Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.019408

ARTICLE

Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Set with Their Application to Solve Multi-Attribute Group Decision Making Problem

Rana Muhammad Zulqarnain1, Imran Siddique2, Aiyared Iampan3 and Dumitru Baleanu4,5,6,*

1Department of Mathematics, University of Management and Technology, Sialkot Campus, Lahore, 54770, Pakistan
2Department of Mathematics, University of Management and Technology, Lahore, 54000, Pakistan
3Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao, 56000, Thailand
4Department of Mathematics, Cankaya University, Balgat Ankara, 06530, Turkey
5Institute of Space Sciences, Magurele-Bucharest, 077125, Romania
6Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40447, Taiwan
*Corresponding Author: Dumitru Baleanu. Email: dumitru.baleanu@gmail.com
Received: 23 September 2021; Accepted: 15 November 2021

1  Introduction

MAGDM is considered as the most appropriate technique to find the finest alternative from all possible alternatives, following criteria or attributes. Conventionally, it is supposed that all information that accesses the alternative in terms of attributes and their corresponding weights are articulated in the form of crisp numbers. On the other hand, in real-life circumstances, most of the decisions are taken in situations where the objectives and limitations are usually indefinite or ambiguous. To overcome such ambiguities and anxieties, Zadeh offered the notion of the fuzzy set (FS) [1], a prevailing tool to handle the obscurities and uncertainties in decision making (DM). Such a set allocates to all objects a membership value ranging from 0 to 1. Mostly, experts consider membership and a non-membership value in the DM process which cannot be handled by FS. Atanassov [2] introduced the idea of the intuitionistic fuzzy set (IFS) to overcome the aforementioned limitation. In 2011, Wang et al. [3] presented numerous operations on IFS, such as Einstein product, Einstein sum, etc., and constructed some novel AOs. They also discussed some important properties of these operators and utilized their proposed operators to resolve multi-attribute decision making (MADM). Atanassov [4] presented a generalized form of IFS in the light of ordinary interval values, called interval-valued intuitionistic fuzzy set (IVIFS). Garg et al. [5] extended the concept of IFS and presented a novel concept of the cubic intuitionistic fuzzy set (CIFS) which is a successful tool to represent vague data by embedding both IFS and IVIFS directly. They also discussed several desirable properties of CIFS.

The above-mentioned models have been well-recognized by the specialists but the existing IFS is unable to handle the inappropriate and vague data because it is considered to envision the linear inequality between the membership and non-membership grades. For example, if decision-makers choose membership and non-membership values 0.9 and 0.6 respectively, then the above-mentioned IFS theory is unable to deal with it because 0.9 + 0.6 ≥ 1. To resolve the aforesaid limitation, Yager [6] presented the idea of the Pythagorean fuzzy set (PFS) by amending the basic condition κ+δ≤1 to κ2+δ2≤1 and developed some results associated with score function and accuracy function. Rahman et al. [7] developed the Pythagorean fuzzy Einstein weighted geometric operator and presented a MAGDM methodology utilizing their proposed operator. Zang et al. [8] developed some basic operational laws and prolonged the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method to resolve multi-criteria decision-making (MCDM) complications for PFS information. Wei et al. [9] offered the Pythagorean fuzzy power AOs along with basic characteristics, they also established a DM technique to resolve MADM difficulties based on presented operators. Wang et al. [10] offered the interaction operational laws for Pythagorean fuzzy numbers (PFNs) and developed power Bonferroni mean operators. To assess the professional health risk, IIbahar et al. [11] offered the Pythagorean fuzzy proportional risk assessment technique. Zhang [12] proposed a novel DM approach based on similarity measures to resolve multi-criteria group decision making (MCGDM) difficulties for the PFS.

All of the aforementioned techniques have a wide range of applications, but owing to their ineffectiveness, they have several restrictions with the parameterization tool. Presenting a solution to this type of uncertainty and obfuscation Molodtsov [13] established the idea of soft sets (SS) and described some basic operations with their characteristics to handle the above-mentioned confusion and ambiguity. Maji et al. [14] expanded the concept of SS and developed many basic and binary operations for it. Maji et al. [15] developed the fuzzy soft set with some desirable properties by merging two existing notions FS and SS. Maji et al. [16] developed the notion of the intuitionistic fuzzy soft set (IFSS) and some fundamental operations with their necessary properties. Garg et al. [5] presented the cubic IFSS and established some AOs for cubic IFSS. They also planned a DM technique based on their developed operators. Zulqarnain et al. [17] planned the TOPSIS method based on the correlation coefficient for interval-valued IFSS to solve MADM problems. Jiang et al. [18] introduced the notion of the interval-valued intuitionistic fuzzy soft set (IVIFSS) and discussed some of their basic properties. Narayanamoorthy et al. [19] proposed the score function for a normal wiggly hesitant fuzzy set and utilized it to expose the deepest ideas hidden in the thought-level of the decision-makers. Narayanamoorthy et al. [20] introduced the hesitant fuzzy subjective and objective weight integrated method to find weights under hesitant fuzzy information. They also presented a novel ranking methodology for hesitant fuzzy sets. Ramya et al. [21] developed the interval-valued hesitant Pythagorean fuzzy set under the normal wiggly mathematical methodology and used it to solve the MCDM problem. Peng et al. [22] merged two well-known theories PFS and SS and offered the concept of Pythagorean fuzzy soft set (PFSS). Zulqarnain et al. [23] developed the AOs for PFSS with their application for the green supplier chain management. Zulqarnain et al. [24] introduced an advanced form of AOs considering the interaction and construct a DM approach based on their developed interactive AOs. Smarandache [25] prolonged the idea of SS to hypersoft sets (HSS) by substituting the single-parameter function f with a multi-parameter (sub-attribute) function. He privileges that HSS proficiently contracts with inexact data comparative to SS.

MAGDM is a very effective and well-known tool to examine fuzzy data more effectively. Therefore, it is obvious from the published literature that the interval-valued structures are more general and increase more consideration in decision-making difficulties. The choice of vehicle is a key part of real-life and will advise on complications of MAGDM. Lack of thinking about the ambiguity of alternative associations will be the core motivation for some MAGDM concerns about the undesirable consequences. By using a wealth of existing content, it contains previous criticisms and suppressed sensitivities. Many logical and scientific tools/procedures are offended in the literature for choosing the most suitable vehicle. As far as we know, there is currently no work on the AOs of IVPFSS. Therefore, this article proposes some operational laws for interval-valued Pythagorean fuzzy soft numbers (IVPFSN). The presented IVPFSN is well worth observing the inaccurate information that occurs in the complications of daily life. Therefore, the main purpose of this work is to propose new IVPFSWA and IVPFSWG operators based on the established operational laws. An algorithm based on the proposed operators to solve the decision-making problem is proposed. To prove the effectiveness of the proposed decision-making method, we use a numerical example to illustrate it. The main benefit of the proposed operator is that the proposed operator can reduce to IVIFSS and IVFSS operators under some specific conditions of unconfidence. The organization of this paper is given as follows: Section 2 of this paper consists of some basic concepts which help us to develop the structure of the following research. In Section 3, some novel operational laws for IVPFSN have been proposed. Also, in the same section, IVPFSWA and IVPFSWG operators have been introduced based on our developed operators with their basic properties. In Section 4, a MAGDM approach has been constructed based on the proposed AOs. To ensure the practicality of the developed approach a numerical example has been presented for the selection of the best vehicle in Section 5.

2  Preliminaries

This section consists of some basic definitions which will provide a structure to form the following work.

Definition 2.1 [1]

Let U be a collection of objects then a fuzzy set (FS) A over U is defined as

A={(t,κ(t))|t∈U}

where, κA(t):X→[0,1] is a membership grade function.

Definition 2.2 [26]

Let U be a collection of objects then an interval-valued fuzzy set (IVFS) A over U is defined as

A={(t,[κl(t),κu(t)])|t∈U}

where, κl(t), κu(t) ∈ [0,1] and represents the lower and upper bounds of the membership value.

Definition 2.3 [4]

Let U be a collection of objects then an interval-valued intuitionistic fuzzy set (IVIFS) A over U is defined as

A={(t,([κAl(t),κAu(t)],[δAl(t),δAu(t)]))|t∈U}

where, [κAl(t),κAu(t)] and [δAl(x),δAu(x)] are intervals for membership and non-membership grades, respectively, whereas [κAl(t),κAu(t)]and[δAl(t),δAu(t)]⊆[0,1], 0≤κAl(t),κAu(t),δAl(t),δAu(t)≤1, and 0≤κAu(x)+δAu(x)≤1.

Definition 2.4 [27]

Let U be a collection of objects then an interval-valued Pythagorean fuzzy set (IVPFS) A over U is defined as

A={(x,([κAl(t),κAu(t)],[δAl(t),δAu(t)]))|t∈U}

where, [κAl(t),κAu(t)] and [δAl(t),δAu(t)] represents the intervals for membership and non-membership grades, respectively. Furthermore, 0≤(κAu(t))2+(δAu(t))2≤1 and [κAl(t),κAu(t)]⊆[0,1] and [δAl(t),δAu(t)]⊆[0,1].

Definition 2.5 [13]

Let U be a universal set and N={t1,t2,t3,…,tm} be set of attributes then a pair (F,N) is called a soft set (SS) over U where F:N→KU is a mapping and KU is known as a collection of all subsets of universal set U.

Definition 2.6 [19]

Let U be a universal set and N be a set of attributes then a pair (Ω,N) is called an interval-valued intuitionistic fuzzy soft set (IVIFSS) over U. Where Ω:N→IKU is a mapping and IKU is known as a collection of all interval-valued intuitionistic fuzzy subsets of universal set U and A⊂N.

(Ω,A)={t,([κAl(t),κAu(t)],[δAl(t),δAu(t)])|t∈A}

where, [κAl(t),κAu(t)]and[δAl(t),δAu(t)] are intervals for membership grade and non-membership functions respectively with 0≤κAu(t)+δAu(t)≤1.

Definition 2.7

Let U be a universal set and N be set of attributes then a pair ((Ω,N) is called an interval-valued Pythagorean fuzzy soft set (IVPFSS) over U where Ω:N→℘KU is a mapping and ℘KU is known as the collection of all interval-valued Pythagorean fuzzy subsets of universal set U.

(Ω,A)={t,([κAl(t),κAu(t)],[δAl(t),δAu(t)])|t∈A}

where, [κAl(t),κAu(t)],[δAl(t),δAu(t)] are intervals for membership grade and non-membership grade, respectively with 0≤(κAu(t))2+(δAu(t))2≤1 and A⊂N.

Definition 2.8

Let Me=([κl,κu],[δl,δu]) be an interval-valued Pythagorean fuzzy soft number (IVPFSN), then the score function is defined as follows:

S(Me)=(κl)2+(κu)2−(δl)2−(δu)22

Definition 2.9

Let Me=([κl,κu],[δl,δu]) be an IVPFSN, then accuracy function is defined as follows:

S(Me)=(κl)2+(κu)2+(δl)2+(δu)22

3  Aggregation Operators for Interval Valued Pythagorean Fuzzy Soft Sets

In this section, we are going to define operational laws under IVPFSNs. Based on these operational laws, we shall also present interval-valued Pythagorean fuzzy soft weighted average (IVPFSWA) and interval-valued Pythagorean fuzzy soft geometric (IVPFSWG) operators.

3.1 Operational Laws for Interval Valued Pythagorean Fuzzy Soft Numbers

Let Me=([κl,κu],[δl,δu]), Me11=([κ11l,κ11u],[δ11l,δ11u]), and

Me12=([κ12l,κ12u],[δ12l,δ12u]) be three interval-valued Pythagorean fuzzy soft numbers and β be a positive real number, and by algebraic norms, we have

1.    Me11⊕Me12=([κ11l2+κ12l2−κ11l2κ12l2,κ11u2+κ12u2−κ11u2κ12u2],[δ11lδ12l,δ11uδ12u])

2.    Me11⊗Me12=([κ11lκ12l,κ11uκ12u],[δ11l2+δ12l2−δ11l2δ12l2,δ11u2+δ12u2−δ11u2δ11u2])

3.    βMe=([1−(1−κl2)β,1−(1−κu2)β],[δlβ,δuβ])=(1−(1−[κl,κu]2)β,[δlβ,δuβ])

4.    Meβ=([κlβ,κuβ],[1−(1−δl2)β,1−(1−δu2)β])=([κlβ,κuβ],1−(1−[δl,δu]2)β)

3.2 Interval Valued Pythagorean Fuzzy Soft Weighted Average Operator

Let Meij=([κijl,κiju],[δijl,δiju]) be a collection of interval-valued Pythagorean fuzzy soft numbers (IVPFSNs), and ωi and νj are the weight vector for experts and parameters, respectively, with given conditions ωi>0,∑i=1nωi=1;νj>0,∑j=1mνj=1. Then, the IVPFSWA operator is defined as IVPFSWA: Ψn⟶Ψ

IVPFSWA(Me11,Me12,…,Menm)=⊕j=1mνj(⊕i=1nωiMeij)

Theorem 3.1

Let Meij=([κijl,κiju],[δijl,δiju]) be a collection of IVPFSNs, where (i=1,2,3…,n and j=1,2,3,…m), and the aggregated value is also an IVPFSN, such as

IVPFSWA(Me11,Me12,…,Menm)

=(1−∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj,∏j=1m(∏i=1n([δijl,δiju])ωi)νj)

where ωi and νj are weight vector for expert’s and attributes respectively with given conditions ωi>0,∑i=1nωi=1;νj>0,∑j=1mνj=1.

Proof. We shall prove the IVPFSWA operator by utilizing the principle of mathematical induction:

For n=1, we get ω1=1. Then, we have

IVPFSWA(Me11,Me12,…,Me1m)=⊕j=1mνjMe1j

IVPFSWA(Me11,Me12,…,Menm)=(1−∏j=1m(1−[κ1jl,κ1ju]2)νj,∏j=1m([δ1jl,δ1ju])νj)

=(1−∏j=1m(∏i=11(1−[κijl,κiju]2)ωi)νj,∏j=1m(∏i=11([δijl,δiju])ωi)νj).

For m=1, we get ν1=1. Then, we have

IVPFSWA(Me11,Me21,…,Men1)=⊕i=1nωiMei1

=(1−∏i=1n(1−[κi1l,κi1u]2)ωi,∏i=1n([δi1l,δ1iu])ωi)

=(1−∏j=11(∏i=1n(1−[κijl,κiju]2)ωi)νj,∏j=11(∏i=1n([δijl,δiju])ωi)νj)

This shows that the above theorem holds for n=1 and m=1. Now, consider the above theorem also holds for m=α1+1,n=α2 and m=α1,n=α2+1, such as

⊕j=1α1+1νj(⊕i=1α2ωiMeij)=(1−∏j=1α1+1(∏i=1α2(1−[κijl,κiju]2)ωi)νj,∏j=1α1+1(∏i=1α2([δijl,δiju])ωi)νj)

⊕j=1α1νj(⊕i=1α2+1ωiMeij)=(1−∏j=1α1(∏i=1α2+1(1−[κijl,κiju]2)ωi)νj,∏j=1α1(∏i=1α2+1([δijl,δiju])ωi)νj)

For m=α1+1 and n=α2+1, we have

⊕j=1α1+1νj(⊕i=1α2+1ωiMeij)=⊕j=1α1+1νj(⊕i=1α2ωiMeij⊕ωα2+1Me(α2+1)j)

=⊕j=1α1+1⊕i=1α2νjωiMeij⊕j=1α1+1νjωα2+1Me(α2+1)j

=(1−∏j=1α1+1(∏i=1α2(1−[κijl,κiju]2)ωi)νj⊕1−∏j=1α1+1((1−[κ(α2+1)jl,κ(α2+1)ju]2)ωα2+1)νj,

∏j=1α1+1(∏i=1α2([δijl,δiju])ωi)νj⊕∏j=1α1+1(([δ(α2+1)jl,δ(α2+1)ju])ω(α2+1))νj)

=(1−∏j=1α1+1(∏i=1α2+1(1−[κijl,κiju]2)ωi)νj,∏j=1α1+1(∏i=1α2+1([δijl,δiju])ωi)νj)

Therefore, it holds for m=α1+1 and n=α2+1. So, we can judge that the above theorem also holds for all values of m and n.

Example 3.1

Let χ={x1,x2,x3} be the set of specialists with weights ωi=(0.38,0.45,0.17)T who wants to choose a bike under some defined set of properties φ={e1=Resale Value,e2=Mileage,e3=Cost of bike} with weights νj=(0.25,0.45,0.3)T. We suppose the rating values of the specialists for each property in the form of IVPFSNs (M,φ)=([κijl,κiju],[δijl,δiju])3×3 is given as

(M,φ)=[([0.3,0.8],[0.4,0.5])([0.4,0.6],[0.3,0.7])([0.5,0.8],[0.5,0.6])([0.1,0.5],[0.2,0.3])([0.3,0.8],[0.5,0.7])([0.2,0.4],[0.2,0.3])([0.2,0.9],[0.2,0.3])([0.5,0.7],[0.2,0.6])([0.2,0.4],[0.2,0.8])]

By using the above theorem, we have

IVPFSWA(Me11,Me12,…,Me33)=(1−∏j=13(∏i=13(1−[κijl,κiju]2)ωi)νj,∏j=13(∏i=13([δijl,δiju])ωi)νj)

=(1−({[0.36,0.91]0.38[0.64,0.84]0.45[0.36,0.75]0.17}0.17{[0.75,0.99]0.38[0.36,0.91]0.45[0.75,0.99]0.17}0.45{[0.19,0.96]0.38[0.51,0.75]0.45[0.84,0.96]0.17}0.3),({[0.4,0.5]0.38[0.3,0.7]0.45[0.5,0.6]0.17}0.25{[0.2,0.7]0.38[0.5,0.7]0.45[0.2,0.3]0.17}0.45{[0.2,0.3]0.38[0.2,0.6]0.45[0.2,0.8]0.17}0.3))

=(1−({[0.6783,0.9648][0.8181,0.9245][0.8406,0.9523]}0.25{[0.8964,0.9962][0.6314,0.9584][0.9523,0.9983]}0.45{[0.5320,0.9846][0.7386,0.8786][0.9708,0.9931]}0.3),({[0.7060,0.7486][0.5817,0.8517][0.8888,0.9168]}0.25{[0.5425,0.8732][0.7320,0.8517][0.7606,0.8149]}0.45{[0.5425,0.6329][0.4847,0.7946][0.7606,0.9628]}0.3))

=(1−([0.4665,0.8494]0.25[0.5390,0.9531]0.45[0.3815,0.8591]0.3),([0.3650,0.5857]0.25[0.3020,0.6060]0.45[0.2000,0.4842]0.3))

=(1−([0.8264,0.9600][0.7572,0.9786][0.7489,0.9555]),([0.7773,0.8748][0.5834,0.7982][0.6170,0.8045]))

=(1−[0.7773,0.8977],[0.2798,0.5617])

=([0.1023,0.2227],[0.2798,0.5617])

=([0.3198,0.4719],[0.2798,0.5617])

3.3 Properties of PFSWA Operator

3.3.1 Idempotency

If Meij=Me=([κijl,κiju],[δijl,δiju])∀i,j, then,

IVPFSWA (Me11,Me12,…,Menm)=Me

Proof: As we know that all Meij=Me=([κijl,κiju],[δijl,δiju]),then, we have

IVPFSWA(Me11,Me12,…,Menm)

=(1−∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj,∏j=1m(∏i=1n([δijl,δiju])ωi)νj)

=(1−((1−[κijl,κiju]2)∑i=1nωi)∑j=1mνj,(([δijl,δiju])∑i=1nωi)∑j=1mνj)

As ∑j=1mνj=1 and ∑i=1nωi=1, then we have

=(1−(1−[κijl,κiju]2),[δijl,δiju])

=([κijl,κiju],[δijl,δiju])

=Me

Hence proved.

3.3.2 Boundedness

Let Meij be a collection of PFSNs where Meij−=(minj mini{[κijl,κiju]},maxj maxi{[δijl,δiju]}) and Meij+=(maxj maxi{[κijl,κiju]},minj mini{[δijl,δiju]}), then

Meij−≤IVPFSWA (Me11,Me12,…,Menm)≤Meij+

Proof. As we know that Meij=⟨[κijl,κiju],[δijl,δiju]⟩ be an IVPFSN, then

minj mini{[κijl,κiju]2}≤[κijl,κiju]2≤maxj maxi{[κijl,κiju]2}

⇒1−maxj maxi{[κijl,κiju]2}≤1−[κijl,κiju]2≤1−minj mini{[κijl,κiju]2}

⇔(1−maxj maxi{[κijl,κiju]2})ωi≤(1−[κijl,κiju]2)ωi≤(1−minj mini{[κijl,κiju]2})ωi

⇔(1−maxj maxi{[κijl,κiju]2})∑i=1nωi≤∏i=1n(1−[κijl,κiju]2)ωi≤(1−minj mini{[κijl,κiju]2})∑i=1nωi

⇔(1−maxj maxi{[κijl,κiju]2})∑j=1nνj≤∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj≤(1−minj mini{[κijl,κiju]2})∑j=1nνj

⇔1−maxj maxi{[κijl,κiju]2}≤∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj≤1−minj mini{[κijl,κiju]2}

⇔minjmini{[κijl,κiju]2}≤1−∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj≤maxj maxi{[κijl,κiju]2}

⇔minj mini{[κijl,κiju]}≤1−∏j=1m(∏i=1n(1−[κijl,κiju]2)ωi)νj≤maxj maxi{[κijl,κiju]} (a)

Similarly, we can prove that

minj mini{[δijl,δiju]}≤∏j=1m(∏i=1n([δijl,δiju])ωi)νj≤maxj maxi{[δijl,δiju]} (b)

Let IVPFSWA (Me11,Me12,…,Menm)=⟨[κσl,κσu],[δσl,δσu]⟩=Mσ, then inequalities (a) and (b) can be transferred into the form: minj mini{[κijl,κiju]}≤Mσ≤maxj maxi{[κijl,κiju]} and minj mini {[δijl,δiju]}≤Mσ≤maxj maxi{[δijl,δiju]}, respectively.