Complex q-Rung Orthopair Fuzzy N-Soft Sets: A New Model with Applications

&is research article expands the formal representation of human thinking to a most generalized hybrid theory, namely, complex q-rung orthopair fuzzy N-soft set. It is able to capture a great deal of graded imprecision and vagueness, which so often appear together in human interpretations. &is model renders a parameterized mathematical tool for the ranking-based fuzzy modeling of two-dimensional paradoxical data. To that purpose, the proposed theory integrates complex q-rung orthopair fuzzy sets with the parametric structure of N-soft sets. &e framework that arises captures information beyond the confined space of complex intuitionistic fuzzy N-soft sets and complex Pythagorean fuzzy N-soft sets, with the assistance of a parameter q. We establish the basic set-theoretical operations of this model and prove some of its fundamental properties. &e Einstein and other elementary algebraic operations on complex q-rung orthopair fuzzy N-soft values shall be introduced to broaden the mathematical toolbox of this field. Its relationships with contemporary approaches shall demonstrate its outstanding flexibility. Moreover, we establish two competent multicriteria decision-making algorithms that capture the nuances of periodical inconsistent data. &eir feasibility shall be demonstrated with an explicit application to the selection of optimum aerospace technology required for the economic development of the Mexican space agency. A comparative analysis of both strategies with the prevailing techniques substantiates their rationality. In addition, we illustrate this comparative study with an explicative bar chart that shows the compatibility of their outcomes. Finally, we examine the functionality of the proposed model and compare it with alternative theories.


Introduction
Decision-making plays a substantial role in almost every discipline of actual life, from engineering to the social sciences or medicine. It is an analytical process whereby realworld problems are modeled and their best solutions are identified after careful examination of all rational alternatives. In this competitive era, decision-making has become a quite challenging task due to the presence of multiple criteria for the appraisement of the alternatives. is has resulted into a growing interest in multiple criteria decision-making (MCDM) as a thriving branch of operation research. In recent decades, MCDM methodologies have gained prestige in numerous territories, including information technology, robotics, automotive industries, social sciences, business management, and various other disciplines of science and technology. In classical MCDM techniques, the assessments of the performance of the alternatives regarding their characteristics are entirely precise and exact in nature. However, most of the actual-world decision-making problems cannot be modeled with such clear-cut interpretations because of inherent vagueness and ambiguity of human perceptions. e limitations that this imprecision impose on the traditional approaches has lead the researchers to establish novel theories with more flexible structures. eir target is the incorporation of ample forms of inconsistent knowledge into the umbrella of MCDM.
Zadeh [1] initially put forward the pioneer solution of such difficulty by establishing the foundations of fuzzy sets (FSs). In a fuzzy set, each element is endowed with a membership degree lying within the unit interval [0, 1]. Fuzzy set theory provides a simple but powerful tool for coping with certain types of imprecision of inexact information. Atanassov [2] extended the idea of FS into intuitionistic fuzzy set (IFS) by adding a nonmembership degree (Θ) to the membership degree (Ψ) of FS. For reasons of consistency, he imposed that that the sum of both degrees should not exceed 1. Later on, Yager [3] introduced the notion of Pythagorean fuzzy set (PFS) which relaxes the aforementioned constraint condition of IFSs to Θ 2 + Ψ 2 ≤ 1. Although both FSs and its extension by IFSs have succeeding in producing remarkable applications, these theories restrict the opinions of the decision-makers since they must be confined to a fixed boundary. To overcome this complexity, Yager [4] proposed the idea of q-rung orthopair fuzzy set (q-ROFS) in which the sum of the q-th power of both membership and nonmembership degrees lies between 0 and 1. Nowadays, the q-ROFS is regarded as a most generalized model of the partial membership approach to uncertainty. Concerning decision-making, Akram and Shahzadi [5] developed a hybrid decision-making technique based on q-rung orthopair fuzzy soft information and illustrated its application in the field of medical diagnosis. Shaheen et al. [6] proposed an alternative algorithm by employing relative frequency distributions to generate membership and nonmembership grades of q-ROFSs. Recently, Yang et al. [7,8] have discussed new decision-making methods in this context. ese traditional and recent models that expand fuzzy set theory are limited to address nonperiodic information. To address periodic phenomena, Ramot et al. [9] remodeled the notion of fuzzy sets by introducing the novel theory of complex fuzzy sets (CFSs) in which the range of membership function is extended from the unit interval [0, 1] to the complex unit disc w | w ∈ C, |w| ≤ 1 { }. Alkouri and Salleh [10] developed the concept of complex intuitionistic fuzzy set (CIFS) by proposing the complex-valued membership (Θe iΩ ) and nonmembership (Ψe iΥ ) degrees. Ullah et al. [11] presented the idea of complex Pythagorean fuzzy sets (CPFSs) and improved the restricted conditions of CIFSs.
Liu et al. [12] have established the most generalized extension of complex fuzzy sets, namely, the theory of complex q-rung orthopair fuzzy sets (C q -ROFSs) which significantly relaxed the constraints in terms of a parameter q. e C q -ROFS provides an effective tool for capturing both vagueness and periodicity within the semantics of human assessments. e remarkable potentialities of C q -ROFS have sparked the attention of many researchers who have implemented this outstanding theory in various scientific areas, including image processing, electromagnetism, pattern recognition, fluid dynamics, networking, relativity, and thermodynamics. Mahmood and Ali [13] developed a novel TOPSIS method that uses the correlation coefficient of C q -ROFS and showed its applications for the evaluation of firewall productions and security assessment of computer systems. Garg et al. [14] successfully established AHP and TOPSIS methods for complex interval-valued q-rung orthopair fuzzy sets (CIV q -ROFSs). Jana et al. [15] developed a remarkable MCDM method based on some novel aggregation operators of single-valued trapezoidal neutrosophic (SVTN) numbers for the prioritization of the best commercial software systems.
Despite the benefits and applications of the models explained above, including FSs [1], CFSs [9], CIFSs [10], CPFSs [11], and C q -ROFSs [12], one major common limitation is their inadequacy for parameterized descriptions. An entirely novel escape was provided by the theory of soft sets (SSs) which presents a competent mathematical tool integrating parametrization for the description of the alternatives [16] and their comparisons [17]. e idea of soft set theory was further hybridized with other fuzzy mathematical structures to develop new models such as fuzzy soft sets (FSSs) [18], intuitionistic fuzzy soft sets (IFSSs) [19], Pythagorean fuzzy soft sets (PFSSs) [20], and q-rung orthopair fuzzy soft sets (q-ROFSSs) [21]. Altogether they produce a rich variety of environments for the fuzzy modeling of parameterized noncrisp data. Later on, Chinram et al. [22] established a competent multicriteria decision-making technique by employing novel geometric aggregation operators to diagnose the ailments of patients. Garg et al. [23] presented the concepts of generalized dice similarity measures for C q -ROFSs and demonstrated its empirical application in pattern recognition. Wang et al. [24] developed an advanced MABAC method under q-rung orthopair fuzzy information for selection of the best construction project. Yang and Pang [25] proposed a novel q-rung orthopair hesitant fuzzy decision-making technique based on TOPSIS and linear programming for the inspection of self-service book sterilizer problems. Jana et al. [26] employed novel Dombi aggregation operators of q-rung orthopair fuzzy numbers for the production of an advanced decision-making strategy. Furthermore, they demonstrated its potential applicability in the emerging field of information technology. Wang et al. [27] established the Pythagorean fuzzy interactive Hamacher power aggregation operators along with a novel MCDM technique.
is procedure was applied to the evaluation of the quality of an express service.
A routine inspection of the research work on soft-setinspired theories will show that they tended to prioritize the use of binary interpretations (either 0 or 1) or else real numbers lying within the unit interval [0, 1]. In spite of this fact, numerous decision-making settings are equipped with multinary, discrete-type data. erefore, a generic modelization of soft set theory was required for coping with such daily-life parameterized ranking-based systems. All these practical concerns lead Fatimah et al. [28] to introduce the now thriving theory of N-soft sets (NSSs) along with appropriate decision-making techniques. Real examples were used to highlight the importance of ordered grades and ranking-based annotations in practical applications. Akram et al. [29,30] soon launched hybrid theories such as hesitant N-soft set (HNSS) and fuzzy N-soft set (FNSS) by amalgamating the idea of N-soft set theory with specific mathematical declinations such as hesitancy and fuzziness, respectively. Beyond these first attempts, Akram et al. [31] established another hybrid theory of intuitionistic fuzzy N-soft set (IFNSS) and proposed decision-making algorithms under such a versatile environment. en, Akram et al. [32] developed the more advanced idea of complex Pythagorean fuzzy N-soft sets (CPFNSSs) which integrated N-soft sets with the very general complex Pythagorean fuzzy sets. e literature on NSS theory was further extended after analyzing its rich potentiality for empirical applications in various directions such as optimization theory, data analysis, forecasting, algebraic structures, information systems, and mathematical analysis and in sundry other decision-making complications. Shahzadi et al. [33] proposed an innovative MCDM strategy based on novel Hamacher interactive aggregation operators of Fermatean fuzzy numbers to properly classify the air quality (AQ) of Guangzhou for 16th Asian Olympic Games. Feng et al. [34] proposed Minkowski score function and a novel decision-making algorithm for the appraisement of sophisticated benchmark problems. Lin et al. [35] developed the linguistic q-rung orthopair fuzzy sets (L q -ROFSs) and their interactional partitioned Heronian mean aggregation operators in order to evaluate the credibility of cloud service productions. For the other notation and application, the reader are referred to [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55] e following arguments and evidences condense the motivation and significance of the theory that shall be studied in this work: (i) e C q -ROFS theory has succeeded to model both the uncertainty and periodicity of source data at the same time. However, it has some limitations because of its inadequacy in situations with graded parametrizations, which cannot be related with this theory as it stands. Our proposed theory is designed to address such parameterized fuzziness of 2-dimensional vague data, which makes it superior to the C q -ROFS model since its inception. (ii) e spirit of the NSS model is exclusively concerned with the grading-based uncertainty of the inspected universe of objects, but it has no abilities to cope with other sources of inexactness. In this regard, our proposed C q -ROFNSS theory renders a more generalized and constructive mathematical framework that allows for parameterized modeling of periodical and fuzzy data. (iii) Although the existing CIFSS and CPFSS theories have great abilities to deal with the parameterized vagueness of 2-dimensional problems, their boundary ranges are restricted by some rigid conditions. Our developed model relaxes them, and at the same time, it enables us to capture the graded imprecision embodied in some parameterized uncertain environments. (iv) From another position, also the q-ROFNSS model provides a proficient mathematical structure for addressing the uncertainty of a parameterized dataset. However, it is constrained by a one-dimensional purpose. Our theory generalizes the q-ROFNSS model by the inclusion of periodical fuzzy interpretations of inconsistent information.
In conjunction with the previous point, both advantages demonstrate the robust generalization ability of the proposed concept.
Motivated by all these facts, this research article establishes an advanced hybrid model, namely, complex q-rung orthopair fuzzy N-soft set (C q -ROFNSS). To do so, it merges the advantageous features of C q -ROFS with outstanding mathematical theory of NSS. In this way, the proposed mathematical framework is designed to capture the graded evaluations of parameterized decision-making problems.
e C q -ROFNSS model is highly competent to express a great deal of two-dimensional ambiguous human evaluations. It is a robust generalization of CIFNSSs and CPFNSSs that widens their restricted boundary space by relaxing their constraint conditions in terms of an adjustable parameter q. Moreover, the fundamental set-theoretic operations and mathematical properties of this theory are elaborated. e remarkable flexibility of the proposed model is briefly illustrated with a description of its relationship with existing competing approaches. en, the Einstein operators and some other elementary algebraic operations of C q -ROFNSVs are properly formulated. We develop two advanced decision-making algorithms and demonstrate their effectivity by investigating a heuristic application for the prioritization of aerospace technologies of the Mexican space agency. At the end, a comparative analysis along with an explicative bar chart is presented to vindicate the reliability and functionality of the proposed strategies.
In a nutshell, the paramount contributions of this article can be summarized as follows: (i) is research article systematically expands the literature by introducing a multiskilled and most generalized hybrid model C q -ROFNSS. It is designed for the correct modelization of rankingbased fuzzy modeling of two-dimensional parameterized inconsistent data. (ii) e rationality and accountability of the proposed techniques are substantiated by an empirical application to the discipline of heavy aerospace industries. (iii) We present a comparative analysis with existing MCDM techniques based on q-ROFS ft WG, C q -ROFWA and C q -ROFWG operators. It proves both the feasibility of the strategies developed in this paper and the compatibility of their final outcomes. (iv) e merits and potentiality of the MCDM methodologies here formulated shed light on their flexibility, competency, and prominence over the contemporary decision-making techniques.
e remainder of this article is organized as follows: Section 2 briefly recalls some fundamental preliminary concepts and terminologies preceding the target theory. Section 3 establishes the framework and basic set-theoretic operations of the proposed C q -ROFNSS model. Section 4 demonstrates the Einstein and other algebraic operational laws on C q -ROFNSVs. Section 5 introduces two decisionmaking algorithms and illustrates their applicability by means of a potential application for the selection of aerospace technology. Furthermore, Section 6 substantiates the versatility of the proposed strategies by conducting a Complexity 3 comparative study with existing MCDM techniques based on q-ROFS ft WG, C q -ROFWA, and C q -ROFWG operators. Section 7 expounds the merits and superiority of the developed C q -ROFNSS theory over other contemporary decision-making models. Finally, Section 8 elaborates the concluding remarks of the article and raises some future research questions. e entire contribution of this research article is graphically interpreted in Figure 1.

Preliminaries
In this section, we briefly explicate some basic rudimentary concepts and terminologies required for the proficient modeling of elementary ideas of coming sections.
Definition 1 (see [2]). Let W be a universe of discourse. An intuitionistic fuzzy set T over the universe W is an object having the following form: where real-valued functions Θ T : W ⟶ [0, 1] and Ψ T : W ⟶ [0, 1] specify the membership and nonmembership degrees of an element w ∈ W, satisfying the con- represents the degree of indeterminacy. e pair of membership and nonmembership degrees (Θ T (w), Ψ T (w)) can be referred as an intuitionistic fuzzy number (IFN).
Definition 2 (see [4]). Let W be a universe of discourse. A q-rung orthopair fuzzy set T over the universe W can be characterized as where real-valued functions Θ T : W ⟶ [0, 1] and Ψ T : W ⟶ [0, 1] indicate membership and nonmembership degrees of an element w ∈ W, subjected to the condition represents the degree of indeterminacy. e pair of membership and nonmembership degrees (Θ T (w), Ψ T (w)) is called a q-rung orthopair fuzzy number (q-ROFN).
Definition 4 (see [21]). Let W be a universe of discourse and J be a collection of parameters, M⊆J. A pair T � (I, M) is said to be a q-rung orthopair fuzzy soft set over W if I: M ⟶ q − ROF(W), where q − ROF(W) is the family of all q-ROFSs over the universe W. A q-ROFSS T over the universe W can be represented as follows: where I(m α ) � (w η , Θ ηα (w η ), Ψ ηα (w η )) | w η ∈ W represents the q-ROFS over the universe W. e membership Θ ηα (w η ) and nonmembership Ψ ηα (w η ) degrees belong to the unit interval represents the degree of indeterminacy, for all w ∈ W. e pair of membership and nonmembership degrees (Θ ηα (w), Ψ ηα (w)) is called a q-rung orthopair fuzzy soft number (q-ROFSN).

Complex q-Rung Orthopair Fuzzy N-Soft Sets
Definition 8. Let W be a universe of discourse and J be a collection of attributes. Let M⊆J and K � 0, 1, . . . , N − 1 { } be a set of ordered grades with N ∈ 2, 3, . . .
is the collection of all complex q-rung orthopair fuzzy sets over W × K. e C q -ROFNSS (p q , Y, N) can be characterized in the following decorum: where p q (m α ) � ((w η , k ηα ), Θ ηα (w η , k ηα )e iΩ ηα (w η ,k ηα ) , Ψ ηα (w η , k ηα )e iΥ ηα (w η ,k ηα ) ) | (w η , k ηα ) ∈ W × K} represents the C q -ROFS over W × K. e amplitude terms Θ ηα (w η , k ηα ), Ψ ηα (w η , k ηα ) belong to unit interval [0, 1] satisfying the condition and phase terms Ω ηα (w η , k ηα ), Υ ηα (w η , k ηα ) belong to closed interval [0, 2π] subjected to the condition . For all (w η , k ηα ) ∈ W × K, the degree of indeterminacy can be explicated as follows: In other words, C q -ROFNSS is a ranking-based parameterized family of complex q-rung orthopair fuzzy sets of W × K; that is, the mapping p q assigns each parameter m α ∈ M to a complex q-rung orthopair fuzzy set p q (m α ) of W × K. Now, we present tabular representation of general complex q-rung orthopair fuzzy N-soft set: Let W be a collection of ℘ objects and M be a set of Z attributes, then the tabular characterization of C q -ROFNSS is illustrated in Table 1.
Definition 11. e accuracy function Q for any C q -ROFNSV, V ηα � k ηα , (Θ ηα e iΩ ηα , Ψ ηα e iΥ ηα ) can be explicated as follows:  Complexity where Definition 12. Let V 11 � k 11 , (Θ 11 e iΩ 11 , Ψ 11 e iΥ 11 ) and V 12 � k 12 , (Θ 12 e iΩ 12 , Ψ 12 e iΥ 12 ) be any two C q -ROFNSVs, then the comparison between V 11 and V 12 can be determined as follows: e motivation and prominence of this newly established model is clearly demonstrated by the following reallife practical application. In order to develop intuition for this advanced theory, this application has been kept comparatively simple. is empirical application interprets the deficiencies of the existing q-rung orthopair fuzzy N-soft model, proposed by Zhang et al. [36] and authenticates the remarkable effectuality of the proposed model in the current sophisticated decision-making problems.
Example 1. Spillways are the hydraulic structures constructed on the hydroelectric dams to provide a safe pathway for surplus floodwater to discharge from a reservoir to a downstream area. ese are the most important part of dams that significantly protects the infrastructural integrity of the dam. Nowadays, the selection of the most efficient spillway is one of the most crucial decision-making problems. Suppose that a spillway has to be constructed on the rockfill dam in the north of Greece for the irrigation purpose. An appropriate spillway is determined by star rankings and ratings awarded by the designated experts of the selection board. ese ratings are based on the proficiency of spillways in the last 5 years. Let W � w 1 � Chute spillway, w 2 � Shaft spillway, w 3 � Ogee spillway, w 4 � Siphon spillway, and w 5 � Labyrinth spillway} be a collection of five spillways and M � m 1 � Topography, m 2 � Construction cost, m 3 � Discharge rate, and m 4 � Reservoir capacity} be a set of four parameters which are employed to assign grades to these spillways.
A 5-soft set can easily be obtained from Table 2, where (i) Four stars represent "Excellent" (ii) ree stars represent "Superb" (iii) Two stars represent "Good" (iv) One star represents "Average" (v) Big inverted delta represents "Poor" e grades K � 0, 1, 2, 3, 4 { } can be definitely associated with the ranking-based evaluation carried out by stars as follows: (i) 0 stands for " " (ii) 1 stands for " " (iii) 2 stands for " " (iv) 3 stands for " " (v) 4 stands for " " e appointed experts of the panel rank each spillway on the basis of their overall competency which is further presented in Table 2. Now, the tabular representation of its corresponding identified 5-soft set is provided in Table 3. e selection panel comprehensively evaluate all the identified spillways to determine their ratings by virtue of both membership (Θ ηα ) and nonmembership degrees (Ψ ηα ). So, we extract a 3-rung orthopair fuzzy 5-soft set (3-ROF5SS) by following a certain grading criteria which is given as follows.
For all w η ∈ W and q � 3, 0 en, 3-ROF5SS can be explicated as follows: where Table 1: Tabular representation of C q -ROFNSS. Complexity e tabular representation of 3-ROF5SS is demonstrated in Table 4. e experts of the selection panel analyze all the inspected spillways and assign membership and nonmembership degrees Θ ηα (w η , k ηα ), Ψ ηα (w η , k ηα ), respectively, to each spillway. Now, suppose an expert interprets that " e discharge rate of a spillway w 1 is very low in the first 3 years but dramatically increases in the last 2 years because of rapid climate change and global warming." en, the values of Θ 11 (w 1 , 4) � 0.96 and Ψ 11 (w 1 , 4) � 0.01 are paradoxical and all the information regarding the time frame of discharge rate would be lost which may badly collapse the infrastructure of spillway as well as corresponding dam also. To avoid such possible disaster, it is better to assign complexvalued membership and nonmembership Ψ 11 (w 1 , 4) degrees for the instant elaboration of all the information scrutinized by the expert. Hence, we establish complex 3-rung orthopair fuzzy 5-soft set (C 3 -ROF5SS) rather than 3-rung orthopair fuzzy 5-soft set (3-ROF5SS) for the proper settlement of such deficiency.Now, the following values may be allocated to Θ 11 (w 1 , 4) and Ψ 11 (w 1 , 4) in the light of redefined grading criteria: Here, phase term illustrates all the time-dependent information regarding discharge rate of spillway under consideration. erefore, C 3 -ROF5SS (p 3 , Y, 5) is proposed by integrating the peculiarities of both 3-ROFS and 5-soft set theories for the two-dimensional ranking-based assessment of these investigated spillways. Now, we reshape the grading criteria on the basis of score degrees of C 3 -ROF5SVs , which can be defined as follows: Finally, an advanced C 3 -ROF5SS can be interpreted as follows: where  Table 3: Associated 5-soft set.
Evidently, C 3 -ROF5SS can be easily demonstrated in tabular form by Table 5.

Remark 1
(1) Any C q -ROF2SS (p q , Y, 2) can be definitely associated with complex q-rung orthopair fuzzy soft set (C q -ROFSS). We can generally identify the , which is characterized as follows: for every m α ∈ M.
It does not interpret that there is insufficient information or lack of proper evaluations.
We now proceed to elaborate the various fundamental notions of complementarity in the advanced framework of complex q-rung orthopair fuzzy N-soft sets.
Definition 13. Let (p q , Y, N) be a C q -ROFNSS over the universe W, where Y � (P, M, N) be its corresponding NSS on W. en, the complex q-rung orthopair fuzzy complement (p c q , Y, N) of (p q , Y, N) can be explicated as where Definition 14. Let (p q , Y, N) be a C q -ROFNSS over the universe W, where Y � (P, M, N) be its respective NSS on W. en, the weak complex q-rung orthopair fuzzy complement (p c q , Y c , N) of (p q , Y, N) is defined as where en, its complex 3-rung orthopair fuzzy complement (p c 3 , Y, 5) and weak complex 3-rung orthopair fuzzy complement (p c 3 , Y c , 5) are provided in Tables 6 and 7, respectively.
be the corresponding NSS on W. en, its top complex q-rung orthopair fuzzy weak complement, represented by (p t q , Y t , N), can be demarcated as Definition 16. Let (p q , Y, N) be a C q -ROFNSS over the universe W, where Y � (P, M, N)is the respective NSS on W. en, its bottom complex q-rung orthopair fuzzy weak complement, represented by (p b q , Y b , N), can be interpreted as Example 3. Consider the C 3 -ROF5SS as provided in Example 1. en, its top complex 3-rung orthopair fuzzy weak complement (p t 3 , Y t , 5) and bottom complex 3-rung orthopair fuzzy weak complement Tables 8 and 9, respectively.
We now interpret two significant practicable approaches for the union and intersection of novel complex q-rung orthopair fuzzy N-soft sets.
en, their restricted intersection symbolized by (N a , N b )) can be characterized as follows: , and S and T are C q -ROFSs on p a q (m α ) and p b q (m α ), respectively.
en, their extended intersection represented by S and T are C q − ROFSs on p a q m α and p b q m α , respectively.  Table 7: Weak complex 3-rung orthopair fuzzy complement.  Table 9: Bottom complex 3-rung orthopair fuzzy weak complement. Tables 10 and 11, respectively, where Y a � (P a , F, 6) and Y b � (P b , G, 4) are 6-soft set and 4-soft set over the universe W, respectively. en, their restricted intersection Tables 12 and 13, respectively. (N a , N b )) can be explicated as follows: , and S and T are C q -ROFSs on p a q (m α ) and p b q (m α ), respectively.
Definition 20. Let (p a q , Y a , N a ) and (p b q , Y b , N b ) be two C q -ROFNSSs over the universe of discourse W, where Y a � (P a , F, N a ) and Y b � (P b , G, N b ) are respective NSSs on W.
en, (N a , N b )) is interpreted as follows: S and T are C q − ROFSs on p a q m α and p b q m α , respectively.
Example 5. Consider the C 2 -ROF6SS and C 2 -ROF4SS as defined in Example 4. en, their restricted union (z 2 , Y a ∪ R Y b , 6) and extended union (d 2 , Y a ∪ E Y b , 6) are illustrated in Tables 14 and 15, respectively.

Remark 2.
e Definitions 17-20 of extended (or restricted) union and intersection will suffice the defined grading criteria if and only if N a � N b .
e notion of C q -ROFNSS can be easily identified with contemporary theories including complex q-rung orthopair fuzzy soft set (C q -ROFSS), q-rung orthopair fuzzy N-soft set, N-soft set, and soft set. Now, we are going to derive C q -ROFSS and SS from the proposed C q -ROFNSS by employing the following definitions: M, N) be its respective NSS on W. Let 0 < H < N be a threshold. en, C q -ROFSS over the universe W associated with (p q , Y, N) and H is (p H q , M), which can be interpreted as follows: if k ηα , Θ ηα e iΩ ηα , Ψ ηα e iΥ ηα ∈ p q m α and k ηα ≥ H,

Complexity
Specifically, (p 1 q , M) is said to be the bottom C q -ROFSS and (p N− 1 q , M) is said to be the top C q -ROFSS relative to C q -ROFNSS. and (H, σ) is represented by (p (H,σ) q , M) and can be explicated by the assignment: Example 6. Consider the C 3 -ROF5SS (p 3 , Y, 5) as represented by Table 5. We have 0 < H < 5, from Definition 21. e C 3 -ROFSS associated with feasible threshold H � 3 is summarized in Table 16. Meanwhile, while taking (H, σ) � (3, 1.5), we can define the soft set (p (3,1.5) q , M) which is illustrated in Table 17.
In view of above analysis, it is contemplated that C q -ROFNSSs can be transformed into C q -ROFSSs and SSs under specific conditions. In other words, C q -ROFNSSs are the potent generalization of C q -ROFSSs and SSs.

Remark 3
(1) Let (p q , Y, N) be a C q -ROFNSS over the universe W and J be a set of parameters with M⊆J. en, the NSS is simple assignment illustrates that C q -ROFNSS generalizes NSS as well as SS too.
(2) Every complex intuitionistic fuzzy N-soft set (CIFNSS) and complex Pythagorean fuzzy N-soft set (CPFNSS) are also a C q -ROFNSS for all q ≥ 3, but converse is not true, because is implies that C q -ROFNSS significantly extrapolates the ideal concepts of CIFNSS and CPFNSS.
erefore, we can also claim that C q -ROFNSSs admirably generalizes all these extant models.

Operations
We now elaborate some rudimentary operational laws on complex q-rung orthopair fuzzy N-soft values.

Decision-Making Algorithms
In this section, we develop two eloquent MCDM algorithms that implemented on the proposed multiskilled C q -ROFNSS model. ese strategies are especially designed for the identification of the optimal alternative of real-life heuristic problems that are characterized by C q -ROFNSSs. e underlying methodology of these established decision-making algorithms can be interpreted as follows (Algorithms 1 and 2).

Application.
In this section, we inspect an empirical MCDM problem related to selection of the most proficient industrial robot for automation, by executing our newly formulated decision-making algorithms in order to demonstrate the compatibility and functionality of the proposed strategies.

Selection of Most Proficient Aerospace Technology.
Aerospace industry is a multidisciplinary branch of science and technology that incredibly overlaps aeronautical and astronautical engineering for the manufacturing of high standard military aircrafts, cosmic shuttles, gliders, jets, space stations, and all other related units. Its fascinating technological progress has revolutionized the entire world by accomplishing their iconic achievements such as landing astronauts on the Moon, remodelling of advanced transportation systems and generation of technical spinoffs. e rapid advancement of this fabulous industry has sparked the attention of many practitioners and space enthusiasts to reveal more about galaxies, stars, planets, and other astronomical phenomena. Nowadays, spacecraft are one of the most important aerospace technologies designed to make significant contributions in telecommunication, navigation systems, space colonization, meteorology, planetary exploration, and the manipulation of information-transfer networks. Its remarkable technical capabilities play a substantial role in the prestigious economic and political development of a sovereign country. Acquisition of competent spacecraft is one of the most tricky decision-making problems because of its massive capital investment and availability of a wide range of such amazing technologies with various dominant specifications in global markets. To accomplish the current need, suppose that a Mexican space agency is intended to launch an outstanding spacecraft for the exclusive operation of large-scale global missions. e collection of seven spacecraft W � w 1 , w 2 , . . . , w 7 is considered after scrutinizing their versatile potentialities, where w 1 is the penetrator spacecraft: this high-quality spacecraft is a bullet-shaped projectile that remarkably penetrates the surface of a moon, planet, or any other such comet in space. e penetrator spacecraft monitors the inherent peculiarities of the penetrated surface to determine whether life exists on other cosmic planets of the solar system. w 2 is the orbiter spacecraft: this spacecraft is designed for traveling to a distant celestial body with considerable propulsive capacity in order to decelerate it at the right moment for proper insertion of spacecraft in their orbits. e orbiter spacecraft can significantly analyze the same planet from different heights without landing on its surface. w 3 is the flyby spacecraft: this ideal spacecraft has potential to spin continuously and utilize thrusters or reaction wheels to be stabilized in 3 axes. ese highspeed flyby spacecraft productively utilized their onboard optical instruments to critically observe the apparent motion of its target at the highest possible temporal resolution. w 4 is the lander spacecraft: the lander spacecraft reaches the surface of an astronomical body and remains long enough to measure the electrical, thermal, and mechanical properties of its inner surface. It has an incredible capability to perform geological and meteorological experiments and telemeter such precise information back to the Earth. w 5 is the atmospheric spacecraft: these competent spacecraft are especially designed for comparatively short-term exploration missions to take accurate measurements of temperature, clouds content, lightning, pressure, density, and atmospheric composition of a planet or satellite with the assistance of their advanced scientific instruments. w 6 is the rover spacecraft: this special type of spacecraft can move across planetary celestial bodies and other solid surfaces of cosmic planets.
is surface exploration device is mainly developed to gather detailed information about the terrain of astronomical objects and to take their different crust specimens such as rocks, soil, dust, and even liquids. w 7 is the observatory spacecraft: an observatory spacecraft is launched to provide amazing capabilities for the clearer observation of distant galaxies, planets, and other important terrestrial events. Such spacecraft competently perform their operations twenty-four hours a day at low temperature, and their observing tasks are free from blurring and obscuring impacts of the Earth's atmosphere.
All the aforementioned spacecraft are analyzed on the basis of four identified decisive factors M � m 1 , m 2 , m 3 , m 4 } that strongly influence the selection process of spacecraft will serve as decision-criteria for this inspected MCDM problem, where m 1 is the propulsion system: this significant feature is employed to accelerate the speed of spacecraft by providing required propelling force. While launching a spacecraft from the Earth, a competent propulsion system continually changes the velocity of the spacecraft and overcomes the high gravitational pull of Earth.
Ψ ηα e iΥ ηα ) and the Einstein addition of two V ηα is interpreted as follows: for each w η ∈ W, and choose the alternative having maximum accuracy degree. else Identify the alternatives w ξ j for which L ξ j � max η L η . (9) e alternative with maximum score or accuracy degree will be selected. (1) Input W � w 1 , w 2 , . . . , w ℘ as a universe of ℘ objects.
, for each w η ∈ W and choose the alternative having maximum accuracy degree. else Identify the alternatives w ξ j for which L ξ j � max η L η . (11) e alternative with maximum score or accuracy degree will be selected. e main purpose of this analysis is the determination of the best spacecraft for the remarkable strategic and economic advancement of Mexican space agency. e characteristic selection of ideal spacecraft with mind-blowing features will provide space-based solutions to the complicated Earth-bound problems by the accomplishment of ground-breaking scientific discoveries and plays a leading role in exploration of various indispensable cosmological phenomena which in turn dramatically optimizes the longterm profit of aerospace industries. e framework for the identification of the scrutinized MCDM problem is demonstrated in Figure 2.
Now, an identified 7-soft set can easily be obtained from Table 18, where (i) six-spade suit represents "Outstanding" (ii) five-spade suit represents "Superb" (iii) four-spade suit represents "Excellent" (iv) three-spade suit represents "Very Good" (v) two-spade suit represents "Good" (vi) one-spade suit represents "Average" (vii) big lozenge represents "Poor" e grades K � 0, 1, 2, 3, 4, 5, 6 { } can be definitely interrelated with the ranking-based evaluations carried out by the allocated spade suit as follows: (i) 0 serves as " " (ii) 1 serves as " " (iii) 2 serves as " " (iv) 3 serves as " " (v) 4 serves as " " (vi) 5 serves as " " (vii) 6 serves as " " Each space exploration technology is ranked on the basis of their overall competency which is further presented in Table 18. Now, the tabular representation of its corresponding specified 7-soft set is illustrated in Table 19.
In this respect, the grading criteria based on the score degrees of C 5 -ROF7SVs are defined as follows: where erefore, by Definition 8, C 5 -ROF7SS can be interpreted as follows: where

Complexity
Now, the C 5 -ROF7SS can be easily demonstrated in tabular form by Table 20.

Choice Values of C 5 -ROF7SS.
e quantified choice values of C 5 -ROF7SS for the determination of best spacecraft are summarized in Table 21.
e spacecraft are further categorized in an ascending order of their respective choice values. e hierarchical ranking of inspected space exploration technologies is highlighted in Table 22.
Consequently, we infer that the lander spacecraft (w 4 ) will be selected as the most competent space exploration technology for the strategic and economic development of Mexican space agency.  Table 24.

H-Choice
Hence, we conclude that the lander spacecraft (w 4 ) will be selected as the most ideal spacecraft for the exploration of various cosmological phenomena.

Comparative Analysis
In this section, a comparative study of our proposed decision-making techniques with contemporary MCDM technique based on the q-rung orthopair fuzzy soft weighted geometric (q-ROFS ft WG) operator is presented. We investigate the pragmatic application named "selection of the most proficient aerospace technology" by applying existing technique to authenticate the flexibility and sustainability of our established strategies.
6.1. q-ROFS ft WG Operator-Based MCDM Strategy. We now simplify the practical application 5.2 by implementing the systematic methodology of q-ROFS ft WG operator-based MCDM technique, proposed by Chinram et al. [22].
Step 1. e first step is to construct the 5-rung orthopair fuzzy soft decision matrix (5-ROFS ft DM) by scrutinizing the credibility of each aerospace technology relative to specific decision-criteria. e 5-rung orthopair fuzzy soft entries of 5-ROFS ft DM I � (F ηα ) 7×4 are obtained by omitting the grades and phase terms of all C 5 -ROF7SVs in Table 20. e calibrated 5-ROFS ft -entries F ηα � Θ, Ψ 〈 〉 of 5-ROFS ft DM regarding proficiency of spacecraft are summarized in Table 25.
Step 2. Normalize the evaluated 5-ROFS ft DM I � (F ηα ) 7×4 according to the nature and specification of each identified decision-criterion by utilizing the following expression: for cost type parameter, F ηα , for benefit type parameter, where F c ηα � Ψ, Θ 〈 〉 represents the complement of F ηα � Θ, Ψ 〈 〉. Here, the normalized 5-rung orthopair fuzzy soft decision matrix E � (E ηα ) 7×4 is the same as presented in Table 25 because all the specified decisive parameters of inspected problem are of the same benefit type.
Step 3. Aggregate the evaluated interpretations of each aerospace technology corresponding to identified decisioncriterion by employing the q-ROFS ft WG operator, which is defined as follows: where k ζ indicates the weights of decision-makers and z α denotes the relative weights of decisive parameters satisfying the conditions that k ζ , z α ∈ [0, 1] with τ ζ�1 k ζ and Z α�1 z α . Now, the appointed expert assigns weights z � (0.32, 0.20, 0.33, 0.15) T to the identified decision-criteria. e aggregated 5-ROFS ft Ns of each spacecraft corresponding to decisive criteria are presented as follows: Step 4. e next step is to evaluate the score degrees (L η ) of aggregated 5-ROFS ft Ns E η � Θ η , Ψ η of each aerospace technology which is interpreted as follows: where η � 1, 2, . . . , 7 and π η � represents the hesitancy degree of q-ROFS ft Ns. e quantified score degrees of E η for each spacecraft are clearly illustrated in Table 26.
Step 5. All the aerospace technologies of the inspected MCDM problem are hierarchically ranked in an ascending Complexity order corresponding to their calibrated score degrees and further demonstrated their systematic ranking in Table 27.
Hence, we conclude that lander spacecraft (w 4 ) will be selected as the most proficient aerospace technology having maximum score degree for the accomplishment of largescale global missions.

Comprehensive Discussion
(1) We exhibit a comparison of developed methodologies with contemporary MCDM techniques based on q-ROFS ft WG, C q -ROFWA, and C q -ROFWG operators to substantiate the accountability and feasibility of the proposed strategies. e methodological implications of both developed and compared strategies are persuasively spotlighted in Table 28. (2) All the proposed and compared strategies prioritize lander spacecraft (w 4 ) as the most productive aerospace technology for the prestigious economic advancement of Mexican space agency which demonstrate the compatibility and enforceability of remarkable decision-making skills of our established techniques in the sophisticated MCDM problems. (3) An explicative bar chart is illustrated in Figure 3 in order to visualize the comparative final outcomes of both proposed and compared MCDM techniques regarding hierarchical ordering of aerospace technologies, which in turn authenticate the rationality and plausibility of our developed methodologies. (4) e hierarchical ordering of inspected aerospace technologies differ slightly in compared and proposed strategies which is due to the appraisement of distinct fuzzy environments as proposed methodologies significantly overcome all the deficiencies of compared strategies by capturing the graded obscurity of two-dimensional parameterized information. In spite of all these, all the compared and proposed approaches interpret the similar end-results which illuminate the effectuality of the proposed strategies. (5) e q-ROFS ft WG operator is not capable enough to deal with C q -ROFNS information because of its limitations that arise due to the inadequacy of multinary assessment grades. However, our proposed systematic techniques have convincingly resolved these shortcomings of compared technique by capturing the graded evaluations of such paradoxical and uncertain data. is remarkable flexibility of our proposed strategies makes them more powerful as compared to the existing techniques. (6) Our developed strategies have an edge over the compared techniques that are directly based on C q -ROFWA and C q -ROFWG operators as it fantastically addresses the vagueness and periodicity of imprecise data at the same time, but the compared techniques are constrained to model the ambiguity of nonperiodic information only, which may lead to the inconsistent and illogical decisions. is incredible peculiarity of our proposed methodologies stimulates them as the most generalized and sustainable decision-making strategies for real-world MCDM complications.

Merits of the C q -ROFNSS Model
e next bullet points summarize the main advantages of the model proposed in this paper: (1) We have proposed a C q -ROFNSS model that provides a multiskilled and most proficient mathematical framework for the fuzzy representation of parameterized vague data. e underlying features of both NSSs and C q -ROFSs are inherited by this extension. In fact, we believe that the theory arising from this model has significantly resolved all the flaws and limitations of existing models for the representation of ambiguous knowledge. (2) A paramount advantage of the proposed model is its powerful ability to model two-dimensional imprecise information. It can address the periodicity and vagueness of parameterized data at the same time in a convincing manner. However, not only is the idea of the proposed model restricted to the two-dimensional case, but also it can handle one-dimensional uncertain data with similar accuracy and precision. (3) In the modern digital era, grading systems became massively popular as they are standard tools for collecting users' inputs about their opinions on electronic services, web pages, online products, marketing sites, movies, and countless other online applications. Our theory is especially crafted with an aim to capture the graded assessments of a variety of informational sources in order to meet future challenges.  Complexity (4) e remarkable decision-making skills of C q -ROFNSS excellently increase the modeler's ability to express their two-dimensional ambiguous assessments. e outstanding generic structure of our proposed model broadens the confined space of traditional fuzzy models by relaxing their constraint conditions. e proposed hybrid model also renders the most generalized tool for ranking-based competent modeling of parameterized inexact data of periodic and nonperiodic nature as well.

Conclusion
In this research article, we have accomplished our goal by establishing the foundations of a multiskilled hybrid model called C q -ROFNSS. It can capture a wide range of imprecision embedded in human cognition. is highly competent and most flexible theory sets up a mathematical framework for the representation of two-dimensional inexact information. It has extended several contemporary models by the integration of the expertise of C q -ROFSs and the remarkable parametric nature of NSSs.
is theory means a powerful generalization of both CIFNSSs and CPFNSSs. It broadens their limited boundary space by relaxing their constraint conditions in terms of an adjustable parameter q. To this purpose, we have elaborated the formal definition of C q -ROFNSS and its basic fundamental set-theoretic operations. We have proposed the Einstein and some other algebraic operations of C q -ROFNSVs. We have demonstrated the rationality of the proposed model with a concise analysis of its relationships with existing theories.
Another remarkable contribution of this study is the development of two systematic MCDM algorithms for the identification of a most favorable alternative on the basis of their intrinsic characteristics. We have tested their feasibility with a real-life application for the evaluation of the most productive aerospace technology. We have conducted a comparative analysis with contemporary MCDM techniques to justify that our strategies are reasonable extensions. We have accompanied the comparative study with an interpretative bar chart in order to illustrate the compatibility and veracity of the final outcomes. Finally, we have investigated the dynamic features of the proposed model in order to throw light on its merits and preeminence over existing decision-making theories.
Although we can argue that the developed model has an edge over the contemporary approaches, this theory does not lack for limitations. Its structure cannot capture the abstinence and refusal aspect of inexact human expressions. Also, the strategies that we used to solve MCDM problems may be computationally costly, as they involve rather cumbersome and tedious calculations. us, in the future, we are committed to reduce the massive calculations of these MCDM techniques with the assistance of some computer programming and to develop graphical representations of the proposed model in order to elaborate this proficient concept more effectively. Additionally, we are planning to extend our research work by establishing more advanced MCGDM strategies, including the C q -ROFNS-PROMETHEE method, C q -ROFNS-VIKOR method, C q -ROFNS-AHP method, C q -ROFNS-TOPSIS method, and C q -ROFNS-ELECTRE method. We intend to explore the scope of the potential applications under the versatile environment of the C q -ROFNS model.

Data Availability
No data were used to support this study.

Ethical Approval
is article does not contain any studies with human participants or animals performed by any of the authors.

Complexity 27
Conflicts of Interest e authors declare that they have no conflicts of interest.