# Intuitionistic Fuzzy Prioritized Aggregation Operators Based on Priority Degrees with Application to Multicriteria Decision-Making

Authors: Ahmad N. Al-Kenani; Rukhshanda Anjum; Sahidul Islam
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In practise, intuitionistic fuzzy numbers (IFNs) are particularly useful for describing ambiguous data. We look at multicriteria decision-making (MCDM) problems with a prioritising relationship between the parameters. The concept of a priority degree is presented. The aggregation operators (AOs) are formed by assigning nonnegative real numbers to stringent priority levels, known as priority degrees. As a result, we construct “intuitionistic fuzzy prioritized averaging operator with priority degrees” and “intuitionistic fuzzy prioritized geometric operator with priority degrees,” which are both prioritized operators. The attributes of the existing method are frequently compared to those of other current approaches, stressing the superiority of the provided work over other methods now in use. In addition, the impact of priority degrees on the overall result is thoroughly investigated. Furthermore, in the intuitionistic fuzzy set (IFS) context, a decision-making strategy is proposed based on these operators. To highlight the efficacy of the proposed approach, an illustrative example relating to the selection of the best choice is considered.


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## Body

## 1. Introduction

Aggregation operators (AOs) are used in a large number of practical multicriteria decision-making (MCDM) situations. Many systems rely on data aggregation and fusion, including machine learning, decision-making, image processing, and pattern recognition. The aggregation strategy, in a broader sense, combines various bits of data to arrive at a result or judgement. It has also been revealed that modelling working situations in human cognition mechanisms using simple data handling algorithms based on crisp integers is problematic. As a result of these techniques, decision-makers (DMs) are left with cloudy conclusions and confusing decisions. As a result, in order to cope with unclear and fuzzy circumstances that occur in the world, DMs demand a new ideology that allows them to comprehend ambiguous data values and sustain their decision making requirements in accordance with the situation. In this regard, Zadeh [1] has revolutionized the use of a fuzzy set theory to represent ambiguous data. Atanassov [2] revealed the notion of the intuitionistic fuzzy set (IFS).Aggregation of data is important for decision-making corporate, administrative, social, medical, technological, psychological, and artificial intelligence fields. Awareness of the alternative has traditionally been seen as a crisp number or linguistic number. However, due to its uncertainty, the data cannot easily be aggregated. AOs, in fact, have a significant role in the context of MCDM issues the main goal of which is to aggregate a series of input to a single number. Ye [3] introduced the operational laws of single-valued neutrosophic sets (SVNSs) and suggested geometric and averaging AOs for SVNNs in this direction. Peng et al. [4] proposed upgraded SVNN operations and established their associated AOs. Nancy and Garg [5] established AOs by employing Frank operations. Liu et al. [6] created some AOs for SVNNs based on Hamacher operations. Zhang et al. [7] provided the AOs in the context of an interval-valued neutrosophic set. Li et al. [8] presented the novel idea of generalized simplified neutrosophic Einstein AOs. Wei and Wei [9] developed Dombi prioritized AOs for SVNSs. Liu [10] gave the idea of AOs based on archimedean t-norm and t-conorm for SVNSs. Garg and Nancy [11] gave the novel idea of prioritized muirhead mean AOs under NSs. AOs such as averaging and geometric operators for IFSs were proposed by Xu et al. [12–14]. Many studies extended aggregation operators to various fuzzy sets: Mahmood et al. [15], Wei et al. [16], Jose and Kuriaskose [17], Feng et al. [18], and Wang and Liu [19]. Liu and Liu [20] initiated the idea of q-rung orthopair (q-ROF) Bonferroni mean AOs. Liu et al. [21] proposed the idea of q-ROF Heronian mean AOs and application related to MCDM. Garg and Rani [22] constructed Bonferroni mean AOs for complex IFS and applied them to MCDM. Akram et al. [23] invented the linguistic q-ROF Einstein graph and applied it to real-world problems. Yager [24] introduced many prioritized AOs. Li and Xu [25] gave a novel idea of prioritized AOs based on the PDs. Wang et al. [26] gave the notion of power Heronian mean AOs related to q-ROFSs and their application towards MCDM. Rani and Garg [27] initiated the concept of complex intuitionistic fuzzy power AOs and their application to MCDM. Liang et al. [28] developed MULTIMOORA with interval-valued PFSs. Liu and Qin [29] introduced Maclaurin symmetric mean AOs related to IFSs. Gul [30] developed the notion of Fermatean fuzzy SAW, ARAS, and VIKOR with applications in COVID-19 testing laboratory selection problem. Ye et al. [31] introduced MCDM method based on fuzzy rough sets. Mu et al. [32] developed power Maclaurin symmetric mean AOs based on interval-valued Pythagorean fuzzy set. Batool et al. [33] gave the idea of Pythagorean probabilistic hesitant fuzzy AOs. Riaz et al. [34] introduced novel approach for third-party reverse logistic provider selection process under linear Diophantine fuzzy framework. Khan et al. [35] proposed new ranking technique for q-ROFSs based on the novel score function. Kamaci [36] proposed the idea of linguistic single-valued neutrosophic soft sets. Ashraf and Abdullah [37] presented some AOs related to the spherical fuzzy set. Karaaslan and Ozlu [38] introduced some correlation coefficients of dual type-2 hesitant fuzzy sets.In our daily lives, we come across numerous situations where a mathematical function capable of reducing a collection of numbers to a single number is needed. As a result, the AO inquiry plays a significant role in MCDM problems. Because of their broad use in fields, many researchers have recently focused on how to aggregate data. However, we often come across cases where the points to be aggregated have a strict prioritization relationship. For example, if we want to buy a plot of land to build a house based on the parameters of utility accessP1, location P2, and cost P3, we do not want to pay utility access for location and location for cost. That is, in this situation, there is strict prioritization among parameters, such as P1>P2>P3, where > indicates preferred to. To deal this type of problem, Yu and Xu [39] introduced prioritized AOs with IFSs.The concept of deciding priority degree (PD) among priority orders expands the versatility of the prioritized operators. The DM should choose the PD vector based on his or her preferences and the nature of the problem. Consider the preceding example of purchasing a plot to further illustrate the concept of PDs. Each priority level will be assigned a PD, which will be a true nonnegative number. SinceP1>P2>P3in the preceding case, the first priority order P1>P2 is given a PD d1 where 0<d1<∞ and this prioritization relationship is written as P1>d1P2. Correspondingly, the PD d2 is allocated to the second priority order P2>d2P3 and 0<d2<∞. As a result, a two-dimensional vector d=d1,d2 is assigned to the prioritized criteria P1>P2>P3, and the relationship is expressed as P1>d1P2>d2P3. Now, we will look at three particular situations involving PDs:
(1)
If the first parameter is to be given top priority, the first PDd1 should be given a large value. Furthermore, we will illustrate in this paper that when d1⟶∞, the consolidated value is calculated solely by the first criterion, with the other criterion values being ignored(2)
If we consider the PD vector to be zero, we can see that all of the parameters become equally as important, and no prioritization among the parameters remains(3)
There is natural prioritization among the parameters if each PD is equal to one. We will show Yu and Xu [39] proposed AOs and our proposed AOs based on PD is sameTaking into consideration the superiority of the IFNs set over the other sets (as discussed above) for dealing with MCDM issues, there is a need to build some new prioritized AOs based on PDs. To the best of our knowledge, no work has been performed in the context of establishing such operators that take PDs into account among strict priority levels in a IFS framework.The rest of this article is arranged as follows. Section2 contains several fundamental IFS notions. In Section 3, we looked at how the IF prioritized AOs based on the priority vector are working. In Section 4, we offer an approach for solving MCDM problems based on new AOs. In Section 5, you will find an application for selecting the agriculture land. Section 6 concludes with some final thoughts and recommendations for the future.


## 2. Certain Fundamental Concepts

In this section of the paper, we keep in mind a few basics and operational principles of IFNs.Definition 1 (see [2]).
Assume IFST~ in Q is defined as
(1)T=ς,ηγTς,ℏITς:ς∈Q,where ηγT,ℏIT:Q⟶0,1 defines the MD and NMD of the alternative ς∈Q and ∀ς; we have
(2)0≤ηγTς+ℏITς≤1.Definition 2 (see [2]).
Letξδ1=ηγ1,ℏI1 and ξδ2=ηγ2,ℏI2 be IFNs. σ>0, Then,
(3)ξδ1c=ℏI1,ηγ1,ξδ1∨ξδ2=maxηγ1,ℏI1,minηγ2,ℏI2,ξδ1∧ξδ2=minηγ1,ℏI1,maxηγ2,ℏI2,ξδ1⊕ξδ2=ηγ12+ηγ22−ηγ12ηγ22,ℏI1ℏI2,ξδ1⊗ξδ2=ηγ1ηγ2,ℏI12+ℏI22−ℏI12ℏI22,σξδ1=1−1−ηγ12σ,ℏI1σ,ξδ1σ=ηγ1σ,1−1−ℏI12σ.Definition 3 (see [2]).
Letξδ=ηγ,ℏI be the IFN, score function Ξ of ξδ is defined as
(4)Ξξδ=ηγ−ℏI,
whereΞξδ∈−1,1. The IFN score shall decide its ranking, i.e., the maximum score shall determine the high IFN priority. In certain situations, the score function is not really beneficial for IFN. It is therefore not sufficient to use the score function to evaluate the IFNs. We are adding an additional function, i.e., an accuracy function.Definition 4 (see [2]).
Letξδ=ηγ,ℏI be the IFN, then an accuracy function H of ξδ is defines as
(5)Hξδ=ηγ+ℏI,Hξδ∈0,1.Definition 5.
Considerξδ=ηγξδ,ℏIξδ and β=ηγβ,ℏIβ are two IFNs, and Ξξδ,Ξβ are the score function of ξδ and β, and Hξδ,Hβ are the accuracy function of ξδ and β, respectively, then
(a)
IfΞξδ>Ξβ, then ξδ>β(b)
IfΞξδ=Ξβ, then
IfHξδ>Hβ then ξδ>β, and if Hξδ=Hβ, then ξδ=β.It should always be noticed that the value of score function is between –1 and 1. We introduce another score function, to support this type of research,Ξ˘R=1+ηγR−ℏIR/2. We can see that 0≤Ξ˘R≤1. This new score function satisfies all properties of score function defined in [2].Definition 6 (see [12]).
Assume thatξδg=ηγg,ℏIg is a family of IFNs, and IFWA:Λn⟶Λ, if
(6)IFWAξδ1,ξδ2,⋯ξδu=∑g=1uY^gξδg=Y^1ξδ1⊕Y^2ξδ2⊕⋯,Y^uξδu,where Λn is the set of all IFNs, and Y^=Y∧1,Y∧2,⋯,Y∧uT is the weight vector of ξδ1,ξδ2,⋯,ξδu, such that 0≤Y^u≤1 and ∑g=1uY^u=1. Then, the IFWA is called the intuitionistic weighted average operator.Based on IFN operational rules, we can also consider IFWA by the theorem below.Theorem 7 (see [12]).
Letξδg=ηγg,ℏIg be the family of IFNs,we can find IFWG by
(7)IFWAξδ1,ξδ2,⋯ξδu=1−∏g=1u1−ηγgY∧g,∏g=1uℏIgY^g.Definition 8 (see [13]).
Assume thatξδg=ηγg,ℏIg is the family of IFN, and IFWG:Λn⟶Λ, if
(8)IFWGξδ1,ξδ2,⋯ξδu=∑g=1uξδgY^g=ξδ1Y^1⊗ξδ2Y^2⊗⋯,ξδuY^u,where Λn is the set of all IFNs, and Y^=Y∧1,Y∧2,⋯,Y∧uT is weight vector of ξδ1,ξδ2,⋯,ξδu, such that 0≤Y^u≤1 and ∑g=1uY^u=1. Then, the IFWG is called the intuitionistic weighted geometric operator.Based on IFNs operational rules, we can also consider IFWG by the theorem below.Theorem 9 (see [13]).
Letξδg=ηγg,ℏIg be the family of IFNs, we can find IFWG by
(9)IFWGξδ1,ξδ2,⋯ξδu=∏g=1uηγgY^g,1−∏g=1u1−ℏIgY∧g.Definition 10 (see [39]).
Letξδg=ηγg,ℏIg be the family of IFNs, and IFPWA:Λn⟶Λ, be a n-dimension mapping. If
(10)IFPWAξδ1,ξδ2,⋯ξδu=u^˘1∑g=1uu^˘gξδ1⊕u^˘2∑g=1uu^˘gξδ2⊕⋯,⊕u^˘u∑g=1uu^˘gξδu,
then the mapping IFPWA is called intuitionistic prioritized weighted averaging (IFPWA) operator, whereu^˘j=∏k=1j−1Ξ˘ξδkj=2⋯,n, u^˘1=1 and Ξ˘˘ξδk is the score of kth IFN.Definition 11 (see [39]).
Letξδp=ηγg,ℏIp be the family of IFNs, and IFPWG:Λn⟶Λ, be a n-dimension mapping. If
(11)IFPWGξδ1,ξδ2,⋯ξδu=ξδ1u^˘1/∑g=1uu^˘g⊗ξδ2u^˘2/∑g=1uu^˘g⊗⋯,⊗ξδuu^˘2/∑g=1uu^˘g,then the mapping IFPWG is called intuitionistic prioritized weighted geometric (IFPWG) operator, where u^˘j=∏k=1j−1Ξ˘ξδkj=2⋯,n, u^˘1=1 and Ξ˘˘ξδk is the score of kth IFN.


## 3. Intuitionistic Fuzzy Prioritized Aggregation Operators with PDs

Within this section, we present the notion of intuitionistic prioritized averaging (IFPAd) operator with PDs and intuitionistic prioritized geometric (IFPGd) operator with PDs.


### 3.1.IFPAd Operator

Assumeξδg=ηγg,ℏIgg=1,2⋯,u is the assemblage of IFNs, there is a prioritization among these IFNs expressed by the strict priority orders ξδ1≻d1ξδ2≻d2⋯≻du−1ξδu−1, where ξδu≻duξδu+1 indicates that the IFN ξδu has du higher priority than ξδu+1. d=d1,d2,⋯,du−1 is the u−1-dimensional vector of PDs. The assemblage of such IFNs with strict priority orders and PDs is denoted by Rd.Definition 12.
AIFPAd operator is a mapping from Rdu to Rd and defined as
(12)IFPAdξδ1,ξδ2,⋯,ξδu=ζ1dξδ1⊕ζ2dξδ2,⋯,ζudξδu,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u, and T1=1. Then, IFPAd is called intuitionistic prioritized averaging operators with PDs.Theorem 13.
Assumeξδg=ηγg,ℏIg is the assemblage of IFNs, we can also find IFPAd by
(13)IFPAdξδ1,ξδ2,⋯,ξδu=ζ1dξδ1⊕ζ2dξδ2,⋯,ζudξδu=1−∏g=1u1−ηγgζgd,∏g=1uℏIgζgd,
whereζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u, and T1=1.Proof.
To prove this theorem, we use mathematical induction.
Foru=2,
(14)ζ1dξδ1=1−1−ηγ1ζ1d,ℏI1ζ1d,ζ2dξδ2=1−1−ηγ2ζ2d,ℏI2ζ2d.
Then,(15)ζ1dξδ1⊕ζ2dξδ2=1−1−ηγ1ζ1d,ℏI1ζ1d⊕1−1−ηγ2ζ2d,ℏI2ζ2d=1−1−ηγ1ζ1d+1−1−ηγ2ζ2d−1−1−ηγ1ζ1d1−1−ηγ2ζ2d,ℏI1ζ1d.ℏI2ζ2d=1−1−ηγ1ζ1d+1−1−ηγ2ζ2d−1−1−ηγ2ζ2d−1−ηγ1ζ1d+1−ηγ2ζ1d1−ηγ1ζ1d,ℏI1ζ1d.ℏI2ζ2d=1−1−ηγ1ζ1d1−ηγ2ζ2d,ℏI1ζ1d.ℏI2ζ2d=1−∏g=11−ηγgζgd,∏g=1uℏIgζgd.
This shows that Equation (13) is true for u=2; now, let that Equation (13) holds for u=b, i.e.,
(16)IFPAdξδ1,ξδ2,⋯ξδb=1−∏g=1b1−ηγgζgd,∏g=1bℏIgζgd.
Nowu=b+1, by operational laws of IFNs, we have
(17)IFPAdξδ1,ξδ2,⋯ξδb+1=IFPAdξδ1,ξδ2,⋯ξδb⊕ξδb+1=1−∏g=1b1−ηγgζgd,∏g=1bℏIgζgd⊕1−1−ηγb+1ζb+1d+1,ℏIb+1ζb+1d+1=1−∏g=1b1−ηγgζgd+1−1−ηγb+1ζb+1d+1−1−∏g=1b1−ηγgζgd1−1−ηγb+1ζb+1d+1,∏g=1bℏIgζgd.ℏIb+1ζb+1d+1=1−∏g=1b+11−ηγgζgd,∏g=1b+1ℏIgζgd.
This shows that foru=b+1, Equation (13) holds. Then,
(18)IFPAdξδ1,ξδ2,⋯ξδu=1−∏g=1u1−ηγgζgd,∏g=1uℏIgζgd.
☐☐Example 14.
Letξδ1=0.73,0.54, ξδ2=0.53,0.75,ξδ3=0.82,0.25, and ξδ4=0.35,0.64 be the four IFNs, there is strict prioritized relation in considered IFNs, such that ξδ1≻d1ξδ2≻d2ξδ3≻d3ξδ4. Priority vector d=d1,d2,d3 is given as 5,1,1, by Equation (13), and get
(19)1−∏g=141−ηγgζgd=0.719519,∏g=14ℏIgζgd=0.544041,IFPAdξδ1,ξδ2,ξδ3,ξδ4=1−∏g=141−ηγgζgd,∏g=1uℏIgζgd=0.719519,0.544041.Furthermore, the suggestedIFPAd operator is examined to ensure that it has idempotency and boundary properties. Their explanations are as follows.Theorem 15.
Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs, and
(20)ξδ−=mingηγg,maxgℏIg,ξδ+=maxgηγg,mingℏIg.
Then,(21)ξδ−≤IFPAdξδ1,ξδ2,⋯ξδn≤ξδ+,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u and T1=1.Proof.
Since,(22)mingηγg≤ηγg≤maxgηγg,(23)mingℏIg≤ℏIg≤maxgℏIg.
From Equation (22), we have
(24)mingηγg≤ηγg≤maxgηγg⇔mingηγg≤ηγg≤maxgηγg⇔1−maxgηγg≤1−ηγg≤1−mingηγg⇔1−maxgηγgζgd≤1−ηγgζgd≤1−mingηγgζgd⇔∏g=1u1−maxgηγgζgd≤∏g=1u1−ηγgζgd≤∏g=1u1−mingηγgζgd⇔1−maxgηγg≤∏g=1u1−ηγgζgd≤1−mingηγg⇔−1+minjηγg≤−∏g=1u1−ηγgζgd≤−1+maxgηγg⇔1−1+minjηγg≤1−∏g=1u1−ηγgζgd≤1−1+maxgηγg⇔minjηγg≤1−∏g=1u1−ηγgζgd≤maxgηγg⇔minjηγg≤1−∏g=1u1−ηγgζgd≤maxgηγg.
From Equation (23), we have
(25)mingℏIg≤ℏIg≤maxgℏIg⇔mingℏIgζgd≤ℏIgζgd≤maxgℏIgζgd⇔∏g=1umingℏIgζgd≤∏g=1uℏIgζgd≤∏g=1umaxgℏIgζgd⇔mingℏIgζgd≤∏g=1uℏIgζgd≤maxgℏIgζgd.
Let(26)IFPAdξδ1,ξδ2,⋯ξδn=ξδ=ηγ,ℏI.
Then,Ξ˘ξδ=ηγ−ℏI≤maxgηγ−minjℏI=Ξ˘ξδmax So, Ξ˘ξδ≤Ξ˘ξδmax.
Again,Ξ˘ξδ=ηγ−ℏI≥mingηγ−maxjℏI=Ξ˘ξδmin. So, Ξ˘ξδ≥Ξ˘ξδmin.
If,Ξ˘ξδ≤Ξ˘ξδmax and Ξ˘ξδ≥Ξ˘ξδmin, then
(27)ξδmin≤IFPAdξδ1,ξδ2,⋯ξδn≤ξδmax.
IfΞ˘ξδ=Ξ˘ξδmax, then ηγ−ℏI=maxgηγ−minjℏI(28)⇔ηγ−ℏI=maxgηγ−mingℏI,⇔ηγ=maxgηγ,ℏI=mingℏI,⇔ηγ=maxgηγ,ℏI=mingℏI.
Now,Hξδ=ηγ+ℏI=maxgηγ+mingℏI=Hξδmax(29)IFPAdξδ1,ξδ2,⋯ξδn=ξδmax.
IfΞ˘ξδ=Ξ˘ξδmin, then ηγ−ℏI=mingηγ−maxjℏI(30)⇔ηγ−ℏI=mingηγ−maxgℏI,⇔ηγ=mingηγ,ℏI=maxgℏI,⇔ηγ=mingηγ,ℏI=maxgℏI.
Now,Hξδ=ηγ+ℏI=mingηγ+maxgℏI=Hξδmax(31)IFPAdξδ1,ξδ2,⋯ξδn=ξδmin.
Thus, from Equations (27), (29), and (31), we get
(32)ξδ−≤IFPAdξδ1,ξδ2,⋯ξδn≤ξδ+.☐Theorem 16.
Assume that ifξδ⋄ is a IFN satisfied the property, ξδg=ξδ⋄,∀g then
(33)IFPAdξδ1,ξδ2,⋯ξδu=ξδ⋄.Proof.
Letξδ⋄=ηγ⋄,ℏI⋄ be the IFN. Then, by assumption, we have ξδg=ξδ⋄,∀g gives ηγg=ηγ⋄ and ℏIg=ℏI⋄∀g. By Definition 12, we have ∑g=1uζgd. Then, by using Theorem 13, we get
(34)IFPAdξδ1,ξδ2,⋯ξδu=1−∏g=1u1−ηγ⋄ζgd,∏g=1uℏI⋄ζgd=1−1−ηγ⋄∑g=1uζgd,ℏI⋄∑g=1uζgd=ηγ⋄,ℏI⋄=ξδ⋄.☐Corollary 17.
Ifξδg=ηγg,ℏIg is the assemblage of largest IFNs, i.e., ξδg=1,0 for all g, then
(35)IFPAdξδ1,ξδ2,⋯ξδu=1,0.Proof.
We can easily obtain a corollary similar to Theorem16.☐Corollary 18.
Ifξδ1=ηγ1,ℏI1 is the smallest IFN, i.e., ξδ1=0,1, then
(36)IFPAdξδ1,ξδ2,⋯ξδu=0,1.Proof.
Here,ξδ1=0,1 then by definition of the score function, we have
(37)Ξ˘ξδ1=0.
Since,(38)Tgd=∏q=1g−1Ξ˘ξδqdq,foreachg=2,3,⋯,uandT1=1.
We have(39)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1=0.
From Definition 3, we have(40)IFPAdξδ1,ξδ2,⋯ξδu=ζ1dξδ1⊕ζ2dξδ2,⋯,ζudξδu=1ξδ1⊕0ξδ2⊕⋯0ξδn=ξδ1=0,1.☐Theorem 19.
Assume thatξδg=ηγg,ℏIg and βg=ϕg,φg are two assemblages of IFNs, if r>0 and β=ηγβ,ℏIβ is a IFN, then
(1)
IFPAdξδ1⊕β,ξδ2⊕β,⋯ξδu⊕β=IFPAdξδ1,ξδ2,⋯ξδu⊕β(2)
IFPAdrξδ1,rξδ2,⋯rξδu=rIFPAdξδ1,ξδ2,⋯ξδu(3)
IFPAdξδ1⊕β1,ξδ2⊕β2,⋯ξδu⊕βn=IFPAdξδ1,ξδ2,⋯ξδn⊕IFPAdβ1,β2,⋯βu(4)
IFPAdrξδ1⊕β,rξδ2⊕β,⋯⊕rξδu⊕β=rIFPAdξδ1,ξδ2,⋯ξδu⊕βProof.
This is trivial by definition.☐IFPAd operator satisfied following properties.Property:1Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs, then we have
(41)limd1,d2,⋯,du−1⟶1,1,⋯,1IFPAdξδ1,ξδ2,⋯ξδu=IFPWAξδ1,ξδ2,⋯ξδu.Proof.
Given thatd1,d2,⋯,du−1⟶1,1,⋯,1, from this, we have
(42)Tgd=∏q=1g−1Ξ˘ξδqdq⟶∏q=1g−1Ξ˘ξδq=Tg,by this, we obtain ζgd⟶ζg(43)limd1,d2,⋯,du−1⟶1,1,⋯,1IFPAdξδ1,ξδ2,⋯ξδu=limd1,d2,⋯,du−1⟶1,1,⋯,1ζ1dξδ1⊕ζ2dξδ2,⋯,ζudξδu=ζ1ξδ1⊕ζ2ξδ2,⋯,ζuξδu=IFPWAξδ1,ξδ2,⋯ξδu.☐Remark.
Whend1=d2=⋯,=du−1=1, Property:1 states that the existing IFPWA operator is a particular situation of the suggested IFPAd operator. As a result, the IFPAd operator is more generic than the IFPWA operator.Property:2Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδg≠0 for all g, then we have
(44)limd1,d2,⋯,du−1⟶0,0,⋯,0IFPAdξδ1,ξδ2,⋯ξδu=1uξδ1⊕ξδ2⊕,⋯,⊕ξδu.Proof.
Given thatd1,d2,⋯,du−1⟶0,0,⋯,0, from this, we have
(45)Tgd=∏q=1g−1Ξ˘ξδqdq=1,and ζgd=Tgd/∑g=1uTgd=1/n. Hence,
(46)limd1,d2,⋯,du−1⟶0,0,⋯,0IFPAdξδ1,ξδ2,⋯ξδu=1uξδ1⊕ξδ2⊕,⋯,⊕ξδu.☐Property:3Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδ1≠0 or 1, then we have
(47)limd1⟶+∞IFPAdξδ1,ξδ2,⋯ξδu=ξδ1.Proof.
Here,d1⟶+∞ for each g=2,3,⋯,u, we have
(48)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1+∞Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1=0,
because0<Ξ˘ξδ1<1, ∑g=1uTgd=T1d=1⇒ζ1d=T1d/∑g=1uT1d=1 and ζgd=Tgd/∑g=1uTgd for each g=2,3,⋯,u. Hence,
(49)limd1⟶∞IFPAdξδ1,ξδ2,⋯ξδu=ξδ1.☐Remark.
According to Property:3, whend1⟶+∞, the PD d1 of IFN ξδ1 is very high in comparison to the PDs of other IFNs. It indicates that IFN ξδ1 is extremely essential. As a result, ξδ1 determines the aggregation result obtained by using the proposed operator IFPAd in this case.Example 20.
Letξδ1=0.73,0.54, ξδ2=0.53,0.75,ξδ3=0.82,0.25, and ξδ4=0.35,0.64 be the four IFNs, it can easily compute that, Ξ˘1=0.6158, Ξ˘2=0.3635, Ξ˘3=0.7679, and Ξ˘4=0.3904. There is strict prioritized relation in considering IFNs, such that ξδ1≻d1ξδ2≻d2ξδ3≻d3ξδ4. In the corresponding portion, we will aggregate the IFNs for 4 distinct priority vectors d=d1,d2,d3, keeping the values of PDs d2,d3 constant while varying the value of d1 and discussing its effect on the aggregated results.Case 1.
whend=1,1,1,
(50)IFPAdξδ1,ξδ2,ξδ3,ξδ4=0.684742,0.556094.Case 2.
whend=5,1,1,
(51)IFPAdξδ1,ξδ2,ξδ3,ξδ4=0.719519,0.544041.Case 3.
whend=8,1,1,
(52)IFPAdξδ1,ξδ2,ξδ3,ξδ4=0.727349,0.541027.Case 4.
whend=13,1,1,
(53)IFPAdξδ1,ξδ2,ξδ3,ξδ4=0.729769,0.540074.The consolidated findings from the preceding 4 cases show that as the PDd1 referring to IFN ξδ1 rises, the aggregated value approaches the IFN ξδ1 ranking values.Property:4Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδg≠0 for all g=1,2,⋯,k+1, and Ξ˘ξδk+1≠1, then we have
(54)limd1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞IFPAdξδ1,ξδ2,⋯ξδu=1k+1ξδ1⊕ξδ2⊕,⋯,⊕ξδk+1.Proof.
Given thatd1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞. So,
(55)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1⟶Ξ˘ξδ10Ξ˘ξδ20⋯Ξ˘ξδg−10=1,for each g=2,3,⋯,k+1.
(56)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1⟶Ξ˘ξδ10Ξ˘ξδ20⋯Ξ˘ξδk0Ξ˘ξδk+1+∞⋯Ξ˘ξδg−1dg−1=0,∀g=k+2,k+3,⋯,u.
So,(57)∑g=1uTgd=T1d=k+1andζgd=Tgd∑g=1uTgd⟶1k+1foreachg=1,2,3,⋯,k+1,ζgd=Tgd∑g=1uTgd⟶0k+1=0foreachg=k+2,k+3,⋯,u.
Hence,(58)limd1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞IFPAdξδ1,ξδ2,⋯ξδu=1k+1ξδ1⊕ξδ2⊕,⋯,⊕ξδk+1.☐Remark.
Whend1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞, it means there is no prioritization association between the IFNs ξδ1,ξδ2,⋯,ξδk+1 and that all of these IFNs ξδ1ξδ2,⋯,ξδk+1 have a much higher priority than the IFNs ξδk+2ξδk+3,⋯,ξδu. As a result, the aggregated value is solely dependent on IFNs ξδ1ξδ2,⋯,ξδk+1, and these IFNs ξδ1ξδ2,⋯,ξδk+1 have similar weightage in the aggregation method.Property:5Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδk+1≠1 or 0 then, we have
(59)limd1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞IFPAdξδ1,ξδ2,⋯ξδu=IFPWAξδ1⊕ξδ2⊕,⋯,⊕ξδk+1.Proof.
Given thatd1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞. So,
(60)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1⟶Ξ˘ξδ1Ξ˘ξδ2⋯Ξ˘ξδg−1=Tg,for each g=2,3,⋯,k+1.
(61)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1⟶Ξ˘ξδ1Ξ˘ξδ2⋯Ξ˘ξδkΞ˘ξδk+1+∞⋯Ξ˘ξδg−1dg−1=0,∀g=k+2,k+3,⋯,u.
So,
∑g=1uTgd⟶∑g=1k+1Tg and ζgd=Tgd/∑g=1uTgd⟶Tg/∑g=1k+1Tg for each g=1,2,3,⋯,k+1.
ζgd=Tgd/∑g=1uTgd⟶0/∑g=1k+1Tg=0 for each g=k+2,k+3,⋯,u.
Hence,(62)limd1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞IFPAdξδ1,ξδ2,⋯ξδu=IFPWAξδ1⊕ξδ2⊕,⋯,⊕ξδk+1.☐Remark.
Whend1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞, it means there is normal prioritization association between the IFNs ξδ1,ξδ2,⋯,ξδk+1 and that all of these IFNs ξδ1ξδ2,⋯,ξδk+1 have a much higher priority than the IFNs ξδk+2ξδk+3,⋯,ξδu. As a result, the aggregated value is solely dependent on IFNs ξδ1ξδ2,⋯,ξδk+1.


### 3.2.IFPGd Operator

Assumeξδg=ηγg,ℏIgg=1,2⋯,u is the assemblage of IFNs, there is a prioritization among these IFNs expressed by the strict priority orders ξδ1≻d1ξδ2≻d2⋯≻du−1ξδu−1, where ξδu≻duξδu+1 indicates that the IFN ξδu has du higher priority than ξδu+1. d=d1,d2,⋯,du−1 is the u−1-dimensional vector of PDs. The assemblage of such IFNs with strict priority orders and PDs is denoted by Rd.Definition 21.
AIFPGd operator is a mapping from Rdu to Rd and defined as
(63)IFPGdξδ1,ξδ2,⋯,ξδu=ξδ1ζ1d⊕ξδ2ζ2d,⋯,ξδuζud,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u and T1=1. Then, IFPGd is called intuitionistic prioritized geometric operator with PDs.Theorem 22.
Assumeξδg=ηγg,ℏIg is the assemblage of IFNs, we can also find IFPGd by
(64)IFPGdξδ1,ξδ2,⋯,ξδu=ξδ1ζ1d⊕ξδ2ζ2d,⋯,ξδuζud=∏g=1uηγgζgd,1−∏g=1u1−ℏIgζgd,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u and T1=1.Proof.
To prove this theorem, we use mathematical induction.
Foru=2,
(65)ξδ1ζ1d=ηγ1ζ1d,1−1−ℏI1ζ1d,ξδ2ζ2d=ηγ2ζ2d,1−1−ℏI2ζ2d.
Then,(66)ξδ1ζ1d⊗ξδ2ζ2d=ηγ1ζ1d,1−1−ℏI1ζ1d⊗ηγ2ζ2d,1−1−ℏI2ζ2d=ηγ1ζ1d.ηγ2ζ2d,1−1−ℏI1ζ1d+1−1−ℏI2ζ2d−1−1−ℏI1ζ1d1−1−ℏI2ζ2d=ηγ1ζ1d.ηγ2ζ2d,1−1−ℏI1ζ1d+1−1−ℏI2ζ2d−1−1−ℏI2ζ2d−1−ℏI1ζ1d+1−ℏI2ζ1d1−ℏI1ζ1d=ηγ1ζ1d.ηγ2ζ2d,1−1−ℏI1ζ1d1−ℏI2ζ2d=∏g=1uηγgζgd,1−∏g=11−ℏIgζgd.
This shows that Equation (64) is true for u=2; now, let that Equation (64) holds for u=b, i.e.,
(67)IFPGdξδ1,ξδ2,⋯ξδb=∏g=1bηγgζgd,1−∏g=1b1−ℏIgζgd.
Nowu=b+1, by operational laws of IFNs, we have
(68)IFPGdξδ1,ξδ2,⋯ξδb+1=IFPGdξδ1,ξδ2,⋯ξδb⊗ξδb+1=∏g=1bηγgζgd,1−∏g=1b1−ℏIgζgd⊗ηγb+1ζb+1d+1,1−1−ℏIb+1ζb+1d+1=∏g=1bηγgζgd.ηγb+1ζb+1d+1,1−∏g=1b1−ℏIgζgd+1−1−ℏIb+1ζb+1d+1−1−∏g=1b1−ℏIgζgd1−1−ℏIb+1ζb+1d+1=∏g=1b+1ηγgζgd,1−∏g=1b+11−ℏIgζgd.
This shows that foru=b+1, Equation (64) holds. Then,
(69)IFPGdξδ1,ξδ2,⋯ξδu=∏g=1uηγgζgd,1−∏g=1u1−ℏIgζgd.☐Example 23.
Letξδ1=0.73,0.54, ξδ2=0.53,0.75,ξδ3=0.82,0.25 and ξδ4=0.35,0.64 be the four IFNs, there is strict prioritized relation in considered IFNs, such that ξδ1≻d1ξδ2≻d2ξδ3≻d3ξδ4. Priority vector d=d1,d2,d3 is given as 5,1,1, by Equation (64)
(70)∏g=14ηγgζgd=0.7032081−∏g=141−ℏIgζgd=0.565078,IFPGdξδ1,ξδ2,ξδ3,ξδ4=∏g=1uηγgζgd,1−∏g=141−ℏIgζgd,=0.703208,0.565078.Furthermore, the suggested IFPGd operator is examined to ensure that it has idempotency and boundary properties. Their explanations are as follows.Theorem 24.
Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs, and
(71)ξδ−=mingηγg,maxgℏIg,ξδ+=maxgηγg,mingℏIg.
Then,(72)ξδ−≤IFPGdξδ1,ξδ2,⋯ξδn≤ξδ+,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u and T1=1.Proof.
Proof is same as Theorem15.☐Theorem 25.
Assume that ifξδ⋄ is a IFN satisfied the property, ξδg=ξδ⋄,∀g then
(73)IFPGdξδ1,ξδ2,⋯ξδu=ξδ⋄.Proof.
Letξδ⋄=ηγ⋄,ℏI⋄ be the IFN. Then, by assumption, we have ξδg=ξδ⋄,∀g gives ηγg=ηγ⋄ and ℏIg=ℏI⋄∀g. By Definition 21, we have ∑g=1uζgd. Then, by using Theorem 22, we get
(74)IFPGdξδ1,ξδ2,⋯ξδu=∏g=1uηγ⋄ζgd,1−∏g=1u1−ℏI⋄ζgd=ηγ⋄∑g=1uζgd,1−1−ℏI⋄∑g=1uζgd=ηγ⋄,ℏI⋄=ξδ⋄.☐Corollary 26.
Ifξδg=ηγg,ℏIg is the assemblage of largest IFNs, i.e., ξδg=1,0 for all g, then
(75)IFPGdξδ1,ξδ2,⋯ξδu=1,0.Proof.
We can easily obtain a corollary similar to Theorem25.☐Corollary 27.
Ifξδ1=ηγ1,ℏI1 is the smallest IFN, i.e., ξδ1=0,1, then
(76)IFPGdξδ1,ξδ2,⋯ξδu=0,1.Proof.
Here,ξδ1=0,1 then by definition of the score function, we have
(77)Ξ˘ξδ1=0.
Since,(78)Tgd=∏q=1g−1Ξ˘ξδqdq,foreachg=2,3,⋯,uandT1=1.
We have(79)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1=0.
From Definition 3, we have(80)IFPGdξδ1,ξδ2,⋯ξδu=ξδ1ζ1d⊗ξδ2ζ2d,⋯,ξδuζud=ξδ11⊗ξδ20⊗⋯ξδu0=ξδ1=0,1.☐Theorem 28.
Assume thatξδg=ηγg,ℏIg and βg=ϕg,φg are two assemblages of IFNs, if r>0 and β=ηγβ,ℏIβ is a IFN, then
(81)IFPGdξδ1⊕β,ξδ2⊕β,⋯ξδu⊕β=IFPGdξδ1,ξδ2,⋯ξδu⊕β,IFPGdrξδ1,rξδ2,⋯rξδu=rIFPGdξδ1,ξδ2,⋯ξδu,IFPGdξδ1⊕β1,ξδ2⊕β2,⋯ξδu⊕βn=IFPGdξδ1,ξδ2,⋯ξδn⊕IFPGdβ1,β2,⋯βu,IFPGdrξδ1⊕β,rξδ2⊕β,⋯⊕rξδu⊕β=rIFPGdξδ1,ξδ2,⋯ξδu⊕β.Proof.
This is trivial by definition.☐IFPGd operator also satisfied following properties.Property:1Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs, then we have
(82)limd1,d2,⋯,du−1⟶1,1,⋯,1IFPGdξδ1,ξδ2,⋯ξδu=IFPWGξδ1,ξδ2,⋯ξδu.Property:2Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδg≠0 for all g, then we have
(83)limd1,d2,⋯,du−1⟶0,0,⋯,0IFPGdξδ1,ξδ2,⋯ξδu=1uξδ1⊗ξδ2⊗,⋯,⊗ξδu.Property:3Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδ1≠0 or 1, then we have
(84)limd1⟶+∞IFPGdξδ1,ξδ2,⋯ξδu=ξδ1.Property:4Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδg≠0 for all g=1,2,⋯,k+1, and Ξ˘ξδk+1≠1 then we have
(85)limd1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞IFPGdξδ1,ξδ2,⋯ξδu=1k+1ξδ1⊗ξδ2⊗,⋯,⊗ξδk+1.Property:5Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδk+1≠1 or 0 then we have
(86)limd1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞IFPGdξδ1,ξδ2,⋯ξδu=IFPWGξδ1⊗ξδ2⊗,⋯,⊗ξδk+1.


## 3.1.IFPAd Operator

Assumeξδg=ηγg,ℏIgg=1,2⋯,u is the assemblage of IFNs, there is a prioritization among these IFNs expressed by the strict priority orders ξδ1≻d1ξδ2≻d2⋯≻du−1ξδu−1, where ξδu≻duξδu+1 indicates that the IFN ξδu has du higher priority than ξδu+1. d=d1,d2,⋯,du−1 is the u−1-dimensional vector of PDs. The assemblage of such IFNs with strict priority orders and PDs is denoted by Rd.Definition 12.
AIFPAd operator is a mapping from Rdu to Rd and defined as
(12)IFPAdξδ1,ξδ2,⋯,ξδu=ζ1dξδ1⊕ζ2dξδ2,⋯,ζudξδu,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u, and T1=1. Then, IFPAd is called intuitionistic prioritized averaging operators with PDs.Theorem 13.
Assumeξδg=ηγg,ℏIg is the assemblage of IFNs, we can also find IFPAd by
(13)IFPAdξδ1,ξδ2,⋯,ξδu=ζ1dξδ1⊕ζ2dξδ2,⋯,ζudξδu=1−∏g=1u1−ηγgζgd,∏g=1uℏIgζgd,
whereζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u, and T1=1.Proof.
To prove this theorem, we use mathematical induction.
Foru=2,
(14)ζ1dξδ1=1−1−ηγ1ζ1d,ℏI1ζ1d,ζ2dξδ2=1−1−ηγ2ζ2d,ℏI2ζ2d.
Then,(15)ζ1dξδ1⊕ζ2dξδ2=1−1−ηγ1ζ1d,ℏI1ζ1d⊕1−1−ηγ2ζ2d,ℏI2ζ2d=1−1−ηγ1ζ1d+1−1−ηγ2ζ2d−1−1−ηγ1ζ1d1−1−ηγ2ζ2d,ℏI1ζ1d.ℏI2ζ2d=1−1−ηγ1ζ1d+1−1−ηγ2ζ2d−1−1−ηγ2ζ2d−1−ηγ1ζ1d+1−ηγ2ζ1d1−ηγ1ζ1d,ℏI1ζ1d.ℏI2ζ2d=1−1−ηγ1ζ1d1−ηγ2ζ2d,ℏI1ζ1d.ℏI2ζ2d=1−∏g=11−ηγgζgd,∏g=1uℏIgζgd.
This shows that Equation (13) is true for u=2; now, let that Equation (13) holds for u=b, i.e.,
(16)IFPAdξδ1,ξδ2,⋯ξδb=1−∏g=1b1−ηγgζgd,∏g=1bℏIgζgd.
Nowu=b+1, by operational laws of IFNs, we have
(17)IFPAdξδ1,ξδ2,⋯ξδb+1=IFPAdξδ1,ξδ2,⋯ξδb⊕ξδb+1=1−∏g=1b1−ηγgζgd,∏g=1bℏIgζgd⊕1−1−ηγb+1ζb+1d+1,ℏIb+1ζb+1d+1=1−∏g=1b1−ηγgζgd+1−1−ηγb+1ζb+1d+1−1−∏g=1b1−ηγgζgd1−1−ηγb+1ζb+1d+1,∏g=1bℏIgζgd.ℏIb+1ζb+1d+1=1−∏g=1b+11−ηγgζgd,∏g=1b+1ℏIgζgd.
This shows that foru=b+1, Equation (13) holds. Then,
(18)IFPAdξδ1,ξδ2,⋯ξδu=1−∏g=1u1−ηγgζgd,∏g=1uℏIgζgd.
☐☐Example 14.
Letξδ1=0.73,0.54, ξδ2=0.53,0.75,ξδ3=0.82,0.25, and ξδ4=0.35,0.64 be the four IFNs, there is strict prioritized relation in considered IFNs, such that ξδ1≻d1ξδ2≻d2ξδ3≻d3ξδ4. Priority vector d=d1,d2,d3 is given as 5,1,1, by Equation (13), and get
(19)1−∏g=141−ηγgζgd=0.719519,∏g=14ℏIgζgd=0.544041,IFPAdξδ1,ξδ2,ξδ3,ξδ4=1−∏g=141−ηγgζgd,∏g=1uℏIgζgd=0.719519,0.544041.Furthermore, the suggestedIFPAd operator is examined to ensure that it has idempotency and boundary properties. Their explanations are as follows.Theorem 15.
Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs, and
(20)ξδ−=mingηγg,maxgℏIg,ξδ+=maxgηγg,mingℏIg.
Then,(21)ξδ−≤IFPAdξδ1,ξδ2,⋯ξδn≤ξδ+,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u and T1=1.Proof.
Since,(22)mingηγg≤ηγg≤maxgηγg,(23)mingℏIg≤ℏIg≤maxgℏIg.
From Equation (22), we have
(24)mingηγg≤ηγg≤maxgηγg⇔mingηγg≤ηγg≤maxgηγg⇔1−maxgηγg≤1−ηγg≤1−mingηγg⇔1−maxgηγgζgd≤1−ηγgζgd≤1−mingηγgζgd⇔∏g=1u1−maxgηγgζgd≤∏g=1u1−ηγgζgd≤∏g=1u1−mingηγgζgd⇔1−maxgηγg≤∏g=1u1−ηγgζgd≤1−mingηγg⇔−1+minjηγg≤−∏g=1u1−ηγgζgd≤−1+maxgηγg⇔1−1+minjηγg≤1−∏g=1u1−ηγgζgd≤1−1+maxgηγg⇔minjηγg≤1−∏g=1u1−ηγgζgd≤maxgηγg⇔minjηγg≤1−∏g=1u1−ηγgζgd≤maxgηγg.
From Equation (23), we have
(25)mingℏIg≤ℏIg≤maxgℏIg⇔mingℏIgζgd≤ℏIgζgd≤maxgℏIgζgd⇔∏g=1umingℏIgζgd≤∏g=1uℏIgζgd≤∏g=1umaxgℏIgζgd⇔mingℏIgζgd≤∏g=1uℏIgζgd≤maxgℏIgζgd.
Let(26)IFPAdξδ1,ξδ2,⋯ξδn=ξδ=ηγ,ℏI.
Then,Ξ˘ξδ=ηγ−ℏI≤maxgηγ−minjℏI=Ξ˘ξδmax So, Ξ˘ξδ≤Ξ˘ξδmax.
Again,Ξ˘ξδ=ηγ−ℏI≥mingηγ−maxjℏI=Ξ˘ξδmin. So, Ξ˘ξδ≥Ξ˘ξδmin.
If,Ξ˘ξδ≤Ξ˘ξδmax and Ξ˘ξδ≥Ξ˘ξδmin, then
(27)ξδmin≤IFPAdξδ1,ξδ2,⋯ξδn≤ξδmax.
IfΞ˘ξδ=Ξ˘ξδmax, then ηγ−ℏI=maxgηγ−minjℏI(28)⇔ηγ−ℏI=maxgηγ−mingℏI,⇔ηγ=maxgηγ,ℏI=mingℏI,⇔ηγ=maxgηγ,ℏI=mingℏI.
Now,Hξδ=ηγ+ℏI=maxgηγ+mingℏI=Hξδmax(29)IFPAdξδ1,ξδ2,⋯ξδn=ξδmax.
IfΞ˘ξδ=Ξ˘ξδmin, then ηγ−ℏI=mingηγ−maxjℏI(30)⇔ηγ−ℏI=mingηγ−maxgℏI,⇔ηγ=mingηγ,ℏI=maxgℏI,⇔ηγ=mingηγ,ℏI=maxgℏI.
Now,Hξδ=ηγ+ℏI=mingηγ+maxgℏI=Hξδmax(31)IFPAdξδ1,ξδ2,⋯ξδn=ξδmin.
Thus, from Equations (27), (29), and (31), we get
(32)ξδ−≤IFPAdξδ1,ξδ2,⋯ξδn≤ξδ+.☐Theorem 16.
Assume that ifξδ⋄ is a IFN satisfied the property, ξδg=ξδ⋄,∀g then
(33)IFPAdξδ1,ξδ2,⋯ξδu=ξδ⋄.Proof.
Letξδ⋄=ηγ⋄,ℏI⋄ be the IFN. Then, by assumption, we have ξδg=ξδ⋄,∀g gives ηγg=ηγ⋄ and ℏIg=ℏI⋄∀g. By Definition 12, we have ∑g=1uζgd. Then, by using Theorem 13, we get
(34)IFPAdξδ1,ξδ2,⋯ξδu=1−∏g=1u1−ηγ⋄ζgd,∏g=1uℏI⋄ζgd=1−1−ηγ⋄∑g=1uζgd,ℏI⋄∑g=1uζgd=ηγ⋄,ℏI⋄=ξδ⋄.☐Corollary 17.
Ifξδg=ηγg,ℏIg is the assemblage of largest IFNs, i.e., ξδg=1,0 for all g, then
(35)IFPAdξδ1,ξδ2,⋯ξδu=1,0.Proof.
We can easily obtain a corollary similar to Theorem16.☐Corollary 18.
Ifξδ1=ηγ1,ℏI1 is the smallest IFN, i.e., ξδ1=0,1, then
(36)IFPAdξδ1,ξδ2,⋯ξδu=0,1.Proof.
Here,ξδ1=0,1 then by definition of the score function, we have
(37)Ξ˘ξδ1=0.
Since,(38)Tgd=∏q=1g−1Ξ˘ξδqdq,foreachg=2,3,⋯,uandT1=1.
We have(39)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1=0.
From Definition 3, we have(40)IFPAdξδ1,ξδ2,⋯ξδu=ζ1dξδ1⊕ζ2dξδ2,⋯,ζudξδu=1ξδ1⊕0ξδ2⊕⋯0ξδn=ξδ1=0,1.☐Theorem 19.
Assume thatξδg=ηγg,ℏIg and βg=ϕg,φg are two assemblages of IFNs, if r>0 and β=ηγβ,ℏIβ is a IFN, then
(1)
IFPAdξδ1⊕β,ξδ2⊕β,⋯ξδu⊕β=IFPAdξδ1,ξδ2,⋯ξδu⊕β(2)
IFPAdrξδ1,rξδ2,⋯rξδu=rIFPAdξδ1,ξδ2,⋯ξδu(3)
IFPAdξδ1⊕β1,ξδ2⊕β2,⋯ξδu⊕βn=IFPAdξδ1,ξδ2,⋯ξδn⊕IFPAdβ1,β2,⋯βu(4)
IFPAdrξδ1⊕β,rξδ2⊕β,⋯⊕rξδu⊕β=rIFPAdξδ1,ξδ2,⋯ξδu⊕βProof.
This is trivial by definition.☐IFPAd operator satisfied following properties.Property:1Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs, then we have
(41)limd1,d2,⋯,du−1⟶1,1,⋯,1IFPAdξδ1,ξδ2,⋯ξδu=IFPWAξδ1,ξδ2,⋯ξδu.Proof.
Given thatd1,d2,⋯,du−1⟶1,1,⋯,1, from this, we have
(42)Tgd=∏q=1g−1Ξ˘ξδqdq⟶∏q=1g−1Ξ˘ξδq=Tg,by this, we obtain ζgd⟶ζg(43)limd1,d2,⋯,du−1⟶1,1,⋯,1IFPAdξδ1,ξδ2,⋯ξδu=limd1,d2,⋯,du−1⟶1,1,⋯,1ζ1dξδ1⊕ζ2dξδ2,⋯,ζudξδu=ζ1ξδ1⊕ζ2ξδ2,⋯,ζuξδu=IFPWAξδ1,ξδ2,⋯ξδu.☐Remark.
Whend1=d2=⋯,=du−1=1, Property:1 states that the existing IFPWA operator is a particular situation of the suggested IFPAd operator. As a result, the IFPAd operator is more generic than the IFPWA operator.Property:2Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδg≠0 for all g, then we have
(44)limd1,d2,⋯,du−1⟶0,0,⋯,0IFPAdξδ1,ξδ2,⋯ξδu=1uξδ1⊕ξδ2⊕,⋯,⊕ξδu.Proof.
Given thatd1,d2,⋯,du−1⟶0,0,⋯,0, from this, we have
(45)Tgd=∏q=1g−1Ξ˘ξδqdq=1,and ζgd=Tgd/∑g=1uTgd=1/n. Hence,
(46)limd1,d2,⋯,du−1⟶0,0,⋯,0IFPAdξδ1,ξδ2,⋯ξδu=1uξδ1⊕ξδ2⊕,⋯,⊕ξδu.☐Property:3Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδ1≠0 or 1, then we have
(47)limd1⟶+∞IFPAdξδ1,ξδ2,⋯ξδu=ξδ1.Proof.
Here,d1⟶+∞ for each g=2,3,⋯,u, we have
(48)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1+∞Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1=0,
because0<Ξ˘ξδ1<1, ∑g=1uTgd=T1d=1⇒ζ1d=T1d/∑g=1uT1d=1 and ζgd=Tgd/∑g=1uTgd for each g=2,3,⋯,u. Hence,
(49)limd1⟶∞IFPAdξδ1,ξδ2,⋯ξδu=ξδ1.☐Remark.
According to Property:3, whend1⟶+∞, the PD d1 of IFN ξδ1 is very high in comparison to the PDs of other IFNs. It indicates that IFN ξδ1 is extremely essential. As a result, ξδ1 determines the aggregation result obtained by using the proposed operator IFPAd in this case.Example 20.
Letξδ1=0.73,0.54, ξδ2=0.53,0.75,ξδ3=0.82,0.25, and ξδ4=0.35,0.64 be the four IFNs, it can easily compute that, Ξ˘1=0.6158, Ξ˘2=0.3635, Ξ˘3=0.7679, and Ξ˘4=0.3904. There is strict prioritized relation in considering IFNs, such that ξδ1≻d1ξδ2≻d2ξδ3≻d3ξδ4. In the corresponding portion, we will aggregate the IFNs for 4 distinct priority vectors d=d1,d2,d3, keeping the values of PDs d2,d3 constant while varying the value of d1 and discussing its effect on the aggregated results.Case 1.
whend=1,1,1,
(50)IFPAdξδ1,ξδ2,ξδ3,ξδ4=0.684742,0.556094.Case 2.
whend=5,1,1,
(51)IFPAdξδ1,ξδ2,ξδ3,ξδ4=0.719519,0.544041.Case 3.
whend=8,1,1,
(52)IFPAdξδ1,ξδ2,ξδ3,ξδ4=0.727349,0.541027.Case 4.
whend=13,1,1,
(53)IFPAdξδ1,ξδ2,ξδ3,ξδ4=0.729769,0.540074.The consolidated findings from the preceding 4 cases show that as the PDd1 referring to IFN ξδ1 rises, the aggregated value approaches the IFN ξδ1 ranking values.Property:4Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδg≠0 for all g=1,2,⋯,k+1, and Ξ˘ξδk+1≠1, then we have
(54)limd1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞IFPAdξδ1,ξδ2,⋯ξδu=1k+1ξδ1⊕ξδ2⊕,⋯,⊕ξδk+1.Proof.
Given thatd1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞. So,
(55)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1⟶Ξ˘ξδ10Ξ˘ξδ20⋯Ξ˘ξδg−10=1,for each g=2,3,⋯,k+1.
(56)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1⟶Ξ˘ξδ10Ξ˘ξδ20⋯Ξ˘ξδk0Ξ˘ξδk+1+∞⋯Ξ˘ξδg−1dg−1=0,∀g=k+2,k+3,⋯,u.
So,(57)∑g=1uTgd=T1d=k+1andζgd=Tgd∑g=1uTgd⟶1k+1foreachg=1,2,3,⋯,k+1,ζgd=Tgd∑g=1uTgd⟶0k+1=0foreachg=k+2,k+3,⋯,u.
Hence,(58)limd1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞IFPAdξδ1,ξδ2,⋯ξδu=1k+1ξδ1⊕ξδ2⊕,⋯,⊕ξδk+1.☐Remark.
Whend1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞, it means there is no prioritization association between the IFNs ξδ1,ξδ2,⋯,ξδk+1 and that all of these IFNs ξδ1ξδ2,⋯,ξδk+1 have a much higher priority than the IFNs ξδk+2ξδk+3,⋯,ξδu. As a result, the aggregated value is solely dependent on IFNs ξδ1ξδ2,⋯,ξδk+1, and these IFNs ξδ1ξδ2,⋯,ξδk+1 have similar weightage in the aggregation method.Property:5Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδk+1≠1 or 0 then, we have
(59)limd1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞IFPAdξδ1,ξδ2,⋯ξδu=IFPWAξδ1⊕ξδ2⊕,⋯,⊕ξδk+1.Proof.
Given thatd1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞. So,
(60)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1⟶Ξ˘ξδ1Ξ˘ξδ2⋯Ξ˘ξδg−1=Tg,for each g=2,3,⋯,k+1.
(61)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1⟶Ξ˘ξδ1Ξ˘ξδ2⋯Ξ˘ξδkΞ˘ξδk+1+∞⋯Ξ˘ξδg−1dg−1=0,∀g=k+2,k+3,⋯,u.
So,
∑g=1uTgd⟶∑g=1k+1Tg and ζgd=Tgd/∑g=1uTgd⟶Tg/∑g=1k+1Tg for each g=1,2,3,⋯,k+1.
ζgd=Tgd/∑g=1uTgd⟶0/∑g=1k+1Tg=0 for each g=k+2,k+3,⋯,u.
Hence,(62)limd1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞IFPAdξδ1,ξδ2,⋯ξδu=IFPWAξδ1⊕ξδ2⊕,⋯,⊕ξδk+1.☐Remark.
Whend1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞, it means there is normal prioritization association between the IFNs ξδ1,ξδ2,⋯,ξδk+1 and that all of these IFNs ξδ1ξδ2,⋯,ξδk+1 have a much higher priority than the IFNs ξδk+2ξδk+3,⋯,ξδu. As a result, the aggregated value is solely dependent on IFNs ξδ1ξδ2,⋯,ξδk+1.


## 3.2.IFPGd Operator

Assumeξδg=ηγg,ℏIgg=1,2⋯,u is the assemblage of IFNs, there is a prioritization among these IFNs expressed by the strict priority orders ξδ1≻d1ξδ2≻d2⋯≻du−1ξδu−1, where ξδu≻duξδu+1 indicates that the IFN ξδu has du higher priority than ξδu+1. d=d1,d2,⋯,du−1 is the u−1-dimensional vector of PDs. The assemblage of such IFNs with strict priority orders and PDs is denoted by Rd.Definition 21.
AIFPGd operator is a mapping from Rdu to Rd and defined as
(63)IFPGdξδ1,ξδ2,⋯,ξδu=ξδ1ζ1d⊕ξδ2ζ2d,⋯,ξδuζud,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u and T1=1. Then, IFPGd is called intuitionistic prioritized geometric operator with PDs.Theorem 22.
Assumeξδg=ηγg,ℏIg is the assemblage of IFNs, we can also find IFPGd by
(64)IFPGdξδ1,ξδ2,⋯,ξδu=ξδ1ζ1d⊕ξδ2ζ2d,⋯,ξδuζud=∏g=1uηγgζgd,1−∏g=1u1−ℏIgζgd,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u and T1=1.Proof.
To prove this theorem, we use mathematical induction.
Foru=2,
(65)ξδ1ζ1d=ηγ1ζ1d,1−1−ℏI1ζ1d,ξδ2ζ2d=ηγ2ζ2d,1−1−ℏI2ζ2d.
Then,(66)ξδ1ζ1d⊗ξδ2ζ2d=ηγ1ζ1d,1−1−ℏI1ζ1d⊗ηγ2ζ2d,1−1−ℏI2ζ2d=ηγ1ζ1d.ηγ2ζ2d,1−1−ℏI1ζ1d+1−1−ℏI2ζ2d−1−1−ℏI1ζ1d1−1−ℏI2ζ2d=ηγ1ζ1d.ηγ2ζ2d,1−1−ℏI1ζ1d+1−1−ℏI2ζ2d−1−1−ℏI2ζ2d−1−ℏI1ζ1d+1−ℏI2ζ1d1−ℏI1ζ1d=ηγ1ζ1d.ηγ2ζ2d,1−1−ℏI1ζ1d1−ℏI2ζ2d=∏g=1uηγgζgd,1−∏g=11−ℏIgζgd.
This shows that Equation (64) is true for u=2; now, let that Equation (64) holds for u=b, i.e.,
(67)IFPGdξδ1,ξδ2,⋯ξδb=∏g=1bηγgζgd,1−∏g=1b1−ℏIgζgd.
Nowu=b+1, by operational laws of IFNs, we have
(68)IFPGdξδ1,ξδ2,⋯ξδb+1=IFPGdξδ1,ξδ2,⋯ξδb⊗ξδb+1=∏g=1bηγgζgd,1−∏g=1b1−ℏIgζgd⊗ηγb+1ζb+1d+1,1−1−ℏIb+1ζb+1d+1=∏g=1bηγgζgd.ηγb+1ζb+1d+1,1−∏g=1b1−ℏIgζgd+1−1−ℏIb+1ζb+1d+1−1−∏g=1b1−ℏIgζgd1−1−ℏIb+1ζb+1d+1=∏g=1b+1ηγgζgd,1−∏g=1b+11−ℏIgζgd.
This shows that foru=b+1, Equation (64) holds. Then,
(69)IFPGdξδ1,ξδ2,⋯ξδu=∏g=1uηγgζgd,1−∏g=1u1−ℏIgζgd.☐Example 23.
Letξδ1=0.73,0.54, ξδ2=0.53,0.75,ξδ3=0.82,0.25 and ξδ4=0.35,0.64 be the four IFNs, there is strict prioritized relation in considered IFNs, such that ξδ1≻d1ξδ2≻d2ξδ3≻d3ξδ4. Priority vector d=d1,d2,d3 is given as 5,1,1, by Equation (64)
(70)∏g=14ηγgζgd=0.7032081−∏g=141−ℏIgζgd=0.565078,IFPGdξδ1,ξδ2,ξδ3,ξδ4=∏g=1uηγgζgd,1−∏g=141−ℏIgζgd,=0.703208,0.565078.Furthermore, the suggested IFPGd operator is examined to ensure that it has idempotency and boundary properties. Their explanations are as follows.Theorem 24.
Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs, and
(71)ξδ−=mingηγg,maxgℏIg,ξδ+=maxgηγg,mingℏIg.
Then,(72)ξδ−≤IFPGdξδ1,ξδ2,⋯ξδn≤ξδ+,where ζgd=Tgd/∑g=1uTgd, Tgd=∏q=1g−1Ξ˘ξδqdq, for each g=2,3,⋯,u and T1=1.Proof.
Proof is same as Theorem15.☐Theorem 25.
Assume that ifξδ⋄ is a IFN satisfied the property, ξδg=ξδ⋄,∀g then
(73)IFPGdξδ1,ξδ2,⋯ξδu=ξδ⋄.Proof.
Letξδ⋄=ηγ⋄,ℏI⋄ be the IFN. Then, by assumption, we have ξδg=ξδ⋄,∀g gives ηγg=ηγ⋄ and ℏIg=ℏI⋄∀g. By Definition 21, we have ∑g=1uζgd. Then, by using Theorem 22, we get
(74)IFPGdξδ1,ξδ2,⋯ξδu=∏g=1uηγ⋄ζgd,1−∏g=1u1−ℏI⋄ζgd=ηγ⋄∑g=1uζgd,1−1−ℏI⋄∑g=1uζgd=ηγ⋄,ℏI⋄=ξδ⋄.☐Corollary 26.
Ifξδg=ηγg,ℏIg is the assemblage of largest IFNs, i.e., ξδg=1,0 for all g, then
(75)IFPGdξδ1,ξδ2,⋯ξδu=1,0.Proof.
We can easily obtain a corollary similar to Theorem25.☐Corollary 27.
Ifξδ1=ηγ1,ℏI1 is the smallest IFN, i.e., ξδ1=0,1, then
(76)IFPGdξδ1,ξδ2,⋯ξδu=0,1.Proof.
Here,ξδ1=0,1 then by definition of the score function, we have
(77)Ξ˘ξδ1=0.
Since,(78)Tgd=∏q=1g−1Ξ˘ξδqdq,foreachg=2,3,⋯,uandT1=1.
We have(79)Tgd=∏q=1g−1Ξ˘ξδqdq=Ξ˘ξδ1d1Ξ˘ξδ2d2⋯Ξ˘ξδg−1dg−1=0.
From Definition 3, we have(80)IFPGdξδ1,ξδ2,⋯ξδu=ξδ1ζ1d⊗ξδ2ζ2d,⋯,ξδuζud=ξδ11⊗ξδ20⊗⋯ξδu0=ξδ1=0,1.☐Theorem 28.
Assume thatξδg=ηγg,ℏIg and βg=ϕg,φg are two assemblages of IFNs, if r>0 and β=ηγβ,ℏIβ is a IFN, then
(81)IFPGdξδ1⊕β,ξδ2⊕β,⋯ξδu⊕β=IFPGdξδ1,ξδ2,⋯ξδu⊕β,IFPGdrξδ1,rξδ2,⋯rξδu=rIFPGdξδ1,ξδ2,⋯ξδu,IFPGdξδ1⊕β1,ξδ2⊕β2,⋯ξδu⊕βn=IFPGdξδ1,ξδ2,⋯ξδn⊕IFPGdβ1,β2,⋯βu,IFPGdrξδ1⊕β,rξδ2⊕β,⋯⊕rξδu⊕β=rIFPGdξδ1,ξδ2,⋯ξδu⊕β.Proof.
This is trivial by definition.☐IFPGd operator also satisfied following properties.Property:1Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs, then we have
(82)limd1,d2,⋯,du−1⟶1,1,⋯,1IFPGdξδ1,ξδ2,⋯ξδu=IFPWGξδ1,ξδ2,⋯ξδu.Property:2Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδg≠0 for all g, then we have
(83)limd1,d2,⋯,du−1⟶0,0,⋯,0IFPGdξδ1,ξδ2,⋯ξδu=1uξδ1⊗ξδ2⊗,⋯,⊗ξδu.Property:3Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδ1≠0 or 1, then we have
(84)limd1⟶+∞IFPGdξδ1,ξδ2,⋯ξδu=ξδ1.Property:4Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδg≠0 for all g=1,2,⋯,k+1, and Ξ˘ξδk+1≠1 then we have
(85)limd1,d2,⋯,dk,dk+1⟶0,0,⋯,0,+∞IFPGdξδ1,ξδ2,⋯ξδu=1k+1ξδ1⊗ξδ2⊗,⋯,⊗ξδk+1.Property:5Assume thatξδg=ηγg,ℏIg is the assemblage of IFNs and Ξ˘ξδk+1≠1 or 0 then we have
(86)limd1,d2,⋯,dk,dk+1⟶1,1,⋯,1,+∞IFPGdξδ1,ξδ2,⋯ξδu=IFPWGξδ1⊗ξδ2⊗,⋯,⊗ξδk+1.


## 4. Methodology for MCDM Using Proposed AOs

Let∐¯=∐¯1,∐¯2⋯,∐¯m be the assemblage of alternatives and ∐∧′=∐^1′,∐^2′,⋯,∐^n′ is the assemblage of criterions, priorities are assigned between the criterions provided by strict priority orientation. ∐^′1≻d1∐^′2≻d2∐^′3⋯≻dn−1∐^′n indicates criteria ∐^′J has a high priority than ∐^′J+1 with degree dq for q∈1,2,⋯,n−1. K=K1,K2,⋯,Kp is a assemblage of decision-makers (DMs). Priorities are assigned between the DMs provided by strict priority orientation, K1≻d1′K2≻d2′K3⋯≻dp−1′Kp. DMs give a matrix according to their own standpoints Dp=Bijpm×n, where Bijp is given for the alternatives ∐¯i∈∐¯ with respect to the attribute ∐^j′∈∐∧′ by Kp DM. If all performance criteria are the same kind, there is no need for normalization; however, since MCGDM has two different types of evaluation criteria (benefit kind attributes τb and cost kinds attributes τc), the matrix Dp has been transformed into a normalize matrix using the normalization formula Yp=Pijpm×n,
(87)Pijpm×n=Bijpc;j∈τc,Bijp;j∈τb.where Bijpc show the compliment of Bijp.The suggested operators will be implemented to the MCGDM, which will require the preceding steps.


### 4.1. Algorithm

Step 1.
Obtain the decision matrixDp=Bijpm×n in the format of IFNs from DMs.
(88)4751835.eq-inline.001Step 2.
Two kinds of criterion are described in the decision matrix:τc cost type indicators and τb benefit type indicators. There is no need for normalization if all indicators are of the same kind, but in MCGDM, there may be two types of criteria. The matrix was updated to the transforming response matrix in this case Yp=Pijpm×n using the normalization formula Equation (87).Step 3.
Using one of provided AOs to combine all of the independent IF decision matricesYp=Pijpm×n into one combined evaluation matrix of the alternatives Wp=χ~ijm×n.
(89)χ~ij=IFPAdPij1,Pij2,⋯Pijp=1−∏z=1p1−ηγijzζijz,∏z=1pℏIijzζijz(90)orχ~ij=IFPGdPij1,Pij2,⋯Pijp=∏z=1pηγijzζijz,1−∏z=1p1−ℏIijzζijz.Step 4.
Aggregate the IF valuesχ~ij for each alternative ∐¯i by the IFPAd (or IFPGd) operator.
(91)χ~ij=IFPAdPi1,Pi2,⋯Pin=1−∏j=1n1−ηγijζij,∏j=1nℏIijζij(92)orχ~ij=IFPGdPi1,Pi2,⋯Pin=∏j=1nηγijζij,1−∏j=1n1−ℏIijζij.Step 5.
Analyze the score for all cumulative alternative assessments.Step 6.
The alternatives were classified by the score function, and the most suitable alternative was selected.


### 4.2. Numerical Illustration

Consider a decision-making problem of finding out the most appropriate agriculture land. Assume the assemblage of alternatives,∐¯1, ∐¯2, ∐¯3, and ∐¯4. There are five criterions for evaluation of these alternatives ∐^′1= soil (chemical and physical), ∐^′2= cost, ∐^′3= irrigation (water and canal), ∐^′4= processing industry and market, and ∐^′5= social-economic. Assume that the criterions have been prioritized in strict priority order ∐^′1>d1∐^′2>d2∐^′3>d3∐^′4>d1∐^′5. The three-dimensional vector of PDs is d=2,1,3,2. Here, three DMs K1, K2, and K3 are involved; they have been prioritized in a strict priority order K1>d1′K2>d2′ and K3, where d′=3,4.


## 4.1. Algorithm

Step 1.
Obtain the decision matrixDp=Bijpm×n in the format of IFNs from DMs.
(88)4751835.eq-inline.001Step 2.
Two kinds of criterion are described in the decision matrix:τc cost type indicators and τb benefit type indicators. There is no need for normalization if all indicators are of the same kind, but in MCGDM, there may be two types of criteria. The matrix was updated to the transforming response matrix in this case Yp=Pijpm×n using the normalization formula Equation (87).Step 3.
Using one of provided AOs to combine all of the independent IF decision matricesYp=Pijpm×n into one combined evaluation matrix of the alternatives Wp=χ~ijm×n.
(89)χ~ij=IFPAdPij1,Pij2,⋯Pijp=1−∏z=1p1−ηγijzζijz,∏z=1pℏIijzζijz(90)orχ~ij=IFPGdPij1,Pij2,⋯Pijp=∏z=1pηγijzζijz,1−∏z=1p1−ℏIijzζijz.Step 4.
Aggregate the IF valuesχ~ij for each alternative ∐¯i by the IFPAd (or IFPGd) operator.
(91)χ~ij=IFPAdPi1,Pi2,⋯Pin=1−∏j=1n1−ηγijζij,∏j=1nℏIijζij(92)orχ~ij=IFPGdPi1,Pi2,⋯Pin=∏j=1nηγijζij,1−∏j=1n1−ℏIijζij.Step 5.
Analyze the score for all cumulative alternative assessments.Step 6.
The alternatives were classified by the score function, and the most suitable alternative was selected.


## 4.2. Numerical Illustration

Consider a decision-making problem of finding out the most appropriate agriculture land. Assume the assemblage of alternatives,∐¯1, ∐¯2, ∐¯3, and ∐¯4. There are five criterions for evaluation of these alternatives ∐^′1= soil (chemical and physical), ∐^′2= cost, ∐^′3= irrigation (water and canal), ∐^′4= processing industry and market, and ∐^′5= social-economic. Assume that the criterions have been prioritized in strict priority order ∐^′1>d1∐^′2>d2∐^′3>d3∐^′4>d1∐^′5. The three-dimensional vector of PDs is d=2,1,3,2. Here, three DMs K1, K2, and K3 are involved; they have been prioritized in a strict priority order K1>d1′K2>d2′ and K3, where d′=3,4.


## 5. Algorithm

Step 1.
Obtain the decision matrixDp=Bijpm×n in the format of IFNs from DMs. The judgement values, given by three DMs, are described in Table 1.Step 2.
Normalize the decision matrixes acquired by DMs using Equation (87). In Table 1, there are two types of criterions. ∐^′2 is cost type criteria and others are benefit type criterions. Normalized IF decision matrix is given in Table 2.Step 3.
Using IFPAd opeartor to combine all of the independent IF decision matrices Yp=Pijpm×n into one combined evaluation matrix of the alternatives Wp=χ~ijm×n given in Table 3. First, we find Tij1Tij2 and Tij3, which are used in the calculation of IFPAd operator.
(93)Tij1=11111111111111111111,Tij2=0.48840.30350.38490.47270.44290.41120.26450.05900.21590.16280.29480.12730.19990.16540.17160.15680.24030.16450.14080.1955,Tij3=0.10080.00030.14110.17760.14510.04790.12720.00040.00210.00940.02610.01060.01290.03510.00060.00010.16510.00450.00270.0010.Step 4.
Aggregate the IF valuesχ~ij for each alternative ∐¯1 by the IFPAd operator using Equation (91) given in Table 4.
(94)Tij=10.61090.38380.15540.087410.49750.34470.01960.006410.41860.22550.05110.015910.26910.19050.03690.0099.Step 5.
Compute the score for all IF-aggregated valuesχ~i.
(95)Ξ˘χ~1=0.493599,Ξ˘χ~2=0.440998,Ξ˘χ~3=0.333649,Ξ˘χ~4=0.211889.Step 6.
Ranks according to score values.(96)χ~1≻χ~2≻χ~3≻χ~4,∐¯1≻∐¯2≻∐¯3≻∐¯4.
∐¯1 is the best alternative among all other alternatives.Table 1
Rating given by DMs.
ExpertsAlternatives∐^′1∐^′2∐^′3∐^′4∐^′5K1∐¯1(0.80, 0.10)(0.24, 0.71)(0.77, 0.12)(0.73, 0.23)(0.80, 0.19)∐¯2(0.78, 0.18)(0.42, 0.71)(0.43, 0.67)(0.61, 0.30)(0.50, 0.32)∐¯3(0.74, 0.42)(0.45, 0.46)(0.62, 0.41)(0.58, 0.46)(0.55, 0.38)∐¯4(0.43, 0.29)(0.44, 0.69)(0.47, 0.20)(0.45, 0.37)(0.57, 0.29)K2∐¯1(0.66, 0.33)(0.80, 0.18)(0.83, 0.15)(0.83, 0.16)(0.66, 0.33)∐¯2(0.56, 0.19)(0.34, 0.89)(0.37, 0.78)(0.11, 0.72)(0.17, 0.29)∐¯3(0.48, 0.27)(0.56, 0.63)(0.20, 0.10)(0.77, 0.17)(0.53, 0.27)∐¯4(0.06, 0.93)(0.40, 0.58)(0.55, 0.44)(0.55, 0.45)(0.67, 0.23)K3∐¯1(0.88, 0.11)(0.77, 0.22)(0.73, 0.13)(0.87, 0.11)(0.84, 0.13)∐¯2(0.76, 0.14)(0.22, 0.75)(0.35, 0.63)(0.63, 0.36)(0.57, 0.36)∐¯3(0.74, 0.23)(0.45, 0.49)(0.69, 0.41)(0.58, 0.39)(0.57, 0.33)∐¯4(0.46, 0.21)(0.41, 0.61)(0.43, 0.28)(0.47, 0.33)(0.54, 0.26)Table 2
Normalized IF decision matrix.
ExpertsAlternatives∐^′1∐^′2∐^′3∐^′4∐^′5K1∐¯1(0.80, 0.10)(0.71, 0.24)(0.77, 0.12)(0.73, 0.23)(0.80, 0.19)∐¯2(0.78, 0.18)(0.71, 0.42)(0.43, 0.67)(0.61, 0.30)(0.50, 0.32)∐¯3(0.74, 0.42)(0.46, 0.45)(0.62, 0.41)(0.58, 0.46)(0.55, 0.38)∐¯4(0.43, 0.29)(0.69, 0.44)(0.47, 0.20)(0.45, 0.37)(0.57, 0.29)K2∐¯1(0.66, 0.33)(0.18, 0.80)(0.83, 0.15)(0.83, 0.16)(0.66, 0.33)∐¯2(0.56, 0.19)(0.89, 0.34)(0.37, 0.78)(0.11, 0.72)(0.17, 0.29)∐¯3(0.48, 0.27)(0.63, 0.56)(0.20, 0.10)(0.77, 0.17)(0.53, 0.27)∐¯4(0.06, 0.93)(0.58, 0.40)(0.55, 0.44)(0.55, 0.45)(0.67, 0.23)K3∐¯1(0.88, 0.11)(0.22, 0.77)(0.73, 0.13)(0.87, 0.11)(0.84, 0.13)∐¯2(0.76, 0.14)(0.75, 0.22)(0.35, 0.63)(0.63, 0.36)(0.57, 0.36)∐¯3(0.74, 0.23)(0.49, 0.45)(0.69, 0.41)(0.58, 0.39)(0.57, 0.33)∐¯4(0.46, 0.21)(0.61, 0.41)(0.43, 0.28)(0.47, 0.33)(0.54, 0.26)Table 3
Combined evaluation matrix.
∐^′1∐^′2∐^′3∐^′4∐^′5∐¯1(0.7861, 0.2172)(0.6624, 0.3240)(0.7844, 0.1455)(0.6958, 0.2845)(0.8019, 0.1904)∐¯2(0.7469, 0.1813)(0.6071, 0.3835)(0.4271, 0.6757)(0.5760, 0.3505)(0.4788, 0.3160)∐¯3(0.6543, 0.3412)(0.5169, 0.3925)(0.6249, 0.2907)(0.5196, 0.4547)(0.5472, 0.4289)∐¯4(0.4233, 0.3371)(0.6909, 0.2893)(0.5569, 0.2467)(0.4654, 0.4035)(0.5761, 0.3020)Table 4
IF-aggregated valuesχ~i.
χ~1(0.741187, 0.247588)χ~2(0.719510, 0.278512)χ~3(0.647383, 0.313734)χ~4(0.568093, 0.356204)


## 6. Conclusion

In the current study, IFSs are used to handle ambiguity in data utilising MDs and NMDs. The IFS paradigm is extended by the IF framework. By considering stringent priority orders, we established the notions of intuitionistic prioritized averaging and intuitionistic prioritized geometric operators with PDs. Many theories about PD have been thoroughly researched, and they will be valuable in merging multiple IF data sets. A group MCDM strategy based on the proposed prioritized AOs has been established within the IF framework. An analogy is used to illustrate the proposed technique, and the methodology results are compared to several current AOs. Aside from that, the effect of PDs on aggregated outcomes is thoroughly explained. Furthermore, the impact of PDs on outcomes makes the proposed solution more robust since the DM can choose the PD vector based on his or her priorities and the complexity of the problem. We apply the proposed group MCDM approach on a case study of selection of agriculture land.



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*Source: 4751835-2022-01-17.xml*