Spherical fuzzy sets and their applications in multi-attribute decision making problems

Abstract

The key objective of the present proposed work in this paper is introduced a new version of picture fuzzy set so called spherical fuzzy sets (SFS). spherical fuzzy set is a new extension of picture fuzzy sets and Pythagorean fuzzy sets. In spherical fuzzy sets, membership degrees are gratifying the condition 0 ⩽ P² (x) + I² (x) + N² (x) ⩽1 instead of 0 ⩽ P (x) + I (x) + N (x) ⩽1 as is in picture fuzzy sets. In this paper, we investigate the basic operations of spherical fuzzy sets and discuss some related results. We extend operational laws to aggregation operators and introduce weighted averaging and weighted geometric aggregation operators based on spherical fuzzy number’s. Further a multi attribute decision making method is developed and these aggregation operators are utilized. Finally, we constructed a numerical approach for implementation of proposed technique.

Keywords

Spherical fuzzy sets aggregation operators multi-attribute decision making problem

1. Introduction

To address the issues of difficulties of acquiring sufficient and accurate data for real decision making due to the imprecision and ambiguity of socioeconomics, fuzzy set theory is one of the most powerful track for treating the multi-attribute decision making problems. Based on fuzziness circumstances fuzzy sets (FSs), developed by Zadeh [31], was initially used. In FSs each element x of the domain set contains only one index namely as degree of membership P (x) which oscillate from 0 to 1. Non-membership degree for the FS is straightforward equivalent to 1 - P (x). However, sometime FS has some drawbacks for example, it has no ability to show the neutral state (which neither favor nor disfavor). However, Intuitionistic fuzzy set (IFS) developed by Atanassov [3], to apprehend the uncertainties or inexact information about degree of membership. Atanassov’s IFSs are the generalization of Zadeh’s FSs. He utilized two indices (membership degree P (x) and non-membership degree N (x)) to define the IFS with the condition that 0 ⩽ P (x) + N (x) ⩽1. From last few decades, the IFS has been explored by many researchers and successfully applied to many practical fields like medical diagnosis, clustering analysis, decision making and pattern recognition [1 , 34]. Later, the degree of membership and non-membership in IFSs may be denoted as interval values alternatively by crisp numbers. So, the interval valued intuitionistic fuzzy sets (IVIFSs), was developed by Atanassov and Gargov [4]. IVIFSs are an extension of FSs & IFSs. The IFS and IVIFS have been explored by many researchers like as [4, 5], and widely applied to many areas, such as group decision making [22, 27], similarity measures [11], multi-criteria decision making [24]. In 2017, Lui [19] introduced the Heronian aggregation operators of intuitionistic fuzzy numbers, also defined interaction partitioned Bonferroni mean operators [18] and Interval-valued intuitionistic fuzzy power Bonferroni aggregation operators [17] for intuitionistic fuzzy numbers. In 2018, Lui [20] introduced the Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group decision making problems.

Since in some real life decision theory the decision makers deal with the situation of particular attributes where values of their summation of membership degrees exceeds 1. In such condition, IFSs has no ability to obtain any satisfactory result. To overcome this situation Yager [28] developed the idea of Pythagorean fuzzy set (PyFS) as a generalization of IFS, which satisfies that the value of square summation of its membership degrees is less then or equals to 1. Now the situation where the neutral membership degree calculate independently in real life problems, the IFS and PyFS fail to attain any satisfactory result. Based on these circumstances, to overcome this situation, Cuong and Kreinovich [9] initiated the idea of picture fuzzy set (PFS). He utilized three index (membership degree P (x), neutral-membership degree I (x), and non-membership degree N (x)) in PFS with the condition that is 0 ⩽ P (x) + I (x) + N (x) ⩽ 1. Obviously PFSs is more suitable than IFS and PyFS to deal with fuzziness and vagueness. H. Garg [12] introduced picture fuzzy weighted averaging operator, Picture fuzzy ordered weighted averaging operator, Picture fuzzy hybrid averaging operator under picture fuzzy environment. From last few decades, the PFS has been explored by many researchers and successfully applied to many practical fields like strategic decision making, Attribute decision making and pattern recognition [9 , 26].

Sometimes in real life, we face many problems which cannot be handled by using PFS for example when P (x) + I (x) + N (x) > 1. In such condition, PFS has no ability to obtain any satisfactory result. To state this condition, we give an example: for support and against the degree of membership of an alternative are $\frac{1}{5}, \frac{3}{5}$ and $\frac{3}{5}$ respectively. This satisfies the condition that their sum exceeds 1 and are not presented for PFS. Based on these circumstances, the idea of spherical fuzzy sets (SFSs) is introduced as a generalization of PFS. In SFS, membership degrees are gratifying the condition. $0 < P^{2} (x) + I^{2} (x) + N^{2} (x) < 1 .$

The objectives of this paper are: (1) to introduce the spherical fuzzy set (2) to define the spherical fuzzy numbers (SFNs) and related basic operational identities, (3) to suggest score, accuracy and certainty functions for comparison, (4) to propose spherical aggregation operators and some debate on their properties, (5) to demonstrate a MADM method based on these aggregation operators under spherical fuzzy information.

The superfluity of this paper is planned as follows. In Section 2, the basic notion of Intuitionistic fuzzy sets, PFSs and their properties are presented. In Section 3, we introduce the concept of SFSs and their operational properties. Section 4, consist of the study of aggregation operators to aggregate the spherical fuzzy information. In Sections 5, we proposed the MADM method to deal with spherical fuzzy information and a descriptive example is illustrated to express the effectiveness and reliability of the suggested technique. The last Section 6, consist of the conclusion of the paper.

2. Preliminaries

The paper gives brief discussion on basic ideas associated to IFS and PFS with their operations and operators. We also discuss, more familiarized ideas which is utilized in following analysis

Definition 1. [3] Let the universe set be R. Then $A = {〈 r, P_{A} (r), N_{A} (r) | r \in R 〉},$ is said to be intuitionistic fuzzy set of R, where P_A : R → [0, 1] and N_A : R → [0, 1] are said to be degree of positive-membership of r in R and negative-membership degree of r in R espectively. Also P_A and N_A satisfy the condition: (∀r ∈ R) (0 ⩽ P_A (r) + N_A (r) ⩽1).

Definition 2. [9] Let the universe set be R. Then the set $A = {〈 r, P_{A} (r), I_{A} (r), N_{A} (r) | r \in R 〉},$ is said to be picture fuzzy set of R, where P_A : R → [0, 1], I_A : R → [0, 1] and N_A : R → [0, 1] are said to be degree of positive-membership of r in R, neutral-membership degree of r in R and negative-membership degree of r in R respectively. Also P_A, I_A and N_A satisfy the condition: $(\forall r \in R) (0 ⩽ P_{A} (r) + I_{A} (r) + N_{A} (r) ⩽ 1) .$

Then for 1 - (P_A (r) + I_A (r) + N_A (r)) is said to be degree of refusal-membership of r in R.

3. Spherical fuzzy sets and their operations

In this section, the idea of spherical fuzzy set (SFS), their operations and operators are introduced. The structure of SFSs give shields to the elements, which satisfies or dissatisfies the condition that values must oscillate from 0 to 1. We see in the figure : 1, which specifies the points where structure of Spherical fuzzy set give shield to the elements. Also, we familiarized with the contour of this graph in figure : 3.

Spherical fuzzy set is a direct generalization of Pythagorean fuzzy set and picture fuzzy set. An interesting scenario emerge when Pythagorean fuzzy sets (PyFS) and picture fuzzy sets both failed to handle the situation. Therefore, there is need the spherical fuzzy set to tackle this situation. The main difference between PyFSs and SFSs is that in SFSs we study the neutral degree, where as in PyFSs it doesn’t. In PFSs the connection of positive, neutral and negative grades of an object is given in unit close interval but in some cases the sum of the positive, neutral and negative grades of the object is greater than 1, so this situation leads us toward spherical fuzzy set.

Now using an example to understand that why spherical fuzzy set is suitable as compare to existing structure like IFS, PyFS and PFS. Cuong structure of PFS is prominence as it has the capability to deal with human opinion competently. In PFSs, we observed that the condition on membership degree belongs to unit interval. i.e. 0 ⩽ P (x) + I (x) + N (x) ⩽1. Therefore, it is observed that, we have restricted to assign values to membership degrees by own choice. Considering these points, an innovative structure of SFS is proposed which somehow give strength to the structure of PFS by expanding the space of membership degrees. i.e. 0 ⩽ P² (x) + I² (x) + N² (x) ⩽1. For example, if we choose P (x) =0.7, I (x) =0.3 and N (x) =0.5. Then in this case 0 ⩽ P (x) + I (x) + N (x) =1.5 > 1. But by squaring P (x), I (x) and N (x) their sum becomes less than or equal to one i.e. 0 ⩽ (0.7) ² + (0.3) ² + (0.5) ² ⩽ 1. This example indicate that SFS is more influential tool than PFS. Also observe that PyFS is also failed to deal such situation because PyFS does not deal with neutral membership. We conclude that all existing structure failed to deal such type of information, so this situation leads us to propose novel structure of SFSs. This example give strength to the novelty and effectiveness of proposed SFSs.

Now we see the contour of this graph.

when we are combining the both graph and their contour, we have.

Definition 3. Let the universe set be R. Then the set $A = {〈 r, P_{A} (r), I_{A} (r), N_{A} (r) | r \in R 〉},$

is said to be spherical fuzzy set, where P_A : R → [0, 1], I_A : R → [0, 1] and N_A : R → [0, 1] are said to be degree of positive-membership of r in R, neutral-membership degree of r in R and negative-membership degree of r in R respectively. Also P_A, I_A and N_A satisfy the following condition: $(\forall r \in R) (0 ⩽ (P_{A} (r))^{2} + (I_{A} (r))^{2} + (N_{A} (r))^{2} ⩽ 1) .$

For SFS {〈 r, P_A (r), I_A (r), N_A (r) |r ∈ R 〉 }, which is triple components $〈 P_{A} (r), I_{A} (r), N_{A} (r) 〉$

are said to SFN and each SFN can be denoted by e =〈 P_e, I_e, N_e 〉, where P_e, I_e and N_e ∈ [0, 1], with condition that $0 ⩽ P_{e}^{2} + I_{e}^{2} + N_{e}^{2} ⩽ 1$ .

Definition 4. Assuming that e_j =〈 P_{e
_j}, I_{e
_j}, N_{e
_j} 〉 and e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 be any two SFNs. Then union, intersection and compliment are describes as;

e_j ⊆ e_k iff ∀r ∈ R, P_{e
_j} ⩽ P_{e
_k}, I_{e
_j} ⩽ I_{e
_k} and N_{e
_j}⩾ N_{e
_k};

e_j = e_k iff e_j ⊆ e_k and e_k⊆ e_j;

e_j ∪ e_k = 〈 max(P_{e
_j}, P_{e
_k}), min(I_{e
_j}, I_{e
_k}), min(N_{e
_j}, N_{e
_k})〉;

e_j ∩ e_k = 〈 min(P_{e
_j}, P_{e
_k}), min(I_{e
_j}, I_{e
_k}), max(N_{e
_j}, N_{e
_k})〉;

$e_{j}^{c} = 〈 N_{e_{j}}, I_{e_{j}}, P_{e_{j}} 〉 .$

Definition 5. Suppose that e_j =〈 P_{e
_j}, I_{e
_j}, N_{e
_j} 〉 and e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 are any two SFNs and τ ⩾ 0. Then the operations of SFNs can be denotes as

$τ e_{j} = 〈 \sqrt{1 - (1 - P_{e_{j}}^{2})^{τ}}, (I_{e_{j}})^{τ}, (N_{e_{j}})^{τ} 〉;$

$e_{j} + e_{k} = 〈 \sqrt{P_{e_{j}}^{2} + P_{e_{k}}^{2} - P_{e_{j}}^{2} \cdot P_{e_{k}}^{2}}, I_{e_{j}} \cdot I_{e_{k}},$ N_{e
_j}· N_{e
_k} 〉;

$\sqrt{N_{e_{j}}^{2} + N_{e_{k}}^{2} - N_{e_{j}}^{2} \cdot N_{e_{k}}^{2}} 〉;$

$e_{j}^{τ} = 〈 (P_{e_{j}})^{τ}, (I_{e_{j}})^{τ}, \sqrt{1 - (1 - N_{e_{j}}^{2})^{τ}} 〉 .$

Theorem 1. Assuming that e_j =〈 P_{e
_j}, I_{e
_j}, N_{e
_j} 〉, e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 and e_l =〈 P_el, I_{e
_l}, N_{e
_l} 〉 be any three SFNs and τ ⩾ 0. Then the following identities are satisfies.

e_j+ e_k = e_k + e_j;

e_j× e_k = e_k × e_j;

(e_j+ e_k) + e_l = e_j + (e_k + e_l);

(e_j× e_k) × e_l = e_j × (e_k × e_l);

τe_j + τe_k = τ (e_j + e_k), τ⩾ 0;

τ_je_j + τ_ke_j = (τ_j + τ_k) e_j, τ_j ⩾ 0 and τ_k⩾ 0;

${(e_{j} \times e_{k})}^{τ} = e_{j}^{τ} \times e_{k}^{τ},$ τ⩾ 0;

$e_{j}^{τ_{j}} \times e_{j}^{τ_{k}} = e_{j}^{τ_{j} + τ_{k}},$ τ_j ⩾ 0 and τ_k ⩾ 0.

Proof. (1) We have to show that e_j + e_k = e_k + e_j. Consider

$\begin{matrix} L . H . S = e_{j} + e_{k} \\ = 〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉 + 〈 P_{e_{k}}, I_{e_{k}}, N_{e_{k}} 〉 \\ = 〈 \sqrt{P_{e_{j}}^{2} + P_{e_{k}}^{2} - P_{e_{j}}^{2} \cdot P_{e_{k}}^{2}}, I_{e_{j}} \cdot I_{e_{k}}, N_{e_{j}} \cdot N_{e_{k}} 〉 \\ = 〈 \sqrt{P_{e_{k}}^{2} + P_{e_{j}}^{2} - P_{e_{k}}^{2} \cdot P_{e_{j}}^{2}}, I_{e_{k}} \cdot I_{e_{j}}, N_{e_{k}} \cdot N_{e_{j}} 〉 \\ = e_{k} + e_{j} = R . H . S \end{matrix}$

Hence, we prove this.

(2) We have to show that e_j × e_k = e_k × e_j. Consider $\begin{matrix} L . H . S = e_{j} \times e_{k} \\ = 〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉 \times 〈 P_{e_{k}}, I_{e_{k}}, N_{e_{k}} 〉 \\ = 〈 P_{e_{j}} \cdot P_{e_{k}}, I_{e_{j}} \cdot I_{e_{k}}, \sqrt{N_{e_{j}}^{2} + N_{e_{k}}^{2} - N_{e_{j}}^{2} \cdot N_{e_{k}}^{2}} 〉 \\ = 〈 P_{e_{k}} \cdot P_{e_{j}}, I_{e_{k}} \cdot I_{e_{j}}, \sqrt{N_{e_{k}}^{2} + N_{e_{j}}^{2} - N_{e_{k}}^{2} \cdot N_{e_{j}}^{2}} 〉 \\ = e_{k} \times e_{j} = R . H . S \end{matrix}$

Hence, we prove this.

The proofs of (3) and (4) are straightforward as (1) and (2).

(5) We have to show that τe_j + τe_k = τ (e_j + e_k), τ ⩾ 0. Consider

$\begin{matrix} L . H . S = τ e_{j} + τ e_{k} \\ = τ \cdot 〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉 + τ \cdot 〈 P_{e_{k}}, I_{e_{k}}, N_{e_{k}} 〉 \\ = 〈 \sqrt{1 - (1 - P_{e_{j}}^{2})^{τ}}, (I_{e_{j}})^{τ}, (N_{e_{j}})^{τ} 〉 \\ + 〈 \sqrt{1 - (1 - P_{e_{k}}^{2})^{τ}}, (I_{e_{k}})^{τ}, (N_{e_{k}})^{τ} 〉 \\ = 〈 \sqrt{1 - {(1 - P_{e_{j}}^{2})}^{τ} {(1 - P_{e_{k}}^{2})}^{τ}}, {(I_{e_{j}} \cdot I_{e_{k}})}^{τ}, \\ {(N_{e_{j}} \cdot N_{e_{k}})}^{τ} 〉 \\ = 〈 \sqrt{1 - {(1 - (P_{e_{j}}^{2} + P_{e_{k}}^{2} - P_{e_{j}}^{2} \cdot P_{e_{k}}^{2}))}^{τ}}, \\ {(I_{e_{j}} \cdot I_{e_{k}})}^{τ}, {(N_{e_{j}} \cdot N_{e_{k}})}^{τ} 〉 \end{matrix}$

$\begin{matrix} R . H . S = τ (e_{j} + e_{k}) \\ = τ \cdot 〈 \sqrt{P_{e_{j}}^{2} + P_{e_{k}}^{2} - P_{e_{j}}^{2} \cdot P_{e_{k}}^{2}}, I_{e_{j}} \cdot I_{e_{k}}, N_{e_{j}} \cdot N_{e_{k}} 〉 \\ = 〈 \sqrt{1 - {(1 - (P_{e_{j}}^{2} + P_{e_{k}}^{2} - P_{e_{j}}^{2} \cdot P_{e_{k}}^{2}))}^{τ}}, \\ {(I_{e_{j}} \cdot I_{e_{k}})}^{τ}, {(N_{e_{j}} \cdot N_{e_{k}})}^{τ} 〉 \end{matrix}$

Hence, we prove this.

(6) We have to show that τ_je_j + τ_ke_j = (τ_j + τ_k) e_j, τ_j ⩾ 0 and τ_k ⩾ 0. Consider

$\begin{matrix} L . H . S = τ_{j} e_{j} + τ_{k} e_{j}, where τ_{j} ⩾ 0 and τ_{k} ⩾ 0 \\ = τ_{j} \cdot 〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉 + τ_{k} \cdot 〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉 \\ = 〈 \sqrt{1 - (1 - P_{e_{j}}^{2})^{τ_{j}}}, (I_{e_{j}})^{τ_{j}}, (N_{e_{j}})^{τ_{j}} 〉 \\ + 〈 \sqrt{1 - (1 - P_{e_{j}}^{2})^{τ_{k}}}, (I_{e_{j}})^{τ_{k}}, (N_{e_{j}})^{τ_{k}} 〉 \\ = 〈 \sqrt{\begin{matrix} (1 - (1 - P_{e_{j}}^{2})^{τ}) + (1 - (1 - P_{e_{k}}^{2})^{τ}) \\ - (1 - (1 - P_{e_{j}}^{2})^{τ}) (1 - (1 - P_{e_{k}}^{2})^{τ}) \end{matrix}}, \\ (I_{e_{j}})^{τ} \cdot (I_{e_{k}})^{τ}, (N_{e_{j}})^{τ} \cdot (N_{e_{k}})^{τ} 〉 \\ = 〈 \sqrt{1 - {(1 - P_{e_{j}}^{2})}^{τ_{j}} {(1 - P_{e_{j}}^{2})}^{τ_{k}}}, {(I_{e_{j}})}^{τ_{j} + τ_{k}}, \\ {(N_{e_{j}})}^{τ_{j} + τ_{k}} 〉 \\ = 〈 \sqrt{(1 - {(1 - P_{e_{j}}^{2})}^{τ_{j} + τ_{k}})}, {(I_{e_{j}})}^{τ_{j} + τ_{k}}, \\ {(N_{e_{j}})}^{τ_{j} + τ_{k}} 〉 \end{matrix}$

Furthermore

$\begin{matrix} R . H . S = (τ_{j} + τ_{k}) e_{j}, where τ_{j} ⩾ 0 and τ_{k} ⩾ 0 \\ = (τ_{j} + τ_{k}) \cdot 〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉 \\ = 〈 \sqrt{(1 - {(1 - P_{e_{j}}^{2})}^{τ_{j} + τ_{k}})}, {(I_{e_{j}})}^{τ_{j} + τ_{k}}, \\ {(N_{e_{j}})}^{τ_{j} + τ_{k}} 〉 \end{matrix}$

Hence, we prove this.

(7) We have to show that ${(e_{j} \times e_{k})}^{τ} = e_{j}^{τ} \times e_{k}^{τ},$ τ ⩾ 0. Consider $\begin{matrix} L . H . S = {(e_{j} \times e_{k})}^{τ}, τ ⩾ 0 \\ = (〈 P_{e_{j}} \cdot P_{e_{k}}, I_{e_{j}} \cdot I_{e_{k}}, \\ {\sqrt{N_{e_{j}}^{2} + N_{e_{k}}^{2} - N_{e_{j}}^{2} \cdot N_{e_{k}}^{2}} 〉)}^{τ} \\ = 〈 {(P_{e_{j}} \cdot P_{e_{k}})}^{τ}, {(I_{e_{j}} \cdot I_{e_{k}})}^{τ}, \\ \sqrt{1 - {(1 - (N_{e_{j}}^{2} + N_{e_{k}}^{2} - N_{e_{j}}^{2} \cdot N_{e_{k}}^{2}))}^{τ}} 〉 \end{matrix}$

Now $\begin{matrix} R . H . S = e_{j}^{τ} \times e_{k}^{τ}, τ ⩾ 0 \\ = (〈 (P_{e_{j}})^{τ}, (I_{e_{j}})^{τ}, \sqrt{1 - (1 - N_{e_{j}}^{2})^{τ}} 〉) \\ \times (〈 (P_{e_{k}})^{τ}, (I_{e_{k}})^{τ}, \sqrt{1 - (1 - N_{e_{k}}^{2})^{τ}} 〉) \\ = 〈 (P_{e_{j}})^{τ} \cdot (P_{e_{k}})^{τ}, (I_{e_{j}})^{τ} \cdot (I_{e_{k}})^{τ}, \\ \sqrt{\begin{matrix} (1 - (1 - N_{e_{j}}^{2})^{τ}) + (1 - (1 - N_{e_{k}}^{2})^{τ}) \\ - (1 - (1 - N_{e_{j}}^{2})^{τ}) (1 - (1 - N_{e_{k}}^{2})^{τ}) \end{matrix}} 〉 \\ = 〈 {(P_{e_{j}} \cdot P_{e_{k}})}^{τ}, {(I_{e_{j}} \cdot I_{e_{k}})}^{τ}, \\ \sqrt{1 - {(1 - N_{e_{j}}^{2})}^{τ} {(1 - N_{e_{k}}^{2})}^{τ}} 〉 \\ = 〈 {(P_{e_{j}} \cdot P_{e_{k}})}^{τ}, {(I_{e_{j}} \cdot I_{e_{k}})}^{τ}, \\ \sqrt{1 - {(1 - (N_{e_{j}}^{2} + N_{e_{k}}^{2} - N_{e_{j}}^{2} \cdot N_{e_{k}}^{2}))}^{τ}} 〉 \end{matrix}$

Hence, we prove this.

(8) We have to show that $e_{j}^{τ_{j}} \times e_{j}^{τ_{k}} = e_{j}^{τ_{j} + τ_{k}},$ τ_j ⩾ 0 and τ_k ⩾ 0. Consider $\begin{matrix} L . H . S = e_{j}^{τ_{j}} \times e_{j}^{τ_{k}}, τ_{j} ⩾ 0 and τ_{k} ⩾ 0 \\ = {(〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉)}^{τ_{j}} \times {(〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉)}^{τ_{k}} \\ = 〈 (P_{e_{j}})^{τ_{j}}, (I_{e_{j}})^{τ_{j}}, \sqrt{1 - (1 - N_{e_{j}}^{2})^{τ_{j}}} 〉 \\ \times 〈 (P_{e_{j}})^{τ_{k}}, (I_{e_{j}})^{τ_{k}}, \sqrt{1 - (1 - N_{e_{j}}^{2})^{τ_{k}}} 〉 \\ = 〈 (P_{e_{j}})^{τ_{j}} \cdot (P_{e_{j}})^{τ_{k}}, (I_{e_{j}})^{τ_{j}} \cdot (I_{e_{j}})^{τ_{k}}, \\ \sqrt{\begin{matrix} (1 - (1 - N_{e_{j}}^{2})^{τ_{j}}) + (1 - (1 - N_{e_{j}}^{2})^{τ_{k}}) \\ - (1 - (1 - N_{e_{j}}^{2})^{τ_{j}}) (1 - (1 - N_{e_{j}}^{2})^{τ_{k}}) \end{matrix}} 〉 \\ = 〈 {(P_{e_{j}})}^{τ_{j} + τ_{k}}, {(I_{e_{j}})}^{τ_{j} + τ_{k}}, \\ \sqrt{1 - {(1 - N_{e_{j}}^{2})}^{τ_{j}} {(1 - N_{e_{j}}^{2})}^{τ_{k}}} 〉 \\ = 〈 {(P_{e_{j}})}^{τ_{j} + τ_{k}}, {(I_{e_{j}})}^{τ_{j} + τ_{k}}, \\ \sqrt{1 - {(1 - N_{e_{j}}^{2})}^{τ_{j} + τ_{k}}} 〉 \end{matrix}$

Furthermore $\begin{matrix} R . H . S = e_{j}^{τ_{j} + τ_{k}}, τ_{j} ⩾ 0 and τ_{k} ⩾ 0 \\ = {(〈 P_{e_{j}}, I_{e_{j}}, N_{e_{j}} 〉)}^{τ_{j} + τ_{k}} \\ = 〈 {(P_{e_{j}})}^{τ_{j} + τ_{k}}, {(I_{e_{j}})}^{τ_{j} + τ_{k}}, \sqrt{1 - {(1 - N_{e_{j}}^{2})}^{τ_{j} + τ_{k}}} 〉 \end{matrix}$

Hence, we prove this.

3.2. Comparison rules for SFNs

In this section we introduce some functions which play an important role for the ranking of SFNs are;

Definition 6. Let e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 be any SFNs. Then (1) $sc (e_{k}) = \frac{(P_{e_{k}} + 1 - I_{e_{k}} + 1 - N_{e_{k}})}{3} = \frac{1}{3} (2 + P_{e_{k}} -$ I_{e
_k} - N_{e
_k}) which denoted as score function.

(2) ac (e_k) = P_{e
_k} - N_{e
_k} which denoted as accuracy function.

(3) cr (e_k) = P_{e
_k} which denoted as certainty function. Idea takes from Definition [3.2.1.], is the technique which using for equating the SFNs can be described as

Definition 7. Let e_j =〈 P_{e
_j}, I_{e
_j}, N_{e
_j} 〉 and e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 be any two SFNs. Then by using the Definition [3.2.1.], equating technique can be described as, (a) If sc (e_j) > sc (e_k), then e_j > e_k.

(b) If sc (e_j) = sc (e_k) and ac (e_j) > ac (e_k), then e_j > e_k.

(d) If sc (e_j) = sc (e_k), ac (e_j) = ac (e_k) and cr (e_j) = cr (e_k), then e_j = e_k.

4. Spherical aggregation operators

This section gives some discussion about weighted aggregated operators according to defined operational properties of SFNs.

4.1. Spherical fuzzy number weighted averaging aggregation operators

This section describes the spherical weighted averaging aggregation (SFNWAA) operators by utilizing the defined operational properties of SFNs.

Definition 8. Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs and SFNWAA : SFNⁿ → SFN, then SFNWAA describe as, $S F N O W G A_{w} (e_{1}, e_{2}, …, e_{n}) = \prod_{k = 1}^{n} e_{η (k)}^{τ_{k}},$

In which τ ={ τ₁, τ₂, …, τ_n } be the weight vector of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, k ∈ N, with τ_k ⩾ 0 and $\sum_{k = 1}^{n} τ_{k} = 1 .$

Theorem 2. Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. Then by utilizing the Definition [4.1.1.] and operational properties of SFNs, we can obtain the following outcome. $\begin{matrix} SFNWAA (e_{1}, e_{2}, \dots, e_{n}) = 〈 (\sqrt{1 - Π_{k = 1}^{n} (1 - P_{e_{k}}^{2})^{τ_{k}}}), \\ Π_{k = 1}^{n} (I_{e_{k}})^{τ_{k}}, Π_{k = 1}^{n} (N_{e_{k}})^{τ_{k}} 〉, \end{matrix}$

Where τ ={ τ₁, τ₂, …, τ_n } be the weight vector of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, k ∈ N, with τ_k ⩾ 0 and $\sum_{k = 1}^{n} τ_{k} = 1 .$

Proof. We done the prove by utilizing the technique of mathematical induction. Therefore, we follow as

(a) For n = 2, since $τ_{1} e_{1} = 〈 \sqrt{(1 - (1 - P_{e_{1}}^{2})^{τ_{1}}}, {(I_{e_{1}})}^{τ_{1}}, {(N_{e_{1}})}^{τ_{1}}) 〉$

and $τ_{2} e_{2} = 〈 \sqrt{(1 - (1 - P_{e_{2}}^{2})^{τ_{2}}}, {(I_{e_{2}})}^{τ_{2}}, {(N_{e_{2}})}^{τ_{2}}) 〉$

Then $\begin{matrix} SFNWAA (e_{1}, e_{2}) = τ_{1} e_{1} + τ_{2} e_{2} \\ = 〈 \sqrt{(1 - (1 - P_{e_{1}}^{2})^{τ_{1}}}, {(I_{e_{1}})}^{τ_{1}}, {(N_{e_{1}})}^{τ_{1}}) 〉 \\ + 〈 \sqrt{(1 - (1 - P_{e_{2}}^{2})^{τ_{2}}}, {(I_{e_{2}})}^{τ_{2}}, {(N_{e_{2}})}^{τ_{2}}) 〉 \\ = 〈 \sqrt{\begin{matrix} (1 - (1 - P_{e_{1}}^{2})^{τ_{1}}) + (1 - (1 - P_{e_{2}}^{2})^{τ_{2}}) - \\ (1 - (1 - P_{e_{1}}^{2})^{τ_{1}}) (1 - (1 - P_{e_{2}}^{2})^{τ_{2}}) \end{matrix}}, \\ (I_{e_{1}})^{τ_{1}} \cdot (I_{e_{2}})^{τ_{2}}, (N_{e_{1}})^{τ_{1}} \cdot (N_{e_{2}})^{τ_{2}} 〉 \\ = 〈 \sqrt{1 - {(1 - P_{e_{1}}^{2})}^{τ_{1}} {(1 - P_{e_{2}}^{2})}^{τ_{2}}}, \\ (I_{e_{1}})^{τ_{1}} \cdot (I_{e_{2}})^{τ_{2}}, (N_{e_{1}})^{τ_{1}} \cdot (N_{e_{2}})^{τ_{2}} 〉 \\ = 〈 \sqrt{1 - Π_{k = 1}^{2} (1 - P_{e_{k}}^{2})^{τ_{k}}}, Π_{k = 1}^{2} {(I_{e_{k}})}^{τ_{k}}, \\ Π_{k = 1}^{2} (N_{e_{k}})^{τ_{k}} 〉 \end{matrix}$

(b) Suppose that outcome is true for n = z that is, $\begin{matrix} SFNWAA (e_{1}, e_{2}, \dots, e_{z}) = \\ 〈 \begin{matrix} \sqrt{1 - Π_{k = 1}^{z} (1 - P_{e_{k}}^{2})^{τ_{k}}}, Π_{k = 1}^{z} {(I_{e_{k}})}^{τ_{k}}, \\ Π_{k = 1}^{z} (N_{e_{k}})^{τ_{k}} \end{matrix} 〉, \end{matrix}$

$\begin{matrix} SFNWAA (e_{1}, e_{2}, \dots, e_{z}, e_{z + 1}) = \sum_{k = 1}^{z} τ_{k} e_{k} + τ_{z + 1} e_{z + 1} \\ = 〈 \sqrt{1 - Π_{k = 1}^{z} (1 - P_{e_{k}}^{2})^{τ_{k}}}, Π_{k = 1}^{z} {(I_{e_{k}})}^{τ_{k}}, Π_{k = 1}^{z} (N_{e_{k}})^{τ_{k}} 〉 + \\ 〈 \sqrt{1 - (1 - P_{e_{z + 1}}^{2})^{τ_{z + 1}}}, {(I_{e_{z + 1}})}^{τ_{z + 1}}, {(N_{e_{z + 1}})}^{τ_{z + 1}} 〉 \\ = 〈 \begin{matrix} \sqrt{\begin{matrix} (1 - Π_{k = 1}^{z} (1 - P_{e_{k}}^{2})^{τ_{k}}) + (1 - (1 - P_{e_{z + 1}}^{2})^{τ_{z + 1}}) \\ - (1 - Π_{k = 1}^{z} (1 - P_{e_{k}}^{2})^{τ_{k}}) (1 - (1 - P_{e_{z + 1}}^{2})^{τ_{z + 1}}) \end{matrix}}, \\ Π_{k = 1}^{z} {(I_{e_{k}})}^{τ_{k}} \cdot {(I_{e_{z + 1}})}^{τ_{z + 1}}, Π_{k = 1}^{z} (N_{e_{k}})^{τ_{k}} \cdot {(N_{e_{z + 1}})}^{τ_{z + 1}} \end{matrix} 〉 \\ = 〈 \begin{matrix} \sqrt{1 - Π_{k = 1}^{z} (1 - P_{e_{k}}^{2})^{τ_{k}} (1 - P_{e_{z + 1}}^{2})^{τ_{z + 1}}}, Π_{k = 1}^{z} {(I_{e_{k}})}^{τ_{k}} \\ \cdot {(I_{e_{z + 1}})}^{τ_{z + 1}, Π_{k = 1}^{z} (N_{e_{k}})^{τ_{k}} \cdot {(N_{e_{z + 1}})}^{τ_{z + 1}}} \end{matrix} 〉 \\ = 〈 \sqrt{1 - Π_{k = 1}^{z + 1} (1 - P_{e_{k}}^{2})^{τ_{k}}}, Π_{k = 1}^{z + 1} {(I_{e_{k}})}^{τ_{k}}, Π_{k = 1}^{z + 1} (N_{e_{k}})^{τ_{k}} 〉 \end{matrix}$

Outcome is satisfying for n = z + 1. Thus, outcome is satisfied for whole n. Therefore

$\begin{matrix} SFNWAA (e_{1}, e_{2}, \dots, e_{n}) \\ = 〈 \begin{matrix} \sqrt{1 - Π_{k = 1}^{n} (1 - P_{e_{k}}^{2})^{τ_{k}}}, Π_{k = 1}^{n} (I_{e_{k}})^{τ_{k}}, Π_{k = 1}^{n} (N_{e_{k}})^{τ_{k}} \end{matrix} 〉 \end{matrix}$

which done the proof.

Properties. There are some properties which are fulfilled by the SFNWAA operator obviously.

(a) Idempotency: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. If all of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) are identical. Then there is $SFNWAA (e_{1}, e_{2}, \dots, e_{n}) = e .$

(b) Boundedness: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. Assuming that $e_{k}^{-} = 〈 {min}_{k}^{P_{e_{k}}}, {min}_{k}^{I_{e_{k}}}, {max}_{k}^{N_{e_{k}}} 〉$ and $e_{k}^{+} = 〈 {max}_{k}^{P_{e_{k}}}, {min}_{k}^{I_{e_{k}}}, {min}_{k}^{N_{e_{k}}} 〉$ for whole k ∈ N, therefore $e_{k}^{-} \subseteq SFNOWAA (e_{1}, e_{2}, \dots, e_{n}) = e_{k}^{+} .$ $e_{k}^{-} \subseteq SFNWAA (e_{1}, e_{2}, \dots, e_{n}) = e_{k}^{+} .$

(c) Monotonically: Let ${\underset{¸}{e}}_{k} = 〈 P_{\underset{¸}{e} k}, I_{\underset{¸}{e} k}, N_{\underset{¸}{e} k} 〉,$ (k ∈ N) be any collection of SFNs. If it satisfies that e_k ⊆ _k for whole k ∈ N, then $SFNWAA (e_{1}, e_{2}, \dots, e_{n}) = SFNWAA ({\underset{¸}{e}}_{1}, {\underset{¸}{e}}_{2}, \dots, {\underset{¸}{e}}_{n}) .$

Definition 9. Let e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 (kɛN) be a collection of SFNs, and SFNOWAA : SFNⁿ → SFN, then the SFNOWAA describe as, ${SFNOWAA}_{w} (e_{1}, e_{2}, \dots, e_{n}) = Π_{k = 1}^{n} τ_{k} e_{η (k)}$

with dimensions n, where kth biggest weighted value is e_η(k) consequently by total order e_η(1) ⩾ e_η(2) ⩾ … ⩾ e_η(n). τ = {τ₁, τ₂, …, τ_n} is the weight vectors such that τ_k ⩾ 0 (k ∈ N) and $Σ_{k = 1}^{n} τ_{k} = 1$ .

Theorem 3. Let e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 (k ∈ N) be a collection of SFNs. Then by utilizing the definition [4.1.4.] and operational properties of SFNs, we can obtain the following outcome. $\begin{matrix} {SFNOWAA}_{w} (e_{1}, e_{2}, \dots, e_{n}) \\ = 〈 \begin{matrix} \sqrt{1 - Π_{k = 1}^{n} (1 - P_{e_{η (k)}}^{2})^{τ_{k}}}, Π_{j = 1}^{n} {(I_{e_{η (k)}})}^{τ_{k}}, \\ Π_{j = 1}^{n} {(N_{e_{η (k)}})}^{τ_{k}} \end{matrix} 〉 \end{matrix}$

where e_η(k) is kth largest value consequently by total order e_η(1) ⩾ e_η(2) ⩾ … ⩾ e_η(n).

Proof. Theorem [4.1.5.] takes the form by utilizing the technique of mathematical induction on n, and procedure is eliminated here.

Properties. There are some properties which are fulfilled by the SFNOWAA operator obviously.

(a) Idempotency: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. If all of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) are identical. Then there is $SFNOWAA (e_{1}, e_{2}, \dots, e_{n}) = e .$

(c) Monotonically: Let ${\underset{¸}{e}}_{k} = 〈 P_{{\underset{¸}{e}}_{k}}, I_{{\underset{¸}{e}}_{k}}, N_{{\underset{¸}{e}}_{k}} 〉,$ (k ∈ N) be any collection of SFNs. If it satisfies that $e_{k} \subseteq {\underset{¸}{e}}_{k}$ for whole k ∈ N, then $SFNWAA (e_{1}, e_{2}, \dots, e_{n}) = SFNOWAA ({\underset{¸}{e}}_{1}, {\underset{¸}{e}}_{2}, \dots, {\underset{¸}{e}}_{n}) .$

The spherical fuzzy weighted averaging operator only considers importance of the aggregated spherical fuzzy sets themselves. The spherical fuzzy OWA operator only concerns with position importance of the ranking order of the aggregated spherical fuzzy sets. To overcome the disadvantages of the aforementioned two spherical fuzzy aggregation operators, we may define the following spherical fuzzy hybrid weighted averaging operator.

Definition 10. Let e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 (kɛN) be a collection of SFNs, and SFNHOWAA : SFNⁿ → SFN, then the SFNHOWAA describe as, ${SFNHOWAA}_{w} (e_{1}, e_{2}, \dots e_{n}) = Π_{k = 1}^{n} τ_{k} e_{η (k)}^{'},$

with dimensions n, where kth biggest weighted value is e_η(k) and $e_{k}^{'} (e_{k}^{'} = n τ_{k} e_{k}, k ɛ N),$ τ = {τ₁, τ₂, …, τ_n} is the weight vectors such that τ_k ⩾ 0 (kɛN) and $Σ_{k = 1}^{n} τ_{k} = 1$ . Also ω = (ω₁, ω₂, …, ω_n) is the associated weight vector such that ω_k ⩾ 0 (k ∈ N) and $Σ_{k = 1}^{n} ω_{k} = 1$ , and balancing coefficient is n.

Theorem 4. Let e_k =〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉 (k ∈ N) be a collection of SFNs. Then by utilizing the definition [4.1.7.] and operational properties of SFNs, we can obtain the following outcome. $\begin{matrix} {SFNHOWAA}_{w} (e_{1}, e_{2}, \dots, e_{n}) = \\ 〈 \begin{matrix} \sqrt{1 - Π_{k = 1}^{n} {(1 - P_{e_{η (k)}^{'}}^{2})}^{ω_{k}}}, \\ Π_{j = 1}^{n} {(I_{e_{η (k)}^{'}})}^{ω_{k}}, Π_{j = 1}^{n} {(N_{e_{η (k)}^{'}})}^{ω_{k}} \end{matrix} 〉 \end{matrix}$

where e_η(k) is kth biggest value consequently by total order e_η(1) ⩾ e_η(2) ⩾ … ⩾ e_η(n) and kth biggest weighted value is $e_{η (k)}^{'}$ , $e_{k}^{'} (e_{k}^{'} = n τ_{k} e_{k}, k ɛ N) .$

Proof. Theorem [4.1.8.] takes the form by utilizing the technique of mathematical induction on n and procedure is eliminated here.

Properties. There are some properties which are fulfilled by the SFNHOWAA operator obviously.

Idempotency: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. If all of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) are identical. Then there is

$SFNHOWAA (e_{1}, e_{2}, \dots, e_{n}) = e .$

Boundedness: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. Assuming that $e_{k}^{-} = 〈 {min}_{k}^{P_{e_{k}}}, {min}_{k}^{I_{e_{k}}}, {max}_{k}^{N_{e_{k}}} 〉$ and $e_{k}^{+} = 〈 {max}_{k}^{P_{e_{k}}}, {min}_{k}^{I_{e_{k}}}, {min}_{k}^{N_{e_{k}}} 〉$ for whole k ∈ N, therefore

$e_{k}^{-} \subseteq SFNHOWAA (e_{1}, e_{2}, \dots, e_{n}) = e_{k}^{+} .$

Monotonically: Let $N_{{\underset{¸}{e}}_{k}} 〉,$ (k ∈ N) be any collection of SFNs. If it satisfies that $e_{k} \subseteq {\underset{¸}{e}}_{k}$ for whole k ∈ N, then

$\begin{matrix} SFNHOWAA (e_{1}, e_{2}, \dots, e_{n}) \\ = SFNHOWAA ({\underset{¸}{e}}_{1}, {\underset{¸}{e}}_{2}, \dots, {\underset{¸}{e}}_{n}) . \end{matrix}$

4.2. Spherical fuzzy number weighted geometric aggregation operators

This section described the spherical weighted geometric aggregation (SFNWGA) operators by utilizing the defined operational properties of SFNs.

Definition 11. Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs and SFNWGA : SFNⁿ → SFN, then SFNWGA operator describe as, $SFNWGA (e_{1}, e_{2}, \dots, e_{n}) = \prod_{k = 1}^{n} e_{k}^{τ_{k}},$

In which τ ={ τ₁, τ₂, …, τ_n } be the weight vector of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, k ∈ N, with τ_k ⩾ 0 and $\sum_{k = 1}^{n} τ_{k} = 1 .$

Theorem 5. Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. Then by utilizing the Definition [4.2.1.] and operational properties of SFNs, we can obtain the following outcome. $\begin{matrix} SFNWGA (e_{1}, e_{2}, \dots, e_{n}) \\ = 〈 \begin{matrix} Π_{k = 1}^{n} (P_{e_{k}})^{τ_{k}}, Π_{k = 1}^{n} (I_{e_{k}})^{τ_{k}}, \\ \sqrt{1 - Π_{k = 1}^{n} (1 - N_{e_{k}}^{2})^{τ_{k}}} \end{matrix} 〉, \end{matrix}$

Where τ ={ τ₁, τ₂, …, τ_n } be the weight vector of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, k ∈ N, with τ_k ⩾ 0 and $\sum_{k = 1}^{n} τ_{k} = 1 .$

Proof. We done the prove by utilizing the technique of mathematical induction. Therefore, we follow as

(a) For n = 2, since $e_{1}^{τ_{1}} = 〈 {(P_{e_{1}})}^{τ_{1}}, {(I_{e_{1}})}^{τ_{1}}, \sqrt{(1 - (1 - N_{e_{1}}^{2})^{τ_{1}}}) 〉$

and $e_{2}^{τ_{2}} = 〈 (P_{e_{2}})^{τ_{2}}, {(I_{e_{2}})}^{τ_{2}}, \sqrt{(1 - (1 - N_{e_{2}}^{2})^{τ_{2}}}) 〉$

Then $\begin{matrix} \begin{matrix} SFNWGA (e_{1}, e_{2}) = e_{1}^{τ_{1}} + e_{2}^{τ_{2}} \\ = 〈 {(P_{e_{1}})}^{τ_{1}}, {(I_{e_{1}})}^{τ_{1}}, \sqrt{(1 - (1 - N_{e_{1}}^{2})^{τ_{1}}}) 〉 \\ + 〈 (P_{e_{2}})^{τ_{2}}, {(I_{e_{2}})}^{τ_{2}}, \sqrt{(1 - (1 - N_{e_{2}}^{2})^{τ_{2}}}) 〉 \\ = 〈 (P_{e_{1}})^{τ_{1}} \cdot (P_{e_{2}})^{τ_{2}}, (I_{e_{1}})^{τ_{1}} \cdot (I_{e_{2}})^{τ_{2}}, \\ \sqrt{\begin{matrix} (1 - (1 - N_{e_{1}}^{2})^{τ_{1}}) + (1 - (1 - N_{e_{2}}^{2})^{τ_{2}}) \\ - (1 - (1 - N_{e_{1}}^{2})^{τ_{1}}) (1 - (1 - N_{e_{2}}^{2})^{τ_{2}}) \end{matrix}} 〉 \\ = 〈 (P_{e_{1}})^{τ_{1}} \cdot (P_{e_{2}})^{τ_{2}}, (I_{e_{1}})^{τ_{1}} \cdot (I_{e_{2}})^{τ_{2}}, \\ \sqrt{1 - {(1 - N_{e_{1}}^{2})}^{τ_{1}} {(1 - N_{e_{2}}^{2})}^{τ_{2}}} 〉 \\ = 〈 Π_{k = 1}^{2} {(P_{e_{k}})}^{τ_{k}}, Π_{k = 1}^{2} {(I_{e_{k}})}^{τ_{k}}, \\ \sqrt{1 - Π_{k = 1}^{2} (1 - N_{e_{k}}^{2})^{τ_{k}}} 〉 \end{matrix} \end{matrix}$

(b) Suppose that outcome is true for n = z that is, $\begin{matrix} SFNWGA (e_{1}, e_{2}, \dots, e_{z}) \\ = 〈 Π_{k = 1}^{z} {(P_{e_{k}})}^{τ_{k}}, Π_{k = 1}^{z} {(I_{e_{k}})}^{τ_{k}}, \\ \sqrt{1 - Π_{k = 1}^{z} (1 - N_{e_{k}}^{2})^{τ_{k}}} 〉 \end{matrix}$

(c) Now we have to prove that outcome is true for n = z + 1, by utilizing the (a) & (b) we have $\begin{matrix} SFNWGA (e_{1}, e_{2}, \dots, e_{z}, e_{z + 1}) = \sum_{k = 1}^{z} e_{k}^{τ_{k}} + e_{z + 1}^{τ_{z + 1}} \\ = 〈 Π_{k = 1}^{z} {(P_{e_{k}})}^{τ_{k}}, Π_{k = 1}^{z} {(I_{e_{k}})}^{τ_{k}}, \sqrt{1 - Π_{k = 1}^{z} (1 - N_{e_{k}}^{2})^{τ_{k}}} 〉 + \\ 〈 {(P_{e_{1}})}^{τ_{1}}, {(I_{e_{1}})}^{τ_{1}}, \sqrt{(1 - (1 - N_{e_{1}}^{2})^{τ_{1}}}) 〉 \\ = 〈 \begin{matrix} Π_{k = 1}^{z} {(P_{e_{k}})}^{τ_{k}} \cdot {(P_{e_{z + 1}})}^{τ_{z + 1}}, Π_{k = 1}^{z} {(I_{e_{k}})}^{τ_{k}} \cdot {(I_{e_{z + 1}})}^{τ_{z + 1}}, \\ \sqrt{\begin{matrix} (1 - Π_{k = 1}^{z} (1 - N_{e_{k}}^{2})^{τ_{k}}) + (1 - (1 - N_{e_{z + 1}}^{2})^{τ_{z + 1}}) \\ - (1 - Π_{k = 1}^{z} (1 - N_{e_{k}}^{2})^{τ_{k}}) (1 - (1 - N_{e_{z + 1}}^{2})^{τ_{z + 1}}) \end{matrix}} \end{matrix} 〉 \\ = 〈 \begin{matrix} Π_{k = 1}^{z} {(P_{e_{k}})}^{τ_{k}} \cdot {(P_{e_{z + 1}})}^{τ_{z + 1}}, Π_{k = 1}^{z} {(I_{e_{k}})}^{τ_{k}} \cdot {(I_{e_{z + 1}})}^{τ_{z + 1}}, \\ \sqrt{1 - Π_{k = 1}^{z} (1 - N_{e_{k}}^{2})^{τ_{k}} (1 - N_{e_{z + 1}}^{2})^{τ_{z + 1}}} \end{matrix} 〉 \\ = 〈 Π_{k = 1}^{z + 1} {(P_{e_{k}})}^{τ_{k}}, Π_{k = 1}^{z + 1} {(I_{e_{k}})}^{τ_{k}}, \sqrt{1 - Π_{k = 1}^{z + 1} (1 - N_{e_{k}}^{2})^{τ_{k}}} 〉 \end{matrix}$

Outcome is satisfying for n = z + 1. Thus, outcome is satisfied for whole n. Therefore $\begin{matrix} SFNWGA (e_{1}, e_{2}, \dots, e_{n}) \\ = 〈 Π_{k = 1}^{n} (P_{e_{k}})^{τ_{k}}, Π_{k = 1}^{n} (I_{e_{k}})^{τ_{k}}, \\ \sqrt{1 - Π_{k = 1}^{n} (1 - N_{e_{k}}^{2})^{τ_{k}}} 〉 \end{matrix}$

which done the proof.

Properties. There are some properties which are fulfilled by the SFNWGA operator obviously.

(a) Idempotency: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. If all of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) are identical. Then there is $SFNWGA (e_{1}, e_{2}, \dots, e_{n}) = e .$

(c) Monotonically: Let ${\underset{¸}{e}}_{k} = 〈 P_{{\underset{¸}{e}}_{k}}, I_{{\underset{¸}{e}}_{k}}, N_{{\underset{¸}{e}}_{k}} 〉,$ (k ∈ N) be any collection of SFNs. If it satisfies that $e_{k} \subseteq {\underset{¸}{e}}_{k}$ for whole k ∈ N, then $\begin{matrix} SFNWGA (e_{1}, e_{2}, \dots, e_{n}) \\ = SFNWGA ({\underset{¸}{e}}_{1}, {\underset{¸}{e}}_{2}, \dots, {\underset{¸}{e}}_{n}) . \end{matrix}$

Definition 12. Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs and SFNOWGA : SFNⁿ → SFN, then SFNOWGA operator describe as, $S F N O W G A_{w} (e_{1}, e_{2}, …, e_{n}) = \prod_{k = 1}^{n} e_{η (k)}^{τ_{k}},$

with dimensions n, where kth biggest weighted value is e_η(k) consequently by total order e_η(1) ⩾ e_η(2) ⩾ … ⩾ e_η(n). τ ={ τ₁, τ₂, …, τ_n } be the weight vector of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, k ∈ N, with τ_k ⩾ 0 and $\sum_{k = 1}^{n} τ_{k} = 1 .$

Theorem 6. Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. Then by utilizing the definition [4.2.4.] and operational properties of SFNs, we can obtain the following outcome. $\begin{matrix} {SFNOWGA}_{w} (e_{1}, e_{2}, \dots, e_{n}) \\ = 〈 \begin{matrix} Π_{k = 1}^{n} {(P_{e_{η (k)}})}^{τ_{k}}, Π_{k = 1}^{n} (I_{e_{η (k)}})^{τ_{k}}, \\ \sqrt{1 - Π_{k = 1}^{n} (1 - N_{e_{η (k)}}^{2})^{τ_{k}}} \end{matrix} 〉 \end{matrix}$

where e_η(k) is kth largest value consequently by total order e_η(1) ⩾ e_η(2) ⩾ … ⩾ e_η(n).

Proof. Theorem [4.2.5.] takes the form by utilizing the technique of mathematical induction on n, and procedure is eliminated here.

Properties. There are some properties which are fulfilled by the SFNOWGA operator obviously.

Idempotency: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. If all of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) are identical. Then there is

$SFNOWGA (e_{1}, e_{2}, \dots, e_{n}) = e .$

Boundedness: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of FNs. Assuming that $e_{k}^{-} = 〈 {min}_{k}^{P_{e_{k}}}, {min}_{k}^{I_{e_{k}}}, {max}_{k}^{N_{e_{k}}} 〉$ and $e_{k}^{+} = 〈 {max}_{k}^{P_{e_{k}}}, {min}_{k}^{I_{e_{k}}}, {min}_{k}^{N_{e_{k}}} 〉$ for whole k ∈ N, therefore

$e_{k}^{-} \subseteq SFNOWGA (e_{1}, e_{2}, \dots, e_{n}) = e_{k}^{+} .$

Monotonically: Let $N_{{\underset{¸}{e}}_{k}} 〉,$ (k ∈ N) be any collection of SFNs. If it satisfies that $e_{k} \subseteq {\underset{¸}{e}}_{k}$ for whole k ∈ N, then

$\begin{matrix} SFNOWGA (e_{1}, e_{2}, \dots, e_{n}) \\ = SFNOWGA ({\underset{¸}{e}}_{1}, {\underset{¸}{e}}_{2}, \dots, {\underset{¸}{e}}_{n}) . \end{matrix}$

Definition 13. Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs and SFNHOWGA : SFNⁿ → SFN, then SFNOWGA operator describe as, ${SFNHOWGA}_{w} (e_{1}, e_{2}, \dots e_{n}) = Π_{j = 1}^{n} e_{η (k)}^{' τ_{k}},$

with dimensions n, where kth biggest weighted value is e_η(k) and $e_{k}^{'} (e_{k}^{'} = n τ_{k} e_{k}, k ɛ N),$ τ ={ τ₁, τ₂, …, τ_n } be the weight vector of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, k ∈ N, with τ_k ⩾ 0 and $\sum_{k = 1}^{n} τ_{k} = 1 .$ Also ω = (ω₁, ω₂, …, ω_n) is the associated weight vector such that ω_j ⩾ 0 (jɛN) and $Σ_{j = 1}^{n} ω_{j} = 1$ , and balancing coefficient is n.

Theorem 7. Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. Then by utilizing the definition [4.2.7.] and operational properties of SFNs, we can obtain the following outcome. $\begin{matrix} {SFNHOWGA}_{w} (e_{1}, e_{2}, \dots, e_{n}) \\ = 〈 \begin{matrix} Π_{k = 1}^{n} {(P_{e_{η (k)}^{'}})}^{ω_{k}}, Π_{k = 1}^{n} {(I_{e_{η (k)}^{'}})}^{ω k}, \\ \sqrt{1 - Π_{k = 1}^{n} {(1 - N_{e_{η (k)}^{'}}^{2})}^{ω_{k}}} \end{matrix} 〉 \end{matrix}$

where e_η(k) is kth biggest value consequently by total order e_η(1) ⩾ e_η(2) ⩾ … ⩾ e_η(n) and kth biggest weighted value is $e_{η (k)}^{'}$ , $e_{k}^{'} (e_{k}^{'} = n τ_{k} e_{k}, k ɛ N) .$

Proof. Theorem [4.2.8.] takes the form by utilizing the technique of mathematical induction on n. and procedure is eliminated here.

Properties. There are some properties which are fulfilled by the SFNHOWGA operator obviously.

Idempotency: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of SFNs. If all of e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) are identical. Then there is

$SFNHOWGA (e_{1}, e_{2}, \dots, e_{n}) = e .$

(b) Boundedness: Let e_k = 〈 P_{e
_k}, I_{e
_k}, N_{e
_k} 〉, (k ∈ N) be any collection of FNs. Assuming that $e_{k}^{-} = 〈 {min}_{k}^{P_{e_{k}}}, {min}_{k}^{I_{e_{k}}}, {max}_{k}^{N_{e_{k}}} 〉$ and $e_{k}^{+} = 〈 {max}_{k}^{P_{e_{k}}}, {min}_{k}^{I_{e_{k}}}, {min}_{k}^{N_{e_{k}}} 〉$ for whole k ∈ N, therefore $e_{k}^{-} \subseteq SFNHOWGA (e_{1}, e_{2}, \dots, e_{n}) = e_{k}^{+} .$

(c) Monotonically: Let ${\underset{¸}{e}}_{k} = 〈 P_{{\underset{¸}{e}}_{k}}, I_{{\underset{¸}{e}}_{k}}, N_{{\underset{¸}{e}}_{k}} 〉,$ (k ∈ N) be any collection of SFNs. If it satisfies that $e_{k} \subseteq {\underset{¸}{e}}_{k}$ for whole k ∈ N, then $\begin{matrix} SFNHOWGA (e_{1}, e_{2}, \dots, e_{n}) \\ = SFNHOWGA ({\underset{¸}{e}}_{1}, {\underset{¸}{e}}_{2}, \dots, {\underset{¸}{e}}_{n}) . \end{matrix}$

5. MADM method utilizing spherical aggregation operators

This section proposes the technique to solve the MADM problems by utilizing the spherical aggregation operators. For a MADM problem, assuming that C ={ c₁, c₂, …, c_m } be any finite collection of m alternatives, G ={ g₁, g₂, …, g_n } be any finite collection of n attributes and T ={ t₁, t₂, …, t_p } be any collection of p DMs. If the zth (z = 1, 2, …, p) DM deliver the assessment of the alternative c_i (i = 1, 2, …, m) on the attribute g_i (i = 1, 2, …, n) under any discrete term set. Let $B = [b_{jk}] = {[〈 P_{e_{jk}}, I_{e_{jk}}, N_{e_{jk}} 〉]}_{m \times n}$ $\begin{matrix} g_{1} g_{2} \dots g_{n} \\ c_{1} 〈 P_{e_{11}}, I_{e_{11}}, N_{e_{11}} 〉〈 P_{e_{12}}, I_{e_{12}}, N_{e_{12}} 〉 \dots 〈 P_{e_{1 n}}, I_{e_{1 n}}, N_{e_{1 n}} 〉 \\ c_{2} 〈 P_{e_{21}}, I_{e_{21}}, N_{e_{21}} 〉〈 P_{e_{22}}, I_{e_{22}}, N_{e_{22}} 〉 \dots 〈 P_{e_{2 n}}, I_{e_{2 n}}, N_{e_{2 n}} 〉 \\ . . . . . \\ . . . . . \\ . . . . . \\ c_{m} 〈 P_{e_{m 1}}, I_{e_{m 1}}, N_{e_{m 1}} 〉〈 P_{e_{m 2}}, I_{e_{m 2}}, N_{e_{m 2}} 〉 \dots 〈 P_{e_{mn}}, I_{e_{mn}}, N_{e_{mn}} 〉 \end{matrix}$

be the DM, where 〈P_{e
_jk}, I_{e
_jk}, N_{e
_jk}〉 are the collection of SFNs and represents the evaluation information of every alternative c_i (i = 1, 2, …, m) on attribute g_i (i = 1, 2, …, n). If τ ={ τ₁, τ₂, …, τ_n } be the weight vector of attribute, with τ_k ⩾ 0, $Σ_{k = 1}^{n} τ_{k} = 1$ and the weight vector of DMs is Q = (q₁, q₂, …, q_p), with q_k ⩾ 0 and $Σ_{k = 1}^{p} q_{k} = 1 .$

Then, listed below the main technique of handling the MADM problems:

Step 1. In extensively, we have two kinds of criterion one is said to be positive criteria and other one said to be negative criteria. For negative criterion, we need to modify the negative criteria into positive criteria. In this step we construct the spherical fuzzy decision making matrices, $D^{s} = {[b_{jk}^{s}]}_{m \times n}$ (s = 1, 2, … n) for decision. If the criteria have two types, such as positive (benefit) criteria and negative (cost) criteria, then the spherical fuzzy decision matrices, $D^{s} = {[b_{jk}^{s}]}_{m \times n}$ can be converted into the normalized spherical fuzzy decision matrices, R $R^{s} = {[r_{jk}^{s}]}_{m \times n}$ (s = 1, 2, … n) where $r_{jk}^{s} = {\begin{matrix} b_{jk}^{s} & for benefit criteria \\ \bar{b_{jk}^{s}} & for cost criteria \end{matrix}$ and $\bar{b_{jk}^{s}}$ is represents the compliment of the $b_{jk}^{s}$ . If all the criteria have the same type, then there is no need of normalization.

Step 2. We accumulate the SFNs for every decision maker which are given, then use the SFNWAA and SFNWGA operators, discussed in Theorem [4.1.2. & 4.2.2.] to aggregate the spherical fuzzy information. Which help us to choose the best alternate in the set of alternatives.

Step 3. Calculating the score, accuracy and certainty function values respectively by utilizing the Definition [3.2.1.] to helpful for ranking the given alternatives.

Step 4. Arrange the values calculated by utilizing the comparison technique in Definition [3.1.2.] of the all alternatives in the form of ascending order and select that alternative which has the highest value. Alternative which has the highest value that will be our finest result or suitable alternative according to decision makers.

5.1. A Numerical example

This section describes a MADM problem, which is utilized to illustrate the pertinence and effectiveness for the procedure of decision making problems.

There are four manageable alternatives, (a) c₁ is car company; (b) c₂ is food company; (c) c₃ is a computer company; (d) c₄ is an arms company. According to the attributes company takes the decision, (a) g₁ is the risk; (b) g₂ is the growth; (c) g₃ is the environmental impact. The weight vector of the attributes is τ = (0.35, 0.25, 0.40), and corresponding associated weight are ω = { 0.5, 0.2, 0.3 }. Now we can calculate the following spherical fuzzy number decision matrix as $\begin{matrix} g_{1} g_{2} g_{3} \\ c_{1} 〈 0.6, 0.6, 0.2 〉〈 0.8, 0.4, 0.1 〉〈 0.8, 0.2, 0.3 〉 \\ c_{2} 〈 0.8, 0.3, 0.2 〉〈 0.5, 0.4, 0.3 〉〈 0.4, 0.3, 0.4 〉 \\ c_{3} 〈 0.4, 0.8, 0.2 〉〈 0.6, 0.5, 0.2 〉〈 0.5, 0.3, 0.6 〉 \\ c_{4} 〈 0.7, 0.1, 0.7 〉〈 0.7, 0.5, 0.1 〉〈 0.7, 0.3, 0.5 〉 \end{matrix}$

Step 1. Since the attributes are same type, so there is no need to be normalized.

Step 2. (Case:1) Utilizing the SFNWAA Operator to find out every value of the alternative c_i as, $\begin{matrix} e_{1} = 〈 0.748, 0.349, 0.197 〉 \\ e_{2} = 〈 0.627, 0.322, 0.292 〉 \\ e_{3} = 〈 0.499, 0.480, 0.310 〉 \\ e_{4} = 〈 0.700, 0.232, 0.376 〉 \end{matrix}$

(Case:2) Utilizing the SFNOWAA Operator to find out every value of the alternative as, Firstly we need score function of the each alternative $\begin{matrix} g_{1} g_{2} g_{3} \\ c_{1} sc (c_{1} g_{1}) = 0.60 sc (c_{1} g_{2}) = 0.76 sc (c_{1} g_{3}) = 0.76 \\ c_{2} sc (c_{2} g_{1}) = 0.76 sc (c_{2} g_{2}) = 0.60 sc (c_{2} g_{3}) = 0.56 \\ c_{3} sc (c_{3 g_{1}}) = 0.46 sc (c_{3} g_{2}) = 0.63 sc (c_{3} g_{3}) = 0.53 \\ c_{4} sc (c_{4 g_{1}}) = 0.63 sc (c_{4} g_{2}) = 0.70 sc (c_{4} g_{3}) = 0.63 \end{matrix}$

Now we rank the alternative in descending order according to the score values as, $\begin{matrix} g_{1} g_{2} g_{3} \\ e_{η_{(1)}} 〈 0.8, 0.3, 0.2 〉〈 0.8, 0.4, 0.1 〉〈 0.8, 0.2, 0.3 〉 \\ e_{η_{(2)}} 〈 0.7, 0.1, 0.7 〉〈 0.7, 0.5, 0.1 〉〈 0.7, 0.3, 0.5 〉 \\ e_{η_{(3)}} 〈 0.6, 0.6, 0.2 〉〈 0.6, 0.5, 0.2 〉〈 0.4, 0.3, 0.4 〉 \\ e_{η_{(4)}} 〈 0.4, 0.8, 0.2 〉〈 0.5, 0.4, 0.3 〉〈 0.5, 0.3, 0.6 〉 \end{matrix}$

Now by utilizing the SFNOWAA Operator to find out every values of the alternative as, $\begin{matrix} e_{1} = 〈 0.800, 0.274, 0.197 〉 \\ e_{2} = 〈 0.700, 0.232, 0.376 〉 \\ e_{3} = 〈 0.535, 0.434, 0.263 〉 \\ e_{4} = 〈 0.468, 0.454, 0.343 〉 \end{matrix}$

(Case:3) Utilizing the SFNHOWAA Operator to find out every value of the alternative as, Firstly we need an order decision matrices. by utilizing following formula $e_{η (k)}^{'}$ , $e_{k}^{'} (e_{k}^{'} = n τ_{k} e_{k}, k ɛ N) .$ where n is the dimension of alternative and (τ_k : k ∈ N) are corresponding weights. $\begin{matrix} g_{1} g_{2} g_{3} \\ e_{η (1)}^{'} 3 \times 0.5 \times 〈 0.6, 0.6, 0.2 〉 3 \times 0.2 \times 〈 0.8, 0.4, 0.1 〉 3 \times 0.3 \times 〈 0.8, 0.2, 0.3 〉 \\ e_{η (2)}^{'} 3 \times 0.5 \times 〈 0.8, 0.3, 0.2 〉 3 \times 0.2 \times 〈 0.5, 0.4, 0.3 〉 3 \times 0.3 \times 〈 0.4, 0.3, 0.4 〉 \\ e_{η (3)}^{'} 3 \times 0.5 \times 〈 0.4, 0.8, 0.2 〉 3 \times 0.2 \times 〈 0.6, 0.5, 0.2 〉 3 \times 0.3 \times 〈 0.5, 0.3, 0.6 〉 \\ e_{η (4)}^{'} 3 \times 0.5 \times 〈 0.7, 0.1, 0.7 〉 3 \times 0.2 \times 〈 0.7, 0.5, 0.1 〉 3 \times 0.3 \times 〈 0.7, 0.3, 0.5 〉 \end{matrix}$

We multiply these scaler as in define multiplication $\begin{matrix} g_{1} g_{2} g_{3} \\ e_{η (1)}^{'} 〈 0.69, 0.46, 0.08 〉〈 0.67, 0.57, 0.25 〉〈 0.77, 0.23, 0.33 〉 \\ e_{η (2)}^{'} 〈 0.88, 0.16, 0.08 〉〈 0.39, 0.57, 0.48 〉〈 0.38, 0.33, 0.43 〉 \\ e_{η (3)}^{'} 〈 0.47, 0.71, 0.08 〉〈 0.48, 0.65, 0.38 〉〈 0.47, 0.33, 0.63 〉 \\ e_{η (4)}^{'} 〈 0.79, 0.03, 0.58 〉〈 0.57, 0.65, 0.25 〉〈 0.67, 0.33, 0.53 〉 \end{matrix}$

Now by utilizing the SFNHOWAA Operator to find out every value of the alternative as, $\begin{matrix} e_{1} = 〈 0.721, 0.367, 0.187 〉 \\ e_{2} = 〈 0.681, 0.293, 0.245 〉 \\ e_{3} = 〈 0.472, 0.511, 0.269 〉 \\ e_{4} = 〈 0.702, 0.168, 0.453 〉 \end{matrix}$

Step 3. (Case:1) Now we find out the score, accuracy and certainty function values respectively by utilizing the Definition [3.2.1.] as, $\begin{matrix} sc (e_{1}) = 0.73 & ac (e_{1}) = 0.55 cr (e_{1}) = 0.75 \\ sc (e_{2}) = 0.66 ac (e_{2}) = 0.34 cr (e_{2}) = 0.63 \\ sc (e_{3}) = 0.57 ac (e_{3}) = 0.19 cr (e_{3}) = 0.50 \\ sc (e_{4}) = 0.70 ac (e_{4}) = 0.32 cr (e_{4}) = 0.70 \end{matrix}$

(Case:2) Now we find out the score, accuracy and certainty function values respectively by utilizing the Definition [3.2.1.] as, $\begin{matrix} sc (e_{1}) = 0.77 & ac (e_{1}) = 0.60 cr (e_{1}) = 0.80 \\ sc (e_{2}) = 0.69 ac (e_{2}) = 0.32 cr (e_{2}) = 0.70 \\ sc (e_{3}) = 0.61 ac (e_{3}) = 0.27 cr (e_{3}) = 0.53 \\ sc (e_{4}) = 0.55 ac (e_{4}) = 0.12 cr (e_{4}) = 0.46 \end{matrix}$

(Case:3) Now we find out the score, accuracy and certainty function values respectively by utilizing the Definition [3.2.1.] as, $\begin{matrix} sc (e_{1}) = 0.72 & ac (e_{1}) = 0.53 cr (e_{1}) = 0.72 \\ sc (e_{2}) = 0.71 ac (e_{2}) = 0.43 cr (e_{2}) = 0.68 \\ sc (e_{3}) = 0.56 ac (e_{3}) = 0.20 cr (e_{3}) = 0.47 \\ sc (e_{4}) = 0.69 ac (e_{4}) = 0.24 cr (e_{4}) = 0.70 \end{matrix}$

Step 4. (Case:1) Now rank the all the alternative by utilizing the comparison technique in Definition [3.2.2.] for SFNWAA operator as, $\begin{matrix} sc (e_{1}) = 0.73 > sc (e_{4}) & = & 0.70 > sc (e_{2}) \\ = & 0.66 > sc (e_{3}) = 0.57 . \end{matrix}$

Thus, according to the scoring function ranking method of spherical fuzzy sets for SFNWAA operator, the ranking order of the suppliers e_i (i = 1, 2, 3, 4) is generated as follows: e₁ > e₄ > e₂ > e₃. The best supplier for the enterprise is e₁.

(Case:2) Now rank the all the alternative by utilizing the comparison technique in Definition [3.2.2.] for SFNOWAA operator as, $\begin{matrix} sc (e_{1}) = 0.77 > sc (e_{2}) & = & 0.69 > sc (e_{3}) \\ = & 0.61 > sc (e_{4}) = 0.55 \end{matrix}$

so, by utilizing the Definition [3.2.2.], we obtained the result which is $e_{1} > e_{2} > e_{3} > e_{4}$

and by using the SFNOWAA, e₁ is the best choose.

(Case:3) Now rank the all the alternative by utilizing the comparison technique in Definition [3.2.2.] for SFNHOWAA operator as,

sc (e₁) =0.72 > sc (e₂) =0.71 > sc (e₄) =0.66 > sc (e₃) =0.56 we obtained the result which is $e_{1} > e_{2} > e_{4} > e_{3}$

and by using the SFNHOWAA, e₁ is the best choose. $\begin{matrix} Averaging AggregationOperators \\ Geometric AggregationOperators \\ Weighted e_{1} > e_{4} > e_{2} > e_{3} e_{1} > e_{4} > e_{2} > e_{3} \\ Ordered e_{1} > e_{2} > e_{3} > e_{4} e_{1} > e_{2} > e_{3} > e_{4} \\ Hybrid Ordered e_{1} > e_{2} > e_{4} > e_{3} e_{1} > e_{2} > e_{4} > e_{3} \end{matrix}$

It is observing that using proposed aggregation operators of SFSs to aggregate the spherical fuzzy information. To choose the best alternative which is e₁ from the set of alternatives. As SFSs is generalized as compared to FSs, IFSs, PyFS and PFSs and have the ability to deal with real life problems more effectively than the existing concepts as it is discussed to evaluation investment company to invest money problem is effectively applied.

5.2. Validity and reliability test

In particularly, to track the best capable alternative from given group decision matrices are not probable in practically. Wang and Triantaphyllou [26] initiated the test to evaluate the reliability and validity of the MADM techniques. These testing criteria are follow as

Test Criteria No:1. An operative and valid MADM technique is that if we are substituting a non-optimal alternative in worse alternative by showing without any change then this indication of the best capable alternative without shifting the comparative status of each decision criteria.

Test Criteria No:2. An operative and valid MADM method should be satisfied the transitive property.

Test Criteria No:3. When an MCGDM problem is transformed into smaller problems. To rank the alternative, we apply similar technique on smaller problems which utilized in MADM problem, a collective alternative rank should be alike with original rank of un-decomposed problem.

The reliability and validity of the suggested aggregation operator constructed by MADM technique is test by utilizing these testing criteria.

Validity and Reliability Test for suggested approach utilizing Test Criteria No:1.

To examine effectiveness of the suggested approach utilizing test criterion no 1, then these decision matrices are attained by exchanging the degree of the membership (non-optimal alternative) and non-membership (worse alternative) in the original decision matrices as follows, after that original decision matrices are transform $\begin{matrix} 〈 0.6, 0.6, 0.2 〉 & 〈 0.1, 0.4, 0.8 〉〈 0.8, 0.2, 0.3 〉 \\ 〈 0.8, 0.3, 0.2 〉〈 0.3, 0.4, 0.5 〉〈 0.4, 0.4, 0.4 〉 \\ 〈 0.2, 0.8, 0.4 〉〈 0.6, 0.5, 0.2 〉〈 0.6, 0.3, 0.5 〉 \\ 〈 0.7, 0.1, 0.7 〉〈 0.7, 0.5, 0.1 〉〈 0.5, 0.3, 0.7 〉 \end{matrix}$

Step 1. Since the attributes are uniform so there is no need to normalize.

Step 2. Utilizing the SFNWAA operator to find out every value of the alternative c_i as, $\begin{matrix} e_{1} = 〈 0.658, 0.349, 0.332 〉 \\ e_{2} = 〈 0.602, 0.361, 0.331 〉 \\ e_{3} = 〈 0.512, 0.480, 0.367 〉 \\ e_{4} = 〈 0.636, 0.232, 0.430 〉 \end{matrix}$

Step 3. Now we find out the score, accuracy and certainty function values respectively by utilizing the Definition [3.2.1.] as, $\begin{matrix} sc (e_{1}) = 0.66 & ac (e_{1}) = 0.32 cr (e_{1}) = 0.65 \\ sc (e_{2}) = 0.63 ac (e_{2}) = 0.27 cr (e_{2}) = 0.60 \\ sc (e_{3}) = 0.55 ac (e_{3}) = 0.14 cr (e_{3}) = 0.51 \\ sc (e_{4}) = 0.65 ac (e_{4}) = 0.20 cr (e_{4}) = 0.63 \end{matrix}$

Step 4. Now rank the all the alternative by utilizing the comparison technique in Definition [3.2.2.] for SFNWAA operator as,

sc (e₁) =0.66 > sc (e₄) =0.65 > sc (e₂) =0.63 > sc (e₃) =0.55 we obtained the result which is $e_{1} > e_{4} > e_{2} > e_{3}$

and by using the SFNWAA, e₁ is the best choose.

Validity and Reliability Test for suggested approach utilizing Test Criteria No:2 & Test Criteria 3.

To test validity and reliability of suggested technique by utilizing test criterion 2 and 3, we transformed MADM technique into three smaller sub-problems as {e₁, e₂, e₃}, {e₂, e₃, e₄} and {e₂, e₄, e₁}. Then by following stages of the suggested technique, accordingly to each sub-problem we suggest rank to their correspondences as e₁ > e₄ > e₂, e₄ > e₂ > e₃ and e₁ > e₂ > e₃ respectively. After aggregating the inclusive rank, we attain e₁ > e₄ > e₂ > e₃ that is similar with original MADM technique. Hence it displays the transitive property. So, under test criterion 2 and 3 recognized by Wang and Triantaphyllou [26] the suggested technique is valid and reliable.

5.3. Comparative analysis

In order to verify the validity and effectiveness of the suggested techniques, a comparative approach is conductive using Pythagorean fuzzy TOPSIS method [30] and aggregation operators of PFS [12].

A Comparison analysis with the Pythagorean fuzzy TOPSIS.

These two approaches are valid to solve the MADM problems, TOPSIS method which proposed by Hwang and Yoon [13] simple and useful method to solve the MADM problems with crisp numbers, which aims at choosing the alternative with the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS).

According to Zhang and Xu [33], the first step is to identify the Pythagorean fuzzy positive ideal solution (PIS) and the Pythagorean fuzzy negative ideal solution (NIS) of the decision matrix, which are $\begin{matrix} P^{+} = {〈 0.7665, 0.4530 〉, 〈 0.8185, 0.4125 〉, \\ 〈 0.7635, 0.6165 〉, 〈 0.8176, 0.5144 〉} \end{matrix}$ $\begin{matrix} P^{-} = {〈 0.5375, 0.7254 〉, 〈 0.4976, 0.7996 〉, \\ 〈 0.4124, 0.7598 〉, 〈 0.4884, 0.8439 〉} \end{matrix}$

The bigger is ζ_i is, the better the alternatives and the most desirable alternative is ζ₂ as given below.

ζ₁ = 0 : 5865; ζ₂ = 0 : 8115; ζ₃ = 0 : 4758; ζ₄ = 0 : 4164; ζ₅ = 0 : 5700

The decision matrix utilized in this paper are not be solve as this above technique because in this paper we utilize positive, neutral and negative membership degrees that’s square sum is less or equal to 1. Pythagorean fuzzy TOPSIS method fail to solve such problems. So, our proposed technique is reliable and valid for such types of alternatives.

A Comparison analysis with the Picture fuzzy Sets.

Picture fuzzy set initialized by B. C. Cuong [9] and Picture fuzzy aggregation operators which were introduced by Harish Garg [12] are listed as, (1) Picture fuzzy weighted averaging operator, (2) Picture fuzzy ordered weighted averaging operator, (3) Picture fuzzy hybrid averaging operator. By utilizing these aggregation operators, a decision making technique proposed for picture fuzzy environment. finally chose the best one alternative from decision matrices.

The above proposed aggregation operators for PFS deals with that real-life problem where sum of membership degrees are equals to 1. In situation where sum of membership degrees exceeds 1, PFS fails to give satisfactory results in such situation. On the other hand, technique proposed in this paper can give satisfactory results in such situations where sum of membership degrees is greater than 1, and chose the best alternative from decision making problems.

6. Conclusion

Information aggregation procedure plays a dynamic role throughout the decision making procedure and hence in this track, the comparative importance of the criteria remains unreformed in modified problems, then we proposed SFNWAA and SFNWGA operators has been executed to invention the best alternative. Almost all the researchers have operated the IFS by considering the positive and negative membership degrees only. But, it has been detected that in some circumstances, like in situation of voting, human thoughts including more degrees as, yes, neutral, no, refusal, and in situation where the sum of membership degree is greater than 1, which cannot be exactly characterized in IFS and PFS respectively. For this circumstance, Spherical fuzzy set, which is an extension of the PFS, has been utilized in the paper and correspondingly aggregation operators have been defined and used. Several required properties of these operators have also been explored in comprehensibly. Finally, based on these operators, a decision-making method has been established for ranking the dissimilar alternatives by utilizing spherical fuzzy environment. The suggested technique has been demonstrated with a descriptive example for viewing their effectiveness as well as reliability. A test to check the reliability and validity has also been conducted for viewing the supremacy of the suggested technique. Thus, the suggested operators provide a new direction to the information measure theory and give a new easier track to grip the uncertainties throughout the decision-making procedure.

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