# Assessment of the Business Proposals Using Frank Aggregation Operators Based on Interval-Valued T-Spherical Fuzzy Information

Authors: Amir Hussain; Kifayat Ullah; Haolun Wang; Mehwish Bari
Journal: Journal of Function Spaces
 (2022)

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## Abstract

An interval-valued T-spherical fuzzy set (IVTSFS) is an efficient framework to treat ambiguous and vague information more efficiently. It has four functions consisting of the information in intervals. With the help of these functions, IVTSFS is a more reliable framework than the other existing frameworks. This article aims to establish algebraic operations of IVTSFSs named Frank sum, Frank multiplication, and Frank scalar multiplication stranded on Frank t-norm and t-conorm. Further, some averaging and geometric aggregation operators (AOs), i.e., interval-valued T-spherical fuzzy Frank weighted averaging (IVTSFFWA) and interval-valued T-spherical fuzzy Frank weighted geometric (IVTSFFWG) operators are established based on the defined operations, and then the properties of these operators are observed. The proposed approach is supported by an illustrative example, in which we have applied our proposed approach to a multiattribute group decision-making (MAGDM) problem. For the significance of the proposed approach, a comparative study is done. The various effects of the variation of parameters are also investigated which plays an important role in the final results.


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## Body

## 1. Introduction

Mathematicians have been dealing with uncertainties and vagueness in data for many years. For this purpose, Zadeh [1] introduced the concept of fuzzy set (FS), where a membership grade (MG) is used to describe the uncertainty of an object, where m∈0,1. Atanassov [2] introduced the model of an intuitionistic fuzzy set (IFS), in which he used two grades, MG m and a nonmembership grade (NMG)n to describe the uncertain situation. Yager [3] extended this concept to Pythagorean fuzzy set (PyFS), where the sole purpose was to increase the range of the IFS by allowing the sum of the squares of m and n in 0,1. Despite, the frame of PyFS provides a wider range for m and n to be taken, but still, it has limitations in some cases and hence does not guarantee the independency of the MG and the NMG. To enlarge the space for assigning m and n, Yager [4] proposed the notion of q-rung orthopair fuzzy set (qROFS) where the range for assigning the MG and the NMG is taken to a limitless level using a variable parameterq.The literature in [1–4] only used two grades, i.e., the MG and NMG of an object for its description in an uncertain environment. But in many real-life problems, opinion about a phenomenon or an object may not be of two types only as discussed in the layout of IFS, PyFS, and qROFS. Cuong [5] observed that opinion about someone or something may have four aspects including favor, abstinence, disfavor, and refusal. To express such kind of opinion, Cuong [5] introduced the notion of picture fuzzy set (PFS) where he portrayed uncertain information using four types of memberships known by an MG, an abstinence grade (AG), NMG, and a refusal grade (RG), hence generalizing the notion IFSs by covering larger information than IFSs. The frame of PFS greatly reduces information loss by taking into account the AG and the RG, yet it has a certain restriction on the MG, NMG, and the AG. Mahmood et al. [6] observed that the frame of PFS has a limited choice for decision-makers to dispense the values of their concern to MG, AG, and NMG. To overcome this problematic situation, Mahmood et al. [6] introduced the model of the spherical fuzzy set (SFS) and analogously T-spherical fuzzy sets (TSFS) with the independent choice to assign the values to MG, AG, and NMG. TSFSs covered a big loss of information. However, Ullah et al. [7] introduced interval-valued TSFS (IVTSFS) having four functions with intervals to cover more information.MAGDM became a very significant concept in FS theory due to its effects in the fields of engineering, management, and social sciences. FS theory has been applied to treat MAGDM problems for many years. It had started in 1970 when Bellman [8] used FS in decision-making. To treat MAGDM problems, IFS was used by Xu [9] with its aggregation tools. Rahman et al. [10] and Yager et al. [11] presented some AOs for PyFSs with a larger range for MG and NMG and used these AOs to deal with multiattribute decision-making (MADM) problems. In [12, 13], the authors developed AOs in the framework of PFSs and used them in MADM. Mahmood et al. [6] developed AOs for SFSs and TSFSs having no restrictions for MG, AG, and NMG. Ullah et al. [7] established AOs in the framework of interval-valued TSFSs (IVTSFSs) and solved the financial policy evaluation problem. Guleria and Bajaj [14] introduced AOs for T-spherical fuzzy (TSF) soft sets and utilized these AOs to solve multiattribute decision-making (MADM) problems. Karaaslan and Dawood [15] established complex (TSF) Dombi aggregation operators and used them in MAGDM. Xu et al. [16] solved the MADM problem by introducing an interval-valued q-Rung dual hesitant uncertain linguistic set. In [17] the authors developed AOs in an environment of IFS to solve the MADM problem. Wang and Li [18] developed AOs for hesitant PFS and used them in MADM. Akram et al. [19] introduced complex picture fuzzy Hamacher AOs and studied their application in MADM. In [20] Riaz and Hashmi developed the AOs in the environment of cubic m-polar fuzzy sets and applied them to solve MAGDM. Peng and Yang [21] developed AOs the phenomenon of interval-valued Pythagorean fuzzy (PyFS) and applied in MAGDM. Fahmi et al. [22] introduced cubic fuzzy Einstein AOs and applied them to the MAGDM problem. Ullah et al. [23] introduced TSF Hamacher AOs and applied them in the evaluation of the performance of robots. Garg et al. [24] developed TSF power AOs and applied them in MAGDM. Akram et al. [25] introduced IVTSFS Bonferroni mean AOs and applied them in the evaluation of solar energy cells.All the described AOs are grounded on some t-norm and t-conorm. The frequently used t-norm and t-conorm is the product and probabilistic sum [26] because computation becomes very laidback by using this dual. But the deficiency of flexibility and robustness are the major negatives of these triangular norms. So, some AOs are familiarized with other t-norm and t-conorms to astound these drawbacks. Xia et al. [27] proposed AOs for IFSs information grounded on Archimedean t-norm and t-conorms. Liu and Wang [28] proposed a family of some AOs in the environment of IFSs based on Einstein t-norm and t-conorms. Shahzadi [29] inspected Hamacher hybrid AOs in the framework of Fermatean fuzzy number and applied them in MAGDM. Jin et al. [30] developed the aggregation operators for IVTSFS based on Hamacher t-norm and t-conorm. Jana [31] explored some MADM problems based on SVNTH methods. Jana et al. [32] introduced AOs in the environment of qROFS using Dombi t-norm and t-conorm and applied in MAGDM. Seikh and Mandal [33] introduced AOs in the framework of IFS using Dombi t-norms and t-conorms and applied them in MAGDM. Jana et al. [34] developed bipolar fuzzy Dombi AOs and applied them in MAGDM. Ullah et al. [35] developed IVTSF Dombi AOs and studied their application in MAGDM. Ullah [36] introduced picture fuzzy (PF) Maclaurin symmetric mean operators and applied them in MADM problems. Some remarkable work can be seen in [37–39].By the use of the aforementioned t-norms and t-conorms, the level of robustness enlarged. However, the abovementioned t-norm and t-conorm are not much flexible to include all opinions (information) to model real-world MAGDM problems. To overcome the gap, Frank [40] generalized the Lukasiewicz and probabilistic t-norm and t-conorm to propose Frank t-norm and t-conorm, a flexible family of triangular norms. The inclusion of parameters in Frank t-norm and t-conorm makes them more malleable in the fusion of information and more appropriate to model the DM problems. Sarkoci [41] studied the comparison between Hamacher and Frank t-norms and found the same family of two different t-norms.After the introduction of Frank t-norm and t-conorms, the mathematicians developed AOs by using this pair of norms. Zhang [42] introduced Frank AOs (FAOs) for IVIFSs and observed their application in MAGDM problems. Qin et al. [43] proposed FAOs for triangular interval type-2 fuzzy information based on Frank t-norms and t-conorms and applied them to MAGDM problems. In [44] the FAOs are developed in probabilistic hesitant fuzzy sets and their application in MAGDM. Tang et al. [45] proposed the FAOs in the framework of dual hesitant fuzzy information.After the introduction of IVTSFSs, the decision-makers have become independent in expressing their information using the MGs, AGs, and NMGs in intervals with greater flexibility. Due to this property of IVTSFSs, and the significance of Frank t-norm and t-conorm, the concept FAOs in the framework of IVTSFSs is eminent and has some desirable properties. It is also noted that there is a lack of study of the effects of changing the behavior of the parameters of the FAOs in MAGDM problems. This fact motivated me to study the family of FAOs and their properties in the framework of IVTSFSs, grounded on frank t-norm and t-conorm, and consequently to apply to the MAGDM problem.We divide the remaining paper into 9 further sections. We stated some basic concepts of IVTSFS and Frank t-norm and t-conorm in Section2. In Section 3, we defined Frank operations of sum and multiplication in the framework of IVTSFSs. In Section 4, we defined the IVTSFFWA operator and investigate some basic properties of the IVTSFFWA operator. In Section 5, we define the IVTSFFWG operator and observe its basic properties. In Section 6, we planned an approach to apply IVTSFFWA and IVTSFFWG operators to a MAGDM problem. In Section 7, we applied proposed operators in the MAGDM problem with help of an illustrative example and observed the influence of parameters in Section 8. In Section 9, we investigated a comparative study with other AOs. In Section 10, we concluded our article.


## 2. Preliminaries

In this section, first, we will introduce fundamental concepts Frank t-norm and t-conorm and IVTSFFs which are used to understand the proposed approach. IVTSFS is a stimulating technique to deal with data, especially in uncertain and ambiguous situations. Due to the involvement of intervals, IVTSFS grabs more information from real-life situations whenever human opinions evolved and hence greatly reduces information loss. In Definition1, we define IVTSFS as follows:Definition 1 (see [7]).
LetX be the universe of discourse, an IVTSFS T in X is defined as
(1)T=x,sx,ix,dx∣x∈X,where s,i,d:X⟶0,1 are MG, AG, and NMG such that s=sl,su,i=il,iu and d=dl,du with condition 0≤suq+iuq+duq≤1. The RG of IVTSFS is defined as r=rl,ru=1−suq+iuq+duqq,1−slq+ilq+dlqq and s,i,d is IVTSF number (IVTSFN).Ullah et al. [7] defined the score function for IVTSFNs given as follows.Definition 2.
Letscα denotes the score value of IVTSFN α, and score function can be defined as
(2)scα=slq1−ilq−dlq+suq1−iuq−duq3.All the operators defined on the FSs have been playing an important role since starting. These operators are grounded on the general concept of t-norms and t-conorms [27]. The definitions of the t-norm and t-conorm introduced by Frank [36] are the following:Definition 3.
The Frank t-norms and t-conorms are functions from0,12 to 0,1 which are below:
(3)TFα,β=logθ1+θα−1θβ−1θ−1,α,β∈0,1,θ>1,SFα,β=1−logθ1+θ1−α−1θ1−β−1θ−1,α,β∈0,1,θ>1.Based on t-norms and t-conorms, we can define the generalized union and generalized intersection of two IVTSFSs as follows:Definition 4.
LetTj=a,sTjal,sTjau,iTjal,iTjau,dTjal,dTjau∣a∈X for j=1,2 be two IVTSFSs, and then the generalized union and intersection can be defined as
(4)T1∪T,ST2=a,SsT1al,sT1au,sT2al,sT2au,TiT1al,iT1au,iT2al,iT2au,TdT1al,dT1au,dT2al,dT2au∣a∈X,T1∩T,ST2=a,TsT1al,sT1au,sT2al,sT2au,SiT1al,iT1au,iT2al,iT2au,SdT1al,dT1au,dT2al,dT2au∣a∈X,where we can utilize any pair of t-norm and t-conorm.Definition 5.
Lett1=s1l,s1u,i1l,i1u,d1l,d1u and t2=s2l,s2u,i2l,i2u,d2l,d2u be two IVTSFNs; the generalized union and intersection between t1 and t2 can be defined as
(5)t1⨁T,St2=Ss1l,s1u,s2l,s2u,Ti1l,i1u,i2l,i2u,Td1l,d1u,d2l,d2u,t1⨂T,St2=Ts1l,s1u,s2l,s2u,Si1l,i1u,i2l,i2u,Sd1l,d1u,d2l,d2u,where we can utilize any pair of t-norm and t-conorm.


## 3. Frank Aggregation Operators for IVTSFNs

In this section, we will describe the introduction of some operations for IVTSFNs grounded on the Frank t-norm and t-conorm and investigate some properties of these operations.AssumeT denotes the t-norm and S denotes t-conorm, and let TF is theFrank product and SF is the Frank sum, and then the generalized union and intersection of two IVTSFNs μ1=s1l,s1u,i1l,i1u,d1l,d1u and μ2=s2l,s2u,i2l,i2u,d2l,d2u become the Frank sum and Frank product, symbolically denoted by μ1⨁Fμ2 and μ1⨂Fμ2, respectively. The sum and product are as follows:
(6)μ1⨁Fμ2=1−logθ1+θ1−s1lq−1θ1−s2lq−1θ−1q,1−logθ1+θ1−s1uq−1θ1−s2uq−1θ−1q,logθ1+θi1lq−1θi2lq−1θ−1,logθ1+θi1uq−1θi2uq−1θ−1,logθ1+θd1lq−1θd2lq−1θ−1,logθ1+θd1uq−1θd2uq−1θ−1,whereθ>1(7)μ1⨂Fμ2=logθ1+θs1lq−1θs2lq−1θ−1,logθ1+θs1uq−1θs2uq−1θ−1,1−logθ1+θ1−i1lq−1θ1−i2lq−1θ−1q,1−logθ1+θ1−i1uq−1θ1−i2uq−1θ−1q,1−logθ1+θ1−d1lq−1θ1−d2lq−1θ−1q,1−logθ1+θ1−d1uq−1θ1−d2uq−1θ−1q,whereθ>1.Now, we describe some properties of IVTSFNs in the environment of Frank t-norm and t-conorm. It is observed that the sum of two IVTSFNs is also an IVTSFN. In Theorem6, we will prove that the sum of two IVTSFNs is again an IVTSFN.Theorem 6.
Letμ1=s1l,s1u,i1l,i1u,d1l,d1u and μ2=s2l,s2u,i2l,i2u,d2l,d2u be two IVTSFNs, and let μ3=μ1⨂Fμ2 and μ4=μ1⨁Fμ2, then both μ3 and μ4 are also IVTSFNs.Proof.
Since0≤sj,ij,dj≤1 and 0≤sjq+ijq+djq≤1j=1,2 for both lower and upper intervals values, then we have
(8)0=logθ1+θ0−1θ0−1θ−1q≤logθ1+θs1lq−1θs2lq−1θ−1q≤logθ1+θ1q−1θ1q−1θ−1q=1,0=logθ1+θ0−1θ0−1θ−1q≤logθ1+θs1uq−1θs2uq−1θ−1q≤logθ1+θ1q−1θ1q−1θ−1q=1,0=1−logθ1+θ1−0−1θ1−0−1θ−1≤1−logθ1+θ1−i1lq−1θ1−i2lq−1θ−1≤1−logθ1+θ1−1−1θ1−1−1θ−1=1,0=1−logθ1+θ1−0−1θ1−0−1θ−1≤1−logθ1+θ1−i1uq−1θ1−i2uq−1θ−1≤1−logθ1+θ1−1−1θ1−1−1θ−1=1,0=1−logθ1+θ1−0−1θ1−0−1θ−1≤1−logθ1+θ1−d1lq−1θ1−d2lq−1θ−1≤1−logθ1+θ1−1−1θ1−1−1θ−1=1,0=1−logθ1+θ1−0−1θ1−0−1θ−1≤1−logθ1+θ1−d1uq−1θ1−d2uq−1θ−1≤1−logθ1+θ1−1−1θ1−1−1θ−1=1,(9)0≤logθ1+θs1lq−1θs2lq−1θ−1q≤1−logθ1+θ1−i1lq−1θ1−i2lq−1θ−1≤1−logθ1+θ1−d1lq−1θ1−d2lq−1θ−1≤1.
Similarly(10)0≤logθ1+θs1uq−1θs2uq−1θ−1q≤1−logθ1+θ1−i1uq−1θ1−i2uq−1θ−1≤1−logθ1+θ1−d1uq−1θ1−d2uq−1θ−1≤1.which, indicates that μ3 is an IVTSFN.Accordingly, we can also prove thatμ4 is also IVTSFN. Thus, proof of Theorem 6 is completed.We can also multiply a positive number to an IVTSFN, which results another IVTSFN. In Theorem7, we will describe scalar multiplication of IVTSFN with a positive integer.Theorem 7.
Ifn is any positive integer and μ=sl,su,il,iu,dl,du is anIVTSFN, then n.Fμ is a mapping from Z+×μtoμ:(11)n.Fμ=1−logθ1+θ1−slq−1nθ−1n−1q,1−logθ1+θ1−suq−1nθ−1n−1q,logθ1+θilq−1nθ−1n−1logθ1+θiuq−1nθ−1n−1,logθ1+θdlq−1nθ−1n−1logθ1+θduq−1nθ−1n−1where n.Fμ=μ⊕Fμ⊕Fμ⊕F⋯μ⊕Fntimes. Moreover, n.Fμ is an IVTSFN for every positive real numbern.Proof.
First, we can prove this equation true forZ+ by mathematical induction as follows:
Forn=1(12)L.H.S=1.Fμ=μ,R.H.S=1−logθ1+θ1−slq−11θ−11−1q,1−logθ1+θ1−suq−11θ−11−1q,logθ1+θilq−11θ−11−1logθ1+θiuq−11θ−11−1,logθ1+θdlq−11θ−11−1,logθ1+θduq−11θ−11−1=sl,su,il,iu,dl,du=μ.The equation is true forn=1.Now we have to prove forn=k+1 by assuming true for n=k, i.e.
(13)k+1.Fμ=1−logθ1+θ1−slq−1k+1θ−1k+1−1q,1−logθ1+θ1−suq−1k+1θ−1k+−1q,logθ1+θilq−1k+1θ−1k+1−1,logθ1+θiuq−1k+1θ−1k+1−1,logθ1+θdlq−1k+1θ−1k+1−1,logθ1+θduq−1k+1θ−1k+1−1.Using the hypothesis that equation is true forn=k, then the left-hand side can be written as k+1.Fμ=k.Fμ⨁Fμ and on the basis of the Frank sum operation of two IVTSFNs as follows:
(14)k.Fμ⨁Fμ=1−logθ1+θ1−slq−1k+1θ−1k+1−1q,1−logθ1+θ1−suq−1k+1θ−1k+−1q,logθ1+θilq−1k+1θ−1k+1−1,logθ1+θiuq−1k+1θ−1k+1−1,logθ1+θdlq−1k+1θ−1k+1−1,logθ1+θduq−1k+1θ−1k+1−1⨁Fsl,su,il,iu,dl,du,=1−logθ1+θ1−1−logθ1+θ1−slq−1k/θ−1k−1−1θ1−slq−1θ−1q,1−logθ1+θ1−1−logθ1+θ1−suq−1k/θ−1k−1−1θ1−suq−1θ−1qlogθ1+θlogθ1+θilq−1k/θ−1k−1−1θilq−1θ−1,logθ1+θlogθ1+θiuq−1k/θ−1k−1−1θiuq−1θ−1logθ1+θlogθ1+θdlq−1k/θ−1k−1−1θdlq−1θ−1,logθ1+θlogθ1+θduq−1k/θ−1k−1−1θduq−1θ−1=1−logθ1+θ1−slq−1k+1θ−1kq,1−logθ1+θ1−suq−1k+1θ−1kq,logθ1+θilq−1k+1θ−1k,logθ1+θiuq−1k+1θ−1k,logθ1+θdlq−1k+1θ−1k,logθ1+θduq−1k+1θ−1k.The equation is true forn=k+1. Thus, it is true for all positive integers.Further, we can prove(15)0≤1−logθ1+θ1−0q−1nθ−1n−1q≤1−logθ1+θ1−slq−1nθ−1n−1q≤1−logθ1+θ1−1q−1nθ−1n−1q=1,0≤logθ1+θ0q−1nθ−1n−1q≤logθ1+θilq−1nθ−1n−1≤logθ1+θ1q−1nθ−1n−1=1,(16)0≤logθ1+θ0q−1nθ−1n−1≤logθ1+θdlq−1nθ−1n−1≤logθ1+θ1q−1nθ−1n−1=1.Evenn∈R+.Since0≤sl,il,dl≤1 and 0≤slq+ilq+dlq≤1, thus we have
(17)0≤1−logθ1+θ1−slq−1nθ−1n−1q,logθ1+θilq−1nθ−1n−1,logθ1+θdlq−1nθ−1n−1≤1.Similarly, we can easily prove for the upper values of interval. Hence, it is verified thatn.Fμ is IVTSFN for any positive real numbern.On the basis of Theorem7, we can define scalar multiplication for any IVTSFN μ=sl,su,il,iu,dl,du and any positive real number ϵ>0 as follows:
(18)ϵ.Fμ=1−logθ1+θ1−slq−1ϵθ−1ϵ−1q,1−logθ1+θ1−suq−1ϵθ−1ϵ−1q,logθ1+θilq−1ϵθ−1ϵ−1,logθ1+θiuq−1ϵθ−1ϵ−1,logθ1+θdlq−1ϵθ−1ϵ−1,logθ1+θduq−1ϵθ−1ϵ−1,θ>1.In Theorem8, we define an operation for any real number to the power of the IVTSFN as follows:Theorem 8.
Ifn∈Z+ and μ=sl,su,il,iu,dl,du is an IVTSFN, then the exponential operation μ∧Fn is a mappingZ+×μ⟶μ:
(19)μ∧Fn=logθ1+θslq−1nθ−1n−1logθ1+θsuq−1nθ−1n−1,1−logθ1+θ1−ilq−1nθ−1n−1q,1−logθ1+θ1−iuq−1nθ−1n−1q1−logθ1+θ1−dlq−1nθ−1n−1q,1−logθ1+θ1−duq−1nθ−1n−1q,,where μ∧Fn=μ⨂Fμ⨂F⋯⨂Fμ (n times).Moreover,μ∧Fn is an IVTSFN for very positive realn.On the basis of Theorem8, for any real number ϵ>0 and an IVTSFN μ=sl,su,il,iu,dl,du, we can define exponential operation μ∧Fϵ as
(20)μ∧Fn=logθ1+θslq−1ϵθ−1ϵ−1,logθ1+θsuq−1ϵθ−1ϵ−1,1−logθ1+θ1−ilq−1ϵθ−1ϵ−1q,1−logθ1+θ1−iuq−1ϵθ−1ϵ−1q,1−logθ1+θ1−dlq−1ϵθ−1ϵ−1q,1−logθ1+θ1−duq−1ϵθ−1ϵ−1q,θ>1.In addition, we investigated the following properties and some interesting laws in Theorem9, as follows:Theorem 9.
For any two IVTSFNsμ1 and μ2, we have the following properties:
(21)μ1⨁Fμ2=μ2⨁Fμ1,μ1⨂Fμ2=μ2⨂Fμ1,ϵ.Fμ1⨁Fμ2=ϵ.Fμ1⨁Fϵ.Fμ2ϵ>0,ϵ1+ϵ2.Fμ=ϵ1.Fμ⨁Fϵ1.Fμ,ϵ1,ϵ2>0,μ1⨂Fμ2∧Fϵ=μ1∧Fϵ⨂Fμ2∧Fϵ,ϵ>0,μ∧Fϵ1+ϵ2=μ∧Fϵ1⨂Fμ2∧Fϵ,ϵ>0,ϵ1ϵ2.Fμ=ϵ1.Fϵ2.Fμ,μ∧Fϵ1ϵ2=μ∧Fϵ1∧Fϵ2.


## 4. IVTSF Frank Averaging Aggregation Operators

In this section, we will define IVTSFFWA operator and investigate some basic properties.Definition 10.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be set of IVTSFNs, having w=w1,w2,⋯,wnT be the weight vector of μjj=1,2,3,⋯,n with wj∈0,1 and ∑j=1nwj=1 (note that we shall use w in next discussion for weight vector having same characteristics). Then, IVTSFFWA operator is a mapping μn⟶μ such that
(22)IVTSFFWAμ1,μ2,⋯,μn=w1.Fμ1⨁Fw2.Fμ2⨁F⋯⨁Fwn.Fμn.Particularly, ifw=1/w1,1/w2,⋯,1/wnT, then the IVTSFFWA operator reduces to the IVTSFFA operator:
(23)IVTSFFAμ1,μ2,⋯,μn=1n.Fμ1⨁Fμ2⊕F⋯⨁Fμn.It is noted that the value obtained after the aggregation of IVTSFNs is also an IVTSFN. In Theorem11, we will state that the number obtained after aggregation by IVTSFFWA operator is also an IVTSFN.Theorem 11.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n set of IVTSFNs, and let w be the weight vector, and then the value obtained after aggregation by using the IVTSFFWA operator is also IVTSFN, and
(24)IVTSFFWAμ1,μ2,⋯,μn=1−logθ1+∏j=1nθ1−sjlq−1wjq,1−logθ1+∏j=1nθ1−sjuq−1wjq,logθ1+∏j=1nθijlq−1wj,logθ1+∏j=1nθijuq−1wj,logθ1+∏j=1nθdjlq−1wj,logθ1+∏j=1nθdjuq−1wj.Theorem11 can be easily proved by operations defined above. The proof is skipped to avoid the extra length of this article.Now, we shall discuss the properties of IVTSFFWA operator in the following. As we know that the aggregation operators hold three basic properties idempotency, monotonicity, and boundedness. In Theorem12, we will prove the idempotency of IVTSFFWA operator, as follows:Theorem 12.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n is set of IVTSFNs; if all μj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n are same, i.e., μj=μ=sl,su,il,iu,dl,du for all j, then IVTSFFWAμ1,μ2,⋯,μn=IVTSFFWAμ,μ,⋯,μ=μ which is called idempotency.Proof.
(25)IVTSFFWAμ1,μ2,⋯,μn=1−logθ1+∏j=1nθ1−sjlq−1wjq,1−logθ1+∏j=1nθ1−sjuq−1wjq,logθ1+∏j=1nθijlq−1wj,logθ1+∏j=1nθijuq−1wj,logθ1+∏j=1nθdjlq−1wj,logθ1+∏j=1nθdjuq−1wj=1−logθ1+∏j=1nθ1−slq−1wjq,1−logθ1+∏j=1nθ1−suq−1wjq,logθ1+∏j=1nθilq−1wj,logθ1+∏j=1nθiuq−1wj,logθ1+∏j=1nθdlq−1wj,logθ1+∏j=1nθduq−1wj=1−logθ1+θ1−slq−1∑jnwjq,1−logθ1+θ1−suq−1∑jnwjq,logθ1+θilq−1∑jnwj,logθ1+θilq−1∑jnwj,logθ1+θdlq−1∑jnwj,logθ1+θduq−1∑jnwj=1−logθθ1−slq,1−logθθ1−suq,logθθil,logθθiu,logθθdl,logθθdl=sl,su,il,iu,dl,du.
This completes proof.In Theorem13, we will prove monotonicity of IVTSFFWA operator, as follows:Theorem 13.
Letμj=sjl,sju,ijl,iju,djl,dju and φj=ajl,aju,bjl,bju,cjl,cjuj=1,2,3,⋯,n be two collections of IVTSFNs; if sjl≤ajl,ijl≥bjl, and djl≥cjl, for all j, then IVTSFFWAμ1,μ2,⋯,μn≤IVTSFFWAφ1,φ2,⋯,φn which is called monotonicity.Proof.
For two collections of IVTSFNsμj=sjl,sju,ijl,iju,djl,dju and φj=ajl,aju,bjl,bju,cjl,cju, if sjl≤ajl,ijl≥bjl and djl≥cjl, for allj, then
(26)1−logθ1+∏j=1nθ1−sjlq−1wjq≤1−logθ1+∏j=1nθ1−ajlq−1wjq,logθ1+∏j=1nθijlq−1wj≥logθ1+∏j=1nθbjlq−1wj,logθ1+∏j=1nθdjlq−1wj≥logθ1+∏j=1nθcjlq−1wj.
Similarly,(27)1−logθ1+∏j=1nθ1−sjuq−1wjq≤1−logθ1+∏j=1nθ1−ajuq−1wjq,logθ1+∏j=1nθijuq−1wj≥logθ1+∏j=1nθbjuq−1wj,logθ1+∏j=1nθdjuq−1wj≥logθ1+∏j=1nθcjuq−1wj.
By definition of score function(28)IVTSFFWAμ1,μ2,⋯,μnS=1−logθ1+∏j=1nθ1−sjlq−1wjq1−logθ1+∏j=1nθijlq−1wj−logθ1+∏j=1nθdjlq−1wj+1−logθ1+∏j=1nθ1−sjuq−1wjq1−logθ1+∏j=1nθijuq−1wj−logθ1+∏j=1nθdjuq−1wj3,IVTSFFWAφ1,φ2,⋯,φnS=1−logθ1+∏j=1nθ1−ajlq−1wjq1−logθ1+∏j=1nθbjlq−1wj−logθ1+∏j=1nθcjlq−1wj+1−logθ1+∏j=1nθ1−ajuq−1wj1−logθ1+∏j=1nθbjuq−1wj−logθ1+∏j=1nθcjuq−1wj3.
By definition(1)
IfIVTSFFWAμ1,μ2,⋯,μnS<IVTSFFWAφ1,φ2,⋯,φnS then IVTSFFWAμ1,μ2,⋯,μn<IVTSFFWAφ1,φ2,⋯,φn(2)
IfIVTSFFWAμ1,μ2,⋯,μnS>IVTSFFWAφ1,φ2,⋯,φnS then IVTSFFWAμ1,μ2,⋯,μn>IVTSFFWAτφ1,φ2,⋯,φn(3)
IfIVTSFFWAμ1,μ2,⋯,μnS=IVTSFFWAφ1,φ2,⋯,φnS then IVTSFFWAμ1,μ2,⋯,μn=IVTSFFWAφ1,φ2,⋯,φn
Then by equations(29)1−logθ1+∏j=1nθ1−sjlq−1wjq=1−logθ1+∏j=1nθ1−ajlq−1wj,qlogθ1+∏j=1nθijlq−1wj=logθ1+∏j=1nθijlq−1wj,logθ1+∏j=1nθdjlq−1wj=logθ1+∏j=1nθdjlq−1wj.
Similarly,(30)1−logθ1+∏j=1nθ1−sjuq−1wjq=1−logθ1+∏j=1nθ1−ajuq−1wj,qlogθ1+∏j=1nθijuq−1wj=logθ1+∏j=1nθijuq−1wj,logθ1+∏j=1nθdjuq−1wj=logθ1+∏j=1nθdjuq−1wj,which shows that IVTSFFWAμ1,μ2,⋯,μn=IVTSFFWAφ1,φ2,⋯,φn. Hence proof is completed.Note that the proofs of further theorems are skipped to control the length of the article.Theorem 14.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be the collection of IVTSFNs and
(31)μ−=minjsjl,minjsju,maxjijl,maxjiju,maxjdjl,maxjdju,μ+=maxjsjl,maxjsju,minjijl,minjiju,minjdjl,minjdju,then
(32)μ−≤IVTSFFWAμ1,μ2,⋯,μn≤μ+,which is called the boundedness.Theorem 15.
Letμj=sjl,sju,ijl,iju,djl,dju is set of IVTSFNs, and let w be the weight vector of μjj=1,2,3,⋯,n; if r>1 is real number, then
(33)IVTSFFWAr.Fμ1,r.Fμ2,⋯,r.Fμn=r.FIVTSFFWAμ1,μ2,⋯,μn.Theorem 16.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be set of IVTSFNs, and let w is weight vector; if μ=sl,su,il,iu,dl,du is a IVTSFN, then
(34)IVTSFFWAμ1⨁Fμ,μ2⊕Fμ,⋯μn⨁Fμ=IVTSFFWAμ1,μ2,⋯,μn⨁Fμ.
By Theorem15 and Theorem 16, we can obtain Theorem 17 as follows:Theorem 17.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be collection of IVTSFNs, and let w is weight vector; if r>1 is real number, then
(35)IVTSFFWAr.Fμ1⨁Fμ,r.Fμ2⊕Fμ,⋯,r.Fμn⨁Fμ=r.FIVTSFFWAμ1,μ2,⋯,μn⨁Fμ.Theorem 18.
Letμj=sjl,sju,ijl,iju,djl,dju and φj=ajl,aju,bjl,bju,cjl,cjuj=1,2,3,⋯,n be two collections of IVTSFNs, and let w be weight vector, thenIVTSFFWAμ1⨁Fφ1,μ2⊕Fφ2,⋯μn⨁Fφn=IVTSFFWAμ1,μ2,⋯,μn⨁FIVTSFFWAφ1,φ2,⋯,φn
Now we shall investigate some special cases of IVTSFFWA operator in following.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be collection of IVTSFNs, and letw then
(1)
The IVTSFFWA operator is reduced to IVTSFWA by applyingθ⟶1.(36)limθ⟶1IVTSFFWAμ1,μ2,⋯,μn=IVTSFWAμ1,μ2,⋯,μn=1−∏j=1n1−sjlqwjq,1−∏j=1n1−sjuqwjq,∏j=1nijlqwj,∏j=1nijuqwj,∏j=1ndjqwj,∏j=1ndjqwj.(2)
The IVTSFFWA operator is reduced into traditional arithmetic weighted average operator whenθ⟶+∞:
(37)limθ⟶+∞IVTSFFWAμ1,μ2,⋯,μn=∑j=1nwjsjlqq,∑j=1nwjsjuqq,∑j=1nwjijlq,∑j=1nwjijuq,∑j=1nwjdjlq,∑j=1nwjdjuq.


## 5. IVTSF Frank Geometric Aggregation Operator

In this section, on the basis of the IVTSFFWA operator and the geometric mean, we define IVTSFFWG operator, as follows:Definition 19.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be collection of IVTSFNs, and let w be weight vector. Then, IVTSFFWG operator is a mapping Gn⟶G such that
(38)IVTSFFWGμ1,μ2,⋯,μn=μ1∧Fw1⨂Fμ2∧Fw2⨂F⋯⨂Fμ2∧Fw2.
Particularly, ifw=1/w1,1/w2,⋯,1/wnT, then the IVTSFFWG operator reduces to the IVTSFFG operator:
(39)IVTSFFGμ1,μ2,⋯,μn=μ1⨂Fμ2⨂F⋯⨂Fμn.
Similar to the IVTSFFWA operator, the IVTSFFWG operator have some basic properties, which are discussed in following. Note that most of the properties of IVTSFFWG operator are similar to the IVTSFFWA operator. That is why, we state only the properties which are different.Theorem 20.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be collection of IVTSFNs, and let w be the weight vector; if r>1 is real number, then
(40)IVTSFFWGμ1∧Fr,μ2∧Fr,⋯,μn∧Fr=IVTSFFWGμ1,μ2,⋯,μn∧Fr.Theorem 21.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be collection of IVTSFNs, and let w be weight vector; if μ=sl,su,il,iu,dl,du is a IVTSFN, then
(41)IVTSFFWGμ1⨂Fμ,μ2⨂Fμ,⋯μn⨂Fμ=IVTSFFWGμ1,μ2,⋯,μn⨂Fμ.Theorem 22.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be collection of IVTSFNs, and let w be weight vector; if r>1 is real number, then
(42)IVTSFFWGμ1∧Fr⨂Fμ,μ2∧Fr⨂Fμ,⋯,μn∧Fr⨂Fμ=IVTSFFWGμ1,μ2,⋯,μn∧Fr⨂Fμ.Theorem 23.
Letμj=sjl,sju,ijl,iju,djl,dju and φj=ajl,aju,bjl,bju,cjl,cjuj=1,2,3,⋯,n be two collections of IVTSFNs, and let w be the weight vector, then IVTSFFWGμ1⨂Fφ1,μ2⨂Fφ2,⋯μn⨂Fφn=IVTSFFWGμ1,μ2,⋯,μn⨂FIVTSFFWGφ1,φ2,⋯,φn.
We next give some special cases of IVTSFFWG operator, which are similar to the IVTSFFWA operator, as follows:
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be collection of IVTSFNs, and let w be the weight vector, then
(1)
The IVTSFFWG operator is reduced to IVTSFWG operator by applyingθ⟶1.
(43)limθ⟶1IVTSFFWGμ1,μ2,⋯,μn=IVTSFWGμ1,μ2,⋯,μn=∏j=1nsjlqwj,∏j=1nsjuqwj,1−∏j=1n1−ijlqwjq,1−∏j=1n1−ijuqwjq,1−∏j=1n1−djlqwjq,1−∏j=1n1−djuqwjq.(2)
The IVTSFFWG operator is reduced into traditional weighted geometric operator whenθ⟶+∞:
(44)limθ⟶+∞IVTSFFWGμ1,μ2,⋯,μn=∑j=1nwjsjlq,∑j=1nwjsjuq,∑j=1nwjijlqq,∑j=1nwjijuqq,∑j=1nwjdjlqq,∑j=1nwjdjuqq.
In the last of this section, we give relationship between the IVTSFFWA and the IVTSFFWG operator, which is given asTheorem 24.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be set of IVTSFNs, and let w be the weight vector, then
(45)IVTSFFWAμ1p,μ2p,⋯,μnp=IVTSFFWGμ1,μ2,⋯,μnp,IVTSFFWGμ1p,μ2p,⋯,μnp=IVTSFFWAμ1,μ2,⋯,μnp.Lemma 25.
Letzj>0,δj>0,j=1,2,3,⋯,n and∑j=1nδj=1, then
(46)∏j=1nzjδj≤∑j=1nδjzj,where ∏j=1nzjδj=∑j=1nδjzj if and only ifz1=z2=⋯=zn.Theorem 26.
Letμj=sjl,sju,ijl,iju,djl,djuj=1,2,3,⋯,n be set of IVTSFNs, and let w the weight vector, then
(47)IVTSFWGμ1,μ2,⋯,μn≤IVTSFWAμ1,μ2,⋯,μn.


## 6. Application of Proposed Approach in MAGDM

In this section, we will explore the application of IVTSFFAOs in MAGD problem with IVTSF information.For a MAGDM problem with IVTSF information, letZ=z1,z2,⋯,zm is set of m alternatives to be selected; let S=s1,s2,⋯,sm be set of attributes having weight vector w and D=d1,d2,⋯,dk are k decision-makers having weight vector ω. Let B~=b~ghm×n be the Atanassov’s IVTSF decision matrix, where the attribute value provided by decision-maker dk is b~gh=s~ghl,s~ghu,i~ghl,i~ghu,d~ghl,d~ghu denoted by anIVTSFN for the alternative zj with respect to attribute sl, where the degree range of satisfaction of attribute sl by the alternative zj is denoted by s~gh, d~gh is degree range that an alternative does not satisfy an attribute, and i~gh is degree range that of hesitancy for an alternative with the condition0≤s~ghlq+i~ghlq+d~ghlq≤1,g=1,2,⋯,m,h=1,2,..n.Generally, there are two types of attributes, benefit attributes and cost attributes in MAGDM problems. Different types of attributes values have different impact. Different impact of attribute values should be eliminated by transformation of attributes into same type. We transform the cost attributes values in benefit attribute values without loss of generality, and we obtainRE=r~ghm×n, where
(48)r~gh=b~ghforbenefitattributeb~ghcforcostattribute.Then we develop an approach to MAGDM to IVTSF information by utilizing IVTSFFWA/IVTSFFWG operator, which has following steps:Step 1.
We investigate the IVTSF information for parameterq.Step 2.
We take the decision information from given decision matrix and utilize IVTSFFWA/IVTSFFWG operator; the individual total IVTSF preference valuer~jk of alternative zj is derived as follows; r~ghk=s~ghl,s~ghu,i~ghl,i~ghu,d~ghl,d~ghu=IVTSFFWAr~gh1,r~gh2,⋯,r~ghn or r~ghk=s~ghl,s~ghu,i~ghl,i~ghu,d~ghl,d~ghu=IVTSFFWGr~gh1,r~gh2,⋯,r~ghn for aggregation of all individual IVTSF decision matrices into collective IVTSF decision matrix.Step 3.
In this step, we aggregate individual all the preference values to obtain collective overall preference value using IVTSFFWA or IVTSFFWG operator.Step 4.
By using ranking method, we rank all the obtained values.Step 5.
Finally, we select the best alternative by ranking all of them, the best one in ranking, the better the alternative.


## 7. Illustrative Example

In this section of our article, we use an illustrative example for the selection of a business proposal to support proposed approach. We shall solve this example to select the best business proposal among some business proposals.Example 1.
A company desires to select a new business proposal in order to invest in new field. After initial screening, four alternativeszi=i=1,2,3,4 have remained in list. Three experts εk=1,2,3 form a decision-maker committee having weight vector w=0.25,0.40,0.35T. Four considered attributes are (1) time duration ci, (2) cost of investment c2, (3) requirement of equipment c3, and (4) expected rate of return c4 having ω=0.25,0.3,0.35,0.2T. Three IVTSF decision matrix are obtained after evaluating proposals by experts. Table 1 has been obtained by expert 1, Table 2 by expert 2, and Table 3 by expert 3, as follows;Table 1
Decision matrix D1 provided by expert 1.
z1z2slsuiliudlduslsuiliudlduc10.230.350.430.50.550.60.50.620.340.450.30.56c20.450.50.680.70.770.80.550.70.240.360.30.43c30.230.450.340.390.490.610.430.540.330.360.20.45c40.670.70.340.430.770.810.540.650.360.490.20.6z3z4slsuiliudlduslsuiliudlduc10.50.720.20.40.40.50.70.80.340.650.320.45c20.440.670.30.70.40.560.60.750.430.460.440.56c30.320.540.50.60.20.390.40.560.530.650.430.57c40.50.760.40.60.30.650.50.760.40.560.570.67Table 2
Decision matrix D2 provided by expert 2.
z1z2slsuiliudlduslsuiliudlduc10.60.70.20.320.20.50.50.70.320.460.20.34c20.50.60.30.320.30.350.40.60.440.520.50.56c30.20.40.350.360.60.70.350.450.310.620.60.71c40.40.50.40.470.40.520.530.670.370.620.30.35z3z4slsuiliudlduslsuiliudlduc10.50.40.50.30.440.50.20.520.40.450.340.45c20.50.80.810.430.530.20.540.630.350.540.450.56c30.50.70.730.450.560.540.40.520.340.540.440.67c40.50.50.620.530.650.430.150.420.360.560.540.74Table 3
Decision matrix D3 provided by expert 3.
z1z2slsuiliudlduslsuiliudlduc10.60.70.30.550.20.340.40.490.430.540.20.43c20.50.60.40.530.30.430.430.560.230.430.50.6c30.20.50.290.350.60.670.420.620.420.540.60.8c40.40.50.340.460.40.510.50.670.520.720.30.6z3z4slsuiliudlduslsuiliudlduc10.40.550.20.350.50.650.30.470.260.50.30.45c20.450.560.250.450.20.450.40.550.290.490.230.47c30.540.670.520.70.20.540.420.560.310.450.340.43c40.390.540.290.440.30.440.320.510.330.390.410.57Attributec2 is of the cost type, and all others are benefit type, the attributes have two different types, and thus, we need to normalize the decision matrix (see Tables 4, 5, and 6). To normalize the matrix, we take compliment of the cost type attribute. In Tables 4, 5, and 6, we took the compliment of the attributec2.Table 4
Decision matrix R1.
z1z2slsuiliudlduslsuiliudlduc10.230.350.430.50.550.60.50.620.340.450.30.56c20.770.80.680.70.450.50.30.430.240.360.550.7c30.230.450.340.390.490.610.430.540.330.360.20.45c40.670.70.340.430.770.810.540.650.360.490.20.6z3z4slsuiliudlduslsuiliudlduc10.50.720.20.40.40.50.70.80.340.650.320.45c20.40.560.30.70.440.670.60.750.430.460.440.56c30.320.540.50.60.20.390.40.560.530.650.430.57c40.50.760.40.60.30.650.50.760.40.560.570.67Table 5
Decision matrix R2.
z1z2slsuiliudlduslsuiliudlduc10.60.70.20.320.20.50.50.70.320.460.20.34c20.30.350.30.320.50.60.50.560.440.520.40.65c30.20.40.350.360.60.70.350.450.310.620.60.71c40.40.50.40.470.40.520.530.670.370.620.30.35z3z4slsuiliudlduslsuiliudlduc10.40.50.30.440.50.650.20.520.40.450.340.45c20.20.450.430.530.80.810.50.560.350.540.540.63c30.70.730.450.560.540.650.40.520.340.540.440.67c40.50.620.530.650.430.550.150.420.360.560.540.74Table 6
Decision matrix R3.
z1z2slsuiliudlduslsuiliudlduc10.60.70.30.550.20.340.40.490.430.540.20.43c20.30.430.40.530.50.60.50.60.230.430.430.56c30.20.50.290.350.60.670.420.620.420.540.60.8c40.40.50.340.460.40.510.50.670.520.720.30.6z3z4slsuiliudlduslsuiliudlduc10.40.550.20.350.50.650.70.80.340.650.320.45c20.20.450.250.450.450.560.60.750.430.460.440.56c30.540.670.520.70.20.540.30.470.260.50.30.45c40.390.540.290.440.30.440.410.570.330.390.320.51


### 7.1. Rank the Alternatives by the Developed Approach

In this section, we use both IVTSFFWA and IVTSFFWG operators to solve the MADM problem and see the impact of the variable parameters on ranking.


#### 7.1.1. By Using IVTSFFWA Operator

In the following, IVTSFFWA operator is used to obtain optimal alternative, which involves following steps:Step 1.
First of all, we aggregate all the individual values of the IVTSF decision matrix without loss of generality by using IVTSFFWA operator (letθ=2). The preferences values are obtained, given in Table 7:Step 2.
Secondly, we apply IVTSFFWA operator on preferences values to get collective preference valuerj for the alternative zj which are given as follows in Table 8:Step 3.
We calculate the score values of obtained collective preference values by using Definition2 of Section 2, as follows:
(49)scz1=0.158,scz2=0.147,scz3=0.146andscz4=0.144.
Sincescz1>scz2>scz3>scz4, thus z1≻z2≻z3≻z4, where "≻" denotes “be superior to.” Therefore, z1 is the best proposal.Table 7
Decision matrix obtained after IVTSFFWA operator.
z1z2slsuiliudlduslsuiliudlduc10.5620.8200.2850.4590.2450.4280.45600.79590.3750.4960.2160.423c20.5040.7820.4090.4830.4890.5780.47320.7830.2820.4390.4420.612c30.2060.7560.3160.3600.5760.6660.40390.78430.3650.5200.4850.692c40.4860.7810.3570.4560.4580.5600.51760.82200.4360.6390.2760.512z3z4slsuiliudlduslsuiliudlduc10.42430.79100.22590.38510.47830.61740.45180.79500.31230.51120.31550.45c20.26860.75910.300.51750.53640.65090.60700.71730.371300.34310c30.5790.82400.49410.63560.27040.53610.41020.77930.35520.51230.38510.5212c40.45230.8050.37130.52760.33440.5090.3500.78640.35200.46810.47620.6378Table 8
Collective preferences values.
z1z2slsuiliudlduslsuiliudlduri0.46690.79430.3220.43040.42970.55430.46670.80630.33360.51690.3450.5596z3z4slsuiliudlduslsuiliudlduri0.4780.80960.33240.53220.36300.5600.49380.78050.33000.35410


#### 7.1.2. By Using IVTSFFWG Operator

In the following, we IVTSFFWG operator is used to get optimal alternative, which involves following steps:Step 1.
In the first step, we aggregate all the individual values of the IVTSF decision matrix without loss of generality by using IVTSFFWG operator (letθ=2). The preferences values are obtained in Table 9:Step 2.
In the second step, we apply IVTSFFWG operator on preferences values to get collective preference valuerj for the alternative zj which are given in Table 10, as follows:Step 3.
We calculate the score values of obtained collective preference values by Definition2 of Section 2, as follows:
(50)scz1=0.032713,scz2=0.054076,scz3=0.045683andscz4=0.039016.
Sincescz2>scz3>scz4>scz1, thus z2≻z3≻z4≻z1,where "≻" denotes “be superior to.” Therefore, z2 is the best proposal.Table 9
Decision matrix obtained after IVTSFFWG operator.
z1z2slsuiliudlduslsuiliudlduc10.49830.61320.31760.49230.34550.46770.44740.57300.38620.50200.22770.4441c20.36500.46080.47650.54090.49090.58320.4520.5500.32650.4510.45270.622c30.20560.45800.32050.36160.58180.66900.39960.54860.37640.54450.56170.7345c40.4440.53550.36690.45740.53610.61380.51670.66590.45730.65950.28520.5503z3z4slsuiliudlduslsuiliudlduc10.41840.5650.23940.39130.48350.62690.31660.5410.33010.52750.31700.45c20.22990.47030.33420.54650.61970.68570.56100.386400.34580c30.52820.65930.4970.6460.37950.55990.40980.54770.38550.53120.39420.5553c40.44180.60390.41370.55550.35020.53110.27980.52310.35510.49160.49290.6537Table 10
Collective preferences values.
z1z2slsuiliudlduslsuiliudlduri0.3140.4810.3920.4760.5250.6150.4250.5580.3910.5520.4620.645z3z4slsuiliudlduslsuiliudlduri0.3670.5550.4160.5770.4990.6200.3780.0000.3770.4750.4010.517From the above value, we obtained same ranking orders of alternatives by using IVTSFFWA and IVTSFFWG operators. We can see that the IVTSFFWA operator is based on the arithmetic average, which emphasize on the group opinion about the alternative, but the IVTSFFWG operator is based on the geometric mean which mainly emphases on the individual opinion between arguments. Hence, the IVTSFFWG gives the better results.


## 7.1. Rank the Alternatives by the Developed Approach

In this section, we use both IVTSFFWA and IVTSFFWG operators to solve the MADM problem and see the impact of the variable parameters on ranking.


### 7.1.1. By Using IVTSFFWA Operator

In the following, IVTSFFWA operator is used to obtain optimal alternative, which involves following steps:Step 1.
First of all, we aggregate all the individual values of the IVTSF decision matrix without loss of generality by using IVTSFFWA operator (letθ=2). The preferences values are obtained, given in Table 7:Step 2.
Secondly, we apply IVTSFFWA operator on preferences values to get collective preference valuerj for the alternative zj which are given as follows in Table 8:Step 3.
We calculate the score values of obtained collective preference values by using Definition2 of Section 2, as follows:
(49)scz1=0.158,scz2=0.147,scz3=0.146andscz4=0.144.
Sincescz1>scz2>scz3>scz4, thus z1≻z2≻z3≻z4, where "≻" denotes “be superior to.” Therefore, z1 is the best proposal.Table 7
Decision matrix obtained after IVTSFFWA operator.
z1z2slsuiliudlduslsuiliudlduc10.5620.8200.2850.4590.2450.4280.45600.79590.3750.4960.2160.423c20.5040.7820.4090.4830.4890.5780.47320.7830.2820.4390.4420.612c30.2060.7560.3160.3600.5760.6660.40390.78430.3650.5200.4850.692c40.4860.7810.3570.4560.4580.5600.51760.82200.4360.6390.2760.512z3z4slsuiliudlduslsuiliudlduc10.42430.79100.22590.38510.47830.61740.45180.79500.31230.51120.31550.45c20.26860.75910.300.51750.53640.65090.60700.71730.371300.34310c30.5790.82400.49410.63560.27040.53610.41020.77930.35520.51230.38510.5212c40.45230.8050.37130.52760.33440.5090.3500.78640.35200.46810.47620.6378Table 8
Collective preferences values.
z1z2slsuiliudlduslsuiliudlduri0.46690.79430.3220.43040.42970.55430.46670.80630.33360.51690.3450.5596z3z4slsuiliudlduslsuiliudlduri0.4780.80960.33240.53220.36300.5600.49380.78050.33000.35410


### 7.1.2. By Using IVTSFFWG Operator

In the following, we IVTSFFWG operator is used to get optimal alternative, which involves following steps:Step 1.
In the first step, we aggregate all the individual values of the IVTSF decision matrix without loss of generality by using IVTSFFWG operator (letθ=2). The preferences values are obtained in Table 9:Step 2.
In the second step, we apply IVTSFFWG operator on preferences values to get collective preference valuerj for the alternative zj which are given in Table 10, as follows:Step 3.
We calculate the score values of obtained collective preference values by Definition2 of Section 2, as follows:
(50)scz1=0.032713,scz2=0.054076,scz3=0.045683andscz4=0.039016.
Sincescz2>scz3>scz4>scz1, thus z2≻z3≻z4≻z1,where "≻" denotes “be superior to.” Therefore, z2 is the best proposal.Table 9
Decision matrix obtained after IVTSFFWG operator.
z1z2slsuiliudlduslsuiliudlduc10.49830.61320.31760.49230.34550.46770.44740.57300.38620.50200.22770.4441c20.36500.46080.47650.54090.49090.58320.4520.5500.32650.4510.45270.622c30.20560.45800.32050.36160.58180.66900.39960.54860.37640.54450.56170.7345c40.4440.53550.36690.45740.53610.61380.51670.66590.45730.65950.28520.5503z3z4slsuiliudlduslsuiliudlduc10.41840.5650.23940.39130.48350.62690.31660.5410.33010.52750.31700.45c20.22990.47030.33420.54650.61970.68570.56100.386400.34580c30.52820.65930.4970.6460.37950.55990.40980.54770.38550.53120.39420.5553c40.44180.60390.41370.55550.35020.53110.27980.52310.35510.49160.49290.6537Table 10
Collective preferences values.
z1z2slsuiliudlduslsuiliudlduri0.3140.4810.3920.4760.5250.6150.4250.5580.3910.5520.4620.645z3z4slsuiliudlduslsuiliudlduri0.3670.5550.4160.5770.4990.6200.3780.0000.3770.4750.4010.517From the above value, we obtained same ranking orders of alternatives by using IVTSFFWA and IVTSFFWG operators. We can see that the IVTSFFWA operator is based on the arithmetic average, which emphasize on the group opinion about the alternative, but the IVTSFFWG operator is based on the geometric mean which mainly emphases on the individual opinion between arguments. Hence, the IVTSFFWG gives the better results.


## 7.1.1. By Using IVTSFFWA Operator

In the following, IVTSFFWA operator is used to obtain optimal alternative, which involves following steps:Step 1.
First of all, we aggregate all the individual values of the IVTSF decision matrix without loss of generality by using IVTSFFWA operator (letθ=2). The preferences values are obtained, given in Table 7:Step 2.
Secondly, we apply IVTSFFWA operator on preferences values to get collective preference valuerj for the alternative zj which are given as follows in Table 8:Step 3.
We calculate the score values of obtained collective preference values by using Definition2 of Section 2, as follows:
(49)scz1=0.158,scz2=0.147,scz3=0.146andscz4=0.144.
Sincescz1>scz2>scz3>scz4, thus z1≻z2≻z3≻z4, where "≻" denotes “be superior to.” Therefore, z1 is the best proposal.Table 7
Decision matrix obtained after IVTSFFWA operator.
z1z2slsuiliudlduslsuiliudlduc10.5620.8200.2850.4590.2450.4280.45600.79590.3750.4960.2160.423c20.5040.7820.4090.4830.4890.5780.47320.7830.2820.4390.4420.612c30.2060.7560.3160.3600.5760.6660.40390.78430.3650.5200.4850.692c40.4860.7810.3570.4560.4580.5600.51760.82200.4360.6390.2760.512z3z4slsuiliudlduslsuiliudlduc10.42430.79100.22590.38510.47830.61740.45180.79500.31230.51120.31550.45c20.26860.75910.300.51750.53640.65090.60700.71730.371300.34310c30.5790.82400.49410.63560.27040.53610.41020.77930.35520.51230.38510.5212c40.45230.8050.37130.52760.33440.5090.3500.78640.35200.46810.47620.6378Table 8
Collective preferences values.
z1z2slsuiliudlduslsuiliudlduri0.46690.79430.3220.43040.42970.55430.46670.80630.33360.51690.3450.5596z3z4slsuiliudlduslsuiliudlduri0.4780.80960.33240.53220.36300.5600.49380.78050.33000.35410


## 7.1.2. By Using IVTSFFWG Operator

In the following, we IVTSFFWG operator is used to get optimal alternative, which involves following steps:Step 1.
In the first step, we aggregate all the individual values of the IVTSF decision matrix without loss of generality by using IVTSFFWG operator (letθ=2). The preferences values are obtained in Table 9:Step 2.
In the second step, we apply IVTSFFWG operator on preferences values to get collective preference valuerj for the alternative zj which are given in Table 10, as follows:Step 3.
We calculate the score values of obtained collective preference values by Definition2 of Section 2, as follows:
(50)scz1=0.032713,scz2=0.054076,scz3=0.045683andscz4=0.039016.
Sincescz2>scz3>scz4>scz1, thus z2≻z3≻z4≻z1,where "≻" denotes “be superior to.” Therefore, z2 is the best proposal.Table 9
Decision matrix obtained after IVTSFFWG operator.
z1z2slsuiliudlduslsuiliudlduc10.49830.61320.31760.49230.34550.46770.44740.57300.38620.50200.22770.4441c20.36500.46080.47650.54090.49090.58320.4520.5500.32650.4510.45270.622c30.20560.45800.32050.36160.58180.66900.39960.54860.37640.54450.56170.7345c40.4440.53550.36690.45740.53610.61380.51670.66590.45730.65950.28520.5503z3z4slsuiliudlduslsuiliudlduc10.41840.5650.23940.39130.48350.62690.31660.5410.33010.52750.31700.45c20.22990.47030.33420.54650.61970.68570.56100.386400.34580c30.52820.65930.4970.6460.37950.55990.40980.54770.38550.53120.39420.5553c40.44180.60390.41370.55550.35020.53110.27980.52310.35510.49160.49290.6537Table 10
Collective preferences values.
z1z2slsuiliudlduslsuiliudlduri0.3140.4810.3920.4760.5250.6150.4250.5580.3910.5520.4620.645z3z4slsuiliudlduslsuiliudlduri0.3670.5550.4160.5770.4990.6200.3780.0000.3770.4750.4010.517From the above value, we obtained same ranking orders of alternatives by using IVTSFFWA and IVTSFFWG operators. We can see that the IVTSFFWA operator is based on the arithmetic average, which emphasize on the group opinion about the alternative, but the IVTSFFWG operator is based on the geometric mean which mainly emphases on the individual opinion between arguments. Hence, the IVTSFFWG gives the better results.


## 8. The Influence of Parameters

The AOs discussed above depends upon two parametersθ and q. In this section, we observed the influence of these parameters on our results.


### 8.1. The Influence ofθ

It can be observed that we consideredθ=2 in our example. However, value of θ can be changed by the interest of the decision maker. In the following, Table 11 and Table 12 represent very interesting behavior of variation of ranking orders in IVTSFFWA and IVTSFFWG operator, respectively.Table 11
Variation of ranking withθ in IVTSFFWA operator.
θRanking of score values2scz1>scz2>scz3>scz43scz3>scz4>scz1>scz24scz3>scz4>scz2>scz15scz3>scz2>scz4>scz16scz3>scz2>scz4>scz17scz3>scz2>scz4>scz18scz3>scz2>scz4>scz19scz3>scz2>scz4>scz110scz3>scz2>scz4>scz115scz3>scz2>scz4>scz130scz3>scz2>scz4>scz150scz3>scz2>scz4>scz1100scz3>scz2>scz4>scz1Table 12
Variation of ranking withθ in IVTSFFG operator.
θRanking of score values2scz2>scz3>scz4>scz13scz2>scz3>scz4>scz14scz2>scz3>scz4>scz15scz2>scz3>scz4>scz16scz2>scz3>scz4>scz17scz2>scz3>scz4>scz18scz2>scz3>scz4>scz19scz2>scz3>scz1>scz410scz2>scz3>scz1>scz415scz2>scz3>scz1>scz430scz2>scz3>scz1>scz450scz2>scz3>scz1>scz4100scz2>scz3>scz1>scz4We can observe the behavior of variation of ranking values by changingθ in IVTSFFWA operator in Table 11. When we increase value of θ from 2, we note that the IVTSFFWA operator gives same ranking order. Hence, we recommend the decision-maker to use value of θ greater than2.We also plotted the variation in ranking values by changing inθ in Figure 1. It can be observed from Figure 1 that there is linear change in score values, as follows:Figure 1
Variation of ranking byθ and its effect on ranking.From above tables, it is noted that the ranking of score values do not fluctuate with the values of parameterθ also in IVTSFFWG operator. We varied θ up to 100, but we found the same ranking of alternatives as in Table 12. The decision-makers are independent to use any value of θ while using IVTSFFWG operator. In Figure 2, we can observe the stability of score values graphically, as given in following.Figure 2
Variation in ranking byθ and its effect on ranking.


### 8.2. The Influence ofq

The variation in ranking of score values is not influenced only byθ; it also depends upon the value of q. The variation of ranking by changing the value of q is given in following tables:In Table13, we can observe the variation in ranking of score values of alternatives using IVTSFFWA operator. As the framework of IVTSFS has a variable parameter "q", and the value of "q" can be assigned the by decision makers subject to the staisfaction of essential conditions. However, we may get some stable results after q=6. It is also noted that if decision-maker assigns significantly large value, then the results can be impossible.Table 13
Variation of ranking withq in IVTSFFWA operator.
qRanking of score values3scz1>scz2>scz3>scz44scz4>scz3>scz2>scz15scz4>scz3>scz1>scz26scz4>scz1>scz3>scz27scz4>scz1>scz3>scz28scz4>scz1>scz3>scz29scz4>scz1>scz3>scz210scz4>scz1>scz3>scz215scz4>scz1>scz3>scz2The variation behavior of ranking of score values of alternatives can be presented in Figure3 graphically, as follows:Figure 3
Variation in ranking byq and its effect on ranking.Unlike to variation in IVTSFFWA operator, the variations of ranking values in IVTSFFWG operator byq are observed in Table 14. It is observed that in IVTSFFWG operator, the results are stable for all values of q. However, we also observed these results can be impossible on significantly large value of q. The variation of ranking of score values is presented graphically in Figure 4, as the following:Table 14
Variation of ranking withq in IVTSFFWG operator.
qRanking of score values3scz2>scz3>scz4>scz14scz2>scz3>scz1>scz45scz2>scz3>scz1>scz46scz2>scz3>scz1>scz47scz2>scz3>scz1>scz48scz2>scz3>scz1>scz49scz2>scz3>scz1>scz410scz2>scz3>scz1>scz415scz2>scz3>scz1>scz4Figure 4
Variation ofq and its effect on ranking.


## 8.1. The Influence ofθ

It can be observed that we consideredθ=2 in our example. However, value of θ can be changed by the interest of the decision maker. In the following, Table 11 and Table 12 represent very interesting behavior of variation of ranking orders in IVTSFFWA and IVTSFFWG operator, respectively.Table 11
Variation of ranking withθ in IVTSFFWA operator.
θRanking of score values2scz1>scz2>scz3>scz43scz3>scz4>scz1>scz24scz3>scz4>scz2>scz15scz3>scz2>scz4>scz16scz3>scz2>scz4>scz17scz3>scz2>scz4>scz18scz3>scz2>scz4>scz19scz3>scz2>scz4>scz110scz3>scz2>scz4>scz115scz3>scz2>scz4>scz130scz3>scz2>scz4>scz150scz3>scz2>scz4>scz1100scz3>scz2>scz4>scz1Table 12
Variation of ranking withθ in IVTSFFG operator.
θRanking of score values2scz2>scz3>scz4>scz13scz2>scz3>scz4>scz14scz2>scz3>scz4>scz15scz2>scz3>scz4>scz16scz2>scz3>scz4>scz17scz2>scz3>scz4>scz18scz2>scz3>scz4>scz19scz2>scz3>scz1>scz410scz2>scz3>scz1>scz415scz2>scz3>scz1>scz430scz2>scz3>scz1>scz450scz2>scz3>scz1>scz4100scz2>scz3>scz1>scz4We can observe the behavior of variation of ranking values by changingθ in IVTSFFWA operator in Table 11. When we increase value of θ from 2, we note that the IVTSFFWA operator gives same ranking order. Hence, we recommend the decision-maker to use value of θ greater than2.We also plotted the variation in ranking values by changing inθ in Figure 1. It can be observed from Figure 1 that there is linear change in score values, as follows:Figure 1
Variation of ranking byθ and its effect on ranking.From above tables, it is noted that the ranking of score values do not fluctuate with the values of parameterθ also in IVTSFFWG operator. We varied θ up to 100, but we found the same ranking of alternatives as in Table 12. The decision-makers are independent to use any value of θ while using IVTSFFWG operator. In Figure 2, we can observe the stability of score values graphically, as given in following.Figure 2
Variation in ranking byθ and its effect on ranking.


## 8.2. The Influence ofq

The variation in ranking of score values is not influenced only byθ; it also depends upon the value of q. The variation of ranking by changing the value of q is given in following tables:In Table13, we can observe the variation in ranking of score values of alternatives using IVTSFFWA operator. As the framework of IVTSFS has a variable parameter "q", and the value of "q" can be assigned the by decision makers subject to the staisfaction of essential conditions. However, we may get some stable results after q=6. It is also noted that if decision-maker assigns significantly large value, then the results can be impossible.Table 13
Variation of ranking withq in IVTSFFWA operator.
qRanking of score values3scz1>scz2>scz3>scz44scz4>scz3>scz2>scz15scz4>scz3>scz1>scz26scz4>scz1>scz3>scz27scz4>scz1>scz3>scz28scz4>scz1>scz3>scz29scz4>scz1>scz3>scz210scz4>scz1>scz3>scz215scz4>scz1>scz3>scz2The variation behavior of ranking of score values of alternatives can be presented in Figure3 graphically, as follows:Figure 3
Variation in ranking byq and its effect on ranking.Unlike to variation in IVTSFFWA operator, the variations of ranking values in IVTSFFWG operator byq are observed in Table 14. It is observed that in IVTSFFWG operator, the results are stable for all values of q. However, we also observed these results can be impossible on significantly large value of q. The variation of ranking of score values is presented graphically in Figure 4, as the following:Table 14
Variation of ranking withq in IVTSFFWG operator.
qRanking of score values3scz2>scz3>scz4>scz14scz2>scz3>scz1>scz45scz2>scz3>scz1>scz46scz2>scz3>scz1>scz47scz2>scz3>scz1>scz48scz2>scz3>scz1>scz49scz2>scz3>scz1>scz410scz2>scz3>scz1>scz415scz2>scz3>scz1>scz4Figure 4
Variation ofq and its effect on ranking.


## 9. Comparison with Other Operators

Ullah et al. [7] developed aggregation operators for the environment of IVTSFSs based on which they investigated the evaluation of an investment policy. Jin et al. [30] proposed Hamacher aggregation operator for the environment of IVTSFSs and applied in MAGDM. Ullah et al. [35] introduced aggregation operator for IVTSFSs based on Dombi t-norm and t-conorm. In the following, we will study comparison of our proposed AOs with approaches mentioned above, and then we will mention advantages of our proposed AOs. To do this, let us solve Example 1 of Section 7 by AOs in [7, 30, 35]; we get the ranking of score values of alternatives which is given in Table 15 and Table 16. By using AOs in [7, 35], we note z3 alternative is the best. According to AOs in [30] z1 is the best alternative. But the ranking of alternatives depends upon the value of parameters and changed up to specific value. While using our proposed approach, it is noted that the ranking of alternatives is more stable as in Tables 11, 12, and 13 and Table 14. This describes the significance of our proposed AOs. In addition, the proposed approach has following advantages:
(1)
The AOs defined in [7] simply aggregate the IVTSF information, but IVTSFFWA and IVTSFFWG operators are based on Frank t-norm and t-conorm; hence, these AOs have more flexibility(2)
As Frank t-norm and t-conorm are more flexible than the Dombi and Hamacher pair of norms, hence IVTSFFWA and IVTSFFWG operators are the more appropriate AOs to aggregate IVTSF information(3)
Since IVTSFFWA and IVTSFFWG operators have two parametersθ and q, hence the decision-makers can assign the value of their own interest (studied in Subsection 8.1).Table 15
Comparison with IVTSFFWA operator.
OperatorRanking of score valuesIVTSFWAscz3>scz2>scz1>scz4IVTSFHWAscz1>scz4>scz3>scz2IVTSFDWAscz3>scz2>scz1>scz4Table 16
Comparison with IVTSFFWG operator.
OperatorRanking of score valuesIVTSFWGscz2>scz3>scz1>scz4IVTSFHWGscz2>scz3>scz1>scz4IVTSFDWGscz2>scz3>scz1>scz4The comparison of IVTSFFWA operator with IVTSF Hamacher (IVTSFHWA), IVTSF Dombi (IVTSFDWA), and IVTSF weightage averaging (IVTSFWA) operator is given following in Table15.Similarly, the comparison of the geometric operators is given following in Table16:We also plot the score values obtained by different operators mentioned in Tables15 and 16, as follows in Figures 5 and 6:Figure 5
Graph between averaging operators.Figure 6
Graph between geometric operators.


## 10. Conclusion

In this paper, we have studied Frank AOs for the atmosphere of IVTSFSs in detail. To do this, first, we prolonged the Frank t-norm and t-conorm in the environment of IVTSFSs, defined some new operations for the environment of IVTSFNs, and studied their properties in detail. We introduced new AOs such as IVTSFFWA operator and IVTSFFWG operator which are based on Frank t-norm t-conorm. Numerous properties and the relation between these AOs have investigated in detail. These AOs have applied to the MAGDM problem with the help of an illustrative example. The variations of ranking of alternatives have observed by the change in values of parametersq and θ, i.e.:
(i)
When we change the value ofθ, the ranking of the score values obtained by the IVTSFFWA operator remains unchanged after θ=2. The ranking of alternatives slightly differs atθ=2(ii)
Interestingly, the IVTSFFWG operator gives the same ranking of score values for all values ofθ(iii)
When we change the values ofq, the ranking of score values of alternatives slightly differ up to q=5. For the values of q>5, the ranking remains the same. So q>5 is recommended value for decision-makers, while using the IVTSFFWA operator. However, it is also noted that the score values approach zero for a significantly large value ofq(iv)
We noted that the ranking of score values of alternatives remain unchanged when we varied the value ofq in the IVTSFFWG operator. Hence the decision-makers are independent of the values of q while using the IVTSFFWG operator. However, the score values also approach zero for significantly large values ofqThe behavior of variation in the ranking of alternatives with the change of parameters has also been represented graphically. In the last, these AOs have been compared to Dombi, Hamacher, and traditional AOs. In the future we aim to extend this idea of Frank AOs in the environment of PFS, spherical fuzzy sets [6], complex TSFSs [46], and some decision theories [47]. In the future, we shall also study the application of IVTSFFAOs in other fields except for MAGDM like game theory, uncertain programming, and pattern recognition due to their flexibility and interesting properties.



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*Source: 2880340-2022-04-08.xml*