# Multiple Attribute Group Decision-Making Based on Intuitionistic Fuzzy Schweizer-Sklar Generalized Power Aggregation Operators

Authors: Qaisar Khan; Hizbullah Khattak; Ahmad Ali AlZubi; Jazem Mutared Alanazi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

An intuitionistic fuzzy set (IFS) is a valuable tool to execute uncertain and indeterminate information. IFSs are more suitable to identify decision-maker’s evaluation data in decision-making problems. Intuitionistic fuzzy aggregation operators (AOs) are of enormous consequences in multiple attribute group decision-making (MAGDM) problems with an intuitionistic fuzzy environment. Consequently, the main impacts of this article are: firstly, to instigate various new novel generalized power AOs based on Schweizer-Sklar operational rules for IFS. Secondly, the study aims to discuss characteristics and particular cases of AOs. The core edge of proposed AOs is that they can eradicate the influence of uncomfortable data which could be too high or too low, making them more admirable for efficiently solving more and more complex MAGDM problems. Thirdly, we instigate two new algorithms to deal with MAGDM established on the generalized Schweizer-Sklar power AOs. Lastly, we appertain the anticipated method and algorithms to health care waste treatment technology selection (HCW-TT) to show the competence of the anticipated method and algorithms. The prevailing novelties of these items are duplex. Firstly, new generalized AOs established on Schweizer-Sklar operational rules are initiated for IFNs. Secondly, two new approaches for IF MAGDM are initiated, one for known decision-makers (DMs) and attributes weights, while the other for unknown DMs and attributes weights.


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## Body

## 1. Introduction

Nowadays, multiple attribute decision-making (MADM) or MAGDM problems are one of the most essential topics in decision theory. In MADM or MAGDM problems, the most favorable alternative is selected from a group of limited alternatives established on the inclusive data. Naturally, the information given about the alternatives is in the form of real numbers. Due to the complexity arising day by day in MADM and MAGDM problems, it is tough for DMs to provide assessment values about the objects in the form of crisp values owing to ambiguity and inadequate information. To some extent, it has been considered suitable that the valuations are specified by fuzzy set (FS) or its enlarged form. IFS [1, 2] is one of the strong extensions of FS [3] to pact with ambiguity by incorporating an equal falsity degree into the investigation. IFSs have excellent competence to clarify and clear DMs fuzzy decision information in MAGDM problems. Plenty of studies by disparate scholars were performed on IFS in various fields, such as operational rules for IF numbers (IFNs) [4], distance and similarity measures among IFNs [5–9], information entropies among IFNs [10–13], correlation measure among IFNs [14–16], etc. Furthermore, a huge number of methods have been presented and been applied to resolve MAGDM problems [17–24].The aggregation operators (AOs) are very significant tools for handling MADM or MAGDM problems [25, 26], and these aggregation operators have additional advantages than various conventional approaches, such as TOPSIS [27, 28], VIKOR [29–31], TODIM [32, 33], and so on. Based on conventional aggregation operators [34–38], such as operators WA, OWA, OWG, etc., several expanded aggregation operators have been developed to sustain decision-making, such as IF AOs [39–41] and generalized AOs [42, 43]. Yet, most expanded AOs do not adequately take the supportive correlations between input arguments. Hence, Yager [44] proposed the concepts of power average (PA) operator and power ordered weighted average (POWA) operator, which have the capability of eradicating the bad impact of awful data offered by DMs and can also consider the attribute values to support each other. Moreover, Xu and Yager [45] introduced the concepts of power geometric (PG) operator and power ordered weighted geometric (POWG) operator. These AOs are further extended by several scholars under different environments, such as Xu [46] proposed the concept of intuitionistic fuzzy power AOs, Wan and Dong [47] presented the concept of power geometric AOs under trapezoidal intuitionistic fuzzy environment, Liu and Wang [48] proposed generalized power AOs for linguistic intuitionistic fuzzy numbers, and Zhang [49] presented generalized intuitionistic fuzzy power geometric operator.For aggregating IFNs, several AOs are proposed utilizing different operational laws based on disparate T-norms (TN) and T-conorms (TCN), such as Wang and Liu [50, 51] proposed several AOs based on Einstein operational rules concerning IF information. Garg [52] developed some generalized geometric AOs established on Einstein TN and TCN and apply these AOs to deal with MADM problems under the IF environment. Huang [53] proposed some IF Hamacher AOs. Zhang et al. [54] presented a few Frank power AOs for IFNs and consider these AOs to concern with MAGDM problems under the IF environment. Some other AOs are proposed using Dombi operational laws such as, Khan et al. [55] proposed Dombi power Bonferroni mean for interval neutrosophic numbers and Li et al. [56] proposed IF Dombi Hamy mean operator and applied them to solve MAGDM problems. Generally, the Archimedean TN (ATN) and Archimedean TCN (ATCN) are the expansions of different TNs and TCNs, such as algebraic, Einstein, Hamacher, Frank, and Dombi TNs and TCNs. ATN and ATCN are the enlargements of numerous TNs and TCNs, which have several cases preferred to articulate the union and intersection of IFS [57]. Schweizer-Sklar (SS) operations [58] are the cases from ATN and ATCN; they are with a general parameter, so they are extra flexible and better than the former operations. However, most of the scholars mainly determine the elementary theory and types of Schweizer-Sklar TN (SSTN) and TCN (SSTCN) [59, 60]. Recently, Liu et al. [61, 62] and Zhang [63] merge SS operations with interval-valued IFS (IVIFS) and IFS, and established power average/geometric operators and weighted averaging operators, Maclaurin symmetric mean (MSM) operator for IVIFSs and IFSs, respectively. The MSM operator can take the correlation among multiple input arguments. Liu et al. [64] further proposed Schweizer-Sklar operational laws for single valued neutrosophic number (SVNN), proposed some prioritized AOs established on these operational laws and applied them to deal MADM problems with SVN information. Moreover, Zhang et al. [65] developed Schweizer-Sklar Muirhead mean operator and gave its application in MADM.From the above literature, we can notice that all the aggregation operators proposed on Schweizer-Sklar operational laws for aggregating different types of fuzzy information can consider interrelationship among input arguments or can consider priority among the attributes. Yet, there is no research to combine Schweizer-Sklar operational laws and generalized power AOs developed by Yager. So, there is a need to initiate some generalized power aggregation operators based on Schweizer-Sklar operational laws. The main advantages of these aggregation operators are that they consist of generalized parameters in which one parameter can take values and can remove the effect of awkward data from the final ranking results. Therefore, to achieve the following, the main contributions of our article are:(1)
To develop some generalized power aggregation operators based on Schweizer-Sklar operational laws for IFNs.(2)
To discuss some properties and some special cases of the developed aggregation operators.(3)
To develop two MAGDM algorithms based on these newly developed aggregation operators. The develop aggregation operators have some advantages:(i)
These newly developed AOs can have the facility of eradicating the influence of awful data.(ii)
They consist of two parameters, both taking values from infinite set, which makes the decision-making process additionally supple.(4)
To initiate two MAGDM models based on these aggregation operators to deal with MAGDM problems under the IF environment.To do so, this article is planned as follows. In Section2, several fundamental definitions concerning IFSs, Schweizer-Sklar TN and TCN, PA operators, and correlated properties are investigated. In Section 3, based on Schweizer-Sklar operational rules, we initiated several generalized intuitionistic fuzzy Schweizer-Sklar power AOs, with associated properties and particular cases being conferred. In Section 4, two novel MAGDM models are presented and established on these newly initiated AOs. In Section 5, a numerical example is specified to confirm the efficacy of the initiated MAGDM approaches and comparison of the initiated approaches with several approaches being given. In the last part, conclusion, future work, and references are provided.


## 2. Preliminaries

In this section, several essential notions of IFSs, operational laws of IFNs, the PA operator and associated properties are investigated.


### 2.1. IFS and Their Operations

Definition 1 (see [1]).
Letฝ be the reference set with common element in ฝ represented by ล. An IFS И⌣ in the reference set ฝ is expressed mathematically as follows:(1)Й=ล,ÜИ⌣ล,ÕИ⌣ล|ล∈ฝ,where ÜИ⌣ล and ÕИ⌣ล symbolize, respectively, the MS and the NM degrees of an element ล∈ฝ to the IFS И⌣, respectively, ÜИ⌣ล, ÕИ⌣ล∈0,1 with the restriction 0≤ÜИ⌣ล+ÕИ⌣ล≤1. The hesitancy degree ℵИ⌣ล of an element ล belonging to IFS И⌣ is described by ℵИ⌣ล=1−ÜИ⌣ล+ÕИ⌣ล.
The pairÜИ⌣,ÕИ⌣ in the IFS Й=ล,ÜИ⌣ล,ÕИ⌣ล|ล∈ฝ is said to be an IFN. To be easily understood, we shall represent IFN by Б=ÜБ,ÕБ, where ÜБ,ÕБ∈0,1 and 0≤ÜБ+ÕБ≤1.Definition 2 (see [39]).
LetБ1=Ü1,Õ1 and Б2=Ü2,Õ2 be any two arbitrary IFNs, ℘>0 . Subsequently, the operational rules for IFNs are explained as follows:(2)1.Б1⊕Б2=Ü1+Ü2−Ü1Ü2,Õ1Õ2;2.Б1⊕Б2=Ü1Ü2,Õ1+Õ2−Õ1Õ2;3.℘Б1=1−1−Ü1℘,Õ1℘;4.Б1℘=Ü1℘,1−1−Õ1℘.Definition 3 (see [17]).
LetБ=Ü,Õ be an IFN. The score function SE¯¯ of the IFN ъ is explained as follows:(3)SC¯¯Б=Ü−Õ,where the score values of IFN lies among the closed interval −1,1. The higher the score value SC¯¯Б, the better the IFN ъ.Definition 4 (see [18]).
Letъ=Ü,Õ be an IFN. The accuracy function AY¯¯ of the IFN ъ is explained as follows:(4)AY¯¯Б=Ü+Õ;,where the accuracy values of IFN lies among the closed interval 0,1. The bigger the accuracy value AY¯¯Б, the bigger the IFN ъ.Definition 5 (see [41]).
Letъ1=Ü1,Õ1 and ъ2=Ü2,Õ2 be any two arbitrary IFNs. Then, for comparing two IFNs, Xu [41] classified the judgment rules, which are explained as follows:1. IfSC¯¯Б1<SC¯¯Б2, then Б1<Б22. IfSC¯¯Б1=SC¯¯Б2, then(a)
IfAY¯¯Б1<AY¯¯Б2, afterward Б1<Б2(b)
IfAY¯¯Б1=AY¯¯Б2, afterward Б1<Б2Definition 6 (see [9]).
Letъ1=Ü1,Õ1 and ъ2=Ü2,Õ2 be any two IFNs. Then, the similarity measure between Б1 and Б2 is classified as follows.(5)SM˜Б1,Б2=1−KБ1−KБ2;,where KБ1 and KБ2 are, respectively, the knowledge measures of ъ1 and ъ2 and are described as follows:(6)KБ1=Ü12+Õ12+1−ℵ122,KБ2=Ü22+Õ22+1−ℵ222.Definition 7 (see [5]).
Let the two IFNs beБ1=Ü1,Õ1 and Б2=Ü2,Õ2. Then, the Hamming distance measure amongst Б1 and Б2 is classified as follows.(7)DSБ1,Б2=12Ü1−Ü2+Õ1−Õ2.


### 2.2. The PA Operator

PA operator initiated by Yager [44] is one of the imperative AOs. The PA operator reduces several unconstructive influences of unreasonably high or unreasonably low arguments given by DMs. The conservative PA operator can contract with real numbers, and is identified as follows.Definition 8 (see [44]).
LetЛ1,Л2,…,Лz be a faction of positive real numbers. A PA operator is classified as follows:(8)PAЛ1,Л2,…,Лz=∑i=1z1+TЛiЛi∑i=1z1+TЛi,where TЛi=∑i=1,j≠izSupЛi,Лj, and SupЛi,Лj is the support degree (SPD) for лi from лj satisfying the following axioms:
(1)SupЛi,Лj∈0,1; (2) SupЛi,Лj=SupЛj,Лi; (3) SupЛi,Лj≥SuppЛp,Лq, if Лi,Лj<Лp,Лq.Definition 9 (see [45]).
LetЛ1,Л2,…,Лz be a faction of positive real numbers. A PG operator is described as follows:(9)PGЛ1,Л2,…,Лz=∏i=1zЛi1+TЛi/∑i=1z1+TЛi,where TЛi=∑i=1,j≠izSupЛi,Лj and SupЛi,Лj is the SPD for Лi from Лj satisfying the above axioms.Definition 10 (see [44]).
A POWA operator of dimensionz is a function Uz⟶U, representing by the following formula:(10)POWAΘЛ1,Л2,…,Лz=∑g=1zΘgЛσg,where(11)Θg=kUgL−kUg−1L,Ug=⊕j=1gQρj,L=⊕g=1sQρg,Qρg=1+TЛρg,TЛρg=⊕j=1,g≠jssupЛρg,Лρj.
Here,Лρg represents the g−th largest arguments among Л1,Л2,…,Лz; k:0,1⟶0,1 is a fundamental BUM mapping [38] which must hold the following axioms: k0=0,k1=1,ka≥kb if a>b.Definition 11 (see [45]).
A POWG operator of dimensionz is a function Uz⟶U , represented by the following formula:(12)POWGΘЛ1,Л2,…,Лz=∏g=1zЛσgΘg.Definition 12 (see [36]).
LetЛ1,Л2,…,Лz be a faction of positive real numbers. A WGPA operator is described as follows:(13)WGPAЛ1,Л2,…,Лz=∑i=1z1+TЛiЛiA¯¯∑i=1z1+TЛi1/A¯¯.Where TЛG=∑G=1,H≠GZSupЛG,ЛH, and SupЛG,ЛH is the SPD for лG from лH satisfying the following axioms:Definition 13 (see [36]).
LetЛ1,Л2,…,Лz be a group of positive real numbers. A WGPOWA operator is depicted as follows:(14)GPOWAЛ1,Л2,…,Лz=∑i=1zΘiЛiA¯¯1/A¯¯.


### 2.3. Schweizer-Sklar Operational Laws for IFNs

The Schweizer-Sklar (SS) operational rules for IFNs consist of SS product and SS sum, which are some particular cases of ATT, respectively.Definition 14 (see [58]).
Assume thatИ⌣1=Ü1,Õ1 and И⌣2=Ü2,Õ2 be any two arbitrary IFSs, then the generalized union and intersection are identified as follows:(15)И⌣1∪ϒ,ϒ∗И⌣2=И⌣,ϒ∗Ü1Л,Ü2Л,ϒÕ1Л,Õ2Л,л∈ฝ,И⌣1∪ϒ,ϒ∗И⌣2=И⌣,ϒÜ1Л,Ü2Л,ϒ∗Õ1Л,Õ2Л,л∈ฝ.where ϒ and ϒ∗, respectively, represent TN and TCN.
The Schweizer-Sklar T-norm (SSTN) and T-conorm (TCN) are explained as follows:(16)ϒSS,ςЛ1,Л2=Л1+Л2−11/ς,ϒSS,ς∗Л1,Л2=1−1−Л1ς+1−Л2ς−11/ς.where ς<0,Л1,Л2∈0,1.
Moreover, whenς=0 , SSTN and SSTCN perverted into algebraic TN and TCN.
On the established TN and TCN of SS operations, we can offer the following Definition10 about SS operations for IFNs.Definition 15 (see [62]).
LetБ1=Ü1,Õ1 and Б2=Ü2,Õ2 be any two arbitrary IFNs, then based on SS operations, the following generalized union and generalized intersection are developed:(17)1Б1⊕ϒ∗,ϒБ2=ϒ∗Ü1,Ü2,ϒÕ1,Õ2,2Б1⊗ϒ∗,ϒБ2=ϒÜ1,Ü2,ϒ∗Õ1,Õ2.
Based on Definitions1 and 2, the SS operational laws for IFNs are initiated as follows ς<0:(18)1Б1⊕SSБ2=1−1−Ü1ς+1−Ü2ς−11/ς,Õ1ς+Õ2ς−11/ς,2Б1⊗SSБ2=Ü1ς+Ü2ς−11/ς,1−1−Õ1ς+1−Õ2ς−11/ς,3ξ¯¯Б=1−ξ¯¯1−Üς−ξ¯¯−11/ς,ξ¯¯Õς−ξ¯¯−11/ς,ξ¯¯>0;a2+b24Бξ¯¯=ξ¯¯Üς−ξ¯¯−11/ς,1−ξ¯¯1−Õς−ξ¯¯−11/ς,ξ¯¯>0.0,1.Theorem 1 (see [62]).
LetБ=Ü,Õ,Б1=Ü1,Õ1 and Б2=Ü2,Õ2 be any three IFNs. Then,(19)1Б1⊕SSБ2=Б2⊕SSБ1,2Б1⊗SSБ2=Б2⊗SSБ1,3ξБ1⊕SSБ2=ξБ1⊕SSξБ2,ξ≥0,4ξ1Б⊕SSξ2Б=ξ1+ξ2Б,ξ1,ξ2≥0,5Бξ1⊗SSБξ2=Бξ1+ξ2,ξ1,ξ2≥0,6Б1ξ⊗SSБ2ξ=Б1⊗SSБ2ξ,ξ>0.


## 2.1. IFS and Their Operations

Definition 1 (see [1]).
Letฝ be the reference set with common element in ฝ represented by ล. An IFS И⌣ in the reference set ฝ is expressed mathematically as follows:(1)Й=ล,ÜИ⌣ล,ÕИ⌣ล|ล∈ฝ,where ÜИ⌣ล and ÕИ⌣ล symbolize, respectively, the MS and the NM degrees of an element ล∈ฝ to the IFS И⌣, respectively, ÜИ⌣ล, ÕИ⌣ล∈0,1 with the restriction 0≤ÜИ⌣ล+ÕИ⌣ล≤1. The hesitancy degree ℵИ⌣ล of an element ล belonging to IFS И⌣ is described by ℵИ⌣ล=1−ÜИ⌣ล+ÕИ⌣ล.
The pairÜИ⌣,ÕИ⌣ in the IFS Й=ล,ÜИ⌣ล,ÕИ⌣ล|ล∈ฝ is said to be an IFN. To be easily understood, we shall represent IFN by Б=ÜБ,ÕБ, where ÜБ,ÕБ∈0,1 and 0≤ÜБ+ÕБ≤1.Definition 2 (see [39]).
LetБ1=Ü1,Õ1 and Б2=Ü2,Õ2 be any two arbitrary IFNs, ℘>0 . Subsequently, the operational rules for IFNs are explained as follows:(2)1.Б1⊕Б2=Ü1+Ü2−Ü1Ü2,Õ1Õ2;2.Б1⊕Б2=Ü1Ü2,Õ1+Õ2−Õ1Õ2;3.℘Б1=1−1−Ü1℘,Õ1℘;4.Б1℘=Ü1℘,1−1−Õ1℘.Definition 3 (see [17]).
LetБ=Ü,Õ be an IFN. The score function SE¯¯ of the IFN ъ is explained as follows:(3)SC¯¯Б=Ü−Õ,where the score values of IFN lies among the closed interval −1,1. The higher the score value SC¯¯Б, the better the IFN ъ.Definition 4 (see [18]).
Letъ=Ü,Õ be an IFN. The accuracy function AY¯¯ of the IFN ъ is explained as follows:(4)AY¯¯Б=Ü+Õ;,where the accuracy values of IFN lies among the closed interval 0,1. The bigger the accuracy value AY¯¯Б, the bigger the IFN ъ.Definition 5 (see [41]).
Letъ1=Ü1,Õ1 and ъ2=Ü2,Õ2 be any two arbitrary IFNs. Then, for comparing two IFNs, Xu [41] classified the judgment rules, which are explained as follows:1. IfSC¯¯Б1<SC¯¯Б2, then Б1<Б22. IfSC¯¯Б1=SC¯¯Б2, then(a)
IfAY¯¯Б1<AY¯¯Б2, afterward Б1<Б2(b)
IfAY¯¯Б1=AY¯¯Б2, afterward Б1<Б2Definition 6 (see [9]).
Letъ1=Ü1,Õ1 and ъ2=Ü2,Õ2 be any two IFNs. Then, the similarity measure between Б1 and Б2 is classified as follows.(5)SM˜Б1,Б2=1−KБ1−KБ2;,where KБ1 and KБ2 are, respectively, the knowledge measures of ъ1 and ъ2 and are described as follows:(6)KБ1=Ü12+Õ12+1−ℵ122,KБ2=Ü22+Õ22+1−ℵ222.Definition 7 (see [5]).
Let the two IFNs beБ1=Ü1,Õ1 and Б2=Ü2,Õ2. Then, the Hamming distance measure amongst Б1 and Б2 is classified as follows.(7)DSБ1,Б2=12Ü1−Ü2+Õ1−Õ2.


## 2.2. The PA Operator

PA operator initiated by Yager [44] is one of the imperative AOs. The PA operator reduces several unconstructive influences of unreasonably high or unreasonably low arguments given by DMs. The conservative PA operator can contract with real numbers, and is identified as follows.Definition 8 (see [44]).
LetЛ1,Л2,…,Лz be a faction of positive real numbers. A PA operator is classified as follows:(8)PAЛ1,Л2,…,Лz=∑i=1z1+TЛiЛi∑i=1z1+TЛi,where TЛi=∑i=1,j≠izSupЛi,Лj, and SupЛi,Лj is the support degree (SPD) for лi from лj satisfying the following axioms:
(1)SupЛi,Лj∈0,1; (2) SupЛi,Лj=SupЛj,Лi; (3) SupЛi,Лj≥SuppЛp,Лq, if Лi,Лj<Лp,Лq.Definition 9 (see [45]).
LetЛ1,Л2,…,Лz be a faction of positive real numbers. A PG operator is described as follows:(9)PGЛ1,Л2,…,Лz=∏i=1zЛi1+TЛi/∑i=1z1+TЛi,where TЛi=∑i=1,j≠izSupЛi,Лj and SupЛi,Лj is the SPD for Лi from Лj satisfying the above axioms.Definition 10 (see [44]).
A POWA operator of dimensionz is a function Uz⟶U, representing by the following formula:(10)POWAΘЛ1,Л2,…,Лz=∑g=1zΘgЛσg,where(11)Θg=kUgL−kUg−1L,Ug=⊕j=1gQρj,L=⊕g=1sQρg,Qρg=1+TЛρg,TЛρg=⊕j=1,g≠jssupЛρg,Лρj.
Here,Лρg represents the g−th largest arguments among Л1,Л2,…,Лz; k:0,1⟶0,1 is a fundamental BUM mapping [38] which must hold the following axioms: k0=0,k1=1,ka≥kb if a>b.Definition 11 (see [45]).
A POWG operator of dimensionz is a function Uz⟶U , represented by the following formula:(12)POWGΘЛ1,Л2,…,Лz=∏g=1zЛσgΘg.Definition 12 (see [36]).
LetЛ1,Л2,…,Лz be a faction of positive real numbers. A WGPA operator is described as follows:(13)WGPAЛ1,Л2,…,Лz=∑i=1z1+TЛiЛiA¯¯∑i=1z1+TЛi1/A¯¯.Where TЛG=∑G=1,H≠GZSupЛG,ЛH, and SupЛG,ЛH is the SPD for лG from лH satisfying the following axioms:Definition 13 (see [36]).
LetЛ1,Л2,…,Лz be a group of positive real numbers. A WGPOWA operator is depicted as follows:(14)GPOWAЛ1,Л2,…,Лz=∑i=1zΘiЛiA¯¯1/A¯¯.


## 2.3. Schweizer-Sklar Operational Laws for IFNs

The Schweizer-Sklar (SS) operational rules for IFNs consist of SS product and SS sum, which are some particular cases of ATT, respectively.Definition 14 (see [58]).
Assume thatИ⌣1=Ü1,Õ1 and И⌣2=Ü2,Õ2 be any two arbitrary IFSs, then the generalized union and intersection are identified as follows:(15)И⌣1∪ϒ,ϒ∗И⌣2=И⌣,ϒ∗Ü1Л,Ü2Л,ϒÕ1Л,Õ2Л,л∈ฝ,И⌣1∪ϒ,ϒ∗И⌣2=И⌣,ϒÜ1Л,Ü2Л,ϒ∗Õ1Л,Õ2Л,л∈ฝ.where ϒ and ϒ∗, respectively, represent TN and TCN.
The Schweizer-Sklar T-norm (SSTN) and T-conorm (TCN) are explained as follows:(16)ϒSS,ςЛ1,Л2=Л1+Л2−11/ς,ϒSS,ς∗Л1,Л2=1−1−Л1ς+1−Л2ς−11/ς.where ς<0,Л1,Л2∈0,1.
Moreover, whenς=0 , SSTN and SSTCN perverted into algebraic TN and TCN.
On the established TN and TCN of SS operations, we can offer the following Definition10 about SS operations for IFNs.Definition 15 (see [62]).
LetБ1=Ü1,Õ1 and Б2=Ü2,Õ2 be any two arbitrary IFNs, then based on SS operations, the following generalized union and generalized intersection are developed:(17)1Б1⊕ϒ∗,ϒБ2=ϒ∗Ü1,Ü2,ϒÕ1,Õ2,2Б1⊗ϒ∗,ϒБ2=ϒÜ1,Ü2,ϒ∗Õ1,Õ2.
Based on Definitions1 and 2, the SS operational laws for IFNs are initiated as follows ς<0:(18)1Б1⊕SSБ2=1−1−Ü1ς+1−Ü2ς−11/ς,Õ1ς+Õ2ς−11/ς,2Б1⊗SSБ2=Ü1ς+Ü2ς−11/ς,1−1−Õ1ς+1−Õ2ς−11/ς,3ξ¯¯Б=1−ξ¯¯1−Üς−ξ¯¯−11/ς,ξ¯¯Õς−ξ¯¯−11/ς,ξ¯¯>0;a2+b24Бξ¯¯=ξ¯¯Üς−ξ¯¯−11/ς,1−ξ¯¯1−Õς−ξ¯¯−11/ς,ξ¯¯>0.0,1.Theorem 1 (see [62]).
LetБ=Ü,Õ,Б1=Ü1,Õ1 and Б2=Ü2,Õ2 be any three IFNs. Then,(19)1Б1⊕SSБ2=Б2⊕SSБ1,2Б1⊗SSБ2=Б2⊗SSБ1,3ξБ1⊕SSБ2=ξБ1⊕SSξБ2,ξ≥0,4ξ1Б⊕SSξ2Б=ξ1+ξ2Б,ξ1,ξ2≥0,5Бξ1⊗SSБξ2=Бξ1+ξ2,ξ1,ξ2≥0,6Б1ξ⊗SSБ2ξ=Б1⊗SSБ2ξ,ξ>0.


## 3. Some Generalized Power Aggregation Operators for IFNs

In this segment, we exploit a variety of generalized power AOs launched on the initiated SS operational rules for IFNs.


### 3.1. Weighted Generalized Intuitionistic Fuzzy Schweizer-Sklar Power Aggregation Operator

In this subpart, we advanced intuitionistic fuzzy Schweizer-Sklar generalized power average (IFSSGPA) operator, weighted intuitionistic fuzzy Schweizer-Sklar generalized power average (WIFSSGPA) operator, intuitionistic fuzzy Schweizer-Sklar generalized power ordered weighted averaging (GIFSSPOWA) operator, and conferred their privileged properties and various cases.Definition 16.
For a faction of IFNsБgg=1,2,…,s, IFSSGPA operator is a function ℑs⟶ℑ,(20)IFSSGPAББ1,Б2,…,Бs=⊕SSg=1s1+TБgБgБ⊕g=1s1+TБg1/Б,where TБg=⊕h=1,g≠hssupБg,Бh, parameters Љ∈0,+∞ and Supъg,ъh are the SPD for ъg from ъh with the following constraint:
(1)Supъg,ъh∈0,1; (2) SupБg,Бh=SupБh,Бg; (3) SupБg,Бh≥SupБu,Бv if DSБg,Бh<DSБu,Бv, where DS is the distance measure among two IFNs.
To compose (20) in unsophisticated mode, we have(21)Жg=1+TБg⊕g=1s1+TБg.
So, equations (21) and (20) become(22)IFSSgPAЉБ1,Б2,…,Бs=⊕SSg=1sЖgif¯¯gЉ1/Љ,Theorem 2.
LetБgg=1,2,…,s be a faction of IFNs, then the value aggregated utilizing Definition 16 is still IFN, and we have(23)IFSSGPAЉБ1,Б2,…,Бs=1Љ1−⊕g=1sЖg1−ЉÜgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς.Proof.
In the succeeding, firstly, we can prove the subsequent:(24)Ж1ъ1Љ+Ж2ъ2Љ+...+ЖsъsЉ=1−⊕g=1sЖg1−ЉÜgς−Љ−11ςς−⊕g=1sЖg+11ς⊕g=1sЖg1−Љ1−Õgς−Љ−11ςς−⊕g=1sЖg+11ς.
By utilizing mathematical induction ons
Fors=2.
From the operational rules stated for IFNs in Definition15, we have(25)Љ1Љ=ЉÜ1ς−Љ−11/ς,1−Љ1−Õ1ς−Љ−11/ς.and(26)Ж1Б1Љ=1−Ж11−ЉÜ1ς−Љ−11/ςς−Ж1−11/ς,Ж11−Љ1−Õ1ς−Љ−11/ςς−Ж1−11/ς,
Similarly,(27)Ж2Б2Љ=1−Ж21−ЉÜ2ς−Љ−11/ςς−Ж2−11/ς,Ж21−Љ1−Õ2ς−Љ−11/ςς−Ж2−11/ς.
Then,(28)Ж1Б1Љ+Ж2Б2Љ=1−1−1−Ж11−ЉÜ1ς−Љ−11/ςς−Ж1−11/ςς+1−1−Ж21−ЉÜ2ς−Љ−11/ςς−Ж2−11/ςς−11/ςЖ11−Љ1−Õ1ς−Љ−11/ςς−Љ1−11/ςς+Ж21−Љ1−Õ2ς−Љ−11/ςς−Ж2−11/ςς−11/ς;=1−Ж11−ЉÜ1ς−Љ−11/ςς−Ж1−11/ςς+Ж21−ЉÜ2ς−Љ−11/ςς−Ж2−11/ςς−11/ςЖ11−Љ1−Õ1ς−Љ−11/ςς−Ж1−1+Ж21−Љ1−Õ2ς−Љ−11/ςς−Ж2−1−11/ς;Ж1Б1Љ+Ж2Б2Љ=1−⊕g=12Жg1−ЉÜgς−Љ−11/ςς−⊕g=12Жg+11/ς⊕g=12Жg1−Љ1−Õgς−Љ−11/ςς−⊕g=12Жg+11/ς;
If equation (24) secures for s=m(29)Ж1Б1Љ+Ж2Б2Љ+⋯+ЖmБmЉ=1−⊕g=1mЖg1−ЉÜgς−Љ−11/ςς−⊕g=1mЖg+11/ς⊕g=1mЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1mЖg+1.1/ς.
Then, whiles=m+1, by the operational rules provided in Definition 15, we have(30)Ж1Б1Љ+Ж2Б2Љ+⋯+Жm+1Бm+1Љ=Ж1Б1Љ+Ж2Б2Љ+⋯+ЖmБmЉ+Жm+1Бm+1Љ=1−⊕g=1mЖg1−ЉÜgς−Љ−11/ςς−⊕g=1mЖg+11/ς⊕g=1mЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1mЖg+11/ς+1−Жm+11−ЉÜm+1ς−Љ−11/ςς−Жm+1−11/ςЖm+11−Љ1−Õm+1ς−Љ−11/ςς−Жm+1−11/ς,Ж1Б1Љ+Ж2Б2Љ+⋯+Жm+1Бm+1Љ=1−⊕g=1m+1Жg1−ЉÜgς−Љ−11/ςς−⊕g=1m+1Жg+11/ς⊕g=1m+1Жg1−Љ1−Õgς−Љ−11/ςς−⊕g=1m+1Жg+11/ς.
That is, equation (24) is true for s=m+1. So, equation (24) is true for all g. Then,(31)⊕g=1sЖgБgЉ1/Љ=1Љ1−⊕g=1sЖg1−ЉÜgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς.
Therefore,(32)IFSSGPAЉБ1,Б2,…,Бs=1Љ1−⊕g=1sЖg1−ЉÜgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς,which completes the proof of the Theorem 2.Theorem 3.
The IFSSGPA operator has the following properties:(1)
Commutativity: LetБ1∗,Б2∗,…,Бs∗ be any permutation of Б1,Б2,…,Бs , then(33)IFSSGPAЉБ1∗,Б2∗,...,Бs∗=IFSSGPAЉБ1,Б2,…,Бs.(2)
Idempotency: If allБg=Бg=1,2,…,s then(34)IFSSGPAЉБ1,Б2,...,Бs=Б.(3)
Bonudedness: The IFSSGPA operator lies among the min and max operators(35)Б−≤IFSSGPAЉБ1,Б2,...,Бs≤Б+.Proof.
(1)
Suppose thatБ1∗,Б2∗,...,Бs∗ is any permutation of Б1,Б2,...,Бs, then for each ъg there exists one and only one Бh∗ such that Бg=Бh∗ and vice versa. And, TБg=TБh∗. Then, based on Theorem (2), we have(36)IFSSGPAЉБ1,Б2,...,Бs=⊕g=1sЖgБgЉ1/Љ=⊕h=1sЖh∗Љh∗Љ1/Љ=IFSSGPAЉБ1∗,Б2∗,...,Бs∗.(2)
SinceБg=Б for all g=1,2,...,s, thus(37)IFSSGPAЉБ1,Б2,...,Бs=1Љ1−⊕g=1sЖg1−ЉÜgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς,=1Љ1−1−ЉÜς−Љ−11/ςς1/ςς−1Љ−11/ς1−1Љ1−1−Љ1−Õς−Љ−11/ςς1/ςς−1Љ−11/ς=Ü,Õ=Б.(3)
LetБ−=Ü−,Õ+,Б+=Ü+,Õ−, where Ü−=min1≤g≤sÜ,Ü+=max1≤g≤sÜ,Õ−=min1≤g≤sÕ,Õ+=max1≤g≤sÕ for all g=1,2,...,s, we have(38)1Љ1−⊕g=1sЖg1−ЉÜ−ς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς≤1Љ1−⊕g=1sЖg1−ЉÜgς−Ж−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς≤1Љ1−⊕g=1sЖg1−ЉÜ+ς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς.
And, meanwhile,(39)1−1Љ1−⊕g=1sЖg1−Љ1−Õ−ς−Љ−11ςς−⊕g=1sЖg+11ςς−1Љ−11/ς≤1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11ςς−⊕g=1sЖg+11ςς−1Љ−11/ς≤1−1Љ1−⊕g=1sЖg1−Љ1−Õ+ς−Љ−11ςς−⊕g=1sЖg+11ςς−1Љ−11/ς.
So, we have(40)1љ1−⊕g=1sЖg1−љÜ−ς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς−1−1љ1−⊕g=1sЖg1−љ1−Õ+ς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς≤1љ1−⊕g=1sЖg1−љÜgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς−1−1љ1−⊕g=1sЖg1−љ1−Õgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς≤1љ1−⊕g=1sЖg1−љÜ+ς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς−1−1љ1−⊕g=1sЖg1−љ1−Õ−ς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς.
Then, according to Definition3, Definition 5, and Theorem 2, we have(41)Ъ−≤IFSSGPAљЪ1,Ъ2,…,Ъs≤Ъ+.
Furthermore, we shall discuss a few cases of the initiated AO with respect to the parameterљ, which is listed as follows:(1)
IfЉ=1, then the IFSSGPA operator reduces into IFSS power average (PA) operator:(42)IFSSGPAЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪg=1−⊕g=1sЖg1−Üg−⊕g=1sЖg+11/ς,⊕g=1sЖgÕgς−⊕g=1sЖg+11/ς.(2)
Ifљ=1 and ς=0, then the IFSSGPA operator reduces into IF power average operator based on algebraic operation. That is,(43)IFSSGPAς=0Ъ1,Ъ2,…,Ъs=⊕g=1sЖgЪg=1−⊗g=1s1−ÜgЖg,⊗g=1sÕgЖg.(3)
Ifљ=1, supЪg,Ъh=b (i.e., a constant) and ς=0, then the IFSSGPA operator reduces into IF average operator based on algebraic operation. That is,(44)IFSSGPAς→0Ъ1,Ъ2,…,Ъs=1s⊕g=1sЖg=1−⊗g=1s1−Üg1/s,⊗g=1sÕg1/s.Definition 17.
For a group of IFNsЪgg=1,2,…,s, a weighted intuitionistic fuzzy Schweizer-Sklar generalized power average (WIFSSGPA) operator is a function ℑs⟶ℑ,(45)WIFSSGPAϕ,љЪ1,Ъ2,…,Ъs=⊕ssg=1sϕg1+TЪgЪgљ⊕g=1sϕg1+TЪg1/љ.where TЪg=⊕h=1,g≠hsϕhSupЪg,Ъh,ϕ=ϕ1,ϕ2,...,ϕsT is the weight vector for Ъgg=1,2,…,s such that ϕg∈0,1 and ⊕g=1sϕg=1, parameter љ∈0,+∞, and SupЪg,Ъh is the support for ъg from ъh with the following constraint:
(1)SupЪg,Ъh∈0,1; (2) SupЪg,Ъh=SupЪh,Ъg; (3) SupЪg,Ъh≥SupЪu,Ъv if DSЪg,Ъh<DSЪu,Ъv, where DS is the distance measure among two IFNs.
To write (43) in an uncomplicated way, we have(46)Ћg=ϕg1+TЪg⊕g=1sϕg1+TЪg.
So, from equations (46) and (45) ,we have(47)WIFSSGPAϕ,љЪ1,Ъ2,…,Ъs=⊕ssg=1sЋgЪgљ1/љ.Theorem 4.
LetЪgg=1,2,...,s be a group of IFNs, then the value aggregated utilizing Definition 17 is still IFN, and we have(48)WIFSSGPAљ,ϕЪ1,Ъ2,…,Ъs=1љ1−⊕g=1sЖg1−љÜgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς,1−1љ1−⊕g=1sЖg1−љ1−Õgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς.Proof.
Proof of the Theorem is the same as Theorem2.Theorem 5.
For a faction of IFNsЪgg=1,2,…,s, Љ is a parameter and љ>0. ϕ=ϕ1,ϕ2,…,ϕsT is the weighting vector for Ъgg=1,2,…,s, ϕg∈0,1 and ⊕g=1sϕg=1. If ϕ=1/s,1/s,…,1/sT, then WIFSSGPA degenerates into the IFSSGPA operator:(49)IFSSGPAљЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪgљ1/љ=1љ1−⊕g=1sЖg1−љÜgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς,1−1љ1−⊕g=1sЖg1−љ1−Õgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς.where Ћg=1+TЪg/⊕g=1s1+TЪg is the power weight vector. Furthermore, we shall discuss a few cases of the initiated AO with respect to the parameters Љ and ϕ , which are listed below:(1)
Ifљ=1, ϕ=1/s,1/s,…,1/sT, then the WIFSSGPA operator reduces into the IFSS power average (PA) operator:(50)IFSSPAЪ1,Ъ2,…,Ъs=⊕g=1sЋgЪg=1−⊕g=1sЋg1−Üg−⊕g=1sЋg+11/ς,⊕g=1sЋgÕgς−⊕g=1sЋg+11/ς.(2)
IfЉ=1, ϕ=1/s,1/s,…,1/sT and ς=0, then the WIFSSGPA operator reduces into the IFPA operator based on algebraic operation. That is,(51)WIFSSGPAς=0Ъ1,Ъ2,…,Ъs=⊕g=1sЋgЪg=1−⊗g=1s1−ÜgЋg,⊗g=1sÕgЋg.(3)
IfЉ=1, ϕ=1/s,1/s,…,1/sT, supЪg,Ъh=b (i.e., a constant) and ς=0, then the WIFSSGPA operator reduces into IF average operator based on algebraic operation. That is,(52)WIFSSGPAς=0Ъ1,Ъ2,…,Ъs=1s⊕g=1sЪg=1−⊗g=1s1−Üg1/s,⊗g=1sÕg1/s.Similar to IFSSGPA operator, the WIFSSGPA operator comprises the characteristics of idempotency and boundedness. The WIFSSGPA operator does not have the property of commutativity.Similar to IFSSGPA operator, the WIFSSGPA operator comprises the characteristics of idempotency and boundedness. The WIFSSGPA operator does not have the property of commutativity.Definition 18.
For a faction of IFNsЪgg=1,2,..,s, parameter љ∈0,+∞ and if¯¯ρg be the gth largest of them. Then, the IFSSG power ordered weighted averaging (IFSSGPOWA) operator is a function ℑs⟶ℑ,(53)IFSSGPOWAЪ1,Ъ2,…,Ъs=⊕g=1sЖgЋρgљ1/љ,where Жg=kUg/L−kUg−1/L with Жg∈0,1 and ⊕g=1sЖg=1,(54)Ug=⊕j=1gQρj,L=⊕g=1sQρg,Qρg=1+TЪρg,TЪρg=⊕j=1,g≠jssupЪρg,Ъρj.
In (51), TЪρg expresses support degree that the gth largest argument takes from all other arguments, supЪρg,Ъρj denotes support for Ъρg from Ъρj. k:0,1⟶0,1 is a basic BUM mapping [38], which must hold the following axioms:(55)1k0=0,2k1=1,3ka≥kb,ifa>b.Theorem 6.
For a faction of IFNsЪgg=1,2,…,s, ζ is a parameter and љ>0. Then, the result aggregated utilizing WIFSSGPOWA is still IFN, and we have(56)IFSSGPOWAΘ,љЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪρgљ1/љ=1љ1−⊕g=1sЖg1−љÜρgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς,1−1љ1−⊕g=1sЖg1−љ1−Õρgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς.where Жg satisfies (51) and Ъρg be the gth largest of Ъgg=1,2,.…,s and Љ is the parameter such that љ∈0,+∞.Proof.
Proof is same as Theorem2.
Now, we will discuss some special cases of the IFSSGPOWA operator with respect to the parametersЉ and ς.(1)
Ifљ=1, then the IFSSGPOWA operator degenerates into the following formula:(57)FSSGPOWAΘ,љЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪg=1−⊕g=1sЖg1−Üρg−⊕g=1sЖg+11/ς,⊕g=1sЖgÕρgς−⊕g=1sЖg+11/ς.(2)
Ifљ=1 and ς=0, then the IFSSGPOWA operator degenerates into the following formula:(58)IFSSGPOWAЖ,љЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪg=1−⊗g=1s1−ÜρgЖg,⊗g=1sÕρgЖg.
This is the IFPOWA operator based on algebraic operational laws for IFNs.


### 3.2. Some Intuitionistic Fuzzy Schweizer-Sklar Generalized Power Geometric Aggregation Operators

In this subpart, the intuitionistic fuzzy Schweizer-Sklar generalized power geometric average operator, weighted intuitionistic fuzzy Schweizer-Sklar generalized power geometric average operator, and intuitionistic fuzzy Schweizer-Sklar generalized power ordered weighted geometric average operator discuss their desirable properties and some core cases with respect to two generalized parameters.Definition 19.
For a faction of IFNsЪgg=1,2,…,s, intuitionistic fuzzy Schweizer-Sklar generalized power geometric average (IFSSGPGA) operator is a function ℑs→ℑ,(59)IFSSGPGAљЪ1,Ъ2,…,Ъs=1љ⊗g=1sљЪg1+TЪg⊕h=1s1+TЪh,where TЪg=⊕h=1,g≠hsϕhSupЪg,Ъh, parameter љ∈0,+∞, and SupЪg,Ъh is the support for ъg from ъh with the following constraint:
(1)SupЪg,Ъh∈0,1; (2) SupЪg,Ъh=SupЪh,Ъg;
(3)SupЪg,Ъh≥SupЪu,Ъv if DSЪg,Ъh<DSЪu,Ъv, where DS is the distance measure among two IFNs.
To write (56) in an uncomplicated way, we have(60)Ћg=1+TЪg⊕g=1s1+TЪg.
So, from equations (59) and (59), we have(61)IFSSGPGAљЪ1,Ъ2,…,Ъs=1љ⊗g=1sљЪgљg.Theorem 7.
LetЪgg=1,2,...,s be a faction of IFNs, then the value aggregated utilizing Definition 19 is still IFN, and we have(62)IFSSGPGљЪ1,Ъ2,…,Ъs=1−1Ъ1−⊕g=1sЋg1−љ1−Ügς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς,1љ1−⊕g=1sЋg1−ЪÕgς−Ъ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς.Proof.
In the following, we first prove(63)љЪ1Ψ1×љЪ2Ψ2×⋯×љЪsΨs=⊕g=1sЋg1−Ъ1−Ügς−љ−11/ςς−⊕g=1sЋg+11/ς,1−⊕g=1sЋg1−ЪÕgς−љ−11/ςς−⊕g=1sЋg+11/ς.
By utilizing mathematical induction ons:
Fors=2.
From the operational rules defined for IFNs in Definition15, we have(64)љЪ1=1−љ1−Ü1ς−љ−11/ς,љÕ1ς−љ−11/ς.
And,(65)љЪ1Ћ1=Ћ11−љ1−Ü1ς−љ−11/ςς−Ћ1−11/ς,1−Ћ11−љÕ1ς−љ−11/ςς−Ћ1−11/ς.
Similarly,(66)љЪ2Ћ2=Ћ21−љ1−Ü2ς−љ−11/ςς−Ћ2−11/ς,1−Ћ21−љÕ2ς−љ−11/ςς−Ћ2−11ς.
Then,(67)љЪ1Ћ1×љЪ2Ћ2=Ћ11−љ1−Ü1ς−љ−11/ςς−Ћ1−11/ςς+Ћ21−љ1−Ü2ς−љ−11/ςς−Ћ2−11/ςς−11/ς,1−1−1−Ћ11−љÕ1ς−љ−11/ςς−Ћ1−11/ςς+1−1−Ћ21−љÕ2ς−љ−11/ςς−Ћ2−11/ςς−11/ς;=Ћ11−љ1−Ü1ς−љ−11/ςς−Ћ1−1+Ћ21−љ1−Ü2ς−љ−11/ςς−Ћ2−1−11/ς,1−Ћ11−љÕ1ς−љ−11/ςς−Ћ1−11/ςς+Ћ21−љÕ2ς−љ−11/ςς−Ћ2−11/ςς−11/ς;љЪ1Ћ1×љЪ2Ћ2=⊕g=12Ћg1−љ1−Ügς−љ−11/ςς−⊕g=12Ћg+11/ς,1−⊕g=12Ћg1−љÕgς−љ−11/ςς−⊕g=12Ћg+11/ς;.
If (60) holds for s=m(68)љЪ1Ψ1×љЪ2Ψ2×⋯×љЪmΨm=⊕g=1mΨg1−љ1−Ügς−љ−11/ςς−⊕g=1mΨg+11/ς,1−⊕g=1mΨg1−љÕgς−љ−11/ςς−⊕g=1mΨg+11/ς.
Then, whens=m+1, by the operational laws specified in Definition 15, we have(69)љЪ1Ћ1×љЪ2Ћ2×⋯×љЪm+1Ћm+1=љЪ1Ћ1×љЪ2Ћ2×⋯×љЪmЋm×љЪm+1Ћm+1=⊕g=1mЋg1−љ1−Ügς−љ−11/ςς−⊕g=1mЋg+11/ς,1−⊕g=1mЋg1−љÕgς−љ−11/ςς−⊕g=1mЋg+11/ς+Ћm+11−љ1−Üm+1ς−љ−11/ςς−љm+1−11/ς,1−Ћm+11−љÕm+1ς−љ−11/ςς−Ћm+1−11/ς,љЪ1Ћ1×љЪ2Ћ2×⋯×љЪm+1Ћm+1=⊕g=1m+1Ћg1−љ1−Ügς−љ−11/ςς−⊕g=1m+1Ћg+11/ς,1−⊕g=1m+1Ћg1−љÕgς−љ−11/ςς−⊕g=1m+1Ћg+11/ς.
That is, (60) is true for g=m+1. So (60) is true for all g. Then,(70)1љ⊗g=1sљЪgЋg=1−1љ1−⊕g=1sЋg1−љ1−Ügς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς,1љ1−⊕g=1sЋg1−љÕgς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς,
Therefore,(71)IFSSGPGAљЪ1,Ъ2,…,Ъs=1−1љ1−⊕g=1sЋg1−љ1−Ügς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς,1љ1−⊕g=1sЋg1−љÕgς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς.
This completes the proof of the Theorem7.
Furthermore, we shall discuss a few cases of the initiated AO with respect to the parametersљ and ϕ , which are listed as follows:(1)
Ifљ=1, then the IFSSGPGA operator reduces into the IFSSP geometric average operator:(72)IFSSGPGAЪ1,Ъ2,…,Ъs=⊗g=1sЪgЋg=⊕g=1sЋgÜgς−⊕g=1sЋg+11/ς,1−⊕g=1sЋg1−Õg−⊕g=1sЋg+11/ς.(2)
Ifљ=1 and ς=0, then the IFSSGPGA operator reduces into the IFP geometric average operator based on algebraic operation. That is,(73)IFSSGPGAς=0Ъ1,Ъ2,…,Ъs=⊗g=1sЪgЋg=⊗g=1sÜgЋg,1−⊗g=1s1−ÕgЋg.(3)
Ifљ=1, supЪg,Ъh=b (i.e., a constant), and ς=0, then the IFSSGPGA operator reduces into the IF geometric average operator based on algebraic operation. That is,(74)IFSSGPGAς=0Ъ1,Ъ2,…,Ъs=⊗g=1sЋg1/s=⊗g=1sÜg1/s,1−⊗g=1s1−Õg1/s.
The IFSSGPGA operator has the following properties of commutativity, idempotency, and boundedness.Definition 20.
For a faction of IFNsЪgg=1,2,…,s, WIFSSGPGA operator is a function ℑs→ℑ,(75)WIFSSGPGAϕ,љЪ1,Ъ2,…,Ъs=1љ⊗g=1sљЪgϕg1+Tљg⊕h=1sϕh1+Tљh,where TЪg=⊕h=1,g≠hsϕhSupЪg,Ъh is the weight vector for Ъgg=1,2,...,s such that ϕg∈0,1 and ⊕g=1sϕg=1, parameter љ∈0,+∞ and SupЪg,Ъh is the support for ъg from ъh with the following constraint:
(1)SupЪg,Ъh∈0,1; (2) SupЪg,Ъh=SupЪh,Ъg; (3) SupЪg,Ъh≥SupЪu,Ъv if DSЪg,Ъh<DSЪu,Ъv, where DS is the distance measure among two IFNs.
To write (72) in an uncomplicated way, we have(76)Жg=ϕg1+TЪg⊕g=1sϕg1+TЪg.
So, from equations (67) and (66), we have(77)WIFSSGPGAϕ,љЪ1,Ъ2,…,Ъs=1љ⊗g=1sљЪgЖg.Theorem 8.
LetЪgg=1,2,…,s be a faction of IFNs, then the value aggregated utilizing Definition 20 is still IFN, and we have(78)WIFSSGPGAљ,ϕЪ1,Ъ2,…,Ъs=1−⊕g=1sЖg1љ−1−1−Ügς−1љ−11/ς,⊕g=1sЖg1љ−1−Õgς−1љ−11/ς.
Furthermore, we shall discuss a few cases of the initiated AO with respect to the parametersЉ and ϕ , which are listed as follows:(1)
IfЉ=1, ϕ=1/s,1/s,...,1/sT, then the WIFSSGPGA operator reduces into the IFSSP average (IFSSPA) operator:(79)WIFSSWPGAЪ1,Ъ2,…,Ъs=⊗g=1sЖgЖg=⊕g=1sЪgÜgς−⊕g=1sЖg+11/ς,1−⊕g=1sЖg1−Õg−⊕g=1sЖg+11/ς.(2)
IfЉ=1, ϕ=1/s,1/s,…,1/sT, and ς=0, then the WIFSSGPGA operator reduces into the IFP geometric average operator based on algebraic operation. That is,(80)WIFSSGPGAς=0Ъ1,Ъ2,…,Ъs=⊗g=1sЪgЖg=⊗g=1sÜgЖg,1−⊗g=1s1−ÕgЖg.(3)
Ifљ=1, ϕ=1/s,1/s,…,1/sT, supъg,ъh=b (i.e., a constant), and ς=0, then the WIFSSGPGA operator reduces into the IFG average operator based on algebraic operation. That is,(81)WIFSSGPGAς=0Ъ1,Ъ2,…,Ъs=⊗g=1sЪg1/s=⊗g=1sÜg1/s,1−⊗g=1s1−Õg1/s.
The WIFSSGPGA operator has the properties of idempotency and boundedness. But the WGIFSSPGA operator does not have the property of commutativity.Definition 21.
For a faction of IFNsЪgg=1,2,..,s, parameters љ∈0,+∞ and ъρg be the gth largest of them. Then, the IFSSG power ordered weighted geometric averaging (IFSSGPOWGA) operator is a function ℑs→ℑ,(82)IFSSGPOWGAЪ1,Ъ2,...,Ъs=1љ⊗g=1sЪρgЋg,where Ћg=kUg/L−kUg−1/L with Ћg∈0,1 and ⊕g=1sЂg=1,Ug=⊕j=1gQρj,L=⊕g=1sQρg,Qρg=1+TЪρg.(83)TЪρg=⊕j=1,g≠jssupЪρg,Ъρj
In (74), TЪρg expresses support degree that the gth largest argument takes from all other arguments, supЪρg,Ъρj denotes support for Ъρg from Ъρj. k:0,1→0,1 is a basic BUM mapping [38], which must hold the following axioms:
(1)k0=0, (2) k1=1 and (3) ka≥kb if a>b .Theorem 9.
For a faction of IFNsъgg=1,2,...,s, Љ is a parameter and Љ>0. Then, the result aggregated utilizing WIFSSGPOWGA is still IFN, and we have(84)IFSSGPOWGAΘ,љЪ1,Ъ2,...,Ъs=1љ⊗g=1sљЪρgЋ=1−⊕g=1sЋg1љ−1−1−Üρgς−1љ−11/ς,⊕g=1sЋg1љ−1−Õρgς−1љ−11/ς.where Ћg satisfies (51) and ъρg be the gth largest of Ъgg=1,2,....,s and Љ is the parameter such that љ∈0,+∞.Proof.
Proof is same as Theorem7.
Now, we will discuss some special cases of the GIFSSPOWA operator with respect to the parametersζ and ς.(1)
Ifљ=1, then the IFSSGPOWGA operator degenerates into the following formula:(85)IFSSGPOWGAЋ,љЪ1,Ъ2,...,Ъs=⊗g=1sЪρgЋg=⊕g=1sЋgÜρgς−⊕g=1sЋg+11/ς,1−⊕g=1sЋg1−Õρg−⊕g=1sЋg+11/ς.(2)
Ifљ=1 and ς=0, then the IFSSGPOWGA operator degenerates into the following formula:(86)IFSSGPOWGAΘ,љЪ1,Ъ2,...,Ъs=⊗g=1sЪρgЋg=⊗g=1sÜρgЋg,1−⊗g=1s1−ÕρgЋg.
This is the IFPOWGA operator based on algebraic operational laws for IFNs.


## 3.1. Weighted Generalized Intuitionistic Fuzzy Schweizer-Sklar Power Aggregation Operator

In this subpart, we advanced intuitionistic fuzzy Schweizer-Sklar generalized power average (IFSSGPA) operator, weighted intuitionistic fuzzy Schweizer-Sklar generalized power average (WIFSSGPA) operator, intuitionistic fuzzy Schweizer-Sklar generalized power ordered weighted averaging (GIFSSPOWA) operator, and conferred their privileged properties and various cases.Definition 16.
For a faction of IFNsБgg=1,2,…,s, IFSSGPA operator is a function ℑs⟶ℑ,(20)IFSSGPAББ1,Б2,…,Бs=⊕SSg=1s1+TБgБgБ⊕g=1s1+TБg1/Б,where TБg=⊕h=1,g≠hssupБg,Бh, parameters Љ∈0,+∞ and Supъg,ъh are the SPD for ъg from ъh with the following constraint:
(1)Supъg,ъh∈0,1; (2) SupБg,Бh=SupБh,Бg; (3) SupБg,Бh≥SupБu,Бv if DSБg,Бh<DSБu,Бv, where DS is the distance measure among two IFNs.
To compose (20) in unsophisticated mode, we have(21)Жg=1+TБg⊕g=1s1+TБg.
So, equations (21) and (20) become(22)IFSSgPAЉБ1,Б2,…,Бs=⊕SSg=1sЖgif¯¯gЉ1/Љ,Theorem 2.
LetБgg=1,2,…,s be a faction of IFNs, then the value aggregated utilizing Definition 16 is still IFN, and we have(23)IFSSGPAЉБ1,Б2,…,Бs=1Љ1−⊕g=1sЖg1−ЉÜgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς.Proof.
In the succeeding, firstly, we can prove the subsequent:(24)Ж1ъ1Љ+Ж2ъ2Љ+...+ЖsъsЉ=1−⊕g=1sЖg1−ЉÜgς−Љ−11ςς−⊕g=1sЖg+11ς⊕g=1sЖg1−Љ1−Õgς−Љ−11ςς−⊕g=1sЖg+11ς.
By utilizing mathematical induction ons
Fors=2.
From the operational rules stated for IFNs in Definition15, we have(25)Љ1Љ=ЉÜ1ς−Љ−11/ς,1−Љ1−Õ1ς−Љ−11/ς.and(26)Ж1Б1Љ=1−Ж11−ЉÜ1ς−Љ−11/ςς−Ж1−11/ς,Ж11−Љ1−Õ1ς−Љ−11/ςς−Ж1−11/ς,
Similarly,(27)Ж2Б2Љ=1−Ж21−ЉÜ2ς−Љ−11/ςς−Ж2−11/ς,Ж21−Љ1−Õ2ς−Љ−11/ςς−Ж2−11/ς.
Then,(28)Ж1Б1Љ+Ж2Б2Љ=1−1−1−Ж11−ЉÜ1ς−Љ−11/ςς−Ж1−11/ςς+1−1−Ж21−ЉÜ2ς−Љ−11/ςς−Ж2−11/ςς−11/ςЖ11−Љ1−Õ1ς−Љ−11/ςς−Љ1−11/ςς+Ж21−Љ1−Õ2ς−Љ−11/ςς−Ж2−11/ςς−11/ς;=1−Ж11−ЉÜ1ς−Љ−11/ςς−Ж1−11/ςς+Ж21−ЉÜ2ς−Љ−11/ςς−Ж2−11/ςς−11/ςЖ11−Љ1−Õ1ς−Љ−11/ςς−Ж1−1+Ж21−Љ1−Õ2ς−Љ−11/ςς−Ж2−1−11/ς;Ж1Б1Љ+Ж2Б2Љ=1−⊕g=12Жg1−ЉÜgς−Љ−11/ςς−⊕g=12Жg+11/ς⊕g=12Жg1−Љ1−Õgς−Љ−11/ςς−⊕g=12Жg+11/ς;
If equation (24) secures for s=m(29)Ж1Б1Љ+Ж2Б2Љ+⋯+ЖmБmЉ=1−⊕g=1mЖg1−ЉÜgς−Љ−11/ςς−⊕g=1mЖg+11/ς⊕g=1mЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1mЖg+1.1/ς.
Then, whiles=m+1, by the operational rules provided in Definition 15, we have(30)Ж1Б1Љ+Ж2Б2Љ+⋯+Жm+1Бm+1Љ=Ж1Б1Љ+Ж2Б2Љ+⋯+ЖmБmЉ+Жm+1Бm+1Љ=1−⊕g=1mЖg1−ЉÜgς−Љ−11/ςς−⊕g=1mЖg+11/ς⊕g=1mЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1mЖg+11/ς+1−Жm+11−ЉÜm+1ς−Љ−11/ςς−Жm+1−11/ςЖm+11−Љ1−Õm+1ς−Љ−11/ςς−Жm+1−11/ς,Ж1Б1Љ+Ж2Б2Љ+⋯+Жm+1Бm+1Љ=1−⊕g=1m+1Жg1−ЉÜgς−Љ−11/ςς−⊕g=1m+1Жg+11/ς⊕g=1m+1Жg1−Љ1−Õgς−Љ−11/ςς−⊕g=1m+1Жg+11/ς.
That is, equation (24) is true for s=m+1. So, equation (24) is true for all g. Then,(31)⊕g=1sЖgБgЉ1/Љ=1Љ1−⊕g=1sЖg1−ЉÜgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς.
Therefore,(32)IFSSGPAЉБ1,Б2,…,Бs=1Љ1−⊕g=1sЖg1−ЉÜgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς,which completes the proof of the Theorem 2.Theorem 3.
The IFSSGPA operator has the following properties:(1)
Commutativity: LetБ1∗,Б2∗,…,Бs∗ be any permutation of Б1,Б2,…,Бs , then(33)IFSSGPAЉБ1∗,Б2∗,...,Бs∗=IFSSGPAЉБ1,Б2,…,Бs.(2)
Idempotency: If allБg=Бg=1,2,…,s then(34)IFSSGPAЉБ1,Б2,...,Бs=Б.(3)
Bonudedness: The IFSSGPA operator lies among the min and max operators(35)Б−≤IFSSGPAЉБ1,Б2,...,Бs≤Б+.Proof.
(1)
Suppose thatБ1∗,Б2∗,...,Бs∗ is any permutation of Б1,Б2,...,Бs, then for each ъg there exists one and only one Бh∗ such that Бg=Бh∗ and vice versa. And, TБg=TБh∗. Then, based on Theorem (2), we have(36)IFSSGPAЉБ1,Б2,...,Бs=⊕g=1sЖgБgЉ1/Љ=⊕h=1sЖh∗Љh∗Љ1/Љ=IFSSGPAЉБ1∗,Б2∗,...,Бs∗.(2)
SinceБg=Б for all g=1,2,...,s, thus(37)IFSSGPAЉБ1,Б2,...,Бs=1Љ1−⊕g=1sЖg1−ЉÜgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς,=1Љ1−1−ЉÜς−Љ−11/ςς1/ςς−1Љ−11/ς1−1Љ1−1−Љ1−Õς−Љ−11/ςς1/ςς−1Љ−11/ς=Ü,Õ=Б.(3)
LetБ−=Ü−,Õ+,Б+=Ü+,Õ−, where Ü−=min1≤g≤sÜ,Ü+=max1≤g≤sÜ,Õ−=min1≤g≤sÕ,Õ+=max1≤g≤sÕ for all g=1,2,...,s, we have(38)1Љ1−⊕g=1sЖg1−ЉÜ−ς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς≤1Љ1−⊕g=1sЖg1−ЉÜgς−Ж−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς≤1Љ1−⊕g=1sЖg1−ЉÜ+ς−Љ−11/ςς−⊕g=1sЖg+11/ςς−1Љ−11/ς.
And, meanwhile,(39)1−1Љ1−⊕g=1sЖg1−Љ1−Õ−ς−Љ−11ςς−⊕g=1sЖg+11ςς−1Љ−11/ς≤1−1Љ1−⊕g=1sЖg1−Љ1−Õgς−Љ−11ςς−⊕g=1sЖg+11ςς−1Љ−11/ς≤1−1Љ1−⊕g=1sЖg1−Љ1−Õ+ς−Љ−11ςς−⊕g=1sЖg+11ςς−1Љ−11/ς.
So, we have(40)1љ1−⊕g=1sЖg1−љÜ−ς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς−1−1љ1−⊕g=1sЖg1−љ1−Õ+ς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς≤1љ1−⊕g=1sЖg1−љÜgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς−1−1љ1−⊕g=1sЖg1−љ1−Õgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς≤1љ1−⊕g=1sЖg1−љÜ+ς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς−1−1љ1−⊕g=1sЖg1−љ1−Õ−ς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς.
Then, according to Definition3, Definition 5, and Theorem 2, we have(41)Ъ−≤IFSSGPAљЪ1,Ъ2,…,Ъs≤Ъ+.
Furthermore, we shall discuss a few cases of the initiated AO with respect to the parameterљ, which is listed as follows:(1)
IfЉ=1, then the IFSSGPA operator reduces into IFSS power average (PA) operator:(42)IFSSGPAЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪg=1−⊕g=1sЖg1−Üg−⊕g=1sЖg+11/ς,⊕g=1sЖgÕgς−⊕g=1sЖg+11/ς.(2)
Ifљ=1 and ς=0, then the IFSSGPA operator reduces into IF power average operator based on algebraic operation. That is,(43)IFSSGPAς=0Ъ1,Ъ2,…,Ъs=⊕g=1sЖgЪg=1−⊗g=1s1−ÜgЖg,⊗g=1sÕgЖg.(3)
Ifљ=1, supЪg,Ъh=b (i.e., a constant) and ς=0, then the IFSSGPA operator reduces into IF average operator based on algebraic operation. That is,(44)IFSSGPAς→0Ъ1,Ъ2,…,Ъs=1s⊕g=1sЖg=1−⊗g=1s1−Üg1/s,⊗g=1sÕg1/s.Definition 17.
For a group of IFNsЪgg=1,2,…,s, a weighted intuitionistic fuzzy Schweizer-Sklar generalized power average (WIFSSGPA) operator is a function ℑs⟶ℑ,(45)WIFSSGPAϕ,љЪ1,Ъ2,…,Ъs=⊕ssg=1sϕg1+TЪgЪgљ⊕g=1sϕg1+TЪg1/љ.where TЪg=⊕h=1,g≠hsϕhSupЪg,Ъh,ϕ=ϕ1,ϕ2,...,ϕsT is the weight vector for Ъgg=1,2,…,s such that ϕg∈0,1 and ⊕g=1sϕg=1, parameter љ∈0,+∞, and SupЪg,Ъh is the support for ъg from ъh with the following constraint:
(1)SupЪg,Ъh∈0,1; (2) SupЪg,Ъh=SupЪh,Ъg; (3) SupЪg,Ъh≥SupЪu,Ъv if DSЪg,Ъh<DSЪu,Ъv, where DS is the distance measure among two IFNs.
To write (43) in an uncomplicated way, we have(46)Ћg=ϕg1+TЪg⊕g=1sϕg1+TЪg.
So, from equations (46) and (45) ,we have(47)WIFSSGPAϕ,љЪ1,Ъ2,…,Ъs=⊕ssg=1sЋgЪgљ1/љ.Theorem 4.
LetЪgg=1,2,...,s be a group of IFNs, then the value aggregated utilizing Definition 17 is still IFN, and we have(48)WIFSSGPAљ,ϕЪ1,Ъ2,…,Ъs=1љ1−⊕g=1sЖg1−љÜgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς,1−1љ1−⊕g=1sЖg1−љ1−Õgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς.Proof.
Proof of the Theorem is the same as Theorem2.Theorem 5.
For a faction of IFNsЪgg=1,2,…,s, Љ is a parameter and љ>0. ϕ=ϕ1,ϕ2,…,ϕsT is the weighting vector for Ъgg=1,2,…,s, ϕg∈0,1 and ⊕g=1sϕg=1. If ϕ=1/s,1/s,…,1/sT, then WIFSSGPA degenerates into the IFSSGPA operator:(49)IFSSGPAљЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪgљ1/љ=1љ1−⊕g=1sЖg1−љÜgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς,1−1љ1−⊕g=1sЖg1−љ1−Õgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς.where Ћg=1+TЪg/⊕g=1s1+TЪg is the power weight vector. Furthermore, we shall discuss a few cases of the initiated AO with respect to the parameters Љ and ϕ , which are listed below:(1)
Ifљ=1, ϕ=1/s,1/s,…,1/sT, then the WIFSSGPA operator reduces into the IFSS power average (PA) operator:(50)IFSSPAЪ1,Ъ2,…,Ъs=⊕g=1sЋgЪg=1−⊕g=1sЋg1−Üg−⊕g=1sЋg+11/ς,⊕g=1sЋgÕgς−⊕g=1sЋg+11/ς.(2)
IfЉ=1, ϕ=1/s,1/s,…,1/sT and ς=0, then the WIFSSGPA operator reduces into the IFPA operator based on algebraic operation. That is,(51)WIFSSGPAς=0Ъ1,Ъ2,…,Ъs=⊕g=1sЋgЪg=1−⊗g=1s1−ÜgЋg,⊗g=1sÕgЋg.(3)
IfЉ=1, ϕ=1/s,1/s,…,1/sT, supЪg,Ъh=b (i.e., a constant) and ς=0, then the WIFSSGPA operator reduces into IF average operator based on algebraic operation. That is,(52)WIFSSGPAς=0Ъ1,Ъ2,…,Ъs=1s⊕g=1sЪg=1−⊗g=1s1−Üg1/s,⊗g=1sÕg1/s.Similar to IFSSGPA operator, the WIFSSGPA operator comprises the characteristics of idempotency and boundedness. The WIFSSGPA operator does not have the property of commutativity.Similar to IFSSGPA operator, the WIFSSGPA operator comprises the characteristics of idempotency and boundedness. The WIFSSGPA operator does not have the property of commutativity.Definition 18.
For a faction of IFNsЪgg=1,2,..,s, parameter љ∈0,+∞ and if¯¯ρg be the gth largest of them. Then, the IFSSG power ordered weighted averaging (IFSSGPOWA) operator is a function ℑs⟶ℑ,(53)IFSSGPOWAЪ1,Ъ2,…,Ъs=⊕g=1sЖgЋρgљ1/љ,where Жg=kUg/L−kUg−1/L with Жg∈0,1 and ⊕g=1sЖg=1,(54)Ug=⊕j=1gQρj,L=⊕g=1sQρg,Qρg=1+TЪρg,TЪρg=⊕j=1,g≠jssupЪρg,Ъρj.
In (51), TЪρg expresses support degree that the gth largest argument takes from all other arguments, supЪρg,Ъρj denotes support for Ъρg from Ъρj. k:0,1⟶0,1 is a basic BUM mapping [38], which must hold the following axioms:(55)1k0=0,2k1=1,3ka≥kb,ifa>b.Theorem 6.
For a faction of IFNsЪgg=1,2,…,s, ζ is a parameter and љ>0. Then, the result aggregated utilizing WIFSSGPOWA is still IFN, and we have(56)IFSSGPOWAΘ,љЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪρgљ1/љ=1љ1−⊕g=1sЖg1−љÜρgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς,1−1љ1−⊕g=1sЖg1−љ1−Õρgς−љ−11/ςς−⊕g=1sЖg+11/ςς−1љ−11/ς.where Жg satisfies (51) and Ъρg be the gth largest of Ъgg=1,2,.…,s and Љ is the parameter such that љ∈0,+∞.Proof.
Proof is same as Theorem2.
Now, we will discuss some special cases of the IFSSGPOWA operator with respect to the parametersЉ and ς.(1)
Ifљ=1, then the IFSSGPOWA operator degenerates into the following formula:(57)FSSGPOWAΘ,љЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪg=1−⊕g=1sЖg1−Üρg−⊕g=1sЖg+11/ς,⊕g=1sЖgÕρgς−⊕g=1sЖg+11/ς.(2)
Ifљ=1 and ς=0, then the IFSSGPOWA operator degenerates into the following formula:(58)IFSSGPOWAЖ,љЪ1,Ъ2,…,Ъs=⊕g=1sЖgЪg=1−⊗g=1s1−ÜρgЖg,⊗g=1sÕρgЖg.
This is the IFPOWA operator based on algebraic operational laws for IFNs.


## 3.2. Some Intuitionistic Fuzzy Schweizer-Sklar Generalized Power Geometric Aggregation Operators

In this subpart, the intuitionistic fuzzy Schweizer-Sklar generalized power geometric average operator, weighted intuitionistic fuzzy Schweizer-Sklar generalized power geometric average operator, and intuitionistic fuzzy Schweizer-Sklar generalized power ordered weighted geometric average operator discuss their desirable properties and some core cases with respect to two generalized parameters.Definition 19.
For a faction of IFNsЪgg=1,2,…,s, intuitionistic fuzzy Schweizer-Sklar generalized power geometric average (IFSSGPGA) operator is a function ℑs→ℑ,(59)IFSSGPGAљЪ1,Ъ2,…,Ъs=1љ⊗g=1sљЪg1+TЪg⊕h=1s1+TЪh,where TЪg=⊕h=1,g≠hsϕhSupЪg,Ъh, parameter љ∈0,+∞, and SupЪg,Ъh is the support for ъg from ъh with the following constraint:
(1)SupЪg,Ъh∈0,1; (2) SupЪg,Ъh=SupЪh,Ъg;
(3)SupЪg,Ъh≥SupЪu,Ъv if DSЪg,Ъh<DSЪu,Ъv, where DS is the distance measure among two IFNs.
To write (56) in an uncomplicated way, we have(60)Ћg=1+TЪg⊕g=1s1+TЪg.
So, from equations (59) and (59), we have(61)IFSSGPGAљЪ1,Ъ2,…,Ъs=1љ⊗g=1sљЪgљg.Theorem 7.
LetЪgg=1,2,...,s be a faction of IFNs, then the value aggregated utilizing Definition 19 is still IFN, and we have(62)IFSSGPGљЪ1,Ъ2,…,Ъs=1−1Ъ1−⊕g=1sЋg1−љ1−Ügς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς,1љ1−⊕g=1sЋg1−ЪÕgς−Ъ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς.Proof.
In the following, we first prove(63)љЪ1Ψ1×љЪ2Ψ2×⋯×љЪsΨs=⊕g=1sЋg1−Ъ1−Ügς−љ−11/ςς−⊕g=1sЋg+11/ς,1−⊕g=1sЋg1−ЪÕgς−љ−11/ςς−⊕g=1sЋg+11/ς.
By utilizing mathematical induction ons:
Fors=2.
From the operational rules defined for IFNs in Definition15, we have(64)љЪ1=1−љ1−Ü1ς−љ−11/ς,љÕ1ς−љ−11/ς.
And,(65)љЪ1Ћ1=Ћ11−љ1−Ü1ς−љ−11/ςς−Ћ1−11/ς,1−Ћ11−љÕ1ς−љ−11/ςς−Ћ1−11/ς.
Similarly,(66)љЪ2Ћ2=Ћ21−љ1−Ü2ς−љ−11/ςς−Ћ2−11/ς,1−Ћ21−љÕ2ς−љ−11/ςς−Ћ2−11ς.
Then,(67)љЪ1Ћ1×љЪ2Ћ2=Ћ11−љ1−Ü1ς−љ−11/ςς−Ћ1−11/ςς+Ћ21−љ1−Ü2ς−љ−11/ςς−Ћ2−11/ςς−11/ς,1−1−1−Ћ11−љÕ1ς−љ−11/ςς−Ћ1−11/ςς+1−1−Ћ21−љÕ2ς−љ−11/ςς−Ћ2−11/ςς−11/ς;=Ћ11−љ1−Ü1ς−љ−11/ςς−Ћ1−1+Ћ21−љ1−Ü2ς−љ−11/ςς−Ћ2−1−11/ς,1−Ћ11−љÕ1ς−љ−11/ςς−Ћ1−11/ςς+Ћ21−љÕ2ς−љ−11/ςς−Ћ2−11/ςς−11/ς;љЪ1Ћ1×љЪ2Ћ2=⊕g=12Ћg1−љ1−Ügς−љ−11/ςς−⊕g=12Ћg+11/ς,1−⊕g=12Ћg1−љÕgς−љ−11/ςς−⊕g=12Ћg+11/ς;.
If (60) holds for s=m(68)љЪ1Ψ1×љЪ2Ψ2×⋯×љЪmΨm=⊕g=1mΨg1−љ1−Ügς−љ−11/ςς−⊕g=1mΨg+11/ς,1−⊕g=1mΨg1−љÕgς−љ−11/ςς−⊕g=1mΨg+11/ς.
Then, whens=m+1, by the operational laws specified in Definition 15, we have(69)љЪ1Ћ1×љЪ2Ћ2×⋯×љЪm+1Ћm+1=љЪ1Ћ1×љЪ2Ћ2×⋯×љЪmЋm×љЪm+1Ћm+1=⊕g=1mЋg1−љ1−Ügς−љ−11/ςς−⊕g=1mЋg+11/ς,1−⊕g=1mЋg1−љÕgς−љ−11/ςς−⊕g=1mЋg+11/ς+Ћm+11−љ1−Üm+1ς−љ−11/ςς−љm+1−11/ς,1−Ћm+11−љÕm+1ς−љ−11/ςς−Ћm+1−11/ς,љЪ1Ћ1×љЪ2Ћ2×⋯×љЪm+1Ћm+1=⊕g=1m+1Ћg1−љ1−Ügς−љ−11/ςς−⊕g=1m+1Ћg+11/ς,1−⊕g=1m+1Ћg1−љÕgς−љ−11/ςς−⊕g=1m+1Ћg+11/ς.
That is, (60) is true for g=m+1. So (60) is true for all g. Then,(70)1љ⊗g=1sљЪgЋg=1−1љ1−⊕g=1sЋg1−љ1−Ügς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς,1љ1−⊕g=1sЋg1−љÕgς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς,
Therefore,(71)IFSSGPGAљЪ1,Ъ2,…,Ъs=1−1љ1−⊕g=1sЋg1−љ1−Ügς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς,1љ1−⊕g=1sЋg1−љÕgς−љ−11/ςς−⊕g=1sЋg+11/ςς−1љ−11/ς.
This completes the proof of the Theorem7.
Furthermore, we shall discuss a few cases of the initiated AO with respect to the parametersљ and ϕ , which are listed as follows:(1)
Ifљ=1, then the IFSSGPGA operator reduces into the IFSSP geometric average operator:(72)IFSSGPGAЪ1,Ъ2,…,Ъs=⊗g=1sЪgЋg=⊕g=1sЋgÜgς−⊕g=1sЋg+11/ς,1−⊕g=1sЋg1−Õg−⊕g=1sЋg+11/ς.(2)
Ifљ=1 and ς=0, then the IFSSGPGA operator reduces into the IFP geometric average operator based on algebraic operation. That is,(73)IFSSGPGAς=0Ъ1,Ъ2,…,Ъs=⊗g=1sЪgЋg=⊗g=1sÜgЋg,1−⊗g=1s1−ÕgЋg.(3)
Ifљ=1, supЪg,Ъh=b (i.e., a constant), and ς=0, then the IFSSGPGA operator reduces into the IF geometric average operator based on algebraic operation. That is,(74)IFSSGPGAς=0Ъ1,Ъ2,…,Ъs=⊗g=1sЋg1/s=⊗g=1sÜg1/s,1−⊗g=1s1−Õg1/s.
The IFSSGPGA operator has the following properties of commutativity, idempotency, and boundedness.Definition 20.
For a faction of IFNsЪgg=1,2,…,s, WIFSSGPGA operator is a function ℑs→ℑ,(75)WIFSSGPGAϕ,љЪ1,Ъ2,…,Ъs=1љ⊗g=1sљЪgϕg1+Tљg⊕h=1sϕh1+Tљh,where TЪg=⊕h=1,g≠hsϕhSupЪg,Ъh is the weight vector for Ъgg=1,2,...,s such that ϕg∈0,1 and ⊕g=1sϕg=1, parameter љ∈0,+∞ and SupЪg,Ъh is the support for ъg from ъh with the following constraint:
(1)SupЪg,Ъh∈0,1; (2) SupЪg,Ъh=SupЪh,Ъg; (3) SupЪg,Ъh≥SupЪu,Ъv if DSЪg,Ъh<DSЪu,Ъv, where DS is the distance measure among two IFNs.
To write (72) in an uncomplicated way, we have(76)Жg=ϕg1+TЪg⊕g=1sϕg1+TЪg.
So, from equations (67) and (66), we have(77)WIFSSGPGAϕ,љЪ1,Ъ2,…,Ъs=1љ⊗g=1sљЪgЖg.Theorem 8.
LetЪgg=1,2,…,s be a faction of IFNs, then the value aggregated utilizing Definition 20 is still IFN, and we have(78)WIFSSGPGAљ,ϕЪ1,Ъ2,…,Ъs=1−⊕g=1sЖg1љ−1−1−Ügς−1љ−11/ς,⊕g=1sЖg1љ−1−Õgς−1љ−11/ς.
Furthermore, we shall discuss a few cases of the initiated AO with respect to the parametersЉ and ϕ , which are listed as follows:(1)
IfЉ=1, ϕ=1/s,1/s,...,1/sT, then the WIFSSGPGA operator reduces into the IFSSP average (IFSSPA) operator:(79)WIFSSWPGAЪ1,Ъ2,…,Ъs=⊗g=1sЖgЖg=⊕g=1sЪgÜgς−⊕g=1sЖg+11/ς,1−⊕g=1sЖg1−Õg−⊕g=1sЖg+11/ς.(2)
IfЉ=1, ϕ=1/s,1/s,…,1/sT, and ς=0, then the WIFSSGPGA operator reduces into the IFP geometric average operator based on algebraic operation. That is,(80)WIFSSGPGAς=0Ъ1,Ъ2,…,Ъs=⊗g=1sЪgЖg=⊗g=1sÜgЖg,1−⊗g=1s1−ÕgЖg.(3)
Ifљ=1, ϕ=1/s,1/s,…,1/sT, supъg,ъh=b (i.e., a constant), and ς=0, then the WIFSSGPGA operator reduces into the IFG average operator based on algebraic operation. That is,(81)WIFSSGPGAς=0Ъ1,Ъ2,…,Ъs=⊗g=1sЪg1/s=⊗g=1sÜg1/s,1−⊗g=1s1−Õg1/s.
The WIFSSGPGA operator has the properties of idempotency and boundedness. But the WGIFSSPGA operator does not have the property of commutativity.Definition 21.
For a faction of IFNsЪgg=1,2,..,s, parameters љ∈0,+∞ and ъρg be the gth largest of them. Then, the IFSSG power ordered weighted geometric averaging (IFSSGPOWGA) operator is a function ℑs→ℑ,(82)IFSSGPOWGAЪ1,Ъ2,...,Ъs=1љ⊗g=1sЪρgЋg,where Ћg=kUg/L−kUg−1/L with Ћg∈0,1 and ⊕g=1sЂg=1,Ug=⊕j=1gQρj,L=⊕g=1sQρg,Qρg=1+TЪρg.(83)TЪρg=⊕j=1,g≠jssupЪρg,Ъρj
In (74), TЪρg expresses support degree that the gth largest argument takes from all other arguments, supЪρg,Ъρj denotes support for Ъρg from Ъρj. k:0,1→0,1 is a basic BUM mapping [38], which must hold the following axioms:
(1)k0=0, (2) k1=1 and (3) ka≥kb if a>b .Theorem 9.
For a faction of IFNsъgg=1,2,...,s, Љ is a parameter and Љ>0. Then, the result aggregated utilizing WIFSSGPOWGA is still IFN, and we have(84)IFSSGPOWGAΘ,љЪ1,Ъ2,...,Ъs=1љ⊗g=1sљЪρgЋ=1−⊕g=1sЋg1љ−1−1−Üρgς−1љ−11/ς,⊕g=1sЋg1љ−1−Õρgς−1љ−11/ς.where Ћg satisfies (51) and ъρg be the gth largest of Ъgg=1,2,....,s and Љ is the parameter such that љ∈0,+∞.Proof.
Proof is same as Theorem7.
Now, we will discuss some special cases of the GIFSSPOWA operator with respect to the parametersζ and ς.(1)
Ifљ=1, then the IFSSGPOWGA operator degenerates into the following formula:(85)IFSSGPOWGAЋ,љЪ1,Ъ2,...,Ъs=⊗g=1sЪρgЋg=⊕g=1sЋgÜρgς−⊕g=1sЋg+11/ς,1−⊕g=1sЋg1−Õρg−⊕g=1sЋg+11/ς.(2)
Ifљ=1 and ς=0, then the IFSSGPOWGA operator degenerates into the following formula:(86)IFSSGPOWGAΘ,љЪ1,Ъ2,...,Ъs=⊗g=1sЪρgЋg=⊗g=1sÜρgЋg,1−⊗g=1s1−ÕρgЋg.
This is the IFPOWGA operator based on algebraic operational laws for IFNs.


## 4. An Application of Generalized Intuitionistic Fuzzy Schweizer-Sklar Power Aggregation Operator to Group Decision-Making

In this part, we consider the aforementioned generalized intuitionistic fuzzy Schweizer-Sklar power AOs to ascertain productive approaches for MAGDM under IF environments. Letale¯¯=ale¯¯1,ale¯¯2,...,ale¯¯g be the faction of detached alternatives, the faction of attributes is articulated by Cri¯¯=Cri¯¯1,Cri¯¯2,...,Cri¯¯h, the weight vector of the attributes is symbolized by W¯¯=wei¯¯1,wei¯¯2,...,wei¯¯hT such that w¯¯eie∈0,1,∑e=1hwei¯¯e=1, and DE¯¯=de¯¯1,de¯¯2,...,de¯¯p signifies the set of p DMs, with weight vector articulated by ϖ=ϖ1,ϖ2,...,ϖpT such that ϖf∈0,1,∑f=1pϖf=1. Assume that DT¯¯f=ъdefg×h is the decision matrix, where ъdef=Üdef,Õdef takes the form of IFN, given by the decision-maker de¯¯f for alternative ale¯¯d with respect to the attribute Cri¯¯e.Then, gamble on factual decision situations where the weight vectors of both DMs and attributes are entirely identified in advance or are unknown. For that reason, we initiate two MAGDM approaches established on the proposed GIFSSPA operators.The first approach is based for known weight vectors of both attributes and decision-makers, and the second approach is based for totally unknown weight vectors of both attributes and decision-makers.


### 4.1. Approach I: MAGDM with Known Weight Vectors of Attributes and Decision-Makers

In this subsection, to deal with real decision situations in which the weighting vectors of both decision-makers and attributes are totally known, we apply the WIFSSGPA operator and the WIFSSGPGA operator to establish the following approach I to solve MAGDM problems under IF environments. To do so, immediately go behind the steps as follows:Step I.1: normalizing each decision matrixDT¯¯f=Ъdefg×h. Generally, there are two types of attributes: (1) cost-type attribute and (2) benefit-type attribute. If the attributes are of the benefit type, there is no need to change them. If there are cost-type of attributes, then the following formula is utilized to change them into benefit-type attributes.(87)Ъdef=Üdef,Õdef=Õdef,Üdeffor costtype of attributeCR¯¯e.Step I.2: find out the support by utilizing the following formula:(88)SupЪdef,Ъdes=SM˜Ъdef,Ъdes=1−KЪdef−KЪdes,f,s=1,2,...,p;d=1,2,...,g;e=1,2,...,h.It assures the axioms for support mappings in Definition (16),SM˜Ъdef,Ъdes is the similarity measure given in Definition 6.Step I.3: find out the support degreeTЪdef that IFN Ъdef receives from other IFNs Ъdess=1,2,...,p;s≠f, where(89)TЪdef=∑s=1,s≠fpsupЪdef,Ъdes.Step I.4: utilize weightsϖff=1,2,....,p for the decision-makers DE¯¯f to determine weights Ξdef associated with the IFN Ъdef,(90)Ξdef=ϖf1+TЪdef∑f=1pϖf1+TЪdef,f=1,2,...,p.whereΞdef≥0 and ∑f=1pΞdef=1.Step I.5: aggregate all the individual decision matricesDT¯¯f=Ъdefg×hf=1,2,...,p into a group decision matrix by utilizing WGIFSSPA or WGIFSSPGA operators, where(91)Ъde=WIFSSGPAљ,ϖЪde1,Ъde2,...,Ъdep=⊕f=1pΞf1љ−1−Üdefς−1љ−11/ς,1−⊕f=1pΞf1љ−1−1−Õdefς−1љ−11/ς,or(92)Ъde=WIFSSGPGAљ,ϖЪde1,Ъde2,...,Ъdep=1−⊕f=1pΞf1љ−1−1−Üdefς−1љ−11/ς,⊕f=1pΞf1љ−1−Õdefς−1љ−11/ς.Step I.6: find out supportSupЪde,Ъdx by the following formula:(93)SupЪde,Ъdx=SM˜Ъde,Ъdx=1−KЪde−KЪdx,d=1,2,...,g,e=1,2,...,h,e≠xwhereKЪde and KЪdx are the knowledge measures and are calculated utilizing (6).Step I.7: find out the support degreeTЪde that IFN Ъde collects from other IFNs Ъdxx=1,2,...,h;e≠x, where(94)TЪde=∑x=1,x≠ehwei¯¯xsupЪde,Ъdx.Step I.8: determine weighting vectorϒded=1,2,....,g,e=1,2,...,h associated with Ъde ,(95)ϒde=w¯¯eie1+TЪde∑e=1hwei¯¯e1+TЪde.Step I.9: utilize WIFSSGPA or WIFSSGPGA operators to collective all evaluation valuesЪded=1,2,....,g,e=1,2,...,h into the overall evaluation value Ъdd=1,2,....,g matching to the alternatives ale¯¯dd=1,2,...,g:(96)Ъd=WIFSSGPAЉ,wei¯¯Ъd1,Ъd2,...,Ъdh=⊕e=1hϒde1Љ−1−Üdeς−1Љ−11/ς,1−⊕e=1hϒde1Љ−1−1−Õdeς−1Љ−11/ςor(97)Ъd=WIFSSGPGAЉ,wei¯¯Ъd1,Ъd2,...,Ъdh=1−⊕e=1hϒde1Љ−1−1−Üdeς−1Љ−11/ς,⊕e=1hϒde1Љ−1−Õdeς−1Љ−11/ς.Step I.10: find out the scoresSC¯¯Ъd for the overall IFN of the alternatives ale¯¯dd=1,2,...,g by exploiting Definition 3.Step I.11: rank all alternativesale¯¯dd=1,2,...,g and select the best one(s) with the ranking order Ъdd=1,2,...,g.


### 4.2. Approach II: MAGDM with Totally Unknown Weight Vectors of Attributes and Decision-Makers

In this section, to deal with real decision situations in which the weighting vectors of both decision-makers and attributes are totally unknown, we apply the WIFSSGPOA operator and the WIFSSGPOGA operator to establish the following approach II to solve MAGDM problems under IF environments. To do so, immediately follow the steps as follows:Step II.1: same as Step I.1.Step II.2: find out the supports exploiting the formula:(98)SupЪdeρf,Ъdeρs=SM˜Ъdeρf,Ъdeρs=1−KЪdeρf−KЪdeρs,f,s=1,2,...,p;d=1,2,...,g;e=1,2,...,h.It gratifies the axioms for support mappings in Definition16, SM˜Ъdeρf,Ъdeρs is the similarity measure specified in Definition 6.Step II.3: find out the support degreeTЪdeρf that IFN Ъdeρf receives from other IFNs Ъdeρss=1,2,...,p;s≠f, where(99)Tъdeρf=∑s=1,s≠fpsupъdeρf,ъdeρs.Step II.4: utilize (54) to determine weights Ξdef associated with the fth largest IFN Ъdef, we have Ξdef=kUf/L−kUf−1/L with Ξdef∈0,1 and ⊕f=1pΞdef=1, where k is the BUM function [38],(100)Uf=⊕s=1fQdeρs,Lde=⊕s=1pQdeρs,Qdeρs=1+TЪdeρs.Step II.5: combine all the entity decision matricesDT¯¯f=Ъdefg×hf=1,2,...,p into a group decision matrix DT˜=Ъdeg×h by utilizing the WIFSSGPOA or WIFSSGPOGA operators, where(101)Ъde=WIFSSGPOAЉЪde1,Ъde2,...,Ъdep=⊕f=1pΞdef1Љ−1−Üdeρfς−1Љ−11/ς,1−⊕f=1pΞdef1Љ−1−1−Õdeρfς−1Љ−11/ςor(102)Ъde=WIFSSGPOGAΞЪde1,Ъde2,...,Ъdep=1−⊕f=1pΞdef1Љ−1−1−Üdeρfς−1Љ−11/ς,⊕f=1pΞdef1Љ−1−Õdeρfς−1Љ−11/ς.Step II.6: find out supportsSupЪdρe,Ъdρx by the following formula:(103)SupЪdρe,Ъdρx=SM˜Ъdρe,Ъdρx=1−KЪdρe−KЪdρx,d=1,2,...,g,e=1,2,...,h,e≠xwhereKЪde and KЪdx are the knowledge measures and are calculated utilizing (6).Step II.7: determine the support degreeTЪdρe that the eth largest IFN ъdρe collects from other IFNs Ъdρxx=1,2,...,h;e≠x, where(104)TЪdρe=∑x=1,x≠ehsupЪdρe,Ъdρx.Step II.8: utilize (54) to determine weights ϒded=1,2,...,g;e=1,2,...,h: ϒde=kUde/L−kUde−1/L with ϒde∈0,1 and ⊕e=1hΞde=1, where k is the BUM function [38].(105)Ude=⊕x=1eQdρx,L=⊕e=1hQdρe,Qdρe=1+TЪdρe.Step II.9: utilize the WIFSSGPOA or WIFSSGPOGA operators to aggregate all evaluation valuesЪded=1,2,....,g,e=1,2,...,h into the overall evaluation value Ъdd=1,2,....,g communicating to the alternatives ale¯¯dd=1,2,...,g:(106)Ъd=WIFSSGPOAЉЪd1,Ъd2,...,Ъdh=⊕e=1hϒde1Љ−1−Üdρeς−1Љ−11/ς,1−⊕e=1hϒde1Љ−1−1−Õdρeς−1Љ−11/ςor(107)Ъd=WIFSSGPOGAЪd1,Ъd2,...,Ъdh=1−⊕e=1hϒde1Љ−1−1−Üdρeς−1Љ−11/ς,⊕e=1hϒde1Љ−1−Õdρeς−1Љ−11/ς.Step II.10: find out the scoresSC¯¯Ъd for the overall IFN of the alternatives ale¯¯dd=1,2,...,g by exploiting Definition 3.Step II.11: rank all alternativesale¯¯dd=1,2,...,g and select the most favorable one(s) with the ranking order Ъdd=1,2,...,g.


## 4.1. Approach I: MAGDM with Known Weight Vectors of Attributes and Decision-Makers

In this subsection, to deal with real decision situations in which the weighting vectors of both decision-makers and attributes are totally known, we apply the WIFSSGPA operator and the WIFSSGPGA operator to establish the following approach I to solve MAGDM problems under IF environments. To do so, immediately go behind the steps as follows:Step I.1: normalizing each decision matrixDT¯¯f=Ъdefg×h. Generally, there are two types of attributes: (1) cost-type attribute and (2) benefit-type attribute. If the attributes are of the benefit type, there is no need to change them. If there are cost-type of attributes, then the following formula is utilized to change them into benefit-type attributes.(87)Ъdef=Üdef,Õdef=Õdef,Üdeffor costtype of attributeCR¯¯e.Step I.2: find out the support by utilizing the following formula:(88)SupЪdef,Ъdes=SM˜Ъdef,Ъdes=1−KЪdef−KЪdes,f,s=1,2,...,p;d=1,2,...,g;e=1,2,...,h.It assures the axioms for support mappings in Definition (16),SM˜Ъdef,Ъdes is the similarity measure given in Definition 6.Step I.3: find out the support degreeTЪdef that IFN Ъdef receives from other IFNs Ъdess=1,2,...,p;s≠f, where(89)TЪdef=∑s=1,s≠fpsupЪdef,Ъdes.Step I.4: utilize weightsϖff=1,2,....,p for the decision-makers DE¯¯f to determine weights Ξdef associated with the IFN Ъdef,(90)Ξdef=ϖf1+TЪdef∑f=1pϖf1+TЪdef,f=1,2,...,p.whereΞdef≥0 and ∑f=1pΞdef=1.Step I.5: aggregate all the individual decision matricesDT¯¯f=Ъdefg×hf=1,2,...,p into a group decision matrix by utilizing WGIFSSPA or WGIFSSPGA operators, where(91)Ъde=WIFSSGPAљ,ϖЪde1,Ъde2,...,Ъdep=⊕f=1pΞf1љ−1−Üdefς−1љ−11/ς,1−⊕f=1pΞf1љ−1−1−Õdefς−1љ−11/ς,or(92)Ъde=WIFSSGPGAљ,ϖЪde1,Ъde2,...,Ъdep=1−⊕f=1pΞf1љ−1−1−Üdefς−1љ−11/ς,⊕f=1pΞf1љ−1−Õdefς−1љ−11/ς.Step I.6: find out supportSupЪde,Ъdx by the following formula:(93)SupЪde,Ъdx=SM˜Ъde,Ъdx=1−KЪde−KЪdx,d=1,2,...,g,e=1,2,...,h,e≠xwhereKЪde and KЪdx are the knowledge measures and are calculated utilizing (6).Step I.7: find out the support degreeTЪde that IFN Ъde collects from other IFNs Ъdxx=1,2,...,h;e≠x, where(94)TЪde=∑x=1,x≠ehwei¯¯xsupЪde,Ъdx.Step I.8: determine weighting vectorϒded=1,2,....,g,e=1,2,...,h associated with Ъde ,(95)ϒde=w¯¯eie1+TЪde∑e=1hwei¯¯e1+TЪde.Step I.9: utilize WIFSSGPA or WIFSSGPGA operators to collective all evaluation valuesЪded=1,2,....,g,e=1,2,...,h into the overall evaluation value Ъdd=1,2,....,g matching to the alternatives ale¯¯dd=1,2,...,g:(96)Ъd=WIFSSGPAЉ,wei¯¯Ъd1,Ъd2,...,Ъdh=⊕e=1hϒde1Љ−1−Üdeς−1Љ−11/ς,1−⊕e=1hϒde1Љ−1−1−Õdeς−1Љ−11/ςor(97)Ъd=WIFSSGPGAЉ,wei¯¯Ъd1,Ъd2,...,Ъdh=1−⊕e=1hϒde1Љ−1−1−Üdeς−1Љ−11/ς,⊕e=1hϒde1Љ−1−Õdeς−1Љ−11/ς.Step I.10: find out the scoresSC¯¯Ъd for the overall IFN of the alternatives ale¯¯dd=1,2,...,g by exploiting Definition 3.Step I.11: rank all alternativesale¯¯dd=1,2,...,g and select the best one(s) with the ranking order Ъdd=1,2,...,g.


## 4.2. Approach II: MAGDM with Totally Unknown Weight Vectors of Attributes and Decision-Makers

In this section, to deal with real decision situations in which the weighting vectors of both decision-makers and attributes are totally unknown, we apply the WIFSSGPOA operator and the WIFSSGPOGA operator to establish the following approach II to solve MAGDM problems under IF environments. To do so, immediately follow the steps as follows:Step II.1: same as Step I.1.Step II.2: find out the supports exploiting the formula:(98)SupЪdeρf,Ъdeρs=SM˜Ъdeρf,Ъdeρs=1−KЪdeρf−KЪdeρs,f,s=1,2,...,p;d=1,2,...,g;e=1,2,...,h.It gratifies the axioms for support mappings in Definition16, SM˜Ъdeρf,Ъdeρs is the similarity measure specified in Definition 6.Step II.3: find out the support degreeTЪdeρf that IFN Ъdeρf receives from other IFNs Ъdeρss=1,2,...,p;s≠f, where(99)Tъdeρf=∑s=1,s≠fpsupъdeρf,ъdeρs.Step II.4: utilize (54) to determine weights Ξdef associated with the fth largest IFN Ъdef, we have Ξdef=kUf/L−kUf−1/L with Ξdef∈0,1 and ⊕f=1pΞdef=1, where k is the BUM function [38],(100)Uf=⊕s=1fQdeρs,Lde=⊕s=1pQdeρs,Qdeρs=1+TЪdeρs.Step II.5: combine all the entity decision matricesDT¯¯f=Ъdefg×hf=1,2,...,p into a group decision matrix DT˜=Ъdeg×h by utilizing the WIFSSGPOA or WIFSSGPOGA operators, where(101)Ъde=WIFSSGPOAЉЪde1,Ъde2,...,Ъdep=⊕f=1pΞdef1Љ−1−Üdeρfς−1Љ−11/ς,1−⊕f=1pΞdef1Љ−1−1−Õdeρfς−1Љ−11/ςor(102)Ъde=WIFSSGPOGAΞЪde1,Ъde2,...,Ъdep=1−⊕f=1pΞdef1Љ−1−1−Üdeρfς−1Љ−11/ς,⊕f=1pΞdef1Љ−1−Õdeρfς−1Љ−11/ς.Step II.6: find out supportsSupЪdρe,Ъdρx by the following formula:(103)SupЪdρe,Ъdρx=SM˜Ъdρe,Ъdρx=1−KЪdρe−KЪdρx,d=1,2,...,g,e=1,2,...,h,e≠xwhereKЪde and KЪdx are the knowledge measures and are calculated utilizing (6).Step II.7: determine the support degreeTЪdρe that the eth largest IFN ъdρe collects from other IFNs Ъdρxx=1,2,...,h;e≠x, where(104)TЪdρe=∑x=1,x≠ehsupЪdρe,Ъdρx.Step II.8: utilize (54) to determine weights ϒded=1,2,...,g;e=1,2,...,h: ϒde=kUde/L−kUde−1/L with ϒde∈0,1 and ⊕e=1hΞde=1, where k is the BUM function [38].(105)Ude=⊕x=1eQdρx,L=⊕e=1hQdρe,Qdρe=1+TЪdρe.Step II.9: utilize the WIFSSGPOA or WIFSSGPOGA operators to aggregate all evaluation valuesЪded=1,2,....,g,e=1,2,...,h into the overall evaluation value Ъdd=1,2,....,g communicating to the alternatives ale¯¯dd=1,2,...,g:(106)Ъd=WIFSSGPOAЉЪd1,Ъd2,...,Ъdh=⊕e=1hϒde1Љ−1−Üdρeς−1Љ−11/ς,1−⊕e=1hϒde1Љ−1−1−Õdρeς−1Љ−11/ςor(107)Ъd=WIFSSGPOGAЪd1,Ъd2,...,Ъdh=1−⊕e=1hϒde1Љ−1−1−Üdρeς−1Љ−11/ς,⊕e=1hϒde1Љ−1−Õdρeς−1Љ−11/ς.Step II.10: find out the scoresSC¯¯Ъd for the overall IFN of the alternatives ale¯¯dd=1,2,...,g by exploiting Definition 3.Step II.11: rank all alternativesale¯¯dd=1,2,...,g and select the most favorable one(s) with the ranking order Ъdd=1,2,...,g.


## 5. Illustrative Example

In the accompanying, a case led in Shanghai, China, is embraced from Ref. [26] to demonstrate the utilization of the proposed IF MAGDM technique for the determination procedure of HCW-TT. As we know, Shanghai is one of the most significant urban communities in China, which has a populace of about 25 million inhabitants. The aggregate sum of therapeutic waste that should be discarded by the burning plant in Shanghai is increasing each year. The current handling limits of the cremation plants in Shanghai are insufficient to totally discard the waste created in Shanghai each year. Thus, new HCW treatment advances must be chosen to help HCW in Shanghai. As per the past examination and screening, joined with the real circumstance of HCWM in Shanghai, the accompanying four HCW transfer techniques were chosen as options:ale¯¯1: high temperature incineration technologyale¯¯2: pressure steam sterilizationale¯¯3: sanitary landfillale¯¯4: microwave sterilizationThese four waste transfer strategies have their own favorable circumstances and burdens. The procedure parameters of these four transfer strategies are extraordinary, and every parameter has its very own qualities. To choose the proper HCW transfer strategies, three specialists structure an assessment council, incorporating a college teacher in the division of mechanical buildingde¯¯1, a specialist in waste administration at an enormous clinic de¯¯2, and an expert and specialized architect in HCW treatment de¯¯3. The weighting vector of specialists is resolved as ϖ=0.4,0.4,0.2T by considering their insight level and area experience. In light of the existing exploration results and master discourse results, the accompanying four elements are recognized as the assessment characteristics: Cri¯¯1: monetary cost, that is, net expense per ton; Cri¯¯2: environmental effect, including clamor, squander residuals, and dangerous gas outflows, and so on.;Cri¯¯3: social influence, that is, open acknowledgment; Cri¯¯4: technical execution, including specialized unwavering quality, process viability, and so forth.The weight vector of the criterion are recognized aswei¯¯=0.25,0.35,0.25,0.15T. To explain the evaluation preferences of the decision-makers, they choose to give their evaluation values about the four alternatives with respect to four attributes, which takes the form of IFNs Ъdef=Üdef,Õdeff=1,2,3;d,e=1,2,...,4. The assessment values are given in Tables 1–3.Step I.1: since all criteria are of the benefit type, there is no need to normalize them.Step I.2: find out the supports by the following formula (79). For ease, we shall symbolize SupЪdef,Ъdes by Sdefsf,s=1,2,3;d,e=1,2,3,4.(108)S1112=S1121=0.9365,S1113=S1131=0.9258,S1123=S1132=0.9661,S1212=S1221=0.9468,S1213=S1231=0.9830,S1223=S1232=0.9297,S1312=S1321=0.9495,S1313=S1331=0.9150;S1323=S1332=0.8645,S1412=S1421=0.9401,S1413=S1431=0.9026,S1423=S1432=0.9625;S2112=S2121=0.915,S2113=S2131=0.9277,S2123=S2132=0.9873,S2212=S2221=0.9190,S2213=S2231=0.9873,S2223=S2232=0.9063,S2312=S2321=0.8667,S2313=S2331=0.9006,S2323=S2332=0.9661,S2412=S2421=0.9525,S2413=S2431=0.9943,S2423=S2432=0.9468;S3112=S3121=0.9026,S3113=S3131=0.8060,S3123=S3132=0.9034,S3212=S3221=0.9026,S3213=S3231=0.9752,S3223=S3232=0.9274,S3312=S3321=0.9830,S3313=S3331=0.9656,S3323=S3332=0.9826,S3412=S3421=0.8323,S3413=S3431=0.9026,S3423=S3432=0.9297;S4112=S4121=0.9873,S4113=S4131=0.9625,S4123=S4132=0.9752,S4212=S4221=0.9873,S4213=S4231=0.9050,S4223=S4232=0.8922,S4312=S4321=0.9274,S4313=S4331=0.9789,S4323=S4332=0.9063,S4412=S4421=0.8308,S4413=S4431=0.9873,S4423=S4432=0.8435.Step I.3. find out the support degreesTЪdef by exploiting formula (80). For ease, we shall symbolize TЪdef by Tdeff=1,2,3;d,e=1,2,3,4.(109)T111=1.8390,T112=1.9026,T113=1.8687,T121=1.9297,T122=1.8765,T123=1.9127,T131=1.8645,T132=1.8140,T133=1.7795,T141=1.8427,T142=1.9026,T143=1.8651,T211=1.8427,T212=1.9023,T213=1.9150,T221=1.9063,T222=1.8253,T223=1.8936,T231=1.7673,T232=1.8328,T233=1.8667,T241=1.9468,T242=1.8993,T243=1.9410,T311=1.7085,T312=1.8060,T313=1.7094,T321=1.8778,T322=1.8299,T323=1.9026,T331=1.9485,T332=1.9656,T333=1.9482,T341=1.7349,T342=1.7620,T343=1.8323,T411=1.9498,T412=1.9625,T413=1.9377,T421=1.8922,T422=1.8795,T423=1.7971,T431=1.9063,T432=1.8337,T433=1.8852,T441=1.8181,T442=1.6743,T443=1.8308.Step I.4: utilize weightsϖff=1,2,3 for decision-makers DE¯¯ff=1,2,3 to determine weights Ξdef by formula (81). We have(110)Ξ111=0.3956,Ξ112=0.4045,Ξ113=0.1999,Ξ121=0.4034,Ξ122=0.3961,Ξ123=0.2005,Ξ131=0.4053,Ξ132=0.3981,Ξ133=0.1966,Ξ141=0.3960,Ξ142=0.4044,Ξ143=0.1996,Ξ211=0.3947,Ξ212=0.4030,Ξ213=0.2024,Ξ221=0.4049,Ξ222=0.3936,Ξ223=0.2016,Ξ231=0.3935,Ξ232=0.4028,Ξ233=0.2038,Ξ241=0.4028,Ξ242=0.3963,Ξ243=0.2010,Ξ311=0.3943,Ξ312=0.4085,Ξ313=0.1972,Ξ321=0.4020,Ξ322=0.3953,Ξ323=0.2027,Ξ331=0.3991,Ξ332=0.4014,Ξ333=0.1995,Ξ341=0.3956,Ξ342=0.3995,Ξ343=0.2049,Ξ411=0.3996,Ξ412=0.4014,Ξ413=0.1990,Ξ421=0.4034,Ξ422=0.4016,Ξ423=0.1951,Ξ431=0.4046,Ξ432=0.3945,Ξ433=0.2009,Ξ441=0.4080,Ξ442=0.3871,Ξ443=0.2049.Step I.5: combined all the entity decision matricesDT¯¯f=Ъdefg×hf=1,2,3;d,e=1,2,3,4 into a group decision matrix DT˜=Ъdeg×h by utilizing WIFSSGPA or WIFSSGPG operators given in formula (82) or formula (83) (assume that ζ=2,ς=−2 ) and they are shown in Tables 4 and 5.Step I.6: find out supportsSupЪde,Ъdx by the following formula (84). To be easily understood, we shall denote SupЪde,Ъdx by Sdedd,e=1,...,4. We have(111)S121=S211=0.9145,S131=S311=0.9208,S141=S411=0.9907,S231=S321=0.9938,S241=S421=0.9052,S341=S431=0.9115;S122=S212=0.9296,S132=S312=0.9458,S142=S412=0.9702,S232=S322=0.9838,S242=S422=0.8998,S342=S432=0.9160;S123=S213=0.9572,S133=S313=0.8553,S143=S413=0.9231,S233=S323=0.8981,S243=S423=0.9659,S343=S433=0.9322,S124=S214=0.9777,S134=S314=0.9675,S144=S414=0.9504,S234=S324=0.9451,S244=S424=0.9281,S344=S434=0.9830or(112)S121=S211=0.9294,S131=S311=0.8765,S141=S411=0.9878,S231=S321=0.9471,S241=S421=0.9172,S341=S431=0.8643,S122=S212=0.9446,S132=S312=0.9982,S142=S412=0.9956,S232=S322=0.9464,S242=S422=0.9402,S342=S432=0.9939,S123=S213=0.9797,S133=S313=0.9319,S143=S413=0.9308,S233=S323=0.9522,S243=S423=0.9512,S343=S433=0.9990;S124=S214=0.9637,S134=S314=0.9585,S144=S414=0.9318,S234=S324=0.9222,S244=S424=0.8954,S344=S434=0.9732.Step I.7: find out the support degreesTъde by the following formula (85). To be easily understood, we shall denote Tif¯¯de by Tde. We have(113)T11=2.826,T12=2.8135,T13=2.826,T14=2.8073,T21=2.8456,T22=2.8132,T23=2.8456,T24=2.7860,T31=2.7356,T32=2.8212,T33=2.6855,T34=2.8212,T41=2.8956,T42=2.8509,T43=2.8956,T44=2.8615or(114)T11=2.7937,T12=2.7937,T13=2.6879,T14=2.7693,T21=2.9385,T22=2.8312,T23=2.9385,T24=2.9297,T31=2.8423,T32=2.8830,T33=2.8830,T34=2.8810,T41=2.8539,T42=2.7812,T43=2.8539,T44=2.8003.Step I.8: find out weighting vectorϒded,e=1,...,4 by utilizing formula (86):(115)ϒ11=0.2505,ϒ12=0.3495,ϒ13=0.2505,ϒ14=0.1495,ϒ21=0.2513,ϒ22=0.3489,ϒ23=0.2513,ϒ24=0.1485,ϒ31=0.2480,ϒ32=0.3551,ϒ33=0.2447,ϒ34=0.1522,ϒ41=0.2513,ϒ42=0.3478,ϒ43=0.2513,ϒ44=0.1495or(116)ϒ11=0.2520,ϒ12=0.3528,ϒ13=0.2450,ϒ14=0.1502,ϒ21=0.2525,ϒ22=0.3439,ϒ23=0.2525,ϒ24=0.1512,ϒ31=0.2481,ϒ32=0.3510,ϒ33=0.2507,ϒ34=0.1503,ϒ41=0.2522,ϒ42=0.3464,ϒ43=0.2521,ϒ44=0.1492.Step I.9: utilize WIFSSGPA or WIFSSPGA operators to combine all assessment valuesЪded=1,...,4,e=1,...,4 into the overall evaluation value Ъdd=1,...,4 matching to the alternatives ale¯¯dd=1,...,4 by formula (87) or formula (88). We have(117)Ъ1=0.3048,0.6254,Ъ2=0.3415,0.5593,Ъ3=0.2469,0.6395,Ъ4=0.3311,0.5475or(118)Ъ1=0.2057,0.6757,Ъ2=0.2108,0.6253,Ъ3=0.2111,0.6670,Ъ4=0.2862,0.5833.Step I.10: find out the scoresSR¯¯Ъd for the overall IFN of the alternatives ale¯¯dd=1,2,...,g by utilizing Definition 3.(119)SR¯¯Ъ1=−0.3206,SR¯¯Ъ2=−0.2179,SR¯¯Ъ3=−0.3927,SR¯¯Ъ4=−0.2164or(120)SR¯¯Ъ1=−0.4700,SR¯¯Ъ2=−0.4145,SR¯¯Ъ3=−0.4560,SR¯¯Ъ4=−0.2971.Step I.11: according to score values, we can get the following ranking orders.(121)ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3or(122)ale¯¯4>ale¯¯2>ale¯¯3>ale¯¯1.Table 1
The intuitionistic fuzzy decision matrixDT¯¯1 provided by de¯¯1.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.2,0.70.4,0.60.5,0.50.2,0.6ale¯¯20.5,0.50.4,0.50.1,0.80.5,0.5ale¯¯30.2,0.70.2,0.70.3,0.70.2,0.6ale¯¯40.5,0.40.3,0.60.2,0.60.3,0.6Table 2
The intuitionistic fuzzy decision matrixDT¯¯2 provided by de¯¯2.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1,0.70.2,0.70.2,0.80.4,0.5ale¯¯20.4,0.50.3,0.50.2,0.60.2,0.7ale¯¯30.2,0.60.2,0.60.4,0.60.3,0.7ale¯¯40.3,0.60.4,0.50.3,0.60.2,0.5Table 3
The intuitionistic fuzzy decision matrixDT¯¯3 provided by de¯¯3.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.2,0.60.3,0.70.4,0.50.2,0.7ale¯¯20.3,0.60.3,0.60.1,0.70.4,0.6ale¯¯30.2,0.50.3,0.60.1,0.80.2,0.7ale¯¯40.2,0.70.3,0.70.3,0.50.4,0.5Table 4
The collective intuitionistic fuzzy decision matrixDT¯¯ utilizing the WIFSSGPA operator.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1627,0.67690.3143,0.65510.3941,0.58190.2977,0.5681ale¯¯20.4306,0.51680.3450,0.51680.1435,0.68560.3941,0.5830ale¯¯30.2,0.60820.2229,0.63580.3179,0.67040.2438,0.6560ale¯¯40.3880,0.50530.3447,0.56810.2632,0.57600.2914,0.5355Table 5
The collective intuitionistic fuzzy decision matrixDT¯¯ utilizing the WIFSSGPGA operator.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1357,0.68540.2648,0.66830.2867,0.70690.2426,0.5988ale¯¯20.4001,0.52520.3318,0.52510.1205,0.73490.2871,0.6255ale¯¯30.2,0.63770.2129,0.64960.2006,0.70720.2555,0.6690ale¯¯40.3069,0.57980.3315,0.59830.2459,0.58410.2565,0.5478Hence, the optimal alternative isale¯¯4 while the worst ones are ale¯¯3 and ale¯¯1.


### 5.1. The Effect of the Parametersς and Љ on Decision-Making Result

In this section, we will further discuss the effect of the parameterς based on WIFSSGPA and WIFSSGPG operators, the ranking results with different values of ς are shown in Figures 1 and 2 while Љ=2 is fixed.Figure 1
Score values for different values of parameterς based on the WIFSSGPA operator.Figure 2
Score values for different values of parameterς based on the WIFSSGPG operator.From Figures1 and 2, one can locate that for different values of the parameter ς the ranking orders are different. Yet, the optimal alternatives are ale¯¯2 or ale¯¯4. From Figures 1 and 2, we can also locate that, when the values of the parameter vector ς decreases the score values based on the WIFSSGPA operator increases, while that based on the WIFSSGPG operator decreases.From Figures3 and 4, one can locate that for different values of the parameter Љ, the ranking orders are different. Yet, the optimal alternatives are ale¯¯2 or ale¯¯4. From Figures 3 and 4, we can also locate that, as the values of the parameter vector Љ increases the score values based on the WIFSSGPA operator decreases, while that based on the WIFSSGPG operator increases. So, the decision-makers may choose the values of the parameters according to their actual needs of the situation.Figure 3
Score values for different values of parameterζ based on the WIFSSGPA operator.Figure 4
Score values for different values of parameterζ based on the WIFSSGPA operator.Case II. When experts and attributes weights are unknown.Assume that the weights of both the attributes and experts are unknown in the above example. To deal with such a situation, we can apply approach II. Firstly, we reordered the decision matrices given in Tables1–3. The reorder decision matrices are given in Tables 6–8Step II.1: same as Step I.1.Step II.2: find out the supports by the subsequent formula (89). To be easily understood, we shall denote SupЪdeρf,Ъdeρs by Sdefsf,s=1,...,3;d,e=1,....,4, we have(123)S1112=S1121=0.9026,S1113=S1131=0.9661,S1123=S1132=0.9365,S1212=S1221=0.9830,S1213=S1231=0.9468,S1223=S1232=0.9297;S1312=S1321=0.9150,S1313=S1331=0.9495,S1323=S1332=0.8645,S1412=S1421=0.9401,S1413=S1431=0.9625,S1423=S1432=0.9026;S2112=S2121=0.915,S2113=S2131=0.9277,S2123=S2132=0.9873,S2212=S2221=0.9190,S2213=S2231=0.9873,S2223=S2232=0.9063,S2312=S2321=0.9661,S2313=S2331=0.8667,S2323=S2332=0.9006,S2412=S2421=0.9942,S2413=S2431=0.9525,S2423=S2432=0.9468;S3112=S3121=0.9034,S3113=S3131=0.8060,S3123=S3132=0.9026,S3212=S3221=0.9074,S3213=S3231=0.9752,S3223=S3232=0.9026;S3312=S3321=0.9830,S3313=S3331=0.9826,S3323=S3332=0.9656,S3412=S3421=0.8323,S3413=S3431=0.9297,S3423=S3432=0.9026;S4112=S4121=0.9873,S4113=S4131=0.9625,S4123=S4132=0.9752,S4212=S4221=0.9873,S4213=S4231=0.8922,S4223=S4232=0.9050;S4312=S4321=0.9063,S4313=S4331=0.9789,S4323=S4332=0.9274,S4412=S4421=0.9873,S4413=S4431=0.8435,S4423=S4432=0.8308.Step II.3: find out the support degreesTЪdeρf by utilizing formula (90). For ease, we shall symbolize TЪdeρf by Tdefd,e=1,...,4,f=1,...,3. We have(124)T111=1.8687,T112=1.8390,T113=1.9026,T121=1.9297,T122=1.9127,T123=1.8765,T131=1.8645,T132=1.7795,T133=1.8140,T141=1.9026,T142=1.8427,T143=1.8651,T211=1.8427,T212=1.9023,T213=1.9150,T221=1.9063,T222=1.8253,T223=1.8936,T231=1.8328,T232=1.8667,T233=1.7673,T241=1.9467,T242=1.9410,T243=1.8993,T311=1.7093,T312=1.8060,T313=1.7085,T321=1.9026,T322=1.8300,T323=1.8778,T331=1.9656,T332=1.9485,T333=1.9482,T341=1.7620,T342=1.7349,T343=1.8323,T411=1.9498,T412=1.9625,T413=1.9377,T421=1.8795,T422=1.8922,T423=1.7971,T431=1.8852,T432=1.8337,T433=1.9063,T441=1.830774,T442=1.8181,T443=1.6742.Step II.4: utilize equation (91) to determine weights Ξdef associated with the fth largest IFN ъdef,f=1,2,3;d,e=1,..,4. We have(125)Ξ111=0.1110,Ξ112=0.3284,Ξ113=0.5606,Ξ121=0.1129,Ξ122=0.3361,Ξ123=0.5510,Ξ131=0.1147,Ξ132=0.3306,Ξ133=0.5547,Ξ141=0.1136,Ξ142=0.3316,Ξ143=0.5548,Ξ211=0.1077,Ξ212=0.3323,Ξ213=0.5599,Ξ221=0.1135,Ξ222=0.3280,Ξ223=0.5584,Ξ231=0.1119,Ξ232=0.3412,Ξ233=0.5468,Ξ241=0.1125,Ξ242=0.3365,Ξ243=0.5510,Ξ311=0.1085,Ξ312=0.3412,Ξ313=0.5502,Ξ321=0.1136,Ξ322=0.3296,Ξ323=0.5567,Ξ331=0.1120,Ξ332=0.3334,Ξ333=0.5547,Ξ341=0.1099,Ξ342=0.3256,Ξ343=0.5645,Ξ411=0.1111,Ξ412=0.3352,Ξ413=0.5537,Ξ421=0.1129,Ξ422=0.3408,Ξ423=0.5463,Ξ431=0.1119,Ξ432=0.3277,Ξ433=0.5604,Ξ441=0.1157,Ξ442=0.3449,Ξ443=0.5394.Step II.5: combine all the entity decision matricesDT¯¯f=Ъdef4×4f=1,2,3 into the group decision matrix DT˜=Ъde4×4 given in Tables 9 and 10 by utilizing WIFSSGPOA or WIFSSGPOGA operators given in formula (92) or formula (93). We haveOrStep II.6: find out supportsSupЪdρe,Ъdρx by the following formula (94), we have(126)S121=S211=0.9208,S131=S311=0.9351,S141=S411=0.9957,S231=S321=0.9858,S241=S421=0.9251,S341=S431=0.9394,S122=S212=0.9594,S132=S312=0.9998,S142=S412=0.9518,S232=S322=0.9596,S242=S422=0.9112,S342=S432=0.9516,S123=S213=0.9748,S133=S313=0.8865,S143=S413=0.9631,S233=S323=0.9117,S243=S423=0.9883,S343=S433=0.9234,S124=S214=0.9563,S134=S314=0.9504,S144=S414=0.9079,S234=S324=0.9067,S244=S424=0.8642,S344=S434=0.9575Or(127)S121=S211=0.9258,S131=S311=0.8673,S141=S411=0.9759,S231=S321=0.9415,S241=S421=0.9500,S341=S431=0.8914,S122=S212=0.9748,S132=S312=0.9690,S142=S412=0.9760,S232=S322=0.9437,S242=S422=0.9508,S342=S432=0.9930;S123=S213=0.9895,S133=S313=0.9404,S143=S413=0.9795,S233=S323=0.9509,S243=S423=0.9900,S343=S433=0.9608,S124=S214=0.9465,S134=S314=0.9333,S144=S414=0.8896,S234=S324=0.8798,S244=S424=0.8361,S344=S434=0.9562.Step II.7: find out the support degreesTъdρe exploiting formula (95), we have(128)T11=2.8516,T12=2.8317,T13=2.8602,T14=2.8602,T21=2.9110,T22=2.8302,T23=2.9110,T24=2.81860,T31=2.8243,T32=2.8748,T33=2.7216,T34=2.8748,T41=2.8146,T42=2.7272,T43=2.8146,T44=2.7296or(129)T11=2.7689,T12=2.8171,T13=2.7001,T14=2.8171,T21=2.9197,T22=2.8693,T23=2.9057,T24=2.9196,T31=2.9095,T32=2.9304,T33=2.8521,T34=2.9304,T41=2.7694,T42=2.6624,T43=2.7694,T44=2.6819.Step II.8: exploit formula (96) to determine weights ϒded,e=1,2,...,4, we have Ξdef=(130)ϒ11=0.0625,ϒ12=0.1863,ϒ13=0.3128,ϒ14=0.4384,ϒ21=0.0639,ϒ22=0.1866,ϒ23=0.3171,ϒ24=0.4324,ϒ31=0.0625,ϒ32=0.1908,ϒ33=0.3041,ϒ34=0.4424,ϒ41=0.0639,ϒ42=0.1860,ϒ43=0.3167,ϒ44=0.4333or(131)ϒ11=0.0623,ϒ12=0.1900,ϒ13=0.3061,ϒ14=0.4416,ϒ21=0.0630,ϒ22=0.1858,ϒ23=0.3121,ϒ24=0.4390,ϒ31=0.0626,ϒ32=0.1892,ϒ33=0.3083,ϒ34=0.4399,ϒ41=0.0641,ϒ42=0.1852,ϒ43=0.3171,ϒ44=0.4336.Step II.9: utilize WIFSSGPOA or WIFSSGPOGA operators to aggregate all assessment values by formula (97) or formula (98) to obtain the overall assessment values Ъded,e=1,2,...,4, we have(132)ale¯¯1=0.2654,0.6430,ale¯¯2=0.2729,0.6360,ale¯¯3=0.2146,0.6790,ale¯¯4=0.2701,0.5685or(133)ale¯¯1=0.2123,0.6983,ale¯¯2=0.1659,0.6909,ale¯¯3=0.1721,0.7094,ale¯¯4=0.2448,0.5934.Step II.10: find out the scoresSC¯¯Ъd for the overall IFN of the alternatives ale¯¯dd=1,2,...,4 by exploiting Definition 3.(134)SR¯¯Ъ1=−0.3776,SR¯¯Ъ2=−0.3630,SR¯¯Ъ3=−0.4643,SR¯¯Ъ4=−0.2984.Otherwise,(135)SR¯¯Ъ1=−0.4860,SR¯¯Ъ2=−0.5250,SR¯¯Ъ3=−0.5373,SR¯¯Ъ4=−0.3486.Step II.11: from score values, we can get the following ranking orders.(136)ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3.Otherwise,(137)ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3.Table 6
The reorder decision matrixDT¯¯σ1.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.2,0.70.3,0.70.4,0.50.2,0.6ale¯¯20.4,0.50.3,0.50.1,0.70.4,0.6ale¯¯30.2,0.60.2,0.60.3,0.70.2,0.6ale¯¯40.3,0.60.3,0.60.3,0.60.3,0.6Table 7
The reorder decision matrixDT¯¯σ2.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.2,0.60.4,0.60.5,0.50.4,0.5ale¯¯20.5,0.50.4,0.50.2,0.60.5,0.5ale¯¯30.2,0.50.3,0.60.4,0.60.3,0.7ale¯¯40.5,0.40.4,0.50.3,0.50.4,0.5Table 8
The reorder decision matrixDT¯¯σ3.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1,0.70.2,0.70.2,0.80.2,0.7ale¯¯20.3,0.60.3,0.60.1,0.80.2,0.7ale¯¯30.2,0.70.2,0.70.1,0.80.2,0.7ale¯¯40.2,0.70.3,0.70.2,0.60.2,0.5Table 9
Collective decision matrix utilizing WIFSSGPOA.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1472,0.68700.2644,0.68670.3257,0.62420.2305,0.6341ale¯¯20.3646,0.55040.3133,0.55020.1126,0.73530.3268,0.6339ale¯¯30.2,0.63440.2130,0.65110.2211,0.73610.2128,0.6633ale¯¯40.2858,0.60900.3133,0.63340.2478,0.58640.2634,0.5296Table 10
Collective decision matrix utilizing WIFSSGPOG.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1230,0.69210.2360,0.69200.2509,0.74160.2099,0.6590ale¯¯20.3402,0.56280.3081,0.56270.1047,0.76370.2514,0.6587ale¯¯30.2,0.65880.2069,0.66470.1315,0.76440.2067,0.6751ale¯¯40.2377,0.65570.3080,0.65820.2312,0.59130.2371,0.5412Hence, the optimal alternative isale¯¯4 while the worst one is ale¯¯3.


### 5.2. Comparison with Some Other Existing Approaches

In this subsection, we compare our developed approaches with some existing approaches based on different aggregation operators.Firstly, we solve the same example by utilizing the intuitionistic fuzzy Schweizer-Sklar weighted averaging operator developed by Zhang [63]. Secondly, we solve the same example by utilizing intuitionistic fuzzy power weighted averaging operator and intuitionistic fuzzy power geometric operator developed by Xu [46]. The score values and ranking order are given in Table 11.Table 11
Comparison with different approaches.
Aggregation operatorsScore valuesRanking orderIFWSSWA operator [63]SC¯¯Ъ1=−0.3098,SC¯¯Ъ2=−0.2075,SC¯¯Ъ3=−0.3874,SC¯¯Ъ4=−0.2090ale¯¯2>ale¯¯4>ale¯¯1>ale¯¯3IFPWA [46]SC¯¯Ъ1=−0.3453,SC¯¯Ъ2=−0.2388,SC¯¯Ъ3=−0.4027,SC¯¯Ъ4=−0.2380ale¯¯2>ale¯¯4>ale¯¯1>ale¯¯3IFPWG [46]SC¯¯Ъ1=−0.4025,SC¯¯Ъ2=−0.3058,SC¯¯Ъ3=−0.4270,SC¯¯Ъ4=−0.2688ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3Proposed GIFSSPWASC¯¯b1=−0.3206,SR¯¯Ъ2=−0.2179,SR¯¯Ъ3=−0.3927,SR¯¯Ъ4=−0.2164ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3Proposed GIFSSPWGSC¯¯Ъ1=−0.4700,SC¯¯Ъ2=−0.4145,SC¯¯Ъ3=−0.4560,SC¯¯Ъ4=−0.2971ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3From Table11, we can see that the ranking order acquired from the AOs developed by Zhang [63] is different from the proposed aggregation operators and that developed by Xu [46]. IFPWA and IFPWG operators are special cases of the proposed aggregation operators by taking Љ=1,ς⟶0. Also, the ranking orders obtained from the aggregation operators developed by Xu [46] and the proposed aggregation operators are the same. This shows the validity of the developed aggregation operators. The proposed generalized Schweizer-Sklar power aggregation operators have some advantage over the existing power aggregation operators for IFS. Firstly, the proposed aggregation operators have two parameters, one takes values from 0,+∞ and the other parameter takes values from −∞,0. These parameters make the decision-making process more supple. Secondly, it can eradicate the effect of awful data from the decision-making result. So, these aggregation operators are more general then the existing aggregation operators.


## 5.1. The Effect of the Parametersς and Љ on Decision-Making Result

In this section, we will further discuss the effect of the parameterς based on WIFSSGPA and WIFSSGPG operators, the ranking results with different values of ς are shown in Figures 1 and 2 while Љ=2 is fixed.Figure 1
Score values for different values of parameterς based on the WIFSSGPA operator.Figure 2
Score values for different values of parameterς based on the WIFSSGPG operator.From Figures1 and 2, one can locate that for different values of the parameter ς the ranking orders are different. Yet, the optimal alternatives are ale¯¯2 or ale¯¯4. From Figures 1 and 2, we can also locate that, when the values of the parameter vector ς decreases the score values based on the WIFSSGPA operator increases, while that based on the WIFSSGPG operator decreases.From Figures3 and 4, one can locate that for different values of the parameter Љ, the ranking orders are different. Yet, the optimal alternatives are ale¯¯2 or ale¯¯4. From Figures 3 and 4, we can also locate that, as the values of the parameter vector Љ increases the score values based on the WIFSSGPA operator decreases, while that based on the WIFSSGPG operator increases. So, the decision-makers may choose the values of the parameters according to their actual needs of the situation.Figure 3
Score values for different values of parameterζ based on the WIFSSGPA operator.Figure 4
Score values for different values of parameterζ based on the WIFSSGPA operator.Case II. When experts and attributes weights are unknown.Assume that the weights of both the attributes and experts are unknown in the above example. To deal with such a situation, we can apply approach II. Firstly, we reordered the decision matrices given in Tables1–3. The reorder decision matrices are given in Tables 6–8Step II.1: same as Step I.1.Step II.2: find out the supports by the subsequent formula (89). To be easily understood, we shall denote SupЪdeρf,Ъdeρs by Sdefsf,s=1,...,3;d,e=1,....,4, we have(123)S1112=S1121=0.9026,S1113=S1131=0.9661,S1123=S1132=0.9365,S1212=S1221=0.9830,S1213=S1231=0.9468,S1223=S1232=0.9297;S1312=S1321=0.9150,S1313=S1331=0.9495,S1323=S1332=0.8645,S1412=S1421=0.9401,S1413=S1431=0.9625,S1423=S1432=0.9026;S2112=S2121=0.915,S2113=S2131=0.9277,S2123=S2132=0.9873,S2212=S2221=0.9190,S2213=S2231=0.9873,S2223=S2232=0.9063,S2312=S2321=0.9661,S2313=S2331=0.8667,S2323=S2332=0.9006,S2412=S2421=0.9942,S2413=S2431=0.9525,S2423=S2432=0.9468;S3112=S3121=0.9034,S3113=S3131=0.8060,S3123=S3132=0.9026,S3212=S3221=0.9074,S3213=S3231=0.9752,S3223=S3232=0.9026;S3312=S3321=0.9830,S3313=S3331=0.9826,S3323=S3332=0.9656,S3412=S3421=0.8323,S3413=S3431=0.9297,S3423=S3432=0.9026;S4112=S4121=0.9873,S4113=S4131=0.9625,S4123=S4132=0.9752,S4212=S4221=0.9873,S4213=S4231=0.8922,S4223=S4232=0.9050;S4312=S4321=0.9063,S4313=S4331=0.9789,S4323=S4332=0.9274,S4412=S4421=0.9873,S4413=S4431=0.8435,S4423=S4432=0.8308.Step II.3: find out the support degreesTЪdeρf by utilizing formula (90). For ease, we shall symbolize TЪdeρf by Tdefd,e=1,...,4,f=1,...,3. We have(124)T111=1.8687,T112=1.8390,T113=1.9026,T121=1.9297,T122=1.9127,T123=1.8765,T131=1.8645,T132=1.7795,T133=1.8140,T141=1.9026,T142=1.8427,T143=1.8651,T211=1.8427,T212=1.9023,T213=1.9150,T221=1.9063,T222=1.8253,T223=1.8936,T231=1.8328,T232=1.8667,T233=1.7673,T241=1.9467,T242=1.9410,T243=1.8993,T311=1.7093,T312=1.8060,T313=1.7085,T321=1.9026,T322=1.8300,T323=1.8778,T331=1.9656,T332=1.9485,T333=1.9482,T341=1.7620,T342=1.7349,T343=1.8323,T411=1.9498,T412=1.9625,T413=1.9377,T421=1.8795,T422=1.8922,T423=1.7971,T431=1.8852,T432=1.8337,T433=1.9063,T441=1.830774,T442=1.8181,T443=1.6742.Step II.4: utilize equation (91) to determine weights Ξdef associated with the fth largest IFN ъdef,f=1,2,3;d,e=1,..,4. We have(125)Ξ111=0.1110,Ξ112=0.3284,Ξ113=0.5606,Ξ121=0.1129,Ξ122=0.3361,Ξ123=0.5510,Ξ131=0.1147,Ξ132=0.3306,Ξ133=0.5547,Ξ141=0.1136,Ξ142=0.3316,Ξ143=0.5548,Ξ211=0.1077,Ξ212=0.3323,Ξ213=0.5599,Ξ221=0.1135,Ξ222=0.3280,Ξ223=0.5584,Ξ231=0.1119,Ξ232=0.3412,Ξ233=0.5468,Ξ241=0.1125,Ξ242=0.3365,Ξ243=0.5510,Ξ311=0.1085,Ξ312=0.3412,Ξ313=0.5502,Ξ321=0.1136,Ξ322=0.3296,Ξ323=0.5567,Ξ331=0.1120,Ξ332=0.3334,Ξ333=0.5547,Ξ341=0.1099,Ξ342=0.3256,Ξ343=0.5645,Ξ411=0.1111,Ξ412=0.3352,Ξ413=0.5537,Ξ421=0.1129,Ξ422=0.3408,Ξ423=0.5463,Ξ431=0.1119,Ξ432=0.3277,Ξ433=0.5604,Ξ441=0.1157,Ξ442=0.3449,Ξ443=0.5394.Step II.5: combine all the entity decision matricesDT¯¯f=Ъdef4×4f=1,2,3 into the group decision matrix DT˜=Ъde4×4 given in Tables 9 and 10 by utilizing WIFSSGPOA or WIFSSGPOGA operators given in formula (92) or formula (93). We haveOrStep II.6: find out supportsSupЪdρe,Ъdρx by the following formula (94), we have(126)S121=S211=0.9208,S131=S311=0.9351,S141=S411=0.9957,S231=S321=0.9858,S241=S421=0.9251,S341=S431=0.9394,S122=S212=0.9594,S132=S312=0.9998,S142=S412=0.9518,S232=S322=0.9596,S242=S422=0.9112,S342=S432=0.9516,S123=S213=0.9748,S133=S313=0.8865,S143=S413=0.9631,S233=S323=0.9117,S243=S423=0.9883,S343=S433=0.9234,S124=S214=0.9563,S134=S314=0.9504,S144=S414=0.9079,S234=S324=0.9067,S244=S424=0.8642,S344=S434=0.9575Or(127)S121=S211=0.9258,S131=S311=0.8673,S141=S411=0.9759,S231=S321=0.9415,S241=S421=0.9500,S341=S431=0.8914,S122=S212=0.9748,S132=S312=0.9690,S142=S412=0.9760,S232=S322=0.9437,S242=S422=0.9508,S342=S432=0.9930;S123=S213=0.9895,S133=S313=0.9404,S143=S413=0.9795,S233=S323=0.9509,S243=S423=0.9900,S343=S433=0.9608,S124=S214=0.9465,S134=S314=0.9333,S144=S414=0.8896,S234=S324=0.8798,S244=S424=0.8361,S344=S434=0.9562.Step II.7: find out the support degreesTъdρe exploiting formula (95), we have(128)T11=2.8516,T12=2.8317,T13=2.8602,T14=2.8602,T21=2.9110,T22=2.8302,T23=2.9110,T24=2.81860,T31=2.8243,T32=2.8748,T33=2.7216,T34=2.8748,T41=2.8146,T42=2.7272,T43=2.8146,T44=2.7296or(129)T11=2.7689,T12=2.8171,T13=2.7001,T14=2.8171,T21=2.9197,T22=2.8693,T23=2.9057,T24=2.9196,T31=2.9095,T32=2.9304,T33=2.8521,T34=2.9304,T41=2.7694,T42=2.6624,T43=2.7694,T44=2.6819.Step II.8: exploit formula (96) to determine weights ϒded,e=1,2,...,4, we have Ξdef=(130)ϒ11=0.0625,ϒ12=0.1863,ϒ13=0.3128,ϒ14=0.4384,ϒ21=0.0639,ϒ22=0.1866,ϒ23=0.3171,ϒ24=0.4324,ϒ31=0.0625,ϒ32=0.1908,ϒ33=0.3041,ϒ34=0.4424,ϒ41=0.0639,ϒ42=0.1860,ϒ43=0.3167,ϒ44=0.4333or(131)ϒ11=0.0623,ϒ12=0.1900,ϒ13=0.3061,ϒ14=0.4416,ϒ21=0.0630,ϒ22=0.1858,ϒ23=0.3121,ϒ24=0.4390,ϒ31=0.0626,ϒ32=0.1892,ϒ33=0.3083,ϒ34=0.4399,ϒ41=0.0641,ϒ42=0.1852,ϒ43=0.3171,ϒ44=0.4336.Step II.9: utilize WIFSSGPOA or WIFSSGPOGA operators to aggregate all assessment values by formula (97) or formula (98) to obtain the overall assessment values Ъded,e=1,2,...,4, we have(132)ale¯¯1=0.2654,0.6430,ale¯¯2=0.2729,0.6360,ale¯¯3=0.2146,0.6790,ale¯¯4=0.2701,0.5685or(133)ale¯¯1=0.2123,0.6983,ale¯¯2=0.1659,0.6909,ale¯¯3=0.1721,0.7094,ale¯¯4=0.2448,0.5934.Step II.10: find out the scoresSC¯¯Ъd for the overall IFN of the alternatives ale¯¯dd=1,2,...,4 by exploiting Definition 3.(134)SR¯¯Ъ1=−0.3776,SR¯¯Ъ2=−0.3630,SR¯¯Ъ3=−0.4643,SR¯¯Ъ4=−0.2984.Otherwise,(135)SR¯¯Ъ1=−0.4860,SR¯¯Ъ2=−0.5250,SR¯¯Ъ3=−0.5373,SR¯¯Ъ4=−0.3486.Step II.11: from score values, we can get the following ranking orders.(136)ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3.Otherwise,(137)ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3.Table 6
The reorder decision matrixDT¯¯σ1.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.2,0.70.3,0.70.4,0.50.2,0.6ale¯¯20.4,0.50.3,0.50.1,0.70.4,0.6ale¯¯30.2,0.60.2,0.60.3,0.70.2,0.6ale¯¯40.3,0.60.3,0.60.3,0.60.3,0.6Table 7
The reorder decision matrixDT¯¯σ2.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.2,0.60.4,0.60.5,0.50.4,0.5ale¯¯20.5,0.50.4,0.50.2,0.60.5,0.5ale¯¯30.2,0.50.3,0.60.4,0.60.3,0.7ale¯¯40.5,0.40.4,0.50.3,0.50.4,0.5Table 8
The reorder decision matrixDT¯¯σ3.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1,0.70.2,0.70.2,0.80.2,0.7ale¯¯20.3,0.60.3,0.60.1,0.80.2,0.7ale¯¯30.2,0.70.2,0.70.1,0.80.2,0.7ale¯¯40.2,0.70.3,0.70.2,0.60.2,0.5Table 9
Collective decision matrix utilizing WIFSSGPOA.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1472,0.68700.2644,0.68670.3257,0.62420.2305,0.6341ale¯¯20.3646,0.55040.3133,0.55020.1126,0.73530.3268,0.6339ale¯¯30.2,0.63440.2130,0.65110.2211,0.73610.2128,0.6633ale¯¯40.2858,0.60900.3133,0.63340.2478,0.58640.2634,0.5296Table 10
Collective decision matrix utilizing WIFSSGPOG.
Cri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.1230,0.69210.2360,0.69200.2509,0.74160.2099,0.6590ale¯¯20.3402,0.56280.3081,0.56270.1047,0.76370.2514,0.6587ale¯¯30.2,0.65880.2069,0.66470.1315,0.76440.2067,0.6751ale¯¯40.2377,0.65570.3080,0.65820.2312,0.59130.2371,0.5412Hence, the optimal alternative isale¯¯4 while the worst one is ale¯¯3.


## 5.2. Comparison with Some Other Existing Approaches

In this subsection, we compare our developed approaches with some existing approaches based on different aggregation operators.Firstly, we solve the same example by utilizing the intuitionistic fuzzy Schweizer-Sklar weighted averaging operator developed by Zhang [63]. Secondly, we solve the same example by utilizing intuitionistic fuzzy power weighted averaging operator and intuitionistic fuzzy power geometric operator developed by Xu [46]. The score values and ranking order are given in Table 11.Table 11
Comparison with different approaches.
Aggregation operatorsScore valuesRanking orderIFWSSWA operator [63]SC¯¯Ъ1=−0.3098,SC¯¯Ъ2=−0.2075,SC¯¯Ъ3=−0.3874,SC¯¯Ъ4=−0.2090ale¯¯2>ale¯¯4>ale¯¯1>ale¯¯3IFPWA [46]SC¯¯Ъ1=−0.3453,SC¯¯Ъ2=−0.2388,SC¯¯Ъ3=−0.4027,SC¯¯Ъ4=−0.2380ale¯¯2>ale¯¯4>ale¯¯1>ale¯¯3IFPWG [46]SC¯¯Ъ1=−0.4025,SC¯¯Ъ2=−0.3058,SC¯¯Ъ3=−0.4270,SC¯¯Ъ4=−0.2688ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3Proposed GIFSSPWASC¯¯b1=−0.3206,SR¯¯Ъ2=−0.2179,SR¯¯Ъ3=−0.3927,SR¯¯Ъ4=−0.2164ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3Proposed GIFSSPWGSC¯¯Ъ1=−0.4700,SC¯¯Ъ2=−0.4145,SC¯¯Ъ3=−0.4560,SC¯¯Ъ4=−0.2971ale¯¯4>ale¯¯2>ale¯¯1>ale¯¯3From Table11, we can see that the ranking order acquired from the AOs developed by Zhang [63] is different from the proposed aggregation operators and that developed by Xu [46]. IFPWA and IFPWG operators are special cases of the proposed aggregation operators by taking Љ=1,ς⟶0. Also, the ranking orders obtained from the aggregation operators developed by Xu [46] and the proposed aggregation operators are the same. This shows the validity of the developed aggregation operators. The proposed generalized Schweizer-Sklar power aggregation operators have some advantage over the existing power aggregation operators for IFS. Firstly, the proposed aggregation operators have two parameters, one takes values from 0,+∞ and the other parameter takes values from −∞,0. These parameters make the decision-making process more supple. Secondly, it can eradicate the effect of awful data from the decision-making result. So, these aggregation operators are more general then the existing aggregation operators.


## 6. Conclusion

In this article, we initiated a number of new generalized power AOs based on Schweizer-Sklar operational laws, such as generalized intuitionistic fuzzy Schweizer-Sklar power averaging operator, weighted generalized intuitionistic fuzzy Schweizer-Sklar power averaging operator, generalized intuitionistic fuzzy Schweizer-Sklar power ordered averaging operator, generalized intuitionistic fuzzy Schweizer-Sklar power geometric averaging operator, weighted generalized intuitionistic fuzzy Schweizer-Sklar power geometric averaging operator, and generalized intuitionistic fuzzy Schweizer-Sklar power ordered geometric averaging operator. Furthermore, some core properties of the initiated AOs and various special cases with respect to generalized parameters are also investigated. Moreover, two MAGDM models are initiated. The first one is for solving MAGDM problems under an intuitionistic fuzzy environment in which the weights of the attributes and decision-makers are known and the second one is for solving MAGDM problems under an intuitionistic fuzzy environment in which the weights of the attributes and decision-makers are totally unknown. Finally, a numerical example is illustrated to show the effectiveness and practicality of the initiated two MAGDM models. We also confirm the advantages of the initiated MAGDM approaches by evaluating with some subsisting MAGDM approaches. The key advantages of the initiated AOs are as follows:(1)
The presented AOs are established on Schweizer-Sklar operational rules, which have a general parameter vector, which makes the information aggregation procedure more supple.(2)
The initiated AOs have two features at a time, firstly, it can negate the effect of awful data by taking the edge of the PA operator, Secondly, the initiated AO has a generalized parameter which takes values from0,+∞. For these motives, the initiated MAGDM method established on these initiated AOs is more general and reasonable.In future research, we will develop power Heronian mean operators established on these Schweizer-Sklar operational laws for IFNs, double-valued neutrosophic sets, q-rung orthopair fuzzy sets, and so on. At the same time, we also study a few applications in energy and supply chain management.



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*Source: 4634411-2022-06-13.xml*