# Spherical Fuzzy Information Aggregation Based on Aczel–Alsina Operations and Data Analysis for Supply Chain

Authors: Muhammad Riaz; Hafiz Muhammad Athar Farid; Dragan Pamucar; Shaista Tanveer
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Spherical fuzzy sets (SFSs) are often made up of membership, nonmembership, and hesitancy grades, and also have the advantage of accurately representing decision makers (DMs) preferences. This article proposes novel spherical fuzzy aggregation operators (AOs) based on Aczel–Alsina (AA) operations, which offer a lot of advantages when tackling real-world situations. We begin by introducing some new SFS operations, such as the Aczel–Alsina product, the Aczel–Alsina sum, the Aczel–Alsina exponent, and the Aczel–Alsina scalar multiplication. We developed many AOs namely, the “spherical fuzzy Aczel–Alsina weighted averaging (SFAAWA) operator,” “spherical fuzzy Aczel–Alsina ordered weighted averaging (SFAAOWA) operator,” “spherical fuzzy Aczel–Alsina hybrid averaging (SFAAHA) operator,” “spherical fuzzy Aczel–Alsina weighted geometric (SFAAWG) operator,” “spherical fuzzy Aczel–Alsina ordered weighted geometric (SFAAOWG) operator,” and “spherical fuzzy Aczel–Alsina hybrid geometric (SFAAHG) operator.” Different attributes of these operators have been defined. The idempotency, boundary, monotonicity, and commutativity of suggested averaging and geometric operators are demonstrated. Then, based on these operators, we propose a novel approach for tackling the “multi-criteria decision-making” (MCDM) problems. We use a agriculture land selection scenario to demonstrate the efficacy of our proposed approach. The outcome confirms the new technique’s applicability and viability. Furthermore, sensitivity analysis and a comparison analysis between the existing approaches and the recommended technique have been provided.


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## Body

## 1. Introduction

Decision-making problems are common in a wide range of fields, including technology, finance, and marketing. Traditionally, it has been assumed that all data on alternate access is kept as discrete integers. Because managing the imprecision and uncertainty inherent in data is crucial in real-world circumstances. There are three alternative reactions or attitudes when it comes to selection: yes, no, and refusal. However, first, the most sophisticated response is “refusal,” which conventional “fuzzy sets” [1] and “intuitionistic fuzzy sets” (IFSs) [2] may not accurately represent. To address such problems, Cuong proposed the idea of “picture fuzzy set” (PFS) [3, 4]. In PFS, each component in the universe of discourse set has varying degrees of “positive membership degree” (PMD), “neutral membership degree” (Nu MD), and “negative membership degree” (Ng MD) with values ranging from 0,1.Fuzzy clustering is a useful technique for pattern detection and information extraction from databases, and it has been used to a wide range of practical issues. Son [5] defined “distributed picture fuzzy clustering method” for PFSs. Singh [6] proposed some “correlation coefficients.” The correlation coefficient is applied to clustering in PFS in this article. The benefits of proposed correlation coefficients as well as the disadvantages of existing correlation coefficients have been examined. Wei [7] proposed some new similarity measures (SMs) between PFSs, such as set-theoretic SMs, weighted set-theoretic cosine SMs, cosine SMs, weighted cosine SMs, grey SMs, and weighted grey SMs. Wei and Gao [8] introduced various innovative dice SMs for PFSs and generalized dice SMs for PFSs, indicating that dice SMs and asymmetric measures are special cases of generalized dice SMs for certain parameter values. Wei et al. [9] proposed some results about “projection model” under the PFSs, the modules of picture fuzzy number (PFN), and PF-ideal point.Over the last few decades, there has been a strong emphasis on information fusion and the development of new AOs. AO’s effectiveness and limitations have been entrenched in decision-making. AO obviously includes a number of operating rules for concatenating a finite set of fuzzy numbers into a single fuzzy number. Data aggregation is essential in decision-making, economy, administration, healthcare, technology, and intelligence areas. In terms of their functions and operating laws, numerous AOs have been established for PFSs. Wei et al. [10] proposed “picture 2-tuple linguistic AOs” with MCDM. Garg [11] proposed some weighted averaging and ordered weighted averaging operators for the aggregation of PFNs. Wei [12] presented “Hamacher” AOs for the PFS, and a realistic example of selecting an enterprise system is provided to validate the established approach and to illustrate its feasibility and efficiency. Jana et al. [13] gave the notion of picture fuzzy Dombi AOs for PFNs with MCDM applications. Tian et al. [14] defined some “picture fuzzy power Choquet ordered geometric AOs” and “picture fuzzy power shapley Choquet ordered geometric AOs” with shapley fuzzy measures-based MCDM. Wang et al. [15] proposed hotel building energy efficiency retrofit project selection under PFSs. Wang et al. [16] introduced “Muirhead mean AOs” for PFNs. Wei established TODIM method for PFSs [17], “picture 2-tuple linguistic Bonferroni mean AOs” [18], and “picture uncertain linguistic Bonferroni mean AOs” [19]. Abdullah et al. [20] proposed some new AOs based on sine trigonometric function with application. Qiyas et al. [21] defined some novel picture fuzzy AOs under the linguistic environment. Farid and Riaz [22] developed several new Einstein interactive geometric AOs for q-rung orthopair fuzzy numbers. Riaz and Farid [23] proposed some proportional distribution based spherical AOs. Farid et al. [24] introduced some AOs for the thermal power equipment supplier selection. Saha et al. [25] introduced the new hybrid hesitant fuzzy weighted AOs for MCDM that are based on Archimedean and Dombi operations. Feng et al. [26] proposed the idea of score functions related to generalized orthopair fuzzy membership grades with application. Akram et al. [27] introduced the idea of prioritized AOs for complex spherical fuzzy sets. Riaz and Farid [28] developed some fairly AOs for PFSs. Riaz et al. [29] proposed some Frank AOs for interval-valued linear Diophantine fuzzy set.Menger [30] introduced the concept of triangle norms in his hypothesis of probabilistic metric spaces. It has been discovered that t-norms and their corresponding t-conorms are fundamental operations in fuzzy sets and structures, such as the product t-norm and probabilistic sum t-conorm [31], Einstein t-norm and t-conorm [32], and the Hamacher t-norm and t-conorm [33]. Klement et al. [34] carried out a thorough analysis of the characteristics and associated elements of triangular norms in recent years. In 1982, Aczél and Alsina [35] introduced new operations named Aczel–Alsina t-norm and Aczel–Alsina t-conorm, which place a high premium on parameter changeability. Based on the Aczel–Alsina triangular norm (AA t-norm), Wang et al. [36] devised a score level fusion technique that simultaneously increases the distance between imposters. Senapati et al. proposed Aczel–Alsina AOs for IFSs [37] and Aczel–Alsina AOs for interval-valued IFSs [38].In everyday life, we might confront a variety of circumstances that PFS cannot resolve, such as when the sum of PMD,Nu MD, and Ng MD > 1. PFS is incapable of producing an appropriate conclusion in such a case. Mahmood et al. [39], Gundogdu and Kahraman [40], and Ashraf et al. [41] separately developed the concept of SFSs in their works. SFS provides the DM with extra freedom when confronted with uncertainty in decision-making situations. Gundogdu and Kahraman developed the SF-TOPSIS [42], SF-WASPAS [43], and SF-VIKOR methods [44]. Ashraf et al. presented AOs for SFSs and the GRA method for the SF-linguistic set [45]. Zeng et al. [46] proposed a TOPSIS-based hybrid covering-based SF-rough set model. The cosine similarity measures for SFSs were proposed by Rafiq et al. [47]. Jin et al. [48] developed AOs for SFSs based on logarithmic functions. Additionally, Ashraf et al. [49] presented several Dombi AOs for SFSs with implementation to group MCDM. Jaller and Otay used [50] when they suggested SF AHP and TOPSIS for assessing efficient vehicle technology for cargo handling. Ali et al. [51] and Ashraf et al. [52] proposed some AOs for interval-valued picture fuzzy set. Kazemitash et al. [53] and Bozanic et al. [54] gave some ideas related to some different extensions of fuzzy set. For other terminologies not discussed in the paper, the readers are referred to [55–58].In light of the foregoing, we recognize that decision-making concerns are becoming increasingly complex in reality. To select the superior alternatives to the MCDM concerns, it is significant to communicate the uncertain information in a far more beneficial approach. Furthermore, it is critical to control the relationship between input contentions. Taking each of these characteristics into account, the primary objective of this informative article is to demonstrate numerous aggregation operators under SF circumstances, referred as SF Aczel–Alsina AOs. Despite the established inventive approaches that have developed previously in this field, we have thoroughly researched every possibility to demonstrate our provided strategy, for it to outperform all previous efforts to comprehend the actual global presented concern.The following information is included in the paper: the next section discusses some fundamental concepts relating to Aczel–Alsina triangular norms and SFSs. Section3 summarizes the Aczel–Alsina operation laws for SFNs. In Section 4, we discuss the “spherical fuzzy Aczel–Alsina weighted averaging (SFAAWA) operator,” “spherical fuzzy Aczel–Alsina ordered weighted averaging (SFAAOWA) operator,” “spherical fuzzy Aczel–Alsina hybrid averaging (SFAAHA) operator,” “spherical fuzzy Aczel–Alsina weighted geometric (SFAAWG) operator,” “spherical fuzzy Aczel–Alsina ordered weighted geometric (SFAAOWG) operator,” and “spherical fuzzy Aczel–Alsina hybrid geometric (SFAAHG) operator” as well as a few advantageous properties. In Section 5, we employ the proposed operators to develop a set of techniques for resolving MCDM difficulties in which the characteristic values are represented as SF data. Section 6 illustrates how to choose agriculture land to showcase the proposed technique. In Section 7, we analyze the effect of a parameter on the alternate raking order. In Section 8, we compare the developed method to current methods to determine the proposed technique’s appropriateness. Finally, Section 9 discusses a few conclusions for future study.


## 2. Preliminaries

Several fundamental concepts associated with SFSs have been addressed in this section of the article.Definition 1 (see [39–41]).
A “spherical fuzzy set” (SFS) inX is defined as follows:(1)χ=⋎⌣,κχ⋎⌣,υχ⋎⌣,τχ⋎⌣|⋎⌣∈X,where κχ⋎⌣,υχ⋎⌣,τχ⋎⌣∈0,1, such that 0≤κχ2⋎⌣+υχ2⋎⌣+τ2χ⋎⌣≤1 for all ⋎⌣∈X and κχ⋎⌣,υχ⋎⌣,τχ⋎⌣ denote degree of membership, nonmembership and hesitancy, respectively, for some ⋎⌣∈X.
We denote this pair asℜγ=κℜγ,υℜγ,τℜγ, throughout this article, and called as SFN with the conditions κℜγ,υℜγ,τℜγ∈0,1 and κℜγ2+υℜγ2+τℜγ2≤1.Definition 2 (see [40]).
When implementing SFNs to real-world situations, it is critical to prioritize them. For this, “score function” (SF) for SFNℜγ=κℜγ,υℜγ,τℜγ can be defined as follows:(2)Sℜγ=κℜγ−τℜγ2−υℜγ−τℜγ2.Example 1.
Consider two SFNsℜ1γ=0.236,0.126,0.175 and ℜ2γ=0.308,0.228,0.482 then by using equation (2), we get Sℜ1γ=0.00119646 and Sℜ2γ=−0.0341963. As Sℜ1γ>Sℜ2γ so, we have ℜ1γ>ℜ2γ.
However, because the aforementioned function appears incapable of classifying the SFNs in a variety of conditions, it is hard to determine which one is largerSℜ1γ=Sℜ2γ. For this, an “accuracy function H” of ℜγ is defined as follows:(3)Hℜγ=κℜγ2+υℜγ2+τℜγ2.
Based on [40], we presented some operational rules to aggregate the SFNs.Definition 3.
Letℜ1γ=κ1,υ1,τ1 and ℜ2γ=κ1,υ1,τ1 be two SFNs, then(4)ℜ1γc=υ1,κ1,τ1,ℜ1γ∨ℜ2γ=maxκ1,κ2,minυ1,υ2,min1−maxκ1,κ22+minυ1,υ2212,maxτ1,τ2,ℜ1γ∧ℜ2γ=minκ1,κ2,maxυ1,υ2,max1−minκ1,κ22+maxυ1,υ2212,minτ1,τ2,ℜ1γ⊕ℜ2γ=κ12+κ22−κ12κ221/2,υ1υ2,1−κ22τ12+1−κ12τ22−τ12τ221/2,ℜ1γ⊗ℜ2γ=κ1κ2,υ12+υ22−υ12υ221/2,1−υ22τ12+1−υ12τ22−τ12τ221/2,ℜ1γσ=μA˜sσ,1−1−υ12σ1/2,1−υ12σ−1−υ12−πA˜s2σ1/2,σ>0,andσ⋅ℜ1γ=1−1−κ12σ1/2,υ1σ,1−κ12σ−1−κ12−τ12σ1/2,σ>0.Definition 4 (see [40]).
Letℜ1γ=κ1,υ1,τ1 and ℜ2γ=κ2,υ2,τ2 be two SFNs and σ,σ1,σ2>0 be the real numbers, then we have,(1)
ℜ1γ⊕ℜ2γ=ℜ2γ⊕ℜ1γ(2)
ℜ1γ⊗ℜ2γ=ℜ2γ⊗ℜ1γ(3)
σℜ1γ⊕ℜ2γ=σℜ1γ⊕σℜ2γ(4)
ℜ1γ⊗ℜ2γσ=ℜ1γσ⊗ℜ2γσ(5)
σ1+σ2ℜ1γ=σ1ℜ1γ⊕σ2ℜ2γ(6)
ℜ1γσ1+σ2=ℜ1γσ1⊗ℜ1γσ1


### 2.1. Basics aboutt-Norm, t-Conorm, and Aczel–Alsina t-Norm

Definition 5 (see [34]).
A function℧λ˘:0,12⟶0,1 is a t-norm, if for all g,h,u∈0,1, the consecutive axioms are fulfilled.(1)
℧λ˘g,h=℧λ˘h,g(2)
℧λ˘g,h≤℧λ˘g,u if h≤u(3)
℧λ˘g,℧λ˘h,u=℧λ˘℧λ˘g,h,u(4)
℧λ˘g,1=g
These axioms are called, symmetry, monotonicity, associativity, and “1” as identity, respectively.Definition 6 (see [34]).
A functionℑ¯:0,12⟶0,1 is a t-conorm, if for all g,h,u∈0,1, the consecutive axioms are fulfilled.(1)
ℑ¯g,h=ℑ¯h,g(2)
ℑ¯g,h≤ℑ¯g,u if h≤u(3)
ℑ¯g,ℑ¯h,u=ℑ¯ℑ¯g,h,u(4)
ℑ¯g,0=g
These axioms are called, symmetry, monotonicity, associativity, and “0” as identity, respectively.Example 2.
Some famoust-norms are given as follows:(i)
℧λ˘Pg,h=g. h (Product t-norm)(ii)
℧λ˘Mg,h=ming,h (Minimum t-norm)(iii)
℧λ˘Lg,h=maxg+h−1,0 (Lukasiewicz t-norm)(iv)
℧λ˘Dg,h=f,ifh=1h,ifg=10,otherwise for all g,h∈0,1 (Drastic t-norm)Example 3.
Some famoust-conorms are given as follows:(i)
ℑ¯Pg,h=g+h−g.h (Probabilistic sum)(ii)
ℑ¯Mg,h=maxg,h (Maximum t-conorm)(iii)
ℑ¯Lg,h=ming+h,1 (Lukasiewicz t-conorm)(iv)
ℑ¯Dg,h=g,ifh=0h,ifg=01,otherwise for all g,h∈0,1 (Drastic t-conorm)Definition 7 (see [35]).
This class oft-norm is originally proposed by Aczel–Alsina in mid-1980s under the condition of functional equations.
The category℧λ˘Aℷℷ∈0,∞ of Aczel–Alsina t-norms is stated by the following equation:(5)℧λ˘Aℷg,h=℧λ˘Dg,h,ifℷ=0,ming,h,ifℷ=∞,e−−loggℷ+−loghℷ1/ℷ,otherwise.
The categoryℑ¯Aℷℷ∈0,∞ of Aczel–Alsina t-conorms is stated by the following equation:(6)ℑ¯Aℷg,h=ℑ¯Dg,h,ifℷ=0,maxg,h,ifℷ=∞,1−e−−log1−gℷ+−log1−hℷ1/ℷ,otherwise.
Limiting Cases:℧λ˘A0=℧λ˘D,℧λ˘A1=℧λ˘P,℧λ˘A∞=min,ℑ¯A0=ℑ¯D,ℑ¯A1=ℑ¯P,andℑ¯A∞=max.
For everyℷ∈0,∞ the t-norm ℧λ˘Aℷ and t-conorm ℑ¯Aℷ are dual to each other. The class of Aczel–Alsina t-norms is strictly increasing and the class of Aczel–Alsina t-conorms is strictly decreasing.


## 2.1. Basics aboutt-Norm, t-Conorm, and Aczel–Alsina t-Norm

Definition 5 (see [34]).
A function℧λ˘:0,12⟶0,1 is a t-norm, if for all g,h,u∈0,1, the consecutive axioms are fulfilled.(1)
℧λ˘g,h=℧λ˘h,g(2)
℧λ˘g,h≤℧λ˘g,u if h≤u(3)
℧λ˘g,℧λ˘h,u=℧λ˘℧λ˘g,h,u(4)
℧λ˘g,1=g
These axioms are called, symmetry, monotonicity, associativity, and “1” as identity, respectively.Definition 6 (see [34]).
A functionℑ¯:0,12⟶0,1 is a t-conorm, if for all g,h,u∈0,1, the consecutive axioms are fulfilled.(1)
ℑ¯g,h=ℑ¯h,g(2)
ℑ¯g,h≤ℑ¯g,u if h≤u(3)
ℑ¯g,ℑ¯h,u=ℑ¯ℑ¯g,h,u(4)
ℑ¯g,0=g
These axioms are called, symmetry, monotonicity, associativity, and “0” as identity, respectively.Example 2.
Some famoust-norms are given as follows:(i)
℧λ˘Pg,h=g. h (Product t-norm)(ii)
℧λ˘Mg,h=ming,h (Minimum t-norm)(iii)
℧λ˘Lg,h=maxg+h−1,0 (Lukasiewicz t-norm)(iv)
℧λ˘Dg,h=f,ifh=1h,ifg=10,otherwise for all g,h∈0,1 (Drastic t-norm)Example 3.
Some famoust-conorms are given as follows:(i)
ℑ¯Pg,h=g+h−g.h (Probabilistic sum)(ii)
ℑ¯Mg,h=maxg,h (Maximum t-conorm)(iii)
ℑ¯Lg,h=ming+h,1 (Lukasiewicz t-conorm)(iv)
ℑ¯Dg,h=g,ifh=0h,ifg=01,otherwise for all g,h∈0,1 (Drastic t-conorm)Definition 7 (see [35]).
This class oft-norm is originally proposed by Aczel–Alsina in mid-1980s under the condition of functional equations.
The category℧λ˘Aℷℷ∈0,∞ of Aczel–Alsina t-norms is stated by the following equation:(5)℧λ˘Aℷg,h=℧λ˘Dg,h,ifℷ=0,ming,h,ifℷ=∞,e−−loggℷ+−loghℷ1/ℷ,otherwise.
The categoryℑ¯Aℷℷ∈0,∞ of Aczel–Alsina t-conorms is stated by the following equation:(6)ℑ¯Aℷg,h=ℑ¯Dg,h,ifℷ=0,maxg,h,ifℷ=∞,1−e−−log1−gℷ+−log1−hℷ1/ℷ,otherwise.
Limiting Cases:℧λ˘A0=℧λ˘D,℧λ˘A1=℧λ˘P,℧λ˘A∞=min,ℑ¯A0=ℑ¯D,ℑ¯A1=ℑ¯P,andℑ¯A∞=max.
For everyℷ∈0,∞ the t-norm ℧λ˘Aℷ and t-conorm ℑ¯Aℷ are dual to each other. The class of Aczel–Alsina t-norms is strictly increasing and the class of Aczel–Alsina t-conorms is strictly decreasing.


## 3. Aczel–Alsina Operations for SFNs

In this section, we will introduce the Aczel–Alsina operations for SFNs and look at some of their basic properties.Definition 8.
Letℜγ=κℜγ,υℜγ,τℜγ,ℜ1γ=κℜ1γ,υℜ1γ,τℜ1γ, and ℜ2γ=κℜ2γ,υℜ2γ,τℜ2γ be three SFNs, ℵ≥1 and ℷ>0. Then, the Aczel–Alsina t-norm and t-conorm operations of SFNs are defined as follows:(1)
ℜ1γ⊕ℜ2γ=1−e−−log1−κℜ1γ2ℵ+−log1−κℜ2γ2ℵ1/ℵ,e−−logυℜ1γ2ℵ+−logυℜ2γ2ℵ1/ℵ,e−−logτℜ1γ2ℵ+−logτℜ2γ2ℵ1/ℵ(2)
ℜ1γ⊗ℜ2γ=e−−logκℜ1γ2ℵ+−logκℜ2γ2ℵ1/ℵ,1−e−−log1−υℜ1γ2ℵ+−log1−υℜ2γ2ℵ1/ℵ,1−e−−log1−τℜ1γ2ℵ+−log1−τℜ2γ2ℵ1/ℵ(3)
ℷℜγ=1−e−ℷ−log1−κℜγ2ℵ1/ℵ,e−ℷ−logυℜγ2ℵ1/ℵ,e−ℷ−logτℜγ2ℵ1/ℵ(4)
ℜγℷ=e−ℷ−logκℜγ2ℵ1/ℵ,1−e−ℷ−log1−υℜγ2ℵ1/ℵ,1−e−ℷ−log1−τℜγ2ℵ1/ℵTheorem 1.
Letℜγ=κℜγ,υℜγ,τℜγ,ℜ1γ=κℜ1γ,υℜ1γ,τℜ1γ, and ℜ2γ=κℜ2γ,υℜ2γ,τℜ2γ be three SFNs, then we have(i)
ℜ1γ⊕ℜ2γ=ℜ2γ⊕ℜ1γ(ii)
ℜ1γ⊗ℜ2γ=ℜ2γ⊗ℜ1γ(iii)
ℷℜ1γ⊕ℜ2γ=ℷℜ1γ⊕ℷℜ2γ,ℷ>0(iv)
ℷ1+ℷ2ℜγ=ℷ1ℜγ⊕ℷ2ℜγ,ℷ1,ℷ2>0(v)
ℜ1γ⊗ℜ2γℷ=ℜ1γℷ⊗ℜ2γℷ,ℷ>0(vi)
ℜγℷ1⊗ℜγℷ2=ℜγℷ1+ℷ2,ℷ1,ℷ2>0Proof.
For the three SFNsℜγ,ℜ1γ, and ℜ2γ, and ℷ,ℷ1,ℷ2>0, we can get the following equations:(i)
(7)ℜ1γ⊕ℜ2γ=1−e−−log1−κℜ1γ2ℵ+−log1−κℜ2γ2ℵ1/ℵ,e−−logυℜ1γ2ℵ+−logυℜ2γ2ℵ1/ℵ,e−−logτℜ1γ2ℵ+−logτℜ2γ2ℵ1/ℵ,=1−e−−log1−κℜ2γ2ℵ+−log1−κℜ1γ2ℵ1/ℵ,e−−logυℜ2γ2ℵ+−logυℜ1γ2ℵ1/ℵ,e−−logτℜ2γ2ℵ+−logτℜ1γ2ℵ1/ℵ,=ℜ2γ⊕ℜ1γ(ii)
(8)ℜ1γ⊗ℜ2γ=e−−logκℜ1γ2ℵ+−logκℜ2γ2ℵ1/ℵ,1−e−−log1−υℜ1γ2ℵ+−log1−υℜ1γ2ℵ1/ℵ,1−e−−log1−τℜ1γ2ℵ+−log1−τℜ2γ2ℵ1/ℵ,=e−−logκℜ2γ2ℵ+−logκℜ1γ2ℵ1/ℵ,1−e−−log1−υℜ2γ2ℵ+−log1−υℜ1γ2ℵ1/ℵ,1−e−−log1−τℜ2γ2ℵ+−log1−τℜ1γ2ℵ1/ℵ,=ℜ2γ⊗ℜ1γ(iii)
Using this, we get(9)ℷℜ1γ⊕ℜ2γ=ℷ1−e−−log1−κℜ1γ2ℵ+−log1−κℜ2γ2ℵ1/ℵ,e−−logυℜ1γ2ℵ+−logυℜ2γ2ℵ1/ℵ,e−−logτℜ1γ2ℵ+−logτℜ2γ2ℵ1/ℵ,=1−e−ℷ−log1−κℜ1γ2ℵ+−log1−κℜ2γ2ℵ1/ℵ,e−ℷ−logυℜ1γ2ℵ+−logυℜ2γ2ℵ1/ℵ,e−ℷ−logτℜ1γ2ℵ+−logτℜ2γ2ℵ1/ℵ,=1−e−ℷ−log1−κℜ1γ2ℵ1/ℵ,e−ℷ−logυℜ1γ2ℵ1/ℵ,e−ℷ−logτℜ1γ2ℵ1/ℵ,⊕1−e−ℷ−log1−κℜ2γ2ℵ1/ℵ,e−ℷ−logυℜ2γ2ℵ1/ℵ,e−ℷ−logτℜ2γ2ℵ1/ℵ,=ℷℜ1γ⊕ℷℜ2γ(iv)
(10)ℷ1ℜγ⊕ℷ2ℜγ=1−e−ℷ1−log1−κℜγ2ℵ1/ℵ,e−ℷ1−logυℜγ2ℵ1/ℵ,e−ℷ1−logτℜγ2ℵ1/ℵ,⊕1−e−ℷ2−log1−κℜγ2ℵ1/ℵ,e−ℷ2−logυℜγ2ℵ1/ℵ,e−ℷ2−logτℜγ2ℵ1/ℵ,=1−e−ℷ1+ℷ2−log1−κℜγ2ℵ1/ℵ,e−ℷ1+ℷ2−logυℜγ2ℵ1/ℵ,e−ℷ1+ℷ2−logτℜγ2ℵ1/ℵ,=ℷ1+ℷ2ℜγ(v)
(11)ℜ1γ⊗ℜ2γℷ=e−−logκℜ1γ2ℵ+−logκℜ2γ2ℵ1/ℵ,1−e−−log1−υℜ1γ2ℵ+−log1−υℜ2γ2ℵ1/ℵ,1−e−−log1−τℜ1γ2ℵ+−log1−τℜ1γ2ℵ1/ℵℷ,=e−ℷ−logκℜ1γ2ℵ+−logκℜ2γ2ℵ1/ℵ,1−e−ℷ−log1−υℜ1γ2ℵ+−log1−υℜ2γ2ℵ1/ℵ,1−e−ℷ−log1−τℜ1γ2ℵ+−log1−τℜ2γ2ℵ1/ℵ,=e−ℷ−logκℜ1γ2ℵ1/ℵ,1−e−ℷ−log1−υℜ1γ2ℵ1/ℵ,1−e−ℷ−log1−τℜ1γ2ℵ1/ℵ,⊗e−ℷ−logκℜ2γ2ℵ1/ℵ,1−e−ℷ−log1−υℜ2γ2ℵ1/ℵ,1−e−ℷ−log1−τℜ2γ2ℵ1/ℵ,=ℜ1γℷ⊗ℜ2γℷ(vi)
(12)ℜγℷ1⊗ℜγℷ2=e−ℷ1−logκℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−υℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−τℜ1γ2ℵ1/ℵ,⊗e−ℷ2−logκℜ2γ2ℵ1/ℵ,1−e−ℷ2−log1−υℜ2γ2ℵ1/ℵ,1−e−ℷ2−log1−τℜ2γ2ℵ1/ℵ,=e−ℷ1+ℷ2−logκℜ2γ2ℵ1/ℵ,1−e−ℷ1+ℷ2−log1−υℜ2γ2ℵ1/ℵ,1−e−ℷ1+ℷ2−log1−τℜ2γ2ℵ1/ℵ,=ℜγℷ1+ℷ2


## 4. Spherical Fuzzy Aczel–Alsina Aggregation Operators

In this section, we present a few SF aggregation operators by means of the Aczel–Alsina operations.


### 4.1. Spherical Fuzzy Aczel–Alsina Averaging AOs

Definition 9.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ,ϕ=1,2,…,ℶ be an accumulation of SFNs and ξζ=ξ1ζ,ξ1ζ,…,ξℶζT be the weight vector (WV) of ℜϕγ, with ξϕζ>0 and ∑ϕ=1ℶξϕζ=1. Then, “spherical fuzzy Aczel–Alsina weighted average (SFAAWA) operator” is a mapping SFAAWA: L∗ℶ⟶L∗, where(13)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ=ξ1ζℜ1γ⊕ξ2ζℜ2γ⊕…,⊕ξℶζℜℶγ.
The following theorem is obtained using Aczel–Alsina operations on SFNs.Theorem 2.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs, then aggregated value of them utilizing the SFAAWA operation is also an SFNs, and(14)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜϕγ=1−e−∑ϕ=1ℶξϕζ−log1−κℜϕγℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logτℜϕγℵ1/ℵ,where ξζ=ξ1ζ,ξ2ζ,…,ξℶζ is the WV of ℜϕγ s.t ξϕζ>0 and ∑ϕ=1ℶξϕζ=1.Proof.
We can derive Theorem2 in the following way using the mathematical induction technique.
Forℶ=2, depend on Aczel–Alsina operations of SFNs, we obtain the following equation:(15)ξ1ζℜ1γ=1−e−ℷ1−log1−κℜ1γ2ℵ1/ℵ,e−ℷ1−logυℜ1γ2ℵ1/ℵ,e−ℷ1−logτℜ1γ2ℵ1/ℵandξ2ζℜ2γ=1−e−ℷ2−log1−κℜ2γ2ℵ1/ℵ,e−ℷ2−logυℜ2γ2ℵ1/ℵ,e−ℷ2−logτℜ2γ2ℵ1/ℵ.
Based on Aczel–Alsina operations of SFNs, we obtain the following equation:(16)SFAAWAℜ1γ,ℜ2γ=1−e−ℷ1−log1−κℜ1γ2ℵ1/ℵ,e−ℷ1−logυℜ1γ2ℵ1/ℵ,e−ℷ1−logτℜ1γ2ℵ1/ℵ,⊕1−e−ℷ2−log1−κℜ2γ2ℵ1/ℵ,e−ℷ2−logυℜ2γ2ℵ1/ℵ,e−ℷ2−logτℜ2γ2ℵ1/ℵ,=1−e−ℷ1−log1−κℜ1γ2ℵ+ℷ2−log1−κℜ2γ2ℵ1/ℵ,e−ℷ1−logυℜ1γ2ℵ+ℷ2−logυℜ2γ2ℵ1/ℵ,e−ℷ1−logτℜ1γ2ℵ+ℷ2−logτℜ2γ2ℵ1/ℵ,and=1−e−∑ϕ=12ξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=12ξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=12ξϕζ−logτℜϕγ2ℵ1/ℵ.
Thus, it is true forℶ=2.
Consider equation (14) is true for ℶ=k, then we have the following equation:(17)SFAAWAℜ1γ,ℜ2γ,…,ℜkγ=1−e−∑ϕ=1kξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=1kξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1kξϕζ−logτℜϕγ2ℵ1/ℵ,we will prove that equation (14) holds for ℶ=k+1.(18)SFAAWAℜ1γ,ℜ2γ,…,ℜkγ,ℜk+1γ=⊕kξϕζℜϕγ⊕ξk+1ζℜk+1γ,=1−e−∑ϕ=1kξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=1kξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1kξϕζ−logτℜϕγ2ℵ1/ℵ,⊕1−e−ℷk+1−log1−κℜk+1γ2ℵ1/ℵ,e−ℷk+1−logυℜk+1γ2ℵ1/ℵ,e−ℷk+1−logτℜk+1γ2ℵ1/ℵ,=1−e−∑ϕ=1k+1ξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=1k+1ξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1k+1ξϕζ−logτℜϕγ2ℵ1/ℵ.
As a result, we can conclude that equation (4.2) stands true for anyℶ.
By applying the SFAAWA operator, we can illustrate the following features efficiently.Theorem 3.
If allℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ are equal, that is, ℜϕγ=ℜγ∀ϕ, then(19)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ=ℜγ.Proof.
Given thatℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ, by equation (4.2) we get the following equation:(20)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ=1−e−∑ϕ=1ℶξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logτℜϕγ2ℵ1/ℵ,=1−e−−log1−κℜγ2ℵ1/ℵ,e−−logυℜγ2ℵ1/ℵ,e−−logτℜγ2ℵ1/ℵ,=1−elog1−κℜγ2,elogυℜγ2,elogτℜγ2,=κℜγ2,υℜγ2,τℜγ2=ℜγ.Theorem 4.
Letℜγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. Let
ℜγ−=minℜ1γ,ℜ1γ,…,ℜℶγ and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ. Then,(21)ℜγ−≤SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+.Proof.
Letℜϕγ=κℜϕγ,υℜϕγ be an accumulation of SFNs. Let ℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ=κℜγ−,υℜγ−,τℜγ− and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ=κℜγ+,υℜγ+,τℜγ+. We have, κℜγ−=minϕκℜϕγ,υℜγ−=maxϕυℜϕγ,
τℜγ−=maxϕτℜϕγ,κℜγ+=maxϕκℜϕγ, υℜγ+=minϕυℜϕγ, and τℜγ+=minϕτℜϕγ Hence, here we have the subsequent inequalities,(22)1−e−∑ϕ=1ℶξϕζ−log1−κℜγ−2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−κℜϕγ2ℵ1/ℵ,≤1−e−∑ϕ=1ℶξϕζ−log1−κℜγ+2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜγ+2ℵ1/ℵ≤e−∑ϕ=1ℶξϕζ−logυℜϕγ2ℵ1/ℵ,≤e−∑ϕ=1ℶξϕζ−logυℜγ−2ℵ1/ℵ,(23)e−∑ϕ=1ℶξϕζ−logτℜγ+2ℵ1/ℵ≤e−∑ϕ=1ℶξϕζ−logτℜϕγ2ℵ1/ℵ,≤e−∑ϕ=1ℶξϕζ−logτℜγ−2ℵ1/ℵ.
Therefore,ℜγ−≤SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+ □Theorem 5.
Letℜϕγ and ℜϕγ′ be two sets of SFNs, if ℜϕγ≤ℜϕγ′∀ϕ, then(24)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ≤SFAAWAℜ1γ′,ℜ2γ′,…,ℜℶγ′,
Now, we present “spherical fuzzy Aczel–Alsina ordered weighted averaging (SFAAOWA) operator.”Definition 10.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. SFAAOWA operator is a mapping SFAAOWA: L∗ℶ⟶L∗ with the corresponding WV ξζ=ξ1ζ,ξ2ζ,…,ξℶζT such that ξϕζ>0, and ∑ϕ=1ℶξϕζ=1, as follows:(25)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜΥϕγ=ξ1ζℜΥ1γ⊕ξ2ζℜΥ2γ⊕,…,⊕ξℶζℜΥℶγ,where Υ1,Υ2,…,Υℶ are the permutation of ϕ=1,2,…,ℶ, including ℜΥϕ−1γ≥ℜΥϕγ∀ϕ=1,2,…,ℶ.
Thus, the following theorem is obtained using Aczel–Alsina operations on SFNs.Theorem 6.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. SFAAOWA operator is a mapping SFAAOWA: L∗ℶ⟶L∗ with the corresponding vector ξζ=ξ1ζ,ξ2ζ,…,ξℶζT such that ξϕζ>0, and ∑ϕ=1ℶξϕζ=1. Then,(26)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜϕγ=1−e−∑ϕ=1ℶξϕζ−log1−κℜΥϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜΥϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logτℜΥϕγ2ℵ1/ℵ,where Υ1,Υ2,…,Υℶ are the permutation of ϕ=1,2,…,ℶ, including ℜΥϕ−1γ≥ℜΥϕγ∀ϕ=1,2,…,ℶ.Proof.
Same as Theorem2.
By applying the SFAAOWA operator, we can illustrate the following features efficiently.Theorem 7.
If allℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ are equal, that is, ℜϕγ=ℜγ∀ϕ, then(27)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ=ℜγProof.
Same as Theorem3.Theorem 8.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. Letℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ and ℜγ+=minℜ1γ,ℜ2γ,…,ℜℶγ. Then,(28)ℜγ−≤SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+.Proof.
Same as Theorem4.Theorem 9.
Letℜϕγ and ℜϕγ′ be two sets of SFNs, if ℜϕγ≤ℜϕγ′∀ϕ, then(29)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ≤SFAAOWAℜ1γ′,ℜ2γ′,…,ℜℶγ′.
It is self-evident that the SFAAWA operator weights only the SFNs, and that the SFAAOWA operator weights only the SFN’s ordered locations. Following that, weights are used to indicate various elements of the SFAAWA and SFAAOWA operators. Nonetheless, both one and the other operators consider only one of these. To address this shortcoming, we will also demonstrate the “spherical fuzzy Aczel–Alsina hybrid averaging (SFAAHA) operator,” which weights all of the given SFN and their appropriate ordered position.Definition 11.
Letℜϕγ be an accumulation of SFNs. SFAAHA operator is a mapping SFAAHA: L∗ℶ⟶L∗, s.t.(30)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜΥϕγ..=ξ1ζℜΥ1γ..⊕ξ2ζℜΥ2γ..⊕,…,⊕ξℶζℜΥℶγ..,where ξζ=ξ1ζ,ξ2ζ,…,ξℶζT is the weighting vector associated with the SFAAHA operator, with ξϕζ∈0,1 and ∑ϕ=1ℶξϕζ=1;ℜϕγ..=ℶδϕℜϕγ,ϕ=1,2,…,ℶ, ℜΥ1γ..,ℜΥ2γ..,…,ℜΥℶγ.. is any permutation of a collection of the weighted SFNs ℜ1γ..,ℜ2γ..,…ℜℶγ.., s.t. ℜYϕ−1γ..≥ℜYϕγ..⋅δ=δ1,δ2,…δℶT is the weight vector of ℜϕγ, with δϕ∈0,1 and ∑ϕ=1ℶδϕ=1, and ℶ is the balancing coefficient, which plays a role of balance.Theorem 10.
Letℜϕγ be the collection of SFNs. Their aggregated value by SFAAHA operator is still an SFN, and(31)SFAAHAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜϕγ=1−e−∑ϕ=1ℶξϕζ−log1−κℜΥϕγ..2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜΥϕγ..2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logτℜΥϕγ..2ℵ1/ℵ.Proof.
Same as Theorem2.Theorem 11.
The SFAAWA and SFAAOWA operators are special cases of the SFAAHA operator.Proof.
(1)
Letξζ=1/ℶ,1/ℶ,…,1/ℶT. Then,(32)SFAAHAℜ1γ,ℜ2γ,…,ℜℶγ=ξ1ζℜℶ1γ.⊕ξ2ζℜℶ2γ.⊕⋯⊕ξℶζℜℶℶγ.,=1ℶℜℶ1γ.⊕ℜℶ2γ.⊕⋯⊕ℜℶℶγ.,=ξ1ζℜ1γ⊕ξ2ζℜ2γ⊕⋯⊕ξℶζℜℶγ,=SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ.(2)
Letξζ=1/ℶ,1/ℶ,…,1/ℶT. Then, ℜϕγ.=ℜϕγ and(33)IFAAHAξζ,ξζℜ1γ,ℜ2γ,…,ℜℶγ=ξ1ζℜℶ1γ.⊕ξ2ζℜℶ2γ.⊕⋯⊕ξℶζℜℶℶγ.,ξ1ζℜℶ1γ.⊕ξ2ζℜℶ2γ.⊕⋯⊕ξℶζℜℶℶℶγ.,SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ.


### 4.2. Spherical Fuzzy Aczel–Alsina Geometric AOs

Definition 12.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs and ξζ=ξ1ζ,ξ2ζ,…,ξℶζT be the weight vector (WV) of ℜϕγ, with ξϕζ>0 and ∑ϕ=1ℶξϕζ=1. Then, “spherical fuzzy Aczel–Alsina weighted geometric (SFAAWG) operator” is a mapping SFAAWG: L∗ℶ⟶L∗, where(34)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=ℜ1γξ1ζ⊗ℜ2γξ2ζ⊗…,⊗ℜℶγξℶζ.
Thus, the following theorem is obtained using Aczel–Alsina operations on SFNs.Theorem 12.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs, then aggregated value of them utilizing the SFAAWG operation is also an SFNs, and(35)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=⊗ϕ=1ℶξϕζℜϕγ,=e−∑ϕ=1ℶξϕζ−logκℜϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜϕγℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−τℜϕγℵ1/ℵ,where ξζ=ξ1ζ,ξ2ζ,…,ξℶζT be the WV of ℜϕγ s.t ξϕζ>0 and ∑ϕ=1ℶξϕζ=1.Proof.
We can derive Theorem12 in the following way using the mathematical induction technique.
Forℶ=2, depend on Aczel–Alsina operations of SFNs, we obtain the following equation:(36)ℜ1γξ1ζ=e−ℷ1−logκℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−υℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−τℜ1γ2ℵ1/ℵandℜ2γξ2ζ=e−ℷ2−logκℜ2γ2ℵ1/ℵ,1−e−ℷ2−log1−υℜ2γ2ℵ2/ℵ,1−e−ℷ2−log1−τℜ2γ2ℵ1/ℵ,
Based on Aczel–Alsina operations of SFNs, we obtain the following equation:(37)SFAAWGℜ1γ,ℜ2γ=e−ℷ1−logκℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−υℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−τℜ1γ2ℵ1/ℵ,⊕e−ℷ2−logκℜ2γ2ℵ1/ℵ,1−e−ℷ2−log1−υℜ2γ2ℵ2/ℵ,1−e−ℷ2−log1−τℜ2γ2ℵ1/ℵ,=e−∑ϕ=12ξϕζ−logκℜϕγ2ℵ1/ℵ,1−e−∑ϕ=12ξϕζ−log1−υℜϕγℵ1/ℵ,1−e−∑ϕ=12ξϕζ−log1−τℜϕγℵ1/ℵ.
Thus, it is true forℶ=2.
Consider equation (4.15) is true forℶ=k, then we have the following equation:(38)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=e−∑ϕ=1kξϕζ−logκℜϕγ2ℵ1/ℵ,1−e−∑ϕ=1kξϕζ−log1−υℜϕγℵ1/ℵ,1−e−∑ϕ=1kξϕζ−log1−τℜϕγℵ1/ℵ,we will prove that equation (4.15) holds for ℶ=k+1.(39)SFAAWGℜ1γ,ℜ2γ,…,ℜkγ,ℜk+1γ=⊕kξϕζℜϕγ⊕ξk+1ζℜk+1γ=e−∑ϕ=1kξϕζ−logκℜϕγ2ℵ1ℵ,1−e−∑ϕ=1kξϕζ−log1−υℜϕγℵ1ℵ,1−e−∑ϕ=1kξϕζ−log1−τℜϕγℵ1ℵ,⊕e−ℷk+1−logκℜk+1γ2ℵ1ℵ,1−e−ℷk+1−log1−υℜk+1γ2ℵ1ℵ,1−e−ℷk+1−log1−τk+1γℵ1ℵ,=e−∑ϕ=1k+1ξϕζ−logκℜϕγ2ℵ1ℵ,1−e−∑ϕ=1k+1ξϕζ−log1−υℜϕγℵ1ℵ,1−e−∑ϕ=1k+1ξϕζ−log1−τℜϕγℵ1ℵ.
As a result, we can conclude that equation (4.15) stands true for anyℶ.
By applying the SFAAWG operator, we can illustrate the following features efficiently.Theorem 13.
If allℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ are equal, that is, ℜϕγ=ℜγ∀ϕ, then(40)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=ℜγ.Proof.
Given thatℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ, by equation (4.15) we get the following equation:(41)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=e−∑ϕ=1ℶξϕζ−logκℜϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−τℜϕγ2ℵ1/ℵ,=e−−logκℜγ2ℵ1/ℵ,1−e−−log1−υℜγ2ℵ1/ℵ,1−e−−log1−τℜγ2ℵ1ℵ,=elogκℜγ21−elog1−υℜγ2,1−elog1−τℜγ2,=κℜγ2,υℜγ2,τℜγ2=ℜγ.Theorem 14.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. Letℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ. Then,(42)ℜγ−≤SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+.Proof.
Letℜϕγ=κℜϕγ,υℜϕγ be an accumulation of SFNs. Let ℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ=κℜγ−,υℜγ−,τℜγ− and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ=κℜγ+,υℜγ+,τℜγ+. We have, κℜγ−=minϕκℜϕγ,υℜγ−=maxϕυℜϕγ, τℜγ−=maxϕτℜϕγ,κℜγ+=maxϕκℜϕγ, υℜγ+=minϕυℜϕγ, and τℜγ+=minϕτℜϕγ Hence, here, we have the subsequent inequalities,(43)e−∑ϕ=1ℶξϕζ−logκℜγ+2ℵ1/ℵ≤e−∑ϕ=1ℶξϕζ−logκℜγϕ2ℵ1/ℵ≤e−∑ϕ=1ℶξϕζ−logκℜγ−2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜγ−2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−υℜγϕ2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−υℜγ+2ℵ1/ℵ,(44)1−e−∑ϕ=1ℶξϕζ−log1−τℜγ−2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−τℜγϕ2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−τℜγ+2ℵ1/ℵ.
Therefore,ℜγ−≤SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+Theorem 15.
Letℜϕγ and ℜϕγ′ be two sets of SFNs, if ℜϕγ≤ℜϕγ′∀ϕ, then(45)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ≤SFAAWGℜ1γ′,ℜ2γ′,…,ℜℶγ′,
Now, we present “spherical fuzzy Aczel–Alsina ordered weighted geometric (SFAAOWG) operator.”Definition 13.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. SFAAOWG operator is a mapping SFAAOWG: L∗ℶ⟶L∗ with the corresponding WV ξζ=ξ1ζ,ξ2ζ,…,ξℶζT such that ξϕζ>0 and ∑ϕ=1ℶξϕζ=1, as follows:(46)SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ=⊗ϕ=1ℶℜΥϕγξϕζ=ℜΥ1γξ1ζ⊕ℜΥ2γξ2ζ⊕,…,⊕ℜΥℶγξℶζ,where Υ1,Υ2,…,Υℶ are the permutation of ϕ=1,2,…,ℶ, including ℜΥϕ−1γ≥ℜΥϕγ∀ϕ=1,2,…,ℶ.
Thus, the following theorem is obtained using Aczel–Alsina operations on SFNs.Theorem 16.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. SFAAOWG operator is a mapping SFAAOWG: L∗ℶ⟶L∗ with the corresponding vector ξζ=ξ1ζ>0,ξ2ζ>0,…,ξℶζT such that ξϕζ>0 and ∑ϕ=1ℶξϕζ=1. Then,(47)SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ=⊗ϕ=1ℶℜϕγξϕζ,=e−∑ϕ=1ℶξϕζ−logκℜΥϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜΥϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−τℜΥϕγ2ℵ1/ℵ,where Υ1,Υ2,…,Υℶ are the permutation of ϕ=1,2,…,ℶ, including ℜΥϕ−1γ≥ℜΥϕγ∀ϕ=1,2,…,ℶ.Proof.
Same as Theorem12.
By applying the SFAAOWG operator, we can illustrate the following features efficiently.Theorem 17.
If allℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ are equal, that is, ℜϕγ=ℜγ∀ϕ, then(48)SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ=ℜγ.Proof.
Same as Theorem13.Theorem 18.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. Let
ℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ. Then,(49)ℜγ−≤SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+.Proof.
Same as Theorem14.Theorem 19.
Letℜϕγ and ℜϕγ′ be two sets of SFNs, if ℜϕγ≤ℜϕγ′∀ϕ, then(50)SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ≤SFAAOWGℜ1γ′,ℜ2γ′,…,ℜℶγ′.
It is self-evident that the SFAAWG operator weights only the SFNs, and that the SFAAOWG operator weights only the SFN’s ordered locations. Following that, weights are used to indicate various elements of the SFAAWG and SFAAOWG operators. Nonetheless, both one and the other operators consider only one of these. To address this shortcoming, we will also demonstrate the “spherical fuzzy Aczel–Alsina hybrid geometric (SFAAHG) operator,” which weights all of the given SFN and their appropriate ordered position.Definition 14.
Letℜϕγ be an accumulation of SFNs. SFAAHG operator is a mapping SFAAHG: L∗ℶ⟶L∗, s.t.(51)SFAAHGℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶℜγΥϕξϕζ..=ξ1ζℜΥ1γ..⊕ξ2ζℜΥ2γ..⊕,…,⊕ξℶζℜΥℶγ..where ξζ=ξ1ζ,ξ1ζ,…,ξℶζT is the weighting vector associated with the SFAAHG operator, with ξϕζ∈0,1 and ∑ϕ=1ℶξϕζ=1;ℜϕγ..=ℶδϕℜϕγ,ϕ=1,2,…,ℶ, ℜΥ1γ..,ℜΥ2γ..,…,ℜΥℶγ.. is any permutation of a collection of the weighted SFNs ℜ1γ..,ℜ2γ..,…ℜℶγ.., s.t. ℜyϕ−1γ..≥ℜyϕγ..⋅δ=δ1,δ2,…δℶT is the weight vector of ℜϕγ, with δϕ∈0,1 and ∑ϕ=1ℶδϕ=1, and ℶ is the balancing coefficient, which plays a role of balance.Theorem 20.
Letℜϕγ be the collection of SFNs. Their aggregated value by SFAAHG operator is still an SFN, and(52)SFAAHGℜ1γ,ℜ2γ,…,ℜℶγ=⊗ϕ=1ℶℜϕγξϕζ,=e−∑ϕ=1ℶξϕζ−logκℜΥϕγ..2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜΥϕγ..2ℵ1/ℵ1−e−∑ϕ=1ℶξϕζ−log1−τℜΥϕγ..2ℵ1/ℵ.Proof.
Same as Theorem12.Theorem 21.
The SFAAWG and SFAAOWG operators are special cases of the SFAAHG operator.


## 4.1. Spherical Fuzzy Aczel–Alsina Averaging AOs

Definition 9.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ,ϕ=1,2,…,ℶ be an accumulation of SFNs and ξζ=ξ1ζ,ξ1ζ,…,ξℶζT be the weight vector (WV) of ℜϕγ, with ξϕζ>0 and ∑ϕ=1ℶξϕζ=1. Then, “spherical fuzzy Aczel–Alsina weighted average (SFAAWA) operator” is a mapping SFAAWA: L∗ℶ⟶L∗, where(13)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ=ξ1ζℜ1γ⊕ξ2ζℜ2γ⊕…,⊕ξℶζℜℶγ.
The following theorem is obtained using Aczel–Alsina operations on SFNs.Theorem 2.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs, then aggregated value of them utilizing the SFAAWA operation is also an SFNs, and(14)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜϕγ=1−e−∑ϕ=1ℶξϕζ−log1−κℜϕγℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logτℜϕγℵ1/ℵ,where ξζ=ξ1ζ,ξ2ζ,…,ξℶζ is the WV of ℜϕγ s.t ξϕζ>0 and ∑ϕ=1ℶξϕζ=1.Proof.
We can derive Theorem2 in the following way using the mathematical induction technique.
Forℶ=2, depend on Aczel–Alsina operations of SFNs, we obtain the following equation:(15)ξ1ζℜ1γ=1−e−ℷ1−log1−κℜ1γ2ℵ1/ℵ,e−ℷ1−logυℜ1γ2ℵ1/ℵ,e−ℷ1−logτℜ1γ2ℵ1/ℵandξ2ζℜ2γ=1−e−ℷ2−log1−κℜ2γ2ℵ1/ℵ,e−ℷ2−logυℜ2γ2ℵ1/ℵ,e−ℷ2−logτℜ2γ2ℵ1/ℵ.
Based on Aczel–Alsina operations of SFNs, we obtain the following equation:(16)SFAAWAℜ1γ,ℜ2γ=1−e−ℷ1−log1−κℜ1γ2ℵ1/ℵ,e−ℷ1−logυℜ1γ2ℵ1/ℵ,e−ℷ1−logτℜ1γ2ℵ1/ℵ,⊕1−e−ℷ2−log1−κℜ2γ2ℵ1/ℵ,e−ℷ2−logυℜ2γ2ℵ1/ℵ,e−ℷ2−logτℜ2γ2ℵ1/ℵ,=1−e−ℷ1−log1−κℜ1γ2ℵ+ℷ2−log1−κℜ2γ2ℵ1/ℵ,e−ℷ1−logυℜ1γ2ℵ+ℷ2−logυℜ2γ2ℵ1/ℵ,e−ℷ1−logτℜ1γ2ℵ+ℷ2−logτℜ2γ2ℵ1/ℵ,and=1−e−∑ϕ=12ξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=12ξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=12ξϕζ−logτℜϕγ2ℵ1/ℵ.
Thus, it is true forℶ=2.
Consider equation (14) is true for ℶ=k, then we have the following equation:(17)SFAAWAℜ1γ,ℜ2γ,…,ℜkγ=1−e−∑ϕ=1kξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=1kξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1kξϕζ−logτℜϕγ2ℵ1/ℵ,we will prove that equation (14) holds for ℶ=k+1.(18)SFAAWAℜ1γ,ℜ2γ,…,ℜkγ,ℜk+1γ=⊕kξϕζℜϕγ⊕ξk+1ζℜk+1γ,=1−e−∑ϕ=1kξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=1kξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1kξϕζ−logτℜϕγ2ℵ1/ℵ,⊕1−e−ℷk+1−log1−κℜk+1γ2ℵ1/ℵ,e−ℷk+1−logυℜk+1γ2ℵ1/ℵ,e−ℷk+1−logτℜk+1γ2ℵ1/ℵ,=1−e−∑ϕ=1k+1ξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=1k+1ξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1k+1ξϕζ−logτℜϕγ2ℵ1/ℵ.
As a result, we can conclude that equation (4.2) stands true for anyℶ.
By applying the SFAAWA operator, we can illustrate the following features efficiently.Theorem 3.
If allℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ are equal, that is, ℜϕγ=ℜγ∀ϕ, then(19)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ=ℜγ.Proof.
Given thatℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ, by equation (4.2) we get the following equation:(20)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ=1−e−∑ϕ=1ℶξϕζ−log1−κℜϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logτℜϕγ2ℵ1/ℵ,=1−e−−log1−κℜγ2ℵ1/ℵ,e−−logυℜγ2ℵ1/ℵ,e−−logτℜγ2ℵ1/ℵ,=1−elog1−κℜγ2,elogυℜγ2,elogτℜγ2,=κℜγ2,υℜγ2,τℜγ2=ℜγ.Theorem 4.
Letℜγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. Let
ℜγ−=minℜ1γ,ℜ1γ,…,ℜℶγ and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ. Then,(21)ℜγ−≤SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+.Proof.
Letℜϕγ=κℜϕγ,υℜϕγ be an accumulation of SFNs. Let ℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ=κℜγ−,υℜγ−,τℜγ− and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ=κℜγ+,υℜγ+,τℜγ+. We have, κℜγ−=minϕκℜϕγ,υℜγ−=maxϕυℜϕγ,
τℜγ−=maxϕτℜϕγ,κℜγ+=maxϕκℜϕγ, υℜγ+=minϕυℜϕγ, and τℜγ+=minϕτℜϕγ Hence, here we have the subsequent inequalities,(22)1−e−∑ϕ=1ℶξϕζ−log1−κℜγ−2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−κℜϕγ2ℵ1/ℵ,≤1−e−∑ϕ=1ℶξϕζ−log1−κℜγ+2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜγ+2ℵ1/ℵ≤e−∑ϕ=1ℶξϕζ−logυℜϕγ2ℵ1/ℵ,≤e−∑ϕ=1ℶξϕζ−logυℜγ−2ℵ1/ℵ,(23)e−∑ϕ=1ℶξϕζ−logτℜγ+2ℵ1/ℵ≤e−∑ϕ=1ℶξϕζ−logτℜϕγ2ℵ1/ℵ,≤e−∑ϕ=1ℶξϕζ−logτℜγ−2ℵ1/ℵ.
Therefore,ℜγ−≤SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+ □Theorem 5.
Letℜϕγ and ℜϕγ′ be two sets of SFNs, if ℜϕγ≤ℜϕγ′∀ϕ, then(24)SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ≤SFAAWAℜ1γ′,ℜ2γ′,…,ℜℶγ′,
Now, we present “spherical fuzzy Aczel–Alsina ordered weighted averaging (SFAAOWA) operator.”Definition 10.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. SFAAOWA operator is a mapping SFAAOWA: L∗ℶ⟶L∗ with the corresponding WV ξζ=ξ1ζ,ξ2ζ,…,ξℶζT such that ξϕζ>0, and ∑ϕ=1ℶξϕζ=1, as follows:(25)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜΥϕγ=ξ1ζℜΥ1γ⊕ξ2ζℜΥ2γ⊕,…,⊕ξℶζℜΥℶγ,where Υ1,Υ2,…,Υℶ are the permutation of ϕ=1,2,…,ℶ, including ℜΥϕ−1γ≥ℜΥϕγ∀ϕ=1,2,…,ℶ.
Thus, the following theorem is obtained using Aczel–Alsina operations on SFNs.Theorem 6.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. SFAAOWA operator is a mapping SFAAOWA: L∗ℶ⟶L∗ with the corresponding vector ξζ=ξ1ζ,ξ2ζ,…,ξℶζT such that ξϕζ>0, and ∑ϕ=1ℶξϕζ=1. Then,(26)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜϕγ=1−e−∑ϕ=1ℶξϕζ−log1−κℜΥϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜΥϕγ2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logτℜΥϕγ2ℵ1/ℵ,where Υ1,Υ2,…,Υℶ are the permutation of ϕ=1,2,…,ℶ, including ℜΥϕ−1γ≥ℜΥϕγ∀ϕ=1,2,…,ℶ.Proof.
Same as Theorem2.
By applying the SFAAOWA operator, we can illustrate the following features efficiently.Theorem 7.
If allℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ are equal, that is, ℜϕγ=ℜγ∀ϕ, then(27)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ=ℜγProof.
Same as Theorem3.Theorem 8.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. Letℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ and ℜγ+=minℜ1γ,ℜ2γ,…,ℜℶγ. Then,(28)ℜγ−≤SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+.Proof.
Same as Theorem4.Theorem 9.
Letℜϕγ and ℜϕγ′ be two sets of SFNs, if ℜϕγ≤ℜϕγ′∀ϕ, then(29)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ≤SFAAOWAℜ1γ′,ℜ2γ′,…,ℜℶγ′.
It is self-evident that the SFAAWA operator weights only the SFNs, and that the SFAAOWA operator weights only the SFN’s ordered locations. Following that, weights are used to indicate various elements of the SFAAWA and SFAAOWA operators. Nonetheless, both one and the other operators consider only one of these. To address this shortcoming, we will also demonstrate the “spherical fuzzy Aczel–Alsina hybrid averaging (SFAAHA) operator,” which weights all of the given SFN and their appropriate ordered position.Definition 11.
Letℜϕγ be an accumulation of SFNs. SFAAHA operator is a mapping SFAAHA: L∗ℶ⟶L∗, s.t.(30)SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜΥϕγ..=ξ1ζℜΥ1γ..⊕ξ2ζℜΥ2γ..⊕,…,⊕ξℶζℜΥℶγ..,where ξζ=ξ1ζ,ξ2ζ,…,ξℶζT is the weighting vector associated with the SFAAHA operator, with ξϕζ∈0,1 and ∑ϕ=1ℶξϕζ=1;ℜϕγ..=ℶδϕℜϕγ,ϕ=1,2,…,ℶ, ℜΥ1γ..,ℜΥ2γ..,…,ℜΥℶγ.. is any permutation of a collection of the weighted SFNs ℜ1γ..,ℜ2γ..,…ℜℶγ.., s.t. ℜYϕ−1γ..≥ℜYϕγ..⋅δ=δ1,δ2,…δℶT is the weight vector of ℜϕγ, with δϕ∈0,1 and ∑ϕ=1ℶδϕ=1, and ℶ is the balancing coefficient, which plays a role of balance.Theorem 10.
Letℜϕγ be the collection of SFNs. Their aggregated value by SFAAHA operator is still an SFN, and(31)SFAAHAℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶξϕζℜϕγ=1−e−∑ϕ=1ℶξϕζ−log1−κℜΥϕγ..2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logυℜΥϕγ..2ℵ1/ℵ,e−∑ϕ=1ℶξϕζ−logτℜΥϕγ..2ℵ1/ℵ.Proof.
Same as Theorem2.Theorem 11.
The SFAAWA and SFAAOWA operators are special cases of the SFAAHA operator.Proof.
(1)
Letξζ=1/ℶ,1/ℶ,…,1/ℶT. Then,(32)SFAAHAℜ1γ,ℜ2γ,…,ℜℶγ=ξ1ζℜℶ1γ.⊕ξ2ζℜℶ2γ.⊕⋯⊕ξℶζℜℶℶγ.,=1ℶℜℶ1γ.⊕ℜℶ2γ.⊕⋯⊕ℜℶℶγ.,=ξ1ζℜ1γ⊕ξ2ζℜ2γ⊕⋯⊕ξℶζℜℶγ,=SFAAWAℜ1γ,ℜ2γ,…,ℜℶγ.(2)
Letξζ=1/ℶ,1/ℶ,…,1/ℶT. Then, ℜϕγ.=ℜϕγ and(33)IFAAHAξζ,ξζℜ1γ,ℜ2γ,…,ℜℶγ=ξ1ζℜℶ1γ.⊕ξ2ζℜℶ2γ.⊕⋯⊕ξℶζℜℶℶγ.,ξ1ζℜℶ1γ.⊕ξ2ζℜℶ2γ.⊕⋯⊕ξℶζℜℶℶℶγ.,SFAAOWAℜ1γ,ℜ2γ,…,ℜℶγ.


## 4.2. Spherical Fuzzy Aczel–Alsina Geometric AOs

Definition 12.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs and ξζ=ξ1ζ,ξ2ζ,…,ξℶζT be the weight vector (WV) of ℜϕγ, with ξϕζ>0 and ∑ϕ=1ℶξϕζ=1. Then, “spherical fuzzy Aczel–Alsina weighted geometric (SFAAWG) operator” is a mapping SFAAWG: L∗ℶ⟶L∗, where(34)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=ℜ1γξ1ζ⊗ℜ2γξ2ζ⊗…,⊗ℜℶγξℶζ.
Thus, the following theorem is obtained using Aczel–Alsina operations on SFNs.Theorem 12.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs, then aggregated value of them utilizing the SFAAWG operation is also an SFNs, and(35)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=⊗ϕ=1ℶξϕζℜϕγ,=e−∑ϕ=1ℶξϕζ−logκℜϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜϕγℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−τℜϕγℵ1/ℵ,where ξζ=ξ1ζ,ξ2ζ,…,ξℶζT be the WV of ℜϕγ s.t ξϕζ>0 and ∑ϕ=1ℶξϕζ=1.Proof.
We can derive Theorem12 in the following way using the mathematical induction technique.
Forℶ=2, depend on Aczel–Alsina operations of SFNs, we obtain the following equation:(36)ℜ1γξ1ζ=e−ℷ1−logκℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−υℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−τℜ1γ2ℵ1/ℵandℜ2γξ2ζ=e−ℷ2−logκℜ2γ2ℵ1/ℵ,1−e−ℷ2−log1−υℜ2γ2ℵ2/ℵ,1−e−ℷ2−log1−τℜ2γ2ℵ1/ℵ,
Based on Aczel–Alsina operations of SFNs, we obtain the following equation:(37)SFAAWGℜ1γ,ℜ2γ=e−ℷ1−logκℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−υℜ1γ2ℵ1/ℵ,1−e−ℷ1−log1−τℜ1γ2ℵ1/ℵ,⊕e−ℷ2−logκℜ2γ2ℵ1/ℵ,1−e−ℷ2−log1−υℜ2γ2ℵ2/ℵ,1−e−ℷ2−log1−τℜ2γ2ℵ1/ℵ,=e−∑ϕ=12ξϕζ−logκℜϕγ2ℵ1/ℵ,1−e−∑ϕ=12ξϕζ−log1−υℜϕγℵ1/ℵ,1−e−∑ϕ=12ξϕζ−log1−τℜϕγℵ1/ℵ.
Thus, it is true forℶ=2.
Consider equation (4.15) is true forℶ=k, then we have the following equation:(38)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=e−∑ϕ=1kξϕζ−logκℜϕγ2ℵ1/ℵ,1−e−∑ϕ=1kξϕζ−log1−υℜϕγℵ1/ℵ,1−e−∑ϕ=1kξϕζ−log1−τℜϕγℵ1/ℵ,we will prove that equation (4.15) holds for ℶ=k+1.(39)SFAAWGℜ1γ,ℜ2γ,…,ℜkγ,ℜk+1γ=⊕kξϕζℜϕγ⊕ξk+1ζℜk+1γ=e−∑ϕ=1kξϕζ−logκℜϕγ2ℵ1ℵ,1−e−∑ϕ=1kξϕζ−log1−υℜϕγℵ1ℵ,1−e−∑ϕ=1kξϕζ−log1−τℜϕγℵ1ℵ,⊕e−ℷk+1−logκℜk+1γ2ℵ1ℵ,1−e−ℷk+1−log1−υℜk+1γ2ℵ1ℵ,1−e−ℷk+1−log1−τk+1γℵ1ℵ,=e−∑ϕ=1k+1ξϕζ−logκℜϕγ2ℵ1ℵ,1−e−∑ϕ=1k+1ξϕζ−log1−υℜϕγℵ1ℵ,1−e−∑ϕ=1k+1ξϕζ−log1−τℜϕγℵ1ℵ.
As a result, we can conclude that equation (4.15) stands true for anyℶ.
By applying the SFAAWG operator, we can illustrate the following features efficiently.Theorem 13.
If allℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ are equal, that is, ℜϕγ=ℜγ∀ϕ, then(40)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=ℜγ.Proof.
Given thatℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ, by equation (4.15) we get the following equation:(41)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ=e−∑ϕ=1ℶξϕζ−logκℜϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−τℜϕγ2ℵ1/ℵ,=e−−logκℜγ2ℵ1/ℵ,1−e−−log1−υℜγ2ℵ1/ℵ,1−e−−log1−τℜγ2ℵ1ℵ,=elogκℜγ21−elog1−υℜγ2,1−elog1−τℜγ2,=κℜγ2,υℜγ2,τℜγ2=ℜγ.Theorem 14.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. Letℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ. Then,(42)ℜγ−≤SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+.Proof.
Letℜϕγ=κℜϕγ,υℜϕγ be an accumulation of SFNs. Let ℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ=κℜγ−,υℜγ−,τℜγ− and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ=κℜγ+,υℜγ+,τℜγ+. We have, κℜγ−=minϕκℜϕγ,υℜγ−=maxϕυℜϕγ, τℜγ−=maxϕτℜϕγ,κℜγ+=maxϕκℜϕγ, υℜγ+=minϕυℜϕγ, and τℜγ+=minϕτℜϕγ Hence, here, we have the subsequent inequalities,(43)e−∑ϕ=1ℶξϕζ−logκℜγ+2ℵ1/ℵ≤e−∑ϕ=1ℶξϕζ−logκℜγϕ2ℵ1/ℵ≤e−∑ϕ=1ℶξϕζ−logκℜγ−2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜγ−2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−υℜγϕ2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−υℜγ+2ℵ1/ℵ,(44)1−e−∑ϕ=1ℶξϕζ−log1−τℜγ−2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−τℜγϕ2ℵ1/ℵ≤1−e−∑ϕ=1ℶξϕζ−log1−τℜγ+2ℵ1/ℵ.
Therefore,ℜγ−≤SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+Theorem 15.
Letℜϕγ and ℜϕγ′ be two sets of SFNs, if ℜϕγ≤ℜϕγ′∀ϕ, then(45)SFAAWGℜ1γ,ℜ2γ,…,ℜℶγ≤SFAAWGℜ1γ′,ℜ2γ′,…,ℜℶγ′,
Now, we present “spherical fuzzy Aczel–Alsina ordered weighted geometric (SFAAOWG) operator.”Definition 13.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. SFAAOWG operator is a mapping SFAAOWG: L∗ℶ⟶L∗ with the corresponding WV ξζ=ξ1ζ,ξ2ζ,…,ξℶζT such that ξϕζ>0 and ∑ϕ=1ℶξϕζ=1, as follows:(46)SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ=⊗ϕ=1ℶℜΥϕγξϕζ=ℜΥ1γξ1ζ⊕ℜΥ2γξ2ζ⊕,…,⊕ℜΥℶγξℶζ,where Υ1,Υ2,…,Υℶ are the permutation of ϕ=1,2,…,ℶ, including ℜΥϕ−1γ≥ℜΥϕγ∀ϕ=1,2,…,ℶ.
Thus, the following theorem is obtained using Aczel–Alsina operations on SFNs.Theorem 16.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. SFAAOWG operator is a mapping SFAAOWG: L∗ℶ⟶L∗ with the corresponding vector ξζ=ξ1ζ>0,ξ2ζ>0,…,ξℶζT such that ξϕζ>0 and ∑ϕ=1ℶξϕζ=1. Then,(47)SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ=⊗ϕ=1ℶℜϕγξϕζ,=e−∑ϕ=1ℶξϕζ−logκℜΥϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜΥϕγ2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−τℜΥϕγ2ℵ1/ℵ,where Υ1,Υ2,…,Υℶ are the permutation of ϕ=1,2,…,ℶ, including ℜΥϕ−1γ≥ℜΥϕγ∀ϕ=1,2,…,ℶ.Proof.
Same as Theorem12.
By applying the SFAAOWG operator, we can illustrate the following features efficiently.Theorem 17.
If allℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ are equal, that is, ℜϕγ=ℜγ∀ϕ, then(48)SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ=ℜγ.Proof.
Same as Theorem13.Theorem 18.
Letℜϕγ=κℜϕγ,υℜϕγ,τℜϕγ be an accumulation of SFNs. Let
ℜγ−=minℜ1γ,ℜ2γ,…,ℜℶγ and ℜγ+=maxℜ1γ,ℜ2γ,…,ℜℶγ. Then,(49)ℜγ−≤SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ≤ℜγ+.Proof.
Same as Theorem14.Theorem 19.
Letℜϕγ and ℜϕγ′ be two sets of SFNs, if ℜϕγ≤ℜϕγ′∀ϕ, then(50)SFAAOWGℜ1γ,ℜ2γ,…,ℜℶγ≤SFAAOWGℜ1γ′,ℜ2γ′,…,ℜℶγ′.
It is self-evident that the SFAAWG operator weights only the SFNs, and that the SFAAOWG operator weights only the SFN’s ordered locations. Following that, weights are used to indicate various elements of the SFAAWG and SFAAOWG operators. Nonetheless, both one and the other operators consider only one of these. To address this shortcoming, we will also demonstrate the “spherical fuzzy Aczel–Alsina hybrid geometric (SFAAHG) operator,” which weights all of the given SFN and their appropriate ordered position.Definition 14.
Letℜϕγ be an accumulation of SFNs. SFAAHG operator is a mapping SFAAHG: L∗ℶ⟶L∗, s.t.(51)SFAAHGℜ1γ,ℜ2γ,…,ℜℶγ=⊕ϕ=1ℶℜγΥϕξϕζ..=ξ1ζℜΥ1γ..⊕ξ2ζℜΥ2γ..⊕,…,⊕ξℶζℜΥℶγ..where ξζ=ξ1ζ,ξ1ζ,…,ξℶζT is the weighting vector associated with the SFAAHG operator, with ξϕζ∈0,1 and ∑ϕ=1ℶξϕζ=1;ℜϕγ..=ℶδϕℜϕγ,ϕ=1,2,…,ℶ, ℜΥ1γ..,ℜΥ2γ..,…,ℜΥℶγ.. is any permutation of a collection of the weighted SFNs ℜ1γ..,ℜ2γ..,…ℜℶγ.., s.t. ℜyϕ−1γ..≥ℜyϕγ..⋅δ=δ1,δ2,…δℶT is the weight vector of ℜϕγ, with δϕ∈0,1 and ∑ϕ=1ℶδϕ=1, and ℶ is the balancing coefficient, which plays a role of balance.Theorem 20.
Letℜϕγ be the collection of SFNs. Their aggregated value by SFAAHG operator is still an SFN, and(52)SFAAHGℜ1γ,ℜ2γ,…,ℜℶγ=⊗ϕ=1ℶℜϕγξϕζ,=e−∑ϕ=1ℶξϕζ−logκℜΥϕγ..2ℵ1/ℵ,1−e−∑ϕ=1ℶξϕζ−log1−υℜΥϕγ..2ℵ1/ℵ1−e−∑ϕ=1ℶξϕζ−log1−τℜΥϕγ..2ℵ1/ℵ.Proof.
Same as Theorem12.Theorem 21.
The SFAAWG and SFAAOWG operators are special cases of the SFAAHG operator.


## 5. MCDM Approach Based on Proposed Aczel–Alsina AOs

With the assistance of suggested AOs, we investigate MCDM problems. Consider the set of possible alternativesΛδ=Λ1δ,Λ2δ,…,Λmδ and ℶℷ=ℶ1ℷ,ℶ2ℷ,…,ℶnℷ is the collection of criterion. DMs gave their opinion matrix D=Pijm×n, where Pij is given for the alternatives Λiδ∈Λδ with respect to the criteria ℶjℷ∈ℶℷ in the form of SFNs. SFN decision matrix denoted by D=Pijm×n.The proposed operators will be applied to the MCDM, which will include the points listed in Algorithm 1.Algorithm 1: A decision making process based on SFAAWA, SFAAOWA, SFAAHA, is proposed in Algorithm 1.
Step 1: acquire the declaration matrixD=Pijm×n in the format of SFNs from the DM.Step 2: the declaration matrix’s attributes are classified into two types, such as the cost form parameterτc and the benefit form parameter τb. If all parameters are of the same form, no normalization is required. However, because the MCDM contains parameters of multiple forms, the D matrix has been converted to a normalization matrix using the normalization formula, YϜ⌣ij.Ϝ⌣ij=Pijc;j∈τc,Pij;j∈τb.wherePijc show the compliment of Pij.Step 3: using one of the proposed operators to evaluate combined evaluations of the alternatives. One can use geometric operators also instead of averaging operators.Ϝ⌣i=SFAAWAϜ⌣i1,Ϝ⌣i2,…,Ϝ⌣in,Ϝ⌣i=SFAAOWAϜ⌣i1,Ϝ⌣i2,…,Ϝ⌣in,orϜ⌣i=SFAAHAϜ⌣i1,Ϝ⌣i2,…,Ϝ⌣in.Step 4: calculate the combined score for all alternative assessments.Step 5: alternatives are ranked first by their scoring function, and the best one can be chosen.


## 6. Numerical Example

This section provides an illustration of how to apply the proposed strategy to the land selection for agriculture purpose.


### 6.1. Explanation of the Problem

Due to the COVID-19 pandemic’s consequences, the E-commerce phenomenon is accelerating, having a huge influence on global supply chains. Thus, logistics management tasks have been elevated in importance in practically every organization that transports physical commodities. There are several methods for businesses to acquire a comparative edge via the out scoured of logistics management processes in today’s diversified and incredibly quickly world. Exporters, distributors, and businesses with distribution networks have all demonstrated that turning to third-party logistics (3PL) providers benefits them. 3PL is a term that refers to the process through which a company outsources its warehouse and transportation activities. A 3PL organization can provide stock control, cross-docking, the door of the house distribution, and packaging material. The market for third-party logistics services has accelerated its expansion as a result of the E-commerce boom and expanded reverse logistics activities. The E-commerce trend includes faster, more dependable delivery, increased inventory turnover, and goods staged in forwarding sites near clients. There has been a large surge of 3PL firms offering a range of services to assist in maintaining this very sophisticated supply chain. 3PLs are frequently requested for assistance with E-commerce fulfilment, warehousing, and delivery facilities, and 3PLs invest in technology for both client service and internal usage. Due to the current worldwide problem, the COVID-19 pandemic, the function of E-commerce has been enhanced and expedited.Due to the features of multidimensional decision-making difficulties, 3PL selection may well be considered a complex MCDM challenge, given the presence of statistical, interpersonal, and numerous factors in the natural decision-making phase. Given the critical nature of sustainable third-party logistics providers, there is a dearth of studies on the 3PL selection challenge in emerging economies. The 3PL sector is growing at a breakneck pace due to the rise of the e-commerce sector. Indeed, the requirement for 3PL services is projected to grow as brands and distributors seek to focus exclusively on their core industries. As a result, they frequently outsource logistical services. In a nutshell, analyzing and choosing optimum third-party logistics providers is a critical component of any business’s long-term goals.Consider a corporation that is looking for the best 3PL provider. Following prescreening, five 3PLs have been identified for further considerationΛηδη=1,2,…,5. You must choose between the following four characteristics: (1) ℶ1ℷ = financial stability, (2) ℶ2ℷ = reliability and delivery time, (3) ℶ3ℷ = reputation, and (4) ℶ4ℷ = green operation. The DM distributes the attribute weight in the following way: ξζ=0.20,0.30,0.25,0.25T. Table 1 evaluates the five candidates Ληδη=1,2,…,5.Table 1
Spherical fuzzy decision matrix.
ℶ1ℷℶ2ℷℶ3ℷℶ4ℷΛ1δ(0.173, 0.144, 0.108)(0.243, 0.050, 0.203)(0.253, 0.011, 0.113)(0.223, 0.133, 0.009)Λ2δ(0.333, 0.244, 0.143)(0.143, 0.223, 0.333)(0.228, 0.151, 0.418)(0.233, 0.433, 0.243)Λ3δ(0.368, 0.263, 0.273)(0.133, 0.208, 0.543)(0.233, 0.150, 0.218)(0.263, 0.223, 0.431)Λ4δ(0.218, 0.463, 0.133)(0.248, 0.413, 0.243)(0.413, 0.150, 0.138)(0.163, 0.443, 0.133)Λ5δ(0.141, 0.152, 0.463)(0.268, 0.456, 0.163)(0.118, 0.413, 0.258)(0.213, 0.113, 0.433)


### 6.2. Proposed Method Based on SFAAWA Operator

To select the best agriculture landΛδηη=1,2,…,υ, we utilize the SFAAWA operator to construct an MCDM premise with SF information, which is usually evaluated as follows:Step 1: declaration matrixD=Pijm×n in the format of SFNs from the DM given in Table 1.Step 2: there is no cost type attributes in the four characteristics: (1)ℶ1ℷ  = financial stability, (2) ℶ2ℷ  = reliability and delivery time, (3) ℶ3ℷ  = reputation, and (4) ℶ4ℷ  = green operation. So, there is no need of normalization in Table 1.Step 3: consider thatℵ=4. We take the SFAAWA operator to calculate the alternative values Kη of five applicants Ληδη=1,2,…,5,Ϝ⌣1=0.23595,0.0331817,0.0329636, Ϝ⌣2=0.27865,0.214317,0.234073Ϝ⌣3=0.308017,0.19699,0.300857,Ϝ⌣4=0.352767,0.189726,0.152533, and Ϝ⌣5=0.23513,0.181907,0.237248.Step 4: evaluate the score functionsSϜ⌣ηη=1,2,…,5 of the SFNs Ϝ⌣ηη=1,2,…,5 as SϜ⌣1=0.0412034,SϜ⌣2=0.00159675,SϜ⌣3=−0.0107371,SϜ⌣4=0.0387103,andSϜ⌣5=−0.00305816.Step 5: classify all five candidates according to the value of the SFN’s score functions asΛ1δ>Λ4δ>Λ2δ>Λ5δ>Λ3δ. Λ5δ is chosen as the most suitable agriculture land.


## 6.1. Explanation of the Problem

Due to the COVID-19 pandemic’s consequences, the E-commerce phenomenon is accelerating, having a huge influence on global supply chains. Thus, logistics management tasks have been elevated in importance in practically every organization that transports physical commodities. There are several methods for businesses to acquire a comparative edge via the out scoured of logistics management processes in today’s diversified and incredibly quickly world. Exporters, distributors, and businesses with distribution networks have all demonstrated that turning to third-party logistics (3PL) providers benefits them. 3PL is a term that refers to the process through which a company outsources its warehouse and transportation activities. A 3PL organization can provide stock control, cross-docking, the door of the house distribution, and packaging material. The market for third-party logistics services has accelerated its expansion as a result of the E-commerce boom and expanded reverse logistics activities. The E-commerce trend includes faster, more dependable delivery, increased inventory turnover, and goods staged in forwarding sites near clients. There has been a large surge of 3PL firms offering a range of services to assist in maintaining this very sophisticated supply chain. 3PLs are frequently requested for assistance with E-commerce fulfilment, warehousing, and delivery facilities, and 3PLs invest in technology for both client service and internal usage. Due to the current worldwide problem, the COVID-19 pandemic, the function of E-commerce has been enhanced and expedited.Due to the features of multidimensional decision-making difficulties, 3PL selection may well be considered a complex MCDM challenge, given the presence of statistical, interpersonal, and numerous factors in the natural decision-making phase. Given the critical nature of sustainable third-party logistics providers, there is a dearth of studies on the 3PL selection challenge in emerging economies. The 3PL sector is growing at a breakneck pace due to the rise of the e-commerce sector. Indeed, the requirement for 3PL services is projected to grow as brands and distributors seek to focus exclusively on their core industries. As a result, they frequently outsource logistical services. In a nutshell, analyzing and choosing optimum third-party logistics providers is a critical component of any business’s long-term goals.Consider a corporation that is looking for the best 3PL provider. Following prescreening, five 3PLs have been identified for further considerationΛηδη=1,2,…,5. You must choose between the following four characteristics: (1) ℶ1ℷ = financial stability, (2) ℶ2ℷ = reliability and delivery time, (3) ℶ3ℷ = reputation, and (4) ℶ4ℷ = green operation. The DM distributes the attribute weight in the following way: ξζ=0.20,0.30,0.25,0.25T. Table 1 evaluates the five candidates Ληδη=1,2,…,5.Table 1
Spherical fuzzy decision matrix.
ℶ1ℷℶ2ℷℶ3ℷℶ4ℷΛ1δ(0.173, 0.144, 0.108)(0.243, 0.050, 0.203)(0.253, 0.011, 0.113)(0.223, 0.133, 0.009)Λ2δ(0.333, 0.244, 0.143)(0.143, 0.223, 0.333)(0.228, 0.151, 0.418)(0.233, 0.433, 0.243)Λ3δ(0.368, 0.263, 0.273)(0.133, 0.208, 0.543)(0.233, 0.150, 0.218)(0.263, 0.223, 0.431)Λ4δ(0.218, 0.463, 0.133)(0.248, 0.413, 0.243)(0.413, 0.150, 0.138)(0.163, 0.443, 0.133)Λ5δ(0.141, 0.152, 0.463)(0.268, 0.456, 0.163)(0.118, 0.413, 0.258)(0.213, 0.113, 0.433)


## 6.2. Proposed Method Based on SFAAWA Operator

To select the best agriculture landΛδηη=1,2,…,υ, we utilize the SFAAWA operator to construct an MCDM premise with SF information, which is usually evaluated as follows:Step 1: declaration matrixD=Pijm×n in the format of SFNs from the DM given in Table 1.Step 2: there is no cost type attributes in the four characteristics: (1)ℶ1ℷ  = financial stability, (2) ℶ2ℷ  = reliability and delivery time, (3) ℶ3ℷ  = reputation, and (4) ℶ4ℷ  = green operation. So, there is no need of normalization in Table 1.Step 3: consider thatℵ=4. We take the SFAAWA operator to calculate the alternative values Kη of five applicants Ληδη=1,2,…,5,Ϝ⌣1=0.23595,0.0331817,0.0329636, Ϝ⌣2=0.27865,0.214317,0.234073Ϝ⌣3=0.308017,0.19699,0.300857,Ϝ⌣4=0.352767,0.189726,0.152533, and Ϝ⌣5=0.23513,0.181907,0.237248.Step 4: evaluate the score functionsSϜ⌣ηη=1,2,…,5 of the SFNs Ϝ⌣ηη=1,2,…,5 as SϜ⌣1=0.0412034,SϜ⌣2=0.00159675,SϜ⌣3=−0.0107371,SϜ⌣4=0.0387103,andSϜ⌣5=−0.00305816.Step 5: classify all five candidates according to the value of the SFN’s score functions asΛ1δ>Λ4δ>Λ2δ>Λ5δ>Λ3δ. Λ5δ is chosen as the most suitable agriculture land.


## 7. Sensitively Analysis

To highlight the influence of different parameterℵ magnitudes, we use different parameter ℵ inside our discussed methodologies to classify the options. The ordering effects of the options based on the SFAAWA operator are reported in Table 2. It is clear that as the magnitude of ℵ for the SFAAWA operator increases, the score values of the alternatives gradually increase, but the optimum alternative remains constant, implying that the proposed techniques have the property of isotonicity, and the DMs can choose the appropriate value based on their preferences.Table 2
Score functions for different values ofℵ.
ℵSϜ⌣1SϜ⌣2SϜ⌣3SϜ⌣4SϜ⌣5Ordering40.04120340.00159675-0.01073710.0387103-0.00305816Λ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δ100.05081290.01425630.000715270.0590490.0000415248Λ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δ150.0532270.0202840.005217210.06482270.00229417Λ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δ200.05454110.0238170.007851350.06789210.00366547Λ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δ400.05680680.02965650.01246970.07272740.00595754Λ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δ600.05768240.03172620.01419570.07443420.00677951Λ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δ800.05813990.03278260.01508820.07532730.00720298Λ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δ1000.05841750.03342330.01563250.07588550.0074612Λ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δAdditionally, we can observe that even though the value ofℵ is varied throughout the demonstration, the results produced from the alternatives appear to be consistent, confirming the proposed operator’s robustness.


## 8. Comparative Analysis

This section compares proposed AOs to several existing AOs. We equate our findings by solving the data with preexisting AOs and get a comparable optimum solution. This demonstrates the AO’s long-term viability and efficacy. The technique outlined here is more practical and superior to a number of previously published AOs. We validate our optimal solution by running it through numerous current operators. We receive the same optimal decision, demonstrating the validity of our proposed AOs. Comparison of proposed AOs with some exiting operators is given in Table3.Table 3
Comparison of proposed AOs with some exiting operators.
AuthorsAOsRanking of alternativesThe optimal alternativeAshraf and Abdullah [59]SFEWAΛ1δ>Λ2δ>Λ5δ>Λ4δ>Λ3δΛ1δSFEWGΛ1δ>Λ2δ>Λ5δ>Λ3δ>Λ4δΛ1δJin et al. [48]LSFWAΛ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δΛ1δLSFWGΛ1δ>Λ2δ>Λ3δ>Λ4δ>Λ5δΛ1δAshraf et al. [49]SFDWAΛ1δ>Λ4δ>Λ2δ>Λ3δ>Λ5δΛ1δSFDWGΛ1δ>Λ4δ>Λ5δ>Λ2δ>Λ3δΛ1δAshraf et al. [41]SFNWAAΛ1δ>Λ2δ>Λ4δ>Λ5δ>Λ3δΛ1δSFNWGAΛ1δ>Λ5δ>Λ2δ>Λ4δ>Λ3δΛ1δProposedSFAAWAΛ1δ>Λ4δ>Λ2δ>Λ4δ>Λ3δΛ1δSFAAWGΛ1δ>Λ4δ>Λ2δ>Λ4δ>Λ3δΛ1δ


## 9. Conclusion

The main contribution of the work is outlined as follows:(1)
The aim of this paper is to present a novel idea about the operational laws and the AOs for the aggregation of SFNs. For this we first extended the Aczel–Alsina t-norm and t-conorm to SF contexts, then established and assessed a number of innovative operational principles for SFNs. The fundamental properties of the proposed laws are discussed in detail.(2)
Based on the proposed laws, we define several AOs to aggregate the SF information, namely the “spherical fuzzy Aczel–Alsina weighted averaging (SFAAWA) operator,” “spherical fuzzy Aczel–Alsina ordered weighted averaging (SFAAOWA) operator,” “spherical fuzzy Aczel–Alsina hybrid averaging (SFAAHA) operator,” “spherical fuzzy Aczel–Alsina weighted geometric (SFAAWG) operator,” “spherical fuzzy Aczel–Alsina ordered weighted geometric (SFAAOWG) operator,” and “spherical fuzzy Aczel–Alsina hybrid geometric (SFAAHG) operator.” The basic axioms of the operators are also satisfied with the proposed work.(3)
The proposed operators have been applied to MCDM approach with SF data, and a numerical model illustrating the decision-making technique has been provided.(4)
To highlight the influence of parameterℵ in the decision-making process, we also added the sensitivity analysis. Moreover, we equate our findings by solving the data with preexisting AOs and get a comparable optimum solution. This demonstrates the AO’s long-term viability and efficacy.We will gradually apply the aforementioned operators and techniques to a variety of multiple applications, including network analysis, risk assessment, cognitive science, reinforcement learning, signal processing, and many domains in ambiguous contexts. Furthermore, the current study does not take into account the interrelationships between the pairs of attributes during the aggregation process, which we will do in future projects. Furthermore, we will attempt to develop some more generalized information metrics in order to better realize the information in our everyday lives.



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*Source: 9657703-2022-10-04.xml*