#### Abstract

In recent years, q-rung orthopair fuzzy sets have been appeared to deal with an increase in the value of , which allows obtaining membership and nonmembership grades from a larger area. Practically, it covers those membership and nonmembership grades, which are not in the range of intuitionistic fuzzy sets. The hybrid form of q-rung orthopair fuzzy sets with soft sets have emerged as a useful framework in fuzzy mathematics and decision-makings. In this paper, we presented group generalized q-rung orthopair fuzzy soft sets (GGq-ROFSSs) by using the combination of q-rung orthopair fuzzy soft sets and q-rung orthopair fuzzy sets. We investigated some basic operations on GGq-ROFSSs. Notably, we initiated new averaging and geometric aggregation operators on GGq-ROFSSs and investigated their underlying properties. A multicriteria decision-making (MCDM) framework is presented and validated through a numerical example. Finally, we showed the interconnection of our methodology with other existing methods.

#### 1. Introduction

Zadeh originated the fuzzy set (FS) as an enlargement of the standard sets by the concept of inclusion of vague human judgements in computing situations [1]. The FS is indicated by the fuzzy information , which gives values from the unit close interval for each prospector . The idea of the FS plays an important role in the domain of soft computing, which manages vagueness, robustness, and partial truth. In some real-world difficulties where humanoid though attains reliable and unreliable information, the FS may not be sufficient to deal with underlying uncertainties.

In 1986, another shape of the FS called intuitionistic fuzzy sets (IFSs) was authorized by Atanassov, which provide a reliable grade and unreliable grade for all in the universe of discourse . The IFSs are characterized by the sum and the degree of indeterminacy [2]. Xu and Yager [3, 4] discussed the intuitionistic fuzzy value (IFV), which is an ordered pair of reliable and unreliable information for a component in the IFS on any . Different rudiments of IFSs have been established such as aggregation operators [4], similarity and distance function [5, 6], and multicriteria decision-makings (MCDM) [7]. The aggregation operators are imperious in the MCDM process, which attains a shape of the measurable information by the accumulation of big data [8–10].

The IFSs enhance FSs in a meaningful approach, which is more capable of overcoming uncertainties, sharpless boundaries caused by the hesitation, and lack of assurance in human cognition. Xu and Zhao [11] extended a meaningful and insightful view on the information synthesis for MCDM using IFSs. To deal with real-life cases of reliable and unreliable information, which do not satisfy inequality , Yager initiated Pythagorean fuzzy sets (PFSs) [12, 13] and q-rung orthopair fuzzy sets (q-ROFSs) [14], which are crucial generalizations of IFSs. The q-ROFSs possess overall anticipation of symmetry of reliable and unreliable information in a larger space [15], that is, . A q-ROFS appears as an IFS (PFS) when . The fundamental score function and operators for q-ROFSs were investigated by Liu and Wang [16]. Several basic properties of PFSs and q-ROFSs can be seen in the literature [17–21]. Notably, researchers around the planet check out hybrid MCDM methods of PFSs and q-ROFSs using TODIM [22], TOPSIS [23, 24], MULTIMOORA [25], MABAC method [26], aggregation operators [17, 27–33], entropy measures [34], and distance measures [35].

A general parametrization model called soft set theory initiated by Molodtsov [36] has a great tendency to cop uncertainties. The soft set is free of inadequacy as it is a classical tool for coping parameters. It is further connected with usual mathematical operations on sets by Maji et al. [37] and Ali et al. [38]. A combination of soft sets and FSs known as fuzzy soft sets was introduced by Maji et al. [39], and it has been applied in various fields [40–46]. An extended form of fuzzy soft sets, known as intuitionistic fuzzy soft sets (IFSSs), was initiated by Maji et al. [47]. Recently, q-rung orthopair fuzzy soft sets (q-ROFSSs) have been introduced by Hamid et al. [48]. The model of q-ROFSS is a valuable tool to deal with vagueness by means of the label of parameters along with reliable and unreliable grades in the larger space [49]. Hussain et al. [50] presented MCDM techniques using averaging operators on q-ROFSSs. The generalized IFSSs (GIFSSs) were investigated by Agarwal et al. [51], and it possesses an important opinion with the model IFSS. A different scenario that overcomes the adequacies [52] of the original concept of GIFSSs was given by Feng et al. [53]. Both the ideas of GIFSSs were extended by several researchers Garg and Arora [54], Hayat et al. [55, 56], and Khan et al. [57, 58]. GGIFSSs produce a deep and meaningful insight in the MCDM problem by merging aggregation operators [56]. Another aspect of GGIFSS-based operators has been investigated by Hayat et al. [59], which handle information in a collected form. On the prospect of group-based GIFSSs (GGIFSSs) [56, 59], it is required to develop underlying operators, which can handle MCDM problems in different scenarios of combinations of information. More importantly, the extended space of q-ROFSs is the general form to deal with any implicit information.

On this prospect, there is a huge capacity to exercise another view of GGIFSS aggregation operators because the q-ROFSs relays the ambiguous information in higher productive ways than the GGIFSSs. Another important and fundamental point is to develop a different study to GGIFSSs that aggregate information concerning attributes until final ranking appears. Thus, we developed the group-based generalized q-ROFSSs (GGq-ROFSSs) and new aggregation operators through entire components in GGq-ROFSSs. By motivations of the above discussion, the purpose and aim of this article are given as follows:(1)To initiate a different form of aggregation operators for GGq-ROFSSs that do not abandon the importance of attributes initially and do not quickly fascinate alternatives(2)To develop an internal mechanism that gently addresses the importance of parameters in aggregation operators for GGq-ROFSSs(3)To address the higher range of reliable and unreliable information in GGq-ROFSSs for possible values of (4)To develop the MCDM method for GGq-ROFSSs environment

In Section 2, we recall basic ideas of IFSs, PFSs, q-ROFSs, soft sets, and q-ROFSSs. In Sections 3 and 4, we discuss the notions of GGq-ROFSSs and their operations. In Section 5, we define new aggregations operators on GGq-ROFSSs. In Section 6, we give a new method of MCDM and a numerical example of real-life applications. Section 7 gives comparisons with other existing methods, and the last section concludes the paper.

#### 2. Preliminaries

In this section, we will recall the concepts of IFSs, PFSs, q-ROFSs, soft sets, and q-ROFSSs. Throughout this section, will represent the collection of alternatives.

##### 2.1. Intuitionistic Fuzzy Sets and Pythagorean Fuzzy Sets

A FS is a mapping , where is membership grade for an element [1]. In several real-life situations, reliable and unreliable information rectifies the proper signification of uncertainties. FSs were not sufficient in such situations; therefore, the concept of IFS was introduced.

*Definition 1 (see [2]). *An IFS is expressed as follows:With the functions , called reliable and unreliable grades of an element of under the condition that the following inequality holds:For an element , is called IF value (IFV) in . In an IFS, the hesitancy of an IFV to is given byThe hesitancy of IFV is also called indeterminacy of the in .

*Definition 2 (see [12, 13]). *A PFS is expressed as follows:With the functions , called reliable and unreliable grades of an element of under the condition that the following inequality holds:For an element , is called PF value (PFV) in . In an PFS, the hesitancy (or indeterminacy) of an PFV to is given by

##### 2.2. q-Rung Orthopair Fuzzy Sets

In 2016, Yager extended the range of double-graded fuzzy models in higher space. It is defined as

*Definition 3 (see [14]). *A q-RFS is defined asWith the functions , called reliable and unreliable grades of an element of under the condition that the following inequality holds:Particularly, the hesitancy degree for q-ROFS is given asThe pair is called q-rung orthopair fuzzy value (q-ROFV) for an object . Let and be two q-ROFSs; then,The study on q-ROFS is extended by Liu and Wang [16] in the following crucial notions:

*Definition 4 (see [16]). *Let be q-ROFN; then, the score function is defined as follows:This notion is effective when we have to transfer a q-ROFN to a real value in interval and therefore we can compare two or more q-ROFNs on their score functions. Consider a special case when , with the condition . Then, ; thus, has an inadequacy for such a case. Therefore, in this study, when fails, we will use the following.

*Definition 5 (see [32]). *Let be q-ROFN; then,

*Definition 6 (see [16]). *Let us take the collection of q-ROFNs , and weight vector ; then, the q-rung orthopair fuzzy weighted averaging operator (q-ROFWA) is indicated as

*Definition 7 (see [16]). *Let us take the collection of q-ROFNs , and weight vector ; then, the q-rung orthopair fuzzy weighted geometric operator (q-ROFWG) is indicated asThe q-ROFWA and q-ROFWG are effective to aggregate data involving large number of q-ROFNs to a single q-ROFN.

##### 2.3. q-Rung Orthopair Fuzzy Soft Sets

Jointly the approach of soft sets [36] with q-ROFSs is known as the q-rung orthopair fuzzy soft set, which is handy in MCDM problems. The notion of soft set is described as follows.

*Definition 8 (see [36]). *Let be a fixed set and be the set of all subsets of . Consider the set of parameters and be the subset of . Let us define a function asThen, pair is called the soft set.

*Definition 9 (see [49]). *Let be a soft universe and . Define a mapping ; then, pair is called the q-rung orthopair fuzzy soft set (q-ROFSS) over , where denotes the collection of all q-ROFSs over . The q-ROFSS can be described aswhere and are reliable and nonreliable grades of fulfilling inequality . Take , , where . Then, the tabular form of q-ROFSS is given in Table 1.

For simplicity, we will represent by .

*Definition 10 (see [49]). *Consider two q-ROFSSs and on . We say if and only if(i)(ii) is q-ROF subset of for all

*Definition 11 (see [49]). *Consider two q-ROFSSs and on . Then, denotes the union of and on , such thatwhere is the union of q-ROFSs .

*Definition 12 (see [49]). *Consider two q-ROFSSs and on . Then, denotes the intersection of and on , such that , where , where is the intersection of q-ROFSs .

#### 3. Group Generalized q-Rung Orthopair Fuzzy Soft Sets

In this section, we will define generalized q-Rung orthopair fuzzy soft sets (Gq-ROFSSs) and group-based generalized q-Rung orthopair fuzzy soft sets (GGq-ROFSSs). First Gq-ROFSS is described as follows.

*Definition 13. *Consider a soft universe and contained in . A triple is called Gq-ROFSS over if is a q-ROFSS over and is a q-ROFS over .

In Gq-ROFSS, only one extra opinion appears, but in many real-life satisfactions, more than one crucial additional opinions are needed. Thus, we define a greater prospect of Gq-ROFSS as GGq-ROFSS as follows.

*Definition 14. *Consider a soft universe and contained in . A triple is called GGq-ROFSS over if is a q-ROFSS over and , where are the parametrized q-ROFSs (Pq-ROFSs) over . In other words, is the group Pq-ROFSs considered by “” number of senior experts/moderators.

*Remark 1. *If in GGq-ROFSSs, then it is Gq-ROFSS. Therefore, Gq-ROFSS is a special case of GGq-ROFSS, and thus, generally, we will focus on GGq-ROFSSs.

A broaden tabular form of GGq-ROFSS (in Definition 14) is given in Table 2.

In Table 2, the light gray part represents q-ROFSS and the brown part represents the group of q-ROFSs in GGq-ROFSS.

#### 4. Operations on Group-Based Generalized q-Rung Orthopair Fuzzy Soft Sets

In this section, we will define subset, union, intersection, and complement of GGq-ROFSSs.

*Definition 15. *Consider a soft universe and contained in . Let two GGq-ROFSSs and on , where , , , respectively, are group Pq-ROFSs related to “” number of senior experts/moderators. The is group q-ROF subset of if and only if , …, , and , …, and for each . It is denoted as .

In the prospect of Definition 15, we introduce GGq-ROFS subsets.

*Definition 16. *Let two GGq-ROFSSs and on , where . Then, is a GGq-ROFS subset of if(i)(ii)

*Definition 17. *Let two GGq-ROFSSs and on , where such that and , . The extended union of and is investigated in the form of GGq-ROFSS. which is given as follows:such that(i).(ii)For each , is defined aswhere .

*Definition 18. *Let two GGq-ROFSSs be and on , where such that and , . The restricted intersection of and is investigated in the form of GGq-ROFSS, which is given assuch that(i)(ii)For each , is defined as and for all , where Now, we will take an example of GGq-ROFSS to clarify the above concepts.

*Example 1. *Let be the universal set consisting of five different kinds of face masks available in market, and the set of attributes is given as , where each , respectively, stands for affordable price, good fabrication, effective comfortable design, capable of stopping viruses and bacteria, and comfortable breathing while wearing the mask. Let the two buyers and , respectively, have the following preferences while buying the face mask:The GGq-ROFSSs and for buyers and are interpreted in Tables 3 and 4, where two senior persons and provide their opinions on q-ROFSSs (given in the light gray parts of Tables 3 and 4). The extra inputs as the group of q-ROFSs of and are interpreted in the brown parts of both the tables. The union and intersection of and are computed in Tables 5 and 6 respectively.

*Definition 19. *Consider a soft universe and contained in . Let GGq-ROFSS be on where . The complement of is denoted by , where is the complement of q-ROFSS and .

*Definition 20. *Let GGq-ROFSS be given in Table 2.(1)If >< = >1,0< and >< = >1,0< for all , and , then is called whole GGq-ROFSS. It is denoted by .(2)If >< = >0,1< and >< = >0,1< for all , and , then is called null GGq-ROFSS. It is denoted by .

Proposition 1. *Let be a GGq-ROFSS over . Then,*(1)*, *(2)*, *(3)*, *

#### 5. Aggregation Operators on GGq-ROFSS

To attain substantial effect, there is an immense need to define better aggregation operators that accurately deal with all components of GGq-ROFSSs.

##### 5.1. GWq-ROFW Operators

*Definition 21. *Consider Definition 14, where GGq-ROFSS is given in Table 2, where the light gray part represents q-ROFSS and the brown part describes the group of q-ROFSs of “” number of senior moderators. Let , , that is to say are q-ROFVs in the light gray part of Table 2. Moreover, , , that is to say are q-ROFVs in the brown part of Table 2. Assume that . A symbolization is given byOn the above fundamental and crucial symbolic notion of the GGq-ROFSS, we contemplate novel averaging aggregation operators.

*Definition 22. *Consider Definition 14, where GGq-ROFSS is given in Table 2. Let be the weighted vector over , such that and . Also take weighted vector such that and , where are weights for the judgements of the “” number of senior moderators and is the weight for each q-ROFV in the light gray part of Table 2. In other words, is the weight of whole data in q-ROFSS (see the light gray part of Table 2). Assume . Then, the generalized weighted q-Rung orthopair fuzzy averaging operator (GWq-ROFA) undertaken by GGq-ROFSS is contemplated as follows:where is the q-rung orthopair fuzzy weighted averaging operator and it operates over the set of criteria, and is the q-rung orthopair fuzzy weighted averaging operator and it operates mutually over q-ROFSS and on the set of senior moderators.

The set of all GWq-ROFA operators for number of alternatives is indicated as . The above novel GWq-ROFA operators are realistic instruments for linear and entire aggregations of q-ROFVs in GGq-ROFSS.

The GWq-ROFA operators have a specific way of incorporating each component of GGq-ROFSS. They entirely compel q-ROFVs in a linear way towards attributes until the final q-ROFV appears.

*Example 2. *Consider GGq-ROFSS indicated in Table 4 in Example 20. Given . We have to calculate the GWq-ROFA operator for . The q-ROFVs are depicted asLet be the weighted vector over . Also take weighted vector for q-ROFSS and judgements of senior person/moderators. For , we haveNow, GWq-ROFA is given by = = . Similarly, GWq-ROFA operators , and can be obtained.

Theorem 1. *Let and be the q-ROFVs in GGq-ROFSS in Table 2, where , and . If we consider , then the GWq-ROFA operator is given by*

*Proof. *Assume that and . Take ; we apply mathematical induction on . By the definition of GGq-ROFSS,Now, , . Thus,Hence, the theorem is valid for . Considering that this result is fine for that isThen, for , we haveHence, by mathematical induction, Theorem 1 satisfy for all positive integer .

Theorem 2. *(idempotency) If and for all , then .*

*Proof. *Given and for all and .

Theorem 3. *(boundedness) If and for all and , then .*

*Proof. *Since ≥ ≥ ≤ ≥ ≥ ≥ ≤ ≤ ≤ .

Similarly, nonmembership part is aggregated as . This concluded the proof of the theorem, .

Theorem 4. *(monotonicity) If and for all , are two IFVs such that , then .*

*Proof. *It can be concluded from Theorem 3.

Proposition 2. *Let be a GGq-ROFSS, given in Table 2. Then,*(1)*If for all and , then .*(2)*If for all and , then *