\tilde{α}}_{2}})}$ , be any three PTF Ns and δ ≥ 0. Then,

${\tilde{α}}_{1} \oplus {\tilde{α}}_{2} = {[p_{1} + p_{2}, q_{1} + q_{2}, r_{1} + r_{2}, s_{1}$ $+ s_{2}]; \sqrt{(Ψ_{{\tilde{α}}_{1}})^{2} + (Ψ_{{\tilde{α}}_{2}})^{2} - (Ψ_{{\tilde{α}}_{1}} Ψ_{{\tilde{α}}_{2}})^{2}}, ϒ_{{\tilde{α}}_{1}}$ $ϒ_{{\tilde{α}}_{2}}}$ ,

${\tilde{α}}_{1} \otimes {\tilde{α}}_{2} = {[p_{1} p_{2}, q_{1} q_{2}, r_{1} r_{2}, s_{1} s_{2}]; Ψ_{{\tilde{α}}_{1}}$ $Ψ_{{\tilde{α}}_{2}}, \sqrt{(ϒ_{{\tilde{α}}_{1}})^{2} + (ϒ_{{\tilde{α}}_{2}})^{2} - (ϒ_{{\tilde{α}}_{1}} ϒ_{{\tilde{α}}_{2}})^{2}}}$ ,

$δ \tilde{α} = {([δ p, δ q, δ r, δ s]; \sqrt{1 - {(1 - Ψ_{\tilde{α}}^{2})}^{δ}};$ $(ϒ_{\tilde{α}})^{δ})}$ ,

${\tilde{α}}^{δ} = {[p^{δ}, q^{δ}, r^{δ}, s^{δ}]; Ψ_{\tilde{α}}^{δ}, \sqrt{1 - {(1 - ϒ_{\tilde{α}}^{2})}^{δ}}}$ .

In the following, we define some operation of Pythagorean trapezoidal uncertain linguistic (PTUL) fuzzy numbers, based on Einstein operational laws as follows;

Definition 7. [34] Let $\tilde{α} = ([p, q, r, s]; [Ψ_{\tilde{α}}, ϒ_{\tilde{α}}])$ be PTFN, a score function s can be defined as follows;

$s (\tilde{α}) = (\frac{p + q + r + s}{4} \cdot (Ψ_{\tilde{α}}^{2} - ϒ_{\tilde{α}}^{2})) s (\tilde{α}) \in [- 1, 1] .$ (4)

Definition 8. [34] Let $\tilde{α} = ([p, q, r, s]; [Ψ_{\tilde{α}}, ϒ_{\tilde{α}}])$ be PTFN, an accuracy function h can be defined as follows:

$h (\tilde{α}) = (\frac{p + q + r + s}{4} . (Ψ_{\tilde{α}}^{2} + ϒ_{\tilde{α}}^{2})) h (\tilde{α}) \in [0, 1] .$ (5)

2.1 Einstein operations with pythagorean trapezoidal fuzzy numbers

The assumption of aggregation operators has an significant function since in the establishment of fuzzy set theory. Einstein operations is a class of assorted t-norms and t-conorms families [26, 27] can be utilized to realize the corresponding intersections and unions of IFSs. Pythagorean Einstein operations consists of the Pythagorean Einstein product and the Pythagorean Einstein sum, respectively, defined as follows.

$\begin{matrix} a \otimes_{ɛ} b = \frac{a . b}{\sqrt{1 + (1 - a^{2}) . (1 - b^{2})}}, \\ a \oplus_{ɛ} b = \frac{\sqrt{(a^{2} + b^{2})}}{\sqrt{(1 + a^{2} . b^{2})}}, for all (a, b) \in {[0, 1]}^{2} \end{matrix}$ (6)

In the following, we define some operation of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy (PTULHF) numbers, based on Einstein operational laws as follows;

Definition 9. Let $\tilde{α} = {(s_{θ (p)}, s_{t (p)}, [p, q, r, s];$ $(γ_{\tilde{α}}, β_{\tilde{α}})}$ , ${\tilde{α}}_{1} = {(s_{θ (p_{1})}, s_{t (p_{1})}, [p_{1}, q_{1}, r_{1}, s_{1}]; (γ_{{\tilde{α}}_{1}},$ $β_{{\tilde{α}}_{1}})}$ , and ${\tilde{α}}_{2} = {(s_{θ (p_{2})}, s_{t (p_{2})}, [p_{2}, q_{2}, r_{2}, s_{2}]; (γ_{{\tilde{α}}_{2}},$ $β_{{\tilde{α}}_{2 s}})}$ , be any three PTULHFNs and δ ≥ 0. Then,

${\tilde{α}}_{1} \oplus_{ɛ} {\tilde{α}}_{2} = {\begin{matrix} (s_{θ (p_{1})} + s_{θ (p_{2})}, s_{t (p_{1})} + s_{t (p_{2})}), \\ [p_{1} + p_{2}, q_{1} + q_{2}, r_{1} + r_{2}, s_{1} + s_{2}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{(Ψ_{{\tilde{α}}_{1}})^{2} + (Ψ_{{\tilde{α}}_{2}})^{2}}}{\sqrt{1 + (Ψ_{{\tilde{α}}_{1}})^{2} (Ψ_{{\tilde{α}}_{2}})^{2}}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{ϒ_{{\tilde{α}}_{1}} ϒ_{{\tilde{α}}_{2}}}{\sqrt{1 + (1 - ϒ_{{\tilde{α}}_{1}}^{2}) (1 - ϒ_{{\tilde{α}}_{2}}^{2})}} \end{matrix}}$ ,

${\tilde{α}}_{1} \otimes_{ɛ} {\tilde{α}}_{2} = {\begin{matrix} (s_{θ (p_{1})} s_{θ (p_{2})}, s_{t (p_{1})} s_{t (p_{2})}), \\ [p_{1} p_{2}, q_{1} q_{2}, r_{1} r_{2}, s_{1} s_{2}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{Ψ_{{\tilde{α}}_{1}} Ψ_{{\tilde{α}}_{2}}}{\sqrt{1 + (1 - Ψ_{{\tilde{α}}_{1}}^{2}) (1 - Ψ_{{\tilde{α}}_{2}}^{2})}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{(ϒ_{{\tilde{α}}_{1}})^{2} + (ϒ_{{\tilde{α}}_{2}})^{2}}}{\sqrt{1 + (ϒ_{{\tilde{α}}_{1}})^{2} (ϒ_{{\tilde{α}}_{2}})^{2}}}, \end{matrix}}$ ,

$δ_{\dot{ɛ}} \tilde{α} = {\begin{matrix} (s_{δ θ (p)}, s_{δ t (p)}) [δ p, δ q, δ r, δ s]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}}{\cup} \frac{\sqrt{(1 + Ψ_{\tilde{α}}^{2})^{δ} - (1 - Ψ_{\tilde{α}}^{2})^{δ}}}{\sqrt{(1 + Ψ_{\tilde{α}}^{2})^{δ} + (1 - Ψ_{\tilde{α}}^{2})^{δ}}}, \underset{ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{2 {(ϒ_{\tilde{α}}^{2})}^{δ}}}{\sqrt{(2 - ϒ_{\tilde{α}}^{2})^{δ} + (ϒ_{\tilde{α}}^{2})^{δ}}}, \end{matrix}}$ ,

${\tilde{α}}^{\land_{ɛ} δ} = {\begin{matrix} (s_{θ {(p)}^{δ}}, s_{t {(p)}^{δ}}), [p^{δ}, q^{δ}, r^{δ}, s^{δ}]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}}{\cup} \frac{\sqrt{2 {(Ψ_{\tilde{α}}^{2})}^{δ}}}{\sqrt{(2 - Ψ_{\tilde{α}})^{δ} + (Ψ_{\tilde{α}})^{δ}}}, \underset{ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(1 + ϒ_{\tilde{α}}^{2})^{δ} - (1 - ϒ_{\tilde{α}}^{2})^{δ}}}{\sqrt{(1 + ϒ_{\tilde{α}}^{2})^{δ} + (1 - ϒ_{\tilde{α}}^{2})^{δ}}} \end{matrix}} .$

Definition 10. Let $\tilde{α} = {s_{θ (p)}, s_{t (p)}, [p, q, r, s];$ $(γ_{\tilde{α}}, β_{\tilde{α}})}$ , be PTULH fuzzy numbers, a score function s can be defined as follows;

$\begin{matrix} s (\tilde{α}) = & \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} {\frac{(s_{θ (p)} + s_{t (p)}) [p + q + r + s]}{8} . (Ψ_{\tilde{α}}^{2} - ϒ_{\tilde{α}}^{2})} \\ s (\tilde{α}) \in [- 1, 1] . \end{matrix}$ (7)

Definition 11. Let $\tilde{α} = {s_{θ (p)}, s_{t (p)}, [p, q, r, s];$ $(γ_{\tilde{α}}, β_{\tilde{α}})}$ , be PTULH fuzzy numbers, a accuracy function h can be defined as follows;

$\begin{matrix} h (\tilde{α}) = & \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} {\frac{(s_{θ (p)} + s_{t (p)}) [p + q + r + s]}{8} . (Ψ_{\tilde{α}}^{2} + ϒ_{\tilde{α}}^{2})} \\ h (\tilde{α}) \in [- 1, 1] . \end{matrix}$ (8)

Theorem 1. Let ${\tilde{α}}_{1}, {\tilde{α}}_{2}$ and $\tilde{α}$ be three Pythagorean trapezoidal uncertain linguistic fuzzy Einstein fuzzy (PTULHFE) numbers and n and m be any positive real number. Then,

${\tilde{α}}_{1} \oplus {\tilde{α}}_{2}$ ,

${\tilde{α}}_{1} \otimes {\tilde{α}}_{2}$ ,

$δ_{\dot{ɛ}} \tilde{α}$ , and

${\tilde{α}}^{\land_{ɛ} δ}$ are also PTULHFE numbers.

Theorem 2. ${\tilde{α}}_{1} = {s_{θ (p_{1})}, s_{t (p_{1})}, [p_{1}, q_{1}, r_{1}, s_{1}];$ $(γ_{{\tilde{α}}_{1}}, β_{{\tilde{α}}_{1}})}$ , and ${\tilde{α}}_{2} = {s_{θ (p_{2})}, s_{t (p_{2})}, [p_{2}, q_{2}, r_{2},$ $s_{2}]; (γ_{{\tilde{α}}_{2}}, β_{{\tilde{α}}_{2 s}})}$ , be any two PTULHFNs. Then, the operational laws between ${\tilde{α}}_{1}$ and ${\tilde{α}}_{2}$ are shown as follows;

${\tilde{α}}_{1} \otimes {\tilde{α}}_{2} = {\tilde{α}}_{2} \otimes {\tilde{α}}_{1}$

$({\tilde{α}}_{1} \otimes {\tilde{α}}_{2}) δ = {\tilde{α}}_{2}^{δ} \otimes {\tilde{α}}_{1}^{δ} δ \geq 0$ ,

${\tilde{α}}^{δ_{1}} \otimes {\tilde{α}}^{δ_{2}} = {\tilde{α}}^{(δ_{1} + δ_{2})} δ_{1}, δ_{2} \geq 0$ ,

3 Pythagorean trapezoidal uncertain linguistic hesitant fuzzy einstein aggregation operators

We define Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein aggregation operators, based on the above Einstein operational laws of PTULHFENs, we will investigate the Pythagorean trapezoidal uncertain linguistic hesitant fuzzy information aggregation operators and give the Definitions of some aggregation operators with the Pythagorean trapezoidal uncertain linguistic hesitant fuzzy numbers based on Einstein operational laws as follows. Let Ω be the set of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein numbers.

3.1 Pythagorean trapezoidal uncertain linguistic hesitant fuzzy einstein geometric aggregation operators

Definition 12. Let ${\tilde{α}}_{j} = {s_{θ (p_{j})}, s_{t (p_{j})}, [p_{j}, q_{j}, r_{j},$ $s_{j}]; (γ_{{\tilde{α}}_{j}}, β_{{\tilde{α}}_{j}})}$ , where j = (1, 2 . . . , n) be a collection of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy numbers. Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein weighted geometric (PTULHFEWG) operator of dimension n is mapping PTULHFEWG : Ωⁿ - > Ω, such that,

$\begin{matrix} PTULFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ = \overset{n}{\underset{j = 1}{\otimes_{ɛ}}} {({\tilde{α}}_{j})}^{w_{j}}, \end{matrix}$ (10) where w = (w₁, w₂, . . . w_n) ^T is the weighting vector of ${\tilde{α}}_{j} (j = 1, 2 . . ., n)$ with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$

$\begin{matrix} PTULFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ = \overset{n}{\underset{j = 1}{\otimes_{ɛ}}} {({\tilde{α}}_{j})}^{w_{j}} \\ = (\begin{matrix} (\prod_{j = 1}^{n} (s_{θ (p_{j})})^{w_{j}}, \prod_{j = 1}^{n} (s_{t (p_{j})})^{w_{j}}), \\ [\prod_{j = 1}^{n} (p_{j})^{w_{j}}, \prod_{j = 1}^{n} (q_{j})^{w_{j}}, \prod_{j = 1}^{n} (r_{j})^{w_{j}}, \prod_{j = 1}^{n} (s_{j})^{w_{j}}]; \\ \underset{Ψ_{{\tilde{α}}_{j}} \in γ_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{2 \prod_{j = 1}^{n} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}, \\ \underset{ϒ_{{\tilde{α}}_{j}} \in β_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{\prod_{j = 1}^{n} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}, \end{matrix}) \end{matrix}$ (11)

where w = (w₁, w₂, . . . w_n) ^T is the weighting vector of ${\tilde{α}}_{j} (j = 1, 2 . . ., n)$ , with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$

Theorem 4. Let ${\tilde{α}}_{j} (j = 1, 2, . . ., n)$ be a collection of PTULHFENs, where w = (w₁, w₂, . . . , w_n,) ^T is the weighting vector of ${\tilde{α}}_{j} (j = 1, 2, . . ., n)$ , with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . Then, the PTULHFEWG operator satisfies the following properties;

Proof. (1) (Idempotency) If all ${\tilde{α}}_{j} (j = 1, 2, . . ., n)$ are equal, i.e. ${\tilde{α}}_{j} = \tilde{α}$ for all j, then

$PTULHFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) = \tilde{α} .$ (12)

By Definition 11, we have $\begin{matrix} PTULHFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ = {\tilde{α}}_{1}^{w_{1}} \otimes_{ɛ} {\tilde{α}}_{2}^{w_{2}} \otimes_{ɛ}, . . ., \otimes_{ɛ} {\tilde{α}}_{n}^{w_{n}}, \\ = {\tilde{α}}^{w_{1}} \otimes_{ɛ} {\tilde{α}}^{w_{2}} \otimes_{ɛ}, . . ., \otimes_{ɛ} {\tilde{α}}^{w_{n}} \\ = \prod_{j = 1}^{n} {\tilde{α}}^{w_{j}} = \tilde{α} . \end{matrix}$

(2) (Boundray) Let ${\tilde{α}}_{min} = min {{\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}}$ and ${\tilde{α}}_{max} = max {{\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}}$ , then

${\tilde{α}}_{min} \leq PTULHFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \leq {\tilde{α}}_{max} .$ (13)

(3) (Monotonicity) Let ${\tilde{α}}_{j}^{*} = {(s_{θ (p_{j})}^{*}, s_{t (p_{j})}^{*})$ $[p_{j}^{*}, q_{j}^{*}, r_{j}^{*}, s_{j}^{*}]; (Ψ_{{\tilde{α}}_{j}}^{*}, ϒ_{{\tilde{α}}_{j}}^{*})}$ (j = 1, 2 . . . , n) be a collection of Pythagorean trapezoidal fuzzy numbers. If ${\tilde{α}}_{j} \leq {\tilde{α}}_{j}^{*}$ , for all j, then

$\begin{matrix} PTULHFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ \leq PTULHEWG ({\tilde{α}}_{1}^{*}, {\tilde{α}}_{2}^{*}, . . ., {\tilde{α}}_{n}^{*}) . \end{matrix}$ (14)

Let $\begin{matrix} PTULHFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) = & \prod_{j = 1}^{n} ({\tilde{α}}_{j})^{w_{j}}, \\ PTULHFEWA ({\tilde{α}}_{1}^{*}, {\tilde{α}}_{2}^{*}, . . ., {\tilde{α}}_{n}^{*}) = & \prod_{j = 1}^{n} ({\tilde{α}}_{j}^{*})^{w_{j}} . \end{matrix}$

Since ${\tilde{α}}_{j} \leq {\tilde{α}}_{j}^{*}$ for all j, then we have ${\tilde{α}}_{j}^{w_{j}} \leq {\tilde{α}}_{j}^{* w_{j}}$ . Therefore, we have $\begin{matrix} PTULHFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ \leq PTULHFEWG ({\tilde{α}}_{1}^{*}, {\tilde{α}}_{2}^{*}, . . ., {\tilde{α}}_{n}^{*}) . \end{matrix}$

Definition 13. Let ${\tilde{α}}_{j} = {s_{θ (p_{j})}, s_{t (p_{j})}, [p_{j}, q_{j}, r_{j},$ $s_{j}]; (γ_{{\tilde{α}}_{j}}, β_{{\tilde{α}}_{j}})}$ , where j = (1, 2 . . . , n) be a collection of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy numbers. The Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein ordered weighted geometric (PTULHFEOWG) operator of dimension n is mapping PTULHFEOWG : Ωⁿ - > Ω, and w = (w₁, w₂, . . . w_n) ^T is the weighting vector of ${\tilde{α}}_{j} (j = 1, 2 . . ., n)$ , with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

$PTULHFEOWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) = \overset{n}{\underset{j = 1}{\otimes_{ɛ}}} {({\tilde{α}}_{σ_{(j)}})}^{w_{j}}$ (15) where (σ (1) , σ (2) , . . . , σ (n)) is a permutation of (1, 2 . . . , n), such that, ${\tilde{α}}_{σ_{(j)}} \leq$ ${\tilde{α}}_{σ_{(j - 1)}}$ for all j = 1, 2, . . . , n. The basic concept of the PTULHFEOWG operator is its re-ordering step, and PTULHFEOWG operator first ranks all the given PTULFNs in descending order, and then additively aggregates these PTULFNs together with the weights of their ordered positions, where the corresponding operations are Einstein operations.

Theorem 5. Let ${\tilde{α}}_{i} (i = 1, 2, . . ., n)$ be a collection of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy numbers, then their aggregated value by using the PTULHFEOWG operator is also and Pythagorean trapezoidal uncertain linguistic hesitant fuzzy number such that,

$\begin{matrix} PTULHFEOWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) = \overset{n}{\underset{j = 1}{\otimes_{ɛ}}} {({\tilde{α}}_{σ_{(j)}})}^{w_{j}} \\ = (\begin{matrix} (\prod_{j = 1}^{n} (s_{θ (p_{σ_{(j)}})})^{w_{j}}, \prod_{j = 1}^{n} (s_{t (p_{σ_{(j)}})})^{w_{j}}), \\ [\prod_{j = 1}^{n} (p_{σ_{(j)}})^{w_{j}}, \prod_{j = 1}^{n} (q_{σ_{(j)}})^{w_{j}}, \prod_{j = 1}^{n} (r_{σ_{(j)}})^{w_{j}}, \prod_{j = 1}^{n} (s_{σ_{(j)}})^{w_{j}}]; \\ \frac{\sqrt{2 \prod_{j = 1}^{n} {(Ψ_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - Ψ_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(Ψ_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}}}}, \\ \frac{\sqrt{\prod_{j = 1}^{n} {(1 + ϒ_{\tilde{α} σ_{(j)}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ϒ_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 + ϒ_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ϒ_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}}}}, \end{matrix}) \end{matrix}$ (16)

where (σ (1) , σ (2) , . . . , σ (n)) is a permutation of (1, 2 . . . , n), such that, ${\tilde{α}}_{σ_{(j)}} \leq$ ${\tilde{α}}_{σ_{(j - 1)}}$ for all j = 1, 2, . . . , n. w = (w₁, w₂, . . . w_n) ^T is weight vector of PTULFEOWG operator, with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

Proof. Similarly as proof of Theorem 3, it is omitted.

Theorem 6. Let ${\tilde{α}}_{i} (i = 1, 2, . . ., n)$ (j = 1, 2 . . . , n) be a collection of PTULHFENs, and w = (w₁, w₂, . . . w_n) ^T is the weighting vector of PTULHFEOWG with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j}$ = 1, which satisfies the following properties;

(Idmpotency) If for all ${\tilde{α}}_{j} (j = 1, 2 . . ., n)$ are equal, i.e. ${\tilde{α}}_{j} = \tilde{α}$ , for all j, then

$PTULHFEOWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) = \tilde{α} .$ (17)

(Boundary) Let ${\tilde{α}}_{min} = min {{\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}}$ and ${\tilde{α}}_{max} = max {{\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}}$ , then

$\begin{matrix} {\tilde{α}}_{min} & \leq PTULHFEOWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ \leq {\tilde{α}}_{max} . \end{matrix}$ (18)

(Monotonicity) Let ${\tilde{α}}_{j}^{*} = {(s_{θ (p_{j})}^{*}, s_{t (p_{j})}^{*}) [p_{j}^{*},$ $q_{j}^{*}, r_{j}^{*}, s_{j}^{*}]; (Ψ_{{\tilde{α}}_{j}}^{*}, ϒ_{{\tilde{α}}_{j}}^{*})}$ (j = 1, 2 . . . , n) be a collection of PTULHF numbers. If ${\tilde{α}}_{j} \leq {\tilde{α}}_{j}^{*}$ , for all j, then

$\begin{matrix} PTULHFEOWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ \leq PTULHFEOWG ({\tilde{α}}_{1}^{*}, {\tilde{α}}_{2}^{*}, . . ., {\tilde{α}}_{n}^{*}) . \end{matrix}$ (19)

(Commutativity) Let ${\tilde{α}}_{j}^{*} = {(s_{θ (p_{j})}^{*}, s_{t (p_{j})}^{*}) [p_{j}^{*},$ $q_{j}^{*}, r_{j}^{*}, s_{j}^{*}]; (Ψ_{{\tilde{α}}_{j}}^{*}, ϒ_{{\tilde{α}}_{j}}^{*})}$ (j = 1, 2 . . . , n) be a collection of PTULF numbers then,

$\begin{matrix} PTULHFEOWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ = PTULHFEOWG ({\tilde{α}}_{1}^{*}, {\tilde{α}}_{2}^{*}, . . ., {\tilde{α}}_{n}^{*}) . \end{matrix}$ (20)

where

({\tilde{α}}_{1}^{*}, {\tilde{α}}_{2}^{*}, . . ., {\tilde{α}}_{n}^{*})

is any permutation of

({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n})

We shall propose Pythagorean trapezoidal uncertain linguistic fuzzy Einstein hybrid geometric (PTULHFEHG) operator as follows.

Definition 14. Let ${\tilde{α}}_{j}$ (j = 1, 2 . . . , n) be a collection of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy numbers. Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein hybrid geometric (PTULHFEHG) operator of dimension n is mapping PTULHFEHG: Ωⁿ - > Ω having weight w = (w₁, w₂, . . . w_n) ^T such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ $PTULHFEHA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) = \overset{n}{\underset{j = 1}{\otimes_{ɛ}}} {({\tilde{α}}_{σ_{(j)}}^{.})}^{w_{j}}$ (21) where ${\tilde{β}}_{σ_{(j)}}$ is the jth largest of the weighted Pythagorean trapezoidal uncertain linguistic hesitant fuzzy numbers ${\tilde{β}}_{j} ({\tilde{β}}_{j} = {\tilde{α}}_{j}^{{nw}_{j}}, j = 1, 2, . . ., n)$ , and w = (w₁, w₂, . . . w_n) ^T such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ where n is balancing coefficient.

Theorem 7. Let ${\tilde{α}}_{j}$ (j = 1, 2 . . . , n) be a collection of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy numbers. then their aggregated value by using the PTULHFEHG operator is also Pythagorean trapezoidal fuzzy number, and

$\begin{matrix} PTULFEHG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) = \overset{n}{\underset{j = 1}{\otimes_{ɛ}}} {({\tilde{α}}_{σ_{(j)}}^{.})}^{w_{j}} \\ = (\begin{matrix} (\prod_{j = 1}^{n} (s_{θ (p_{σ_{(j)}})}^{.})^{w_{j}}, \prod_{j = 1}^{n} (s_{t (p_{σ_{(j)}})}^{.})^{w_{j}}), \\ [\prod_{j = 1}^{n} (p_{σ_{(j)}}^{.})^{w_{j}}, \prod_{j = 1}^{n} (q_{σ_{(j)}}^{.})^{w_{j}}, \prod_{j = 1}^{n} (r_{σ_{(j)}}^{.})^{w_{j}}, \prod_{j = 1}^{n} (s_{σ_{(j)}}^{.})^{w_{j}}]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}}{\cup} \frac{\sqrt{2 \prod_{j = 1}^{n} {({\dot{Ψ}}_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - {\dot{Ψ}}_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {({\dot{Ψ}}_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}}}}, \\ \underset{ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{\prod_{j = 1}^{n} {(1 + {\dot{ϒ}}_{\tilde{α} σ_{(j)}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{ϒ}}_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 + {\dot{ϒ}}_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{ϒ}}_{{\tilde{α}}_{σ_{(j)}}}^{2})}^{w_{j}}}}, \end{matrix}) \end{matrix}$ (22)

Similarly as proof of Theorem 3, it is omitted here.

4 A novel method for MAGDM based on the proposed PTULHFEWG and PTULHFEHAG operators

In this section, we apply the proposed aggregation operators to develop an approach for dealing with multiple attribute group decision making problems under the Pythagorean trapezoidal uncertain linguistic hesitant fuzzy environment.

Algorithm: For a group decision making problem. Consider A = {A₁, A₂, . . . , A_m} be a set of alternatives, C = {C₁, C₂, . . . , C_n} be a set of attributes/characteristic, and w = (w₁, w₂, . . . , w_n) ^T is the weighting vector of the attributes such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . Let M = {m₁, m₂, . . . , m_p} be the set of decision makers and λ = (λ₁, λ₂, . . . , λ_p) ^T be the weighting vector of decision makers λ_k ∈ [0, 1] and $\sum_{k = 1}^{p} λ_{k} = 1$ . Let $R^{(k)} = {(r_{ij}^{(k)})}_{m \times n} (k = 1, 2, . . . p)$ be Pythagorean trapezoidal uncertain linguistic hesitant fuzzy decision matrix, where

$h_{ij}^{(k)} = ((s_{θ_{ij}}^{(k)}, s_{t_{ij}}^{(k)}), [h_{1 ij}^{(k)}, h_{2 ij}^{(k)}, h_{3 ij}^{(k)}, h_{4 ij}^{(k)}]; Ψ_{ij}^{(k)}, ϒ_{ij}^{(k)})$

is Pythagorean trapezoidal uncertain linguistic hesitant fuzzy number provided by decision makers m_k ∈ M, such that $\begin{matrix} 0 \leq & Ψ_{ij}^{(k)} \leq 1, 0 \leq ϒ_{ij}^{(k)} \leq 1, \\ 0 \leq & Ψ_{ij}^{2 (k)} + ϒ_{ij}^{2 (k)} \leq 1, i = 1, 2, . . . m, j = 1, 2, . . . n . \end{matrix}$

In the following, we apply the PTULHFEWG and PTULHFEHG operators to develop an approach to deal with multiple attribute group decision making problems, when decision information is this method consists of the following steps:

Step (1). In this step, we construct the Pythagorean trapezoidal uncertain linguistic hesitant fuzzy matrix $R^{(k)} = (r_{ij}^{(k)})_{m \times n}$ . In which the decision maker give his opinion related to each alternative with respect to each criteria.

Step (2). In this step, we apply the decision information given in the Pythagorean trapezoidal uncertain linguistic hesitant fuzzy matrix $R^{(k)} = (r_{ij}^{(k)})_{m \times n}$ . and the PTULHFEWG operator to derive the individual overall preference values $r_{i}^{(k)}$ of the alternative A_i, where w = (w₁, w₂, . . . , w_n) ^T is the weighting vector of the attributes.

$\begin{matrix} r_{i}^{(k)} = & PTULFEWG (r_{i 1}^{(k)}, r_{i 2}^{(k)}, . . ., r_{in}^{(k)}), \\ i = 1, 2, . . ., m, k = 1, 2, . . . p . \end{matrix}$ (23)

Step (3). In this step, we apply the PTULHFEHG operator to aggregate all the individual overall preference Pythagorean trapezoidal uncertain linguistic fuzzy values $r_{i}^{(k)} (k = 1, 2, . . . p)$ into the collective overall preference Pythagorean trapezoidal uncertain linguistic fuzzy values r_i, where

w = (w₁, w₂, . . . , w_n) ^T is the associated vector of PTULHFEHG operator such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1, λ = {(λ_{1}, λ_{2}, . . ., λ_{p})}^{T}$ be the weighting vector of decision makers, λ_k ∈ [0, 1] and $\sum_{k = 1}^{p} λ_{k} = 1$ .

$\begin{matrix} r_{i} = & PTULHFEHG (r_{i}^{(1)}, r_{i}^{(2)}, . . ., r_{i}^{(p)}), \\ i = 1, 2, . . ., m, k = 1, 2, . . . p . \end{matrix}$ (24)

Step (4). In this step, we calculate the scores to aggregate the value of each alternative A_i. If there is no difference between two or more than two scores then we have must to find out the accuracy degrees of the aggregate value of each alternative.

Step (5). In this step, we arrange all the score values of the alternatives in the form of descending order and select the best alternative A_i which has the highest degree of the score value.

Step (6). End.

5 Example of surface irrigation problem

In this section, we developed a numerical application of proposed method and solved the surface irrigation problem. Consider that the water flows and spreads over the surface of the land. Varied quantities of water are allowed on the fields at different times. Therefore, flow of water under surface irrigation comes under wobbly flow. Consequently, it is very difficult to understand the hydraulics of surface irrigation. However suitable and efficient surface irrigation system can be espoused after taking into consideration different factors which are involved in the hydraulics of surface irrigation.

Therefore, we are going to present an illustrative example of the new approach in a decision making problem. We analyze that the ratio of water flows of the land was increasing day-by-day in Pakistan, therefore the government of Pakistan wanted to control the ratio of this problem. For this purpose they consider four possible alternatives A_i (i = 1, 2, 3, 4) such that,

A₁: Surface slope of the field

A₂: Roughness of the field surface

A₃: Depth of water to be applied

A₄: Field resistance to erosion

To evaluate these alternatives a group of committee take a decision about the best surface land according to the following four attributes C_j (j = 1, 2, 3, 4). Let w = (0.4, 0.3, 0.2, 0.1) are weighting vector of the attributes C_j (j = 1, 2, 3, 4). such that,

C₁: It assists in storing required amount of water in the root-zone-depth

C₂: It reduces the wastage of irrigation water from the field in the form of run-off water

C₃: It reduces the soil erosion to minimum.

C₄: It does not avert use of machinery for land preparation, cultivation and harvesting.

The experts of the government evaluate and offer their own opinions regarding the results obtained with each alternative, suppose the weights of the experts are given as λ = (0.1, 0.2, 0.3, 0.4). As the environment is very uncertain, the group of experts needs to assess the available information by using PTULHFNs. The expected results given in the form of PTULHFNs depending on the characteristic C_j and the alternative A_i are shown Tables 1 –4. Such that, $R^{(k)} = {(r_{ij}^{(k)})}_{4 \times 4} (k = 1, 2, 3, 4)$ .

Step 1. The decision makers give his decision in the following Tables.

Table 1
R⁽¹⁾ Decision matrix of expert -1

1 C₁ C₂

A₁ (S_2, S_2,[0.3,0.4,0.5,0.1] (S_3,S_3,[0.4,0.3,0.4,0.5]

(0.7,0.6) (0.5,0.5)

A₂ (S_4, S_2,[0.4,0.1,0.7,0.8] (S_3,S_3,[0.6,0.5,0.4,0.1]

(0.8,0.3) (0.6,0.5)

A₃ (S_3, S_3,[0.9,0.1,0.3,0.7] (S_4,S_2,[0.3,0.4,0.8,0.9]

(0.6,0.5) (0.8,0.2)

A₄ (S_4, S_4,[0.2,0.1,0.4,0.5] (S_2,S_2,[0.4,0.1,0.2,0.6]

(0.8,0.3) (0.6,0.5)

C₃ C₄

A₁ (S_2, S_3,[0.6,0.5,0.4,0.3] (S_2,S_1,[0.4,0.5,0.6,0.7]

(0.6,0.4) (0.8,0.3)

A₂ (S_2, S_2,[0.3,0.4,0.2,0.5] (S_2,S_4,[0.1,0.5,0.3,0.4]

(0.8,0.4) (0.9,0.2)

A₃ (S_2, S_4,[0.4,0.5,0.1,0.2] (S_2,S_2,[0.3,0.4,0.5,0.6]

(0.9,0.3) (0.9,0.1)

A₄ (S_3, S_2,[0.6,0.4,0.8,0.3] (S_2,S_3,[0.2,0.3,0.4,0.5]

(0.5,0.5) (0.7,0.4)

1	C₁	C₂
A₁	(S_2, S_2,[0.3,0.4,0.5,0.1]	(S_3,S_3,[0.4,0.3,0.4,0.5]
	(0.7,0.6)	(0.5,0.5)
A₂	(S_4, S_2,[0.4,0.1,0.7,0.8]	(S_3,S_3,[0.6,0.5,0.4,0.1]
	(0.8,0.3)	(0.6,0.5)
A₃	(S_3, S_3,[0.9,0.1,0.3,0.7]	(S_4,S_2,[0.3,0.4,0.8,0.9]
	(0.6,0.5)	(0.8,0.2)
A₄	(S_4, S_4,[0.2,0.1,0.4,0.5]	(S_2,S_2,[0.4,0.1,0.2,0.6]
	(0.8,0.3)	(0.6,0.5)
	C₃	C₄
A₁	(S_2, S_3,[0.6,0.5,0.4,0.3]	(S_2,S_1,[0.4,0.5,0.6,0.7]
	(0.6,0.4)	(0.8,0.3)
A₂	(S_2, S_2,[0.3,0.4,0.2,0.5]	(S_2,S_4,[0.1,0.5,0.3,0.4]
	(0.8,0.4)	(0.9,0.2)
A₃	(S_2, S_4,[0.4,0.5,0.1,0.2]	(S_2,S_2,[0.3,0.4,0.5,0.6]
	(0.9,0.3)	(0.9,0.1)
A₄	(S_3, S_2,[0.6,0.4,0.8,0.3]	(S_2,S_3,[0.2,0.3,0.4,0.5]
	(0.5,0.5)	(0.7,0.4)

Table 2

R⁽²⁾. Decision matrix of expert -2

2	C₁	C₂
A₁	(S_6, S_2,[0.3,0.4,0.4,0.5]	(S_3,S_4,[0.4,0.3,0.9,0.6]
	(0.8,0.3)	(0.6,0.5)
A₂	(S_5, S_3,[0.5,0.6,0.3,0.4]	(S_2,S_2,[0.5,0.4,0.4,0.5]
	(0.7,0.4)	(0.8,0.2)
A₃	(S_4, S_2,[0.4,0.3,0.2,0.1]	(S_6,S_3,[0.4,0.5,0.1,0.3]
	(0.6,0.5)	(0.8,0.2)
A₄	(S_2, S₄,[0.3,0.4,0.2,0.5]	(S₂,S₂,[0.5,0.1,0.3,0.4]
	(0.5,0.5)	(0.8,0.4)
	C₃	C₄
A₁	(S_5, S_2,[0.4,0.1,0.2,0.3]	(S_4,S_3,[0.4,0.5,0.6,0.9]
	(0.8,0.4)	(0.9,0.2)
A₂	(S_6, S_3,[0.4,0.5,0.3,0.1]	(S_1,S_2,[0.4,0.5,0.2,0.1]
	(0.9,0.1)	(0.6,0.5)
A₃	(S_3, S_4,[0.5,0.6,0.5,0.7]	(S_4,S_3,[0.3,0.4,0.3,0.2]
	(0.9,0.3)	(0.9,0.1)
A₄	(S_4, S_3,[0.5,0.1,0.9,0.1]	(S_3,S_5,[0.4,0.3,0.1,0.5]
	(0.7,0.6)	(0.8,0.5)

Table 3

R⁽³⁾. Decision matrix of expert -3

3	C₁	C₂
A₁	(S_4, S_3,[0.4,0.6,0.7,0.2]	(S_1,S_3,[0.9,0.4,0.3,0.6]
	(0.8,0.3)	(0.6,0.5)
A₂	(S_2, S_2,[0.3,0.4,0.6,0.3]	(S_3,S_4,[0.6,0.2,0.3,0.7]
	(0.6,0.5)	(0.8,0.2)
A₃	(S_4, S_2,[0.4,0.3,0.4,0.1]	(S_3,S_4,[0.3,0.6,0.5,0.8]
	(0.7,0.6)	(0.5,0.5)
A₄	(S_2, S_4,[0.5,0.6,0.7,0.1]	(S₂₅S_2,[0.4,0.3,0.4,0.3]
	(0.7,0.4)	(0.8,0.2)
	C₃	C₄
A₁	(S_2, S_2,[0.1,0.2,0.4,0.6]	(S_2,S_4,[0.4,0.6,0.8,0.9]
	(0.5,0.5)	(0.7,0.4)
A₂	(S_3, S_2,[0.1,0.2,0.3,0.4]	(S_2,S_3,[0.4,0.2,0.3,0.9]
	(0.9,0.3)	(0.9,0.1)
A₃	(S_2, S_3,[0.1,0.4,0.3,0.2]	(S_2,S_3,[0.3,0.4,0.7,0.2]
	(0.6,0.4)	(0.8,0.3)
A₄	(S_2, S_2,[0.2,0.3,0.1,0.3]	(S_2,S_3,[0.3,0.1,0.5,0.6]
	(0.9,0.1)	(0.6,0.5)

Table 4

R⁽⁴⁾. Decision matrix of expert -4

4	C₁	C₂
A₁	(S_3,S_2,[0.4,0.3,0.5,0.1]	(S_4,S_2,[0.5,0.6,0.7,0.5]
	(0.7,0.6)	(0.5,0.5)
A₂	((S_2,S_3,[0.1,0.2,0.3,0.4]	((S_4,S_3,[0.4,0.5,0.7,0.2]
	(0.8,0.3)	(0.6,0.5)
A₃	(S_4,S_2,[0.4,0.3,0.5,0.6]	(S_6,S_3,[0.3,0.1,0.6,0.5]
	(0.7,0.4)	(0.8,0.2)
A₄	(S_3,S_2,[0.7,0.3,0.8,0.1]	(S_2,S_3,[0.2,0.5,0.7,0.1]
	(0.5,0.5)	(0.8,0.4)
	C₃	C₄
A₁	(S_4,S_2,[0.6,0.7,0.2,0.5]	(S_2,S_5,[0.4,0.2,0.5,0.9]
	(0.6,0.4)	(0.8,0.3)
A₂	(S_2,S_3,[0.3,0.7,0.2,0.1]	(S_3,S_2,[0.5,0.6,0.3,0.4]
	(0.8,0.4)	(0.9,0.2)
A₃	(S_4,S_4,[0.5,0.6,0.3,0.4]	(S_4,S_2,[0.2,0.3,0.6,0.5]
	(0.9,0.1)	(0.6,0.5)
A₄	(S_4,S_3,[0.5,0.6,0.8,0.9]	(S_6,S_4,[0.7,0.4,0.6,0.2]
	(0.7,0.6)	(0.8,0.5)

Step 2. In this step, we apply the decision information given in the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein decision matrix, R^(k) (k = 1, 2, 3, 4), and the PTULHFEWG operator to derive the individual overall preference Pythagorean trapezoidal fuzzy values $r_{i}^{(k)} (i = 1, 2, 3, 4)$ of the alternative A_i,

r_{1}^{(1)}

=(S_2.25, S_2.28, [0.38,0.39,0.45,0.50];(0.74,0.46)

r_{1}^{(2)}

=(S_2.63, S_2.42,[0.37,0.25,0.42,0.36];(0.75,0.34)

r_{1}^{(3)}

=(S_2.89, S_2.70,[0.49,0.24,0.34,0.57];(0.84,0.25)

r_{1}^{(4)}

=(S_2.63, S_2.74, [0.30,0.14,0.37,0.47];(0.62,0.50)

r_{2}^{(1)}

=(S_4.51, S_2.74, [0.35,0.28,0.46,0.50];(0.75,0.34)

r_{2}^{(2)}

=(S_4.51, S_2.56,[0.46,0.50,0.31,0.28];(0.73,0.72)

r_{2}^{(3)}

=(S_4.26, S_2.70,[0.40,0.41,0.20,0.21];(0.84,0.25)

r_{2}^{(4)}

=(S_2.70, S_2.58, [0.39,0.19,0.28,0.33];(0.74,0.51)

r_{3}^{(1)}

=(S_2.63, S_2.84, [0.38,0.42,0.49,0.40];(0.62,0.50)

r_{3}^{(2)}

=(S_2.25, S_2.56,[0.30,0.26,0.39,0.45];(0.84,0.25)

r_{3}^{(3)}

=(S_2.63, S_2.78,[0.27,0.40,0.42,0.22];(0.74,0.46)

r_{3}^{(4)}

=(S_2.63, S_2.74, [0.36,0.35,0.38,0.20];(0.73,0.72)

r_{4}^{(1)}

=(S_3.32, S_2.19, [0.46,0.42,0.46,0.27];(0.74,0.46)

r_{4}^{(2)}

=(S_3.30, S_2.42,[0.22,0.37,0.35,0.24];(0.75,0.34)

r_{4}^{(3)}

=(S_2.56, S_2.55,[0.35,0.24,0.48,0.51];(0.73,0.72)

r_{4}^{(4)}

=(S_3.01, S_2.66, [0.44,0.41,0.74,0.16];(0.74,0.51)

Step 3. We utilize the PTULHFEHG operator to derive the collective overall preference Pythagorean trapezoidal fuzzy numbers.

r₁=(S_3.21, S_2.52, [0.68,0.67,0.73,0.66];(0.87,0.32)

r₂=(S_2.81, S_2.50,[0.33,0.37,0.35,0.32];(0.81,0.27)

r₃=(S_3.14, S_2.69,[0.29,0.25,0.37,0.31];(0.70,0.39)

r₄=(S_2.71, S_2.48, [0.21,0.23,0.29,0.08];(0.56,0.71)

Step 4. In this step, we calculate the score function of the collective overall preference values A_i. $\begin{matrix} S ({\tilde{r}}_{1}) = 0.4540, S ({\tilde{r}}_{2}) = 0.5255, \\ S ({\tilde{r}}_{3}) = 0.2950, S (r_{4}) = - 0.0987 . \end{matrix}$

Step 5. Now we arrange the scores of all alternatives such as, A₂ > A₁ > A₃ > A₄. Thus the most wanted alternative is A₂.

Step 6. End

6 Comparison analysis with existing methods

In way to verify the sagacity and efficiency of the proposed approach, a comparative study is developed with intuitionistic trapezoidal fuzzy numbers W. Jianqiang et al. 2009, Pythagorean trapezoidal fuzzy numbers Fasil et al. 2019 and Pythagorean fuzzy aggregation operator Zhenming et al. 2016, to related expressive example shown in Tables 5 –7.

Table 5
Z⁽¹⁾

C₁ C₂

A₁ ([0.4,0.5,0.6,0.5] ([0.9,0.6,0.3,0.4]

(0.7,0.2) (0.7,0.1)

A₂ ([0.3,0.4,0.5,0.4] ([0.4,0.5,0.4,0.1]

(0.4,0.3) (0.5,0.1)

A₃ ([0.6,0.2,0.3,0.4] ([0.4,0.5,0.6,0.7]

(0.5,0.2) (0.4,0.1)

A₄ ([0.4,0.5,0.6,0.9] ([0.3,0.4,0.5,0.9]

(0.4,0.3) (0.3,0.2)

C₃ C₄

A₁ ([0.4,0.5,0.1,0.2] ([0.3,0.4,0.3,0.5]

(0.5,0.3) (0.6,0.2)

A₂ ([0.7,0.6,0.3,0.2] ([0.8,0.4,0.3,0.4]

(0.4,0.3) (0.3,0.5)

A₃ ([0.4,0.6,0.2,0.3] ([0.4,0.6,0.3,0.4]

(0.3,0.3) (0.7,0.2)

A₄ ([0.6,0.7,0.8,0.6] ([0.4,0.5,0.6,0.5]

(0.5,0.1) (0.7,0.2)

	C₁	C₂
A₁	([0.4,0.5,0.6,0.5]	([0.9,0.6,0.3,0.4]
	(0.7,0.2)	(0.7,0.1)
A₂	([0.3,0.4,0.5,0.4]	([0.4,0.5,0.4,0.1]
	(0.4,0.3)	(0.5,0.1)
A₃	([0.6,0.2,0.3,0.4]	([0.4,0.5,0.6,0.7]
	(0.5,0.2)	(0.4,0.1)
A₄	([0.4,0.5,0.6,0.9]	([0.3,0.4,0.5,0.9]
	(0.4,0.3)	(0.3,0.2)
	C₃	C₄
A₁	([0.4,0.5,0.1,0.2]	([0.3,0.4,0.3,0.5]
	(0.5,0.3)	(0.6,0.2)
A₂	([0.7,0.6,0.3,0.2]	([0.8,0.4,0.3,0.4]
	(0.4,0.3)	(0.3,0.5)
A₃	([0.4,0.6,0.2,0.3]	([0.4,0.6,0.3,0.4]
	(0.3,0.3)	(0.7,0.2)
A₄	([0.6,0.7,0.8,0.6]	([0.4,0.5,0.6,0.5]
	(0.5,0.1)	(0.7,0.2)

Table 6

	C₁	C₂
A₁	([0.4,0.5,0.6,0.5]	([0.9,0.6,0.3,0.4]
	(0.8,0.3)	(0.7,0.5)
A₂	([0.3,0.4,0.5,0.4]	([0.4,0.5,0.4,0.1]
	(0.6,0.5)	(0.5,0.7)
A₃	([0.6,0.2,0.3,0.4]	([0.4,0.5,0.6,0.7]
	(0.5,0.5)	(0.4,0.8)
A₄	([0.4,0.5,0.6,0.9]	([0.3,0.4,0.5,0.9]
	(0.7,0.4)	(0.6,0.5)
	C₃	C₄
A₁	([0.4,0.5,0.1,0.2]	([0.3,0.4,0.3,0.5]
	(0.5,0.6)	(0.6,0.4)
A₂	([0.7,0.6,0.3,0.2]	([0.8,0.4,0.3,0.4]
	(0.4,0.7)	(0.6,0.5)
A₃	([0.4,0.6,0.2,0.3]	([0.4,0.6,0.3,0.4]
	(0.3,0.7)	(0.7,0.4)
A₄	([0.6,0.7,0.8,0.6]	([0.4,0.5,0.6,0.5]
	(0.5,0.6)	(0.8,0.3)

Table 7

	C₁	C₂
A₁	(0.8,0.5)	(0.7,0.4)
A₂	(0.8,0.4)	(0.5,0.6)
A₃	(0.5,0.7)	(0.4,0.7)
A₄	(0.4,0.8)	(0.3,0.7)
	C₃	C₄
A₁	(0.5,0.5)	(0.6,0.5)
A₂	(0.7,0.4)	(0.8,0.3)
A₃	(0.8,0.3)	(0.7,0.7)
A₄	(0.5,0.6)	(0.7,0.4)

6.1 Comparison analysis with intuitionistic trapezoidal fuzzy sets

Step 1. According to the decision information given in the intuitionistic trapezoidal fuzzy number decision matrix $Z^{(k)} = (z_{ij}^{k})$ such that,

Step 2. Utilize the ITFWG operator to aggregate all the decision matrix into single collective decision matrix $r_{i}^{k}$ of the alternative A_i. Suppose that the weighted vector of the alternative are (0.3, 0.4, 0.1, 0.2) such that,

Aggregated values

([0.58,0.52,037,0.43](0.85,0.04)

([0.48,0.46,0.40,0.26](0.70,0.12)

([0.51,0.55,0.32,0.40](0.56,0.06)

([0.18,0.41,0.44,0.65](0.60,0.06)

Step 3. Calculate the scores $S ({\tilde{r}}_{i})$ of all the overall intuitionistic trapezoidal fuzzy values such that, $\begin{matrix} S ({\tilde{r}}_{1}) = 0.3807, S ({\tilde{r}}_{3} {1 pt}_{2}) = 0.2320, \\ S ({\tilde{r}}_{3}) = 0.2225, S (r_{4}) = 0.2268 . \end{matrix}$

Step 4. In this step, we find out the ranking of the alternatives such as, A₁ > A₂ > A₄ > A₃. Thus the most desirable alternative is A₁.

6.2 Comparison analysis with Pythagorean trapezoidal fuzzy sets

Step 1. According to the decision information given in the Pythagorean trapezoidal fuzzy number decision matrix $Z^{(k)} = (z_{ij}^{k})$ such that,

Step 2. Utilize the PTFWG operator to aggregate all the decision matrix into single collective decision matrix $r_{i}^{k}$ of the alternative A_i. Suppose that the weighted vector of the alternative are (0.3, 0.4, 0.1, 0.2) such that,

Aggregated values

([0.58,0.52,037,0.43](0.68,0.43)

([0.48,0.46,0.40,0.26](0.47,0.45)

([0.51,0.55,0.32,0.40](0.46,0.36)

([0.18,0.41,0.44,0.65](0.65,0.46)

Table 8
Comparison analysis with existing methods

Approaches Methods/techniques Score values Ranking outcome

Pythagorean

Trapezoidal S (r1) =0.4540

Uncertain S (r2) =0.5255

Linguistic Proposed S (r3) =0.2950 A₂> A₁ >A₃ > A₄

Hesitant fuzzy Method S (r4) = -0.0987

Einstein A₂ = 0.4540

Information

Intuitionistic

Trapezoidal

Fuzzy ITFWG S (r1) =0.3807

Information operator S (r2) =0.2320 A₁> A₂ >A₄ > A₃

W.Jianqiang S (r3) =0.2225

et al. 2009 S (r4) =0.2268 A₁ = 0.3807

Pythagorean

Trapezoidal PTFWG S (r1) =0.1304

Fuzzy sets pperators S (r2) =0.0073 A₃> A₁ >A₄ > A₃

Fasial et al. S (r3) =0.3605

2019 S (r4) =0.0885 A₃ = 0.1304

Pythagorean

Fuzzy PFWG S (r1) =0.2380

information operator S (r2) =0.1781

Zhenming et al. S (r3) = --0.2044 A₁> A₂ >A₃ > A₄

2016 S (r4) = --0.2376 A₁ = 0.2380

Approaches	Methods/techniques	Score values	Ranking outcome
Pythagorean
Trapezoidal		S (r1) =0.4540
Uncertain		S (r2) =0.5255
Linguistic	Proposed	S (r3) =0.2950	A₂> A₁ >A₃ > A₄
Hesitant fuzzy	Method	S (r4) = -0.0987
Einstein			A₂ = 0.4540
Information
Intuitionistic
Trapezoidal
Fuzzy	ITFWG	S (r1) =0.3807
Information	operator	S (r2) =0.2320	A₁> A₂ >A₄ > A₃
W.Jianqiang		S (r3) =0.2225
et al. 2009		S (r4) =0.2268	A₁ = 0.3807
Pythagorean
Trapezoidal	PTFWG	S (r1) =0.1304
Fuzzy sets	pperators	S (r2) =0.0073	A₃> A₁ >A₄ > A₃
Fasial et al.		S (r3) =0.3605
2019		S (r4) =0.0885	A₃ = 0.1304
Pythagorean
Fuzzy	PFWG	S (r1) =0.2380
information	operator	S (r2) =0.1781
Zhenming et al.		S (r3) = --0.2044	A₁> A₂ >A₃ > A₄
2016		S (r4) = --0.2376	A₁ = 0.2380

Step 3. Calculate the scores $S ({\tilde{r}}_{i})$ of all the overall intuitionistic trapezoidal fuzzy values such that, $\begin{matrix} S ({\tilde{r}}_{1}) = 0.1304, S ({\tilde{r}}_{2}) = 0.0073, \\ S ({\tilde{r}}_{3}) = 0.3605, S (r_{4}) = 0.0885 . \end{matrix}$

Step 4. In this step, we find out the ranking of the alternatives such as, A₃ > A₁ > A₄ > A₃. Thus the most desirable alternative is A₃.

6.3 Comparison analysis with pythagorean fuzzy sets

Step 1. According to the decision information given in the Pythagorean fuzzy number decision matrix $Z^{(k)} = (z_{ij}^{k})$ such that,

Step 2. Utilize the PFWG operator to aggregate all the decision matrix into single collective decision matrix $r_{i}^{k}$ of the alternative A_i. Suppose that the weighted vector of the alternative are (0.3, 0.4, 0.1, 0.2) such that, $r^{(k)} = \begin{matrix} Aggregated values \\ (0.6831, 0.4781) \\ (0.6541, 0.4997) \\ (0.5127, 0.6836) \\ (0.4077, 0.6355) \end{matrix}$

Step 3. Calculate the scores $S ({\tilde{r}}_{i})$ of all the overall intuitionstic trapezoidal fuzzy values such that, $\begin{matrix} S ({\tilde{r}}_{1}) = 0.2380, S ({\tilde{r}}_{2}) = 0.1781, \\ S ({\tilde{r}}_{3}) = - 0.2044, S (r_{4}) = - 0.2376 . \end{matrix}$

Step 4. In this step, we find out the ranking of the alternatives such as, A₁ > A₂ > A₃ > A₄. Thus the most desirable alternative is A₁.

We have given the comparison analysis of the proposed aggregation operators with intuitionistic trapezoidal fuzzy aggregation operators, Pythagorean trapezoidal fuzzy aggregation operators and Pythagorean fuzzy aggregation operator. It is noted that the ranking results obtained by proposed method and existing methods are very different. From the comparison analysis, we concluded that our proposed aggregation operator is much better in finding the optimal solutions. Based on these analysis the proposed aggregation operator is more reliable and effective in real life problem.

7 Conclusions

In this paper, we introduced the initiative of Pythagorean trapezoidal uncertain linguistic hesitant Einstein fuzzy sets. We have defined their basic properties and develop some operational laws for Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein numbers. We have discussed some new types of aggregation operators for Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein numbers consists of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein weighted geometric (PTULHFEWG) operator, Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein ordered weighted geometric (PTULHFEOWG) operator and Pythagorean trapezoidal uncertain linguistic hesitant fuzzy Einstein hybrid geometric (PTULHFEHG) operators. Moreover, we apply these aggregation operators to develop an approach to MAGDM with PTULF. Finally, an illustrative example has been constructed to show the proposed MAGDM method. Our proposed method is different from all the previous techniques for group decision making. So it is efficient and feasible for real-world decision making applications. These operators can be applied to many other fields, such as data mining, symmetric aggregation operator, Dombi aggregation operator, pattern recognition and TODIM-TOPSIS method which may be the possible topic for the future research.

Footnotes

Appendix 1

${\tilde{α}}_{1} \oplus {\tilde{α}}_{2}$ ,

${\tilde{α}}_{1} \otimes {\tilde{α}}_{2}$ ,

$δ_{\dot{ɛ}} \tilde{α}$ , and

${\tilde{α}}^{\land_{ɛ} δ}$ are also PTULHFE numbers.

Proof. (1) and (2) result is obvious.

(3) Let n be any positive integer and $\tilde{α}$ is PTULHFE numbers. Then, (9) $n_{\dot{ɛ}} \tilde{α} = \overset{n}{\overset{︷}{\tilde{α} \oplus_{ɛ} \tilde{α} \oplus_{ɛ} . . . \oplus_{ɛ} \tilde{α}}} .$

By using mathematical induction we prove that Equation (9) holds for all positive integers n. First, we show that Equation (9) holds for n = 2. Since $\begin{matrix} \begin{matrix} \tilde{α} + \tilde{α} = & (\begin{matrix} (s_{θ_{(p)}} + s_{θ_{(p)}}, s_{t_{(p)}} + s_{t_{(p)}}), \\ [p + p, q + q, r + r, s + s]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(Ψ_{\tilde{α}})^{2} + (Ψ_{\tilde{α}})^{2}}}{\sqrt{1 + (Ψ_{\tilde{α}})^{2} (Ψ_{\tilde{α}})^{2}}}, \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{ϒ_{\tilde{α}} ϒ_{\tilde{α}}}{\sqrt{1 + (1 - ϒ_{\tilde{α}}^{2}) (1 - ϒ_{\tilde{α}}^{2})}} \end{matrix}), \\ = & (\begin{matrix} (2 s_{θ_{(p)}}, 2 s_{t_{(p)}}), [2 p, 2 q, 2 r, 2 s]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(1 + Ψ_{\tilde{α}}^{2})^{2} - (1 - Ψ_{\tilde{α}}^{2})^{2}}}{\sqrt{(1 + Ψ_{\tilde{α}}^{2})^{2} + (1 - Ψ_{\tilde{α}}^{2})^{2}}}, \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{2 {(ϒ_{\tilde{α}}^{2})}^{2}}}{\sqrt{(2 - ϒ_{\tilde{α}})^{2} + (ϒ_{\tilde{α}})^{2}}} \end{matrix}), \\ = & 2_{\dot{ɛ}} \tilde{α} . \end{matrix} \end{matrix}$

Therefore, the Equation (9) holds for n = 2. If Equation (9) holds for n = k. Then, $n_{\dot{ɛ}} \tilde{α} = \overset{n}{{\overset{︷}{\tilde{α} \oplus_{ɛ} \tilde{α} \oplus_{ɛ} . . . \oplus_{ɛ} \tilde{α}}}^{.}}$

When n = k + 1, we have $\begin{matrix} \overset{n + 1}{\overset{︷}{\tilde{α} \oplus_{ɛ} \tilde{α} \oplus_{ɛ} . . . \oplus_{ɛ} \tilde{α}}} \\ = \overset{n}{\overset{︷}{\tilde{α} \oplus_{ɛ} \tilde{α} \oplus_{ɛ} . . . \oplus_{ɛ} \tilde{α}} \oplus_{ɛ} \tilde{α} =} n_{\dot{ɛ}} \tilde{α} \oplus_{ɛ} \tilde{α}, \\ = (\begin{matrix} ({ns}_{θ_{(p)}}, {ns}_{t_{(p)}}), [np, nq, nr, ns]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(1 + Ψ_{\tilde{α}}^{2})^{n} - (1 - Ψ_{\tilde{α}}^{2})^{n}}}{\sqrt{(1 + Ψ_{\tilde{α}}^{2})^{n} + (1 - Ψ_{\tilde{α}}^{2})^{n}}}, \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{2 {(ϒ_{\tilde{α}}^{2})}^{n}}}{\sqrt{(2 - ϒ_{\tilde{α}}^{2})^{n} + (ϒ_{\tilde{α}}^{2})^{n}}} \end{matrix}) \\ \oplus_{ɛ} (s_{θ_{(p)}}, s_{t_{(p)}}), [p, q, r, s]; Ψ_{\tilde{α}}, ϒ_{\tilde{α}}), \\ = & (\begin{matrix} (n + 1) s_{θ_{(p)}}, (n + 1) s_{t_{(p)}}), \\ [(n + 1) p, (n + 1) q, (n + 1) r, (n + 1) s]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(1 + Ψ_{\tilde{α}}^{2})^{n + 1} - (1 - Ψ_{\tilde{α}}^{2})^{n + 1}}}{\sqrt{(1 + Ψ_{\tilde{α}}^{2})^{n + 1} + (1 - Ψ_{\tilde{α}}^{2})^{n + 1}}}, \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{2 {(ϒ_{\tilde{α}}^{2})}^{n + 1}}}{\sqrt{(2 - ϒ_{\tilde{α}}^{2})^{n + 1} + (ϒ_{\tilde{α}}^{2})^{n + 1}}} \end{matrix}), \\ = & {(n + 1)}_{\dot{ɛ}} \tilde{α} . \end{matrix}$

Hence Equation (9) holds for n = k + 1, therefore Equation (9) holds for all n.

(4) Let m be any positive integer and $\tilde{α}$ is PTULHF number. Then,

(9) ${\tilde{α}}^{\land_{ɛ} m} = \overset{m}{{\overset{︷}{\tilde{α} \otimes_{ɛ} \tilde{α} \otimes_{ɛ} . . . \otimes_{ɛ} \tilde{α}}}^{.}}$

By using mathematical induction we prove that Equation (9) holds for all positive integers m. First, we show that Equation (9) holds for m = 2.Since

$\begin{matrix} \tilde{α} \otimes_{ɛ} \tilde{α} = & (\begin{matrix} (s_{θ_{(p)}} . s_{θ_{(p)}}, s_{t_{(p)}} . s_{t_{(p)}}), [p . p, q . q, r . r, s . s]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{Ψ_{\tilde{α}} Ψ_{\tilde{α}}}{\sqrt{1 + (1 - Ψ_{\tilde{α}}^{2}) (1 - Ψ_{\tilde{α}}^{2})}}, \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(ϒ_{\tilde{α}})^{2} + (ϒ_{\tilde{α}})^{2}}}{\sqrt{1 + (ϒ_{\tilde{α}})^{2} (ϒ_{\tilde{α}})^{2}}} \end{matrix}), \\ = & (\begin{matrix} (s_{θ_{(p)}^{2}}, s_{t_{(p)}^{2}}), [p^{2}, q^{2}, r^{2}, s^{2}]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{2 {(Ψ_{\tilde{α}}^{2})}^{2}}}{\sqrt{(2 - Ψ_{\tilde{α}}^{2})^{2} + (ϒ_{\tilde{α}}^{2})^{2}}}, \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(1 + ϒ_{\tilde{α}}^{2})^{2} - (1 - ϒ_{\tilde{α}}^{2})^{2}}}{\sqrt{(1 + ϒ_{\tilde{α}}^{2})^{2} + (1 - ϒ_{\tilde{α}}^{2})^{2}}} \end{matrix}), \\ = & {\tilde{α}}^{\land_{ɛ} 2} . \end{matrix}$

Therefore, the Equation (9) holds for m = 2. If Equation (9) holds for m = k, such that ${\tilde{α}}_{1}^{\land_{ɛ} m} = \overset{m}{{\overset{︷}{{\tilde{α}}_{1} \otimes_{ɛ} {\tilde{α}}_{1} \otimes_{ɛ} . . . \otimes_{ɛ} {\tilde{α}}_{1}}}^{.}}$

Then, we have

$\begin{matrix} \overset{m + 1}{\overset{︷}{\tilde{α} \otimes_{ɛ} \tilde{α} \otimes_{ɛ} . . . \otimes_{ɛ} \tilde{α}}} \\ = \overset{m}{{\overset{︷}{\tilde{α} \otimes_{ɛ} \tilde{α} \otimes_{ɛ} . . . \otimes_{ɛ} \tilde{α}}}_{ɛ}^{\otimes} \tilde{α} =} {\tilde{α}}^{\land_{ɛ} m} \otimes_{ɛ} \tilde{α}, \\ = (\begin{matrix} (s_{θ_{(p)}^{m}}, s_{t_{(p)}^{m}}), [p^{m}, q^{m}, r^{m}, s^{m}]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{2 {(Ψ_{\tilde{α}}^{2})}^{m}}}{\sqrt{(2 - Ψ_{\tilde{α}}^{2})^{m} + (Ψ_{\tilde{α}}^{2})^{m}}}, \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(1 + ϒ_{\tilde{α}}^{2})^{m} - (1 - ϒ_{\tilde{α}}^{2})^{m}}}{\sqrt{(1 + ϒ_{\tilde{α}}^{2})^{m} + (1 - ϒ_{\tilde{α}}^{2})^{m}}}, \end{matrix}), \\ \otimes_{ɛ} (s_{θ_{(p)}}, s_{t_{(p)}}), [p, q, r, s]; Ψ_{\tilde{α}}, ϒ_{\tilde{α}}) \\ = (\begin{matrix} (s_{θ_{(p)}^{m + 1}}, s_{t_{(p)}^{m + 1}}) [p^{(m + 1)}, q^{(m + 1)}, r^{(m + 1)}, s^{(m + 1)}]; \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{2 {(Ψ_{\tilde{α}}^{2})}^{m + 1}}}{\sqrt{(2 - Ψ_{\tilde{α}}^{2})^{m + 1} + (Ψ_{\tilde{α}}^{2})^{m + 1}}}, \\ \underset{Ψ_{\tilde{α}} \in γ_{\tilde{α}}, ϒ_{\tilde{α}} \in β_{\tilde{α}}}{\cup} \frac{\sqrt{(1 + ϒ_{\tilde{α}}^{2})^{m + 1} - (1 - ϒ_{\tilde{α}}^{2})^{m + 1}}}{\sqrt{(1 + ϒ_{\tilde{α}}^{2})^{m + 1} + (1 - ϒ_{\tilde{α}}^{2})^{m + 1}}} \end{matrix}), \\ = {\tilde{α}}_{1}^{\land_{ɛ} m + 1} . \end{matrix}$

Hence Equation (9) holds for m = k + 1. Therefore Equation (9) holds for all m.

By the Einstein operational laws of PTULHF numbers satisfies the following properties;

Appendix 2

Theorem 2. ${\tilde{α}}_{1} = {\begin{matrix} (s_{θ (p_{1})}, s_{t (p_{1})}, \\ [p_{1}, q_{1}, r_{1}, s_{1}]; \\ (γ_{{\tilde{α}}_{1}}, β_{{\tilde{α}}_{1}}) \end{matrix}}$ , and ${\tilde{α}}_{2} = {\begin{matrix} (s_{θ (p_{2})}, s_{t (p_{2})}, \\ [p_{2}, q_{2}, r_{2}, s_{2}]; \\ (γ_{{\tilde{α}}_{2}}, β_{{\tilde{α}}_{2}}) \end{matrix}}$ be any two PTULHFNs. Then, the operational laws between ${\tilde{α}}_{1}$ and ${\tilde{α}}_{2}$ are shown as follows;

${\tilde{α}}_{1} \otimes {\tilde{α}}_{2} = {\tilde{α}}_{2} \otimes {\tilde{α}}_{1}$

$({\tilde{α}}_{1} \otimes {\tilde{α}}_{2}) δ = {\tilde{α}}_{2}^{δ} \otimes {\tilde{α}}_{1}^{δ}$ δ ≥ 0,

${\tilde{α}}^{δ_{1}} \otimes {\tilde{α}}^{δ_{2}} = {\tilde{α}}^{(δ_{1} + δ_{2})}$ δ₁, δ₂ ≥ 0,

Proof. (1) result is obvious.

(2) By Einstein operational laws (2) in Definition 9, we have $\begin{matrix} {\tilde{α}}_{1} \otimes {\tilde{α}}_{2} = {\begin{matrix} (s_{θ_{(p_{1})}} s_{θ_{(p_{2})}}, s_{t_{(p_{1})}} s_{t_{(p_{2})}}), \\ [p_{1} p_{2}, q_{1} q_{2}, r_{1} r_{2}, s_{1} s_{2}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{Ψ_{{\tilde{α}}_{1}} Ψ_{{\tilde{α}}_{2}}}{\sqrt{1 + (1 - Ψ_{{\tilde{α}}_{1}}^{2}) (1 - Ψ_{{\tilde{α}}_{2}}^{2})}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{(ϒ_{{\tilde{α}}_{1}})^{2} + (ϒ_{{\tilde{α}}_{2}})^{2}}}{\sqrt{1 + (ϒ_{{\tilde{α}}_{1}})^{2} (ϒ_{{\tilde{α}}_{2}})^{2}}} \end{matrix}} . \end{matrix}$ We can write the above equation into the following form; $\begin{matrix} {\tilde{α}}_{1} \otimes {\tilde{α}}_{2} \\ = {\begin{matrix} (s_{θ_{(p_{1})}} s_{θ_{(p_{2})}}, s_{t_{(p_{1})}} s_{t_{(p_{2})}}), \\ [p_{1} p_{2}, q_{1} q_{2}, r_{1} r_{2}, s_{1} s_{2}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 (Ψ_{{\tilde{α}}_{1}}^{2} Ψ_{{\tilde{α}}_{2}}^{2})}}{\sqrt{(2 - Ψ_{{\tilde{α}}_{1}}^{2}) (2 - Ψ_{{\tilde{α}}_{2}}^{2}) + (Ψ_{{\tilde{α}}_{1}}^{2} Ψ_{{\tilde{α}}_{2}}^{2})}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2}) (1 + ϒ_{{\tilde{α}}_{2}}^{2}) - (1 - ϒ_{{\tilde{α}}_{1}}^{2}) (1 - ϒ_{{\tilde{α}}_{2}}^{2})}}{\sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2}) (1 - ϒ_{{\tilde{α}}_{2}}^{2}) + (1 - ϒ_{{\tilde{α}}_{1}}^{2}) (1 - ϒ_{{\tilde{α}}_{2}}^{2})}}, \end{matrix}} . \end{matrix}$

Let $\begin{matrix} x = & \sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2}) (1 + ϒ_{{\tilde{α}}_{2}}^{2}),} y = \sqrt{(1 - ϒ_{{\tilde{α}}_{1}}^{2}) (1 - ϒ_{{\tilde{α}}_{2}}^{2})}, \\ z = & \sqrt{(Ψ_{{\tilde{α}}_{1}}^{2} Ψ_{{\tilde{α}}_{2}}^{2})}, g = \sqrt{(2 - Ψ_{{\tilde{α}}_{1}}^{2}) (2 - Ψ_{{\tilde{α}}_{2}}^{2})} . \end{matrix}$ Then, ${\tilde{α}}_{1} \otimes {\tilde{α}}_{2} = {\begin{matrix} (s_{θ_{(p_{1})}} s_{θ_{(p_{2})}}, s_{t_{(p_{1})}} s_{t_{(p_{2})}}), \\ [p_{1} p_{2}, q_{1} q_{2}, r_{1} r_{2}, s_{1} s_{2}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z}{g + z}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x - y}{x + y} \end{matrix}} .$ By the Einstein operational law (4) and Definition 9, we have $\begin{matrix} ({\tilde{α}}_{1} \oplus {\tilde{α}}_{2})^{8} \\ = (\begin{matrix} (s_{(θ (p_{1}) θ (p_{2}))^{δ}}, s_{(t (p_{1}) t (p_{2}))^{δ}}), \\ [(p_{1} p_{2})^{δ}, (q_{1} q_{2})^{δ}, (r_{1} r_{2})^{δ}, (s_{1} s_{2})^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} {(\frac{\sqrt{2} z}{g + z})}^{δ}}{{(2 \frac{\sqrt{2} z}{g + z})}^{δ} + {(\frac{\sqrt{2} z}{g + z})}^{δ}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{{(1 + \frac{x - y}{x - y})}^{δ} - {(1 - \frac{x - y}{x - y})}^{δ}}{{(1 + \frac{x - y}{x - y})}^{δ} + {(1 - \frac{x - y}{x - y})}^{δ}}, \end{matrix}) \\ = (\begin{matrix} (s_{(θ (p_{1}) θ (p_{2}))^{δ}}, s_{(t (p_{1}) t (p_{2}))^{δ}}), \\ [(p_{1} p_{2})^{δ}, (q_{1} q_{2})^{δ}, (r_{1} r_{2})^{δ}, (s_{1} s_{2})^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} (z)^{δ}}{(g)^{δ} + (z)^{δ}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{(x)^{δ} - (y)^{δ}}{(x)^{δ} + (y)^{δ}} \end{matrix}) \\ = (\begin{matrix} (s_{(θ (p_{1}) θ (p_{2}))^{δ}}, s_{(t (p_{1}) t (p_{2}))^{δ}}) \\ [(p_{1} p_{2})^{δ}, (q_{1} q_{2})^{δ}, (r_{1} r_{2})^{δ}, (s_{1} s_{2})^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{1}}^{2} Ψ_{{\tilde{α}}_{2}}^{2})}^{δ}}}{\sqrt{{(2 - Ψ_{{\tilde{α}}_{1}}^{2})}^{δ} {(2 - Ψ_{{\tilde{α}}_{2}}^{2})}^{δ} + {(Ψ_{{\tilde{α}}_{1}}^{2})}^{δ} {(Ψ_{{\tilde{α}}_{2}}^{2})}^{δ}}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{{(1 + ϒ_{{\tilde{α}}_{1}}^{2})}^{δ} {(1 + ϒ_{{\tilde{α}}_{2}}^{2})}^{δ} - {(1 - ϒ_{{\tilde{α}}_{1}}^{2})}^{δ} {(1 - ϒ_{{\tilde{α}}_{2}}^{2})}^{δ}}}{\sqrt{{(1 + ϒ_{{\tilde{α}}_{1}}^{2})}^{δ} {(1 + ϒ_{{\tilde{α}}_{2}}^{2})}^{δ} + {(1 - ϒ_{{\tilde{α}}_{1}}^{2})}^{δ} {(1 - ϒ_{{\tilde{α}}_{2}}^{2})}^{δ}}} \end{matrix}), \end{matrix}$ Let $\begin{matrix} {\tilde{α}}_{1}^{δ} = (\begin{matrix} (s_{θ {(p_{1})}^{δ}}, s_{t {(p_{1})}^{δ}}), [p_{1}^{δ}, q_{1}^{δ}, r_{1}^{δ}, s_{1}^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{1}}^{2})}^{δ}}}{\sqrt{(2 - Ψ_{{\tilde{α}}_{1}}^{2})^{δ} + (Ψ_{{\tilde{α}}_{1}}^{2})^{δ}}}, \\ \frac{\sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2})^{δ} - (1 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ}}}{\sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2})^{δ} + (1 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ}}} \end{matrix}), \\ {\tilde{α}}_{2}^{δ} = (\begin{matrix} (s_{θ {(p_{2})}^{δ}}, s_{t {(p_{2})}^{δ}}), [p_{2}^{δ}, q_{2}^{δ}, r_{2}^{δ}, s_{2}^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{2}}^{2})}^{δ}}}{\sqrt{(2 - Ψ_{{\tilde{α}}_{2}}^{2})^{δ} + (Ψ_{{\tilde{α}}_{2}}^{2})^{δ}}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{(1 + ϒ_{{\tilde{α}}_{2}}^{2})^{δ} - (1 - ϒ_{{\tilde{α}}_{2}}^{2})^{δ}}}{\sqrt{(1 + ϒ_{{\tilde{α}}_{2}}^{2})^{δ} + (1 - ϒ_{{\tilde{α}}_{2}}^{2})^{δ}}} \end{matrix}) . \end{matrix}$ We consider $\begin{matrix} x_{1} = \sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2})^{δ}}, y_{1} = \sqrt{(1 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ}}, z_{1} = \sqrt{{(Ψ_{{\tilde{α}}_{1}}^{2})}^{δ}}, \\ x_{2} = \sqrt{(1 + ϒ_{{\tilde{α}}_{2}}^{2})^{δ}}, g_{2} = \sqrt{(2 - Ψ_{{\tilde{α}}_{2}}^{2})^{δ}}, z_{2} = \sqrt{{(Ψ_{{\tilde{α}}_{2}}^{2})}^{δ}} . \end{matrix}$ Then, $\begin{matrix} {\tilde{α}}_{1}^{δ} = (\begin{matrix} (s_{θ {(p_{1})}^{δ}}, s_{t {(p_{1})}^{δ}}), [p_{1}^{δ}, q_{1}^{δ}, r_{1}^{δ}, s_{1}^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{1}}{g_{1} + z_{1}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x_{1} - y_{1}}{x_{1} + y_{1}} \end{matrix}), \\ {\tilde{α}}_{2}^{δ} = (\begin{matrix} (s_{θ {(p_{2})}^{δ}}, s_{t {(p_{2})}^{δ}}), [p_{2}^{δ}, q_{2}^{δ}, r_{2}^{δ}, s_{2}^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{2}}{g_{2} + z_{2}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x_{2} - y_{2}}{x_{2} + y_{2}} \end{matrix}) . \end{matrix}$ By the Einstein operational laws (2) and (4) and Definition 9, we have $\begin{matrix} ({\tilde{α}}_{1} \otimes {\tilde{α}}_{2})^{δ} \\ = (\begin{matrix} (s_{(θ (p_{1})^{δ}}, s_{t (p_{1})^{δ}}), [p_{1}^{δ}, q_{1}^{δ}, r_{1}^{δ}, s_{1}^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{1}}{g_{1} + z_{1}}, \frac{x_{1} - y_{1}}{x_{1} + y_{1}} \end{matrix}) \\ \otimes (\begin{matrix} (s_{(θ (p_{2})^{δ}}, s_{t (p_{2})^{δ}}), [p_{2}^{δ}, q_{2}^{δ}, r_{2}^{δ}, s_{2}^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{2}}{g_{2} + z_{2}}, \frac{x_{2} - y_{2}}{x_{2} + y_{2}} \end{matrix}), \\ = (\begin{matrix} (s_{θ (p_{1})^{δ}} s_{θ (p_{2})^{δ}}, s_{t (p_{1})^{δ}} s_{t (p_{2})^{δ}}), \\ [(p_{1} p_{2})^{δ}, (q_{1} q_{2})^{δ}, (r_{1} r_{2})^{δ}, (s_{1} s_{2})^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\frac{\sqrt{2} z_{1}}{g_{1} + z_{1}} \cdot \frac{\sqrt{2} z_{2}}{g_{2} + z_{2}}}{1 + (1 - \frac{\sqrt{2} z_{1}}{g_{1} + z_{1}}) \cdot (\frac{\sqrt{2} z_{2}}{g_{2} + z_{2}})}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\frac{x_{1} - y_{1}}{x_{1} + y_{1}} - \frac{x_{2} - y_{2}}{x_{2} + y_{2}}}{1 + \frac{x_{1} - y_{1}}{x_{1} + y_{1}} \cdot \frac{x_{2} + y_{2}}{x_{2} + y_{2}}} \end{matrix}), \\ = (\begin{matrix} (s_{(θ (p_{1}) θ (p_{2}))^{δ}}, s_{(t (p_{1}) t (p_{2}))^{δ}}), \\ [(p_{1} p_{2})^{δ}, (q_{1} q_{2})^{δ}, (r_{1} r_{2})^{δ}, (s_{1} s_{2})^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{1}}^{2} Ψ_{{\tilde{α}}_{2}}^{2})}^{δ}}}{\sqrt{{(2 - Ψ_{{\tilde{α}}_{1}}^{2})}^{δ} {(2 - Ψ_{{\tilde{α}}_{2}}^{2})}^{δ} + {(Ψ_{{\tilde{α}}_{1}}^{2})}^{δ} {(Ψ_{{\tilde{α}}_{2}}^{2})}^{δ}}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{{(1 + ϒ_{{\tilde{α}}_{1}}^{2})}^{δ} {(1 + ϒ_{{\tilde{α}}_{2}}^{2})}^{δ} - {(1 - ϒ_{{\tilde{α}}_{1}}^{2})}^{δ} {(1 - ϒ_{{\tilde{α}}_{2}}^{2})}^{δ}}}{\sqrt{{(1 + ϒ_{{\tilde{α}}_{1}}^{2})}^{δ} {(1 + ϒ_{{\tilde{α}}_{2}}^{2})}^{δ} + {(1 - ϒ_{{\tilde{α}}_{1}}^{2})}^{δ} {(1 - ϒ_{{\tilde{α}}_{2}}^{2})}^{δ}}} \end{matrix}) . \end{matrix}$ Therefore its prove that, ${({\tilde{α}}_{1} \otimes {\tilde{α}}_{2})}^{δ} = {\tilde{α}}_{1}^{δ} \otimes {\tilde{α}}_{2}^{δ}$ .

(3) Since $\begin{matrix} {\tilde{α}}_{1}^{δ_{1}} = & (\begin{matrix} (s_{θ_{{(p_{1})}^{δ_{1}}}}, s_{t_{{(p_{1})}^{δ_{1}}}}), \\ [p_{1}^{δ}, q_{1}^{δ}, r_{1}^{δ}, s_{1}^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{1}}^{2})}^{δ_{1}}}}{\sqrt{(2 - Ψ_{{\tilde{α}}_{1}}^{2})^{δ_{1}} + (Ψ_{{\tilde{α}}_{1}}^{2})^{δ_{1}}}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{1}} - (1 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{1}}}}{\sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{1}} + (1 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{1}}}} \end{matrix}), \\ {\tilde{α}}_{1}^{δ_{2}} = & (\begin{matrix} (s_{θ_{{(p_{1})}^{δ_{2}}}}, s_{t_{{(p_{1})}^{δ_{2}}}}), [p_{2}^{δ}, q_{2}^{δ}, r_{2}^{δ}, s_{2}^{δ}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{1}}^{2})}^{δ_{2}}}}{\sqrt{(2 - Ψ_{{\tilde{α}}_{1}}^{2})^{δ_{2}} + (Ψ_{{\tilde{α}}_{1}}^{2})^{δ_{2}}}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{2}} - (1 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{2}}}}{\sqrt{(1 + ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{2}} + (1 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{2}}}} \end{matrix}), \end{matrix}$ where δ₁, δ₁ ≥ 0. Let $\begin{matrix} \begin{matrix} x_{1} = & \sqrt{(1 + Ψ_{{\tilde{α}}_{1}}^{2})^{δ_{1}}}, y_{1} = \sqrt{(1 - Ψ_{{\tilde{α}}_{1}}^{2})^{δ_{1}}}, \\ z_{1} = & \sqrt{{(ϒ_{{\tilde{α}}_{1}}^{2})}^{δ_{1}}}, g_{1} = \sqrt{(2 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{1}}}, \\ x_{2} = & \sqrt{(1 + Ψ_{{\tilde{α}}_{1}}^{2})^{δ_{2}}}, y_{2} = \sqrt{(1 - Ψ_{{\tilde{α}}_{1}}^{2})^{δ_{2}}}, \\ z_{2} = & \sqrt{{(ϒ_{{\tilde{α}}_{1}}^{2})}^{δ_{2}}}, g_{2} = \sqrt{(2 - ϒ_{{\tilde{α}}_{1}}^{2})^{δ_{2}}} . \end{matrix} \end{matrix}$ Then, we have $\begin{matrix} {\tilde{α}}_{1}^{δ_{1}} = & (\begin{matrix} (s_{θ_{{(p_{1})}^{δ_{1}}}}, s_{t_{{(p_{1})}^{δ_{1}}}}), [p_{1}^{δ_{1}}, q_{1}^{δ_{1}}, r_{1}^{δ_{1}}, s_{1}^{δ_{1}}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{1}}{g_{1} + z_{1}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x_{1} - y_{1}}{x_{1} + y_{1}} \end{matrix}), \\ {\tilde{α}}_{1}^{δ_{2}} = & (\begin{matrix} (s_{θ_{{(p_{1})}^{δ_{2}}}}, s_{t_{{(p_{1})}^{δ_{2}}}}), [p_{1}^{δ_{2}}, q_{1}^{δ_{2}}, r_{1}^{δ_{2}}, s_{1}^{δ_{2}}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{2}}{g_{2} + z_{2}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x_{2} - y_{2}}{x_{2} + y_{2}} \end{matrix}) . \end{matrix}$ By the Einstein operational laws (2) and (4) and Definition 9, we have $\begin{matrix} {\tilde{α}}_{1}^{δ_{1}} \otimes {\tilde{α}}_{1}^{δ_{2}} \\ = (\begin{matrix} (s_{θ_{{(p_{1})}^{δ_{1}}}}, s_{t_{{(p_{1})}^{δ_{1}}}}), [p_{1}^{δ_{1}}, q_{1}^{δ_{1}}, r_{1}^{δ_{1}}, s_{1}^{δ_{1}}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{1}}{g_{1} + z_{1}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x_{1} - y_{1}}{x_{1} + y_{1}} \end{matrix}) \\ \oplus (\begin{matrix} (s_{θ_{{(p_{1})}^{δ_{2}}}}, s_{t_{{(p_{1})}^{δ_{2}}}}), [p_{1}^{δ_{2}}, q_{1}^{δ_{2}}, r_{1}^{δ_{2}}, s_{1}^{δ_{2}}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{2}}{g_{2} + z_{2}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x_{2} - y_{2}}{x_{2} + y_{2}} \end{matrix}), \\ = (\begin{matrix} (s_{θ (p_{1})^{δ_{1}}} s_{θ (p_{1})^{δ_{2}}}, s_{t (p_{1})^{δ_{1}}} s_{t (p_{1})^{δ_{2}}}), \\ [p_{1}^{δ_{1}} p_{1}^{δ_{2}}, q_{1}^{δ_{1}} q_{1}^{δ_{2}}, r_{1}^{δ_{1}} r_{1}^{δ_{2}}, s_{1}^{δ_{1}} s_{1}^{δ_{2}}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\frac{\sqrt{2} z_{1}}{g_{1} + z_{1}} \cdot \frac{\sqrt{2} z_{2}}{g_{2} + z_{2}}}{1 + (1 - \frac{\sqrt{2} z_{1}}{g_{1} + z_{1}}) \cdot (1 - \frac{\sqrt{2} z_{2}}{g_{2} + z_{2}})}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\frac{x_{1} - y_{1}}{x_{1} + y_{1}} - \frac{x_{2} - y_{2}}{x_{2} + y_{2}}}{1 + \frac{x_{1} - y_{1}}{x_{1} + y_{1}} \cdot \frac{x_{2} - y_{2}}{x_{2} + y_{2}}}, \end{matrix}) \\ = (\begin{matrix} (s_{θ (p_{1})^{δ_{1} + δ_{1}}}, s_{t (p_{1})^{δ_{1} + δ_{2}}}), \\ [p_{1}^{δ_{1} + δ_{2}}, q_{1}^{δ_{1} + δ_{2}}, r_{1}^{δ_{1} + δ_{2}}, s_{1}^{δ_{1} + δ_{2}}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{1} z_{2}}{g_{1} g_{2} + z_{1} z_{2}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x_{1} x_{2} - y_{1} y_{2}}{x_{1} x_{2} + y_{1} y_{2}} \end{matrix}) \\ = (\begin{matrix} (s_{θ (p_{1})^{δ_{1} + δ_{1}}}, s_{t (p_{1})^{δ_{1} + δ_{2}}}), \\ [p_{1}^{δ_{1} + δ_{2}}, q_{1}^{δ_{1} + δ_{2}}, r_{1}^{δ_{1} + δ_{2}}, s_{1}^{δ_{1} + δ_{2}}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}, Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2} z_{1} z_{2}}{g_{1} g_{2} + z_{1} z_{2}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}, ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{x_{1} x_{2} - y_{1} y_{2}}{x_{1} x_{2} + y_{1} y_{2}} \end{matrix}) . \end{matrix}$

Appendix 3

Theorem 3. Let ${\tilde{α}}_{j} (j = 1, 2, . . ., n)$ be a collection of Pythagorean trapezoidal uncertain linguistic hesitant fuzzy numbers, then their aggregated value by using the PTULHFEWG operator is also and Pythagorean trapezoidal fuzzy number such that; (11) $\begin{matrix} PTULFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ = \overset{n}{\underset{j = 1}{\otimes_{ɛ}}} {({\tilde{α}}_{j})}^{w_{j}} \\ = (\begin{matrix} (\prod_{j = 1}^{n} (s_{θ (p_{j})})^{w_{j}}, \prod_{j = 1}^{n} (s_{t (p_{j})})^{w_{j}}), \\ [\prod_{j = 1}^{n} (p_{j})^{w_{j}}, \prod_{j = 1}^{n} (q_{j})^{w_{j}}, \prod_{j = 1}^{n} (r_{j})^{w_{j}}, \prod_{j = 1}^{n} (s_{j})^{w_{j}}]; \\ \underset{Ψ_{{\tilde{α}}_{j}} \in γ_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{2 \prod_{j = 1}^{n} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}, \\ \underset{ϒ_{{\tilde{α}}_{j}} \in β_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{\prod_{j = 1}^{n} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}, \end{matrix}) \end{matrix}$ where w = (w₁, w₂, . . . w_n) ^T is the weighting vector of ${\tilde{α}}_{j} (j = 1, 2 . . ., n)$ , with w_j ∈ [0, 1] and $\sum_{j = 1}^{n}$ w_j = 1.

Proof. In the following, we prove the second result by using mathematical induction on n.

(1) We first prove that Equation (11) holds for n = 2. Since $\begin{matrix} ({\tilde{α}}_{1})^{w_{1}} \\ = (\begin{matrix} [(s_{θ (p_{1})})^{w_{1}}, (s_{t (p_{1})})^{w_{1}}], [p_{1}^{w_{1}}, q_{1}^{w_{1}}, r_{1}^{w_{1}}, s_{1}^{w_{1}}]; \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{1}}^{2})}^{w_{1}}}}{\sqrt{{(2 - Ψ_{{\tilde{α}}_{1}}^{2})}^{w_{1}} + {(Ψ_{{\tilde{α}}_{1}}^{2})}^{w_{1}}}}, \\ \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}}{\cup} \frac{\sqrt{{(1 + ϒ_{{\tilde{α}}_{1}}^{2})}^{w_{1}} - {(1 - ϒ_{{\tilde{α}}_{1}}^{2})}^{w_{1}}}}{\sqrt{{(1 + ϒ_{{\tilde{α}}_{1}}^{2})}^{w_{1}} + {(1 - ϒ_{{\tilde{α}}_{1}}^{2})}^{w_{1}}}} \end{matrix}), \end{matrix}$ and $\begin{matrix} {\tilde{α}}_{2}^{w_{2}} \\ = (\begin{matrix} [(s_{θ (p_{2})})^{w_{2}}, (s_{t (p_{2})})^{w_{2}}], [p_{2}^{w_{2}}, q_{2}^{w_{2}}, r_{2}^{w_{2}}, s_{2}^{w_{2}}]; \\ \underset{Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{2}}^{2})}^{w_{2}}}}{\sqrt{{(2 - Ψ_{{\tilde{α}}_{2}}^{2})}^{w_{2}} + {(Ψ_{{\tilde{α}}_{2}}^{2})}^{w_{2}}}}, \\ \underset{ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{{(1 + ϒ_{{\tilde{α}}_{2}}^{2})}^{w_{2}} - {(1 - ϒ_{{\tilde{α}}_{2}}^{2})}^{w_{2}}}}{\sqrt{{(1 + ϒ_{{\tilde{α}}_{2}}^{2})}^{w_{2}} + {(1 - ϒ_{{\tilde{α}}_{2}}^{2})}^{w_{2}}}} \end{matrix}) . \end{matrix}$ Let $\begin{matrix} \begin{matrix} x_{1} = & {(1 + ϒ_{{\tilde{α}}_{1}}^{2})}^{w_{1}}, y_{1} = {(1 - ϒ_{{\tilde{α}}_{1}}^{2})}^{w_{1}}, \\ z_{1} = & {(Ψ_{{\tilde{α}}_{1}}^{2})}^{w_{1}}, g_{1} = {(2 - Ψ_{{\tilde{α}}_{1}}^{2})}^{w_{1}}, \\ x_{2} = & {(1 + ϒ_{{\tilde{α}}_{2}}^{2})}^{w_{2}}, y_{2} = {(1 - ϒ_{{\tilde{α}}_{2}}^{2})}^{w_{2}}, \\ z_{2} = & {(Ψ_{{\tilde{α}}_{2}}^{2})}^{w_{2}}, g_{2} = {(2 - Ψ_{{\tilde{α}}_{2}}^{2})}^{w_{2}} . \end{matrix} \end{matrix}$ Then, $\begin{matrix} {\tilde{α}}_{1}^{w_{1}} = (\begin{matrix} [(s_{θ (p_{1})})^{w_{1}}, (s_{t (p_{1})})^{w_{1}}]; \\ [p_{1}^{w_{1}}, q_{1}^{w_{1}}, r_{1}^{w_{1}}, s_{1}^{w_{1}}] \\ \underset{Ψ_{{\tilde{α}}_{1}} \in γ_{{\tilde{α}}_{1}}}{\cup} \frac{\sqrt{2 z_{1}}}{\sqrt{g_{1} + z_{1}}}, \underset{ϒ_{{\tilde{α}}_{1}} \in β_{{\tilde{α}}_{1}}}{\cup} \frac{\sqrt{x_{1} - y_{1}}}{\sqrt{x_{1} + y_{1}}} \end{matrix}), \end{matrix}$ and $\begin{matrix} {\tilde{α}}_{2}^{w_{2}} = (\begin{matrix} [(s_{θ (p_{2})})^{w_{2}}, (s_{t (p_{2})})^{w_{2}}], \\ [p_{2}^{w_{2}}, q_{2}^{w_{2}}, r_{2}^{w_{2}}, s_{2}^{w_{2}}]; \\ \underset{Ψ_{{\tilde{α}}_{2}} \in γ_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{2 z_{2}}}{\sqrt{g_{2} + z_{2}}}, \underset{ϒ_{{\tilde{α}}_{2}} \in β_{{\tilde{α}}_{2}}}{\cup} \frac{\sqrt{x_{2} - y_{2}}}{\sqrt{x_{2} + y_{2}}} \end{matrix}) . \end{matrix}$ Thus, by the Einstein operational law (3) and Definition 9, we have $\begin{matrix} PTULFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}) \\ = {\tilde{α}}_{1}^{w_{1}} \otimes_{ɛ} {\tilde{α}}_{2}^{w_{2}} \\ = (\begin{matrix} (s_{θ {(p_{1})}^{w_{1}}} s_{θ {(p_{2})}^{w_{2}}}, s_{t {(p_{1})}^{w_{1}}} s_{t {(p_{2})}^{w_{2}}}), \\ [p_{1}^{w_{1}} p_{2}^{w_{2}}, q_{1}^{w_{1}} q_{2}^{w_{2}}, r_{1}^{w_{1}} r_{2}^{w_{2}}, s_{1}^{w_{1}} s_{2}^{w_{2}}]; \\ \frac{\sqrt{2 z_{1} z_{2}}}{\sqrt{g_{1} g_{2} + z_{1} z_{2}}}, \frac{\sqrt{x_{1} x_{2} - y_{1} y_{2}}}{\sqrt{x_{1} x_{2} + y_{1} y_{2}}} \end{matrix}), \\ = (\begin{matrix} [\prod_{j = 1}^{2} (s_{θ (p_{j})})^{w_{j}}, \prod_{j = 1}^{2} (s_{t (p_{j})})^{w_{j}}], \\ [\prod_{j = 1}^{2} p_{j}^{w_{j}}, \prod_{j = 1}^{2} q_{j}^{w_{j}}, \prod_{j = 1}^{2} r_{j}^{w_{j}}, \prod_{j = 1}^{2} s_{j}^{w_{j}}]; \\ \underset{Ψ_{{\tilde{α}}_{i}} \in γ_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{2 \prod_{j = 1}^{2} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{2} {(2 - Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{2} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}, \\ \underset{ϒ_{{\tilde{α}}_{j}} \in β_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{\prod_{j = 1}^{2} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{2} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{2} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{2} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}} \end{matrix}) . \end{matrix}$

(2) If Equation (11) holds for n = k, that is $\begin{matrix} PTULFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{k}) \overset{k}{= \underset{j = 1}{\otimes_{ɛ}}} {({\tilde{α}}_{j})}^{w_{j}} \\ = & (\begin{matrix} [\prod_{j = 1}^{2} (s_{θ (p_{j})})^{w_{j}}, \prod_{j = 1}^{2} (s_{t (p_{j})})^{w_{j}}], \\ [\prod_{j = 1}^{2} p_{j}^{w_{j}}, \prod_{j = 1}^{2} q_{j}^{w_{j}}, \prod_{j = 1}^{2} r_{j}^{w_{j}}, \prod_{j = 1}^{2} s_{j}^{w_{j}}]; \\ \underset{Ψ_{{\tilde{α}}_{i}} \in γ_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{2 \prod_{j = 1}^{2} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{2} {(2 - Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{2} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}, \\ \underset{ϒ_{{\tilde{α}}_{j}} \in β_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{\prod_{j = 1}^{2} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{2} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{2} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{2} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}} \end{matrix}), \end{matrix}$ when, n = k + 1, we have $\begin{matrix} PTULFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{k + 1}) \\ = PTULFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{k}) \otimes_{ɛ} ({\tilde{α}}_{k + 1}^{w_{k + 1}}) \\ = PTULFEWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{k}) \otimes_{ɛ} \\ = (\begin{matrix} [(s_{θ {(p_{k + 1})}^{w_{k + 1}}}, s_{t {(p_{k + 1})}^{w_{k + 1}}})], \\ [p_{k + 1}^{w_{k + 1}}, q_{k + 1}^{w_{k + 1}}, r_{k + 1}^{w_{k + 1}}, s_{k + 1}^{w_{k + 1}},]; \\ \underset{Ψ_{{\tilde{α}}_{i}} \in γ_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{2 {(Ψ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{k + 1}}}}{\sqrt{{(2 - Ψ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{k + 1}} + {(Ψ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{k + 1}}}}, \\ \underset{ϒ_{{\tilde{α}}_{j}} \in β_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{{(1 + ϒ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{k + 1}} - {(1 - ϒ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{k + 1}}}}{\sqrt{{(1 + ϒ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{k + 1}} + {(1 - ϒ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{k + 1}}}} \end{matrix}) . \end{matrix}$ Let $\begin{matrix} \begin{matrix} x_{1} = \prod_{j = 1}^{k} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}, y_{1} = \prod_{j = 1}^{k} {(1 - ϒ_{{\tilde{α}}_{j}})}^{w_{j}}, \\ z_{1} = \prod_{j = 1}^{k} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}, g_{1} = \prod_{j = 1}^{k} {(2 - Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}, \\ x_{2} = {(1 + ϒ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{j}}, y_{2} = {(1 - ϒ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{j}}, \\ z_{2} = {(Ψ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{j}}, g_{2} = {(2 - Ψ_{{\tilde{α}}_{k + 1}}^{2})}^{w_{j}} . \end{matrix} \end{matrix}$

Then, by the Einstein operational law (3) of Definition 9, we have $\begin{matrix} PTULFEWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{k + 1}) \\ = PTULFEWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{k}) \otimes_{ɛ} {({\tilde{α}}_{k + 1})}^{w_{k + 1}}, \\ = (\begin{matrix} [\prod_{j = 1}^{2} (s_{θ (p_{j})})^{w_{j}}, \prod_{j = 1}^{2} (s_{t (p_{j})})^{w_{j}}], \\ [\prod_{j = 1}^{2} p_{j}^{w_{j}}, \prod_{j = 1}^{2} q_{j}^{w_{j}}, \prod_{j = 1}^{2} r_{j}^{w_{j}}, \prod_{j = 1}^{2} s_{j}^{w_{j}}]; \\ \underset{Ψ_{{\tilde{α}}_{i}} \in γ_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{2 z_{1}}}{\sqrt{g_{1} + z_{1}}}, \underset{ϒ_{{\tilde{α}}_{j}} \in β_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{x_{1} - y_{1}}}{\sqrt{x_{1} + y_{1}}} \end{matrix}) \end{matrix}$ $\begin{matrix} \oplus_{ɛ} (\begin{matrix} [(s_{θ (p_{k + 1})^{w_{k + 1}}}, s_{t (p_{k + 1})^{w_{k + 1}}})] \\ [p_{k + 1}^{w_{k + 1}}, q_{k + 1}^{w_{k + 1}}, r_{k + 1}^{w_{k + 1}}, s_{k + 1}^{w_{k + 1}},]; \\ \frac{\sqrt{2 z_{2}}}{\sqrt{g_{2} + z_{2}}}, \frac{\sqrt{x_{2} - y_{2}}}{\sqrt{x_{2} + y_{2}}} \end{matrix}), \\ = (\begin{matrix} [\prod_{j = 1}^{k + 1} (s_{θ (p_{j})})^{w_{j}}, \prod_{j = 1}^{k + 1} (s_{t (p_{j})})^{w_{j}},] \\ [\prod_{j = 1}^{k + 1} p_{j}^{w_{j}}, \prod_{j = 1}^{k + 1} q_{j}^{w_{j}}, \prod_{j = 1}^{k + 1} r_{j}^{w_{j}}, \prod_{j = 1}^{k + 1} s_{j}^{w_{j}}]; \\ \underset{Ψ_{{\tilde{α}}_{j}} \in γ_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{2 z_{1} z_{2}}}{\sqrt{g_{1} g_{2} + z_{1} z_{2}}}, \underset{ϒ_{{\tilde{α}}_{j}} \in β_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{x_{1} x_{2} - y_{1} y_{2}}}{\sqrt{x_{1} x_{2} + y_{1} y_{2}}} \end{matrix}) \\ = (\begin{matrix} [\prod_{j = 1}^{k + 1} (s_{θ (p_{j})})^{w_{j}}, \prod_{j = 1}^{k + 1} (s_{t (p_{j})})^{w_{j}},] \\ [\prod_{j = 1}^{k + 1} p_{j}^{w_{j}}, \prod_{j = 1}^{k + 1} q_{j}^{w_{j}}, \prod_{j = 1}^{k + 1} r_{j}^{w_{j}}, \prod_{j = 1}^{k + 1} s_{j}^{w_{j}}]; \\ \underset{Ψ_{{\tilde{α}}_{j}} \in γ_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{2 \prod_{j = 1}^{k + 1} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{k + 1} {(2 - Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k + 1} {(Ψ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}, \\ \underset{ϒ_{{\tilde{α}}_{j}} \in β_{{\tilde{α}}_{j}}}{\cup} \frac{\sqrt{\prod_{j = 1}^{k + 1} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{k + 1} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{k + 1} {(1 + ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k + 1} {(1 - ϒ_{{\tilde{α}}_{j}}^{2})}^{w_{j}}}}, \end{matrix}) \end{matrix}$ such that, Equation (11) holds for n = k + 1. Therefore, Equation (11) holds for all n.

Acknowledgments

The authors extended their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Program under grant number R-G.P2/52/40.

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