# A Novel Approach on Decision Support System Based on the Aczel-Alsina Aggregation Operators and Their Applications to Supplier Selection Problems

Authors: Yong-Long Wang; Ebenezer Bonyah; Mashael Khayyat; Zubair Ahmad; Muhammad Shakeel; Waris Khan
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Induced aggregation operators are more suitable for aggregating the individual preference relations into a collective fuzzy preference relation. Therefore, in this paper, we introduce the notion of some new types of induced aggregation operators, based on the Aczel-Alsina operations. We construct some induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging/geometric (I-IVITrFAAOWA/G) operator, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging/geometric (I-IVITrFAAHA/G) operator. Moreover, some dominant properties of these developed operators are studied in detail. Based on these proposed approaches, a model is a build up for multicriteria decision making (MCDM), and their stepwise algorithm is being presented. Finally, in utilizing the developed approach, an illustrative example is solved with the help of proposed operators. In the end, we utilize an applicable example for supplier selection to prove the proposed methods and compare the result with existing methods, which shows the superiority, competence, and ability of the developed model.


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## Body

## 1. Introduction

Multicriteria group decision-making (MCGDM) problems are of consequence in most kinds of fields such as engineering, economics, medical diagnosis, and management. Traditionally, it has been assumed that the information, which accesses the alternatives in terms of criteria, and weight are expressed in real numbers. Multicriteria group decision making (MCGDM) has develop into tremendously precious in preceding few decades in the learning of decision support systems [1–3]. The convolution of DM problems increases time to time in this world of global antagonism as the socioeconomic structure becomes more dynamic. Therefore, making sensible and successful decisions in this situation is very complicated for a single decision expert. In this real world, group DM models are often used to combine the opinions of team of experienced experts in order to generate highly accurate and optimal values. Hence, MCGDM have an excellent ability and systematic method for improving and evaluating various competing criteria in all aspect of DM in order to achieve more appropriate and possible DM outcomes. In DM issues, the knowledge based about even a fact is often unclear, making the decision-making task more complicated and ambiguous. To resolve this restriction, Zadeh [4] in 1965 first established the idea of fuzzy set (FS) by considering the membership degree MD. This concept has been investigated in a various real-world problems, including clustering analysis [5], decision-making problems (DMPs) [6], and medical diagnosis [7]. Atanassov [8] developed the intuitionistic fuzzy set (IFS) as an extension of the FS, and its constraint is that the sum of the MD and the N-MD will be less than or equal to one. IFS has been a hot research topic for researchers who examined its hybrid structure in a variety of ways. The idea of intuitionistic fuzzy weighted averaging (IFWA) aggregation operators was first investigated by Xu [9]. Xu and Yager [10] developed the definition of intuitionistic fuzzy weighted geometric (IFWG) aggregation operators. The graphical techniques for rating score and accuracy functions were created by Ali et al. [11]. He et al. [12] explored the concept of intuitionistic fuzzy neutral averaging operators (IFNAO). He et al. [13] invented the notions of geometric relationship averaging operator and proposed its implementation in DM. Also Atanassov [14] presented preference relation based on IVIF. Further, Wan and Dong [15] work on extension of best-worst method based on IFS. Zaho et al. [16] were the first to apply the generalized intuitionistic fuzzy weighted averaging (GIFWA), generalized intuitionistic fuzzy order weighted averaging (IFOWA), and generalized intuitionistic fuzzy hybrid averaging (GIFHA) operators to DM. However, it was discovered that these aggregation process was performed by using Archimedean t-norms and t-conorm by operators. As an equivalent to algebraic product and sum, Einstein-based t-norm and t-conorm provides the best estimate for product and sum of intuitionistic fuzzy numbers (IFNs). Wang and Liu and Wang and Zhang [17, 18] provided IF Einstein weighted averaging (IFEWA) and IF Einstein weighted geometric (IFEWG) operators by using the concept of Einstein operation. After that, Abbas et al. [19] developed the idea of intuitionistic fuzzy rough Einstein weighted averaging (IFREWA), intuitionistic fuzzy rough Einstein hybrid averaging (IFREHA), and intuitionistic fuzzy rough ordered weighted averaging (IFREOWA) aggregation operators based on the TOPSIS method. Yager and Filev [20, 21] developed an expansion of the OWA operator, called the induced ordered weighted averaging (IOWA) operator. It takes the argument in pairs (OWA pairs), in which one component is used to induce an ordering over the second component and then aggregates. Wei [22] developed two new aggregation operators: the induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and the induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator. Su et al. [23] also constructed induced intuitionistic fuzzy hybrid averaging (I-IFHA) operator and induced interval-valued intuitionist fuzzy hybrid averaging (I-IIFHA) operator. Shakeel at al [24] developed the induced Pythagorean trapezoidal fuzzy Einstein ordered weighted averaging operator and the induced Pythagorean trapezoidal fuzzy Einstein hybrid averaging (I-PTFEHA) operator. In [25], Klement et al. discussed the triangular norms, and in [26, 27], Aczel-Alsina (AA) presented t-norm as well as t-conorm. After that, some new idea of AA was developed by Senapati et al. in [28]. Senapati et al. introduced the IF Aczel-Alsina weighted averaging (IFAAWA) operator, the IF Aczel-Alsina orders weighted averaging (IFAAOWA) operator, and the IF Aczel-Alsina hybrid averaging (IFAAHA) operator. Senapati et al. [29, 30] developed interval-valued intuitionistic fuzzy aggregation operators, such as the IVIF Aczel-Alsina weighted averaging/geometric operator, the IVIF Aczel–Alsina order weighted averaging/geometric operator, and the IVIF Aczel–Alsina hybrid averaging/geometric operator and built up several features of such operators. Senapati [31] also defined picture fuzzy aggregation operator such as the PF Aczel–Alsina weighted average (PFAAWA) operator, the PF Aczel–Alsina order weighted average (PFAAOWA) operator, and the PF Aczel–Alsina hybrid average (PFAAHA) operator with their properties. Later on, Senapati et al. [32] originated a few new aggregation operators for aggregating hesitant fuzzy information, namely, the HF Aczel-Alsina weighted averaging (HFAAWA) operator, HF Aczel-Alsina ordered weighted averaging (HFAAOWA) operator, HF Aczel-Alsina hybrid averaging (HFAAHA) operator, HF Aczel-Alsina weighted geometric (HFAAWG) operator, HF Aczel-Alsina ordered weighted geometric (HFAAOWG) operator, HF Aczel–Alsina hybrid geometric (HFAAHG) operator, and HF Aczel–Alsina weighted Bonferroni mean (HFAAWBM) operator.After that, Ban [33, 34] initiated the concept of trapezoidal fuzzy numbers (TrFNs) and interval-valued trapezoidal fuzzy numbers (I-VTrFNs). Later on, Wang and Zhang [35] expanded the definition of intuitionistic trapezoidal fuzzy numbers (ITrFNs) and interval-valued intuitionistic trapezoidal fuzzy numbers (IVITrFNs). Wei [36] developed the intuitionistic trapezoidal fuzzy ordered weighted averaging (ITrFOWA) operator and the intuitionistic trapezoidal fuzzy hybrid aggregation (ITrFHA) operator. Wan and Dong [37] defined the anticipation and expectant score of intuitionistic trapezoidal fuzzy numbers from the geometric angle. As for our information and based from the above review, no implementation of the MCGDM approach to review of induced interval-valued intuitionist trapezoidal fuzzy Aczel-Alsina (I-IVITrFAA) averaging/geometric aggregation operators. We proposed the average/geometric aggregation operators such as I-IVITrF Aczel-Alsina ordered weighted averaging (I-IVITrFAAOWA) operator, I-IVITrF Aczel-Alsina hybrid averaging (I-IVITrFAAHA) operator, I-IVITrF Aczel-Alsina ordered weighted geometric (I-IVITrFAAOWG) operator, and I-IVITrF Aczel-Alsina hybrid geometric (I-IVITrFAAHG) operator for solving MCGDM. To solve a MCGDM process, the weight of the attributes plays an important role in making decisions under the aggregation approaches.The paper provided the following: in the next section, we briefly discuss some fundamental ideas of IFSs, I-VIFS, TrFNs, IVITrFNs, IFAA, and I-VIFAA aggregation operators. In Section3, we outline AA operation laws for the I-VITrFNs. In Section 4, we develop I-IVITrFAAOWA operator, I-IVITrFAAHA operator, I-IVITrFAAOWG operator, and I-IVITrFAAOWG operator. In Section 5, we apply the I-IVITrFAAOWA operator to build up certain methodologies for managing MCGDM issues. Section 6 provides an example of choosing greatest alternative of the proposed method. In Section 7, we compare the developed method with the existing methods. At end of this paper, a few conclusions and upcoming research are mentioned in Section 8.


## 2. Preliminaries

In this section, we will define basic definition, results, and operational laws which are important for the improvement of this paper.


### 2.1. IFSs

Atanassov [8] introduced the idea of IFSs which is a generalization of FSs. FSs discussed only M-D where IFSs discussed M-D and NM-D. Moreover, the sum of M-D and NM-D is less than or equal to 1.Definition 1 (see [8]).
Let a setUz be fixed. An IFSlee in Uz is an object having the form
(1)Aee=uz,μleeuz,Ileeuzuz∈Uz,
whereμıee:Uz⟶0,1 and Iıee:Uz⟶0,1 represent the M-D and N-MD such that 0≤μıeeuz+Iıeeuz≤1 for each IFSıee in Uz,πıeeuz=1−μıeeuz−Iıeeuz,foralluz∈Uzπıeeuz is called the degree of indeterminacy of uz to ıee.Definition 2.
Atanassov [8] let ıee=uz,μıeeuz,Iıeeuzuz∈Uz and Gee=uz,μGeeuz,IGeeuzuz∈Uz are two IFSs over the universe Uz, then the operational laws between Aee and Gee are defined as
(i)
ıee⊆Gee, if μıeeuz≤μGeeuz and Iıeeuz≥IGeeuz∀uz∈Uz(ii)
ıee=Gee⇔ıee⊆Gee,Gee⊆ıee(iii)
ıee∪Gee=uz,maxμıeeuz,μGeeuz,minIıeeuz,IGeeuz/uz∈Uz(iv)
ıee∩Gee=uz,minμıeeuz,μGeeuz,maxIıeeuz,IGeeuz/uz∈Uz(v)
ıeec=uz,Iıeeuz,μıeeuz/uz∈Uz(vi)
ıee⊕Gee=uz,μıeeuz+μGeeuz−μıeeuz.μGeeuz,IıeeuzIGeeuz/uz∈Uz(vii)
ıee⊗Gee=uz,μıeeuz.μGeeuz,Iıeeuz+IGeeuz−Iıeeuz.IGeeuz/uz∈UzDefinition 3 (see [26]).
Letıee and Gee be any two IFSs, then the generalized intersection and the generalized union are defined in the following ways:
(2)ıee∩T,SGee=uz,Tμıeeuz,μGeeuz,SIıeeuz,IGeeuz/uz∈Uz,ıee∪S,TGee=uz,Sμıeeuz,μGeeuz,TIıeeuz,IGeeuz/uz∈Uz,
whereT presents a t-norm, and S presents a t-conorm.


### 2.2. I-VIFSs

Definition 4 (see [38]).
Letıee=μıee,Iıee=μıeelr,μıeeurIıeelr,Iıeeur be an I-VIFNs, where μıee=μıeelr,μıeıeur and Iıee=Iıeelr,Iıeeur represent an I-VIFNs, hence μıee⊂0,1 and Iıee⊂0,1, such that 0≤μıee+Iıee≤1.Definition 5.
Letıee=μıee,Iıee=μıeelr,μıeeurIıeelr,Iıeeur,ıe1e=μıe1e,Iıe1e=μıe1elr,μıe1eurIıe1elr,Iıe1eurıe2e=μıe2e,Iıe2e=μıe2elr,μıe2eurIıe2elr,Iıe2eur be any two I‐IVFNs, numbers, and ∂≥0. Then,
(1)
ıe1e⊕ıe2e=μıe1elr+μıe2elr−μıe1elrμıe2elr,Iıe1elrIıe2elr,μıe1eur+μıe2eur−μıe1eurμıe2eur,Iıe1eurIıe2eur(2)
ıe1e⊗ıe2e=μıe1elrμıe2elr,Iıe1elr+Iıe2elr−Iıe1elrIıe2elr,μıe1eurμıe2eur,Iıe1eur+Iıe2eur−Iıe1eurIıe2eur(3)
∂ıee=1−1−μıeelr∂,Iıeelr∂,1−1−μıeeur∂,Iıeeur∂(4)
ıee∂=μıeelr∂,1−1−Iıeelr∂,μıeeur∂,1−1−Iıeeur∂Example 6.
Letıee=0.45,0.04,0.35,0.23,ıe1e=0.34,0.21,0.05,0.34,ıe2e=0.27,0.34,0.21,0.31 be any three I-VIFNs, and ∂=0.4; then, we verify the above results such that,
(1)
ıe1e⊕ıe2e=μıe1elr+μıe2elr−μıe1elrμıe2elr,Iıe1elrIıe2elr,μıe1eur+μıe2eur−μıe1eurμıe2eur,Iıe1eurIıe2eur=0.34+0.27−0.340.27,0.050.21,0.21+0.34−0.210.34,0.340.31=0.61−.0.091,0.0105,0.55−0.0714,0.1054=0.519,0.0105,0.4786,0.1054(2)
ıe1e⊗ıe2e=μıe1elrμıe2elr,Iıe1elr+Iıe2elr−Iıe1elrIıe2elr,μıe1eurμıe2eur,Iıe1eur+Iıe2eur−Iıe1eurIıe2eur=0.340.27,0.05+0.21−0.050.21,0.210.34,0.34+0.31−0.340.31=0.0918,0.26−0.0105,0.0714,0.65−0.1054=0.0918,0.2495,0.0714,0.6394(3)
∂ıee=1−1−μıeelr∂,Iıeelr∂,1−1−μıeeur∂,Iıeeur∂1−1−0.450.4,0.350.4,1−1−0.040.4,0.230.4=1−0.7873,0.6570,1−0.9838,0.5550.=0.2127,0.6570,0.0162,0.5550(4)
ıee∂=μıeelr∂,1−1−Iıeelr∂,μıeeur∂,1−1−Iıeeur∂=0.450.4,1−1−0.350.40.040.4,1−1−0.230.4=0.7265,0.15820.2759,0.0992Definition 7 (see [39]).
Letıee=μıeelr,μıeeurIıeelr,Iıeeur be an I−VIFNs, a Scrıee can be defined as follows:
(3)Scrıee=μıeelr−Iıeelr+μıeeur−Iıeeur2Scrıee∈−1,1.Example 8.
Letıe1e=0.37,0.310.16,0.23,ıe2e=0.20,0.340.40,0.54.
ıe3e=0.46,0.320.32,0.24,ıe4e=0.46,0.210.20,0.10 be I−VIFNs. Then, we verify the above results such that,
(4)Scr1ıe1e=0.37−0.16+0.31−0.232=0.1450,Scr2ıe2e=0.20−0.40+0.34−0.542=−0.2100,Scr3ıe3e=0.46−0.32+0.32−0.242=0.1100,Scr4ıe4e=0.46−0.20+0.21−0.102=0.1910.In Figure1, we supposed some interval-valued of intuitionistic fuzzy numbers and apply them score function to find out the highest score values for ranking process.Definition 9 (see [39]).
Letıee=μıeelr,μıeeurIıeelr,Iıeeur be an I‐VIFNs; an accuracy function Acrıee can be defined as follows:
(5)Acrıee=μıeelr+Iıeelr+μıeeur+Iıeeur2Acrıee∈0,1.Example 10.
Let(6)ıe1e=0.37,0.310.16,0.23,ıe2e=0.20,0.340.40,0.54,ıe3e=0.46,0.320.32,0.24,ıe4e=0.46,0.210.20,0.10,be I−VIFNs. Then, we verify the above results such that,
(7)Acr1ıe1e=0.37+0.16+0.31+0.232=0.5400,Acr2ıe2e=0.20+0.40+0.34+0.542=0.7400,Acr3ıe3e=0.46+0.32+0.32+0.242=0.6700,Acr4ıe4e=0.46+0.20+0.21+0.102=0.4900.Figure 1
Graphical approach of score function.In Figure2, we supposed some interval-valued of intuitionistic fuzzy numbers and apply their accuracy function to find out the highest accuracy values for ranking process.Figure 2
Graphical approach of accuracy function.


### 2.3. ITFNs and I-VITFNs

Definition 11 (see [40]).
Letıee be ITFNs, its membership function
(8)μıeel=x−abba−abμıee,ab≤l≤ba,μıee,ba≤l≤cd,dc−xdc−cdμıeecd≤l≤dc,1,otherwise.
Its nonmembership function is(9)Iıeel=dc−l+Iıeel−abba−abab≤l≤ba,Iıee,ba≤l≤cd,l−cd+Iıeedc−ldc−cd,cd≤l≤dc,0,otherwise,
where0≤μıee≤1;0≤Iıee≤1;0≤μıee+Iıeıe≤1;ab,ba,cd,dc∈R. Then, ıee is called ITFNs. For convenience, Aee=ab,ba,cd,dcμıee,Iıee.Definition 12 (see [40]).
Let(10)ıee=ab,ba,cd,dcμıee,Iıee,ıe1e=a1b,b1a,c1d,d1cμıe1e,Iıe1e,(11)ıe2e=a2b,b2a,c2d,d2cμıe2e,Iıe2ebe the two ITFNs, and ∂≥0. Then,
(1)
ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1a+d2a;μıe1e+μıe2e−μıe1eμıe2e,Iıe1eIıe2e(2)
ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1ad2a;μıe1eμıe2e,Iıe1e+Iıe2e−Iıe1eIıe2e(3)
∂ıee=∂ab,∂ba,∂cd,∂da;1−1−μıee∂;Iıee∂(4)
ıee∂=ab∂,ba∂,cd∂,da∂;μıee,1−1−Iıee∂Definition 13 (see [40]).
Let(12)ıee=ab,ba,cd,dc,μıeelr,μıeeurIıeelr,Iıeeur,ıe1e=a1b,b1a,c1d,d1c,μıe1elr,μıe1eurIıe1elr,Iıe1eur,(13)ıe2e=a2b,b2a,c2d,d2c,μıe2elr,μıe2eurIıe2elr,Iıe2eurbe any three I-VITrNs, and ∂≥0. Then,
(1)
ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1a+d2a,μıe1elr+μıe2elr−μıe1elrμıe2elr,Iıe1elrIıe2elr,μıe1eur+μıe2eur−μıe1eurμıe2eur,Iıe1eurIıe2eur,(2)
ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1ad2a,μıe1elrμıe2elr,Iıe1elr+Iıe2elr−Iıe1elrIıe2elr,μıe1eurμıe2eur,Iıe1eur+Iıe2eur−Iıe1eurIıe2eur,(3)
∂ıee=∂ab,∂ba,∂cd,∂da,1−1−μıeelr∂,Iıeelr∂,1−1−μıeeur∂,Iıeeur∂(4)
ıee∂=ab∂,ba∂,cd∂,da∂,μıeelr∂,1−1−Iıeelr∂,μıeeur∂,1−1−Iıeeur∂Example 14.
Let(14)ıee=0.30,0.40,0.20,0.50,0.40,0.300.50,0.20,ıe1e=0.20,0.30,0.10,0.40,0.50,0.200.30,0.60,(15)ıe2e=0.40,0.50,0.60,0.60,0.40,0.300.50,0.20be any three I-VITrNs, and ∂=2. We verify the above operational laws such that,
(16)ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1a+d2a,μıe1elr+μıe2elr−μıe1elrμıe2elr,Iıe1elrIıe2elrμıe1eur+μıe2eur−μıe1eurμıe2eur,Iıe1eurIıe2eur=0.20+0.40,0.30+0.50,0.10+0.60,0.40+0.60,0.50+0.40−0.500.40,0.300.500.20+0.30−0.200.30,0.600.20=0.60,0.80,0.70,0.10,0.90−0.20,0.150.50−0.06,0.12=0.60,0.80,0.70,0.10,0.70,0.150.44,0.12,ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1ad2a;μıe1elrμıe2elr,Iıe1elr+Iıe2elr−Iıe1elrIıe2elr,μıe1eurμıe2eur,Iıe1eur+Iıe2eur−Iıe1eurIıe2eur=0.200.40,0.300.50,0.100.60,0.400.600.50.40,0.30+0.50−0.300.500.20.30,0.60+0.20−0.600.20=0.08,0.15,0.06,0.240.20,0.80−0.150.06,0.80−0.12=0.08,0.15,0.06,0.240.20,0.650.06,0.68,∂ıee=∂ab,∂ba,∂cd,∂da,1−1−μıeelr∂,Iıeelr∂,1−1−μıeeur∂,Iıeeur∂=20.30,20.40,20.20,20.50,1−1−0.43,0.531−1−0.33,0.23=0.60,0.80,0.40,0.10,0.78,0.130.66,0.01,ıee∂=ab∂,ba∂,cd∂,da∂,μıeelr∂,1−1−Iıeelr∂,μıeeur∂,1−1−Iıeeur∂=0.302,0.402,0.2002,0.502,0.402,1−1−0.5020.32,1−1−0.202=0.09,0.16,0.04,0.25,0.16,0.750.09,0.36.Definition 15 (see [40]).
Let(17)ıee=ab,ba,cd,dc,μıeelr,μıeeurIıeelr,Iıeeurbe I‐VITFNs; a score function Scrıee can be defined as follows:
(18)Scrıee=ab+ba+cd+dc4.μıeelr−Iıeelr+μıeeur−Iıeeur2Scrıee∈−1,1.Definition 16 (see [40]).
Let(19)ıee=ab,ba,cd,dc,μıeelr,μıeeurIıeelr,Iıeeurbe I‐VITFNs; an accuracy function Acrıee can be defined as follows:
(20)Acrıee=ab+ba+cd+dc4.μıeelr+Iıeelr+μıeeur+Iıeeur2Acrıee∈−1,1.


### 2.4. The Aczel-Alsina Operations of IFNs

In this section, we will present A-A operations on IFNs [28]. Let T presents a t-norm and S presents a t-conorm be A-A product TA and A-A sum A, respectively. A-A product ıee⊗Gee and A-A sum ıee⊕Gee over two IFSs ıee and Gee defined in the following way μıeelr,μıeeur:
(21)ıee⊗Gee=uz,TAμıeeuz,μGeeuz,SAIıeeuz,IGeeuz/uz∈Uz,(22)ıee⊕Gee=uz,SAμıeeuz,μGeeuz,TAIıeeuz,IGeeuz/uz∈Uz.Definition 17 (see [28]).
Letıee=μıee,Iıee,ıe1e=μıe1e,Iıe1e and ıe2e=μıe2e,Iıe2e be any three IFNs, ξ≥1, and ∂>0. Then, the A-A t-norm and t-conorm operation of IFNs are defined as
(i)
ıe1e⊕ıe2e=1−e−−log1−μıe1eξ+−log1−μıe2eξ1/ξ,e−−logIıe1eξ+−logIıe2eξ1/ξ(ii)
ıe1e⊗ıe2e=e−−logμıe1eξ+−logμıe2eξ1/ξ,1−e−−log1−Iıe1eξ+−log1−Iıe2eξ1/ξ(iii)
∂ıee=1−e−∂−log1−μıeeξ1/ξ,e−∂−logIıeeξ1/ξ(iv)
ıee∂=e−∂−logμıeeξ1/ξ,1−e−∂−log1−Iıeeξ1/ξ


## 2.1. IFSs

Atanassov [8] introduced the idea of IFSs which is a generalization of FSs. FSs discussed only M-D where IFSs discussed M-D and NM-D. Moreover, the sum of M-D and NM-D is less than or equal to 1.Definition 1 (see [8]).
Let a setUz be fixed. An IFSlee in Uz is an object having the form
(1)Aee=uz,μleeuz,Ileeuzuz∈Uz,
whereμıee:Uz⟶0,1 and Iıee:Uz⟶0,1 represent the M-D and N-MD such that 0≤μıeeuz+Iıeeuz≤1 for each IFSıee in Uz,πıeeuz=1−μıeeuz−Iıeeuz,foralluz∈Uzπıeeuz is called the degree of indeterminacy of uz to ıee.Definition 2.
Atanassov [8] let ıee=uz,μıeeuz,Iıeeuzuz∈Uz and Gee=uz,μGeeuz,IGeeuzuz∈Uz are two IFSs over the universe Uz, then the operational laws between Aee and Gee are defined as
(i)
ıee⊆Gee, if μıeeuz≤μGeeuz and Iıeeuz≥IGeeuz∀uz∈Uz(ii)
ıee=Gee⇔ıee⊆Gee,Gee⊆ıee(iii)
ıee∪Gee=uz,maxμıeeuz,μGeeuz,minIıeeuz,IGeeuz/uz∈Uz(iv)
ıee∩Gee=uz,minμıeeuz,μGeeuz,maxIıeeuz,IGeeuz/uz∈Uz(v)
ıeec=uz,Iıeeuz,μıeeuz/uz∈Uz(vi)
ıee⊕Gee=uz,μıeeuz+μGeeuz−μıeeuz.μGeeuz,IıeeuzIGeeuz/uz∈Uz(vii)
ıee⊗Gee=uz,μıeeuz.μGeeuz,Iıeeuz+IGeeuz−Iıeeuz.IGeeuz/uz∈UzDefinition 3 (see [26]).
Letıee and Gee be any two IFSs, then the generalized intersection and the generalized union are defined in the following ways:
(2)ıee∩T,SGee=uz,Tμıeeuz,μGeeuz,SIıeeuz,IGeeuz/uz∈Uz,ıee∪S,TGee=uz,Sμıeeuz,μGeeuz,TIıeeuz,IGeeuz/uz∈Uz,
whereT presents a t-norm, and S presents a t-conorm.


## 2.2. I-VIFSs

Definition 4 (see [38]).
Letıee=μıee,Iıee=μıeelr,μıeeurIıeelr,Iıeeur be an I-VIFNs, where μıee=μıeelr,μıeıeur and Iıee=Iıeelr,Iıeeur represent an I-VIFNs, hence μıee⊂0,1 and Iıee⊂0,1, such that 0≤μıee+Iıee≤1.Definition 5.
Letıee=μıee,Iıee=μıeelr,μıeeurIıeelr,Iıeeur,ıe1e=μıe1e,Iıe1e=μıe1elr,μıe1eurIıe1elr,Iıe1eurıe2e=μıe2e,Iıe2e=μıe2elr,μıe2eurIıe2elr,Iıe2eur be any two I‐IVFNs, numbers, and ∂≥0. Then,
(1)
ıe1e⊕ıe2e=μıe1elr+μıe2elr−μıe1elrμıe2elr,Iıe1elrIıe2elr,μıe1eur+μıe2eur−μıe1eurμıe2eur,Iıe1eurIıe2eur(2)
ıe1e⊗ıe2e=μıe1elrμıe2elr,Iıe1elr+Iıe2elr−Iıe1elrIıe2elr,μıe1eurμıe2eur,Iıe1eur+Iıe2eur−Iıe1eurIıe2eur(3)
∂ıee=1−1−μıeelr∂,Iıeelr∂,1−1−μıeeur∂,Iıeeur∂(4)
ıee∂=μıeelr∂,1−1−Iıeelr∂,μıeeur∂,1−1−Iıeeur∂Example 6.
Letıee=0.45,0.04,0.35,0.23,ıe1e=0.34,0.21,0.05,0.34,ıe2e=0.27,0.34,0.21,0.31 be any three I-VIFNs, and ∂=0.4; then, we verify the above results such that,
(1)
ıe1e⊕ıe2e=μıe1elr+μıe2elr−μıe1elrμıe2elr,Iıe1elrIıe2elr,μıe1eur+μıe2eur−μıe1eurμıe2eur,Iıe1eurIıe2eur=0.34+0.27−0.340.27,0.050.21,0.21+0.34−0.210.34,0.340.31=0.61−.0.091,0.0105,0.55−0.0714,0.1054=0.519,0.0105,0.4786,0.1054(2)
ıe1e⊗ıe2e=μıe1elrμıe2elr,Iıe1elr+Iıe2elr−Iıe1elrIıe2elr,μıe1eurμıe2eur,Iıe1eur+Iıe2eur−Iıe1eurIıe2eur=0.340.27,0.05+0.21−0.050.21,0.210.34,0.34+0.31−0.340.31=0.0918,0.26−0.0105,0.0714,0.65−0.1054=0.0918,0.2495,0.0714,0.6394(3)
∂ıee=1−1−μıeelr∂,Iıeelr∂,1−1−μıeeur∂,Iıeeur∂1−1−0.450.4,0.350.4,1−1−0.040.4,0.230.4=1−0.7873,0.6570,1−0.9838,0.5550.=0.2127,0.6570,0.0162,0.5550(4)
ıee∂=μıeelr∂,1−1−Iıeelr∂,μıeeur∂,1−1−Iıeeur∂=0.450.4,1−1−0.350.40.040.4,1−1−0.230.4=0.7265,0.15820.2759,0.0992Definition 7 (see [39]).
Letıee=μıeelr,μıeeurIıeelr,Iıeeur be an I−VIFNs, a Scrıee can be defined as follows:
(3)Scrıee=μıeelr−Iıeelr+μıeeur−Iıeeur2Scrıee∈−1,1.Example 8.
Letıe1e=0.37,0.310.16,0.23,ıe2e=0.20,0.340.40,0.54.
ıe3e=0.46,0.320.32,0.24,ıe4e=0.46,0.210.20,0.10 be I−VIFNs. Then, we verify the above results such that,
(4)Scr1ıe1e=0.37−0.16+0.31−0.232=0.1450,Scr2ıe2e=0.20−0.40+0.34−0.542=−0.2100,Scr3ıe3e=0.46−0.32+0.32−0.242=0.1100,Scr4ıe4e=0.46−0.20+0.21−0.102=0.1910.In Figure1, we supposed some interval-valued of intuitionistic fuzzy numbers and apply them score function to find out the highest score values for ranking process.Definition 9 (see [39]).
Letıee=μıeelr,μıeeurIıeelr,Iıeeur be an I‐VIFNs; an accuracy function Acrıee can be defined as follows:
(5)Acrıee=μıeelr+Iıeelr+μıeeur+Iıeeur2Acrıee∈0,1.Example 10.
Let(6)ıe1e=0.37,0.310.16,0.23,ıe2e=0.20,0.340.40,0.54,ıe3e=0.46,0.320.32,0.24,ıe4e=0.46,0.210.20,0.10,be I−VIFNs. Then, we verify the above results such that,
(7)Acr1ıe1e=0.37+0.16+0.31+0.232=0.5400,Acr2ıe2e=0.20+0.40+0.34+0.542=0.7400,Acr3ıe3e=0.46+0.32+0.32+0.242=0.6700,Acr4ıe4e=0.46+0.20+0.21+0.102=0.4900.Figure 1
Graphical approach of score function.In Figure2, we supposed some interval-valued of intuitionistic fuzzy numbers and apply their accuracy function to find out the highest accuracy values for ranking process.Figure 2
Graphical approach of accuracy function.


## 2.3. ITFNs and I-VITFNs

Definition 11 (see [40]).
Letıee be ITFNs, its membership function
(8)μıeel=x−abba−abμıee,ab≤l≤ba,μıee,ba≤l≤cd,dc−xdc−cdμıeecd≤l≤dc,1,otherwise.
Its nonmembership function is(9)Iıeel=dc−l+Iıeel−abba−abab≤l≤ba,Iıee,ba≤l≤cd,l−cd+Iıeedc−ldc−cd,cd≤l≤dc,0,otherwise,
where0≤μıee≤1;0≤Iıee≤1;0≤μıee+Iıeıe≤1;ab,ba,cd,dc∈R. Then, ıee is called ITFNs. For convenience, Aee=ab,ba,cd,dcμıee,Iıee.Definition 12 (see [40]).
Let(10)ıee=ab,ba,cd,dcμıee,Iıee,ıe1e=a1b,b1a,c1d,d1cμıe1e,Iıe1e,(11)ıe2e=a2b,b2a,c2d,d2cμıe2e,Iıe2ebe the two ITFNs, and ∂≥0. Then,
(1)
ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1a+d2a;μıe1e+μıe2e−μıe1eμıe2e,Iıe1eIıe2e(2)
ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1ad2a;μıe1eμıe2e,Iıe1e+Iıe2e−Iıe1eIıe2e(3)
∂ıee=∂ab,∂ba,∂cd,∂da;1−1−μıee∂;Iıee∂(4)
ıee∂=ab∂,ba∂,cd∂,da∂;μıee,1−1−Iıee∂Definition 13 (see [40]).
Let(12)ıee=ab,ba,cd,dc,μıeelr,μıeeurIıeelr,Iıeeur,ıe1e=a1b,b1a,c1d,d1c,μıe1elr,μıe1eurIıe1elr,Iıe1eur,(13)ıe2e=a2b,b2a,c2d,d2c,μıe2elr,μıe2eurIıe2elr,Iıe2eurbe any three I-VITrNs, and ∂≥0. Then,
(1)
ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1a+d2a,μıe1elr+μıe2elr−μıe1elrμıe2elr,Iıe1elrIıe2elr,μıe1eur+μıe2eur−μıe1eurμıe2eur,Iıe1eurIıe2eur,(2)
ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1ad2a,μıe1elrμıe2elr,Iıe1elr+Iıe2elr−Iıe1elrIıe2elr,μıe1eurμıe2eur,Iıe1eur+Iıe2eur−Iıe1eurIıe2eur,(3)
∂ıee=∂ab,∂ba,∂cd,∂da,1−1−μıeelr∂,Iıeelr∂,1−1−μıeeur∂,Iıeeur∂(4)
ıee∂=ab∂,ba∂,cd∂,da∂,μıeelr∂,1−1−Iıeelr∂,μıeeur∂,1−1−Iıeeur∂Example 14.
Let(14)ıee=0.30,0.40,0.20,0.50,0.40,0.300.50,0.20,ıe1e=0.20,0.30,0.10,0.40,0.50,0.200.30,0.60,(15)ıe2e=0.40,0.50,0.60,0.60,0.40,0.300.50,0.20be any three I-VITrNs, and ∂=2. We verify the above operational laws such that,
(16)ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1a+d2a,μıe1elr+μıe2elr−μıe1elrμıe2elr,Iıe1elrIıe2elrμıe1eur+μıe2eur−μıe1eurμıe2eur,Iıe1eurIıe2eur=0.20+0.40,0.30+0.50,0.10+0.60,0.40+0.60,0.50+0.40−0.500.40,0.300.500.20+0.30−0.200.30,0.600.20=0.60,0.80,0.70,0.10,0.90−0.20,0.150.50−0.06,0.12=0.60,0.80,0.70,0.10,0.70,0.150.44,0.12,ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1ad2a;μıe1elrμıe2elr,Iıe1elr+Iıe2elr−Iıe1elrIıe2elr,μıe1eurμıe2eur,Iıe1eur+Iıe2eur−Iıe1eurIıe2eur=0.200.40,0.300.50,0.100.60,0.400.600.50.40,0.30+0.50−0.300.500.20.30,0.60+0.20−0.600.20=0.08,0.15,0.06,0.240.20,0.80−0.150.06,0.80−0.12=0.08,0.15,0.06,0.240.20,0.650.06,0.68,∂ıee=∂ab,∂ba,∂cd,∂da,1−1−μıeelr∂,Iıeelr∂,1−1−μıeeur∂,Iıeeur∂=20.30,20.40,20.20,20.50,1−1−0.43,0.531−1−0.33,0.23=0.60,0.80,0.40,0.10,0.78,0.130.66,0.01,ıee∂=ab∂,ba∂,cd∂,da∂,μıeelr∂,1−1−Iıeelr∂,μıeeur∂,1−1−Iıeeur∂=0.302,0.402,0.2002,0.502,0.402,1−1−0.5020.32,1−1−0.202=0.09,0.16,0.04,0.25,0.16,0.750.09,0.36.Definition 15 (see [40]).
Let(17)ıee=ab,ba,cd,dc,μıeelr,μıeeurIıeelr,Iıeeurbe I‐VITFNs; a score function Scrıee can be defined as follows:
(18)Scrıee=ab+ba+cd+dc4.μıeelr−Iıeelr+μıeeur−Iıeeur2Scrıee∈−1,1.Definition 16 (see [40]).
Let(19)ıee=ab,ba,cd,dc,μıeelr,μıeeurIıeelr,Iıeeurbe I‐VITFNs; an accuracy function Acrıee can be defined as follows:
(20)Acrıee=ab+ba+cd+dc4.μıeelr+Iıeelr+μıeeur+Iıeeur2Acrıee∈−1,1.


## 2.4. The Aczel-Alsina Operations of IFNs

In this section, we will present A-A operations on IFNs [28]. Let T presents a t-norm and S presents a t-conorm be A-A product TA and A-A sum A, respectively. A-A product ıee⊗Gee and A-A sum ıee⊕Gee over two IFSs ıee and Gee defined in the following way μıeelr,μıeeur:
(21)ıee⊗Gee=uz,TAμıeeuz,μGeeuz,SAIıeeuz,IGeeuz/uz∈Uz,(22)ıee⊕Gee=uz,SAμıeeuz,μGeeuz,TAIıeeuz,IGeeuz/uz∈Uz.Definition 17 (see [28]).
Letıee=μıee,Iıee,ıe1e=μıe1e,Iıe1e and ıe2e=μıe2e,Iıe2e be any three IFNs, ξ≥1, and ∂>0. Then, the A-A t-norm and t-conorm operation of IFNs are defined as
(i)
ıe1e⊕ıe2e=1−e−−log1−μıe1eξ+−log1−μıe2eξ1/ξ,e−−logIıe1eξ+−logIıe2eξ1/ξ(ii)
ıe1e⊗ıe2e=e−−logμıe1eξ+−logμıe2eξ1/ξ,1−e−−log1−Iıe1eξ+−log1−Iıe2eξ1/ξ(iii)
∂ıee=1−e−∂−log1−μıeeξ1/ξ,e−∂−logIıeeξ1/ξ(iv)
ıee∂=e−∂−logμıeeξ1/ξ,1−e−∂−log1−Iıeeξ1/ξ


## 3. The Aczel-Alsina Operations of I-VITrFNs

In this section, we will present the operational laws of I-VITrFNs based on A-A aggregation operators.Definition 18.
Let(23)ıee=ab,ba,cd,dc,μıeelr,μıeeurIıeelr,Iıeeur,ıe1e=a1b,b1a,c1d,d1c,μıe1elr,μıe1eurIıe1elr,Iıe1eur,(24)ıe2e=a2b,b2a,c2d,d2c,μıe2elr,μıe2eurIıe2elr,Iıe2eurbe any three I-VITrFNs, ∂>0. Then, the Aczel-Alsina operation laws on I-VITrFNs are defined as
(1)
ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1c+d2c,1−e−−log1−μıe1elrξ+−log1−μıe2elrξ1/ξ,e−−logIıe1elrξ+−logIıe2elrξ1/ξ1−e−−log1−μıe1eurξ+−log1−μıe2eurξ1/ξ,e−−logIıe1eurξ+−logIıe2eurξ1/ξ(2)
ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1cd2c,e−−logμıe1elrξ+−logμıe2elrξ1/ξ,1−e−−log1−Iıe1elrξ+−log1−Iıe2elrξ1/ξe−−logμıe1eurξ+−logμıe2eurξ1/ξ,1−e−−log1−Iıe1eurξ+−log1−Iıe2eurξ1/ξ(3)
∂ıee=∂ab,∂ba,∂cd,∂dc,1−e−∂−log1−μıeelrξ1/ξ,e−∂−logIıeelrξ1/ξ1−e−∂−log1−μıeeurξ1/ξ,e−∂−logIıeeurξ1/ξ(4)
ıee∂=ab∂,ba∂,cd∂,cd∂,e−∂−logμıeelrξ1/ξ,1−e−∂−log1−Iıeelrξ1/ξe−∂−logμıeeurξ1/ξ,1−e−∂−log1−Iıeeurξ1/ξExample 19.
Let(25)ıee=0.2,0.4,0.5,0.6,0.6,0.30.5,0.4,ıe1e=0.3,0.5,0.6,0.7,0.2,0.70.1,0.8,(26)ıe2e=0.4,0.3,0.2,0.1,0.8,0.30.6,0.3be any three I-VITrFNs; then, using Aczel-Alsina operation on I-VITrFNs as defined in above operation laws for ξ=2 and ∂=3, we get
(1)
ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1c+d2c,1−e−−log1−μıe1elrξ+−log1−μıe2elrξ1/ξ,e−−logIıe1elrξ+−logIıe2elrξ1/ξ1−e−−log1−μıe1eurξ+−log1−μıe2eurξ1/ξ,e−−logIıe1eurξ+−logIıe2eurξ1/ξ0.3+0.4,0.5+0.3,0.6+0.2,0.7+0.11−e−−log1−0.22+−log1−0.821/2,e−−log0.12+−log0.621/21−e−−log1−0.72+−log1−0.321/2,e−−log0.82+−log0.321/2=0.7,0.8,0.8,0.9,0.8031,0.09460.7151,0.2939(2)
ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1cd2c,e−−logμıe1elrξ+−logμıe2elrξ1/ξ,1−e−−log1−Iıe1elrξ+−log1−Iıe2elrξ1/ξe−−logμıe1eurξ+−logμıe2eurξ1/ξ,1−e−−log1−Iıe1eurξ+−log1−Iıe2eurξ1/ξ=0.30.4,0.50.3,0.60.2,0.70.1e−−log0.22+−log0.821/2,1−e−−log1−0.12+−log1−0.621/2e−−log0.72+−log0.321/2,1−e−−log1−0.82+−log1−0.321/2=0.12,0.15,0.12,0.07,0.1969,0.60240.2849,0.8077(3)
∂ıee=∂ab,∂ba,∂cd,∂dc,1−e−∂−log1−μıeelrξ1/ξ,e−∂−logIıeelrξ1/ξ1−e−∂−log1−μıeeurξ1/ξ,e−∂−logIıeeurξ1/ξ=0.30.2,0.30.4,0.30.5,0.30.6,1−e−3−log1−0.621/2,e−3−log0.521/21−e−3−log1−0.321/2,e−3−log0.421/2=0.06,0.12,0.15,0.18,0.7955,0.30400.4609,0.2045(4)
ıee∂=ab∂,ba∂,cd∂,cd∂,e−∂−logμıeelrξ1/ξ,1−e−∂−log1−Iıeelrξ1/ξe−∂−logμıeeurξ1/ξ,1−e−∂−log1−Iıeeurξ1/ξ=0.20.3,0.40.3,0.50.3,0.60.3e−3−log0.621/2,1−e−3−log1−0.521/2,e−3−log0.321/2,1−e−3−log1−0.421/2=0.62,0.76,0.81,0.86,0.4128,0.69900.1243,0.5872Theorem 20.
Let(27)ıee=ab,ba,cd,dc;μıeelr,μıeeurIıeelr,Iıeeur,ıe1e=a1b,b1a,c1d,d1cμıe1elr,μıe1eurIıe1elr,Iıe1eur,(28)ıe2e=a2b,b2a,c2d,d2cμıe2elr,μıe2eurIıe2elr,Iıe2eurbe any three I-VITrFNs, ∂>0. Then, we define the following results.
(1)
ıe1e⊕ıe2e=ıe2e⊕ıe1e(2)
ıe1e⊗ıe2e=ıe2e⊗ıe1e(3)
∂ıe1e⊕ıe2e=∂ıe1e⊕∂ıe2e, ∂>0(4)
∂1⊕∂2ıee=∂1ıee⊕∂2ıee,∂1,∂2>0(5)
ıe1e⊗ıe2e∂=ıe1e∂⊗ıe2e∂, ∂>0Proof.
For the threeα~,α~1, and α~2, and ∂,∂1,∂2>0, as in Definition 18 such that
(1)
ıe1e⊕ıe2e=a1b+a2b,b1a+b2a,c1d+c2d,d1c+d2c1−e−−log1−μıe1elrξ+−log1−μıe2elrξ1/ξ,e−−logIıe1elrξ+−logIıe2elrξ1/ξ1−e−−log1−μıe1eurξ+−log1−μıe2eurξ1/ξ,e−−logIıe1eurξ+−logIıe2eurξ1/ξ=a2b+a1b,b2a+b1a,c2d+c1d,d2c+d1c1−e−−log1−μıe2elrξ+−log1−μıe1elrξ1/ξ,e−−logIıe2elrξ+−logIıe1elrξ1/ξ1−e−−log1−μıe2eurξ+−log1−μıe1eurξ1/ξ,e−−logIıe2eurξ+−logIıe1eurξ1/ξ=ıe2e⊕ıe1e(2)
ıe1e⊗ıe2e=a1ba2b,b1ab2a,c1dc2d,d1cd2ce−−logμıe1elrξ+−logμAe2elrξ1/ξ,1−e−−log1−Iıe1elrξ+−log1−IAe2elrξ1/ξe−−logμıe1eurξ+−logμAe2eurξ1/ξ,1−e−−log1−Iıe1eurξ+−log1−IAe2eurξ1/ξ=a2ba1b,b2ab1a,c2dc1d,d2cd1ce−−logμıe2elrξ+−logμıe1elrξ1/ξ,1−e−−log1−Iıe2elrξ+−log1−Iıe1elrξ1/ξe−−logμıe2eurξ+−logμıe1eurξ1/ξ,1−e−−log1−Iıe2eurξ+−log1−Iıe1eurξ1/ξ=ıe2e⊗ıe1e(3)
∂ıe1e⊕ıe2e=∂a1b+a2b,b1a+b2a,c1d+c2d,d1c+d2c1−e−−log1−μıe1elrξ+−log1−μıe2elrξ1/ξ,e−−logIıe1elrξ+−logIıe2elrξ1/ξ1−e−−log1−μıe1eurξ+−log1−μıe2eurξ1/ξ,e−−logIıe1eurξ+−logIıe2eurξ1/ξ=∂a1b+a2b,b1a+b2a,c1d+c2d,d1c+d2c1−e−∂−log1−μıe1elrξ+−log1−μıe2elrξ1/ξ,e−∂−logIıe1elrξ+−logIıe2elrξ1/ξ1−e−∂−log1−μıe1eurξ+−log1−μıe2eurξ1/ξ,e−∂−logIıe1eurξ+−logIıe2eurξ1/ξ=∂a1b,b1a,c1d,d1c+∂a2b,b2a,c2d,d2c1−e−∂−log1−μıe1elrξ+−log1−Iıe1elrξ1/ξ,e−∂−logμıe1elrξ+−logIıe2elrξ1/ξ⊕1−e−∂−log1−μıe1eurξ+−log1−μıe1eurξ1/ξ,e−∂−logIıe1eurξ+−logIıe2eurξ1/ξ=∂ıe1e⊕∂ıe2e(4)
∂1ıee⊕∂2ıee=∂1ab,ba,cd,dc+∂2ab,ba,cd,dc1−e−∂1−log1−μıeelrξ+−log1−μıeeurξ1/ξ,e−∂1−logIıeelrξ+−logIıeeurξ1/ξ⊕1−e−∂2−log1−μıeelrξ+−log1−μıeeurξ1/ξ,e−∂2−logIıeelrξ+−logIıeeurξ1/ξ=∂1+∂2ab,ba,cd,dc1−e−∂1+∂2−log1−μıeelrξ+−log1−μıeeurξ1/ξ,e−∂1+∂2−logIıeelrξ+−logIıeelrξ1/ξ=∂1⊕∂2ıee(5)
ıe1e⊗ıe2e∂=a1ba2b,b1ab2a,c1dc2d,d1cd2c,e−−logμıe1elrξ+−logμıe2elrξ1/ξ,1−e−−log1−Iıe1elrξ+−log1−Iıe2elrξ1/ξe−∂−logμıe1eurξ+−logμıe2eurξ1/ξ,1−e−−log1−Iıe1eurξ+−log1−Iıe2eurξ1/ξ∂=a1ba2b∂,b1ab2a∂,c1dc2d∂,d1cd2c∂,e−∂−logμıe1elrξ1/ξ,1−e−∂−log1−Iıe1elrξ1/ξ,e∂−logμıe2elrξ1/ξ,1−e−∂−log1−Iıe2elrξ1/ξ⊗e−∂−logμıe1eurξ1/ξ,1−e−∂−log1−Iıe1eurξ1/ξ,e−∂−logμıe2eurξ1/ξ,−e−∂−log1−Iıe2eurξ1/ξ=ıe1e∂⊗ıe2e∂


## 4. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel-Alsina Ordered Weighted Averaging (I-IVITrFAAOWA) Operator

Definition 21.
Letueje,ıeje=ueje,ajb,bja,cjd,djcμıejelr,μıejeurIıejelr,Iıejeurj=1,2,⋯,n be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging I−IVIFTrAAOWA operator of dimension n is mapping I−IVIFTrAAOWA,Ωn⟶Ω such that
(29)I‐IVIFTrAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊕j=1nεjııeσne=ε1ııeσ1e⊕ε2ııeσ2e⊕,⋯,⊕εnııeσne.
εı=ε1ı,ε2ı,⋯,εnıT be the weighting vectors of ıejej=1,2,⋯,n with εjı∈0,1 and ∑j=1nεjı=1.
Also,ıeσje is the ıeje value of the I-VIFTrNs, and pairs ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistict trapezoidal fuzzy argument variable.Theorem 22.
Letueje,ıejej=1,2,⋯,n be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging I‐IVIFTrAAOWA operator of dimension n is mapping I−IVITrFAAOWA:Ωn⟶Ω with the corresponding vector εı=ε1ı,ε2ı,⋯,εnıT is the weight vector such that εjı∈0,1 and ∑j=1nεjı=1, such that,
(30)I‐IVIFTrAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊕j=1nεjııeσne∑j=1nεjıaσjb,∑j=1nεjıbσja,∑j=1nεjıcσjd,∑j=1nεjıdσjc1−e−∑j=1nεjı−log1−μıeσjelrξ1/ξ,e−∑j=1nεjı−logIıeσjelrξ1/ξ1−e−∑j=1nεjı−log1−μıeσjeurξ1/ξ,e−∑j=1nεjı−logIıeσjeurξ1/ξ.
Also,ıeσje is the ıeje value of the I-VITrFNs, and pairs ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistic trapezoidal fuzzy argument variable.Proof.
By mathematical induction, we can prove that, Equation (30) holds for all positive integer n. First, we show that Equation (22) holds for n=2, since
(31)ε1ııeσ1e=ε1ıaσ1b,ε1ıbσ1a,ε1ıcσ1d,ε1ıdσ1c;1−e−ε1ı−log1−μıeσ1elrξ1/ξ,e−ε1ı−logIıeσ1elrξ1/ξ1−e−ε1ı−log1−μıeσ1eurξ1/ξ,e−ε1ı−logIıeσ1eurξ1/ξ,ε2ııeσ2e=ε2ıaσ2b,ε2ıbσ2a,ε2ıcσ2d,ε2ıdσ2c;1−e−ε2ı−log1−μıeσ2elrξ1/ξ,e−ε2ı−logIıeσ2elrξ1/ξ1−e−ε2ı−log1−μıeσ2eurξ1/ξ,e−ε2ı−logIıeσ2eurξ1/ξ.
Based on Definition18, we get
(32)I‐IVITrFAAOWAıe1e,ıe2e=ε1ııeσ1e⊕ε2ııeσ2e=ε1ıaσ1b,ε1ıbσ1a,ε1ıcσ1d,ε1ıdσ1c1−e−ε1ı−log1−μıeσ1elrξ1/ξ,e−ε1ı−logIıeσ1elrξ1/ξ1−e−ε1ı−log1−μıeσ1eurξ1/ξ,e−ε1ı−logIıeσ1eurξ1/ξ⊕ε2ıaσ2b,ε2ıbσ2a,ε2ıcσ2d,ε2ıdσ2c1−e−ε2ı−log1−μıeσ2elrξ1/ξ,e−ε2ı−logIıeσ2elrξ1/ξ1−e−ε2ı−log1−μıeσ2eurξ1/ξ,e−ε2ı−logIıeσ2eurξ1/ξ=ε1ıaσ1b+ε2ıaσ2b,ε1ıbσ1a+ε2ıbσ2a,ε1ıcσ1d+ε2ıcσ2d,ε1ıdσ1c+ε2ıdσ2c1−e−ε1ı−log1−μıeσ1elrξ+ε2ı−log1−μıeσ2elrξ1/ξ,e−ε1ı−logIıeσ1elrξ+ε2ı−logIıeσ2elrξ1/ξ1−e−ε1ı−log1−μıeσ1eurξ+ε2ı−log1−μıeσ2eurξ1/ξ,e−ε1ı−logIıeσ1eurξ+ε2ı−logIıeσ2eurξ1/ξ=∑j=12εjıaσjb,∑j=12εjıbσja,∑j=12εjıcσjd,∑j=12εjıdσjc1−e−∑j=12εjı−log1−μıeσjelrξ1/ξ,e−∑j=12εjı−logIıeσjelrξ1/ξ1−e−∑j=12εjı−log1−μıeσjeurξ1/ξ,e−∑j=12εjı−logIıeσjeurξ1/ξ
Hence, (22) is true for n=2.
Let, we want to show that Equation (30) is true for n=k, then we have
(33)I‐IVITrFAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,ueke,ıeke=∑j=1kεjıaσjb,∑j=1kεjıbσja,∑j=1kεjıcσjd,∑j=1kεjıdσjc1−e−∑j=1kεjı−log1−μıeσjelrξ1/ξ,e−∑j=1kεjı−logIıeσjelrξ1/ξ1−e−∑j=1kεjı−log1−μleσjeurξ1/ξ,e−∑j=1kεjı−logIleσjeurξ1/ξ
Now, forn=k+1, we get
(34)I−IVITrFAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,ueke,ıek+1e=∑j=1k+1εjıaσjb,∑j=1k+1εjıbσja,∑j=1k+1εjıcσjd,∑j=1k+1εjıdσjc1−e−∑j=1kεjı−log1−μıeσjelrξ1/ξ,e−∑j=1kεjı−logIıeσjelrξ1/ξ1−e−∑j=1kεjı−log1−μıeσjeurξ1/ξ,e−∑j=1kεjı−logIıeσjeurξ1/ξ⊕=1−e−εk+1ı−log1−μıek+1elrξ1/ξ,e−εk+1ı−logIıeek+1lrξ1/ξ1−e−εk+1ı−log1−μıeek+1urξ1/ξ,e−εk+1ı−logIıek+1eurξ1/ξ=∑j=1k+1εjıaσjb,∑j=1k+1εjıbσja,∑j=1k+1εjıcσjd,∑j=1k+1εjıdσjc1−e−∑j=1k+1εjı−log1−μıeσjelrξ1/ξ,e−∑j=1k+1εjı−logIıeσjelrξ1/ξ1−e−∑j=1k+1εjı−log1−μıeσjeurξ1/ξ,e−∑j=1k+1εjı−logIıeσjeurξ1/ξ
Thus, (30) is legitimate for n=k+1.As consequences, we might come to conclusion that Equation (30) holds for all n.Theorem 23 (Commutativity).
Let(35)I‐IVITrFAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=I‐IVITrFAAOWAεıue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗e,where
(36)ue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗eis any permutation of
(37)ue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene.Proof.
As we know that(38)I‐IVITrFAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ε1ııeσ1e⊕ε2ııeσ2e⊕,⋯,⊕εnııeσne,I‐IVITrFAAOWAεıue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗e=ε1ııeσ1∗e⊕ε2ııeσ2∗e⊕,⋯,⊕εnııeσn∗e,
since(39)ue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗eis any permutation of
(40)ue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene.Theorem 24 (Idempotency).
If all(41)ıeje=ueje,ajb,bja,cjd,djcμıejelr,μıejeurIıejelr,Iıejeurj=1,2,⋯,nare equal, ıeje=ıee for all njj=1,2,⋯,n, then
(42)I‐IVITrFAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıee.Proof.
Sinceıeje=ıee for all j, we have
(43)I−IVITrFAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊕j=1nεjııeσje=ε1ııeσ1e⊕ε2ııeσ2e⊕,⋯,⊕εnııeσne=⊕j=1nεjııee=ε1ııee⊕ε2ııee⊕,⋯,⊕εnııee=εjıaσjb,εjıbσja,εjıcσjd,εjıdσjc,1−e−∑j=1nεjı−log1−μıejelrξ1/ξ,e−∑j=1nεjı−logIıejelrξ1/ξ1−e−∑j=1nεjı−log1−μıejeurξ1/ξ,e−∑j=1nεjı−logIıejeurξ1/ξ=ajb,bja,cjd,djc;1−e−−log1−μıeelrξ1/ξ,e−−logIıeelrξ1/ξ1−e−−log1−μıeeurξ1/ξ,e−−logIıeeurξ1/ξ=ajb,bja,cjd,djc;1−elog1−μıeelr,elogIıeelr1−elog1−μıeeur,elogIıeeur=ajb,bja,cjd,djc;μıeelr,μıeeurIıeelr,Iıeeur=ıee.
Thus,(44)I‐IVITrFAAOWAεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıee.Theorem 25 (Monotonicity).
Ifıeje≤ıeje∗ for all jj=1,2,⋯,n, then
(45)I−IVITrFAAOWAεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene≤I−IVITrFAAOWAεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e.Proof.
Let(46)I‐IVITrFAAOWAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ε1ııeσ1e⊕ε2ııeσ2e⊕,⋯,⊕εnııeσneI‐IVITrFAAOWAεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e=ε1ııeσ1∗e⊕ε2ııeσ2∗e⊕,⋯,⊕εnııeσn∗e
Sinceıeje≤ıeje∗ for all j, it follows that ıeσje≤ıeσj∗ej=1,2,⋯,n, then
(47)I‐IVITrFAAOWAεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene≤I‐IVITrFAAOWAεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e.Example 26.
Consider the following collection of I-IVITrFNs in pairs form such that(48)ue1e,ıe1e=0.1,0.7,0.4,0.5,0.3;0.50,0.200.40,0.20,ue2e,ıe2e=0.8,0.1,0.5,0.2,0.6;0.20,0.600.10,0.20,ue3e,ıe3e=0.3,0.2,0.4,0.2,0.3;0.40,0.300.30,0.20,ue4e,ıe4e=0.5,0.6,0.1,0.5,0.7;0.60,0.200.30,0.10.
Performing the ordering of the I-IVITrF pairs with respect to the first component, we have(49)ue2e,ıe2e=0.8,0.1,0.5,0.2,0.6;0.20,0.600.10,0.20,ue4e,ıe4e=0.5,0.6,0.1,0.5,0.7;0.60,0.200.30,0.10,ue3e,ıe3e=0.3,0.2,0.4,0.2,0.3;0.40,0.300.30,0.20,ue1e,ıe1e=0.1,0.7,0.4,0.5,0.3;0.50,0.200.40,0.20.
This ordering includes the ordered I-VITrF arguments.(50)ueσ1e,ıeσ1e=ue2e,ıe2e=0.8,0.1,0.5,0.2,0.6;0.20,0.600.10,0.20,ueσ2e,ıeσ2e=ue4e,ıe4e=0.5,0.6,0.1,0.5,0.7;0.60,0.200.30,0.10,ueσ3e,ıeσ3e=ue3e,ıe3e=0.3,0.2,0.4,0.2,0.3;0.40,0.300.30,0.20,ueσ4e,ıeσ4e=ue1e,ıe1e=0.1,0.7,0.4,0.5,0.3;0.50,0.200.40,0.20.
If the associated weighting vector isεı=0.10,0.20,0.30,0.40 and N=2 then, we get an aggregate value by Equation (9).
(51)I‐IVITrFAAOWAue1e,ıe1e,ue2e,ıe2e,ue3e,ıe3eue4e,ıe4e=j=14εjıaσjb,j=14εjıbσja,j=14εjıcσjd,j=14εjıdσjc1−e−j=14εjı−log1−μıeσjelrξ1/ξ,e−j=14εjı−logIıeσjelrξ1/ξ1−e−j=14εjı−log1−μıeσjeurξ1/ξ,e−j=14εjı−logIıeσjeurξ1/ξ=0.10.1+0.20.6+0.30.3+0.40.7,0.10.5+0.20.1+0.30.4+0.40.4,0.10.2+0.20.5+0.30.3+0.40.5,0.10.6+0.20.7+0.30.3+0.40.3,1−e−0.1−log1−0.221/2+0.2−log1−0.621/2+0.3−log1−0.421/2+0.4−log1−0.521/2,e−0.1−log0.121/2+0.2−log0.321/2+0.3−log0.321/2+0.4−log0.421/20.47,0.35,0.38,0.41;1−e−0.1−log1−0.621/2+0.2−log1−0.221/2+0.3−log1−0.321/2+0.4−log1−0.221/2,e−0.1−log0.221/2+0.2−log0.121/2+0.3−log0.221/2+0.4−log0.221/2=0.47,0.35,0.38,0.41;0.6984,0.08160.5162,0.0321.


### 4.1. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel Alsina Hybrid Averaging (I-IVITrFAAHA) Operator

Definition 27.
Letueje,ıeje=ueje,ajb,bja,cjd,djc;μlejelr,μlejeurIlejelr,Ilejeurj=1,2,⋯,n be a collection of I-VITrFNs; then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging I‐IVITrFAAHA operator of dimension n is mapping I-IVITrFAAHA, Ωn⟶Ω such that
(52)I‐IVITrFAAHAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊕j=1nεjlıeσn⋅e=ε1ııeσ1⋅e⊕ε2ııeσ2⋅e⊕,⋯,⊕εnııeσn⋅e,
whereıeσje. is the weighted interval-valued intuitionistic trapezoidal fuzzy values, ıeσje., and ıeσje.=nεjııejej=1,2,⋯,n of the interval-valued intuitionistic trapezoidal fuzzy ordered weighted averaging pair ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistic trapezoidal fuzzy argument variable. εı=ε1ı,ε2ı,⋯,εnıT is the weighting vector of ıejej=1,2,⋯,n with εjı∈0,1 and j=1nεjı=1.Theorem 28.
Letueje,ıejej=1,2,⋯,n be a collection of I-VITrFNs. then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging I‐IVITrFAAHA operator is defined as
(53)I‐IVITrFAAHAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊕j=1nεjııeσI⋅e=ε1ııeσ1⋅e⊕ε2ııeσ2⋅e⊕,⋯,⊕εnııeσn⋅ej=1nεjıaσj.b,j=1nεjıbσj.a,j=1nεjıcσj.d,j=1nεjıdσj.c1−e−j=1nεjı−log1−μıeσj.elrξ1/ξ,e−j=1nεjı−logIıeσj.elrξ1/ξ1−e−j=1nεjı−log1−μıeσj.eurξ1/ξ,e−j=1nεjı−logIıeσj.eurξ1/ξ.Theorem 29.
The I-VITrFAAWA operator is the special case of the I-IVITrFAAHA operator.Proof.
Letεı=1/n,1/n,⋯,1/n, then
(54)I‐IVITrFAAHAεı,εıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ε1ııeσ1e.⊕ε2ııeσ2e.,⋯,εnııeσne.=1nıeσ1e.⊕ıeσ2e.,⋯,ıeσne.=ε1ııe1e⊕ε2ııe2e,⋯,εnııene=I‐VITrFAAWAεııe1e,ıe2e,⋯,ıene.Theorem 30.
The I-IVITrFAAOWA operator is the special case of the I-IVITrFAAHA operator.Proof.
Letεı=1/n,1/n,⋯,1/n, then ıeje.=ıejej=1,2,⋯,n(55)I‐IVITrFAAHAεı,εıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ε1ııeσ1e.⊕ε2ııeσ2e.,⋯,εnııeσne.=ε1ııeσ1e⊕ε2ııeσ2e,⋯,εnııeσne=I‐VITrFAAOWAεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene.


### 4.2. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel-Alsina Ordered Weighted Geometric (I-IVITrFAAWG) Operator

Definition 31.
Letueje,ıeje=ueje,ajb,bja,cjd,djc;μıejelr,μıejelrIıejelr,Iıejeur be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina weighted geometric I‐IVITrFAAWG operator of dimension n is mapping I‐IVITrFAAWG, Ωn⟶Ω such that
(56)I‐IVIFTrAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσneεjı=ıeσ1eε1ı⊗ıeσ2eε2ı⊗,⋯,⊗ıeσneεnı.
εı=ε1ı,ε2ı,⋯,εnıT be the weighting vectors of ıejej=1,2,⋯,n with εjı∈0,1 and j=1nεjı=1..
Alsoıeσje is the ıeje value of the I-VIFTrNs, and pairs ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistict trapezoidal fuzzy argument variable.Theorem 32.
Letueje,ıejej=1,2,⋯,n be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging I‐IVIFTrAAOWG operator of dimension n is mapping I−IVITrFAAOWG, Ωn⟶Ω with the corresponding vector εı=ε1ı,ε2ı,⋯,εnıT is the weight vector such that εjı∈0,1 and j=1nεjı=1, such that,
(57)I‐IVIFTrAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσneεjıΠj=1naσjbεjı,Πj=1nbσjaεjı,Πj=1ncσjdεjı,Πj=1ndσjcεjıe−Πj=1n−logμıeσjelrεjıξ1/ξ,1−e−Πj=1nεjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1n−logμıeσjeurεjıξ,1−e−Πj=1nεjı−log1−Iıeσjeurεjıξ1/ξ,.
Alsoıeσje is the ıeje value of the I-VITrFNs, and pairs ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistic trapezoidal fuzzy argument variable.Proof.
By mathematical induction, we can prove that Equation (56) holds for all positive integer n. First, we show that Equation (56) holds for n=2, since
(58)ıeσ1eε1ı=aσ1bε1ı,bσ1aε1ı,cσ1dε1ı,dσ1cε1ı;e−−logμıeσ1elrε1ıξ1/ξ,1−e−−log1−Iıeσ1elrε1ıξ1/ξe−−logμıeσ1eurε1ıξ1/ξ,1−e−−log1−Iıeσ1eurε1ıξ1/ξ,ıeσ2eε2ı=aσ2bε2ı,bσ2aε2ı,cσ2dε2ı,dσ2cε2ı;e−−logμıeσ2elrε2ıξ1/ξ,1−e−−log1−Iıeσ2elrε2ıξ1/ξe−−logμıeσ2eurε2ıξ1/ξ,1−e−−log1−μıeσ2eurε2ıξ1/ξ.
Based on Definition18, we get
(59)I‐IVITrFAAOWGıe1e,ıe2e=ıeσ1eε1ı⊗ıeσ2eε2ı=aσ1bε1ı,bσ1aε1ı,cσ1dε1ı,dσ1cε1ı;e−−logμıeσ1elrε1ıξ1/ξ,1−e−−log1−Iıeσ1elrε1ıξ1/ξe−−logμıeσ1eurε1ıξ1/ξ,1−e−−log1−Iıeσ1eurε1ıξ1/ξ⊗aσ2bε2ı,bσ2aε2ı,cσ2dε2ı,dσ2cε2ı;e−−logμıeσ2elrε2ıξ1/ξ,1−e−−log1−Iıeσ2elrε2ıξ1/ξe−−logμıeσ2eurε2ıξ1/ξ,1−e−−log1−μıeσ2eurε2ıξ1/ξ=aσ1bε1ıaσ2bε2ı,bσ1aε1ıbσ2aε2ı,cσ1dε1ıcσ2dε2ı,dσ1cε1ıdσ2cε2ıe−−logμıeσ1elrε1ıξ−logμıeσ2elrε2ıξ1/ξ,1−e−−log1−Iıeσ1elrε1ıξ−log1−Iıeσ2elrε2ıξ1/ξe−−logμıeσ1eurε1ıξ−logμıeσ2eurε2ıξ1/ξ,1−e−−log1−Iıeσ1eurε1ıξ−log1−Iıeσ2eurε2ıξ1/ξ.=Πj=12aσjbεjı,Πj=12bσjaεjı,Πj=12cσjdεjı,Πj=12dσjcεjıe−Πj=12−logμıeσjelrεjıξ1/ξ,1−e−Πj=12εjı−log1−Iıeσjelrεjıξ1/ξe−Πj=12−logμıeσjeurεjıξ1/ξ,1−e−Πj=12εjı−log1−Iıeσjeurεjıξ1/ξ,
Hence, (17) is true for n=2.
Let we want to show that Equation (17) is true for n=k, then we have
(60)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,ueke,ıeke=Πj=1kaσjbεjı,Πj=1kbσjaεjı,Πj=1kcσjdεjı,Πj=1kdσjcεjıe−Πj=1k−logμıeσjelrεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1k−logμıeσjeurεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjeurεjıξ1/ξ.
Now, forn=k+1, we get
(61)I−IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,ueke,ıek+1e=Πj=1k+1aσjbεjı,Πj=1k+1bσjaεjı,Πj=1k+1cσjdεjı,Πj=1k+1dσjcεjıe−Πj=1k−logμıeσjelrεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1k−logμıeσjeurεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjeurεjıξ1/ξ,⊗e−−logμıeek+1lrεk+1ıξ1/ξ,1−e−−log1−Iıek+1elrεk+1ıξ1/ξe−−logμıek+1eurεk+1ıξ1/ξ,1−e−−log1−Iıeek+1urεk+1ıξ1/ξ=Πj=1k+1aσjbεjı,Πj=1k+1bσjaεjı,Πj=1k+1cσjdεjı,Πj=1k+1dσjcεjıe−Πj=1k+1−logμıeσjelrεjıξ1/ξ,1−e−Πj=1k+1εjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1k+1−logμıeσjeurεjıξ1/ξ,1−e−Πj=1k+1εjı−log1−Iıeσjeurεjıξ1/ξTheorem 33 (Commutativity).
Let(62)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=I‐IVITrFAAOWGεıue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗e,where
(63)ue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗eis any permutation of
(64)ue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene.Proof.
As we know that(65)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıeσ1eε1ı⊗ıeσ2eε2ı⊗,⋯,⊗ıeσneεnı,I‐IVITrFAAOWGεıue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗e=ıeσ1∗eε1ı⊗ıeσ2∗eε2ı⊗,⋯,⊗ıeσn∗eεnı,
since(66)ue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗e
is any permutation of(67)ue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene.Theorem 34 (Idempotency).
If allıeje=ueje,ajb,bja,cjd,djc;μıejelr,μıejeurIıejelr,Iıejeur are equal, ıeje=ıee for all jj=1,2,⋯,n, then
(68)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıee.Proof.
Sinceıeje=ıee for all j we have
(69)I−IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσjeεjı=ıeσ1eε1ı⊗ıeσ2eε2ı⊗,⋯,⊗ıeσneεnı=⊗j=1nıeeεjı=ıeeε1ı⊗ıeeε2ı⊗,⋯,⊗ıeeεnı=aσjbεjı,bσjaεjı,cσjdεjı,dσjcεjıe−Πj=1k−logμıeσjelrεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1k−logμıeσjeurεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjeurεjıξ1/ξ=ajb,bja,cjd,djc;e−−logμıeelrξ1/ξ,1−e−−log1−Iıeelrξ1/ξe−−logμıeeurξ1/ξ,1−e−−log1−Iıeeurξ1/ξ=ajb,bja,cjd,djc;elogμıeelr,1−elog1−Iıeelrelogμıeeur,1−elog1−Iıeeur=ajb,bja,cjd,djc;μıeelr,μıeeurIıeelr,Iıeeur=ıee
Thus,(70)I‐IVITrFAAOWGεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıee.Theorem 35 (Monotonicity).
Ifıeje≤ıeje∗ for all jj=1,2,⋯,n, then
(71)I‐IVITrFAAOWGεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene≤I‐IVITrFAAOWGεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e,Proof.
Let(72)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıeσ1eε1l⊗ıeσ2eε2l⊗,⋯,⊗ıeσneεnl,I‐IVITrFAAOWGεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e=ıeσ1∗eε1l⊗ıeσ2∗eε2l⊗,⋯,⊗ıeσn∗eεnl.
Sinceıeje≤ıeje∗ for all j, it follows that ıeσje≤ıeσj∗ej=1,2,⋯,n, then
(73)I‐IVITrFAAOWGεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene≤I‐IVITrFAAOWGεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e.


### 4.3. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel-Alsina Hybrid Geometric (I-IVITrFAAHG) Operator

Definition 36.
Letueje,ıeje=ueje,ajb,bja,cjd,djc;μıejelr,μıejeurIıejelr,Iıejeurj=1,2,⋯,n be a collection of I-VITrFNs, then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid geometric I‐IVITrFAAHG operator of dimension n is mapping I‐IVITrFAAHAG, Ωn⟶Ω such that
(74)I‐IVITrFAAHGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσn⋅eεjı=ıeσ1⋅eε1ı⊗ıeσ2⋅eε2ı⊗,⋯,⊗ıeσn⋅eεnı,where ıeσje. is the weighted interval-valued intuitionistic trapezoidal fuzzy values ıeσje., and ıeσje.=ıejenεjı.j=1,2,⋯,n of the interval-valued intuitionistic trapezoidal fuzzy ordered weighted averaging pair ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistic trapezoidal fuzzy argument variable. and εı=ε1ı,ε2ı,⋯,εnıT is the weighting vector of ıejej=1,2,⋯,n with εjı∈0,1 and j=1nεjı=1.Theorem 37.
Letueje,ıejej=1,2,⋯,n, be a collection of I-VITrFNs. then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid geometric I‐IVITrFAAHG operator is defined as
(75)I‐IVITrFAAHGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσn⋅eεjı=ıeσ1⋅eε1ı⊗ıeσ2⋅eε2ı⊗,⋯,⊗ıeσn⋅eεnıΠj=1naσj.bεjı,Πj=1nbσj.aεjı,Πj=1ncσj.dεjı,Πj=1ndσj.cεjıe−Πj=1n−logμıeσj.elrεjıξ1/ξ,1−e−Πj=1nεjı−log1−Iıeσj.elrεjıξ1/ξe−Πj=1n−logμıeσj.eurεjıξ1/ξ,1−e−Πj=1nεjı−log1−Iıeσj.eurεjıξ1/ξ.


## 4.1. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel Alsina Hybrid Averaging (I-IVITrFAAHA) Operator

Definition 27.
Letueje,ıeje=ueje,ajb,bja,cjd,djc;μlejelr,μlejeurIlejelr,Ilejeurj=1,2,⋯,n be a collection of I-VITrFNs; then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging I‐IVITrFAAHA operator of dimension n is mapping I-IVITrFAAHA, Ωn⟶Ω such that
(52)I‐IVITrFAAHAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊕j=1nεjlıeσn⋅e=ε1ııeσ1⋅e⊕ε2ııeσ2⋅e⊕,⋯,⊕εnııeσn⋅e,
whereıeσje. is the weighted interval-valued intuitionistic trapezoidal fuzzy values, ıeσje., and ıeσje.=nεjııejej=1,2,⋯,n of the interval-valued intuitionistic trapezoidal fuzzy ordered weighted averaging pair ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistic trapezoidal fuzzy argument variable. εı=ε1ı,ε2ı,⋯,εnıT is the weighting vector of ıejej=1,2,⋯,n with εjı∈0,1 and j=1nεjı=1.Theorem 28.
Letueje,ıejej=1,2,⋯,n be a collection of I-VITrFNs. then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging I‐IVITrFAAHA operator is defined as
(53)I‐IVITrFAAHAue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊕j=1nεjııeσI⋅e=ε1ııeσ1⋅e⊕ε2ııeσ2⋅e⊕,⋯,⊕εnııeσn⋅ej=1nεjıaσj.b,j=1nεjıbσj.a,j=1nεjıcσj.d,j=1nεjıdσj.c1−e−j=1nεjı−log1−μıeσj.elrξ1/ξ,e−j=1nεjı−logIıeσj.elrξ1/ξ1−e−j=1nεjı−log1−μıeσj.eurξ1/ξ,e−j=1nεjı−logIıeσj.eurξ1/ξ.Theorem 29.
The I-VITrFAAWA operator is the special case of the I-IVITrFAAHA operator.Proof.
Letεı=1/n,1/n,⋯,1/n, then
(54)I‐IVITrFAAHAεı,εıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ε1ııeσ1e.⊕ε2ııeσ2e.,⋯,εnııeσne.=1nıeσ1e.⊕ıeσ2e.,⋯,ıeσne.=ε1ııe1e⊕ε2ııe2e,⋯,εnııene=I‐VITrFAAWAεııe1e,ıe2e,⋯,ıene.Theorem 30.
The I-IVITrFAAOWA operator is the special case of the I-IVITrFAAHA operator.Proof.
Letεı=1/n,1/n,⋯,1/n, then ıeje.=ıejej=1,2,⋯,n(55)I‐IVITrFAAHAεı,εıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ε1ııeσ1e.⊕ε2ııeσ2e.,⋯,εnııeσne.=ε1ııeσ1e⊕ε2ııeσ2e,⋯,εnııeσne=I‐VITrFAAOWAεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene.


## 4.2. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel-Alsina Ordered Weighted Geometric (I-IVITrFAAWG) Operator

Definition 31.
Letueje,ıeje=ueje,ajb,bja,cjd,djc;μıejelr,μıejelrIıejelr,Iıejeur be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina weighted geometric I‐IVITrFAAWG operator of dimension n is mapping I‐IVITrFAAWG, Ωn⟶Ω such that
(56)I‐IVIFTrAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσneεjı=ıeσ1eε1ı⊗ıeσ2eε2ı⊗,⋯,⊗ıeσneεnı.
εı=ε1ı,ε2ı,⋯,εnıT be the weighting vectors of ıejej=1,2,⋯,n with εjı∈0,1 and j=1nεjı=1..
Alsoıeσje is the ıeje value of the I-VIFTrNs, and pairs ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistict trapezoidal fuzzy argument variable.Theorem 32.
Letueje,ıejej=1,2,⋯,n be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging I‐IVIFTrAAOWG operator of dimension n is mapping I−IVITrFAAOWG, Ωn⟶Ω with the corresponding vector εı=ε1ı,ε2ı,⋯,εnıT is the weight vector such that εjı∈0,1 and j=1nεjı=1, such that,
(57)I‐IVIFTrAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσneεjıΠj=1naσjbεjı,Πj=1nbσjaεjı,Πj=1ncσjdεjı,Πj=1ndσjcεjıe−Πj=1n−logμıeσjelrεjıξ1/ξ,1−e−Πj=1nεjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1n−logμıeσjeurεjıξ,1−e−Πj=1nεjı−log1−Iıeσjeurεjıξ1/ξ,.
Alsoıeσje is the ıeje value of the I-VITrFNs, and pairs ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistic trapezoidal fuzzy argument variable.Proof.
By mathematical induction, we can prove that Equation (56) holds for all positive integer n. First, we show that Equation (56) holds for n=2, since
(58)ıeσ1eε1ı=aσ1bε1ı,bσ1aε1ı,cσ1dε1ı,dσ1cε1ı;e−−logμıeσ1elrε1ıξ1/ξ,1−e−−log1−Iıeσ1elrε1ıξ1/ξe−−logμıeσ1eurε1ıξ1/ξ,1−e−−log1−Iıeσ1eurε1ıξ1/ξ,ıeσ2eε2ı=aσ2bε2ı,bσ2aε2ı,cσ2dε2ı,dσ2cε2ı;e−−logμıeσ2elrε2ıξ1/ξ,1−e−−log1−Iıeσ2elrε2ıξ1/ξe−−logμıeσ2eurε2ıξ1/ξ,1−e−−log1−μıeσ2eurε2ıξ1/ξ.
Based on Definition18, we get
(59)I‐IVITrFAAOWGıe1e,ıe2e=ıeσ1eε1ı⊗ıeσ2eε2ı=aσ1bε1ı,bσ1aε1ı,cσ1dε1ı,dσ1cε1ı;e−−logμıeσ1elrε1ıξ1/ξ,1−e−−log1−Iıeσ1elrε1ıξ1/ξe−−logμıeσ1eurε1ıξ1/ξ,1−e−−log1−Iıeσ1eurε1ıξ1/ξ⊗aσ2bε2ı,bσ2aε2ı,cσ2dε2ı,dσ2cε2ı;e−−logμıeσ2elrε2ıξ1/ξ,1−e−−log1−Iıeσ2elrε2ıξ1/ξe−−logμıeσ2eurε2ıξ1/ξ,1−e−−log1−μıeσ2eurε2ıξ1/ξ=aσ1bε1ıaσ2bε2ı,bσ1aε1ıbσ2aε2ı,cσ1dε1ıcσ2dε2ı,dσ1cε1ıdσ2cε2ıe−−logμıeσ1elrε1ıξ−logμıeσ2elrε2ıξ1/ξ,1−e−−log1−Iıeσ1elrε1ıξ−log1−Iıeσ2elrε2ıξ1/ξe−−logμıeσ1eurε1ıξ−logμıeσ2eurε2ıξ1/ξ,1−e−−log1−Iıeσ1eurε1ıξ−log1−Iıeσ2eurε2ıξ1/ξ.=Πj=12aσjbεjı,Πj=12bσjaεjı,Πj=12cσjdεjı,Πj=12dσjcεjıe−Πj=12−logμıeσjelrεjıξ1/ξ,1−e−Πj=12εjı−log1−Iıeσjelrεjıξ1/ξe−Πj=12−logμıeσjeurεjıξ1/ξ,1−e−Πj=12εjı−log1−Iıeσjeurεjıξ1/ξ,
Hence, (17) is true for n=2.
Let we want to show that Equation (17) is true for n=k, then we have
(60)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,ueke,ıeke=Πj=1kaσjbεjı,Πj=1kbσjaεjı,Πj=1kcσjdεjı,Πj=1kdσjcεjıe−Πj=1k−logμıeσjelrεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1k−logμıeσjeurεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjeurεjıξ1/ξ.
Now, forn=k+1, we get
(61)I−IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,ueke,ıek+1e=Πj=1k+1aσjbεjı,Πj=1k+1bσjaεjı,Πj=1k+1cσjdεjı,Πj=1k+1dσjcεjıe−Πj=1k−logμıeσjelrεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1k−logμıeσjeurεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjeurεjıξ1/ξ,⊗e−−logμıeek+1lrεk+1ıξ1/ξ,1−e−−log1−Iıek+1elrεk+1ıξ1/ξe−−logμıek+1eurεk+1ıξ1/ξ,1−e−−log1−Iıeek+1urεk+1ıξ1/ξ=Πj=1k+1aσjbεjı,Πj=1k+1bσjaεjı,Πj=1k+1cσjdεjı,Πj=1k+1dσjcεjıe−Πj=1k+1−logμıeσjelrεjıξ1/ξ,1−e−Πj=1k+1εjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1k+1−logμıeσjeurεjıξ1/ξ,1−e−Πj=1k+1εjı−log1−Iıeσjeurεjıξ1/ξTheorem 33 (Commutativity).
Let(62)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=I‐IVITrFAAOWGεıue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗e,where
(63)ue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗eis any permutation of
(64)ue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene.Proof.
As we know that(65)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıeσ1eε1ı⊗ıeσ2eε2ı⊗,⋯,⊗ıeσneεnı,I‐IVITrFAAOWGεıue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗e=ıeσ1∗eε1ı⊗ıeσ2∗eε2ı⊗,⋯,⊗ıeσn∗eεnı,
since(66)ue1∗e,ıe1∗e,ue2∗e,ıe2∗e,⋯,uen∗e,ıen∗e
is any permutation of(67)ue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene.Theorem 34 (Idempotency).
If allıeje=ueje,ajb,bja,cjd,djc;μıejelr,μıejeurIıejelr,Iıejeur are equal, ıeje=ıee for all jj=1,2,⋯,n, then
(68)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıee.Proof.
Sinceıeje=ıee for all j we have
(69)I−IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσjeεjı=ıeσ1eε1ı⊗ıeσ2eε2ı⊗,⋯,⊗ıeσneεnı=⊗j=1nıeeεjı=ıeeε1ı⊗ıeeε2ı⊗,⋯,⊗ıeeεnı=aσjbεjı,bσjaεjı,cσjdεjı,dσjcεjıe−Πj=1k−logμıeσjelrεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjelrεjıξ1/ξe−Πj=1k−logμıeσjeurεjıξ1/ξ,1−e−Πj=1kεjı−log1−Iıeσjeurεjıξ1/ξ=ajb,bja,cjd,djc;e−−logμıeelrξ1/ξ,1−e−−log1−Iıeelrξ1/ξe−−logμıeeurξ1/ξ,1−e−−log1−Iıeeurξ1/ξ=ajb,bja,cjd,djc;elogμıeelr,1−elog1−Iıeelrelogμıeeur,1−elog1−Iıeeur=ajb,bja,cjd,djc;μıeelr,μıeeurIıeelr,Iıeeur=ıee
Thus,(70)I‐IVITrFAAOWGεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıee.Theorem 35 (Monotonicity).
Ifıeje≤ıeje∗ for all jj=1,2,⋯,n, then
(71)I‐IVITrFAAOWGεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene≤I‐IVITrFAAOWGεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e,Proof.
Let(72)I‐IVITrFAAOWGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=ıeσ1eε1l⊗ıeσ2eε2l⊗,⋯,⊗ıeσneεnl,I‐IVITrFAAOWGεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e=ıeσ1∗eε1l⊗ıeσ2∗eε2l⊗,⋯,⊗ıeσn∗eεnl.
Sinceıeje≤ıeje∗ for all j, it follows that ıeσje≤ıeσj∗ej=1,2,⋯,n, then
(73)I‐IVITrFAAOWGεıue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene≤I‐IVITrFAAOWGεıue1e,ıe1∗e,ue2e,ıe2∗e,⋯,uene,ıen∗e.


## 4.3. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel-Alsina Hybrid Geometric (I-IVITrFAAHG) Operator

Definition 36.
Letueje,ıeje=ueje,ajb,bja,cjd,djc;μıejelr,μıejeurIıejelr,Iıejeurj=1,2,⋯,n be a collection of I-VITrFNs, then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid geometric I‐IVITrFAAHG operator of dimension n is mapping I‐IVITrFAAHAG, Ωn⟶Ω such that
(74)I‐IVITrFAAHGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσn⋅eεjı=ıeσ1⋅eε1ı⊗ıeσ2⋅eε2ı⊗,⋯,⊗ıeσn⋅eεnı,where ıeσje. is the weighted interval-valued intuitionistic trapezoidal fuzzy values ıeσje., and ıeσje.=ıejenεjı.j=1,2,⋯,n of the interval-valued intuitionistic trapezoidal fuzzy ordered weighted averaging pair ueje,ıeje having the jıh largest ueje and ueje in ueje,ıeje is referred to as the order inducing variable and ıeje as the interval-valued intuitionistic trapezoidal fuzzy argument variable. and εı=ε1ı,ε2ı,⋯,εnıT is the weighting vector of ıejej=1,2,⋯,n with εjı∈0,1 and j=1nεjı=1.Theorem 37.
Letueje,ıejej=1,2,⋯,n, be a collection of I-VITrFNs. then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid geometric I‐IVITrFAAHG operator is defined as
(75)I‐IVITrFAAHGue1e,ıe1e,ue2e,ıe2e,⋯,uene,ıene=⊗j=1nıeσn⋅eεjı=ıeσ1⋅eε1ı⊗ıeσ2⋅eε2ı⊗,⋯,⊗ıeσn⋅eεnıΠj=1naσj.bεjı,Πj=1nbσj.aεjı,Πj=1ncσj.dεjı,Πj=1ndσj.cεjıe−Πj=1n−logμıeσj.elrεjıξ1/ξ,1−e−Πj=1nεjı−log1−Iıeσj.elrεjıξ1/ξe−Πj=1n−logμıeσj.eurεjıξ1/ξ,1−e−Πj=1nεjı−log1−Iıeσj.eurεjıξ1/ξ.


## 5. MCGDM Techniques on I-IVITrFAAWA Operator

In terms of I-VITrF informations, we suggest an approach for solving MCGDM problems. A decision matrix can be used to express MCGDM, where the attributes (criteria) are expressed by columns, and alternatives are expressed by row. Suppose we have a set which consistsm number of alternatives represented by α=α1,α1,⋯,αm and a set consists n number of attributes (criteria) represented as β=β1,β1,⋯βn. Let M=m1,m2,⋯,mp be the set of decision makers and λ=λ1,λ1,⋯,λpT be the weighting vector of DMs λk∈0,1 and k=1pλk=1. DMs provided their assessment report for each alternative αii=1,2,,..,m against their attributes (criteria) βjj=1,2,⋯,n . Let εjı represent the weight vector for attribute such that j=1nεjı=1 and εjı∈0,1 .The I-IVITrF decision matrix is assumed to be denoted by ıee=ueje,ıeijem×n=ueje,ajb,bja,cjd,djc;μıejelr,μıejeurIıejelr,Iıejeur where μijlr,μijur∈ M-D and Iijlr,Iijur∈ N-MD consider by decision makers such that,
(76)β1β2⋯,βna1ue11e,a11b,b11a,c11d,d11cμ11lr,μ11urI11lr,I11ur,ue12e,a12b,b12a,c12d,d12cμ12lr,μ12urI12lr,I12ur,⋯,ue1ne,a1nb,b1na,c1nd,d1ncμ1nlr,μ1nurI1nlr,I1nura2ue21e,a21b,b21a,c21d,d21cμ21lr,μ21urI21lr,I21ur,ue22e,a22b,b22a,c22d,d22cμ22lr,μ22urI22lr,I22ur,⋯,ue2ne,a2nb,b2na,c2nd,d2ncμ2nlr,μ2nurI2nlr,I2nur⋮⋮⋮⋮amuem1e,am1b,bm1a,cm1d,dm1cμm1lr,μm1urIm1lr,Im1ur,uem1e,am2b,bm2a,cm2d,dm2cμm2lr,μm2urIm2lr,Im2ur,⋯,uemne,amnb,bmna,cmnd,dmncμmnlr,μmnurImnlr,Imnur.The following steps are developed for solving MCGDM problem with I-IVITrFAAOWA operators.Step 1.
In this step, we construct the I-IVITrF matrixHk=zijm×nk in which the decision makers give their opinion related to each alternative with respect to each criteria.Step 2.
In this step, we perform the ordering of the decision matrix with respect to the first component.Step 3.
We utilize the decision information given in the I-IVITrF matrixHk=zijkm×n. Assume that ξ=2, and apply the I-IVITrFAAOWA operator to derive the individual overall preference values zik of the alternative αi, and εjı be the weight vector attribute j=1nεjı=1, εjı∈0,1 such that,
(77)I‐IVITrFAAOWAue1e,ıe1e,⋯,uene,ıene=⊕j=1nεjııeσne=j=1nεjıaσjb,j=1nεjıbσja,j=1nεjıcσjd,j=1nεjıdσjc1−e−j=1nεjı−log1−μıeσjelrξ1/ξ,e−j=1nεjı−logIıeσjelrξ1/ξ1−e−j=1nεjı−log1−μıeσjeurξ1/ξ,e−j=1nεjı−logIıeσjeurξ1/ξStep 4.
We utilize the I-VITrFAAOWA operator to aggregate all the individual overall preference values into the collective overall preference values, whereλ is weighting vector of decision makers such that λk∈0,1 and k=1pλk=1..Step 5.
In this step, we calculate the scores function to aggregate the value of each alternativeαi.Step 6.
In ascending order, the alternative with the greatest value is our best choice.


## 6. Numerical Example

This example is taken from [38], with the help of this example we will explain our proposed method. A company wants to select a supplier and now there are four suppliers as candidates αi=α1,α2,α3,α4, i=1,2,3,4. We evaluate each supplier from four aspects βj=β1,β2,β3,β4j=1,2,3,4 which are the following:β1 is the production cost, β2 is the production quality, β3 is the supplier’s service performance, and β4 is the risk factor.The weight vector of attributes isεı=0.10,0.20,0.30,0.40T. There are four experts, and the weight vector of the experts is λ=0.40,0.30,0.20,0.10T. Then, the decision matrix Hk=hijk4×4k=1,2,3,4 are shown in Tables 1–4, and our goal is to rank four suppliers and select the best one.Step 1.
The decision makers give their decision in the following tables.Table 1
Decision matrix of expert -1 (H1).
1β1β2α1(0.5,[0.5,0.2,0.1,0.4][0.3,0.6][0.4,0.3])(0.3,[0.3,0.2,0.1,0.5][0.4,0.2][0.4,0.3])α2(0.5,[0.6,0.3,0.2,0.5][0.3,0.5][0.3,0.4])(0.6,[0.5,0.2,0.1,0.8][0.3,0.6][0.4,0.3])α3(0.2,[0.1,0.6,0.4,0.3][0.4,0.2][0.3,0.2])(0.4,[0.2,0.7,0.1,0.4][0.3,0.3][0.3,0.4])α4(0.5,[0.2,0.3,0.1,0.4][0.4,0.3][0.5,0.5])(0.6,[0.5,0.6,0.1,0.5][0.4,0.2][0.4,0.1])β3β4α1(0.4,[0.1,0.3,0.6,0.8][0.3,0.2][0.4,0.1])(0.2,[0.2,0.7,0.1,0.4][0.4,0.2][0.3,0.5])α2(0.3,[0.2,0.7,0.1,0.4][0.5,0.7][0.3,0.1])(0.1,[0.1,0.3,0.4,0.5][0.3,0.5][0.4,0.6])α3(0.3,[0.3,0.2,0.8,0.5][0.3,0.3][0.4,0.6])(0.7,[0.5,0.8,0.1,0.3][0.3,0.4][0.6,0.3])α4(0.7,[0.5,0.3,0.1,0.4][0.6,0.4][0.4,0.3])(0.8,[0.6,0.3,0.4,0.2][0.3,0.1][0.5,0.2])Table 2
Decision matrix of expert -2 (H2).
2β1β2α1(0.1,[0.4,0.2,0.1,0.3][0.6,0.4][0.4,0.3])(0.2,[0.2,0.3,0.5,0.4][0.4,0.1][0.3,0.2])α2(0.5,[0.2,0.6,0.3,0.4][0.3,0.5][0.4,0.6])(0.7,[0.1,0.6,0.1,0.8][0.3,0.6][0.5,0.3])α3(0.1,[0.6,0.3,0.1,0.5][0.4,0.1][0.5,0.6])(0.5,[0.6,0.3,0.7,0.4][0.3,0.3][0.3,0.4])α4(0.2,[0.1,0.7,0.3,0.4][0.4,0.2][0.6,0.1])(0.1,[0.3,0.5,0.1,0.8][0.4,0.3][0.4,0.2])β3β4α1(0.4,[0.3,0.4,0.1,0.2][0.4,0.3][0.4,0.2])(0.6,[0.6,0.3,0.5,0.4][0.3,0.6][0.5,0.2])α2(0.2,[0.2,0.3,0.6,0.4][0.3,0.2][0.5,0.6])(0.3,[0.2,0.7,0.1,0.8][0.6,0.3][0.3,0.1])α3(0.3,[0.5,0.4,0.1,0.7][0.8,0.1][0.7,0.2])(0.8,[0.5,0.3,0.5,0.4][0.3,0.4][0.4,0.2])α4(0.4,[0.7,0.6,0.3,0.4][0.3,0.1][0.6,0.5])(0.3,[0.2,0.6,0.1,0.8][0.4,0.2][0.3,0.5])Table 3
Decision matrix of expert -3 (H3).
3β1β2α1(0.6,[0.6,0.3,0.1,0.4][0.4,0.1][0.3,0.2])(0.4,[0.4,0.3,0.1,0.4][0.6,0.4][0.4,0.3])α2(0.2,[0.2,0.5,0.2,0.1][0.3,0.6][0.5,0.3])(0.3,[0.2,0.8,0.1,0.3][0.3,0.5][0.4,0.6])α3(0.4,[0.8,0.3,0.1,0.4][0.8,0.1][0.7,0.2])(0.3,[0.9,0.3,0.7,0.4][0.6,0.1][0.5,0.4])α4(0.5,[0.1,0.2,0.8,0.7][0.6,0.3][0.6,0.1])(0.4,[0.1,0.5,0.1,0.2][0.7,0.3][0.4,0.2])β3β4α1(0.9,[0.3,0.2,0.1,0.5][0.3,0.2][0.5,0.3])(0.7,[0.4,0.3,0.2,0.4][0.3,0.2][0.5,0.6])α2(0.5,[0.1,0.3,0.6,0.3][0.5,0.3][0.3,0.1])(0.1,[0.5,0.7,0.1,0.6][0.4,0.3][0.4,0.2])α3(0.7,[0.7,0.3,0.3,0.4][0.7,0.3][0.4,0.2])(0.6,[0.2,0.8,0.3,0.5][0.4,0.3][0.3,0.4])α4(0.8,[0.2,0.3,0.1,0.4][0.3,0.1][0.6,0.5])(0.6,[0.6,0.3,0.5,0.4][0.4,0.2][0.3,0.5])Table 4
Decision matrix of expert -4 (H4).
4β1β2α1(0.4,[0.5,0.3,0.2,0.4][0.4,0.3][0.6,0.3])(0.5,[0.2,0.3,0.5,0.4][0.4,0.1][0.5,0.2])α2(0.4,[0.2,0.8,0.1,0.3][0.6,0.3][0.5,0.3])(0.1,[0.3,0.2,0.7,0.8][0.4,0.2][0.5,0.3])α3(0.5,[0.8,0.3,0.5,0.6][0.5,0.3][0.6,0.2])(0.7,[0.7,0.3,0.1,0.4][0.4,0.3][0.3,0.4])α4(0.7,[0.2,0.8,0.1,0.4][0.3,0.1][0.6,0.5])(0.4,[0.4,0.3,0.3,0.5][0.3,0.2][0.7,0.1])β3β4α1(0.6,[0.5,0.3,0.1,0.5][0.4,0.3][0.4,0.2])(0.7,[0.8,0.5,0.1,0.5][0.4,0.6][0.6,0.4])α2(0.8,[0.4,0.2,0.1,0.3][0.3,0.6][0.7,0.3])(0.7,[0.2,0.7,0.3,0.8][0.7,0.3][0.4,0.6])α3(0.9,[0.8,0.3,0.6,0.7][0.3,0.4][0.8,0.2])(0.2,[0.3,0.3,0.2,0.4][0.6,0.1][0.7,0.3])α4(0.6,[0.2,0.9,0.1,0.3][0.4,0.1][0.3,0.5])(0.1,[0.8,0.3,0.5,0.4][0.6,0.3][0.5,0.2])Perform the ordering of the decision matrix with respect to the first component in the following tables.From Tables5–8, we introduced induced aggregation operators perform the ordering of the decision matrix with respect to the first component.Step 3.
We utilize the decision information given in the I-VITrF matrixHk=hijkm×nk=1,2,3,4 and the I-IVITrFAAOWA operator to derive the individual overall preference values Hik of the alternative αi.Table 5
Decision matrix of expert -5 (H5).
5β1β2α1(0.5,[0.5,0.2,0.1,0.4][0.3,0.6][0.4,0.3])(0.4,[0.1,0.3,0.6,0.8][0.3,0.2][0.4,0.1])α2(0.6,[0.5,0.2,0.1,0.8][0.3,0.6][0.4,0.3])(0.5,[0.6,0.3,0.2,0.5][0.3,0.5][0.3,0.4])α3(0.7,[0.5,0.8,0.1,0.3][0.3,0.4][0.6,0.3])(0.4,[0.2,0.7,0.1,0.4][0.3,0.3][0.3,0.4])α4(0.8,[0.6,0.3,0.4,0.2][0.3,0.1][0.5,0.2])(0.7,[0.5,0.3,0.1,0.4][0.6,0.4][0.4,0.3])β3β4α1(0.3,[0.3,0.2,0.1,0.5][0.4,0.2][0.4,0.3])(0.2,[0.2,0.7,0.1,0.4][0.4,0.2][0.3,0.5])α2(0.3,[0.2,0.7,0.1,0.4][0.5,0.7][0.3,0.1])(0.1,[0.1,0.3,0.4,0.5][0.3,0.5][0.4,0.6])α3(0.3,[0.3,0.2,0.8,0.5][0.3,0.3][0.4,0.6])(0.2,[0.1,0.6,0.4,0.3][0.4,0.2][0.3,0.2])α4(0.6,[0.5,0.6,0.1,0.5][0.4,0.2][0.4,0.1])(0.5,[0.2,0.3,0.1,0.4][0.4,0.3][0.5,0.5])Table 6
Decision matrix of expert -6 (H6).
6β1β2α1(0.6,[0.6,0.3,0.5,0.4][0.3,0.6][0.5,0.2])(0.4,[0.3,0.4,0.1,0.2][0.4,0.3][0.4,0.2])α2(0.7,[0.1,0.6,0.1,0.8][0.3,0.6][0.5,0.3])(0.5,[0.2,0.6,0.3,0.4][0.3,0.5][0.4,0.6])α3(0.8,[0.5,0.3,0.5,0.4][0.3,0.4][0.4,0.2])(0.5,[0.6,0.3,0.7,0.4][0.3,0.3][0.3,0.4])α4(0.4,[0.7,0.6,0.3,0.4][0.3,0.1][0.6,0.5])(0.3,[0.2,0.6,0.1,0.8][0.4,0.2][0.3,0.5])β3β4α1(0.2,[0.2,0.3,0.5,0.4][0.4,0.1][0.3,0.2])(0.1,[0.4,0.2,0.1,0.3][0.6,0.4][0.4,0.3])α2(0.3,[0.2,0.7,0.1,0.8][0.6,0.3][0.3,0.1])(0.2,[0.2,0.3,0.6,0.4][0.3,0.2][0.5,0.6])α3(0.3,[0.5,0.4,0.1,0.7][0.8,0.1][0.7,0.2])(0.1,[0.6,0.3,0.1,0.5][0.4,0.1][0.5,0.6])α4(0.2,[0.1,0.7,0.3,0.4][0.4,0.2][0.6,0.1])(0.1,[0.3,0.5,0.1,0.8][0.4,0.3][0.4,0.2])Table 7
Decision matrix of expert -7 (H7).
7β1β2α1(0.9,[0.3,0.2,0.1,0.5][0.3,0.2][0.5,0.3])(0.7,[0.4,0.3,0.2,0.4][0.3,0.2][0.5,0.6])α2(0.5,[0.1,0.3,0.6,0.3][0.5,0.3][0.3,0.1])(0.3,[0.2,0.8,0.1,0.3][0.3,0.5][0.4,0.6])α3(0.7,[0.7,0.3,0.3,0.4][0.7,0.3][0.4,0.2])(0.6,[0.2,0.8,0.3,0.5][0.4,0.3][0.3,0.4])α4(0.8,[0.2,0.3,0.1,0.4][0.3,0.1][0.6,0.5])(0.6,[0.6,0.3,0.5,0.4][0.4,0.2][0.3,0.5])β3β4α1(0.6,[0.6,0.3,0.1,0.4][0.4,0.1][0.3,0.2])(0.4,[0.4,0.3,0.1,0.4][0.6,0.4][0.4,0.3])α2(0.2,[0.2,0.5,0.2,0.1][0.3,0.6][0.5,0.3])(0.1,[0.5,0.7,0.1,0.6][0.4,0.3][0.4,0.2])α3(0.4,[0.8,0.3,0.1,0.4][0.8,0.1][0.7,0.2])(0.3,[0.9,0.3,0.7,0.4][0.6,0.1][0.5,0.4])α4(0.5,[0.1,0.2,0.8,0.7][0.6,0.3][0.6,0.1])(0.4,[0.1,0.5,0.1,0.2][0.7,0.3][0.4,0.2])Table 8
Decision matrix of expert -8 (H8).
8β1β2α1(0.7,[0.8,0.5,0.1,0.5][0.4,0.6][0.6,0.4])(0.6,[0.5,0.3,0.1,0.5][0.4,0.3][0.4,0.2])α2(0.8,[0.4,0.2,0.1,0.3][0.3,0.6][0.7,0.3])(0.7,[0.2,0.7,0.3,0.8][0.7,0.3][0.4,0.6])α3(0.9,[0.8,0.3,0.6,0.7][0.3,0.4][0.8,0.2])(0.7,[0.7,0.3,0.1,0.4][0.4,0.3][0.3,0.4])α4(0.7,[0.2,0.8,0.1,0.4][0.3,0.1][0.6,0.5])(0.6,[0.2,0.9,0.1,0.3][0.4,0.1][0.3,0.5])β3β4α1(0.5,[0.2,0.3,0.5,0.4][0.4,0.1][0.5,0.2])(0.4,[0.5,0.3,0.2,0.4][0.4,0.3][0.6,0.3])α2(0.4,[0.2,0.8,0.1,0.3][0.6,0.3][0.5,0.3])(0.1,[0.3,0.2,0.7,0.8][0.4,0.2][0.5,0.3])α3(0.5,[0.8,0.3,0.5,0.6][0.5,0.3][0.6,0.2])(0.2,[0.3,0.3,0.2,0.4][0.6,0.1][0.7,0.3])α4(0.4,[0.4,0.3,0.3,0.5][0.3,0.2][0.7,0.1])(0.1,[0.8,0.3,0.5,0.4][0.6,0.3][0.5,0.2])In Table9, we apply IVITrFAAOWA operators to find out overall preference values.Step 4.
We utilize the IVITrFAAOWA operator to aggregate all the individual overall preference values into the collective overall preference values. Consider thatλ=0.40,0.30,0.20,0.10T.Table 9
Overall preference values.
(a)h1(1)[0.22,0.43,0.25,0.54] [0.4223,0.0918] [0.6447,0.0328]h1(2)[0.26,0.40,0.23,0.53] [0.9992,0.0335] [0.8626,0.0014]h1(3)[0.46,0.45,0.34,0.38] [0.7993,0.0037] [0.9575,0.0001]h1(4)[0.57,0.25,0.22,0.34] [0.9029,0.0002] [0.9886,0.0002](b)h2(1)[0.44,0.33,0.34,0.35] [0.5160,0.1101] [0.6784,0.0220]h2(2)[0.18,0.55,0.25,0.64] [0.7417,0.0183] [0.9064,0.0013]h2(3)[0.53,0.33,0.38,0.50] [0.8946,0.0009] [0.9794,0.0003]h2(4)[0.36,0.59,0.18,0.60] [0.9420,0.0002] [0.9953,0.0001](c)h3(1)[0.39,0.27,0.14,0.43] [0.4918,0.0562] [0.7090,0.0570]h3(2)[0.38,0.58,0.26,0.40] [0.7228,0.0112] [0.9064,0.0016]h3(3)[0.55,0.50,0.36,0.44] [0.9275,0.0051] [0.9794,0.0001]h3(4)[0.33,0.34,0.36,0.39] [0.9718,0.0000] [0.0053,0.0003](d)h4(1)[0.56,0.38,0.19,0.47] [0.4759,0.0968] [0.8105,0.0342]h4(2)[0.21,0.33,0.15,0.64] [0.7963,0.0128] [0.9653,0.0025]h4(3)[0.58,0.30,0.23,0.24] [0.9069,0.0010] [0.9963,0.0001]h4(4)[0.48,0.53,0.38,0.35] [0.9527,0.0002] [0.9993,0.0002]Table10 shows that the collective values by applying the IVITrFAAOWA operators for the ranking process.Step 5.
Calculate score values of alternativesαi.(78)sh1=0.4400,sh2=0.3800,sh3=0.4100,sh4=0.1900.Step 6.
Ranking the score values(79)α1≥α3≥α2≥α1.Table 10
Collective values.
h1[0.3500,0.3600,0.2500,0.8300] [0.9709,0.0000] [0.9915,0.0000]h2[0.2600,0.4700,0.2400,0.5500] [0.9999,0.0000] [0.9905,0.0054]h3[0.5100,0.4100,0.3600,0.4100] [0.0001,0.0020] [0.9908,0.0065]h4[0.4500,0.4000,0.2400,0.4300] [0.9999,0.1002] [0.9986,0.0069]In Figure3, we represent the graphical approach of proposed aggregation operator and using the score function to find out the best alternative.Figure 3
Graphical approach of proposed operator.


## 7. Comparative Analysis

To expand on the advantages of the developed methods, we compare them with the existing methods by solving the same example with same weightλ=0.10,0.20,0.30,0.40T, such as the interval-valued intuitionistic fuzzy weighted average [39] operator, the intuitionistic fuzzy weighted average [39] operator, the intuitionistic fuzzy Einstein weighted average [41] operator, the intuitionistic fuzzy weighted geometric [10] operator, and the intuitionistic trapezoidal fuzzy weighted geometric [41] operator.But in our paper, we take the data in the form of I-IVITrF information and used basic concept of the Aczel-Alsina aggregation operators like, I-IVITrFAAOWA operator, I-IVITrFAAHA operator, I-IVITrFAAOWG operator, and I-IVITrFAAHWG operator to aggregate the MCGDM issues. Furthermore, we have used these aggregation operators for solving the MCGDM in form of I-IVITrFNs environment. If we compare the score values of proposed aggregation operators with existing aggregation operators. We can see that our proposed work is more accurate and more applicable than existing methods due to highest score value of the proposed method. Applying the above mentioned methods, we obtain the comparison results shown in Table11.Table 11
The comparison analysis with the different methods.
ApproachMethodsRanking resultBest option(i)I-VIFWA [39]Existing methodα3≥α2≥α1≥α40.0017(ii)IFWA [39]Existing methodα3≥α2≥α1≥α4−0.0131(iii)IFEWA [41]Existing methodα2≥α1≥α4≥α30.0171(iv)IFWG [10]Existing methodα1≥α4≥α2≥α30.0300(v)ITFWG [41]Existing methodα1≥α2≥α3≥α40.0100(vi)I-IVITrFAAOWAProposed methodα1≥α3≥α2≥α40.4400From Table11, we observe that the optimal ranking results are different of the different methods, even though the score functions are different in different methods. Therefore, the novel methods proposed by us are reasonable and valid. Moreover, compared the proposed I-IVITrFAAOWA method with the existing methods, there is difference between the score functions obtained from the existing methods such that the highest score value of the proposed method is 0.4400 which is shown in Table 11. Therefore, decision-making processing based on the novel methods can be performed in an evaluation system, which implies that the propose methods have greater advantages than the existing methods.In Figure4, we compared all the results of proposed method with existing methods with the help of graphical approach.Figure 4
Graphical presentation of comparison analysis.


## 8. Conclusions

The objective of this paper is to present some induced operators based on I-VITrFAA aggregation operators and apply them to the MCGDM problems. The MCGDM has an enormous prospective and restraint development used for improving and assessing different contradictory criteria in all aspects of DM in order to achieve more suitable and pragmatic DM result. In DM issues, the effectual erudition concerning a fastidious fact makes the decision-making task more complicated and dynamic. Induced interval-valued intuitionistic fuzzy sets are general mathematical method that can easily handle ambiguous and imprecise knowledge. In this article we developed operation laws based on the Aczel-Alsina (A-A) and introduced four new types of aggregations operators such as I-IVITrFAAOWA operator, I-IVITrFAAHA operator, I-IVITrFAAOWG operator and I-IVITrFAAHG operator. Finally, examples are given to show the technique’s prospective application and improvement. Furthermore, the established approach can be expanded for future research by incorporating other existing fuzzy sets and applying them to various MCGDM problems.



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*Source: 2557309-2022-12-28.xml*