# Interval Valued Picture Fuzzy Aczel–Alsina Aggregation

Authors: Khalida Jabeen; Kifayat Ullah; Maria Akram; Izatmand Haleemzai
Journal: Journal of Mathematics
 (2023)

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## Abstract

The engineering and construction sector is vital to a country’s economic
growth, financial activities, and development. These sectors generate
opportunities for the unemployed, unskilled, and skilled workforce. Recently, a
lot of researchers worked on the Aczel–Alsina t-norm (TN) and t-conorm
(TCN), which are generalizing many other t-norms and producing reliable results.
In this article, first, we developed some new aggregation operators (AOs) and
fundamental operational laws of Aczel–Alsina operations, including
Aczel–Alsina product, sum, and scalar multiplication based on the IVPF
information. Furthermore, we introduced an innovative AOs in the form of IVPF
Aczel–Alsina weighted averaging (IVPFAAWA) operators with some basic
characteristics. Moreover, we also generalized Aczel–Alsina operations in
the form of the IVPF Aczel–Alsina weighted geometric (IVPFAAWG) operator.
For the solution of daily life problems by utilizing a multiattribute
decision-making (MADM) approach, we also established an application under the
system of engineering and construction sectors. We illustrated a numerical
example to find the suitable construction material for the engineering and
construction sectors. To find the validity and flexibility of our proposed AOs,
we also studied a comprehensive comparative analysis, in which we compared the
results of exiting AOs with the results of our current invented approaches. At
the end, we sum up our whole article in a single paragraph.


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## Body

## 1. Introduction

Decision making (DM) is an efficient apparatus to overcome the problem of vague and
uncertain data in everyday problems. The multiattribute decision‐making
(MADM) strategy has gotten various considerations from various researchers. MADM
strategy is one of the valuable and certified systems, which is used in
day‐to‐day life. The justification behind its existence is to rank the
restricted course of action of DM issues concerning the fact that they are so
sensitive to decision-makers. To deal with MADM problems under uncertainty, the
theory of fuzzy sets (FSs) was developed by Zadeh [1], where human opinion can be expressed in terms of the membership
value (MV) on a scale of unit intervals, i.e., 0,1. Atanassov [2] extended the
concept of FS in the form of intuitionistic FSs (IFSs), where two aspects of
uncertain information are expressed by using a MV and nonmembership value (NMV) with
a restriction that their sum lies in the interval 0,1. The extension of IFS in the framework of interval-valued IFS
(IVIFS) was discovered by Atanassov and Gargov [3]. The IVIFSs carried two aspects of MV and NMV as lower and upper
terms of both MV and NMV. Cuong [4] introduced
the new concept of picture fuzzy sets (PFS) which is an extension of FS and IFSs
that lies in four values, i.e., MV, abstinence value (AV), NMV, and refusal value
(RV). In PFSs, the sum of MV, NMV, and AV lies in the interval 0,1. Wei [5] extended the theory
of PFSs under similarity measures and their applications. Cuong et al. [6] developed the concepts of TN and TCN based on
PFSs. Dutta [7] explored the system of medical
diagnosis under the PFSs. Khalil et al. [8]
enlarged the theory of PFSs in the environment of IVPFSs. We studied work on fuzzy
sets in a different environment by several researchers [9–11].A triangular norm plays an important role to aggregate statistical information.
Firstly, Menger [12] worked on triangular
norms in the system of statistical information. He also introduced different AOs of
t-norm (TN) and its associative t-conorm (TCN) such as algebraic TN and TCN,
Einstein t- norm and TCN, frank TN and TCN, Archimedean TN and TCN, nilpotent
TN and TCN, Hamacher TN and TCN, Dombi TN and TCN, Lukasiewicz TN and TCN. We
studied previously existing AOs such as AOs of IFWA by the Yu and Xu [13], AOs of IFWG by the Xu and Yager [14], PF Maclaurin symmetric mean operators by
Ullah [15], interval-valued Pythagorean fuzzy
AOs by Garg [16], PF Dombi AOs by Jana et al.
[17], AOs of T-spherical fuzzy Dombi
weighted (TSFDW) averaging, and IVTSFDW geometric by the Ullah et al. [18]. Lin et al. [19] developed an interactional partitioned Heronian mean AOs
based on PFSs. Riaz and Farid [20] provided a
list of new AOs for the selection of logistics third parties under the system of
PFSs. Mahmood [21] introduced innovative
concepts of bipolar fuzzy sets in the form of bipolar complex fuzzy Maclaurin
symmetric mean operators. Mahmood et al. [22]
also discussed a series of new AOs of a complex PF soft set based on the theory of
power AOs to solve real-life problems based under the system of MADM technique. Zhou
et al. [23] explored the interactive
recursive feature to promote and improve the structure of the network under the
system of detection salient. Kovalnogov et al. [24] developed some new tools to solve initial value problems based on
the Runge–Kutta pair technique and a variety of scalar autonomous issues. Ye
et al. [25] proposed some new models of state
damping method under the system of sliding mode control and adaptive control. Liu et
al. [26] provided some new detection
approaches to evaluate the reliability and effectiveness of the thermal flaws under
the application of the vehicle cables. Zheng et al. [27] generalized theory of graphs in the framework of knowledge based
embedding models to verify experimental information. Zheng and Jin [28] evaluated the performance of the offloading
strategy under the stochastic models. Khalaf [29] established a few algorithms and optimal options for the selection
of suitable power plants to improve the infrastructure of the energy system. The
aggregation operators (AOs) play an important role to investigate fuzzy information
in MADM problems. Some Bonferroni mean AOs of interval-valued TSFSs are developed by
Akram et al. [30]. We also studied the
previous research work given in [31, 32].Klement et al. [33] generalized the idea of
the triangular norm and its basic properties. Klement and Navara [34] introduced the characterization of
Lukasiewicz TN and TCN by utilizing the binary operations of TN and TCNs. Wang
[35] explored the theory of TN and TCN in
the form of Drastic TN and TCN. Fodor [36]
enlarged the idea of triangular norms in the frame of Nilpotent minimum TN and TCN.
Gál et al. [37] developed the theory of
TN and TCN in Hamacher product TN and TCN. Xu and Zhang [38] approached the concepts of Dombi TN and TCN in the
environment of the q-rung orthopair fuzzy system. Navara [39] introduced nearly frank TN and TCN by using the basic
operations of TN and TCN and discussed its properties. A lot of researchers utilized
the idea of TN and TCN in the field of research given in [40–42].In 1982, Aczél and Alsina (AA) [43]
discovered new operations, namely, Aczel–Alsina t-norm (AA-TN) and AA-TCN
with basic properties. Richman and Walker [44] examined the families of AA-TN and AA-TCN. The characteristics of AA-TN
and AA-TCN are to analyze the information. Farahbod and Eftekhari [45] examined the best ones is AA-TN and AA-TCN
by evaluating the family of TN and TCN. Senapati et al. [46] discovered new AOs of IFS AA by utilizing some basic
operations of AA-TN and AA-TCN. Hussain et al. [47] enlarged the concepts of AA IFSs in the environment of PyFSs.
Senapati [48] explored the concepts of PFSs
and presented some new AOs with the help of Aczel–Alsina operations. Senapati
et al. [49] extended the concepts of IFS AA
in the environment of IVIFAA. He gave an application to check the effectiveness and
flexibility of discussed AOs.When the abovementioned existing AOs are utilized to handle the uncertain and vague
information, sometimes a piece of information is lost during the investigation
process. The purpose of this article is to cope with ambiguous and unreliable
information by utilizing the operations of AA-TN and AA-TCN based on PFSs. We
proposed some new AOs such as IVPFAAWA and IVPFAAWG. A MADM technique was used to
select suitable concrete for the construction of a building. We established an
application with the help of a numerical example to find the reliability and
effectiveness of current proposed AOs. In the comparative analysis, we compared
existing AOs with current approaches. In the end, we summarized this article.This article is organized in the form: In Section1, the author recalls the previous history to understand the background
of this article. In Section 2, the authors
define the notions of AA-TN and AA-TCN, PFS, IVIFS, IVPFS, and some basic
operations. In Section 3, the author utilizes
the basic operations of AA-TN and AA-TCN in the environment of PFNS. In Section
4, the author presents some AOs in the
form IVPFAAWA and IVPFAAOWA and their properties in the form of Idempotency,
boundedness, and monotonicity. In Section 5,
the author elaborates the AOs of IVPFAAWA and IVPFAAWG with the help of AA-TN and
AA-TCN. The author gave an application with the help of a numerical example to check
the effectiveness and flexibility of proposed AOs.


## 2. Preliminaries

In this section, we recall some basic definitions of IVIFSs and IVPFSs. We also
discussed the fundamental operations of AA-TN and AA-TCN.Now we discuss the notion of IVIFS developed by Atanassov and Gargov [3].Definition 1.
see ([3]). An IVIFS B on the universe W is defined as follows:(1)B=r,uBlr,uBur,vBlr,vBur|r∈W,where uBlr,uBur,vBlr,vBur are subintervals of the 0,1 interval denoting MV and NMV of r∈B provided that 0≤uBur+vBur≤1 and δBr=1−uBur+vBur,1−uBlr+vBlr is called the hesitancy value (HV) of r∈B. Further, uBlr,uBur,vBlr,vBur is known as an interval-valued intuitionistic fuzzy value
(IVIFV).
In the following definition, we recall the concepts of IVPFS discovered by Cuong
and Kreinovich [10].Definition 2.
see ([10]). An IVPFS B on the universe W is defined as follows:(2)B=r,uBlr,uBur,ξBlr,ξBur,vBlr,vBur|r∈W,where uBlr,uBur,ξBlr,ξBur,vBlr,vBur are closed subintervals of the 0,1 interval denoting the MV, abstinence value (AV), and NMV of r∈B respectively, provided that 0≤uBur+ξBur+vBur≤1 and δBr=1−uBur+ξBur+vBur,1−uBlr+ξBlr+vBlr is called the refusal value (RV) of r∈B. Further, uBlr,uBur,ξBlr,ξBur,vBlr,vBur is known as IVPF value (IVPFV).
Now, we discuss some basic operations of IVPFVs in Definition3.Definition 3.
(see [50]). Let us consider three IVPFVs β=uβlr,uβur,ξβlr,ξβur,vβlr,vβur,β1=uβ1lr,uβ1ur,ξβ1lr,ξβ1ur,vβ1lr,vβ1ur, and β2=uβ2lr,uβ2ur,ξβ2lr,ξβ2ur,vβ2lr,vβ2ur. Then(1)
β1⊆β2 iff uβ1l≤uβ2l,uβ1u≤uβ2u,ξβ1l≥ξβ2l,ξβ1u≥ξβ2u,vβ1l≥vβ1l, and vβ1u≥vβ1u.(2)
β1=β2 iffβ1⊆β2 andβ2⊆β1(3)
β1∩β2=minuβ1l,uβ2l,minuβ1u,uβ2u,maxξβ1l,ξβ2l,maxξβ1u,ξβ2u,maxvβ1l,vβ2l,maxvβ1u,vβ2u,(4)
β1∪β2=maxuβ1l,uβ2l,maxuβ1u,uβ2u,minξβ1l,ξβ2l,minξβ1u,ξβ2u,minvβ1l,vβ2l,minvβ1u,vβ2u,(5)
βc=vβlr,vβur,ξβlr,ξβur,uβlr,uβur.
In the following definitions, the idea of TN and TCN are discussed in the
form:Definition 4.
(see [51]). A TN is a function Ⱦ:0,12⟶0,1 that satisfies the below properties, i.e., ∀ã,ç,ᵳ∈0,1:(1)
Ⱦã,ç=Ⱦç,ã(2)
Ⱦã,ç≤Ⱦã,ᵳ,ifã≤ᵳ(3)
Ⱦã,Ⱦç,ᵳ=ȾȾã,ç,ᵳ(4)
Ⱦã,1=ãDefinition 5.
(see [52]). A TCN is a function Ʀ:0,12⟶0,1 that satisfies the below properties, i.e., ∀ã,ç,ᵳ∈0,1:(1)
Ʀã,ç=Ʀç,ã(2)
Ʀã,ç≤Ʀã,ᵳifã≤ᵳ(3)
Ʀã,Ʀã,ᵳ=ƦƦã,ç,ᵳ(4)
Ʀã,0=ã
The theory of AA-TN and AA-TCN is developed by Aczél and Alsina [43]. The AA-TN and TCN are more flexible
and diverse in nature and generalizes many other TNs and TCNs. Farahbod and
Eftekhari [45] investigated the
application of several TNs and TCNs in a classification problem and concluded
the AA-TN and AA-TCN as more appropriate. The AA-TN and AA-TCN are defined as
follows:Definition 6.
(see [43]). For ∀ᵿ∈0,∞, the AA-TN and AA-TCN are defined, respectively, as follows:(3)ȾAᵿã,ç=ȾDã,ç,ifᵿ=0,minã,ç,ifᵿ=0,e−−lnãᵿ+−lnçᵿ1/ᵿ,otherwise,ƦAᵿã,ç=ƦDã,ç,ifᵿ=0,maxã,ç,ifᵿ=∞,1−e−−ln1−ãᵿ+−ln1−çᵿ1/ᵿ,otherwise.
The concept of the score function of IVPFV was introduced by [50].Definition 7.
(see [50]). Suppose β=uβlr,uβur,ξβlr,ξβur,vβlr,vβur be the IVPFV. Then the score function S of β is given by(4)Sβ=uBl1−ξβl−vβl+uBu1−ξβu−vβu3Sβ∈0,1.


## 3. Aczel–Alsina (AA) Operations of IVPFVs

In this section, we will discuss the AA operations including product, sum, scalar
multiplications, and power operations based on AA-TN and AA-TCN for IVPFVs. We also
study their basic properties.SupposeȾA represents the AA-TN and ƦA represents the AA-TCN. Then the AA product B⊗C represents the generalized intersection over two IVPFVs and the AA
sum B⊕C represents the generalized union over two IVPFVs is written as follows:(5)B⊗C=r,ȾAuBlr,uclr,ȾAuBur,ucur,ƦAξBlr,ξclr,ƦAξBur,ξcur,ƦAvBlr,vclr,ƦAvBur,vcur;r∈W,B⊕C=r,ƦAuBlr,uclr,ȾAuBur,ucur,ȾAξBlr,ξclr,ƦAξBur,ξcur,ȾAvBlr,vclr,ƦAvBur,vcur;r∈W.Definition 8.
Suppose three IVPFVs,β=uβlr,uβur,ξβlr,ξβur,vβlr,vβur, β1=uβ1lr,uβ1ur,ξβ1lr,ξβ1ur,vβ1lr,vβ1ur and β2=uβ2lr,uβ2ur,ξβ2lr,ξβ2ur,vβ2lr,vβ2ur, ᵿ≥1 and λ>0. Then(1)
β1⊕β2=1−e−−ln1−uβ1lᵿ+−ln1−uβ2lᵿ1/ᵿ,1−e−−ln1−uβ1uᵿ+−ln1−uβ2uᵿ1/ᵿ,e−−lnξβ1lᵿ+−lnξβ2lᵿ1/ᵿ,e−−lnξβ1uᵿ+−lnξβ2uᵿ1/ᵿ,e−−lnvβ1lᵿ+−lnvβ2lᵿ1/ᵿ,e−−lnvβ1uᵿ+−lnvβ2uᵿ1/ᵿ(2)
β1⊗β2=e−−lnuβ1lᵿ+−lnuβ2lᵿ1/ᵿ,e−−lnuβ1uᵿ+−lnuβ2uᵿ1/ᵿ,1−e−−ln1−ξβ1lᵿ+−ln1−ξβ2lᵿ1/ᵿ,1−e−−ln1−ξβ1uᵿ+−ln1−ξβ2uᵿ1/ᵿ,1−e−−ln1−vβ1lᵿ+−ln1−vβ2lᵿ1/ᵿ,1−e−−ln1−vβ1uᵿ+−ln1−vβ2uᵿ1/ᵿ(3)
λβ=1−e−λ−ln1−uβlᵿ1/ᵿ,1−e−λ−ln1−uβuᵿ1/ᵿ,e−λ−lnξβlᵿ1/ᵿ,e−λ−lnξβuᵿ1/ᵿ,e−λ−lnvβlᵿ1/ᵿ,e−λ−lnvβuᵿ1/ᵿ(4)
βλ=e−λ−lnuβlᵿ1/ᵿ,e−λ−lnuβuᵿ1/ᵿ,1−e−λ−ln1−ξβlᵿ1/ᵿ,1−e−λ−ln1−ξβuᵿ1/ᵿ,1−e−λ−ln1−vβlᵿ1/ᵿ,1−e−λ−ln1−vβuᵿ1/ᵿ
To support the above-discussed operations and to see their working, we present
the following Example.Example 1.
Suppose three IVPFVsβ=0.25,0.28,0.30,0.33,0.36,0.38, β1=0.21,0.24,0.10,0.15,0.50,0.54, and β2=0.32,0.35,0.12,0.16,0.29,0.33, ᵿ=3 and λ=4, by applying Definition 9, we have:(1)
β1⊕β2=1−e−−ln1−0.213+−ln1−0.3231/3,1−e−−ln1−0.243+−ln1−0.3531/3,e−−ln0.103+−ln0.1231/3,e−−ln0.153+−ln0.1631/3,e−−ln0.503+−ln0.2931/3,e−−ln0.543+−ln0.3331/3=0.3384,0.3719,0.0614,0.0953,0.2708,0.3107(2)
β1⊗β2=e−−ln0.213+−ln0.3231/3,e−−ln0.243+−ln0.3531/3,1−e−−ln1−0.103+−ln1−0.1231/3,1−e−−ln1−0.153+−ln1−0.1631/3,1−e−−ln1−0.503+−ln1−0.2931/3,1−e−−ln1−0.543+−ln1−0.3331/3=0.1753,0.2028,0.1378,0.1914,0.5132,0.5554(3)
4β=1−e−4−ln1−0.2531/3,1−e−4−ln1−0.2831/3,e−4−ln0.3031/3,e−4−ln0.3331/3,,e−4−ln0.3631/3,,e−4−ln0.3831/3=0.3666,0.4064,0.0.1479,0.1721,0.1975,0.2153(4)
β4=e−4−ln0.2531/3,e−4−ln0.2831/3,1−e−4−ln1−0.3031/3,1−e−4−ln1−0.3331/3,1−e−4−ln1−0.3631/3,1−e−4−ln1−0.3831/3=0.1107,0.1326,0.4323,0.4704,0.5076,0.5318
Now, we have to prove some fundamental operations of AA sum and product under the
system of IVPFVs by utilizing Definition8.Theorem 1.
Suppose three IVPFVsβ=uβlr,uβur,ξβlr,ξβur,vβlr,vβur, β1=uβ1lr,uβ1ur,ξβ1lr,ξβ1ur,vβ1lr,vβ1ur and β2=uβ2lr,uβ2ur,ξβ2lr,ξβ2ur,vβ2lr,vβ2ur, and λ,λ1,λ2>0. Then(1)
β1⊕β2=β2⊕β1(2)
β1⊗β2=β2⊗β1(3)
λβ1⊕β2=λβ1⊕λβ2(4)
λ1+λ2β=λ1β⊕λ2β(5)
β1⊗β2λ=β1λ⊗β2λ(6)
βλ1⊗βλ2=βλ1+λ2Proof.
Straightforward.


## 4. Interval Valued Picture Fuzzy Aczel–Alsina Averaging Aggregation
Operators

In this section, we aim to introduce some AOs based on Definition8. We introduce the notion of the
IVPFAAWA operators and discuss the scenario where the ordered position of the
alternatives does matter by introducing the IVPFAAOWA operator. From this section
onward, by ω=ω1,ω2,…,ωnT, we denote the weight vector of attributes with ωj>0,∑j=1nωj=1 for j=1,2,…,n.Definition 9.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Then the IVPFAAWA operator is a function IVPFAAWA:Ln⟶L defined as follows:(6)IVPFAAWAβ1,β2,…,βn=⊕j=1nωjβj.
By using some basic fundamental operations defined in Definition8, we proposed a result for the
IVPFAAWA operator in the following Theorem 2.Theorem 2.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Then, the aggregated value of the
IVPFAAWA operator is also an IVPFV given by(7)IVPFAAWAβ1,β2,…,βn=1−e−∑j=1nωj−ln1−uβjlᵿ1/ᵿ,1−e−∑j=1nωj−ln1−uβjuᵿ1/ᵿ,e−∑j=1nωj−lnξβjlᵿ1/ᵿ,e−∑j=1nωj−lnξβjuᵿ1/ᵿ,e−∑j=1nωj−lnvβjlᵿ1/ᵿ,e−∑j=1nωj−lnvβjuᵿ1/ᵿ.Proof.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs, and we prove above theorem by using the
mathematical induction technique.
Forn=2, we have(8)ω1β1=1−e−ω1−ln1−uβ1lᵿ1/ᵿ,1−e−ω1−ln1−uβ1uᵿ1/ᵿ,e−ω1−lnξβ1lᵿ1/ᵿ,e−ω1−lnξβ1uᵿ1/ᵿ,e−ω1−lnvβ1lᵿ1/ᵿ,e−ω1−lnvβ1uᵿ1/ᵿ,ω2β2=1−e−ω2−ln1−uβ2lᵿ1/ᵿ,1−e−ω2−ln1−uβ2uᵿ1/ᵿ,e−ω2−lnξβ2lᵿ1/ᵿ,e−ω2−lnξβ2uᵿ1/ᵿ,e−ω2−lnvβ2lᵿ1/ᵿ,e−ω2−lnvβ2uᵿ1/ᵿ.
Now, from the Definition8 and
Definition 9, we get the following equation:(9)IVPFAAWAβ1,β2=ω1β1⊕ω2β2=1−e−ω1−ln1−uβ1lᵿ1/ᵿ,1−e−ω1−ln1−uβ1uᵿ1/ᵿ,e−ω1−lnξβ1lᵿ1/ᵿ,e−ω1−lnξβ1uᵿ1/ᵿ,e−ω1−lnvβ1lᵿ1/ᵿ,e−ω1−lnvβ1uᵿ1/ᵿ⊗1−e−ω2−ln1−uβ2lᵿ1/ᵿ,1−e−ω2−ln1−uβ2uᵿ1/ᵿ,e−ω2−lnξβ2lᵿ1/ᵿ,e−ω2−lnξβ2uᵿ1/ᵿ,e−ω2−lnvβ2lᵿ1/ᵿ,e−ω2−lnvβ2uᵿ1/ᵿ=1−e−ω1−ln1−uβ1lᵿ+ω2−ln1−uβ2lᵿ1/ᵿ,1−e−ω1−ln1−uβ1uᵿ+ω2−ln1−uβ2uᵿ1/ᵿ,e−ω1−lnξβ1lᵿ+ω2−lnξβ2lᵿ1/ᵿ,e−ω1−lnξβ1uᵿ+ω2−lnξβ2uᵿ1/ᵿ,e−ω1−lnvβ1lᵿ+ω2−lnvβ2lᵿ1/ᵿ,e−ω1−lnvβ1uᵿ+ω2−lnvβ2uᵿ1/ᵿ=1−e−∑j=12ωj−ln1−uβjlᵿ1/ᵿ,1−e−∑j=12ωj−ln1−uβjlᵿ1/ᵿ,e−∑j=12ωj−lnξβjlᵿ1/ᵿ,e−∑j=12ωj−lnξβjuᵿ1/ᵿ,e−∑j=12ωj−lnvβjlᵿ1/ᵿ,e−∑j=12ωj−lnvβjuᵿ1/ᵿ.
Hence, we have shown that equation satisfy all the condition forn=2.
Suppose it is also true forn=k, we have(10)IVPFAAWAβ1,β2,…,βk=⊕j=1kωjβj=1−e−∑j=1kωj−ln1−uβjlᵿ1/ᵿ,1−e−∑j=1kωj−ln1−uβjuᵿ1/ᵿ,e−∑j=1kωj−lnξβjlᵿ1/ᵿ,e−∑j=1kωj−lnξβjuᵿ1/ᵿ,e−∑j=1kωj−lnvβjlᵿ1/ᵿ,e−∑j=1kωj−lnvβjuᵿ1/ᵿ.
Now, we have to show that forn=k+1, then we obtain the following equation:(11)IVPFAAWAβ1,β2,…,βk,βk+1=⊕j=1kωkβk⊕ωk+1βk+1=1−e−∑j=1kωj−ln1−uβjlᵿ1/ᵿ,1−e−∑j=1kωj−ln1−uβjuᵿ1/ᵿ,e−∑j=1kωj−lnξβjlᵿ1/ᵿ,e−∑j=1kωj−lnξβjuᵿ1/ᵿe−∑j=1kωj−lnvβjlᵿ1/ᵿ,e−∑j=1kωj−lnvβjuᵿ1/ᵿ⊕1−e−ωk+1−ln1−uβk+1lᵿ1/ᵿ,1−e−ωk+1−ln1−uβk+1uᵿ1/ᵿ,e−ωk+1−lnξk+1lᵿ1/ᵿ,e−ωk+1−lnξk+1uᵿ1/ᵿe−ωk+1−lnvβk+1lᵿ1/ᵿ,e−ωk+1−lnvβk+1uᵿ1/ᵿ=1−e−∑j=1k+1ωj−ln1−uβjlᵿ1/ᵿ,1−e−∑j=1k+1ωj−ln1−uβjuᵿ1/ᵿ,e−∑j=1k+1ωj−lnξβjlᵿ1/ᵿ,e−∑j=1k+1ωj−lnξβjuᵿ1/ᵿ,e−∑j=1k+1ωj−lnvβjlᵿ1/ᵿ,e−∑j=1k+1ωj−lnvβjuᵿ1/ᵿ.
Hence, proved.Theorem 3 (Idempotency).
If allβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur are identical, i.e., βj=β. Then IVPFAAWAβ1,β2,…,βn=β.Proof.
We know thatβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur=uβlr,uβur,ξβlr,ξβur,vβlr,vβur=β, then we have(12)IVPFAAWAβ1,β2,…,βn=1−e−∑j=1nωj−ln1−uβjlᵿ1/ᵿ,1−e−∑j=1nωj−ln1−uβjuᵿ1/ᵿ,e−∑j=1nωj−lnξβjlᵿ1/ᵿ,e−∑j=1nωj−lnξβjuᵿ1/ᵿ,e−∑j=1nωj−lnvβjlᵿ1/ᵿ,e−∑j=1nωj−lnvβjuᵿ1/ᵿ=1−e−−ln1−uβlᵿ1/ᵿ,1−e−−ln1−uβuᵿ1/ᵿ,e−−lnξβlᵿ1/ᵿ,e−−lnξβuᵿ1/ᵿ,e−−lnvβlᵿ1/ᵿ,e−−lnvβuᵿ1/ᵿ=1−eln1−uβl,1−eln1−uβu,elnξβl,elnξβu,elnvβl,elnvβu=uBlr,uBur,ξβlr,ξβur,vBlr,vBur=β.
Therefore,IVPFAAWAβ1,β2,…,βn=β.Theorem 4.
(Boundedness). Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Suppose β−=minβ1,β2,…,βn and β+=maxβ1,β2,…,βn.Then(13)β−≤IVPFAAWAβ1,β2,…,βn≤β+.Proof.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Suppose β−=minβ1,β2,…,βn=uβ−,ξβ−,vβ− and β+=maãβ1,β2,…,βn=uβ+,ξβ+,vβ+. Then(14)1−e−∑j=1nωj−ln1−uβl−ᵿ1/ᵿ,1−e−∑j=1nωj−ln1−uβu−ᵿ1/ᵿ≤1−e−∑j=1nωj−ln1−uβjlᵿ1/ᵿ,1−e−∑j=1nωj−ln1−uβjuᵿ1/ᵿ≤1−e−∑j=1nωj−ln1−uβl+ᵿ1/ᵿ,1−e−∑j=1nωj−ln1−uβu+ᵿ1/ᵿe−∑j=1nωj−lnξβl−ᵿ1/ᵿ,e−∑j=1nωj−lnξβu−ᵿ1/ᵿ≥e−∑j=1nωj−lnξβjlᵿ1/ᵿ,e−∑j=1nωj−lnξβjuᵿ1/ᵿ≥e−∑j=1nωj−lnξβl+ᵿ1/ᵿ,e−∑j=1nωj−lnξβu+ᵿ1/ᵿe−∑j=1nωj−lnvβl−ᵿ1/ᵿ,e−∑j=1nωj−lnvβu−ᵿ1/ᵿ≥e−∑j=1nωj−lnvβjlᵿ1/ᵿ,e−∑j=1nωj−lnvβjuᵿ1/ᵿ≥e−∑j=1nωj−lnvβl+ᵿ1/ᵿ,e−∑j=1nωj−lnvβu+ᵿ1/ᵿ.
Therefore,β−≤IVPFAAWAβ1,β2,…,βn≤β+.Theorem 5 (Monotonicity).
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. If βj≤βj′. Then(15)PFAAWAβ1,β2,…,βn≤IVPFAAWAβ1′,β2′,…,βn′.Proof.
Straightforward.Now we discuss IVPFAA ordered weighted averaging (IVPFAAOWA) AO by using some basic
operations defined in Definition8
where we keep the ordered position of the alternatives in the aggregation
process.Definition 10.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. The IVPFAAOWA is a function IVPFAAOWA:Ln⟶L defined as:(16)IVPFAAOWAβ1,β2,β3,…,β3=⊕j=1nωjβσj,where σ1,σ2,σ3,…,σn be the permutation of 1,2,…,n such that βσj−1≥βσj.
Now, we establish a relation for using the IVPFAAOWA operator on IVPFVs that
depend upon AA operations.Theorem 6.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. The IVPFAAOWA is a function IVPFAAOWA:Ln⟶L then:(17)IVPFAAOWAβ1,β2,β3,…,β3=1−e−∑j=1nωj−ln1−uβσjlᵿ1/ᵿ,1−e−∑j=1nωj−ln1−uβσjuᵿ1/ᵿ,e−∑j=1nωj−lnξβσjlᵿ1/ᵿ,e−∑j=1nωj−lnξβσjuᵿ1/ᵿ,e−∑j=1nωj−lnvβσjlᵿ1/ᵿ,e−∑j=1nωj−lnvβσjuᵿ1/ᵿ.The IVPFAAOWA operator satisfies the properties discussed in Theorems3, 4, and 5 analogously.Clearly, from Definitions10 and 11, the IVPFAAWA operator and IVPFAAOWA
operator weights only the IVPFVs and the ordered positions of IVPFVs respectively.
By merging these two we get the IVPF AA hybrid averaging (IVPFAAHA) operator which
is discussed as follows:Definition 11.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Then IVPFAAHA operator of n dimension is a function IVPFAAHA:Ln⟶L defined as:(18)IVPFAAHAβ1,β2,…,βn=⊕j=1nζjβ˜σj,where ζ=ζ1,ζ2,…,ζn be the weight vector of PFNs, with ζj∈0,1,∑j=1nζj=1 and β˜j=nωjβj with n as a balancing coefficient.
Now, we can derive the theorem for the IVPFAAHA operator as follows.Theorem 7.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Then, the aggregated value of the
IVPFAAHA operator is also an IVPFV, and(19)IVPFAAHAβ1,β2,…,βn=1−e−∑j=1nζj−ln1−u˜σjlᵿ1/ᵿ,1−e−∑j=1nζj−ln1−u˜σjuᵿ1/ᵿ,e−∑j=1nζj−lnξ˜σjlᵿ1/ᵿ,e−∑j=1nζj−lnξ˜σjuᵿ1/ᵿ,e−∑j=1nζj−lnv˜σjlᵿ1/ᵿ,e−∑j=1nζj−lnv˜σjuᵿ1/ᵿ.Proof.
This proof is similar to Theorem2.Now, we discuss some special cases of IVPFAAHA operators as follows.Theorem 8.
The IVPFAAWA and IVPFAAOWA operators are special cases of the IVPFAAHA
operator.Proof.
Supposeζ=1/n,1/n,…,1/nT, Then:(20)IVPFAAHAβ1,β2,…,βn=ζ1β˜σ1⊕ζ2β˜σ2⊕…⊕ζnβ˜σn=1nβ˜σ1⊕β˜σ2⊕…⊕β˜σn=ω1β1⊕ω1β2,…,ωnβn=IVPFAAWAβ1,β2,…,βn.
Supposeω=1/n,1/n,…,1/nT. Then β˜j=βjj=1,2,…,n and(21)IVPFAAHAβ1,β2,…,βn=ζ1β˜σ1⊕ζ2β˜σ2⊕…⊕ζnβ˜σn=ζ1βσ1⊕ζ2βσ2,⊕…,⊕ζnβσn=IVPFAAOWAβ1,β2,…,βn.
This completes the proof.


## 5. Interval Valued Picture Fuzzy Aczel–Alsina Geometric Aggregation
Operators

In this section, we aim to introduce some AOs based on Definition8. We introduce the notion of the
IVPFAAWG, IVPFAAOWG, and IVPFAAHG operators.Definition 12.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Then the IVPFAAWG operator of n dimension is a function IVPFAAWG:Ln⟶L defined as follows:(22)IVPFAAWGβ1,β2,…,βn=⊗j=1nβjωj.
By using some basic fundamental operations defined in Definition8 and Definition 12, we proposed a result for the
IVPFAAWG operator in the following Theorem 14.Theorem 9.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Then the aggregated value of the
IVPFAAWG operator is also an IVPFV given by(23)IVPFAAWGβ1,β2,…,βn=e−∑j=1n−lnuβjlωjᵿ1/ᵿ,e−∑j=1n−lnuβjuωjᵿ1/ᵿ,1−e−∑j=1n−ln1−ξβjlωjᵿ1/ᵿ,1−e−∑j=1n−ln1−ξβjuωjᵿ1/ᵿ,1−e−∑j=1n−ln1−vβjlωjᵿ1/ᵿ,1−e−∑j=1n−ln1−vβjuωjᵿ1/ᵿProof.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs, we prove this theorem by using
mathematical induction technique.
Forn=2, we have(24)β1ω1=e−−lnuβ1lω1ᵿ1/ᵿ,e−−lnuβ1uω1ᵿ1/ᵿ,1−e−−ln1−ξβ1lω1ᵿ1/ᵿ,1−e−−ln1−ξβ1uω1ᵿ1/ᵿ,1−e−−ln1−vβ1lω1ᵿ1/ᵿ,1−e−−ln1−vβ1uω1ᵿ1/ᵿ,β2ω2=e−−lnuβ2lω2ᵿ1/ᵿ,e−−lnuβ2uω2ᵿ1/ᵿ,1−e−−ln1−ξβ2lω2ᵿ1/ᵿ,1−e−−ln1−ξβ2uω2ᵿ1/ᵿ,1−e−−ln1−vβ2lω2ᵿ1/ᵿ,1−e−−ln1−vβ2uω2ᵿ1/ᵿ,
Now, from the Definition8 and
Definition 12,(25)IVPFAAWGβ1,β2=β1ω1⊗β2ω2=e−−lnuβ1lω1ᵿ1/ᵿ,e−−lnuβ1uω1ᵿ1/ᵿ,1−e−−ln1−ξβ1lω1ᵿ1/ᵿ,1−e−−ln1−ξβ1uω1ᵿ1/ᵿ,1−e−−ln1−vβ1lω1ᵿ1/ᵿ,1−e−−ln1−vβ1uω1ᵿ1/ᵿ⊗e−−lnuβ2lω2ᵿ1/ᵿ,e−−lnuβ2uω2ᵿ1/ᵿ,1−e−−ln1−ξβ2lω2ᵿ1/ᵿ,1−e−−ln1−ξβ2uω2ᵿ1/ᵿ,1−e−−ln1−vβ2lω2ᵿ1/ᵿ,1−e−−ln1−vβ2uω2ᵿ1/ᵿ=e−−lnuβ1lω1ᵿ+−lnuβ2lω2ᵿ1/ᵿ,e−−lnuβ1uω1ᵿ+−lnuβ2uω2ᵿ1/ᵿ,1−e−−ln1−ξβ1lω1ᵿ+−ln1−ξβ2lω2ᵿ1/ᵿ,1−e−−ln1−ξβ1uω1ᵿ+−ln1−ξβ2uω2ᵿ1/ᵿ,1−e−−ln1−vβ1lω1ᵿ+−ln1−vβ2lω2ᵿ1/ᵿ,1−e−−ln1−vβ1uω1ᵿ+−ln1−vβ2uω2ᵿ1/ᵿ=e−∑j=12−lnuβjlωjᵿ1/ᵿ,e−∑j=12−lnuβjuωjᵿ1/ᵿ,1−e−∑j=12−ln1−ξβjlωjᵿ1/ᵿ,1−e−∑j=12−ln1−ξβjuωjᵿ1/ᵿ,1−e−∑j=12−ln1−vβjlωjᵿ1/ᵿ,1−e−∑j=12−ln1−vβjuωjᵿ1/ᵿ
Hence, it is true forn=2.
Suppose it is also true forn=k, we have(26)IVPFAAWGβ1,β2,…,βk=⊗j=1kβjωj=e−∑j=1k−lnuβjlωjᵿ1/ᵿ,e−∑j=1k−lnuβjuωjᵿ1/ᵿ,1−e−∑j=1k−ln1−ξβjlωjᵿ1/ᵿ,1−e−∑j=1k−ln1−ξβjuωjᵿ1/ᵿ1−e−∑j=1k−ln1−vβjlωjᵿ1/ᵿ,1−e−∑j=1k−ln1−vβjuωjᵿ1/ᵿ,.
Now, we have to show that forn=k+1, the result holds:(27)IVPFAAWGβ1,β2,…,βk,βk+1=⊗j=1kβjωj⊗βk+1ωk+1=e−∑j=1k−lnuβjlωjᵿ1/ᵿ,e−∑j=1k−lnuβjuωjᵿ1/ᵿ,1−e−∑j=1k−ln1−ξβjlωjᵿ1/ᵿ,1−e−∑j=1k−ln1−ξβjuωjᵿ1/ᵿ,1−e−∑j=1k−ln1−vβjlωjᵿ1/ᵿ,1−e−∑j=1k−ln1−vβjuωjᵿ1/ᵿ⊗e−−lnuβk+1lωk+1ᵿ1/ᵿ,e−−lnuβk+1uωk+1ᵿ1/ᵿ,1−e−−ln1−ξβk+1lωk+1ᵿ1/ᵿ,1−e−−ln1−ξβk+1uωk+1ᵿ1/ᵿ,1−e−−ln1−vβk+1lωk+1ᵿ1/ᵿ,1−e−−ln1−vβk+1uωk+1ᵿ1/ᵿ=e−∑j=1k+1−lnuβjlωjᵿ1/ᵿ,e−∑j=1k+1−lnuβjuωjᵿ1/ᵿ,1−e−∑j=1k+1−ln1−ξβjlωjᵿ1/ᵿ,1−e−∑j=1k+1−ln1−ξβjuωjᵿ1/ᵿ,1−e−∑j=1k+1−ln1−vβjlωjᵿ1/ᵿ,1−e−∑j=1k+1−ln1−vβjuωjᵿ1/ᵿ.
Hence proved.
Now we prove that the IVPFAAWG operator satisfies the basic properties of an
aggregation function in Theorem10–12.Theorem 10 (Idempotency).
If allβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur are identical i.e. βj=β. Then IVPFAAWGβ1,β2,…,βn=β.Proof.
Straightforward.Theorem 11.
(Boundedness). Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Suppose β−=minβ1,β2,…,βn and β+=maxβ1,β2,…,βn. Then(28)β−≤IVPFAAWGβ1,β2,…,βn≤β+.Proof.
Straightforward.Theorem 12.
(Monotonicity):
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of PFNs. If βj≤βj′. Then(29)IVPFAAWGβ1,β2,…,βn≤IVPFAAWGβ1′,β2′,…,βn′.Proof.
Straightforward.
Now, we discuss IVPF AA ordered weighted geometric (IVPFAAOWG) AO by using some
basic operations defined in Definition9 where we keep the ordered position of the alternatives in the
aggregation process.Definition 13.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. The IVPFAAOWG is a function IVPFAAOWG:Ln⟶L defined as follows:(30)IVPFAAOWGβ1,β2,β3,…,β3=⊗j=1nβσjωj,where σ1,σ2,σ3,…,σn be the permutation of 1,2,…,n such that βσj−1≥βσj.
Now we establish a relation for using the IVPFAAOWG operator on IVPFVs that
depend upon AA operations.Theorem 13.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. The IVPFAAOWG is a function IVPFAAOWG:Ln⟶L then(31)IVPFAAOWGβ1,β2,β3,…,β3=e−∑j=1n−lnuβσjlωjᵿ1/ᵿ,e−∑j=1n−lnuβσjuωjᵿ1/ᵿ,1−e−∑j=1n−ln1−ξβσjlωjᵿ1/ᵿ,1−e−∑j=1n−ln1−ξβσjuωjᵿ1/ᵿ,1−e−∑j=1n−ln1−vβσjlωjᵿ1/ᵿ,1−e−∑j=1n−ln1−vβσjuωjᵿ1/ᵿ.The IVPFAAOWG operator satisfies the properties discussed in Theorem 15, 16, and 17
analogously.Clearly, from the Definitions12 and
13, the IVPFAAWG operator and
IVPFAAOWG operator weights only the IVPFVs and the ordered positions of IVPFVs,
respectively. By merging these two we get the IVPF AA hybrid geometric (IVPFAAHG)
operator which is discussed as follows.Definition 14.
Supposeβj=uβjlr,uβjur,ξβjlr,ξβjur,vβjlr,vβjur be the set of IVPFVs. Then IVPFAAHA operator is a function IVPFAAHG:Ln⟶L defined as follows:(32)IVPFAAHGβ1,β2,…,βn=⊗j=1nβ˜σjζj,where ζ=ζ1,ζ2,…,ζn be the weight vector of IVPFVs, with ζj∈0,1,∑j=1nζj=1 and β˜j=βjnωj with n as the balancing coefficient.
Now, we can derive the theorem for the IVPFAAHG operator as follows.Theorem 14.
Supposeβj=uBjlr,uBjur,ξβjlr,ξβjur,vBjlr,vBjur be the set of IVPFVs. Then the aggregated value of the
IVPFAAHG operator is also an IVPFV, and(33)IVPFAAHAβ1,β2,…,βn=e−∑j=1n−lnu˜σjlζjᵿ1/ᵿ,e−∑j=1n−lnu˜σjuζjᵿ1/ᵿ,1−e−∑j=1n−ln1−ξ˜σjlζjᵿ1/ᵿ,1−e−∑j=1n−ln1−ξ˜σjuζjᵿ1/ᵿ,1−e−∑j=1n−ln1−v˜σjlζjᵿ1/ᵿ,1−e−∑j=1n−ln1−v˜σjuζjᵿ1/ᵿ.Proof.
We can prove this theorem easily by using the steps of Theorem2.


## 6. MADM Technique Based on Aczel–Alsina AOs

In this section, we shall discuss the method of MADM problems by using the IVPFAAWA
and IVPFAAWG operators and apply them to a practical problem.Suppose the alternativesτ=τ1,τ2,…,τℷ to be ranked based on attributes ĵ=ĵ1,ĵ2,…,ĵn and let ω=ω1,ω2,…,ωnT be the weight vector as discussed earlier. Here, the uncertain
information is discussed in the form of IVPFVs where the four aspects of the
uncertain information are elaborated by MV, AV, NMV, and RV in a decision matrix U=Myjℷ×n given by the decision-makers. Each entry of the information of U is denoted by PFNs Myj=uβyjlr,uβyjur,ξβyjlr,ξβyjur,vβyjlr,vβyjur that represents the alternatives’ evaluation values τy for criteria ĵj such that uβyjlr,uβyjur,ξβyjlr,ξβyjur,vβyjlr,vβyjur are the subintervals of 0,1 and 0≤uβyju+ξβyju+vβyju≤1 and y,j denote the indexing terms.The detailed steps of the algorithm based on IVPFAAWA and IVPFAAWG operators are
given as follows:Step 1: This step involves obtaining information from experts about the
alternatives based on IVPFVs in the form of a decision matrixU=Myjℷ×n as discussed earlier.Step 2: We get the normalization matrixU¯=M¯yjℷ×n of the decision matrix by the transformation U=Myjℷ×n.(34)M¯yj=Myj,forbenefitattribute,Myjc,forcostattribute.WhereMyjc be the complement of the decision matrix Myj. If all the attributes are identical (in nature) then
it is not necessary to change them but when the attributes are distinct
(in nature) then it’s necessary to make them identical i.e., to
make all of them benefit attributes.Step 3: This step involves using the IVPFAAWA and IVPFAAWG operators
discussed in Definition9
and Definition 12,
respectively, to aggregate the information obtained in the previous step.(35)IVPFAAWAM¯y1,M¯y2,…,M¯yn=1−e−∑j=1nωj−ln1−uM¯yjlᵿ1/ᵿ,1−e−∑j=1nωj−ln1−uM¯yjuᵿ1/ᵿ,e−∑j=1nωj−lnξM¯yjlᵿ1/ᵿ,e−∑j=1nωj−lnξM¯yjuᵿ1/ᵿ,e−∑j=1nωj−lnvM¯yjlᵿ1/ᵿ,e−∑j=1nωj−lnvM¯yjuᵿ1/ᵿ,IVPFAAWGM¯y1,M¯y2,…,M¯yn=e−∑j=1n−lnuM¯yjlωjᵿ1/ᵿ,e−∑j=1n−lnuM¯yjuωjᵿ1/ᵿ,1−e−∑j=1n−ln1−ξM¯yjlωjᵿ1/ᵿ,1−e−∑j=1n−ln1−ξM¯yjuωjᵿ1/ᵿ,1−e−∑j=1n−ln1−vM¯yjlωjᵿ1/ᵿ,1−e−∑j=1n−ln1−vM¯yjuωjᵿ1/ᵿ,using the score formula defined in Definition7.Step 5: To evaluate the most desirable alternative by using the method of
ranking and ordering technique of obtained score values by our invented
methodologies.


### 6.1. Applications in Construction Engineering

Engineering and construction are comprehensive businesses that play a vital role
growth of the economy of a country. It provides millions of jobs opportunity and
about a 15% growth rate in the GDP. Engineering assumes an important part
in guaranteeing the development and improvement of a country’s economy as
well as in working on the personal satisfaction of citizens inside the country.
Its dispatch is far more extensive than just structures and bridges, crossing
enhancements in sustainable power innovations and arrangements through to
worldwide health challenges. Accordingly, there is a significant connection
between a nation’s design limit and its monetary turn of events. In this
part of the report, we set everything up at a wide macroeconomic level,
featuring the manners by which designing can add to the financial turn of events
before considering the particular markers that add to the Engineering Index in
the remaining areas.Engineering and construction are known as the backbone of economic growth, they
play a major role to uplift economic development, financial development, and
economic activities. The engineering and economic sector considered a big
partner of the Development and construction industry assumes a significant part
in the monetary inspire and improvement of the country. It very well may be
viewed as an instrument of producing the business and extending open positions
to a huge number of the unskilled, semiskilled, and skilled workforce. It
likewise assumes a key part in creating pay in both formal and informal areas.
It supplements the unfamiliar trade profit got from the exchange of development
material and designing administrations.


### 6.2. Numerical Example

Several types of construction or building material are utilized to contract a
building. At the time when we select construction materials, we have to ensure
the safety and long life of a building. In this numerical example, we have to
choose suitable construction material such as concrete from four different types
of concreteτi=i=1,2,3,4 on the based following characteristics ĵi=i=1,2,3.(1)
ĵ1 represents the size and shape of concrete(2)
ĵ2 about the quality of concrete (strength and
hardness)(3)
ĵ3 represents the affordable cost of concreteWe have to select the best concrete material according to the weight vectors of
attributes areω=0.35,0.45,0.2T. These weight vectors are distributed by the decision-maker.
The set of information in the form of IVPFVs is depicted in Table 1.Table 1
Decision matrix of IVPF information.
τ1τ2τ3τ4ĵ10.3,0.35,0.28,0.310.32,0.330.15,0.24,0.22,0.280.33,0.360.3,0.45,0.12,0.160.18,0.230.6,0.62,0.12,0.140.22,0.24ĵ20.23,0.27,0.34,0.380.1,0.150.42,0.45,0.33,0.370.15,0.170.12,0.22,0.23,0.260.35,0.390.23,0.24,0.12,0.160.56,0.59ĵ30.29,0.34,0.38,0.410.1,0.140.42,0.45,0.33,0.360.16,0.180.44,0.45,0.16,0.180.35,0.370.18,0.2,0.22,0.230.56,0.57


### 6.3. The Method Based on IVPFAAWA and IVPFAAWG Operators

We utilize the IVPFAAWA operator to build up MADM theory with IVPF data for the
selection of the most desirable person which is discussed as follows:Step 1. Suppose that forᵿ=1. We calculate the alternative values τy of four applicants τyy=1,2,3,4 by using the IVPFAAWA operator and we have(36)τ1=0.267,0.313,0.325,0.359,0.150,0.195,τ2=0.337,0.384,0.286,0.334,0.200,0.224,τ3=0.258,0.356,0.170,0.204,0.277,0.321,τ4=0.380,0.398,0.135,0.164,0.404,0.428,Step 2. Now we calculate the score valuesSτyy=1,2,3,4 of IVPFVs. Sτ1=0.0933,Sτ2=0.1143,Sτ3=0.1040,Sτ4=0.1124.Step 3. Ranking and order of four applicantsτyy=1,2,3,4 corresponding to the score values Sτyy=1,2,3,4 of the IVPFVs.Step 4. The most desirable choice isτ4.In Table2, we show the consequences of
the alternatives τyy=1,2,3,4 by utilizing the AOs of IVPFAAWA and IVPFAAWG. We observe that
the score values of the alternatives gradually increase and decrease if the
magnitude of ᵿ is increased. It means that the choice maker can decide the
most suitable value according to their preferences.Table 2
Ranking and ordering of alternatives.
Sτ1Sτ2Sτ3Sτ4OrderingIVPFAAWA0.09330.11430.10400.1124τ2≻τ4≻τ3≻τ1IVPFAAWG−0.0192−0.02279−0.0187−0.02275τ3≻τ1≻τ4≻τ2In Figure1, we show the score values of
IVPFAAWA and IVPFAAWG operators as a graphical representation, which are
depicted in Table 2.Figure 1
Representation of IVPFAAWA and IVPFAAWG operators forᵿ=1.


### 6.4. The Impact of the Parameter ᵿ in This Technique

To classify the influence of the different sizes of the boundaryᵿ, we see advantages of unmistakable boundary ᵿ inside our referenced procedures to arrange the other options.
The requested impacts of the other options τii=1,2…,4 in light of IVPFAAWA administrators are introduced in Table
3, and outlined graphically in
Figure 2. It is obvious that when the
greatness of ℵ increments for IVPFAAWA administrator, the score upsides
of the choices increment step by step, however, the comparing requests is
something very similar and it is τ4≻τ2≻τ3≻τ1. By seeing the similarities of the ranking and ordering of the
score values of the different values of ᵿ, we observe that this technique show the property of
isotonicity, the decision‐makers can choose the reasonable worth as
indicated by their inclinations.Table 3
Ranking and score values by IVPFAAWA operator with variousᵿ.
ᵿSτ1Sτ2Sτ3Sτ4Ordering and ranking10.09330.11430.10400.1124τ2≻τ4>τ3>τ130.09920.13080.12670.1662τ4≻τ2≻τ3≻τ150.10340.14020.14210.1968τ4≻τ2≻τ3≻τ1150.11400.15720.17230.2366τ4≻τ2≻τ3≻τ1250.11790.16220.18060.2455τ4≻τ2≻τ3≻τ11050.12380.16850.19090.2565τ4≻τ3≻τ2≻τ11550.12460.16920.19190.2577τ4≻τ3≻τ2≻τ12010.12490.16960.19240.2582τ4≻τ3≻τ2≻τ1Figure 2
Consequences of the IVPFAAWA operator by the variation in the value ofᵿ.In above Table3, we observe that, when
we take ᵿ=1, then the order of score values τ2≻τ4>τ3>τ1. If we take ᵿ=3, then upcoming all the values of the score function will remain
the same. It means that ᵿ=3 is the stability point. It is also a highly observable thing
when we take ᵿ as even then there will be no aggregation result found. So, we
must take ᵿ as an odd number.Figure2 depicts the consequences of the
IVPFAAWA operator when the magnitude value of ᵿ is variate. From Figure 2, we analyze that after ᵿ=3 the consequences of proposed AOs gradually decrease.In Table4, we see that when we change
the values of ᵿ then the order of score values will be changed. When we take ᵿ=25, then remaining all the values of the score function will
remain stable. It means that a stability point exists when ᵿ=3. When we take the value of ᵿ as an even number then the values of the score function do not
exist.Table 4
Ranking and score values by IVPFAAWG operators with various ᵿ.
ᵿSτ1Sτ2Sτ3Sτ4Ordering and ranking1−0.0192−0.2279−0.0187−0.2275τ3≻τ1>τ4>τ23−0.0367−0.0399−0.0324−0.0427τ3≻τ1>τ2>τ45−0.0294−0.0314−0.0239−0.0373τ3≻τ1>τ2>τ415−0.0211−0.0212−0.0157−0.0325τ3≻τ1>τ4>τ225−0.0203−0.0198−0.0153−0.0318τ3≻τ2>τ1>τ4105−0.0200−0.0188−0.0153−0.0310τ3≻τ2>τ1>τ4155−0.0200−0.0188−0.0153−0.0310τ3≻τ2>τ1>τ4201−0.0200−0.0188−0.0153−0.0310τ3≻τ2>τ1>τ4Figure3 shows a graphical representation
of the score values which are obtained by the IVPFAAWG operator for the
different values of the magnitude of ᵿ.Figure 3
Consequences of the IVPFAAWG operator by the variation in the value ofᵿ.


## 6.1. Applications in Construction Engineering

Engineering and construction are comprehensive businesses that play a vital role
growth of the economy of a country. It provides millions of jobs opportunity and
about a 15% growth rate in the GDP. Engineering assumes an important part
in guaranteeing the development and improvement of a country’s economy as
well as in working on the personal satisfaction of citizens inside the country.
Its dispatch is far more extensive than just structures and bridges, crossing
enhancements in sustainable power innovations and arrangements through to
worldwide health challenges. Accordingly, there is a significant connection
between a nation’s design limit and its monetary turn of events. In this
part of the report, we set everything up at a wide macroeconomic level,
featuring the manners by which designing can add to the financial turn of events
before considering the particular markers that add to the Engineering Index in
the remaining areas.Engineering and construction are known as the backbone of economic growth, they
play a major role to uplift economic development, financial development, and
economic activities. The engineering and economic sector considered a big
partner of the Development and construction industry assumes a significant part
in the monetary inspire and improvement of the country. It very well may be
viewed as an instrument of producing the business and extending open positions
to a huge number of the unskilled, semiskilled, and skilled workforce. It
likewise assumes a key part in creating pay in both formal and informal areas.
It supplements the unfamiliar trade profit got from the exchange of development
material and designing administrations.


## 6.2. Numerical Example

Several types of construction or building material are utilized to contract a
building. At the time when we select construction materials, we have to ensure
the safety and long life of a building. In this numerical example, we have to
choose suitable construction material such as concrete from four different types
of concreteτi=i=1,2,3,4 on the based following characteristics ĵi=i=1,2,3.(1)
ĵ1 represents the size and shape of concrete(2)
ĵ2 about the quality of concrete (strength and
hardness)(3)
ĵ3 represents the affordable cost of concreteWe have to select the best concrete material according to the weight vectors of
attributes areω=0.35,0.45,0.2T. These weight vectors are distributed by the decision-maker.
The set of information in the form of IVPFVs is depicted in Table 1.Table 1
Decision matrix of IVPF information.
τ1τ2τ3τ4ĵ10.3,0.35,0.28,0.310.32,0.330.15,0.24,0.22,0.280.33,0.360.3,0.45,0.12,0.160.18,0.230.6,0.62,0.12,0.140.22,0.24ĵ20.23,0.27,0.34,0.380.1,0.150.42,0.45,0.33,0.370.15,0.170.12,0.22,0.23,0.260.35,0.390.23,0.24,0.12,0.160.56,0.59ĵ30.29,0.34,0.38,0.410.1,0.140.42,0.45,0.33,0.360.16,0.180.44,0.45,0.16,0.180.35,0.370.18,0.2,0.22,0.230.56,0.57


## 6.3. The Method Based on IVPFAAWA and IVPFAAWG Operators

We utilize the IVPFAAWA operator to build up MADM theory with IVPF data for the
selection of the most desirable person which is discussed as follows:Step 1. Suppose that forᵿ=1. We calculate the alternative values τy of four applicants τyy=1,2,3,4 by using the IVPFAAWA operator and we have(36)τ1=0.267,0.313,0.325,0.359,0.150,0.195,τ2=0.337,0.384,0.286,0.334,0.200,0.224,τ3=0.258,0.356,0.170,0.204,0.277,0.321,τ4=0.380,0.398,0.135,0.164,0.404,0.428,Step 2. Now we calculate the score valuesSτyy=1,2,3,4 of IVPFVs. Sτ1=0.0933,Sτ2=0.1143,Sτ3=0.1040,Sτ4=0.1124.Step 3. Ranking and order of four applicantsτyy=1,2,3,4 corresponding to the score values Sτyy=1,2,3,4 of the IVPFVs.Step 4. The most desirable choice isτ4.In Table2, we show the consequences of
the alternatives τyy=1,2,3,4 by utilizing the AOs of IVPFAAWA and IVPFAAWG. We observe that
the score values of the alternatives gradually increase and decrease if the
magnitude of ᵿ is increased. It means that the choice maker can decide the
most suitable value according to their preferences.Table 2
Ranking and ordering of alternatives.
Sτ1Sτ2Sτ3Sτ4OrderingIVPFAAWA0.09330.11430.10400.1124τ2≻τ4≻τ3≻τ1IVPFAAWG−0.0192−0.02279−0.0187−0.02275τ3≻τ1≻τ4≻τ2In Figure1, we show the score values of
IVPFAAWA and IVPFAAWG operators as a graphical representation, which are
depicted in Table 2.Figure 1
Representation of IVPFAAWA and IVPFAAWG operators forᵿ=1.


## 6.4. The Impact of the Parameter ᵿ in This Technique

To classify the influence of the different sizes of the boundaryᵿ, we see advantages of unmistakable boundary ᵿ inside our referenced procedures to arrange the other options.
The requested impacts of the other options τii=1,2…,4 in light of IVPFAAWA administrators are introduced in Table
3, and outlined graphically in
Figure 2. It is obvious that when the
greatness of ℵ increments for IVPFAAWA administrator, the score upsides
of the choices increment step by step, however, the comparing requests is
something very similar and it is τ4≻τ2≻τ3≻τ1. By seeing the similarities of the ranking and ordering of the
score values of the different values of ᵿ, we observe that this technique show the property of
isotonicity, the decision‐makers can choose the reasonable worth as
indicated by their inclinations.Table 3
Ranking and score values by IVPFAAWA operator with variousᵿ.
ᵿSτ1Sτ2Sτ3Sτ4Ordering and ranking10.09330.11430.10400.1124τ2≻τ4>τ3>τ130.09920.13080.12670.1662τ4≻τ2≻τ3≻τ150.10340.14020.14210.1968τ4≻τ2≻τ3≻τ1150.11400.15720.17230.2366τ4≻τ2≻τ3≻τ1250.11790.16220.18060.2455τ4≻τ2≻τ3≻τ11050.12380.16850.19090.2565τ4≻τ3≻τ2≻τ11550.12460.16920.19190.2577τ4≻τ3≻τ2≻τ12010.12490.16960.19240.2582τ4≻τ3≻τ2≻τ1Figure 2
Consequences of the IVPFAAWA operator by the variation in the value ofᵿ.In above Table3, we observe that, when
we take ᵿ=1, then the order of score values τ2≻τ4>τ3>τ1. If we take ᵿ=3, then upcoming all the values of the score function will remain
the same. It means that ᵿ=3 is the stability point. It is also a highly observable thing
when we take ᵿ as even then there will be no aggregation result found. So, we
must take ᵿ as an odd number.Figure2 depicts the consequences of the
IVPFAAWA operator when the magnitude value of ᵿ is variate. From Figure 2, we analyze that after ᵿ=3 the consequences of proposed AOs gradually decrease.In Table4, we see that when we change
the values of ᵿ then the order of score values will be changed. When we take ᵿ=25, then remaining all the values of the score function will
remain stable. It means that a stability point exists when ᵿ=3. When we take the value of ᵿ as an even number then the values of the score function do not
exist.Table 4
Ranking and score values by IVPFAAWG operators with various ᵿ.
ᵿSτ1Sτ2Sτ3Sτ4Ordering and ranking1−0.0192−0.2279−0.0187−0.2275τ3≻τ1>τ4>τ23−0.0367−0.0399−0.0324−0.0427τ3≻τ1>τ2>τ45−0.0294−0.0314−0.0239−0.0373τ3≻τ1>τ2>τ415−0.0211−0.0212−0.0157−0.0325τ3≻τ1>τ4>τ225−0.0203−0.0198−0.0153−0.0318τ3≻τ2>τ1>τ4105−0.0200−0.0188−0.0153−0.0310τ3≻τ2>τ1>τ4155−0.0200−0.0188−0.0153−0.0310τ3≻τ2>τ1>τ4201−0.0200−0.0188−0.0153−0.0310τ3≻τ2>τ1>τ4Figure3 shows a graphical representation
of the score values which are obtained by the IVPFAAWG operator for the
different values of the magnitude of ᵿ.Figure 3
Consequences of the IVPFAAWG operator by the variation in the value ofᵿ.


## 7. Comparative Analysis

In this section, to observe the validity and reliability of our discussed AOs in the
form of PFAAWA and PFAAWG operators. For this purpose we compare the experimental
results of our proposed AOs with existing AOs given in [11, 17, 53, 54].
We applied all existing AOs on the information given by the decision-maker depicted
in Table 1, AOs of PFDWA and PFDWG by Jana
et al. [17] and IVIFWA and IVIFWG by Xu and
Gou [54], are failed to aggregate the given
information. AOs of IVPHFWA and IVPHFWG by Kamacı et al. [11], and AOs of IVPFFWA and IVPFFWG by Mahmood
et al. [53] are also applied on the
information given in Table 1. The
consequences of current work and existing AOs are shown in Table 5.Table 5
Comparison of consequences of existing AOs with current AOs.
AOsEnvironmentRankingIVPFAAWA (current work)IVPFSsτ2≻τ4≻τ3≻τ1IVPFAAWG (current work)IVPFSsτ3≻τ1≻τ4≻τ2IVPHFWA [11]IVPFSsτ2≻τ4≻τ3≻τ1IVPHFWG [11]IVPFSsτ2≻τ1≻τ3≻τ2IVPFFWA [53]IVPFSsτ2≻τ4≻τ3≻τ1IVPFFWG [53]IVPFSsτ2≻τ3≻τ1≻τ4PFDWA [17]PFSsFailedPFDWG [17]PFSsFailedIVIFWA [54]IVIFSsFailedIVIFWG [54]IVIFSsFailedThe advantages of IVPFAOs over the existing AOs are as follows.The proposed AOs can eliminate the impact of useless data on the decision results.
The discussed AOs are very useful operators to calculate the wide range for
assigning MV, AV, and NMV. Further, we applied AOs discovered by Kamacı et al.
[11] and Mahmood et al. [53]. The consequences of the AOs discussed in
[11, 43] are similar but our proposed AOs gave more suitable results than
existing AOs.The following graphical representation shows the consequences of existing
methodologies with proposed methodologies in Figure4.Figure 4
Comparison of existing AOs with proposed AOs.


## 8. Conclusion

In this article, we enhanced the idea of IVPFS in the framework of IVPFAA AOs by
utilizing the basic operations of AA-TN and AA-TCN. We have to investigate new
aggregation operators just as the IVPFAAWA operator, IVPFAAOWA operator, IVPFAAHA
operator, IVPFAAWG operator, IVPFFAAOWG operator, and IVPFAAHG operator. We proved
that the IVPFAAWA and IVPFAAWG operators satisfy the basic aggregation properties
including idempotency, boundedness, and monotonicity. One of the most widely
discussed applications of AOs is MADM, and we studied the application of the IVPAAWA
and IVPFAAWG operator to analyze the selection of construction material, where the
information is based on IVPFVs. We also studied the impact of variable parameters
associated with IVPFAAWA and IVPFAAWG operators numerically. To see the reliability
of our proposed AOs, we developed a comparative analysis of our proposed work with
the existing work. In the near future, we aim to extend our proposed work in the
environment of bipolar-valued hesitant fuzzy sets [55, 56]. Moreover, we will also
extend our proposed techniques in the environment of complex q-rung orthopair graphs
and spherical fuzzy sets [57–59].



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*Source: 1707867-2023-03-14.xml*