# q-Rung Orthopair Fuzzy Matroids with Application to Human Trafficking

Authors: Muhammad Asif; Doha A. Kattan; Dragan Pamučar; Ghous Ali
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

The theory ofq-rung orthopair fuzzy sets (q-ROFSs) is emerging for the provision of more comprehensive and useful information in comparison to their counterparts like intuitionistic and Pythagorean fuzzy sets, especially when responding to the models of vague data with membership and non-membership grades of elements. In this study, a significant generalized model q-ROFS is used to introduce the concept of q-rung orthopair fuzzy vector spaces (q-ROFVSs) and illustrated by an example. We further elaborate the q-rung orthopair fuzzy linearly independent vectors. The study also involves the results regarding q-rung orthopair fuzzy basis and dimensions of q-ROFVSs. The main focus of this study is to define the concepts of q-rung orthopair fuzzy matroids (q-ROFMs) and apply them to explore the characteristics of their basis, dimensions, and rank function. Ultimately, to show the significance of our proposed work, we combine these ideas and offer an application. We provide an algorithm to solve the numerical problems related to human flow between particular regions to ensure the increased government response action against frequently used path (heavy path) for the countries involved via directed q-rung orthopair fuzzy graph (q-ROFG). At last, a comparative study of the proposed work with the existing theory of Pythagorean fuzzy matroids is also presented.


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## Body

## 1. Introduction

Graph theory and combinatorial geometry are known to have a lot of common grounds, particularly with regard to their basic concepts. Making use of these similarities, a host of research has been conducted for further exploration and development of these fields. Whitney [1] was the one to initiate the fundamental concept of matroids. By doing so, he laid the foundation of an extremely vast field of matroid theory that connected several basic tools like linear algebra, graph, and combinatorial theory. This matroid theory has been widely applied by researchers in different scientific areas.Zadeh [2], in 1965, for the first time introduced fuzzy logics and defined fuzzy sets (FSs). These sets were known for real-life data, uncertainties, and vague information. Soon after its introduction, fuzzy set theory became popular among researchers and came up as a new field. Later, Attanssov [3] expanded the concept of FSs and introduced the intuitionistic fuzzy sets (IFSs) with the help of membership and non-membership values of elements, the sum of which was not being more than 1. These IFSs are effectively applied in theoretical as well as practical problems such as optimization, decision making, and graphical ones in numerous fields. The idea of these sets was further extended by Feng et al. [4] to give intuitionistic fuzzy soft sets (IFSSs). They also presented several new operations to generalize the concept of intuitionistic fuzzy soft sets (IFSSs). While solving some decision-making problems, it was observed that the sum of both membership and non-membership values of elements exceeded one; however, some of their squares remained less than one. To overcome such issues, Yager [5, 6] put forward the idea of IFSs with the introduction of new Pythagorean fuzzy sets. Some useful notions and results for FSs, IFSs, PFSs, and other types of fuzzy sets have been presented in the literature [7–14]. It seems difficult to solve the problems when the sum of the square of the membership and non-membership values of elements exceeds 1. We are unable to handle such kind of information by means of PFSs. Yager [15] introduced q-ROFSs in which the sum of the qth power of membership and nonmembership values of elements is bounded by one. After that, q-ROFSs are frequently used in decision making as q-ROFSs widened the range of acceptable pairs rather than IFSs and PFSs with the parameter “q” adjustment. Recently, Garg used q-ROFSs to introduce a novel concept of connection number-based q-rung orthopair fuzzy set (CN-q-ROFS), defined some operation laws, and proposed a method to handle multiattribute group decision-making (MAGDM) problems [16]. Subsequently, in [17], he introduced the idea of q-connection numbers for interval-valued q-rung orthopair fuzzy set and used it to develop a method for solving multiattribute group decision-making (MAGDM) problems. At present, several studies paid close attention to the information regarding q-ROFSs and provided different novel methods [18–21].The graphical representation of objects has been a subject of great interest for scientists. Recently, a number of studies have involved both fuzzy and graph theories to deal with the optimization related problems in the presence of vague data. The idea of fuzzy graphs came from Kaufmann [22], while some basic concepts related to fuzzy graphs, such as cycles and paths, were characterized by Rosenfield et al. [23]. Akram and Naz [24] further used these concepts and proposed a new work to find the energy of PFGs with their applications. Their work was mainly focused on operations of fuzzy graphs (FGs), IFGs, PFGs, and their different types. They also provided q-ROF competition graphs and studied their applications. Sitara et al. [25] introduced the notion of q-rung picture fuzzy graph structures and provided an algorithm to describe their proposed model. The refining of the idea of hypergraphs given by Kaufmann was carried out by Lee-Kwang and Lee. In addition, different researchers investigated numerous features of FGs and fuzzy hypergraphs based on different FSs [26–31]. In 1988, the concept of matroids in terms of FSs was linked and defined as G-V fuzzy matroids by Goetschel and Voxman [32]. Later on, bases and circuits of the fuzzy matroids were also defined by them [33–36]. As time progressed, different FSs were used to define different fuzzy matroids, and their properties were also discussed by different researchers [37–42]. Recently, we proposed the idea of Pythagorean fuzzy matroids (PFMs) and described their application to decision making [43]. A lot of work based on FSs, IFSs, and PFSs regarding matroid theory has been discussed in the literature, but matroids based on q-ROFSs are still unattended. The existing models, namely, IFMs and PFMs, are insufficient to deal with different decision-making problems which contain membership and nonmembership values of elements whose sum of their squares is greater than 1. This drawback of existing structures motivates us to present this work.The motivations of our work are as follows:(1)
Theq-ROFS is a generalized form of some existing models, including IFSs and PFSs. On setting q=1 and q=2, we get IFSs and PFSs, respectively, as special cases of q-ROFSs.(2)
The existing IFMs and PFMs fail to deal with the information involving membership and nonmembership values whose sum of their squares is not less than 1.(3)
Due to the more flexible approach ofq-ROFSs, the developed q-ROFMs can solve many decision-making problems and overcome deficiencies of existing models such as IFMs and PFMs.The main contributions of this work are as follows:(1)
Our work illustratesq-ROFVSs with an illustrative example.(2)
Most importantly, the notion ofq-ROFMs is defined and characterized with its basis and dimension.(3)
This study also provides various results regardingq-ROFMs.(4)
Ultimately, an algorithm is developed to find an optimal solution along with a particular application.(5)
To check the validity of our proposed work, a comparative analysis with an existing model is also given.In this work, we present the idea ofq-ROFVSs with a numerical example and discuss their bases and dimensions. We also discuss the q-rung orthopair fuzzy linearly independent vectors. We further combined the q-ROFSs with the fuzzy matroids and named them as q-ROFMs. We investigate the concepts of circuits, basis, and rank for q-ROFMs. Note that for q=1 and q=2, our proposed q-ROFMs are reduced to IFMs and PFMs, respectively. We also proposed an application of our work regarding human trafficking between different regions which supports them to find a heavy path used by the traffickers so that they can increase their government response action against this path by using a directed graph having q-rung orthopair fuzzy information. In the end, we give concluding remarks with some of the future directions.The contents of this article are summarized as follows. In Section2, we recall some fundamental definitions including crisp matroids with rank function, q-ROFSs with their score functions, and some basic operations defined on q-ROFSs. In Section 3, we first propose q-ROFVSs and then q-ROFMs. We also discuss some of their basic properties in this section. In Section 4, we explore an application and develop an algorithm to illustrate the importance of our work. In Section 5, we provide the numerical comparison of our developed algorithm with the existing PFM approach [43]. In Section 6, we provide some conclusive remarks with future directions.


## 2. Preliminaries

Our interest in this section is to discuss the theory of matroids and valuable concepts related to matroid theory to understand the proposed work better. Although matroids are defined differently by using various sets, here we write a simple definition of crisp matroids.Definition 1 (see [1]).
LetA≠ϕ be a set of finite elements and PA denote the power set of A. For ℐ⊂PA, a non-empty family of subsets, the pair ℳ=A,ℐ is called a matroid (or crisp matroid) if it satisfies the following:(1)
ϕ∈ℐ.(2)
IfI2⊂I1 with I1∈ℐ, then I2∈ℐ.(3)
IfI1,I2∈ℐ with I2<I1, then another subset I3∈ℐ exists such that I2⊂I3⊆I1∪I2, where I shows the number of elements of I.
The elementI∈ℐ is called independent set in ℳ. Also, I is known as maximal independent in ℳ if we do not have such I′∈ℐ that contains I.Definition 2 (see [1]).
Suppose thatℳ is a matroid and I∈ℐ. If I is maximal independent in ℳ, then I is called base of ℳ and Bℳ represents the family of all bases.Definition 3.
LetJ∈PA be a subset with J∉I; then, J is called dependent subset. A circuit of ℳ is the subset J∈PA∖ℐ where J is inclusion-wise minimal dependent subset.Definition 4 (see [1]).
Letℳ=A,ℐ be a matroid. Consider a mapping ℛA:PA⟶0,1,2,…,A defined as(1)ℛAA=maxI′:I′⊂A,I∈ℐ.
Then,ℛA is called rank function of ℳ.Definition 5 (see [2]).
Consider a non-empty setX. The fuzzy set F is defined as(2)F=x,σFx|x∈X.
The mappingσF:X⟶0,1 assigns the membership value of x∈X to F and ℱSX represents the family of all fuzzy sets on X.Definition 6 (see [32]).
LetX be a non-empty finite universe of discourse and ℱ⊆ℱSX. For any fuzzy sets F1,F2,F3∈ℱSX, the collection ℱ satisfies the following:(1)
ϕ∈ℱ.(2)
IfF1⊂F2andF2∈ℱ,thenF1∈ℱ,whereF1⊂F2meansF1x<F2x, for all x∈X.(3)
LetF1,F2∈ℱ and suppF1<suppF2 where suppF=x∈X|Fx>0 and F=∑x∈XFx; then, another F3∈ℱ exists satisfying(a)
F1⊂F3⊆F1∪F2, for anyx∈X, and union is defined as F1∪F2x=maxF1x,F2x.(b)
mF3≥minmF1,mF2, and mFi=minFix:x∈suppFi is defined for any x∈X.
The pairℱℳX=X,ℱ is called a fuzzy matroid and ℱ is the subfamily of all independent FSs of the matroid ℱℳX.Definition 7 (see [15]).
LetX be a non-empty fixed set with finite elements. Then, the set ζ=ζ+,ζ− defined on a fixed set X is called q-ROFS if it satisfies(1)
q≥1.(2)
ζ+x∈0,1andζ−x∈0,1 with the property ζ+xq+ζ−xq≤1, for any x∈X.
It can be seen easily that forq=1 and q=2, these fuzzy sets are reduced to IFS and PFS, respectively. The q-rung orthopair degree of hesitance for x∈X is given as(3)ϱζx=1−ζ+q−ζ−qq.Definition 8 (see [15]).
Letζ+,ζ− be a q-ROFS and for x∈X, ζ+x,ζ−x be a q-rung orthopair fuzzy number (q-ROFN). Then, a score function Sq˜ of ζ+x,ζ−x is defined as(4)Sq˜ζ+x,ζ−x=121+ζ+xq−ζ−xq,0≤Sq˜ζ+x,ζ−x≤1.Definition 9 (see [15]).
Letζ=ζ+,ζ− be a q-ROFS and for x∈X, ζ+x,ζ−x be a q-ROFN. Then, for any x∈X, an accuracy function Aq˜ for q-ROFN is defined as(5)Aq˜ζ+x,ζ−x=1−ϱζx.Definition 10 (see [41]).
Letζ1andζ2 be two q-ROFSs. Then, for any x∈X, ζ1∪ζ2=ζ1+,ζ1−∨ζ2+,ζ2− and ζ1∩ζ2=ζ1+,ζ1−∧ζ2+,ζ2− are called union and intersection, respectively, defined as(6)ζ1∪ζ2x=ζ1+,ζ1−x,ifSq˜ζ1+,ζ1−x>Sq˜ζ2+,ζ2−x,ζ1+,ζ1−x,ifSq˜ζ1+,ζ1−x=Sq˜ζ2+,ζ2−xandAq˜ζ1+,ζ1−x≥Aq˜ζ2+,ζ2−x,ζ2+,ζ2−x,ifSq˜ζ1+,ζ1−x=Sq˜ζ2+,ζ2−xandAq˜ζ1+,ζ1−x<Aq˜ζ2+,ζ2−x,ζ1∩ζ2x=ζ2+,ζ2−x,ifSq˜ζ1+,ζ1−x>Sq˜ζ2+,ζ2−x,ζ2+,ζ2−x,ifSq˜ζ1+,ζ1−x=Sq˜ζ2+,ζ2−xandAq˜ζ1+,ζ1−x≥Aq˜ζ2+,ζ2−x,ζ1+,ζ1−x,ifSq˜ζ1+,ζ1−x=Sq˜ζ2+,ζ2−xandAq˜ζ1+,ζ1−x<Aq˜ζ2+,ζ2−x.
Here, we denote the family of allq-ROFSs on X by QX. Let ζ=x,ζ+x,ζ−x|x∈X be a q-ROFS; then, some notions are defined as follows.(1)
ζ=∑x∈Xζ+x,ζ−x.(2)
Suppζ+,ζ−=x∈X:Sq˜ζ+x,ζ−x>0.(3)
Qqζ+,ζ−=x∈X:Sq˜ζ+x,ζ−x≥q, for 0≤q≤1.(4)
ℛ+ζ+,ζ−=Sq˜ζ+x,ζ−x|Sq˜ζ+x,ζ−x>0. A q-ROFS ζ+,ζ− is called elementary if ℛ+ζ+,ζ−=1.(5)
hζ=maxx∈Xζ+x,ζ−x is called height of ζ and the set ζ is called normal q-ROFS for the height hζ=1,0.
Note that we denote the smallest and the largestq-rung orthopair fuzzy elements 0=0,1 and 1=1,0, respectively.


## 3.q-Rung Orthopair Fuzzy Vector Spaces

This section illustrates the concept ofq-ROFVSs with their basis and dimension and presents q-rung orthopair fuzzy linearly dependent and independent vectors. Here, we also present matroids based on q-ROFSs and discuss their properties regarding circuits, basis, and their rank function. Katsaras and Liu [44] introduced the hybrid concept of fuzzy vector spaces and discussed their characteristics. Later, many researchers applied different fuzzy sets to the elementary concepts of vector spaces. Here, we use q-ROFSs to generalize the Pythagorean fuzzy vector spaces [43] and define q-ROFVSs.Definition 11.
LetX be a non-empty finite vector space over the field F. The q-ROFS ζ=ζ+x,ζ−x,forx∈X, is called q-ROFVS over X, if for scalars a,b∈Fand for anyx,y∈X, we have(7)ζ+ax+by≥minζ+x,ζ+y,ζ−ax+by≤maxζ−x,ζ+y,where 0≤ζ+xq+ζ−xq≤1 holds for defined mappings ζ+:X⟶0,1 and ζ−:X⟶0,1.
Here, the set of allq-ROFVSs over X is denoted by the pair X˜=X,ζ.
The following proposition illustrates that membership and non-membership functions assign unchanged values under scalar multiplication inq-ROFVSs.Proposition 1.
LetX˜=X,ζ be a q-ROFVS. The following two properties hold for each x,y∈X:(1)
ζ+0,0=maxx∈Xζ+xandζ−0,0=minx∈Xζ−x.(2)
For any non-zero scalark∈F, ζ+kx=ζ+xandζ−kx=ζ−x.Proof.
The proof of properties (15) and (18) is very straightforward (see Definition 11).Proposition 2.
Letx,y∈X with ζ+x≠ζ+y and ζ−x≠ζ−y; then, we have(8)ζ+x+y≥minζ+x,ζ+y,ζ−x+y≤maxζ−x,ζ−y.Proof.
To prove, from Definition11, let a=b=1 and hence ζ+x+y≥minζ−x,ζ+y and ζ−x+y≤maxζ−x,ζ−y.Definition 12.
LetX be a non-empty finite universe and X˜ be a q-ROFVS over F. Then, the set of vectors x1,x2,x3,…,xr⊂X is called a q-rung orthopair fuzzy (q-ROF) linearly independent in X˜ if(1)
x1,x2,…,xr is linearly independent.(2)
For anya1,a2,…,ar⊂F, we have(9)ζ+∑j=1rajxj=minj=1rζ+ajxj,ζ−∑j=1rajxj=maxj=1rζ−ajxj.Proposition 3.
LetX˜=X,ζ be a q-ROFVS over F. Consider any set of vectors with non-zero elements αii=1r∈X∖0 such that for i≠j, ζαi≠ζαj. Then, the set αii=1r is linearly and q-ROF linearly independent.Proof.
By using the induction onr, the statement is true for r=1. We suppose that the statement is true for r. So, αii=1r is q-ROF linearly independent. Let αii=1r+1∈X,0 such that for i≠j, ζ+αi,ζ−αi≠ζ+αj,ζ−αj and suppose that the set αii=1r+1 is not linearly independent. Thus, for ϕ≠v⊂1,2,…,r, we have αr+1=∑i∈vaiαi where ai≠0 for all i∈v. Then,(10)ζ+αr+1=mini∈vζ+aiαi=mini∈vζ+αi∈ζ+αii=1r,ζ−αr+1=maxi∈vζ−aiαi=maxi∈vζ−αi∈ζ−αii=1r,which gives that ζ+αr+1,ζ−αr+1∈ζ+αi,ζ−αii=1r and contradicts that αii=1r+1 has distinct values and hence is linearly independent. Propositions 1 and 2 show that αii=1r is q-ROF linearly independent.Definition 13.
LetX˜=X,ζ be a q-ROFVS and ℬ=βjj=1r, where each βj∈X. Then, the set ℬ is called q-ROF basis in X˜, if it satisfies(1)
The setℬ is basis in X.(2)
For scalarsa1,a2,…,ar⊂F, we have(11)ζ+∑j=1rajβj=minj=1rζ+ajβj,ζ−∑j=1rajβj=maxj=1rζ−ajβj.Definition 14.
LetX≠ϕ and X˜ be a q-ROFVS having basis ℬ. Then, the dimension of q-ROFVS is given by(12)dimqX˜=supℬ∑x∈ℬζ+x,ζ−x.
It is easy to see thatdimq is a function from the class of all q-ROFVSs to 0,∞∪∞. A q-ROFVS is said to be finite dimensional if and only if dimqX˜=n<∞.Proposition 4.
LetX˜=X,ζ be a q-ROFVS over F. For any x,y∈X, if ζ+x>ζ+y and ζ−x<ζ−y, we have(13)ζ+x+y=ζ+y,ζ−x+y=ζ−y.Proof.
We use Proposition2:(14)ζ+x+y≥minζ+x,ζ+y.
Sinceζ+x>ζ+y, then(15)ζ+x+y≥ζ+y.
Now we write(16)ζ+x+y−x=ζ+y,which implies the result(17)ζ+y≥minζ+x+y,ζ+x.
Sinceζ+x>ζ+y, then(18)ζ+y≥ζ+x+y,which proves that ζ+x+y=ζ+y. Similarly, we use Proposition 2:(19)ζ−x+y≤maxζ−x,ζ−y.
Sinceζ−x<ζ−y, then(20)ζ−x+y≤ζ−y.
Now we write(21)ζ−x+y−x=ζ−y,which implies the result(22)ζ−y≤maxζ−x+y,ζ−x.
Sinceζ−x<ζ−y, then(23)ζ−y≤ζ−x+y,which proves that ζ−x+y=ζ−y.
The following example illustrates Definition11 clearly.Example 1.
LetX=ℝ3 and ζ=ζ+,ζ− be a 5-ROFS defined on X. For any α=x,y,z∈ℝ3, the mappings ζ+:ℝ3⟶0,1 and ζ−:ℝ3⟶0,1 are defined by(24)ζ+α=12,if0,0,0,14,ifℝ∖0,0,0,0,ℝ∖0,0or0,0,ℝ∖0,12,otherwise,ζ−α=0,if0,0,0,14,ifℝ∖0,0,0,0,ℝ∖0,0or0,0,ℝ∖0,otherwise,respectively. It can be easily seen that, for α1=α2=0,0,0, the case is trivial.
For the second case, consider two vectorsα1 and α2 from X with one non-zero component, i.e., α1x,0,0 and α20,y,0; then, we have minζ+α1,ζ+α2=1/4 and maxζ−α1,ζ−α2=1/4. For any a,b∈ℝ, we have(25)ζ+aα1+bα2=ζ+ax,by,0=12,ifa=b=0ora≠0,b≠0,14,ifa=0orb=0,ζ−aα1+bα2=ζ−ax,by,0=0,ifa=b=0ora≠0,b≠0,14,ifa=0orb=0.
Clearly, it satisfies all the conditions of Definition11.
Now, considerα1 and α2∈X with one zero component and two non-zero components, i.e., α1x1,y1,0 and α2x2,0,z2; then, minζ+α1,ζ+α2=1/2 and maxζ−α1,ζ+α2=1/2. For any a,b∈ℝ,(26)ζ+aα1+bα2=ζ+ax1+bx2,ay1,bz2=12,ζ−aα1+bα2=ζ−ax1+bx2,ay1,bz2=0,which satisfies the conditions of Definition 11.Definition 15.
LetX≠ϕ be a finite universe. The subset Q⊆QX is a subfamily of q-ROFSs satisfying(1)
ϕ∈Q.(2)
For anyζ1=ζ1+,ζ1−∈Q,ζ2=ζ2+,ζ2−∈QX where 0≤ζ1+yq+ζ1−yq≤1, 0≤ζ2+yq+ζ2−yq≤1, q≥1, and if ζ2⊂ζ1, then ζ2∈Q, for all y∈X.(3)
For anyζ1,ζ2∈Q and suppζ1<suppζ2, there exists ζ3∈Q such that(a)
ζ1⊂ζ3⊆ζ1∪ζ2, for any y∈X.(b)
mζ3≥infmζ1,mζ2,mζi=infSq˜ζix|x∈suppζi.
Then, the pairQℳX=X,Q is called q−ROFM and the set Q is the subfamily of all independent q-ROFSs of QℳX.Proposition 5.
LetX˜=X,ζ be a q-ROFVS and Q be a subset of QX containing q-ROF linearly independent column vectors in X˜. The pair X,Q is a q-ROFM on X.Proof.
Suppose thatX is a non-empty set containing column labels of a q-ROF matrix, and ζx represents a q-ROF submatrix containing those columns which are labeled in X. Consider a set Q of q-ROF linearly independent column vectors of ζx, i.e.,(27)Q=ζx∈QX|column vectors ofζxareq−rung orthopair fuzzy linearly independent.
For any submatrixζx=aijm×n, we have ζx=∑i=1msupξxai1,ξxai2,…,ξxain. It is easy to see from Definitions 11 and 15 that X,Q is OℳX.
Note thatσ∉Q is called dependent q−ROFS.Definition 16.
LetQℳX be a q−ROFM and Qc be a family of dependent q−ROFSs in QℳX. The minimal dependent (inclusion wise) set σ∈Qc is called q-ROF circuit of QℳX and QcQℳ represents the subclass of all circuits of Qℳ, i.e.,(28)QcQℳ=σ|σ∈Qc,inclusion−wise  minimal.
Note that the elements ofQcQℳ follow the properties:(1)
ϕ∉QcQℳ.(2)
Letσ1andσ2 be q-rung orthopair fuzzy circuits with σ1⊆σ2; then,suppσ1=suppσ2.(3)
Letσ1,σ2∈QcQℳ and α∈QX with αx=infσ1x,σ2x where x∈suppσ1∩σ2. Then, there exists σ3 satisfying(a)
σ3≠ϕ and σ3∈QcQℳ.(b)
σ3⊆σ1∪σ2−x,αx.Definition 17.
LetQℳX=X,Q be a q-ROFM. Consider an element ζi∈Q; then, ζi is called maximal independent set in a matroid QℳX if there does not exist ζj∈Q that contains ζi. A maximal independent set in QℳX is called q-ROF base or basis of QℳX. The collection of all q-ROF basis is defined as(29)ℬPℳ=ζ|ζ∈Q,ξis maximal independent inQℳ.
Note that althoughq-ROF basis contains all the independent sets in QℳX, there exist some QℳX that do not have q-ROF basis.Example 2.
LetQX be a family of all q-ROFSs defined on a non-empty set X. Then, for a positive integer i and X=j with i≤j, the set Q is defined as(30)Q=ζ∈QX:suppζ≤i.
The pairX,Q=Uℱji is called q-ROF uniform matroid. Note that the subfamilies of all q-ROFSs of X with the sizes i+1 and i are called the q-ROF circuits and q-ROF basis of Uℱji, respectively.Definition 18.
LetQℳX be a q-ROFM. The q-ROF rank function QR:QX⟶0,∞×0,∞ is defined as(31)QRμ=supζ:ζ⊆μandζ∈Q,where ζ=∑y∈Xζ+y,ζ−y=∑y∈Xζ+y,∑y∈Xξ−y. Also, ζ1≤ζ2 iff ∑y∈Xζ1+y≤∑y∈Xξ2−y and ∑y∈Xζ1−y≥∑y∈Xζ2−y.
It is observed from definition that theq-ROF rank function follows the following properties:(1)
Ifζ∈QX, then QRζ=ζ.(2)
Ifζ∈QX, then QRζ≤ζ.(3)
Ifζ1,ζ2∈QXandξ1⊆ζ2,thenQRζ1≤QRζ2.Definition 19.
Letq1,q2,…,qn be a set of “n” q-ROFNs. Then, the pair qi=qi′,si′ satisfies the ordering for each 1≤i≤n:(32)qi′≤qi+1′,si′≥si+1′⟺qi≤qi+1.
We sayq instead of q=q′,s′ and 0≤q≤1, where 0=0,1 and 1=1,0 with Sq˜0=0andSq˜1=1, respectively.Definition 20.
Theq-cut level set for 0<q≤1 of a q-ROFS ζ∈QX is a crisp set which is defined as follows:(33)Qqζ=x∈X:ζ+x≥q′andζ−x≤s′.Theorem 1.
LetQℳX=X,Q be a q-ROFM and Qq be a collection of all q-cut levels of q-ROF independent sets where 0<q≤1, i.e.,(34)Qq=Qqξ|ξ∈Q.Then,ℳq=X,Qq is a crisp matroid on X.Proof.
The proof is very straight forward from Definition20, and Qq is a collection of crisp subsets of X. Then, for each 0<q≤1, we have ℳq=X,Qq.Definition 21.
LetX≠ϕ be a finite universe and QℳX be a q-ROFM. Then, we have a finite sequence q1<q2<⋯<qn such that(1)
q0=0,qn≤1.(2)
Qris non−empty,if0<r≤qn,andQris empty,ifr>qn.(3)
Ifqi<r1,r2<qi+1,thenQr1=Qr2,0≤i≤n−1.(4)
Ifqi<r1<qi+1<r2<li+2,thenQr1⊃Qr2,0≤i≤n−2.
The sequenceq0=0,q1,q2,…,qn is called fundamental sequence of QℳX.Corollary 1.
From Theorem1 and Definition 21, for 1≤i≤n, assume that qi¯=1/2qi−1+qi; then, ℳq¯1⊃ℳq¯2⊃⋯⊃ℳq¯n is called a ℳq-induced matroid sequence.Theorem 2.
Let0=q0<q1<q2<⋯<qn≤1 be a finite fundamental sequence and X,Qq1,X,Qq2,…,X,Qqn be finite sequence of crisp matroids regarding this fundamental sequence. For each qi−1<q≤qii=1,2,…,n, we assume Qq=Qqi and for qn<q≤1,Qq=ϕ. Then the pair X,Q^ is a q−ROFM where Q^ is defined as(35)Q^=ζ∈QX|Qqζ∈Qq,0<q≤1.Proof.
It is easy to see thatϕ∈Q^ as Qq=ϕ for qn<q≤1. Now, assume that ζ1∈Q^,andζ2∈QX such that ζ2⊆ζ1. It is clear from definition of Q^ that for each 0<q≤1, Qqζ1∈Qq, so Qqζ2⊆Qqζ1, and since we have that X,Qq is a crisp matroid for Qq, it means Qlζ2∈Ql, and hence ζ2∈Q^ and proves (18) of Definition 15. Now, let ζ1,ζ2∈Q^ with suppζ2<suppζ1. It is known that mζi=infSq˜ζix:x∈suppζi. Let ζ be a q−ROFN defined as(36)η=minmζ1,mζ2.
It is easy to observe from definition ofη that Qη contains the support of both ζ1andζ2. Note that Qη contains independent subsets; then, there exists an independent subset I∈Qη satisfying(1)
I contains suppζ2, for all x∈X.(2)
I⊆suppζ1∩suppζ2.
Let us defineζ3 as(37)ζ3y=η,y∈I−suppζ2,ζ2y,y∈suppζ2,0=0,1,otherwise,which shows that ζ3 is q−ROFS and satisfies (20) of Definition 15. Hence, X,Q^ is q−ROFM.Theorem 3.
LetQℳ=X,Q be a q−ROFM and from Definition 20 and Qq defined in Theorem 1, for each 0<q≤1, ℳq=X,Qq is a crisp matroid. Let Q^=ζ∈QX|Qqζ∈Qq,0<l≤1. Then, Q=Q^.Proof.
It is easy to deduce from definition ofQ^ that Q⊂Q^. For ζ∈Q^, let νii=1t be a non-zero q-rung orthopair fuzzy range with νi=νi∗,νi′ and order ν1>ν2>⋯>νt>0. One can notice that for each 1≤i≤t and ζ∈Q^, we have that Qνiζ∈Qνi. So, if 1≤i≤t−1, then from Definition 21, Qνiζ⊂Qνi+1ζ. To prove Q^⊂Q, we define a q-rung orthopair fuzzy set φ∈QX for each 1≤i≤t and u∈X as(38)φiu=νi=νi∗,νi′,if u is inq−cut forζ,0=0,1,otherwise,with 0<ν∗q+ν′q≤1. Since we have Qνiζ∈Qνi, ∪iφi=ζ for 1≤i≤t that gives φi∈Q. Here to show ζ∈, we use induction process. For each 1≤i≤t, we consider(39)suppφi=ukk=1ni,suppζ=ujj=1nt.
Sinceφ1∈Q is an independent q-ROFS, it is enough to show that if ∪i=1l−1φi∈Q for r>l−1, then for each r>l, we have ∪i=1lφi∈Q. Define(40)Φ1u=νl=νl∗,νl′,ifu∈u1,u2,…,unl−1+1,0=0,1,otherwise,which shows that for each 1<i<l−1,νi>νl, so Φ1⊂φl and Φ1∈Q. Define another q-ROFS λ as(41)Ψ1u=ζunl−1+1=νl∗,νl′,ifunl−1+1,0,1,otherwise.
Using the induction method, we havesupp∪i=1l−1φi=u1,u2,…,unl−1 where ∪i=1l−1φi is an independent set, and m∪i=1l−1φi>νl. So, from Definition 15, ∪i=1lφi∪Ψ1∈Q is independent in Qℳ. If nl−1+1=nl, then we have that ∪i=1lφi is also an independent set in Qℳ. But, if nl−1+1<nl, then to move further, we define q-ROFS as(42)Φ2u=νl∗,νl′,ifu∈u1,u2,…,unl−1+1∪unl−1+2,0,1,otherwise.
From Definition19, Φ2∈QX, and from Φ2⊂φl, Φ2∈Q is an independent set in Qℳ. Similarly, define another q-ROFS as(43)Ψ2u=ζunl−1+2=νl∗,νl′,ifunl−1+2,0,1,otherwise.
Sincesupp∪i=1l−1φi∪Ψ1=u1,u2,…,unl−1∪unl−1+1 and ∪i=1l−1φi∪Ψ1>νl, then again from Definition 15, ∪i=1l−1φi∪Ψ1∪Ψ2 is an independent set in Qℳ. So, ∪i=1l−1φi is also an independent set in Qℳ for nl−1+1=nl. But, if nl−1+1<nl, then to proceed further, we obtain a new q-ROFS χn=∪i=1l−1φi∪Ψ1∪Ψ2⋯∪Ψn and hence χn=∪i=1l−1φi is an independent set in Qℳ.
The next result is the direct consequence of Theorem 1.4 discussed in [34].Corollary 2.
LetQℳ=X,Q be a q−ROFM and ζ∈QX. Then, ζ∈QX if and only if for each q∈ℛ+ζwe haveQqζ∈Qq.Theorem 4.
Letζ∈Q be a q-rung orthopair fuzzy base of a q−ROFMX,Q. Then, for each x∈X, ϱζx=0.Proof.
Suppose on contrary thatxi∈X with ϱζxi=α>0 and take α∗=α/2. Let ζ′∈QX such that, for each x∈X, Sq˜ζ′=Sq˜ζ. Define(44)ϱζ′x=ϱζx,ifx∈Xandx≠xi,α∗,otherwise.
Aq˜ζ=Aq˜ζ′, which show that ζ<ζ′. But, for each q∈ℛ+ζ, we have Qqζ=Qqζ′. Hence, it is easy to see from Corollary 2 that ζ′∈Q, which gives the contradiction and completes the proof.Theorem 5.
Let0=q0<q1<q2⋯<qn≤1 be a fundamental sequence of q−ROFMQℳ=X,Q. Let ζ=ζ+,ζ− be a q-rung orthopair fuzzy basis of Qℳ; then,(45)ℛ+ζ+,ζ−⊆q1,q2,…,qn.Proof.
Letζi+,ζi− be a q-run orthopair fuzzy basis. So, ζi+,ζi−∈Q. For each q∈ℛ+ζi+,ζi−, we have Qqζi+,ζi−. Suppose that there exists t∈ℛ+ζi+,ζi− with qi<t<qi+1 and λ=qi+1−t/2. Assume that a q-rung orthopair fuzzy set ζj=ζj+ζj− is an elementary q-ROFS, defined with the suppζj+,ζj−=Qtζi+,ζi− and ℛ+ζj+,ζj−=t+qi+1/2. Take ζk+,ζk−=ζi+,ζi−∨ζj+,ζj− and for each 0<q≤1, if t<q<t+qi+1/2, then Qqζk+,ζk−=Qtζi+,ζi−∈Qt; otherwise, Qqζk+,ζk−=Qqζi+,ζi−∈Qq. By Corollary 2, ζk+,ζk−∈Q. So, for ζi+,ζi−, there exists xr∈suppζi+,ζi− with Sq˜ζi+xr,ζi−xr=t, and hence Sq˜ζk+xr,ζk−xr=t+qi+1/2, which contradicts that ζi+,ζi− is maximal and a q-rung orthopair fuzzy basis.


## 4. Application

### 4.1. Case Study: Human Trafficking

With a rapid increase in the population of different continents, including Asia and Africa, it is unable to stop human trafficking. Many reasons are behind this, such as poverty, greed for a handsome job, unemployment, low literacy rate, labor, and sexual exploitation. Studies show that millions of people are trapped in this modern slavery, and it is a dilemma that people do not realize they are getting trapped due to unawareness. Still, we are unable to get exact statistics due to unreported cases of human trafficking. An important application is to study human trafficking in different countries and provide a guess about a more suitable way or path that the traffickers can use.Considern and m number of countries, say C1,C2,…,Cn and C1′,C2′,…,Cm′, respectively. The traffickers will move from C1 to any of the other Ci countries. However, C1 is not connected with other Ci’s directly or the human trafficking ratio is almost ignorable between them due to multiple factors. The traffickers move in groups and collect their other victims from the countries Cj′ where j=1,2,…,m, to move any one of the countries Ci where i=2,3,…,n. If C1 is the starting point, they will definitely move to any of the other Ci (where i=2,…,n) for their next station. The state agencies seek to solve this puzzle used by the traffickers for trafficking across different stations. The procedure to track their way or path regarding given information is described in Algorithm 1.Algorithm 1: Selection of the heaviest path.
(1)
Input:(i)
Consider two finite sets of countriesC1,C2,…,Cn and C1′,C2′,…,Cm′, and each Ci and Cj′ mark a vertex of the graph G.(ii)
Mark all the edges and give the direction between the countries regarding givenq-rung orthopair fuzzy information and given problem, that is, for each 1≤i≤mn, aiζi+,ζi− represents a directed edge that represents the flow between two countries. There is no edge between any of the Ci’s and also no edge between Cj’s.(2)
Calculate the score functionSq˜ai=1/21+ζi+q−ζi−q.(3)
Find all the edge sets of lengthmn−1 and remove all the cycles of length mn−1.(4)
FindB=Xl=aikk=1m+n−1|Xlis maximal independent, where 1≤i≤mnandl=1,2,…,nmmn.(5)
DetermineB′ by removing all Xl from B such that Xl’s are not spanning paths (the path runs through each vertex of the graph exactly once).(6)
Reduce the setB′ to B″ by removing all the spanning paths from B′ which are not the cases according to the directions given in the data.(7)
ComputeTi¯=∑Sq˜aikk=1n−1 where each aikk=1n−1∈B″.(8)
FindmaxTi¯.Output: in the last step, the heaviest path will help us to take suitable measures according to flow of human trafficking given in the data.We take some data to find the illegal immigration routes used by traffickers for human trafficking. This model is in the form of directed fuzzy graphs. The edge between the two countries is represented by theq-rung fuzzy information. The membership part shows the ratio of human trafficking from one country to another. The non-membership part shows parameters of measurements taken by the concerned governments to stop human trafficking (government response action to reduce flow). We use the data from the model given by Mordeson and Mathew [45]. The data of illegal human flow between different regions are given in Table 1 [45]. In this study, we consider five regions: the Middle East V1, West Central Europe V2, East Asia and Pacific V3, West Southern Europe V3, North America V4, and Central America V5. V=V1,V2,V3,V4,V5 and E=a1,a2,a3,a4,a5,a6 represent the set of vertices and directed edges of graph G, respectively (see Figure 1). We can analyze the flow of human trafficking by these directed edges. Our task is to find a frequently used path by the traffickers so that relevant governments can take some measures to stop this abuse. We calculate the score functions of all the given q-ROF information (see Table 1). To engage all the five concerned governments, it is easy to see from Figure 1 that four edges need to pass through all vertices, and such total possibilities are 15. However, cycles of length four cannot be the choice while solving such a puzzle. The remaining 12 edge sets are maximal independent sets that can be useful to find the most used path by the traffickers. We say B, i.e., B=a1,a2,a3,a5,a1,a2,a3,a6,a1,a2,a4,a5,a1,a2,a4,a6,a2,a3,a4,a5,a2,a3,a5,a6, a2,a3,a4,a6,a2,a4,a5,a6,a1,a3,a4,a5,a1,a3,a5,a6,a1,a4,a5,a6,a1,a3,a4,a6. In the next step, the set B reduces to B′ by removing maximal independent sets which are not spanning paths, i.e., B′=a1,a2,a3,a6,a1,a2,a4,a5,a2,a3,a4,a5,a2,a3,a5,a6,a1,a4,a5,a6,a1,a3,a4,a6. However, the graph shown in Figure 1 is directed, so four more maximal independent sets are deleted due to absurdity in these cases. So, only T1=a2,a3,a4,a5 and T2=a1,a4,a5,a6 are the cases. Table 2 shows that T2=a1,a4,a5,a6 is the heaviest path with the score function 2.436. We find that given five countries should increase their government response action against this obtained path used frequently by the traffickers. They should take some measures to minimize human trafficking.Table 1
3-Rung orthopair fuzzy information of human trafficking between regions from Figure1.
Serial no.Connectionsaiξi+,ξi−S3˜ζ+x,ζ−x1V3⟶V1(0.33, 0.04)0.67962V3⟶V2(0.25, 0.05)0.50773V4⟶V1(0.26, 0.5)0.44624V2⟶V4(0.13, 0.02)0.50105V1⟶V5(0.04, 0.1)0.49956V5⟶V2(0.8, 0.03)0.7559Figure 1
q-Rung orthopair fuzzy graph representation of human trafficking data.Table 2
Directed spanning paths with the sum of their score functions and weights.
Serial no.Ti=ai∑S3˜aiTi¯1a2,a3,a4,a50.5077+0.4462+0.5010+0.49951.95442a1,a4,a5,a60.6796+0.5010+0.4995+0.75592.436


## 4.1. Case Study: Human Trafficking

With a rapid increase in the population of different continents, including Asia and Africa, it is unable to stop human trafficking. Many reasons are behind this, such as poverty, greed for a handsome job, unemployment, low literacy rate, labor, and sexual exploitation. Studies show that millions of people are trapped in this modern slavery, and it is a dilemma that people do not realize they are getting trapped due to unawareness. Still, we are unable to get exact statistics due to unreported cases of human trafficking. An important application is to study human trafficking in different countries and provide a guess about a more suitable way or path that the traffickers can use.Considern and m number of countries, say C1,C2,…,Cn and C1′,C2′,…,Cm′, respectively. The traffickers will move from C1 to any of the other Ci countries. However, C1 is not connected with other Ci’s directly or the human trafficking ratio is almost ignorable between them due to multiple factors. The traffickers move in groups and collect their other victims from the countries Cj′ where j=1,2,…,m, to move any one of the countries Ci where i=2,3,…,n. If C1 is the starting point, they will definitely move to any of the other Ci (where i=2,…,n) for their next station. The state agencies seek to solve this puzzle used by the traffickers for trafficking across different stations. The procedure to track their way or path regarding given information is described in Algorithm 1.Algorithm 1: Selection of the heaviest path.
(1)
Input:(i)
Consider two finite sets of countriesC1,C2,…,Cn and C1′,C2′,…,Cm′, and each Ci and Cj′ mark a vertex of the graph G.(ii)
Mark all the edges and give the direction between the countries regarding givenq-rung orthopair fuzzy information and given problem, that is, for each 1≤i≤mn, aiζi+,ζi− represents a directed edge that represents the flow between two countries. There is no edge between any of the Ci’s and also no edge between Cj’s.(2)
Calculate the score functionSq˜ai=1/21+ζi+q−ζi−q.(3)
Find all the edge sets of lengthmn−1 and remove all the cycles of length mn−1.(4)
FindB=Xl=aikk=1m+n−1|Xlis maximal independent, where 1≤i≤mnandl=1,2,…,nmmn.(5)
DetermineB′ by removing all Xl from B such that Xl’s are not spanning paths (the path runs through each vertex of the graph exactly once).(6)
Reduce the setB′ to B″ by removing all the spanning paths from B′ which are not the cases according to the directions given in the data.(7)
ComputeTi¯=∑Sq˜aikk=1n−1 where each aikk=1n−1∈B″.(8)
FindmaxTi¯.Output: in the last step, the heaviest path will help us to take suitable measures according to flow of human trafficking given in the data.We take some data to find the illegal immigration routes used by traffickers for human trafficking. This model is in the form of directed fuzzy graphs. The edge between the two countries is represented by theq-rung fuzzy information. The membership part shows the ratio of human trafficking from one country to another. The non-membership part shows parameters of measurements taken by the concerned governments to stop human trafficking (government response action to reduce flow). We use the data from the model given by Mordeson and Mathew [45]. The data of illegal human flow between different regions are given in Table 1 [45]. In this study, we consider five regions: the Middle East V1, West Central Europe V2, East Asia and Pacific V3, West Southern Europe V3, North America V4, and Central America V5. V=V1,V2,V3,V4,V5 and E=a1,a2,a3,a4,a5,a6 represent the set of vertices and directed edges of graph G, respectively (see Figure 1). We can analyze the flow of human trafficking by these directed edges. Our task is to find a frequently used path by the traffickers so that relevant governments can take some measures to stop this abuse. We calculate the score functions of all the given q-ROF information (see Table 1). To engage all the five concerned governments, it is easy to see from Figure 1 that four edges need to pass through all vertices, and such total possibilities are 15. However, cycles of length four cannot be the choice while solving such a puzzle. The remaining 12 edge sets are maximal independent sets that can be useful to find the most used path by the traffickers. We say B, i.e., B=a1,a2,a3,a5,a1,a2,a3,a6,a1,a2,a4,a5,a1,a2,a4,a6,a2,a3,a4,a5,a2,a3,a5,a6, a2,a3,a4,a6,a2,a4,a5,a6,a1,a3,a4,a5,a1,a3,a5,a6,a1,a4,a5,a6,a1,a3,a4,a6. In the next step, the set B reduces to B′ by removing maximal independent sets which are not spanning paths, i.e., B′=a1,a2,a3,a6,a1,a2,a4,a5,a2,a3,a4,a5,a2,a3,a5,a6,a1,a4,a5,a6,a1,a3,a4,a6. However, the graph shown in Figure 1 is directed, so four more maximal independent sets are deleted due to absurdity in these cases. So, only T1=a2,a3,a4,a5 and T2=a1,a4,a5,a6 are the cases. Table 2 shows that T2=a1,a4,a5,a6 is the heaviest path with the score function 2.436. We find that given five countries should increase their government response action against this obtained path used frequently by the traffickers. They should take some measures to minimize human trafficking.Table 1
3-Rung orthopair fuzzy information of human trafficking between regions from Figure1.
Serial no.Connectionsaiξi+,ξi−S3˜ζ+x,ζ−x1V3⟶V1(0.33, 0.04)0.67962V3⟶V2(0.25, 0.05)0.50773V4⟶V1(0.26, 0.5)0.44624V2⟶V4(0.13, 0.02)0.50105V1⟶V5(0.04, 0.1)0.49956V5⟶V2(0.8, 0.03)0.7559Figure 1
q-Rung orthopair fuzzy graph representation of human trafficking data.Table 2
Directed spanning paths with the sum of their score functions and weights.
Serial no.Ti=ai∑S3˜aiTi¯1a2,a3,a4,a50.5077+0.4462+0.5010+0.49951.95442a1,a4,a5,a60.6796+0.5010+0.4995+0.75592.436


## 5. Comparison of Given Model with Intuitionistic and Pythagorean Fuzzy Models

IFSs and PFSs are known to be the special cases ofq-ROFSs. The IFS was first introduced by Atanassov [3], and then PFS was discussed later by Yager [15]. The constructions of IFSs and PFSs show the importance of membership and non-membership functions in various real-life problems. However, there is a limitation in these models; that is, they fail to solve decision-making problems having information in which the sum or sum of the squares of membership and nonmembership values is greater than 1. To overcome these issues in more complicated information, q-ROFSs were introduced by Yager [15]. After introducing vector spaces and matroids based on PFSs in [43], in this study, we propose q-ROFVSs and q-ROFMs. This section provides the comparative analysis with PFMs and Algorithm 1 discussed in [43] (Section 4) to prove the efficiency of q-ROFMs and our proposed Algorithm 1, . It can be seen easily that the exiting method used to solve an application discussed in [43] (see Section 4) fails to solve the developed application in this study (see Section 4). Thus, for comparison, we use the dataset from Application (Section 4, Figure 2) [43]. Then, for q=2,3,4 and q=5, we compute the score functions (see Table 3). From Tables 4–7, it can be easily seen that for any q, the spanning path a1,a3,a6 attains the minimum value. Thus, our proposed technique is more flexible and generalized as it allows the decision makers to choose different values of q according to the given fuzzy information. Moreover, we have used directed graphs, spanning trees, and maximal independent sets to propose a particular algorithm (Algorithm 1, ) that can be helpful in solving the human trafficking-related problems.Table 3
Fuzzy information of connections between cities and their score functions (see Figure 2 in [43]).
Serial no.Connectionsaiξi+,ξi−S2˜S3˜S4˜S5˜1F⟷L(0.3, 0.4)0.4650.4810.49120.49602F⟷M(0.7, 0.5)0.620.6090.58880.56843M⟷N(0.6, 0.7)0.4350.4360.44470.45484N⟷L(0.8, 0.2)0.80.7520.70400.66365F⟷N(0.6, 0.2)0.660.6040.56400.53876L⟷M(0.5, 0.7)0.380.3910.41120.4315Table 4
Spanning paths and sum of the score functions of their entries whenq=2.
Serial no.Ti=ai∑S2˜aiTi¯1a1,a2,a30.465+0.62+0.4351.522a1,a2,a40.465+0.62+0.81.8853a1,a3,a40.465+0.435+0.81.74a1,a3,a50.465+0.435+0.661.565a1,a3,a60.465+0.435+0.381.286a1,a5,a60.465+0.66+0.381.5057a2,a3,a40.62+0.435+0.81.8558a2,a4,a50.62+0.8+0.662.089a2,a4,a60.62+0.8+0.381.810a2,a5,a60.62+0.66+0.381.6611a3,a5,a60.435+0.66+0.381.47512a4,a5,a60.8+0.66+0.381.84Table 5
Spanning paths and sum of the score functions of their entries whenq=3.
Serial no.Ti=ai∑S2˜aiTi¯1a1,a2,a30.481+0.609+0.4361.5262a1,a2,a40.481+0.609+0.7521.8423a1,a3,a40.481+0.436+0.7521.6694a1,a3,a50.481+0.436+0.6041.5215a1,a3,a60.481+0.436+0.3911.3086a1,a5,a60.481+0.604+0.3911.4767a2,a3,a40.609+0.436+0.7521.7978a2,a4,a50.609+0.752+0.6041.9659a2,a4,a60.609+0.752+0.3911.75210a2,a5,a60.609+0.604+0.3911.60411a3,a5,a60.436+0.604+0.3911.43112a4,a5,a60.752+0.604+0.3911.745Table 6
Spanning paths and sum of the score functions of their entries whenq=4.
Serial no.Ti=ai∑S2˜aiTi¯1a1,a2,a30.4912+0.5888+0.44471.52472a1,a2,a40.4912+0.5888+0.7041.7843a1,a3,a40.4912+0.4447+0.7041.63994a1,a3,a50.4912+0.4447+0.70401.49995a1,a3,a60.4912+0.4447+0.41121.34716a1,a5,a60.4912+0.564+0.41121.46647a2,a3,a40.5888+0.4447+0.7041.73758a2,a4,a50.5888+0.704+0.5641.85689a2,a4,a60.5888+0.704+0.41121.70410a2,a5,a60.5888+0.564+0.41121.56411a3,a5,a60.4447+0.564+0.41121.419912a4,a5,a60.704+0.564+0.41121.6792Table 7
Spanning paths and sum of the score functions of their entries whenq=5.
Serial no.Ti=ai∑S2˜aiTi¯1a1,a2,a30.4960+0.5684+0.45841.52282a1,a2,a40.4960+0.5684+0.66361.7283a1,a3,a40.4960+0.4584+0.66361.61744a1,a3,a50.4960+0.4584+0.53871.49315a1,a3,a60.4960+0.4584+0.43151.38596a1,a5,a60.4960+0.5387+0.43151.46627a2,a3,a40.5684+0.4584+0.66361.69048a2,a4,a50.5684+0.6636+0.53871.77079a2,a4,a60.5684+0.6636+0.43151.663510a2,a5,a60.5684+0.5387+0.43151.538611a3,a5,a60.4584+0.5387+0.43151.428612a4,a5,a60.6636+0.5387+0.43151.6338


## 6. Conclusion

The study was carried out to enhance the real-life efficiency of some important models by curbing the issues of imprecise and vague information. Since the fuzzy sets are known to have the capacity to provide different models and tools for handling such information,q-ROFS is more suitable than IFS and PFS as it increases the space containing acceptable orthopair by increasing the value of parameter q. In this study, we have proposed vector spaces based on q-ROFSs and subsequently named them as q-ROFVSs. We have also discussed q-ROFVSs with an illustrative numerical example and developed some relevant results like basis and dimension. The q-ROF linearly independent vectors are also discussed. Furthermore, we have introduced q-ROFMs with their characteristics. We have extended some of the results based on IFM and PFM to q-ROFM. We have also discussed the notions of circuits, basis, and rank function for q-ROFMs. Finally, we have concluded the proposed work with a real-life application of decision making regarding human trafficking between different countries. For that, we used a directed graph with q-rung orthopair fuzzy information and combined it with the concept of maximal independent sets of edges of the graph to find the heaviest path. To enhance the capability of the q-ROFMs, we have provided a comparative analysis with an existing model. We are of the view that the given study would help the concerned countries in deciding the action response in a suitable path for the reduction of human flow. The major limitation of the proposed model is that it fails when the objects are evaluated concerning multiple parameters from more than one expert. In other words, there is no parametrization tool present in the initiated approach. Moreover, we are also interested to broaden our work to (a) q-rung orthopair fuzzy soft matroids, (b) fuzzy N-soft matroids, and (c) spherical fuzzy N-soft matroids. This will illustrate more exclusive results based on the given fuzzy matroids and will be helpful in figuring out more real-life problems.



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*Source: 8261118-2021-12-21.xml*