# Complexq-Rung Orthopair Fuzzy N-Soft Sets: A New Model with Applications

Authors: Muhammad Akram; Faiza Wasim; Ahmad N. Al-Kenani
Journal: Complexity
 (2021)

---

## Abstract

This research article expands the formal representation of human thinking to a most generalized hybrid theory, namely, complexq-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. This 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. The 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. The 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. Their 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.


---

## Body

## 1. 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 real-world 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. This 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. The limitations that this imprecision impose on the traditional approaches has lead the researchers to establish novel theories with more flexible structures. Their 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 Θ2+Ψ2≤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.These 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 ΘeiΩ and nonmembership Ψeiϒ 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 (Cq-ROFSs) which significantly relaxed the constraints in terms of a parameter q. The Cq-ROFS provides an effective tool for capturing both vagueness and periodicity within the semantics of human assessments. The remarkable potentialities of Cq-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 Cq-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 (CIVq-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 Cq-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]. The 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 Cq-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. This procedure was applied to the evaluation of the quality of an express service.A routine inspection of the research work on soft-set-inspired 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 interval0,1. In spite of this fact, numerous decision-making settings are equipped with multinary, discrete-type data. Therefore, 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. Then, 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. The 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 (Lq-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 [36–55]The following arguments and evidences condense the motivation and significance of the theory that shall be studied in this work:(i)
TheCq-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 Cq-ROFS model since its inception.(ii)
The 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 proposedCq-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 theq-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, complexq-rung orthopair fuzzy N-soft set (Cq-ROFNSS). To do so, it merges the advantageous features of Cq-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. The Cq-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. The remarkable flexibility of the proposed model is briefly illustrated with a description of its relationship with existing competing approaches. Then, the Einstein operators and some other elementary algebraic operations of Cq-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)
This research article systematically expands the literature by introducing a multiskilled and most generalized hybrid modelCq-ROFNSS. It is designed for the correct modelization of ranking-based fuzzy modeling of two-dimensional parameterized inconsistent data.(ii)
The 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 onq-ROFSftWG, Cq-ROFWA and Cq-ROFWG operators. It proves both the feasibility of the strategies developed in this paper and the compatibility of their final outcomes.(iv)
The merits and potentiality of the MCDM methodologies here formulated shed light on their flexibility, competency, and prominence over the contemporary decision-making techniques.The remainder of this article is organized as follows: Section2 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 Cq-ROFNSS model. Section 4 demonstrates the Einstein and other algebraic operational laws on Cq-ROFNSVs. Section 5 introduces two decision-making 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 comparative study with existing MCDM techniques based on q-ROFSftWG, Cq-ROFWA, and Cq-ROFWG operators. Section 7 expounds the merits and superiority of the developed Cq-ROFNSS theory over other contemporary decision-making models. Finally, Section 8 elaborates the concluding remarks of the article and raises some future research questions. The entire contribution of this research article is graphically interpreted in Figure 1.Figure 1
Graphical demonstration of proposed work.


## 2. 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]).
LetW be a universe of discourse. An intuitionistic fuzzy set T over the universe W is an object having the following form:(1)T=w,ΘTw,ΨTw|w∈W,where real-valued functions ΘT:W⟶0,1 and ΨT:W⟶0,1 specify the membership and nonmembership degrees of an element w∈W, satisfying the constraint condition 0≤ΘTw+ΨTw≤1. For all w∈W, ATw=1−ΘTw−ΨTw represents the degree of indeterminacy. The pair of membership and nonmembership degrees ΘTw,ΨTw can be referred as an intuitionistic fuzzy number (IFN).Definition 2 (see [4]).
LetW be a universe of discourse. A q-rung orthopair fuzzy set T over the universe W can be characterized as(2)T=w,ΘTw,ΨTw|w∈W,where real-valued functions ΘT:W⟶0,1 and ΨT:W⟶0,1 indicate membership and nonmembership degrees of an element w∈W, subjected to the condition 0≤ΘTqw+ΨTqw≤1. For all w∈W, ATw=1−ΘTqw−ΨTqwq represents the degree of indeterminacy. The pair of membership and nonmembership degrees ΘTw,ΨTw is called a q-rung orthopair fuzzy number (q-ROFN).Definition 3 (see [12]).
LetW be a universe of discourse. A complex q-rung orthopair fuzzy set T over the universe W can be elaborated as follows:(3)T=w,ΘTweiΩTw,ΨTweiϒTw|w∈W,where i=−1, and the amplitude terms ΘTw,ΨTw∈0,1 and phase terms ΩTw,ϒTw∈0,2π subjected to the conditions 0≤ΘTqw+ΨTqw≤1 and 0≤ΩTw/2πq+ϒTw/2πq≤1. For all w∈W, ATw=1−ΘTqw−ΨTqwqei2π1−ΩTw/2πq−ϒTw/2πqq represents the degree of indeterminacy. The pair of membership and nonmembership degrees ΘTweiΩTw,ΨTweiϒTw can be referred as a complex q-rung orthopair fuzzy number (Cq-ROFN).Definition 4 (see [21]).
LetW 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−ROFW, where q−ROFW is the family of all q-ROFSs over the universe W. A q-ROFSS T over the universe W can be represented as follows:(4)T=mα,ℑmα|mα∈M,ℑmα∈q−ROFW,where ℑmα=wη,Θηαwη,Ψηαwη|wη∈W represents the q-ROFS over the universe W. The membership Θηαwη and nonmembership Ψηαwη degrees belong to the unit interval 0,1 satisfying the condition 0≤Θηαqwη+Ψηαqwη≤1. The Aηαw=1−Θηαqw−Ψηαqwq represents the degree of indeterminacy, for all w∈W. The pair of membership and nonmembership degrees Θηαw,Ψηαw is called a q-rung orthopair fuzzy soft number (q-ROFSN).Definition 5 (see [21]).
The score function for anyq-ROFSN, Vηα=Θηα,Ψηα can be characterized as follows:(5)LVηα=Θηq−Ψηq+eΘηq−ΨηqeΘηq−Ψηq+1−12πηq,forq≥1,where L represents the score function of q-ROFSN and LVηα lies inside the closed interval −1,1.Definition 6 (see [21]).
LetV1α=Θ1α,Ψ1αα=1,2 and V=Θ,Ψ be any three q-ROFSNs and ℌ>0 be any real number. Then, the operations based on these three q-ROFSNs can be explicated as follows:(1)
V11⊕V12=Θ11q+Θ12q−Θ11qΘ12q,qΨ11Ψ12(2)
V11⊗V12=Θ11Θ12,Ψ11q+Ψ12q−Ψ11qΨ12qq(3)
ℌV=1−1−Θqℌ,qΨℌ(4)
Vℌ=Θℌ,1−1−ΨqℌqDefinition 7 (see [28]).
LetW 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, where N∈2,3,…. A triplet T=℘,M,N is said to be a N-soft set over the universe W if ℘:M⟶2W×K having property that for each m∈M, there exist an unique dyad w,km∈W×K such that w,km∈℘m, w∈W, and km∈K. The N-soft set T over the universe W can be explicated as follows:(6)T=mα,℘mα|mα∈M,℘mα∈2W×K.


## 3. Complexq-Rung Orthopair Fuzzy N-Soft Sets

Definition 8.
LetW 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,…. A triplet T˜=pq,Y,N is said to be a complex q-rung orthopair fuzzy N-soft set on W, when Y=P,M,N is NSS on the universe W and pq:M⟶Cq−ROℱW×K, where Cq−ROℱW×K is the collection of all complex q-rung orthopair fuzzy sets over W×K. The Cq-ROFNSS pq,Y,N can be characterized in the following decorum:(7)T˜=mα,pqmα|mα∈M,pqmα∈Cq−ROℱW×K,where pqmα=wη,kηα,Θηαwη,kηαeiΩηαwη,kηα,Ψηαwη,kηαeiϒηαwη,kηα|wη,kηα∈W×K represents the Cq-ROFS over W×K. The amplitude terms Θηαwη,kηα, Ψηαwη,kηα belong to unit interval 0,1 satisfying the condition(8)0≤Θηαqwη,kηα+Ψηαqwη,kηα≤1,and phase terms Ωηαwη,kηα, ϒηαwη,kηα belong to closed interval 0,2π subjected to the condition(9)0≤Ωηαwη,kηα2πq+ϒηαwη,kηα2πq≤1,where i=−1. For all wη,kηα∈W×K, the degree of indeterminacy can be explicated as follows:(10)Aηαwη,kηα=1−Θηαqwη,kηα−Ψηαqwη,kηα1/qei2π1−Ωηαwη,kηα/2πq−ϒηαwη,kηα/2πq1/q.
In other words,Cq-ROFNSS is a ranking-based parameterized family of complex q-rung orthopair fuzzy sets of W×K; that is, the mapping pq assigns each parameter mα∈M to a complex q-rung orthopair fuzzy set pqmα of W×K.
Now, we present tabular representation of general complexq-rung orthopair fuzzy N-soft set:
LetW be a collection of ℘ objects and M be a set of ℏ attributes, then the tabular characterization of Cq-ROFNSS is illustrated in Table 1.Table 1
Tabular representation ofCq-ROFNSS.
pq,Y,Nm1m2⋯mℏw1k11,Θ11eiΩ11,Ψ11eiϒ11k12,Θ12eiΩ12,Ψ12eiϒ12⋯k1ℏ,Θ1ℏeiΩ1ℏ,Ψ1ℏeiϒ1ℏw2k21,Θ21eiΩ21,Ψ21eiϒ21k22,Θ22eiΩ22,Ψ22eiϒ22⋯k2ℏ,Θ2ℏeiΩ2ℏ,Ψ2ℏeiϒ2ℏ⋮⋮⋮⋮⋮w℘k℘1,Θ℘1eiΩ℘1,Ψ℘1eiϒ℘1k℘2,Θ℘2eiΩ℘2,Ψ℘2eiϒ℘2⋯k℘ℏ,Θ℘ℏeiΩ℘ℏ,Ψ℘ℏeiϒ℘ℏDefinition 9.
TheCq-ROFNSS can be interpreted as ℘×ℏ table, where ℘=W, ℏ=M whose ηαth element is called a complex q-rung orthopair fuzzy N-soft value (Cq-ROFNSV), and it has the representative form, which is interpreted as follows:(11)Vηα=kηα,ΘηαeiΩηα,Ψηαeiϒηα,where η=1,2,…,℘; α=1,2,…,ℏ.Definition 10.
The score functionL for any Cq-ROFNSV, Vηα=kηα,ΘηαeiΩηα,Ψηαeiϒηα can be demarcated as follows:(12)LVηα=Θηαq−Ψηαq+Ωηα2πq−ϒηα2πq+kηαN−1q,where LVηα∈−2,3.Definition 11.
The accuracy functionQ for any Cq-ROFNSV, Vηα=kηα,ΘηαeiΩηα,Ψηαeiϒηα can be explicated as follows:(13)QVηα=Θηαq+Ψηαq+Ωηα2πq+ϒηα2πq+kηαN−1q,where QVηα∈0,5.Definition 12.
LetV11=k11,Θ11eiΩ11,Ψ11eiϒ11 and V12=k12,Θ12eiΩ12,Ψ12eiϒ12 be any two Cq-ROFNSVs, then the comparison between V11 and V12 can be determined as follows:(1)
IfLV11<LV12, then V11≺V12 (V11 is inferior to V12)(2)
IfLV11>LV12, then V11≻V12 (V11 is superior to V12)(3)
IfLV11=LV12, then(i)
ifQV11<QV12, then V11≺V12 (V11 is inferior to V12)(ii)
ifQV11>QV12, then V11≻V12 (V11 is superior to V12)(iii)
ifQV11=QV12, then V11∼V12 (V11 is equivalent to V12)
The motivation and prominence of this newly established model is clearly demonstrated by the following real-life practical application. In order to develop intuition for this advanced theory, this application has been kept comparatively simple. This empirical application interprets the deficiencies of the existingq-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. These 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. These ratings are based on the proficiency of spillways in the last 5 years. LetW=w1= Chute spillway, w2= Shaft spillway, w3= Ogee spillway, w4= Siphon spillway, and w5= Labyrinth spillway} be a collection of five spillways and M=m1= Topography, m2= Construction cost, m3= Discharge rate, and m4= 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 Table2, where(i)
Four stars represent “Excellent”(ii)
Three stars represent “Superb”(iii)
Two stars represent “Good”(iv)
One star represents “Average”(v)
Big inverted delta represents “Poor”
The gradesK=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 “”
The appointed experts of the panel rank each spillway on the basis of their overall competency which is further presented in Table2.
Now, the tabular representation of its corresponding identified 5-soft set is provided in Table3.
The selection panel comprehensively evaluate all the identified spillways to determine their ratings by virtue of both membershipΘηα and nonmembership degrees Ψηα. 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 allwη∈W and q=3, 0≤Θηα3+Ψηα3≤1 with(14)0.0≤Θηαwη<0.2whenkηα=0,0.2≤Θηαwη<0.4whenkηα=1,0.4≤Θηαwη<0.6whenkηα=2,0.6≤Θηαwη<0.8whenkηα=3,0.8≤Θηαwη≤1.0whenkηα=4.
Then,3-ROF5SS can be explicated as follows:(15)p3,Y,5=m1,p3m1,m2,p3m2,m3,p3m3,m4,p3m4,where(16)p3m1=w1,4,0.96,0.01,w2,2,0.68,0.74,w3,0,0.03,0.97,w4,3,0.86,0.19,w5,2,0.16,0.72,p3m2=w1,3,0.45,0.14,w2,1,0.45,0.57,w3,3,0.68,0.11,w4,4,0.97,0.03,w5,0,0.04,0.97,p3m3=w1,0,0.01,0.99,w2,4,1.00,0.02,w3,2,0.75,0.81,w4,1,0.43,0.55,w5,3,0.84,0.16,p3m4=w1,2,0.56,0.63,w2,3,0.76,0.26,w3,0,0.05,0.98,w4,2,0.57,0.48,w5,4,0.95,0.35.
The tabular representation of3-ROF5SS is demonstrated in Table 4.
The experts of the selection panel analyze all the inspected spillways and assign membership and nonmembership degreesΘηαwη,kηα, Ψηαwη,kηα, respectively, to each spillway. Now, suppose an expert interprets that “The discharge rate of a spillway w1 is very low in the first 3 years but dramatically increases in the last 2 years because of rapid climate change and global warming.” Then, the values of Θ11w1,4=0.96 and Ψ11w1,4=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 complex-valued membership and nonmembership Ψ11w1,4 degrees for the instant elaboration of all the information scrutinized by the expert. Hence, we establish complex 3-rung orthopair fuzzy 5-soft set (C3-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 Θ11w1,4 and Ψ11w1,4 in the light of redefined grading criteria:(17)Θ11w1,4=0.96ei1.98π,Ψ11w1,4=0.01ei0.02π.
Here, phase term illustrates all the time-dependent information regarding discharge rate of spillway under consideration. Therefore,C3-ROF5SS p3,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 C3-ROF5SVs LVηα, where Vηα=kηα,ΘηαeiΩηα,Ψηαeiϒηα, which can be defined as follows:(18)−2.0≤LVηα<−1.0whenkηα=0,−1.0≤LVηα<0.0whenkηα=1,0.0≤LVηα<1.0whenkηα=2,1.0≤LVηα<2.0whenkηα=3,2.0≤LVηα≤3.0whenkηα=4.
Finally, an advancedC3-ROF5SS can be interpreted as follows:(19)p3,Y,5=m1,p3m1,m2,p3m2,m3,p3m3,m4,p3m4,where(20)p3m1=w1,4,0.96ei1.98π,0.01ei0.02π,w2,2,0.68ei1.56π,0.74ei1.47π,w3,0,0.03ei0.02π,0.97ei1.94π,w4,3,0.86ei1.75π,0.19ei0.07π,w5,2,0.16ei1.89π,0.72ei1.66π,p3m2=w1,3,0.45ei1.84π,0.14ei1.17π,w2,1,0.45ei0.34π,0.57ei0.45π,w3,3,0.68ei1.84π,0.11ei0.12π,w4,4,0.97ei1.96π,0.03ei0.04π,w5,0,0.04ei0.02π,0.97ei1.99π,p3m3=w1,0,0.01ei0.04π,0.99ei1.95π,w2,4,1.00ei1.94π,0.02ei0.01π,w3,2,0.75ei1.45π,0.81ei1.37π,w4,1,0.43ei0.42π,0.55ei0.56π,w5,3,0.84ei1.72π,0.16ei0.08π,p3m4=w1,2,0.56ei1.85π,0.63ei1.76π,w2,3,0.76ei1.79π,0.26ei0.04π,w3,0,0.05ei0.03π,0.98ei1.96π,w4,2,0.57ei1.67π,0.48ei1.60π,w5,4,0.95ei1.98π,0.35ei0.08π.
Evidently,C3-ROF5SS can be easily demonstrated in tabular form by Table 5.Table 2
Ranking of spillways.
W/Mm1m2m3m4w1w2w3w4w5Table 3
Associated 5-soft set.
P,M,5m1m2m3m4w14302w22143w30320w43412w52034Table 4
Tabular representation of 3-ROF5SS.
p3,Y,5m1m2m3m4w14,0.96,0.013,0.45,0.140,0.01,0.992,0.56,0.63w22,0.68,0.741,0.45,0.574,1.00,0.023,0.76,0.26w30,0.03,0.973,0.68,0.112,0.75,0.810,0.05,0.98w43,0.86,0.194,0.97,0.031,0.43,0.552,0.57,0.48w52,0.16,0.720,0.04,0.973,0.84,0.164,0.95,0.35Table 5
Tabular representation ofC3-ROF5SS.
p3,Y,5m1m2m3m4w14,0.96ei1.98π,0.01ei0.02π3,0.45ei1.84π,0.14ei1.17π0,0.01ei0.04π,0.99ei1.95π2,0.56ei1.85π,0.63ei1.76πw22,0.68ei1.56π,0.74ei1.47π1,0.45ei0.34π,0.57ei0.45π4,1.00ei1.94π,0.02ei0.01π3,0.76ei1.79π,0.26ei0.04πw30,0.03ei0.02π,0.97ei1.94π3,0.68ei1.84π,0.11ei0.12π2,0.75ei1.45π,0.81ei1.37π0,0.05ei0.03π,0.98ei1.96πw43,0.86ei1.75π,0.19ei0.07π4,0.97ei1.96π,0.03ei0.04π1,0.43ei0.42π,0.55ei0.56π2,0.57ei1.67π,0.48ei1.60πw52,0.16ei1.89π,0.72ei1.66π0,0.04ei0.02π,0.97ei1.99π3,0.84ei1.72π,0.16ei0.08π4,0.95ei1.98π,0.35ei0.08πRemark 1.
(1)
AnyCq-ROF2SS pq,Y,2 can be definitely associated with complex q-rung orthopair fuzzy soft set (Cq-ROFSS). We can generally identify the Cq-ROF2SS pq:M⟶Cq−ROℱW×0,1 through Cq-ROFSS p˜:M⟶Cq−ROFW, which is characterized as follows:(21)p˜mα=wη,ΘηαwηeiΩηαwη,Ψηαwηeiϒηαwη|wη,1,Θηαwη,1eiΩηαwη,1,Ψηαwη,1eiϒηαwη,1∈pqmα,for everymα∈M.(2)
ACq-ROFNSS pq,Y,N over the universe W is said to be efficient if Y=P,M,N is NSS on W and wη,N−1,Θwη,N−1eiΩwη,N−1,Ψwη,N−1eiϒwη,N−1∈pqmα, for some mα∈M, wη∈W.(3)
Grade0∈K in Definition 8 indicates the lowest score. 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 complexq-rung orthopair fuzzy N-soft sets.Definition 13.
Letpq,Y,N be a Cq-ROFNSS over the universe W, where Y=P,M,N be its corresponding NSS on W. Then, the complex q-rung orthopair fuzzy complement pqc,Y,N of pq,Y,N can be explicated as(22)pqc,Y,N=mα,pqcmα|mα∈M,pqcmα∈Cq−ROℱW×K,where(23)pqcmα=wη,kηα,Ψwη,kηαeiϒwη,kηα,Θwη,kηαeiΩwη,kηα|wη,kηα∈W×K.Definition 14.
Letpq,Y,N be a Cq-ROFNSS over the universe W, where Y=P,M,N be its respective NSS on W. Then, the weak complex q-rung orthopair fuzzy complement pqc,Yc,N of pq,Y,N is defined as(24)pqc,Yc,N=mα,pqcmα|mα∈M,pqcmα∈Cq−ROℱW×K,where pqcmα=wη,kηα,Ψwη,kηαeiϒwη,kηα,Θwη,kηαeiΩwη,kηα|wη,kηα∈W×K and for all mα∈M, Pcmα∩Pmα=∅.Example 2.
Consider theC3-ROF5SS as illustrated in Example 1. Then, its complex 3-rung orthopair fuzzy complement p3c,Y,5 and weak complex 3-rung orthopair fuzzy complement p3c,Yc,5 are provided in Tables 6 and 7, respectively.Table 6
Complex 3-rung orthopair fuzzy complement.
p3c,Y,5m1m2m3m4w14,0.01ei0.02π,0.96ei1.98π3,0.14ei1.17π,0.45ei1.84π0,0.99ei1.95π,0.01ei0.04π2,0.63ei1.76π,0.56ei1.85πw22,0.74ei1.47π,0.68ei1.56π1,0.57ei0.45π,0.45ei0.34π4,0.02ei0.01π,1.00ei1.94π3,0.26ei0.04π,0.76ei1.79πw30,0.97ei1.94π,0.03ei0.02π3,0.11ei0.12π,0.68ei1.84π2,0.81ei1.37π,0.75ei1.45π0,0.98ei1.96π,0.05ei0.03πw43,0.19ei0.07π,0.86ei1.75π4,0.03ei0.04π,0.97ei1.96π1,0.55ei0.56π,0.43ei0.42π2,0.48ei1.60π,0.57ei1.67πw52,0.72ei1.66π,0.16ei1.89π0,0.97ei1.99π,0.04ei0.02π3,0.16ei0.08π,0.84ei1.72π4,0.35ei0.08π,0.95ei1.98πTable 7
Weak complex 3-rung orthopair fuzzy complement.
p3c,Yc,5m1m2m3m4w11,0.01ei0.02π,0.96ei1.98π2,0.14ei1.17π,0.45ei1.84π1,0.99ei1.95π,0.01ei0.04π3,0.63ei1.76π,0.56ei1.85πw23,0.74ei1.47π,0.68ei1.56π2,0.57ei0.45π,0.45ei0.34π3,0.02ei0.01π,1.00ei1.94π4,0.26ei0.04π,0.76ei1.79πw32,0.97ei1.94π,0.03ei0.02π4,0.11ei0.12π,0.68ei1.84π0,0.81ei1.37π,0.75ei1.45π1,0.98ei1.96π,0.05ei0.03πw44,0.19ei0.07π,0.86ei1.75π3,0.03ei0.04π,0.97ei1.96π2,0.55ei0.56π,0.43ei0.42π0,0.48ei1.60π,0.57ei1.67πw53,0.72ei1.66π,0.16ei1.89π1,0.97ei1.99π,0.04ei0.02π4,0.16ei0.08π,0.84ei1.72π3,0.35ei0.08π,0.95ei1.98πDefinition 15.
Letpq,Y,N be a Cq-ROFNSS over the universe W, where Y=P,M,N be the corresponding NSS on W. Then, its top complex q-rung orthopair fuzzy weak complement, represented by pqt,Yt,N, can be demarcated as(25)pqt,Yt,N=pqtmα=wη,N−1,Ψwη,kηαeiϒwη,kηα,Θwη,kηαeiΩwη,kηα,ifkηα<N−1,pqtmα=wη,0,Ψwη,kηαeiϒwη,kηα,Θwη,kηαeiΩwη,kηα,ifkηα=N−1.Definition 16.
Letpq,Y,N be a Cq-ROFNSS over the universe W, where Y=P,M,Nis the respective NSS on W. Then, its bottom complex q-rung orthopair fuzzy weak complement, represented by pqb,Yb,N, can be interpreted as(26)pqb,Yb,N=pqbmα=wη,0,Ψwη,kηαeiϒwη,kηα,Θwη,kηαeiΩwη,kηα,ifkηα>0,pqbmα=wη,N−1,Ψwη,kηαeiϒwη,kηα,Θwη,kηαeiΩwη,kηα,ifkηα=0.Example 3.
Consider theC3-ROF5SS as provided in Example 1. Then, its top complex 3-rung orthopair fuzzy weak complement p3t,Yt,5 and bottom complex 3-rung orthopair fuzzy weak complement p3b,Yb,5 are summarized in Tables 8 and 9, respectively.
We now interpret two significant practicable approaches for the union and intersection of novel complexq-rung orthopair fuzzy N-soft sets.Table 8
Top complex 3-rung orthopair fuzzy weak complement.
p3t,Yt,5m1m2m3m4w10,0.01ei0.02π,0.96ei1.98π4,0.14ei1.17π,0.45ei1.84π4,0.99ei1.95π,0.01ei0.04π4,0.63ei1.76π,0.56ei1.85πw24,0.74ei1.47π,0.68ei1.56π4,0.57ei0.45π,0.45ei0.34π0,0.02ei0.01π,1.00ei1.94π4,0.26ei0.04π,0.76ei1.79πw34,0.97ei1.94π,0.03ei0.02π4,0.11ei0.12π,0.68ei1.84π4,0.81ei1.37π,0.75ei1.45π4,0.98ei1.96π,0.05ei0.03πw44,0.19ei0.07π,0.86ei1.75π0,0.03ei0.04π,0.97ei1.96π4,0.55ei0.56π,0.43ei0.42π4,0.48ei1.60π,0.57ei1.67πw54,0.72ei1.66π,0.16ei1.89π4,0.97ei1.99π,0.04ei0.02π4,0.16ei0.08π,0.84ei1.72π0,0.35ei0.08π,0.95ei1.98πTable 9
Bottom complex 3-rung orthopair fuzzy weak complement.
p3b,Yb,5m1m2m3m4w10,0.01ei0.02π,0.96ei1.98π0,0.14ei1.17π,0.45ei1.84π4,0.99ei1.95π,0.01ei0.04π0,0.63ei1.76π,0.56ei1.85πw20,0.74ei1.47π,0.68ei1.56π0,0.57ei0.45π,0.45ei0.34π0,0.02ei0.01π,1.00ei1.94π0,0.26ei0.04π,0.76ei1.79πw34,0.97ei1.94π,0.03ei0.02π0,0.11ei0.12π,0.68ei1.84π0,0.81ei1.37π,0.75ei1.45π4,0.98ei1.96π,0.05ei0.03πw40,0.19ei0.07π,0.86ei1.75π0,0.03ei0.04π,0.97ei1.96π0,0.55ei0.56π,0.43ei0.42π0,0.48ei1.60π,0.57ei1.67πw50,0.72ei1.66π,0.16ei1.89π4,0.97ei1.99π,0.04ei0.02π0,0.16ei0.08π,0.84ei1.72π0,0.35ei0.08π,0.95ei1.98πDefinition 17.
Letpqa,Ya,Na and pqb,Yb,Nb be two Cq-ROFNSS over the universe of discourse W, where Ya=Pa,F,Na and Yb=Pb,G,Nb be corresponding NSSs on W. Then, their restricted intersection symbolized by pqa,Ya,Na∩ℛpqb,Yb,Nb=zq,Ya∩RYb,minNa,Nb, where Ya∩RYb=Z,F∩G,minNa,Nb can be characterized as follows:
∀mα∈F∩G,wη∈W, wη,kηα,X,Y∈zqmα⇔kηα=minkηαa,kηαb, X=minΘSwη,kηαa,ΘTwη,kηαbeiminΩSwη,kηαa,ΩTwη,kηαb, Y=maxΨSwη,kηαa,ΨTwη,kηαbeimaxϒSwη,kηαa,ϒTwη,kηαb, if wη,kηαa,ΘSwη,kηαaeiΩSwη,kηαa,ΨSwη,kηαaeiϒSwη,kηαa∈pqamα and wη,kηαb,ΘTwη,kηαbeiΩTwη,kηαb,ΨTwη,kηαbeiϒTwη,kηαb∈pqbmα, and S and T are Cq-ROFSs on pqamα and pqbmα, respectively.Definition 18.
Letpqa,Ya,Na and pqb,Yb,Nb be two Cq-ROFNSSs over the universe of discourse W, where Ya=Pa,F,Na and Yb=Pb,G,Nb are respective NSSs on W. Then, their extended intersection represented by pqa,Ya,Na∩Epqb,Yb,Nb=dq,Ya∩EYb,maxNa,Nb, where Ya∩EYb=D,F∪G,maxNa,Nb can be delineated as(27)dqmα=pqamα,ifmα∈F−G,pqbmα,ifmα∈G−F,wη,kηα,X,Y,such thatkηα=minkηαa,kηαb,X=minΘSwη,kηαa,ΘTwη,kηαbeiminΩSwη,kηαa,ΩTwη,kηαb,Y=maxΨSwη,kηαa,ΨTwη,kηαbeimaxϒSwη,kηαa,ϒTwη,kηαb,wherewη,kηαa,ΘSwη,kηαaeiΩSwη,kηαa,ΨSwη,kηαaeiϒSwη,kηαa∈pqamα,andwη,kηαb,ΘTwη,kηαbeiΩTwη,kηαb,ΨTwη,kηαbeiϒTwη,kηαb∈pqbmα,SandTareCq−ROFSs onpqamαandpqbmα,respectively.Example 4.
Consider the tabular form ofC2-ROF6SS p2a,Ya,6 and C2-ROF4SS p2b,Yb,4 as presented in Tables 10 and 11, respectively, where Ya=Pa,F,6 and Yb=Pb,G,4 are 6-soft set and 4-soft set over the universe W, respectively. Then, their restricted intersection z2,Ya∩RYb,4 and extended intersection d2,Ya∩EYb,6 are encapsulated in Tables 12 and 13, respectively.Table 10
C2-ROF6SS.
p2a,Ya,6m1m2m3w14,0.96ei1.84π,0.15ei0.12π3,0.74ei1.45π,0.36ei0.69π2,0.53ei1.32π,0.46ei0.83πw25,0.98ei1.97π,0.01ei0.02π1,0.15ei0.28π,0.92ei1.78π0,0.01ei0.03π,0.97ei1.95πw33,0.82ei1.44π,0.38ei0.62π0,0.02ei0.01π,0.99ei1.96π3,0.79ei1.46π,0.42ei0.57πw42,0.58ei1.25π,0.43ei0.76π2,0.65ei1.33π,0.48ei0.86π5,0.95ei1.99π,0.02ei0.03πTable 11
C2-ROF4SS.
p2b,Yb,4m1m2m4w12,0.56ei1.99π,0.12ei0.26π1,0.12ei0.22π,0.88ei0.97π3,0.83ei1.69π,0.24ei0.14πw23,0.91ei1.85π,0.32ei0.06π2,0.68ei1.95π,0.05ei0.24π0,0.04ei0.01π,0.97ei1.98πw30,0.11ei0.04π,0.96ei1.99π2,0.93ei0.97π,0.52ei0.04π2,0.79ei1.78π,0.13ei0.23πw41,0.23ei0.15π,0.96ei1.86π3,0.88ei1.93π,0.16ei0.12π1,0.25ei0.13π,0.91ei1.89πTable 12
Restricted intersection.
z2,Ya∩RYb,4m1m2w12,0.56ei1.84π,0.15ei0.26π1,0.12ei0.22π,0.88ei0.97πw23,0.91ei1.85π,0.32ei0.06π1,0.15ei0.28π,0.92ei1.78πw30,0.11ei0.04π,0.96ei1.99π0,0.02ei0.01π,0.99ei1.96πw41,0.23ei0.15π,0.96ei1.86π2,0.65ei1.33π,0.48ei0.86πTable 13
Extended intersection.
d2,Ya∩EYb,6m1m2m3m4w12,0.56ei1.84π,0.15ei0.26π1,0.12ei0.22π,0.88ei0.97π2,0.53ei1.32π,0.46ei0.83π3,0.83ei1.69π,0.24ei0.14πw23,0.91ei1.85π,0.32ei0.06π1,0.15ei0.28π,0.92ei1.78π0,0.01ei0.03π,0.97ei1.95π0,0.04ei0.01π,0.97ei1.98πw30,0.11ei0.04π,0.96ei1.99π0,0.02ei0.01π,0.99ei1.96π3,0.79ei1.46π,0.42ei0.57π2,0.79ei1.78π,0.13ei0.23πw41,0.23ei0.15π,0.96ei1.86π2,0.65ei1.33π,0.48ei0.86π5,0.95ei1.99π,0.02ei0.03π1,0.25ei0.13π,0.91ei1.89πDefinition 19.
Letpqa,Ya,Na and pqb,Yb,Nb be two Cq-ROFNSSs over the universe of discourse W, where Ya=Pa,F,Na and Yb=Pb,G,Nb be corresponding NSSs on W. Then, their restricted union symbolized by pqa,Ya,Na∪ℛpqb,Yb,Nb=zq,Ya∪RYb,maxNa,Nb, where Ya∪RYb=Z,F∩G,maxNa,Nb can be explicated as follows:
∀mα∈F∩G,wη∈W, wη,kηα,X,Y∈zqmα⇔kηα=maxkηαa,kηαb, X=maxΘSwη,kηαa,ΘTwη,kηαbeimaxΩSwη,kηαa,ΩTwη,kηαb, Y=minΨSwη,kηαa,ΨTwη,kηαbeiminϒSwη,kηαa,ϒTwη,kηαb, if wη,kηαa,ΘSwη,kηαaeiΩSwη,kηαa,ΨSwη,kηαaeiϒSwη,kηαa∈pqamα and wη,kηαb,ΘTwη,kηαbeiΩTwη,kηαb,ΨTwη,kηαbeiϒTwη,kηαb∈pqbmα, and S and T are Cq-ROFSs on pqamα and pqbmα, respectively.Definition 20.
Letpqa,Ya,Na and pqb,Yb,Nb be two Cq-ROFNSSs over the universe of discourse W, where Ya=Pa,F,Na and Yb=Pb,G,Nb are respective NSSs on W. Then, their extended union represented by pqa,Ya,Na∪Epqb,Yb,Nb=dq,Ya∪EYb,maxNa,Nb, where Ya∪EYb=D,F∪G,maxNa,Nb is interpreted as follows:(28)dqmα=pqamα,ifmα∈F−G,pqbmα,ifmα∈G−F,wη,kηα,X,Y,such thatkηα=maxkηαa,kηαb,X=maxΘSwη,kηαa,ΘTwη,kηαbeimaxΩSwη,kηαa,ΩTwη,kηαb,Y=minΨSwη,kηαa,ΨTwη,kηαbeiminϒSwη,kηαa,ϒTwη,kηαb,wherewη,kηαa,ΘSwη,kηαaeiΩSwη,kηαa,ΨSwη,kηαaeiϒSwη,kηαa∈pqamαandwη,kηαb,ΘTwη,kηαbeiΩTwη,kηαb,ΨTwη,kηαbeiϒTwη,kηαb∈pqbmα,SandTareCq−ROFSs onpqamαandpqbmα,respectively.Example 5.
Consider theC2-ROF6SS and C2-ROF4SS as defined in Example 4. Then, their restricted union z2,Ya∪RYb,6 and extended union d2,Ya∪EYb,6 are illustrated in Tables 14 and 15, respectively.Table 14
Restricted union.
z2,Ya∪RYb,6m1m2w14,0.96ei1.99π,0.12ei0.12π3,0.74ei1.45π,0.36ei0.69πw25,0.98ei1.97π,0.01ei0.02π2,0.68ei1.95π,0.05ei0.24πw33,0.82ei1.44π,0.38ei0.62π2,0.93ei0.97π,0.52ei0.04πw42,0.58ei1.25π,0.43ei0.76π3,0.88ei1.93π,0.16ei0.12πTable 15
Extended union.
d2,Ya∪EYb,6m1m2m3m4w14,0.96ei1.99π,0.12ei0.12π3,0.74ei1.45π,0.36ei0.69π2,0.53ei1.32π,0.46ei0.83π3,0.83ei1.69π,0.24ei0.14πw25,0.98ei1.97π,0.01ei0.02π2,0.68ei1.95π,0.05ei0.24π0,0.01ei0.03π,0.97ei1.95π0,0.04ei0.01π,0.97ei1.98πw33,0.82ei1.44π,0.38ei0.62π2,0.93ei0.97π,0.52ei0.04π3,0.79ei1.46π,0.42ei0.57π2,0.79ei1.78π,0.13ei0.23πw42,0.58ei1.25π,0.43ei0.76π3,0.88ei1.93π,0.16ei0.12π5,0.95ei1.99π,0.02ei0.03π1,0.25ei0.13π,0.91ei1.89πRemark 2.
The Definitions17–20 of extended (or restricted) union and intersection will suffice the defined grading criteria if and only if Na=Nb.
The notion ofCq-ROFNSS can be easily identified with contemporary theories including complex q-rung orthopair fuzzy soft set (Cq-ROFSS), q-rung orthopair fuzzy N-soft set, N-soft set, and soft set. Now, we are going to derive Cq-ROFSS and SS from the proposed Cq-ROFNSS by employing the following definitions:Definition 21.
Letpq,Y,N be a Cq-ROFNSS over a universe of discourse W, where Y=P,M,N be its respective NSS on W. Let 0<H<N be a threshold. Then, Cq-ROFSS over the universe W associated with pq,Y,N and H is pqH,M, which can be interpreted as follows:(29)pqHmα=ΘηαeiΩηα,Ψηαeiϒηα,ifkηα,ΘηαeiΩηα,Ψηαeiϒηα∈pqmαandkηα≥H,0.00ei0.00π,0.50ei1.00π,ifkηαN≥0.50,0.00ei0.00π,1.00ei2.00π,ifkηαN<0.50.
Specifically,pq1,M is said to be the bottom Cq-ROFSS and pqN−1,M is said to be the top Cq-ROFSS relative to Cq-ROFNSS.Definition 22.
Let0<H<N and σ∈−2,3 be thresholds. The soft set over the universe W corresponding to pq,Y,N and H,σ is represented by pqH,σ,M and can be explicated by the assignment:(30)pqH,σmα=w∈W:LpqHmα>σ,for eachmα∈M,where LpqHmα is the score function of pqHmα=ΘηαHeiΩηαH,ΨηαHeiϒηαH.Example 6.
Consider theC3-ROF5SS p3,Y,5 as represented by Table 5. We have 0<H<5, from Definition 21. The C3-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 pq3,1.5,M which is illustrated in Table 17.
In view of above analysis, it is contemplated thatCq-ROFNSSs can be transformed into Cq-ROFSSs and SSs under specific conditions. In other words, Cq-ROFNSSs are the potent generalization of Cq-ROFSSs and SSs.Table 16
IdentifiedC3-ROFSS.
p33,Mm1m2m3m4w10.96ei1.98π,0.01ei0.02π0.45ei1.84π,0.14ei1.17π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,0.50ei1.00πw20.00ei0.00π,0.50ei1.00π0.00ei0.00π,1.00ei2.00π1.00ei1.94π,0.02ei0.01π0.76ei1.79π,0.26ei0.04πw30.00ei0.00π,1.00ei2.00π0.68ei1.84π,0.11ei0.12π0.00ei0.00π,0.50ei1.00π0.00ei0.00π,1.00ei2.00πw40.86ei1.75π,0.19ei0.07π0.97ei1.96π,0.03ei0.04π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,0.50ei1.00πw50.00ei0.00π,0.50ei1.00π0.00ei0.00π,1.00ei2.00π0.84ei1.72π,0.16ei0.08π0.95ei1.98π,0.35ei0.08πTable 17
Identified SS.
p33,1.5,Mm1m2m3m4w11100w20011w30100w40100w50001Remark 3.
(1)
Letpq,Y,N be a Cq-ROFNSS over the universe W and J be a set of parameters with M⊆J. Then, the NSS interrelated with Cq-ROFNSS pq,Y,N is Y, where Y=P,M,N. This simple assignment illustrates that Cq-ROFNSS generalizes NSS as well as SS too.(2)
Every complex intuitionistic fuzzyN-soft set (CIFNSS) and complex Pythagorean fuzzy N-soft set (CPFNSS) are also a Cq-ROFNSS for all q≥3, but converse is not true, because(i)
If0≤Θ+Ψ≤1 or 0≤Θ2+Ψ2≤1, then 0≤Θq+Ψq≤1, for all Θ,Ψ∈0,1 and q≥3(ii)
If0≤Ω/2π+ϒ/2π≤1 or 0≤Ω/2π2+ϒ/2π2≤1, then 0≤Ω/2πq+ϒ/2πq≤1, for all Ω,ϒ∈0,2π and q≥3This implies thatCq-ROFNSS significantly extrapolates the ideal concepts of CIFNSS and CPFNSS.(3)
Cq-ROFNSS particularly becomes a q-rung orthopair fuzzy N-soft set by taking phase terms of both membership and nonmembership degrees equivalent to zero; that is, when Ωηα=0=ϒηα in Definition 8, we grasp the definition of q-ROFNSS which is elucidated as follows:(31)pq,Y,N=mα,pqmα|mα∈M,pqmα∈q−ROℱW×K,wherepqmα=wη,kηα,Θηαwη,kηα,Ψηαwη,kηα|wη,kηα∈W×K represents the q-ROFS and q-ROℱW×K denotes the family of all q-rung orthopair fuzzy sets over W×K.
Also, it is understood thatq-ROFNSSs generalize the PFNSSs, IFNSSs, FNSSs, NSSs, PFSSs, IFSSs, and SSs as well. Therefore, we can also claim that Cq-ROFNSSs admirably generalizes all these extant models.


## 4. Operations

We now elaborate some rudimentary operational laws on complexq-rung orthopair fuzzy N-soft values.Definition 23.
Consider any threeCq-ROFNSVs, V=k,ΘeiΩ,Ψeiϒ, V1α=k1α,Θ1αeiΩ1α,Ψ1αeiϒ1αα=1,2 and ℌ>0 be any real number. Then, the elementary algebraic operations relative to these Cq-ROFNSVs can be defined as follows:(1)
V11⊕V12=maxk11,k12,Θ11q+Θ12q−Θ11qΘ12qqei2πΩ11/2πq+Ω12/2πq−Ω11/2πqΩ12/2πqq,Ψ11Ψ12ei2πϒ11/2πϒ12/2π(2)
V11⊗V12=mink11,k12,Θ11Θ12ei2πΩ11/2πΩ12/2π,Ψ11q+Ψ12q−Ψ11qΨ12qqei2πϒ11/2πq+ϒ12/2πq−ϒ11/2πqϒ12/2πqq(3)
ℌV=k,1−1−Θqℌqei2π1−1−Ω/2πqℌq,Ψℌei2πϒ/2πℌ(4)
Vℌ=k,Θℌei2πΩ/2πℌ,1−1−Ψqℌqei2π1−1−ϒ/2πqℌqDefinition 24.
LetV1α=k1α,Θ1αeiΩ1α,Ψ1αeiϒ1αα=1,2 and V=k,ΘeiΩ,Ψeiϒ be any three Cq-ROFNSVs and ℌ>0 be any real number. Then, the Einstein operations corresponding to these Cq-ROFNSVs can be defined as follows:(1)
V11⊕eV12=maxk11,k12,Θ11q+Θ12q/1+Θ11qΘ12qqei2πΩ11/2πq+Ω12/2πq/1+Ω11/2πqΩ12/2πqq,Ψ11Ψ12/1+1−Ψ11q1−Ψ12qqei2πϒ11/2πϒ12/2π/1+1−ϒ11/2πq1−ϒ12/2πqq(2)
V11⊗eV12=mink11,k12,Θ11Θ12/1+1−Θ11q1−Θ12qqei2πΩ11/2πΩ12/2π/1+1−Ω11/2πq1−Ω12/2πqq,Ψ11q+Ψ12q/1+Ψ11qΨ12qqei2πϒ11/2πq+ϒ12/2πq/1+ϒ11/2πqϒ12/2πqq(3)
ℌ⋅eV=k,1+Θqℌ−1−Θqℌ/1+Θqℌ+1−Θqℌqei2π1+Ω/2πqℌ−1−Ω/2πqℌ/1+Ω/2πqℌ+1−Ω/2πqℌq,2qΨℌ/2−Ψqℌ+Ψqℌqei2πq2ϒ/2πℌ/q2−ϒ/2πqℌ+ϒ/2πqℌ(4)
Vℌ=k,2qΘℌ/2−Θqℌ+Θqℌqei2π2qΩ/2πℌ/2−Ω/2πqℌ+Ω/2πqℌq,1+Ψqℌ−1−Ψqℌ/1+Ψqℌ+1−Ψqℌqei2π1+ϒ/2πqℌ−1−ϒ/2πqℌ/1+ϒ/2πqℌ+1−ϒ/2πqℌq
These operations ofCq-ROFNSVs are highly advantageous and expedient in decision-making as compared to other algebraic operations of Cq-ROFNSVs because they are formulated on the basis of incredible Einstein t-norm and t-conorm to provide end-results that demonstrate remarkable accuracy and precision.Theorem 1.
LetV=k,ΘeiΩ,Ψeiϒ, V11=k11,Θ11eiΩ11,Ψ11eiϒ11, and V12=k12,Θ12eiΩ12,Ψ12eiϒ12 be any three Cq-ROFNSVs and ℌ,ℌ1,ℌ2>0 be any three real numbers, then(1)
V11⊕eV12=V12⊕eV11(2)
V11⊗eV12=V12⊗eV11(3)
ℌ.eV11⊕eV12=ℌ.eV11⊕eℌ.eV12(4)
V11⊗eV12ℌ=V11ℌ⊗eV12ℌ(5)
ℌ1.e⊕eℌ2.eV=ℌ1.eV⊕eℌ2.eV(6)
Vℌ1⊗eVℌ2=Vℌ1+ℌ2Proof.
(1)
V11⊕eV12=maxk11,k12,Θ11q+Θ12q/1+Θ11qΘ12qqei2πΩ11/2πq+Ω12/2πq/1+Ω11/2πqΩ12/2πqq,Ψ11Ψ12/1+1−Ψ11q1−Ψ12qqei2πϒ11/2πϒ12/2π/1+1−ϒ11/2πq1−ϒ12/2πqq=maxk12,k11,Θ12q+Θ11q/1+Θ12qΘ11qqei2πΩ12/2πq+Ω11/2πq/1+Ω12/2πqΩ11/2πqq,Ψ12Ψ11/1+1−Ψ12q1−Ψ11qqei2πϒ12/2πϒ11/2π/1+1−ϒ12/2πq1−ϒ11/2πqq=V12⊕eV11.Analogously, we can prove (2) in the similar fashion.(2)
First, consider the L.H.S,V11⊕eV12=maxk11,k12,Θ11q+Θ12q/1+Θ11qΘ12qqei2πΩ11/2πq+Ω12/2πq/1+Ω11/2πqΩ12/2πqq,Ψ11Ψ12/1+1−Ψ11q1−Ψ12qqei2πϒ11/2πϒ12/2π/1+1−ϒ11/2πq1−ϒ12/2πqq.Above equation can be transformed into the following mathematicalexpression:(32)V11⊕eV12=maxk11,k12,1+Θ11q1+Θ12q−1−Θ11q1−Θ12q1+Θ11q1+Θ12q+1−Θ11q1−Θ12qqei2π1+Ω11/2πq1+Ω12/2πq−1−Ω11/2πq1−Ω12/2πq/1+Ω11/2πq1+Ω12/2πq+1−Ω11/2πq1−Ω12/2πqq,2qΨ11Ψ122−Ψ11q2−Ψ12q+Ψ11qΨ12qqei2π2qϒ11/2πϒ12/2π/2−ϒ11/2πq2−ϒ12/2πq+ϒ11/2πqϒ12/2πqq.
Let us consider(33)d=1+Θ11q1+Θ12q,f=1−Θ11q1−Θ12q,g=1+Ω112πq1+Ω122πq,h=1−Ω112πq1−Ω122πq,m=Ψ11qΨ12q,p=2−Ψ11q2−Ψ12q,s=ϒ112πqϒ122πq,r=2−ϒ112πq2−ϒ122πq.
Then,(34)V11⊕eV12=maxk11,k12,d−fd+fqei2πg−h/g+hq,2mp+mqei2π2s/s+rq,ℌ.eV11⊕eV12=maxk11,k12,1+d−f/d+fℌ−1−d−f/d+fℌ1+d−f/d+fℌ+1−d−f/d+fℌqei2π1+g−h/g+hℌ−1−g−h/g+hℌ/1+g−h/g+hℌ+1−g−h/g+hℌq,2q2m/p+mqℌ2−2m/p+mℌ+2m/p+mℌqei2π2q2s/r+sqℌ/2−2s/r+sℌ+2s/r+sℌq=maxk11,k12,dℌ−fℌdℌ+fℌqei2πgℌ−hℌ/gℌ+hℌq,2mℌpℌ+mℌqei2π2sℌ/rℌ+sℌq=maxk11,k12,1+Θ11qℌ1+Θ12qℌ−1−Θ11qℌ1−Θ12qℌ1+Θ11qℌ1+Θ12qℌ+1−Θ11qℌ1−Θ12qℌqei2π1+Ω11/2πqℌ1+Ω12/2πqℌ−1−Ω11/2πqℌ1−Ω12/2πqℌ/1+Ω11/2πqℌ1+Ω12/2πqℌ+1−Ω11/2πqℌ1−Ω12/2πqℌq,2qΨ11ℌΨ12ℌ2−Ψ11qℌ2−Ψ12qℌ+Ψ11qℌΨ12qℌqei2π2qϒ11/2πℌϒ12/2πℌ/2−ϒ11/2πqℌ2−ϒ12/2πqℌ+ϒ11/2πqℌϒ12/2πqℌq.
Now, assume the R.H.S,(35)ℌ.eV11=k11,1+Θ11qℌ−1−Θ11qℌ1+Θ11qℌ+1−Θ11qℌqei2π1+Ω11/2πqℌ−1−Ω11/2πqℌ/1+Ω11/2πqℌ+1−Ω11/2πqℌq,2qΨ11ℌ2−Ψ11qℌ+Ψ11qℌqei2π2qϒ11/2πℌ/2−ϒ11/2πqℌ+ϒ11/2πqℌq,ℌ.eV12=k12,1+Θ12qℌ−1−Θ12qℌ1+Θ12qℌ+1−Θ12qℌqei2π1+Ω12/2πqℌ−1−Ω12/2πqℌ/1+Ω12/2πqℌ+1−Ω12/2πqℌq,2qΨ12ℌ2−Ψ12qℌ+Ψ12qℌqei2π2qϒ12/2πℌ/2−ϒ12/2πqℌ+ϒ12/2πqℌq.
Let us consider(36)d1=1+Θ11qℌ,f1=1−Θ11qℌ,g1=1+Ω112πqℌ,h1=1−Ω112πqℌ,s1=Ψ11qℌ,r1=2−Ψ11qℌ,m1=ϒ112πqℌ,p1=2−ϒ112πqℌ,d2=1+Θ12qℌ,f2=1−Θ12qℌ,g2=1+Ω122πqℌ,h2=1−Ω122πqℌ,s2=Ψ12qℌ,r2=2−Ψ12qℌ,m2=ϒ122πqℌ,p2=2−ϒ122πqℌ.
Then, the above equations can be modified as follows:(37)ℌ.eV11=k11,d1−f1d1+f1qei2πg1−h1/g1+h1q,2s1r1+s1qei2π2m1/p1+m1q,ℌ.eV12=k12,d2−f2d2+f2qei2πg2−h2/g2+h2q,2s2r2+s2qei2π2m2/p2+m2q.
Now,(38)ℌ.eV11⊕eℌ.eV12=maxk11,k12,d1−f1/d1+f1+d2−f2/d2+f21+d1−f1/d1+f1d2−f2/d2+f2qei2πg1−h1/g1+h1+g2−h2/g2+h2/1+g1−h1/g1+h1g2−h2/g2+h2q,4s1s2/r1+s1r2+s21+1−2s1/r1+s11−2s2/r2+s2qei2π4m1m2/p1+m1p2+m2/1+1−2m1/p1+m11−2m2/p2+m2q=maxk11,k12,d1d2−f1f2d1d2+f1f2qei2πg1g2−h1h2/g1g2+h1h2q,2s1s2r1r2+s1s2qei2π2m1m2/p1p2+m1m2q=maxk11,k12,1+Θ11qℌ1+Θ12qℌ−1−Θ11qℌ1−Θ12qℌ1+Θ11qℌ1+Θ12qℌ+1−Θ11qℌ1−Θ12qℌqei2π1+Ω11/2πqℌ1+Ω12/2πqℌ−1−Ω11/2πqℌ1−Ω12/2πqℌ/1+Ω11/2πqℌ1+Ω12/2πqℌ+1−Ω11/2πqℌ1−Ω12/2πqℌq,2qΨ11ℌΨ12ℌ2−Ψ11qℌ2−Ψ12qℌ+Ψ11qℌΨ12qℌqei2π2qϒ11/2πℌϒ12/2πℌ/2−ϒ11/2πqℌ2−ϒ12/2πqℌ+ϒ11/2πqℌϒ12/2πqℌq.
Consequently, verify thatℌ.eV11⊕eV12=ℌ.eV11⊕eℌ.eV12.
Similarly, we can prove (4) in the analogous manner.
(5) Assume thatℌ1, ℌ2>0:(39)ℌ1.e⊕eℌ2.eV==k,1+Θqℌ1+ℌ2−1−Θqℌ1+ℌ21+Θqℌ1+ℌ2+1−Θqℌ1+ℌ2qei2π1+Ω/2πqℌ1+ℌ2−1−Ω/2πqℌ1+ℌ2/1+Ω/2πqℌ1+ℌ2+1−Ω/2πqℌ1+ℌ2q,2qΨℌ1+ℌ22−Ψqℌ1+ℌ2+Ψqℌ1+ℌ2qei2π2qϒ/2πℌ1+ℌ2/2−ϒ/2πqℌ1+ℌ2+ϒ/2πqℌ1+ℌ2q.
Now, consideringR.H.S, we have(40)ℌ1.eV=k,1+Θqℌ1−1−Θqℌ11+Θqℌ1+1−Θqℌ1qei2π1+Ω/2πqℌ1−1−Ω/2πqℌ1/1+Ω/2πqℌ1+1−Ω/2πqℌ1q,2qΨℌ12−Ψqℌ1+Ψqℌ1qei2π2qϒ/2πℌ1/2−ϒ/2πqℌ1+ϒ/2πqℌ1q,ℌ2.eV=k,1+Θqℌ2−1−Θqℌ21+Θqℌ2+1−Θqℌ2qei2π1+Ω/2πqℌ2−1−Ω/2πqℌ2/1+Ω/2πqℌ2+1−Ω/2πqℌ2q,2qΨℌ22−Ψqℌ2+Ψqℌ2qei2π2qϒ/2πℌ2/2−ϒ/2πqℌ2+ϒ/2πqℌ2q.
Let us consider, fork=1,2,(41)dk=1+Θqℌk,fk=1−Θqℌk,gk=1+Ω2πqℌk,hk=1−Ω2πqℌk,sk=Ψqℌk,rk=2−Ψqℌk,mk=ϒ2πqℌk,pk=2−ϒ2πqℌk,ℌ1.eV⊕eℌ2.eV=maxk11,k12,d1−f1/d1+f1+d2−f2/d2+f21+d1−f1/d1+f1d2−f2/d2+f2qei2πg1−h1/g1+h1+g2−h2/g2+h2/1+g1−h1/g1+h1g2−h2/g2+h2q,4s1s2/r1+s1r2+s21+1−2s1/r1+s11−2s2/r2+s2qei2π4m1m2/p1+m1p2+m2/1+1−2m1/p1+m11−2m2/p2+m2q=maxk11,k12,d1d2−f1f2d1d2+f1f2qei2πg1g2−h1h2/g1g2+h1h2q,2s1s2r1r2+s1s2qei2π2m1m2/p1p2+m1m2q=k,1+Θqℌ1+ℌ2−1−Θqℌ1+ℌ21+Θqℌ1+ℌ2+1−Θqℌ1+ℌ2qei2π1+Ω/2πqℌ1+ℌ2−1−Ω/2πqℌ1+ℌ2/1+Ω/2πqℌ1+ℌ2+1−Ω/2πqℌ1+ℌ2q,2qΨℌ1+ℌ22−Ψqℌ1+ℌ2+Ψqℌ1+ℌ2qei2π2qϒ/2πℌ1+ℌ2/2−ϒ/2πqℌ1+ℌ2+ϒ/2πqℌ1+ℌ2q.
Hence, verify thatℌ1.e⊕eℌ2.eV=ℌ1.eV⊕eℌ2.eV.
Analogously, we can prove (6) in the similar decorum.


## 5. Decision-Making Algorithms

In this section, we develop two eloquent MCDM algorithms that implemented on the proposed multiskilledCq-ROFNSS model. These strategies are especially designed for the identification of the optimal alternative of real-life heuristic problems that are characterized by Cq-ROFNSSs. The underlying methodology of these established decision-making algorithms can be interpreted as follows (Algorithms 1 and 2).Algorithm 1: The algorithm of choice values of Cq-ROFNSS.
(1)
InputW=w1,w2,…,w℘ as a universe of ℘ objects.(2)
InputM=m1,m2,…,mℏ as a collection of ℏ attributes.(3)
Input theN-soft set P,M,N with K=0,1,…,N−1, where N∈2,3,…. Then, for each wη∈W, mα∈M, there exist unique kηα∈K.(4)
InputCq-ROFNSS pq,Y,N, where pq:M⟶Cq−ℛOℱW×K and Y=P,M,N.(5)
CalculateUη=∑α=1ℏVηα, where Vηα=kηα,ΘηαeiΩηα,Ψηαeiϒηα and the Einstein addition of two Vηα is interpreted as follows:V11⊕eV12=maxk11,k12,Θ11q+Θ12q/1+Θ11qΘ12qqei2πΩ11/2πq+Ω12/2πq/1+Ω11/2πqΩ12/2π/qq,Ψ11Ψ12/1+1−Ψ11q1−Ψ12qqei2πϒ11/2πϒ12/2π/1+1−ϒ11/2πq1−ϒ12/2πqq.(6)
Evaluate its choice valueLηUη=Θηq−Ψηq+Ωη/2πq−ϒη/2πq+kη/N−1q, for each wη∈W and η=1,2,…,℘.(7)
Compute all the indicesξj for which Lξj=maxηLη.(8)
ifLξ1=Lξ2 for some ξ1,ξ2∈1,2,…,℘, thenEvaluate its accuracy degreeQηUη=Θηq+Ψηq+Ωη/2πq+ϒη/2πq+kη/N−1q, for each wη∈W, and choose the alternative having maximum accuracy degree.elseIdentify the alternativeswξj for which Lξj=maxηLη.(9)
The alternative with maximum score or accuracy degree will be selected.Algorithm 2: The algorithm of H-choice values of Cq-ROFNSS.
(1)
InputW=w1,w2,…,w℘ as a universe of ℘ objects.(2)
InputM=m1,m2,…,mℏ as a collection of ℏ attributes.(3)
Input theN-soft set P,M,N with K=0,1,…,N−1, where N∈2,3,… Then, for each wη∈W, mα∈M, there exist unique kηα∈K.(4)
InputCq-ROFNSS pq,Y,N, where pq:M⟶Cq−ℛOℱW×K and Y=P,M,N.(5)
InputH threshold.(6)
ComputepqHmαwη=ΘηαeiΩηα,Ψηαeiϒηα,ifkηα,ΘηαeiΩηα,Ψηαeiϒηα∈pqmαandkηα≥H0.00ei0.00π,0.50ei1.00πifkηα/N≥0.500.00ei0.00π,1.00ei2.00πifkηα/N<0.50(7)
CalculateUηH=∑α=1ℏVηαH, where VηαH=ΘηαHeiΩηαH,ΨηαHeiϒηαH and Einstein addition of two VηαH is presented as follows:V11H⊕eV12H=Θ11Hq+Θ12Hq/1+Θ11HqΘ12Hqqei2πΩ11H/2πq+Ω12H/2πq/1+Ω11H/2πqΩ12H/2πqq,Ψ11HΨ12H/1+1−Ψ11Hq1−Ψ12Hqqei2πϒ11H/2πϒ12H/2π/1+1−ϒ11H/2πq1−ϒ12H/2πqq.(8)
Evaluate itsH-choice value LηHUηH=ΘηHq−ΨηHq+ΩηH/2πq−ϒηH/2πq, for each wη, and η=1,2,…,℘.(9)
Compute all the indicesξj for which LξjH=maxηLηH.(10)
ifLξ1H=Lξ2H for some ξ1,ξ2∈1,2,…,℘, thenEvaluate its accuracy degreeQηHUηH=ΘηHq+ΨηHq+ΩηH/2πq+ϒηH/2πq, for each wη∈W and choose the alternative having maximum accuracy degree.elseIdentify the alternativeswξj for which Lξj=maxηLη.(11)
The alternative with maximum score or accuracy degree will be selected.


### 5.1. 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.


### 5.2. 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. The 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. The collection of seven spacecraftW=w1,w2,…,w7 is considered after scrutinizing their versatile potentialities, wherew1 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. The penetrator spacecraft monitors the inherent peculiarities of the penetrated surface to determine whether life exists on other cosmic planets of the solar system.w2 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. The orbiter spacecraft can significantly analyze the same planet from different heights without landing on its surface.w3 is the flyby spacecraft: this ideal spacecraft has potential to spin continuously and utilize thrusters or reaction wheels to be stabilized in 3 axes. These high-speed flyby spacecraft productively utilized their on-board optical instruments to critically observe the apparent motion of its target at the highest possible temporal resolution.w4 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.w5 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.w6 is the rover spacecraft: this special type of spacecraft can move across planetary celestial bodies and other solid surfaces of cosmic planets. This 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.w7 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 factorsℳ=m1,m2,m3,m4 that strongly influence the selection process of spacecraft will serve as decision-criteria for this inspected MCDM problem, wherem1 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.m2 is the communication and navigation: all spacecraft require a good communication and navigation system to avoid constraints on future space missions. These potent capabilities of spacecraft provide a secure environment for constructive global telecommunication and ensure the accurate tracking of their trajectories.m3 is the payload capacity: the payload capacity is the most important consideration in designing the infrastructure of spacecraft. This measurable parameter usually quantifies the carrying capacity of spacecraft which may include cargo, flight crew, munitions, and other scientific equipment by analyzing the nature of their flight missions.m4 is the cosmic radiation protection: a broad spectrum of ionizing radiation can penetrate spacesuits, solar equipment, and spacecraft and can badly harm the biological health of astronauts. The cosmic radiation protection plays an indispensable role in the proper execution of large-scale space missions. The adequate shielding of spacecraft can minimize the risk of the exposure of such harmful radiation.The main purpose of this analysis is the determination of the best spacecraft for the remarkable strategic and economic advancement of Mexican space agency. The 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 long-term profit of aerospace industries. The framework for the identification of the scrutinized MCDM problem is demonstrated in Figure2.Figure 2
Specification of investigated problem.Now, an identified 7-soft set can easily be obtained from Table18, 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”Table 18
Rating of spacecraft.
P,M,7m1m2m3m4w1w2w3w4w5w6w7The gradesK=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 Table18.Now, the tabular representation of its corresponding specified 7-soft set is illustrated in Table19.Table 19
Identified 7-soft set.
P,M,7m1m2m3m4w14321w21216w33502w46465w52042w63150w75223In this respect, the grading criteria based on the score degrees ofC5-ROF7SVs are defined as follows:(42)−2.00≤LVηα<−1.29whenkηα=0,−1.29≤LVηα<−0.58whenkηα=1,−0.58≤LVηα<0.13whenkηα=2,0.13≤LVηα<0.84whenkηα=3,0.84≤LVηα<1.55whenkηα=4,1.55≤LVηα<2.26whenkηα=5,2.26≤LVηα≤3.00whenkηα=6,where LVηα=Θηαq−Ψηαq+Ωηα/2πq−ϒηα/2πq+kηα/N−1q and Vηα=kηα,ΘηαeiΩηα,Ψηαeiϒηα; η=1,2,3,4,5,6,7; α=1,2,3,4.Therefore, by Definition8, C5-ROF7SS can be interpreted as follows:(43)p5,Y,7=m1,p5m1,m2,p5m2,m3,p5m3,m4,p5m4,where(44)p5m1=w1,4,0.78ei1.89π,0.44ei1.55π,w2,1,0.14ei0.12π,0.77ei1.82π,w3,3,0.45ei1.68π,0.29ei1.57π,w4,6,0.95ei1.95π,0.13ei0.09π,w5,2,0.54ei1.96π,0.99ei1.24π,w6,3,0.34ei1.64π,0.22ei1.53π,w7,5,0.89ei1.86π,0.21ei0.02π,p5m2=w1,3,0.49ei1.77π,0.35ei1.62π,w2,2,0.98ei1.08π,0.62ei1.97π,w3,5,0.81ei1.93π,0.20ei1.03π,w4,4,0.75ei1.78π,0.16ei1.18π,w5,0,0.07ei0.05π,0.89ei1.92π,w6,1,0.12ei0.09π,0.74ei1.75π,w7,2,0.68ei1.34π,0.82ei1.63π,p5m3=w1,2,0.78ei1.55π,0.89ei1.67π,w2,1,0.18ei0.11π,0.82ei1.76π,w3,0,0.04ei0.06π,0.96ei1.89π,w4,6,0.93ei1.97π,0.05ei0.04π,w5,4,0.76ei1.86π,0.11ei1.12π,w6,5,0.85ei1.89π,0.01ei0.05π,w7,2,0.53ei1.45π,0.64ei1.56π,p5m4=w1,1,0.17ei0.13π,0.84ei1.85π,w2,6,0.94ei1.96π,0.03ei0.12π,w3,2,0.61ei1.13π,0.85ei1.54π,w4,5,0.87ei1.87π,0.23ei0.04π,w5,2,0.55ei1.23π,0.67ei1.57π,w6,0,0.01ei0.03π,0.99ei1.96π,w7,3,0.39ei1.81π,0.27ei1.72π.Now, theC5-ROF7SS can be easily demonstrated in tabular form by Table 20.Table 20
Tabular representation ofC5-ROF7SS.
p5,Y,7m1m2m3m4w14,0.78ei1.89π,0.44ei1.55π3,0.49ei1.77π,0.35ei1.62π2,0.78ei1.55π,0.89ei1.67π1,0.17ei0.13π,0.84ei1.85πw21,0.14ei0.12π,0.77ei1.82π2,0.98ei1.08π,0.62ei1.97π1,0.18ei0.11π,0.82ei1.76π6,0.94ei1.96π,0.03ei0.12πw33,0.45ei1.68π,0.29ei1.57π5,0.81ei1.93π,0.20ei1.03π0,0.04ei0.06π,0.96ei1.89π2,0.61ei1.13π,0.85ei1.54πw46,0.95ei1.95π,0.13ei0.09π4,0.75ei1.78π,0.16ei1.18π6,0.93ei1.97π,0.05ei0.04π5,0.87ei1.87π,0.23ei0.04πw52,0.54ei1.96π,0.99ei1.24π0,0.07ei0.05π,0.89ei1.92π4,0.76ei1.86π,0.11ei1.12π2,0.55ei1.23π,0.67ei1.57πw63,0.34ei1.64π,0.22ei1.53π1,0.12ei0.09π,0.74ei1.75π5,0.85ei1.89π,0.01ei0.05π0,0.01ei0.03π,0.99ei1.96πw75,0.89ei1.86π,0.21ei0.02π2,0.68ei1.34π,0.82ei1.63π2,0.53ei1.45π,0.64ei1.56π3,0.39ei1.81π,0.27ei1.72π


### 5.3. Choice Values ofC5-ROF7SS

The quantified choice values ofC5-ROF7SS for the determination of best spacecraft are summarized in Table 21.Table 21
Choice values ofC5-ROF7SS.
p5,Y,7m1m2m3m4UηLηw14,0.78ei1.89π,0.44ei1.55π3,0.49ei1.77π,0.35ei1.62π2,0.78ei1.55π,0.89ei1.67π1,0.17ei0.13π,0.84ei1.85π4,0.88ei1.98π,0.08ei0.77π1.6300w21,0.14ei0.12π,0.77ei1.82π2,0.98ei1.08π,0.62ei1.97π1,0.18ei0.11π,0.82ei1.76π6,0.94ei1.96π,0.03ei0.12π6,0.99ei1.96π,0.01ei0.08π2.8966w33,0.45ei1.68π,0.29ei1.57π5,0.81ei1.93π,0.20ei1.03π0,0.04ei0.06π,0.96ei1.89π2,0.61ei1.13π,0.85ei1.54π5,0.84ei1.97π,0.03ei0.45π1.7741w46,0.95ei1.95π,0.13ei0.09π4,0.75ei1.78π,0.16ei1.18π6,0.93ei1.97π,0.05ei0.04π5,0.87ei1.87π,0.23ei0.04π6,0.99ei1.99π,0.00ei0.00π2.9262w52,0.54ei1.96π,0.99ei1.24π0,0.07ei0.05π,0.89ei1.92π4,0.76ei1.86π,0.11ei1.12π2,0.55ei1.23π,0.67ei1.57π4,0.81ei1.99π,0.05ei0.40π1.4575w63,0.34ei1.64π,0.22ei1.53π1,0.12ei0.09π,0.74ei1.75π5,0.85ei1.89π,0.01ei0.05π0,0.01ei0.03π,0.99ei1.96π5,0.85ei1.94π,0.001ei0.02π1.7280w75,0.89ei1.86π,0.21ei0.02π2,0.68ei1.34π,0.82ei1.63π2,0.53ei1.45π,0.64ei1.56π3,0.39ei1.81π,0.27ei1.72π5,0.92ei1.98π,0.02ei0.01π2.0375The spacecraft are further categorized in an ascending order of their respective choice values. The hierarchical ranking of inspected space exploration technologies is highlighted in Table22.Table 22
Hierarchical ranking of spacecraft.
Spacecraftw1w2w3w4w5w6w7Ranking6241753Consequently, we infer that the lander spacecraftw4 will be selected as the most competent space exploration technology for the strategic and economic development of Mexican space agency.


### 5.4.H-Choice Values of C5-ROF7SS

The calibratedH-choice values of C5-ROF7SS are encapsulated in Table 23, where we select H=3.Table 23
3-choice values ofC5-ROF7SS.
p53,Mm1m2m3m4Uη3Lη3w10.78ei1.89π,0.44ei1.55π0.49ei1.77π,0.35ei1.62π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.79ei1.96π,0.13ei1.16π1.1680w20.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.94ei1.96π,0.03eι0.12π0.94ei1.96π,0.03ei0.12π1.6378w30.45ei1.68π,0.29ei1.57π0.81ei1.93π,0.20ei1.03π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.81ei1.97π,0.05ei0.72π1.2880w40.95ei1.95π,0.13ei0.09π0.75ei1.78π,0.16ei1.18π0.93ei1.97π,0.05ei0.04π0.87ei1.87π,0.23ei0.04π0.99ei1.99π,0.00ei0.00π1.9262w50.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.76ei1.86π,0.11ei1.12π0.00ei0.00π,1.00ei2.00π0.76ei1.86π,0.11ei1.12π0.8941w60.34ei1.64π,0.22ei1.53π0.00ei0.00π,1.00ei2.00π0.85ei1.89π,0.01ei0.05π0.00ei0.00π,1.00ei2.00π0.85ei1.94π,0.001ei0.03π1.3261w70.89ei1.86π,0.21ei0.02π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.39ei1.81π,0.27ei1.72π0.89ei1.96π,0.04ei0.01π1.4805The spacecraft are eventually organized in an ascending order of their respectiveH-choice values, where H=3. The hierarchical ranking of inspected aerospace technologies is clearly demonstrated in Table 24.Table 24
Hierarchical ranking of spacecraft.
Spacecraftw1w2w3w4w5w6w7Ranking6251743Hence, we conclude that the lander spacecraftw4 will be selected as the most ideal spacecraft for the exploration of various cosmological phenomena.


## 5.1. 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.


## 5.2. 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. The 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. The collection of seven spacecraftW=w1,w2,…,w7 is considered after scrutinizing their versatile potentialities, wherew1 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. The penetrator spacecraft monitors the inherent peculiarities of the penetrated surface to determine whether life exists on other cosmic planets of the solar system.w2 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. The orbiter spacecraft can significantly analyze the same planet from different heights without landing on its surface.w3 is the flyby spacecraft: this ideal spacecraft has potential to spin continuously and utilize thrusters or reaction wheels to be stabilized in 3 axes. These high-speed flyby spacecraft productively utilized their on-board optical instruments to critically observe the apparent motion of its target at the highest possible temporal resolution.w4 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.w5 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.w6 is the rover spacecraft: this special type of spacecraft can move across planetary celestial bodies and other solid surfaces of cosmic planets. This 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.w7 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 factorsℳ=m1,m2,m3,m4 that strongly influence the selection process of spacecraft will serve as decision-criteria for this inspected MCDM problem, wherem1 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.m2 is the communication and navigation: all spacecraft require a good communication and navigation system to avoid constraints on future space missions. These potent capabilities of spacecraft provide a secure environment for constructive global telecommunication and ensure the accurate tracking of their trajectories.m3 is the payload capacity: the payload capacity is the most important consideration in designing the infrastructure of spacecraft. This measurable parameter usually quantifies the carrying capacity of spacecraft which may include cargo, flight crew, munitions, and other scientific equipment by analyzing the nature of their flight missions.m4 is the cosmic radiation protection: a broad spectrum of ionizing radiation can penetrate spacesuits, solar equipment, and spacecraft and can badly harm the biological health of astronauts. The cosmic radiation protection plays an indispensable role in the proper execution of large-scale space missions. The adequate shielding of spacecraft can minimize the risk of the exposure of such harmful radiation.The main purpose of this analysis is the determination of the best spacecraft for the remarkable strategic and economic advancement of Mexican space agency. The 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 long-term profit of aerospace industries. The framework for the identification of the scrutinized MCDM problem is demonstrated in Figure2.Figure 2
Specification of investigated problem.Now, an identified 7-soft set can easily be obtained from Table18, 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”Table 18
Rating of spacecraft.
P,M,7m1m2m3m4w1w2w3w4w5w6w7The gradesK=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 Table18.Now, the tabular representation of its corresponding specified 7-soft set is illustrated in Table19.Table 19
Identified 7-soft set.
P,M,7m1m2m3m4w14321w21216w33502w46465w52042w63150w75223In this respect, the grading criteria based on the score degrees ofC5-ROF7SVs are defined as follows:(42)−2.00≤LVηα<−1.29whenkηα=0,−1.29≤LVηα<−0.58whenkηα=1,−0.58≤LVηα<0.13whenkηα=2,0.13≤LVηα<0.84whenkηα=3,0.84≤LVηα<1.55whenkηα=4,1.55≤LVηα<2.26whenkηα=5,2.26≤LVηα≤3.00whenkηα=6,where LVηα=Θηαq−Ψηαq+Ωηα/2πq−ϒηα/2πq+kηα/N−1q and Vηα=kηα,ΘηαeiΩηα,Ψηαeiϒηα; η=1,2,3,4,5,6,7; α=1,2,3,4.Therefore, by Definition8, C5-ROF7SS can be interpreted as follows:(43)p5,Y,7=m1,p5m1,m2,p5m2,m3,p5m3,m4,p5m4,where(44)p5m1=w1,4,0.78ei1.89π,0.44ei1.55π,w2,1,0.14ei0.12π,0.77ei1.82π,w3,3,0.45ei1.68π,0.29ei1.57π,w4,6,0.95ei1.95π,0.13ei0.09π,w5,2,0.54ei1.96π,0.99ei1.24π,w6,3,0.34ei1.64π,0.22ei1.53π,w7,5,0.89ei1.86π,0.21ei0.02π,p5m2=w1,3,0.49ei1.77π,0.35ei1.62π,w2,2,0.98ei1.08π,0.62ei1.97π,w3,5,0.81ei1.93π,0.20ei1.03π,w4,4,0.75ei1.78π,0.16ei1.18π,w5,0,0.07ei0.05π,0.89ei1.92π,w6,1,0.12ei0.09π,0.74ei1.75π,w7,2,0.68ei1.34π,0.82ei1.63π,p5m3=w1,2,0.78ei1.55π,0.89ei1.67π,w2,1,0.18ei0.11π,0.82ei1.76π,w3,0,0.04ei0.06π,0.96ei1.89π,w4,6,0.93ei1.97π,0.05ei0.04π,w5,4,0.76ei1.86π,0.11ei1.12π,w6,5,0.85ei1.89π,0.01ei0.05π,w7,2,0.53ei1.45π,0.64ei1.56π,p5m4=w1,1,0.17ei0.13π,0.84ei1.85π,w2,6,0.94ei1.96π,0.03ei0.12π,w3,2,0.61ei1.13π,0.85ei1.54π,w4,5,0.87ei1.87π,0.23ei0.04π,w5,2,0.55ei1.23π,0.67ei1.57π,w6,0,0.01ei0.03π,0.99ei1.96π,w7,3,0.39ei1.81π,0.27ei1.72π.Now, theC5-ROF7SS can be easily demonstrated in tabular form by Table 20.Table 20
Tabular representation ofC5-ROF7SS.
p5,Y,7m1m2m3m4w14,0.78ei1.89π,0.44ei1.55π3,0.49ei1.77π,0.35ei1.62π2,0.78ei1.55π,0.89ei1.67π1,0.17ei0.13π,0.84ei1.85πw21,0.14ei0.12π,0.77ei1.82π2,0.98ei1.08π,0.62ei1.97π1,0.18ei0.11π,0.82ei1.76π6,0.94ei1.96π,0.03ei0.12πw33,0.45ei1.68π,0.29ei1.57π5,0.81ei1.93π,0.20ei1.03π0,0.04ei0.06π,0.96ei1.89π2,0.61ei1.13π,0.85ei1.54πw46,0.95ei1.95π,0.13ei0.09π4,0.75ei1.78π,0.16ei1.18π6,0.93ei1.97π,0.05ei0.04π5,0.87ei1.87π,0.23ei0.04πw52,0.54ei1.96π,0.99ei1.24π0,0.07ei0.05π,0.89ei1.92π4,0.76ei1.86π,0.11ei1.12π2,0.55ei1.23π,0.67ei1.57πw63,0.34ei1.64π,0.22ei1.53π1,0.12ei0.09π,0.74ei1.75π5,0.85ei1.89π,0.01ei0.05π0,0.01ei0.03π,0.99ei1.96πw75,0.89ei1.86π,0.21ei0.02π2,0.68ei1.34π,0.82ei1.63π2,0.53ei1.45π,0.64ei1.56π3,0.39ei1.81π,0.27ei1.72π


## 5.3. Choice Values ofC5-ROF7SS

The quantified choice values ofC5-ROF7SS for the determination of best spacecraft are summarized in Table 21.Table 21
Choice values ofC5-ROF7SS.
p5,Y,7m1m2m3m4UηLηw14,0.78ei1.89π,0.44ei1.55π3,0.49ei1.77π,0.35ei1.62π2,0.78ei1.55π,0.89ei1.67π1,0.17ei0.13π,0.84ei1.85π4,0.88ei1.98π,0.08ei0.77π1.6300w21,0.14ei0.12π,0.77ei1.82π2,0.98ei1.08π,0.62ei1.97π1,0.18ei0.11π,0.82ei1.76π6,0.94ei1.96π,0.03ei0.12π6,0.99ei1.96π,0.01ei0.08π2.8966w33,0.45ei1.68π,0.29ei1.57π5,0.81ei1.93π,0.20ei1.03π0,0.04ei0.06π,0.96ei1.89π2,0.61ei1.13π,0.85ei1.54π5,0.84ei1.97π,0.03ei0.45π1.7741w46,0.95ei1.95π,0.13ei0.09π4,0.75ei1.78π,0.16ei1.18π6,0.93ei1.97π,0.05ei0.04π5,0.87ei1.87π,0.23ei0.04π6,0.99ei1.99π,0.00ei0.00π2.9262w52,0.54ei1.96π,0.99ei1.24π0,0.07ei0.05π,0.89ei1.92π4,0.76ei1.86π,0.11ei1.12π2,0.55ei1.23π,0.67ei1.57π4,0.81ei1.99π,0.05ei0.40π1.4575w63,0.34ei1.64π,0.22ei1.53π1,0.12ei0.09π,0.74ei1.75π5,0.85ei1.89π,0.01ei0.05π0,0.01ei0.03π,0.99ei1.96π5,0.85ei1.94π,0.001ei0.02π1.7280w75,0.89ei1.86π,0.21ei0.02π2,0.68ei1.34π,0.82ei1.63π2,0.53ei1.45π,0.64ei1.56π3,0.39ei1.81π,0.27ei1.72π5,0.92ei1.98π,0.02ei0.01π2.0375The spacecraft are further categorized in an ascending order of their respective choice values. The hierarchical ranking of inspected space exploration technologies is highlighted in Table22.Table 22
Hierarchical ranking of spacecraft.
Spacecraftw1w2w3w4w5w6w7Ranking6241753Consequently, we infer that the lander spacecraftw4 will be selected as the most competent space exploration technology for the strategic and economic development of Mexican space agency.


## 5.4.H-Choice Values of C5-ROF7SS

The calibratedH-choice values of C5-ROF7SS are encapsulated in Table 23, where we select H=3.Table 23
3-choice values ofC5-ROF7SS.
p53,Mm1m2m3m4Uη3Lη3w10.78ei1.89π,0.44ei1.55π0.49ei1.77π,0.35ei1.62π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.79ei1.96π,0.13ei1.16π1.1680w20.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.94ei1.96π,0.03eι0.12π0.94ei1.96π,0.03ei0.12π1.6378w30.45ei1.68π,0.29ei1.57π0.81ei1.93π,0.20ei1.03π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.81ei1.97π,0.05ei0.72π1.2880w40.95ei1.95π,0.13ei0.09π0.75ei1.78π,0.16ei1.18π0.93ei1.97π,0.05ei0.04π0.87ei1.87π,0.23ei0.04π0.99ei1.99π,0.00ei0.00π1.9262w50.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.76ei1.86π,0.11ei1.12π0.00ei0.00π,1.00ei2.00π0.76ei1.86π,0.11ei1.12π0.8941w60.34ei1.64π,0.22ei1.53π0.00ei0.00π,1.00ei2.00π0.85ei1.89π,0.01ei0.05π0.00ei0.00π,1.00ei2.00π0.85ei1.94π,0.001ei0.03π1.3261w70.89ei1.86π,0.21ei0.02π0.00ei0.00π,1.00ei2.00π0.00ei0.00π,1.00ei2.00π0.39ei1.81π,0.27ei1.72π0.89ei1.96π,0.04ei0.01π1.4805The spacecraft are eventually organized in an ascending order of their respectiveH-choice values, where H=3. The hierarchical ranking of inspected aerospace technologies is clearly demonstrated in Table 24.Table 24
Hierarchical ranking of spacecraft.
Spacecraftw1w2w3w4w5w6w7Ranking6251743Hence, we conclude that the lander spacecraftw4 will be selected as the most ideal spacecraft for the exploration of various cosmological phenomena.


## 6. Comparative Analysis

In this section, a comparative study of our proposed decision-making techniques with contemporary MCDM technique based on theq-rung orthopair fuzzy soft weighted geometric (q-ROFSftWG) 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-ROFSftWG Operator-Based MCDM Strategy

We now simplify the practical application 5.2 by implementing the systematic methodology ofq-ROFSftWG operator-based MCDM technique, proposed by Chinram et al. [22].Step 1.
The first step is to construct the 5-rung orthopair fuzzy soft decision matrix (5-ROFSftDM) by scrutinizing the credibility of each aerospace technology relative to specific decision-criteria. The 5-rung orthopair fuzzy soft entries of 5-ROFSftDM ℑ=Fηα7×4 are obtained by omitting the grades and phase terms of all C5-ROF7SVs in Table 20. The calibrated 5-ROFSft-entries Fηα=Θ,Ψ of 5-ROFSftDM regarding proficiency of spacecraft are summarized in Table 25.Table 25
Tabular representation of5-ROFSftDM.
ℑm1m2m3m4w10.78,0.440.49,0.350.78,0.890.17,0.84w20.14,0.770.98,0.620.18,0.820.94,0.03w30.45,0.290.81,0.200.04,0.960.61,0.85w40.95,0.130.75,0.160.93,0.050.87,0.23w50.54,0.990.07,0.890.76,0.110.55,0.67w60.34,0.220.12,0.740.85,0.010.01,0.99w70.89,0.210.68,0.820.53,0.640.39,0.27Step 2.
Normalize the evaluated5-ROFSftDM ℑ=Fηα7×4 according to the nature and specification of each identified decision-criterion by utilizing the following expression:(45)Eηα=Fηαc,for cost type parameter,Fηα,for benefit type parameter,where Fηαc=Ψ,Θ represents the complement of Fηα=Θ,Ψ. Here, the normalized 5-rung orthopair fuzzy soft decision matrix E=Eηα7×4 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 decision-criterion by employing theq-ROFSftWG operator, which is defined as follows:(46)E˜η=q−ROFSftWGℌEη11,Eη22,Eη33,…,Eηℏτ=⊗α=1ℏ⊗ζ=1τEηαζkζzα=∏α=1ℏ∏ζ=1τΘηαζkζzα,1−∏α=1ℏ∏ζ=1τ1−Ψηαζkζzαq,where kζ indicates the weights of decision-makers and zα denotes the relative weights of decisive parameters satisfying the conditions that kζ,zα∈0,1 with ∑ζ=1τkζ and ∑α=1ℏzα. Now, the appointed expert assigns weights z=0.32,0.20,0.33,0.15T to the identified decision-criteria. The aggregated 5-ROFSftNs of each spacecraft corresponding to decisive criteria are presented as follows:(47)E˜1=0.56554,0.78625,E˜2=0.29868,0.75104,E˜3=0.23834,0.86201,E˜4=0.88799,0.16690,E˜5=0.40281,0.92670,E˜6=0.22009,0.83050,E˜7=0.62802,0.65689.Step 4.
The next step is to evaluate the score degreesLη of aggregated 5-ROFSftNs E˜η=Θ˜η,Ψ˜η of each aerospace technology which is interpreted as follows:(48)LE˜η=Θηq−Ψηq+eΘηq−ΨηqeΘηq−Ψηq+1−12πηq,forq≥1,where η=1,2,…,7 and πη=1−Θq+Ψqq represents the hesitancy degree of q-ROFSftNs. The quantified score degrees of E˜η for each spacecraft are clearly illustrated in Table 26.Table 26
Score degrees ofE˜η
Lη/wηw1w2w3w4w5w6w7Lη−0.29785−0.29230−0.577640.66663−0.80079−0.48265−0.03048Step 5.
All the aerospace technologies of the inspected MCDM problem are hierarchically ranked in an ascending order corresponding to their calibrated score degrees and further demonstrated their systematic ranking in Table27.
Hence, we conclude that lander spacecraftw4 will be selected as the most proficient aerospace technology having maximum score degree for the accomplishment of large-scale global missions.Table 27
Hierarchical ranking of spacecraft.
Spacecraftw1w2w3w4w5w6w7Ranking4361752


### 6.2. Comprehensive Discussion

(1)
We exhibit a comparison of developed methodologies with contemporary MCDM techniques based onq-ROFSftWG, Cq-ROFWA, and Cq-ROFWG operators to substantiate the accountability and feasibility of the proposed strategies. The methodological implications of both developed and compared strategies are persuasively spotlighted in Table 28.(2)
All the proposed and compared strategies prioritize lander spacecraftw4 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 Figure3 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)
The 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)
Theq-ROFSftWG operator is not capable enough to deal with Cq-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. This 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 onCq-ROFWA and Cq-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. This incredible peculiarity of our proposed methodologies stimulates them as the most generalized and sustainable decision-making strategies for real-world MCDM complications.Table 28
Comparative analysis.
MethodsRankingBest spacecraftProposed choice values ofCq-ROFNSS methodw4≻w2≻w7≻w3≻w6≻w1≻w5w4ProposedH-choice values of Cq-ROFNSS methodw4≻w2≻w7≻w6≻w3≻w1≻w5w4q-ROFSftWG operator-based MCDM method [22]w4≻w7≻w2≻w1≻w6≻w3≻w5w4Cq-ROFWA operator-based MADM method [12]w4≻w7≻w2≻w5≻w6≻w1≻w3w4Cq-ROFWG operator-based MADM method [12]w4≻w5≻w3≻w2≻w6≻w1≻w7w4Figure 3
Comparative analysis.


## 6.1.q-ROFSftWG Operator-Based MCDM Strategy

We now simplify the practical application 5.2 by implementing the systematic methodology ofq-ROFSftWG operator-based MCDM technique, proposed by Chinram et al. [22].Step 1.
The first step is to construct the 5-rung orthopair fuzzy soft decision matrix (5-ROFSftDM) by scrutinizing the credibility of each aerospace technology relative to specific decision-criteria. The 5-rung orthopair fuzzy soft entries of 5-ROFSftDM ℑ=Fηα7×4 are obtained by omitting the grades and phase terms of all C5-ROF7SVs in Table 20. The calibrated 5-ROFSft-entries Fηα=Θ,Ψ of 5-ROFSftDM regarding proficiency of spacecraft are summarized in Table 25.Table 25
Tabular representation of5-ROFSftDM.
ℑm1m2m3m4w10.78,0.440.49,0.350.78,0.890.17,0.84w20.14,0.770.98,0.620.18,0.820.94,0.03w30.45,0.290.81,0.200.04,0.960.61,0.85w40.95,0.130.75,0.160.93,0.050.87,0.23w50.54,0.990.07,0.890.76,0.110.55,0.67w60.34,0.220.12,0.740.85,0.010.01,0.99w70.89,0.210.68,0.820.53,0.640.39,0.27Step 2.
Normalize the evaluated5-ROFSftDM ℑ=Fηα7×4 according to the nature and specification of each identified decision-criterion by utilizing the following expression:(45)Eηα=Fηαc,for cost type parameter,Fηα,for benefit type parameter,where Fηαc=Ψ,Θ represents the complement of Fηα=Θ,Ψ. Here, the normalized 5-rung orthopair fuzzy soft decision matrix E=Eηα7×4 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 decision-criterion by employing theq-ROFSftWG operator, which is defined as follows:(46)E˜η=q−ROFSftWGℌEη11,Eη22,Eη33,…,Eηℏτ=⊗α=1ℏ⊗ζ=1τEηαζkζzα=∏α=1ℏ∏ζ=1τΘηαζkζzα,1−∏α=1ℏ∏ζ=1τ1−Ψηαζkζzαq,where kζ indicates the weights of decision-makers and zα denotes the relative weights of decisive parameters satisfying the conditions that kζ,zα∈0,1 with ∑ζ=1τkζ and ∑α=1ℏzα. Now, the appointed expert assigns weights z=0.32,0.20,0.33,0.15T to the identified decision-criteria. The aggregated 5-ROFSftNs of each spacecraft corresponding to decisive criteria are presented as follows:(47)E˜1=0.56554,0.78625,E˜2=0.29868,0.75104,E˜3=0.23834,0.86201,E˜4=0.88799,0.16690,E˜5=0.40281,0.92670,E˜6=0.22009,0.83050,E˜7=0.62802,0.65689.Step 4.
The next step is to evaluate the score degreesLη of aggregated 5-ROFSftNs E˜η=Θ˜η,Ψ˜η of each aerospace technology which is interpreted as follows:(48)LE˜η=Θηq−Ψηq+eΘηq−ΨηqeΘηq−Ψηq+1−12πηq,forq≥1,where η=1,2,…,7 and πη=1−Θq+Ψqq represents the hesitancy degree of q-ROFSftNs. The quantified score degrees of E˜η for each spacecraft are clearly illustrated in Table 26.Table 26
Score degrees ofE˜η
Lη/wηw1w2w3w4w5w6w7Lη−0.29785−0.29230−0.577640.66663−0.80079−0.48265−0.03048Step 5.
All the aerospace technologies of the inspected MCDM problem are hierarchically ranked in an ascending order corresponding to their calibrated score degrees and further demonstrated their systematic ranking in Table27.
Hence, we conclude that lander spacecraftw4 will be selected as the most proficient aerospace technology having maximum score degree for the accomplishment of large-scale global missions.Table 27
Hierarchical ranking of spacecraft.
Spacecraftw1w2w3w4w5w6w7Ranking4361752


## 6.2. Comprehensive Discussion

(1)
We exhibit a comparison of developed methodologies with contemporary MCDM techniques based onq-ROFSftWG, Cq-ROFWA, and Cq-ROFWG operators to substantiate the accountability and feasibility of the proposed strategies. The methodological implications of both developed and compared strategies are persuasively spotlighted in Table 28.(2)
All the proposed and compared strategies prioritize lander spacecraftw4 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 Figure3 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)
The 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)
Theq-ROFSftWG operator is not capable enough to deal with Cq-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. This 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 onCq-ROFWA and Cq-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. This incredible peculiarity of our proposed methodologies stimulates them as the most generalized and sustainable decision-making strategies for real-world MCDM complications.Table 28
Comparative analysis.
MethodsRankingBest spacecraftProposed choice values ofCq-ROFNSS methodw4≻w2≻w7≻w3≻w6≻w1≻w5w4ProposedH-choice values of Cq-ROFNSS methodw4≻w2≻w7≻w6≻w3≻w1≻w5w4q-ROFSftWG operator-based MCDM method [22]w4≻w7≻w2≻w1≻w6≻w3≻w5w4Cq-ROFWA operator-based MADM method [12]w4≻w7≻w2≻w5≻w6≻w1≻w3w4Cq-ROFWG operator-based MADM method [12]w4≻w5≻w3≻w2≻w6≻w1≻w7w4Figure 3
Comparative analysis.


## 7. Merits of theCq-ROFNSS Model

The next bullet points summarize the main advantages of the model proposed in this paper:(1)
We have proposed aCq-ROFNSS model that provides a multiskilled and most proficient mathematical framework for the fuzzy representation of parameterized vague data. The underlying features of both NSSs and Cq-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.(4)
The remarkable decision-making skills ofCq-ROFNSS excellently increase the modeler’s ability to express their two-dimensional ambiguous assessments. The outstanding generic structure of our proposed model broadens the confined space of traditional fuzzy models by relaxing their constraint conditions. The 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.


## 8. Conclusion

In this research article, we have accomplished our goal by establishing the foundations of a multiskilled hybrid model calledCq-ROFNSS. It can capture a wide range of imprecision embedded in human cognition. This 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 Cq-ROFSs and the remarkable parametric nature of NSSs. This 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 Cq-ROFNSS and its basic fundamental set-theoretic operations. We have proposed the Einstein and some other algebraic operations of Cq-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. Thus, 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 theCq-ROFNS-PROMETHEE method, Cq-ROFNS-VIKOR method, Cq-ROFNS-AHP method, Cq-ROFNS-TOPSIS method, and Cq-ROFNS-ELECTRE method. We intend to explore the scope of the potential applications under the versatile environment of the Cq-ROFNS model.



---

*Source: 3690597-2021-09-21.xml*