#### Abstract

The theory of -rung orthopair fuzzy sets (-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 -ROFS is used to introduce the concept of -rung orthopair fuzzy vector spaces (-ROFVSs) and illustrated by an example. We further elaborate the -rung orthopair fuzzy linearly independent vectors. The study also involves the results regarding -rung orthopair fuzzy basis and dimensions of -ROFVSs. The main focus of this study is to define the concepts of -rung orthopair fuzzy matroids (-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 -rung orthopair fuzzy graph (-ROFG). At last, a comparative study of the proposed work with the existing theory of Pythagorean fuzzy matroids is also presented.

#### 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 -ROFSs in which the sum of the *q*th power of membership and nonmembership values of elements is bounded by one. After that, -ROFSs are frequently used in decision making as -ROFSs widened the range of acceptable pairs rather than IFSs and PFSs with the parameter “” adjustment. Recently, Garg used -ROFSs to introduce a novel concept of connection number-based -rung orthopair fuzzy set (CN--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 -connection numbers for interval-valued -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 -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 -ROF competition graphs and studied their applications. Sitara et al. [25] introduced the notion of -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 -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)The -ROFS is a generalized form of some existing models, including IFSs and PFSs. On setting and , we get IFSs and PFSs, respectively, as special cases of -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 of -ROFSs, the developed -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 illustrates -ROFVSs with an illustrative example.(2)Most importantly, the notion of -ROFMs is defined and characterized with its basis and dimension.(3)This study also provides various results regarding -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 of -ROFVSs with a numerical example and discuss their bases and dimensions. We also discuss the -rung orthopair fuzzy linearly independent vectors. We further combined the -ROFSs with the fuzzy matroids and named them as -ROFMs. We investigate the concepts of circuits, basis, and rank for -ROFMs. Note that for and , our proposed -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 -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 Section 2, we recall some fundamental definitions including crisp matroids with rank function, -ROFSs with their score functions, and some basic operations defined on -ROFSs. In Section 3, we first propose -ROFVSs and then -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]). *Let be a set of finite elements and denote the power set of . For , a non-empty family of subsets, the pair is called a matroid (or crisp matroid) if it satisfies the following:(1).(2)If with , then .(3)If with , then another subset exists such that , where shows the number of elements of .The element is called independent set in . Also, is known as maximal independent in if we do not have such that contains .

*Definition 2 (see [1]). *Suppose that is a matroid and . If is maximal independent in , then is called base of and represents the family of all bases.

*Definition 3. *Let be a subset with ; then, is called dependent subset. A circuit of is the subset where is inclusion-wise minimal dependent subset.

*Definition 4 (see [1]). *Let be a matroid. Consider a mapping defined asThen, is called rank function of .

*Definition 5 (see [2]). *Consider a non-empty set . The fuzzy set is defined asThe mapping assigns the membership value of to and represents the family of all fuzzy sets on .

*Definition 6 (see [32]). *Let be a non-empty finite universe of discourse and . For any fuzzy sets , the collection satisfies the following:(1).(2)If , for all .(3)Let and where and ; then, another exists satisfying(a), , and union is defined as .(b), and is defined for any .The pair is called a fuzzy matroid and is the subfamily of all independent FSs of the matroid .

*Definition 7 (see [15]). *Let be a non-empty fixed set with finite elements. Then, the set defined on a fixed set is called -ROFS if it satisfies(1).(2) with the property , for any .It can be seen easily that for and , these fuzzy sets are reduced to IFS and PFS, respectively. The -rung orthopair degree of hesitance for is given as

*Definition 8 (see [15]). *Let be a -ROFS and for , be a -rung orthopair fuzzy number (-ROFN). Then, a score function of is defined as

*Definition 9 (see [15]). *Let be a -ROFS and for , be a -ROFN. Then, for any , an accuracy function for -ROFN is defined as

*Definition 10 (see [41]). *Let be two -ROFSs. Then, for any , and are called union and intersection, respectively, defined asHere, we denote the family of all -ROFSs on by . Let be a -ROFS; then, some notions are defined as follows.(1).(2).(3), for .(4). A -ROFS is called elementary if .(5) is called height of and the set is called normal -ROFS for the height .Note that we denote the smallest and the largest -rung orthopair fuzzy elements and , respectively.

#### 3. -Rung Orthopair Fuzzy Vector Spaces

This section illustrates the concept of -ROFVSs with their basis and dimension and presents -rung orthopair fuzzy linearly dependent and independent vectors. Here, we also present matroids based on -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 -ROFSs to generalize the Pythagorean fuzzy vector spaces [43] and define -ROFVSs.

*Definition 11. *Let be a non-empty finite vector space over the field . The -ROFS , is called -ROFVS over , if for scalars , we havewhere holds for defined mappings and .

Here, the set of all -ROFVSs over is denoted by the pair .

The following proposition illustrates that membership and non-membership functions assign unchanged values under scalar multiplication in -ROFVSs.

Proposition 1. *Let be a -ROFVS. The following two properties hold for each :*(1)*.*(2)*For any non-zero scalar , .*

*Proof. *The proof of properties (15) and (18) is very straightforward (see Definition 11).

Proposition 2. *Let with and ; then, we have*

*Proof. *To prove, from Definition 11, let and hence and .

*Definition 12. *Let be a non-empty finite universe and be a -ROFVS over . Then, the set of vectors is called a -rung orthopair fuzzy (-ROF) linearly independent in if(1) is linearly independent.(2)For any , we have

Proposition 3. *Let be a -ROFVS over . Consider any set of vectors with non-zero elements such that for , . Then, the set is linearly and -ROF linearly independent.*

*Proof. *By using the induction on , the statement is true for . We suppose that the statement is true for . So, is -ROF linearly independent. Let such that for , and suppose that the set is not linearly independent. Thus, for , we have where for all . Then,which gives that and contradicts that has distinct values and hence is linearly independent. Propositions 1 and 2 show that is -ROF linearly independent.

*Definition 13. *Let be a -ROFVS and , where each . Then, the set is called -ROF basis in , if it satisfies(1)The set is basis in .(2)For scalars , we have

*Definition 14. *Let and be a -ROFVS having basis . Then, the dimension of -ROFVS is given byIt is easy to see that is a function from the class of all -ROFVSs to . A -ROFVS is said to be finite dimensional if and only if .

Proposition 4. *Let be a -ROFVS over . For any , if and , we have*

*Proof. *We use Proposition 2:Since , thenNow we writewhich implies the resultSince , thenwhich proves that . Similarly, we use Proposition 2:Since , thenNow we writewhich implies the resultSince , thenwhich proves that .

The following example illustrates Definition 11 clearly.

*Example 1. *Let and be a 5-ROFS defined on . For any , the mappings and are defined byrespectively. It can be easily seen that, for , the case is trivial.

For the second case, consider two vectors and from with one non-zero component, i.e., and ; then, we have and . For any , we haveClearly, it satisfies all the conditions of Definition 11.

Now, consider and with one zero component and two non-zero components, i.e., and ; then, and . For any ,which satisfies the conditions of Definition 11.

*Definition 15. *Let be a finite universe. The subset is a subfamily of -ROFSs satisfying(1).(2)For any where , , , and if , then , for all .(3)For any and , there exists such that(a), for any .(b).Then, the pair is called and the set is the subfamily of all independent -ROFSs of .

Proposition 5. *Let be a -ROFVS and be a subset of containing -ROF linearly independent column vectors in . The pair is a -ROFM on .*

*Proof. *Suppose that is a non-empty set containing column labels of a -ROF matrix, and represents a -ROF submatrix containing those columns which are labeled in . Consider a set of -ROF linearly independent column vectors of , i.e.,For any submatrix , we have . It is easy to see from Definitions 11 and 15 that is .

Note that is called dependent .

*Definition 16. *Let be a and be a family of dependent in . The minimal dependent (inclusion wise) set is called -ROF circuit of and represents the subclass of all circuits of , i.e.,Note that the elements of follow the properties:(1).(2)Let be -rung orthopair fuzzy circuits with .(3)Let and with where . Then, there exists satisfying(a) and .(b).

*Definition 17. *Let be a -ROFM. Consider an element ; then, is called maximal independent set in a matroid if there does not exist that contains . A maximal independent set in is called -ROF base or basis of . The collection of all -ROF basis is defined asNote that although -ROF basis contains all the independent sets in , there exist some that do not have -ROF basis.

*Example 2. *Let be a family of all -ROFSs defined on a non-empty set . Then, for a positive integer and with , the set is defined asThe pair is called -ROF matroid. Note that the subfamilies of all -ROFSs of with the sizes and are called the -ROF circuits and -ROF basis of , respectively.

*Definition 18. *Let be a -ROFM. The -ROF rank function is defined aswhere . Also, iff and .

It is observed from definition that the -ROF rank function follows the following properties:(1)If , then .(2)If , then .(3)If .

*Definition 19. *Let be a set of “” -ROFNs. Then, the pair satisfies the ordering for each :We say instead of and , where and with , respectively.

*Definition 20. *The -cut level set for of a -ROFS is a crisp set which is defined as follows:

Theorem 1. *Let be a -ROFM and be a collection of all -cut levels of -ROF independent sets where , i.e.,*

Then, is a crisp matroid on .

*Proof. *The proof is very straight forward from Definition 20, and is a collection of crisp subsets of . Then, for each , we have .

*Definition 21. *Let be a finite universe and be a -ROFM. Then, we have a finite sequence such that(1).(2).(3)If .(4)If .The sequence is called fundamental sequence of .

Corollary 1. *From Theorem 1 and Definition 21, for , assume that ; then, is called a -induced matroid sequence.*

Theorem 2. *Let be a finite fundamental sequence and be finite sequence of crisp matroids regarding this fundamental sequence. For each , we assume and for . Then the pair is a where is defined as*

*Proof. *It is easy to see that as for . Now, assume that such that . It is clear from definition of that for each , , so , and since we have that is a crisp matroid for , it means , and hence and proves (18) of Definition 15. Now, let with . It is known that . Let be a defined asIt is easy to observe from definition of that contains the support of both . Note that contains independent subsets; then, there exists an independent subset satisfying(1) contains , for all .(2).Let us define aswhich shows that is and satisfies (20) of Definition 15. Hence, is .

Theorem 3. *Let be a and from Definition 20 and defined in Theorem 1, for each , is a crisp matroid. Let . Then, .*

*Proof. *It is easy to deduce from definition of that . For , let be a non-zero -rung orthopair fuzzy range with and order . One can notice that for each and , we have that . So, if , then from Definition 21, . To prove , we define a -rung orthopair fuzzy set for each and aswith . Since we have , for that gives . Here to show , we use induction process. For each , we considerSince is an independent -ROFS, it is enough to show that if for , then for each , we have . Definewhich shows that for each , so and . Define another -ROFS asUsing the induction method, we have where is an independent set, and . So, from Definition 15, is independent in . If , then we have that is also an independent set in . But, if , then to move further, we define -ROFS asFrom Definition 19, , and from , is an independent set in . Similarly, define another -ROFS asSince and , then again from Definition 15, is an independent set in . So, is also an independent set in for . But, if , then to proceed further, we obtain a new -ROFS and hence is an independent set in .

The next result is the direct consequence of Theorem 1.4 discussed in [34].

Corollary 2. *Let be a and . Then, if and only if for each *