Group Generalized q-Rung Orthopair Fuzzy Soft Sets: New Aggregation Operators and Their Applications

Hayat, Khizar; Shamim, Raja Aqib; AlSalman, Hussain; Gumaei, Abdu; Yang, Xiao-Peng; Azeem Akbar, Muhammad

doi:https://doi.org/10.1155/2021/5672097

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5672097 | https://doi.org/10.1155/2021/5672097

Group Generalized q-Rung Orthopair Fuzzy Soft Sets: New Aggregation Operators and Their Applications

Khizar Hayat,¹Raja Aqib Shamim,¹Hussain AlSalman,²Abdu Gumaei,³Xiao-Peng Yang,⁴and Muhammad Azeem Akbar⁵

Academic Editor: Giuseppe D'Aniello

Received10 Jul 2021

Revised21 Nov 2021

Accepted25 Nov 2021

Published31 Dec 2021

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 .(3)If for all and for all , then .(4)If for all and for all then .

Definition 23. 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 geometric operator (GWq-ROFG) undertook by GGq-ROFSS is contemplated as follows:where is the q-rung orthopair fuzzy weighted geometric operator and it operates over the set of criteria, and is the q-rung orthopair fuzzy weighted geometric operator and it operates mutually over q-ROFSS and on the set of senior moderators.
The set of all GWq-ROFG operators for number of alternatives is indicated as . The above novel GWq-ROFG operators are realistic instruments for linear and entire aggregations of q-ROFVs in GGq-ROFSS.
The GWq-ROFG 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 3. Consider GGq-ROFSS indicated in Table 4 in Example 3. Given , we have to calculate the GWq-ROFG 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-ROFG is given bySimilarly, GWq-ROFG operators , and can be obtained.

Theorem 5. Let and be the q-ROFVs in GGq-ROFSS in Table 2, where , , and . If we consider , then GWq-ROFG operator is given by

Proof. Same as the proof of Theorem 1.

Theorem 6. (idempotency) If and for all , then .

Proof. Same as the proof of Theorem 2.

Theorem 7. (boundedness) If and for all and , then.

Proof. Same as the proof of Theorem 3.

Theorem 8. (monotonicity) If and for all , are two IFVs such that , then

Proof. It can be concluded from Theorem 4.

Proposition 3. Let be a GGq-ROFSS, given in Table 2. Then,(1)If for all and , then .(2)If for all and , then .(3)If for all and for all then .(4)If for all and for all then .

6. Multicriteria Decision-Making Method

In this section, a methodology on proposed operators is introduced, and for application, a numerical application is investigated.

6.1. Methodology

Let be the set of alternatives for a problem in which we have to choose the best alternative. Every alternative possesses the specific q-ROFVs of a set of criteria/attributes . Let . An expert’s committee give the q-ROFVs on set comprising attributes in the form of q-ROFSS. The final examination of the q-ROFSS is provided by some senior experts/moderators in the form of q-ROFs over . This forms GGq-ROFSS given in Table 2 in Definition 14. In real-life problem, usually, the importance of attributes is given in the form of a weighted vector. Let be the weighted vector over , such that . Also take weighted vector such that where are the 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). Next, the normalization of the q-ROFVs in is obtained by using the following equation:

This designs a new GGq-ROFSS . Then, we compute GWq-ROFA or GWq-ROFG operators on and calculate score function on each GWq-ROFA or GWq-ROFG. Finally, rank the alternatives on the real values of score functions.

In order to evaluate most suitable alternative, we numerate the steps of the MCDM method in an algorithm as follows.

(1)	Consider a set of alternatives say their attributes as a set . .
(2)	Constitute number of committees of experts comprising sets of attributes respectively, where .
(3)	Obtain number q-ROFSSs on the respectively.
(4)	Obtain curial opinions of number moderators on each q-ROFSSs. This formulate a GGq-ROFSSs .
(5)	Take union which is equal to , where
(6)	Normalize , which gives normalized GGq-ROFSS .
(7)	Obtain the weighted vector over set of attributes. Obtain the weighted vector comprising q-ROFSS and opinions of moderators.
(8)	Compute GGq-ROFA (or GGq-ROFG) operators on normalized GGq-ROFSS .
(9)	Get score function or on each GGq-ROFA (or GGq-ROFG) operator or using Definition 4.
(10)	Classify alternatives and foremost alternative is on the maximum score.

This algorithm is shown in Figure 1.

6.2. Numerical Example

An automation company wants to choose a director for their Research and Development (R and D) department from four applicants , and . The crucial attributes for evaluation are : personality : self-confidence : score obtained in a college degree : administrative skills : experience : abilities in research areas

In order to evaluate most suitable candidate, two different committees of experts, namely, and , are constituted by the company. The attribute specifications for and are and , respectively. The q-ROFV-based evaluations are q-ROFSSs (in the light gray part of Table 7) and (in the light gray part of Table 8). To finalize q-ROFV-based data, two senior experts provided their opinion as q-ROFSs on the q-ROFSSs (see the brown part of Tables 7 and 8). The following steps are adopted to finalize decision-making process:(1)Take union on GGq-ROFSSs and , and obtain GGq-ROFSS in Table 9.(2)Take be the weighted vector over . Also take weighted vector for q-ROFSS and judgements of senior experts.(3)Calculate GWq-ROFA operators; we use and obtain , and .(4)Calculate score function; we obtain , , and .(5)Final ranking of candidates is given by . Thus, is the most appropriate candidate for the position of director in the company.

7. Comparative Study

In this part of the article, we compare our method with existing frameworks. The primely advantage of new aggregation operators is to characterize a weighted vector on GGq-ROFSS, where are the weights on q-ROFSs of number of moderators respectively, and is a weight for q-ROFSS in GGq-ROFSS. Since assessments of moderators on q-ROFSS are the crucial component in GGq-ROFSS, in many real-life problems, their weights should be greater than . On the other hand, in some real-life situations, weight on q-ROFSS is also important, the fact that it denotes a perception on fundamental data (or q-ROFSS) given by a committee of experts on crucial judgements on alternatives. Notably, we consider following example.

Example 4. Consider GGq-ROFSS as depicted in Table 6, where is set of attributes with weighted vector and being the set of alternatives. is q-ROFS for moderator’s assessments on q-ROFSS in Table 10.
Let and a weighted vector , where is the weight for q-ROFSS and is the weight for q-ROFS . By our new operators on GGq-ROFSSs, we obtain , , and .
By the operators in Hayat et al. [56], we obtain and .
By Feng et al. [54], .
By the operators in Hayat et al. [55, 59], we obtain . A comparison for different methods on this example is shown in Table 11.
The main advantage our framework is that it aggregates information or data by concerning attributes until final ranking appears. It develop an internal mechanism that gently addresses the importance of parameters in aggregation operators for GGq-ROFSSs. The different components of the above methods are discussed in Table 12.

8. Conclusions

In this article, we have presented GGq-ROFSS and investigated some basic operations. Mainly, we have initiated new averaging and geometric aggregation operators on GGq-ROFSSs and investigated the underlying properties of these operators. We have given a MCDM framework and its validation through a numerical example. Finally, we have given comparison of our methodology with other existing methods. In GGq-ROFSSs, expert’s judgements suggest the reliability of the evaluation of the alternatives on criteria. These judgements appear as generalization part in GGq-ROFSSs to fulfil and complete q-ROFSS-based data as some final examination. Therefore, for prospect MCDM, GGq-ROFSS, which indicates in for some suggests a better framework to deal with uncertainty.

The method of computation of q-ROFS-based data from newly introduced ideas can sight wide applications for available data in machine learning, applied intelligence, and supply chain management. Although there are some methods in machine learning, artificial intelligence, and supply chain management, proposed operators’ beauty will be a notable sight in such domains. Our focus will be on these operators’ new rudiments and their supply chain management applications in future work. Furthermore, we will define GGq-ROFSS-based order operators, Shapley Choquet operators, and distance aggregation operator.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This research was supported by the Researchers Supporting Project number (RSP-2021/244), King Saud University, Riyadh, Saudi Arabia.

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Copyright © 2021 Khizar Hayat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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