# A Multiattribute Decision-Making Framework: VIKOR Method with Complex Spherical FuzzyN-Soft Sets

Authors: Muhammad Akram; Maria Shabir; Arooj Adeel; Ahmad N. Al-Kenani
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we set forth a framework for solving a multiattribute group decision-making (MAGDM) problem, namely, the selection of a firm for participation in a Saudi oil refinery project in Pakistan. This project will prove a key success factor for the economic growth of Pakistan due to its enormous economic impact on the energy sector, industrial development, commerce, transportation, and so on. This multiplicity justifies that several intricate components comprising both intrinsic and external attributes should be adequately evaluated for the selection of such a firm, that is, the formulation of this question as a MAGDM problem. Nonbinary evaluation with two-dimensional ambiguity and uncertainty in the parameters are general concerns in modern literature, and they fit into this problem. Within this context, one of the most superior and amenable theories (complex spherical fuzzyN-soft sets, henceforth CSFNSfSs) shall be used to formulate a new comprehensive method, known as complex spherical fuzzy N-soft-VIKOR (CSFNSf-VIKOR) method. According to the general spirit of the benchmark technique, the normalized Euclidean distances and the weights of the attributes are jointly handled, and as consequence, two main features (“maximum group utility” and “minimum individual regret”) are acquired. The coefficient strategy with reference to group utility measure and individual regret measure of opponents are employed for the compromise measure. Armed with this novel tool, we single out the most feasible firm according to the preference order of the alternatives examined by the decision-makers on the subject of linear normalized weights of experts and attributes. Furthermore, a comparative analysis justifies the CSF-VIKOR method, and some results prove its capabilities and validity. Moreover, a sensitivity test certifies the stability of the proposed method.


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## Body

## 1. Introduction

In 1998, the VIKOR approach was drafted by Opricovic [1] as a multiattribute decision-making (MADM) method. In the setting of civil engineering, this system attempted to find a compromise solution based on two dominant principles (group utility measure and individual regret of opponent), where the compromise solution means a decision done by generic involvement. Due to the paradigm of maximum group utility and minimum individual regret, the feasible solution closest to the best values and most distant from the worst value is resolved by the recourse to the Lp-metric as an aggregation function, according to Opricovic. Opricovic and Tzeng [2] extended the VIKOR theory for MADM for the postearthquake reconstruction problem in Central Taiwan and the selection of destination for the mountain climber, respectively. Yazdani-Chamzini et al. [3] employed the modified version of the VIKOR method along with the combination of TOPSIS, MOORA, additive weighting, and ratio assessment techniques in order to address a multicriteria decision-making (MCDM) problem relating to renewable energy resources.People habitually live with real-life properties that are not sufficiently precise and not fully objective, such as “beautiful,” “tall,” or “experienced.” Many decision-related problems must take into account such properties. To contend with such vague and uncertain data, Zadeh [4] presented the notion of fuzzy set. Suh et al. [5] assessed the mobile service quality through the proposed groundbreaking fuzzy-VIKOR method using an integrated weighted approach. Lee et al. [6] extended the VIKOR technique for the MAGDM problem of detection of flood risks under the fuzzy frame. Li and Liu [7] presented the VIKOR with the combination of the QUALIFLEX method on the basis of trapezoidal fuzzy numbers corresponded to each two-dimensional linguistic data. Chang [8] disclosed the situation about the private and governmental hospital agencies in Taiwan with the help of the new fuzzy-VIKOR approach. Wang and Chang [9] resolved the MAGDM problem through fuzzy-VIKOR methodology. Another MAGDM problem, the selection of machine tools evaluated by the fuzzy-VIKOR approach is introduced by Wu et al. [10].In accordance with fuzzy descriptions, the degree of truthiness (or satisfaction), collected from the closed unit interval0,1, denotes the correspondence of an object related to a parameter. Under these interpretations, its inability to represent the nonassociation of the object related to that particular parameter is quite obvious. For this reason, in 1986, Atanassov [11] eradicated this obstacle through the extended concept of intuitionistic fuzzy set (IFS) and confronted the vague opinion through both a degree of truthiness ϕ and a degree of falsity χ, which are jointly subject to the constraint: ϕ+χ≤1. Roostaee et al. [12] developed the theoretical background for an IF-VIKOR method to rank the suppliers following the opinions of decision-makers, and Gupta et al. [13] extended the IF-VIKOR ideology for allocating the best environment for plantation. Krishankumar et al. [14] resolved a personnel selection problem through the extended version of the IF-VIKOR method.Pythagorean fuzzy setsPyFSs evolved as an extension of IFSs. Introduced by Yager [15], PyFS also deal with both degree of truth ϕ and degree of falsity χ, but they are under the modified condition: ϕ2+χ2≤1. Gul et al. [16] extended the VIKOR based approach within the field of PyFSs and evaluated the safety risks in the mine industry. Rani et al. [17] proposed the PyF-VIKOR method within the tool of entropy and divergence measures for the selection of renewable energy technology in India. Ma et al. [18] introduced the group decision-making framework using complex Pythagorean fuzzy information.People habitual nature sometimes has a neutral judgment. This is generally exhibited in certain problems or events like voting situations. Therefore, along with the judgments of yes or no, there is often a need for an abstain part of the opinion (possibly related to the satisfaction of a particular parameter). Since the Pythagorean fuzzy set is not able to handle such part of the decisional attitude, Cuong [19] introduced picture fuzzy set PFS along with the degree of truth ϕ, remain neutral ψ, and degree of falsity χ but with their range limited by an inequality: ϕ+ψ+χ≤1. Meksavang et al. [20] extended the decision-making approach based on the VIKOR methodology for the picture fuzzy environment, which was applied for sustainable suppliers along with an application in the beef industry. Liu and You [21] presented the PF-VIKOR technique for green supplier evaluation and selection.However, PFSs are very reliable to deal with imprecision and fuzziness and still useless when the sum of the degrees of truth, neutrality, and falsity exceeds 1. To get over this drawback, Gundogdu and Kahraman [22–25] worked out a model with a relaxed condition, introduced the idea of spherical fuzzy sets SFSs, and employed them on MADM problems. Later on, Mahmood et al. [26] proposed T-spherical fuzzy sets, as an extension of SFSs. Gundogdu et al. [27, 28] developed the theory for the SF-VIKOR method and applied it to the MADM problem for waste management problems and selection of site, respectively.The aforementioned models of fuzzy knowledge have a one-dimensional structure. This pattern is incompatible with certain types of problems. Ramot et al. [29] developed a fuzzy set model in two-directional frames, known as complex fuzzy set (CFS). In this model, the degree of truth is ϕ=tei2πζ, which is divided into amplitude term t and periodic term ζ that belongs to the unit closed interval. For the sake of decision-making, complex fuzzy aggregation operators are defined by Akram and Bashir [30]. Alkouri and Salleh [31] generalized the ideology to a complex intuitionistic fuzzy set (CIFS), which accounts for both degree of truthiness ϕ=tei2πζ and falsity χ=fei2πρ along with the constraints t+f≤1 and ζ+ρ≤1. Ullah et al. [32] introduced a complex Pythagorean fuzzy set CPyFS with degrees of truth ϕ=tei2πζ and falsity χ=fei2πρ within the complex unit circle and restricted by the constraints t2+f2≤1 and ζ2+ρ2≤1. Currently, Akram et al. [33] presented the model of a complex spherical fuzzy set CSFS representing three degrees of truth ϕ=tei2πζ, neutrality ψ=qei2πϱ, and falsity χ=fei2πρ subjected to conditions t2+q2+f2≤1 and ζ2+ϱ2+ρ2≤1. They also extended the VIKOR method under the CSF environment. The yielding and tractable conditions of the CSF representation make it a privileged framework for the modelization of two-dimensional ambiguous knowledge. In the advantageous CSF-VIKOR technique, decision-makers scrutinize the competencies of the feasible choices with reference to the preferred criteria and indicate initial observation through the use of linguistic information, which further enhanced the virtues of complex spherical fuzzy numbers.ThoughCSFSs are highly competent and skillful, they neglect the possibility of nonbinary parameterized information. In this panorama, Akram et al. [34] presented the concept of complex spherical fuzzy N-soft sets CSFNSfSs within the TOPSIS methodology, which has the ability to look over the modern real-life ranking problems in every field of science. In relation to this, the idea of N-soft sets NSfSs was presented by [35] as an extension of soft set theory [36], which accommodates all kind of attributes. Soon afterwards, Akram et al. [37] explored the scope of N-soft sets and put forward hybrid models, namely, fuzzy N-soft sets FNSfSs, intuitionistic fuzzy N-soft sets IFNSfS, and hesitant N-soft sets. Recently, Zhang et al. [38] combined N-soft theory with PyFSs and proposed the novel model of Pythagorean fuzzy N-soft set PyFNSfS. The CSFNSf model was presented (by Akram et al. [34]) within research conducive to an extension of the TOPSIS method that can deal with MAGDM problems based on CSFNSf information. For other notations and applications, the readers are referred to [39–44].In modern times, ranking evaluation systems (instead of linguistic information) can be in use as a primary source in furtherance of decision-making problems and surveys. TheCSF-VIKOR technique (proposed by Akram et al. [33]) is unqualified for situations comprising parameterized ranking information. Therefore, we develop a novel technique, namely, the CSFNSf-VIKOR method. The delineation and explanation related to this technique do not pertain to any of the existing techniques; hence, it requires a concrete analysis. Of course, the proposed model successfully evaluates decision-making problems related to the modern era. The purpose of our concept is to extend the VIKOR method under the circumstances of CSFNSfSs. Its motivations are given as follows:(i)
The two-dimensional influential technique ofCSF-VIKOR is inadequate to operate with data comprising ordered grades along with two or several parameters.(ii)
The fuzzyN-soft models, along with intuitionistic fuzzy N-soft, Pythagorean fuzzy N-soft, and complex Pythagorean fuzzy N-soft environments provide an array of models that operate under parameterized ranking systems. But they are all useless when it comes to incorporating uncertain data containing neutral opinions.(iii)
TheCSFNSf-TOPSIS method uses a different methodology but ultimately computes a feasible solution, which merely does not take into account the relative importance of distances from the ideal solution.(iv)
These limitations provoke the development ofCSFNSf-VIKOR method that gives us the ability to incorporate information regarding ordered grades among complex valued degrees of truth, neutrality, and falsity.The main aim of our proposed idea is to develop the methodology of VIKOR within the hybrid model ofCSFNSfSs specifically for the solution of MAGDM problems precisely. The contributions of this paper are as follows:(i)
We introduce a hybrid MAGDM VIKOR approach whose structure is based onCSFNSfSs, and it is known as CSFNSf-VIKOR technique. This methodology qualifies for situations that comprise parameterized ranking information; hence, it successfully evaluates a large proportion of existing MAGDM problems, as well as decision-making problems with nonbinary parameterized information as an initial assessment.(ii)
Linear normalized weights of experts and attributes are defined along with normalized Euclidean distance for sake of maximum group utility measure, individual regret, and compromise ranking whose values are arranged in ascending order for final decisions.(iii)
The presented technique is supported through a real-life application from the grounds of economy and business assisted by comparative analysis and sensitivity tests.The rest of the article is structured as follows. Section2 contains preliminaries from CSFNSf model [34] with some operations, comparison rule, and averaging operator. In Section 3, a descriptive theory for CSFNSf-VIKOR method is developed along with a flowchart. Section 4 interpreted a real-life MAGDM problem of firm’s selection for the Saudi oil refinery project. Section 5 narrated the proposed method by a sensitivity test. In Section 6, we compare our proposed method with the existing techniques. In Section 7, the merits of the CSFNSf-VIKOR method are clarified. In the end, we present concluding remarks and mention some future research work in Section 8.


## 2. Complex Spherical FuzzyN-Soft Sets

Definition 1 (see [34]).
LetJ be a nonempty set and X⊆T, whereas T be a set of parameters (or attributes), and C=0,1,2,…,N−1 be a set of grades level with N∈2,3,…,. Then a complex spherical fuzzy N-soft set CSFNSfS on X is denoted by a triplet E¨H,X,N, defined as follows:(1)E¨Hxa=E¨xa,Hxa:xa∈X=jb,cab,ϕba,ψba,χba=jb,cab,twsei2πζws,qbaei2πϱba,fbaei2πρba,along with the assumption of E¨:X⟶2J×H, where NSfS be defined on J, and H is a function from X to ℂSFℕ. The notation ℂSFℕ denotes the collection of all complex spherical fuzzy numbers of J, and tba, ζba, qba, ϱba, fba, and ρba will be taken from unit closed interval, with constraint(2)0≤tba2+qba2+fba2≤1,0≤ζba2+ϱba2+ρba2≤1.
The termcab denotes the ranking of attributes for the alternative jb, for all jb∈J.Definition 2 (see [34]).
LetE¨Hxa=jb,cab,tbaei2πζba,qbaei2πϱbafbaei2πρba be a CSFNSfS. Then the complex spherical fuzzy N-soft number CSFNSfN is defined as follows:(3)Ϝba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba,and the hesitancy degree is defined as follows:(4)ΩϜba=1−tba2+qba2+fba2ei2π1−ζba2+ϱba2+ρba2.Definition 3.
(see [34]). Let Ϝba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba be CSFNSfN. The score function ScϜba is defined as follows:(5)ScϜba=cabN−12+tba2−qba2−fba2+ζba2−ϱba2−ρba2,where ScϜba∈−2,3. The accuracy function AcϜba is defined as follows:(6)AcϜba=cabN−12+tba2+qba2+fba2+ζba2+ϱba2+ρba2,where AcϜba∈0,3.Definition 4 (see [34]).
LetϜbl=clb,tljei2πζbl,qblei2πϱbl,fblei2πρbl and Ϝba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba be two CSFNSfNs.(1)
IfScϜbl<ScϜba, then Ϝbl≺Ϝba (Ϝbl is inferior to Ϝba(2
)IfScϜbl>ScϜba, then Ϝbl≻Ϝba (Ϝbl is superior to Ϝba(3
)IfScϜbl=ScϜba, then(i)
AcϜbl<AcϜba, then Ϝbl≺Ϝba (Ϝbl is inferior to Ϝba(ii)
AcϜbl>AcϜba, then Ϝbl≻Ϝba (Ϝbl is superior to Ϝba(iii)
AcϜbl=AcϜba, then Ϝbl∼Ϝba (Ϝbl is equivalent to ϜbaDefinition 5 (see [34]).
LetϜbl=clb,tblei2πζbl,qblei2πϱbl,fblei2πρbl and Ϝba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba be two CSFNSfNs and λ>0. Some operations for CSFNSfNs are as follows:(7)λϜbl=clb,1−1−tbl2λei2π1−1−ζbl2λ,qblλei2πϱblλ,fblλei2πρblλ,Ϝblλ=clb,tblλei2πζblλ,1−1−vbl2λei2π1−1−ϱbl2λ,1−1−fbl2λei2π1−1−ρbl2λ,Ϝbl⊕Ϝba=maxclb,cab,tbl2+tba2−tbl2tba2ei2πζbl2+ζba2−ζbl2ζba2,qblqbaei2πϱblϱba,fblfbaei2πρblρba,Ϝbl⊗Ϝba=minclb,cab,tbltbaei2πζblζba,vlj2+vkj2−vlj2vkj2ei2πϱbl2+ϱba2−ϱbl2ϱba2,flj2+fkj2−flj2fkj2ei2πρbl2+ρba2−ρbl2ρba2.Proposition 1 (see [34]).
LetϜba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba and Ϝbl=clb,tblei2πζbl,qblei2πϱbl,fblei2πρbl be two CSFNSfNs and λ>0, then the following properties hold:(1)
Ϝba⊕Ϝbl=Ϝbl⊕Ϝba(2)
Ϝba⊗Ϝbl=Ϝbl⊗Ϝba(3)
λϜba⊕λϜbl=λϜbl⊕Ϝba, λ>0(4)
λ1Ϝba⊕λ2Ϝba=λ1+λ2Ϝba, λ1, λ2>0(5)
Ϝbaλ⊗Ϝblλ=Ϝbl⊗Ϝbaλ, λ>0(6)
Ϝbaλ1⊗Ϝbaλ2=Ϝbaλ1+λ2⋅λ1, λ2>0Proof.
(8)1Ϝba⊕Ϝbl=maxcab,clb,tba2+tbl2−tba2tbl2ei2πζba2+ζbl2−ζba2ζbl2,qbaqblei2πϱbaϱbl,fbafblei2πρbaρbl,=maxclb,cab,tbl2+tba2−tbl2tba2ei2πζbl2+ζba2−ζbl2ζba2,qblqbaei2πϱblϱba,fblfbaei2πρblρba,=Ϝbl⊕Ϝba.(9)2Ϝba⊗Ϝbl=mincab,clb,tbatblei2πζbaζbl,qba2+qbl2−qba2qbl2ei2πϱba2+ϱbl2−ϱba2ϱbl2,fba2+fbl2−fba2fbl2ei2πρba2+ρbl2−ρba2ρbl2=minclb,cab,tbltbaei2πζblζba,qbl2+qba2−qbl2qba2ei2πϱbl2+ϱba2−ϱbl2ϱba2,fbl2+fba2−fbl2fba2ei2πρbl2+ρba2−ρbl2ρba2=Ϝbl⊗Ϝba.(10)3λϜba⊕λϜbl=cab,1−1−tba2λei2π1−1−ζba2λ,qbaλei2πϱbaλ,fbaλei2πρbaλ⊕clb,1−1−tbl2λei2π1−1−ζbl2λ,qblλei2πϱblλ,fblλei2πρblλ=maxcab,clb,1−1−tba2λ+1−1−tbl2λ−1−1−tba2λ1−1−tbl2λei2π1−1−ζba2λ+1−1−ζbl2λ−1−1−ζba2λ1−1−ζbl2λ,qbaλqblλei2πϱbaλϱblλ,fbaλfblλei2πρbaλρblλ=maxcab,clb,1−1−tba2+tbl2−tba2tbl2λei2π1−1−ζba2+ζbl2−ζba2ζbl2λ,qbaqblλei2πϱbaϱblλ,fbafblλei2πρbaρblλ=λmaxcab,clb,tba2+tbl2−tba2tbl2ei2πζba2+bl2−ζba2ζbl2,qbaqble2πϱbaϱbl,fbafble2πρbaρbl=λϜba⊕Ϝbl.(11)4λ1Ϝba⊕λ2Ϝba=cab,1−1−tba2λ1ei2π1−1−ζba2λ1,qbaλ1ei2πϱbaλ1,fbaλ1ei2πρbaλ1⊕clb,1−1−tba2λ2ei2π1−1−ζba2λ2,qbaλ2ei2πϱbaλ2,fbaλ2ei2πρbaλ2=maxcab,clb,1−1−tba2λ1+1−1−tba2λ2−1−1−tba2λ11−1−tba2λ2ei2π1−1−ζba2λ1+1−1−ζba2λ2−1−1−ζba2λ1−1−ζba2λ2,qbaλ1qbaλ2ei2πϱbaλ1ϱbaλ2,fbaλ1fbaλ2ei2πρbaλ1ρbaλ2=maxcab,cab,1−1−tba2λ1+λ2ei2π1−1−ζba2λ1+λ2,qbaλ1+λ2ei2πϱbaλ1+λ2,fbaλ1+λ2ei2πρbaλ1+λ2=λ1+λ2cab,tbaei2πζba,qbae2πϱba,fbae2πρba=λ1+λ2Ϝba.
Similarly, we can clarify (5) and (6).Definition 6 (see [34]).
LetϜba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρbaa=1,2,…,v be a collection of CSFNSfNs, then complex spherical fuzzy N-soft weighted average CSFNSfWA operator is defined as follows:(12)CSFNSfWAϜb1,Ϝb2,…,Ϝbv=⊕a=1vλaϜba=maxa=1vcab,1−∏a=1v1−tba2λaei2π1−∏a=1v1−ζba2λa,∏a=1vqbaλaei2π∏a=1vϱbaλa,∏a=1vfbaλaei2π∏a=1vρbaλa,where λa=λ1,λ1,…,λvT is a weighted vector of Ϝba with the property that λa>0 and ∑a=1vλa=1, for all b=1,2,…,m.Theorem 1.
LetϜba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρbaa=1,2,…,v be a collection of CSFNSfNs and λ=λ1,λ2,…,λvT be the weight vector then the aggregated value by applying the CSFNSfWA operator is also a CSFNSfN formulated as follows:(13)CSFNSfWAϜb1,Ϝb2,…,Ϝbv=⊕a=1vλaϜba=maxa=1vcab,1−∏a=1v1−tba2λaei2π1−∏a=1v1−ζba2λa,∏a=1vqbaλaei2π∏a=1vϱbaλa,∏a=1vfbaλaei2π∏a=1vρbaλa.Proof.
The proof is followed by mathematical induction.Case 1.
Whena=1, the CSFNSfWA operator gives the following equation:(14)CSFNSfWAϜb1,Ϝb2,…,Ϝbv=λ1Ϝb1=Ϝb1,where λ1=1, and equation (13) becomes(15)=c1b,1−1−tb12ei2π1−1−ζb12,qb1ei2πϱba,fbaei2πρba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba.
The result is true fora=1, as equation (15) is a CSFNSfN.Case 2.
Suppose that the result hold fora=n, n is a natural number, and therefore, equation (13) becomes(16)CSFNSfWAϜb1,Ϝb2,…,Ϝbn=⊕a=1nλaϜba=maxa=1ncab,1−∏a=1n1−tba2λaei2π1−∏a=1n1−ζba2λa,∏a=1nqbaλaei2π∏a=1nϱbaλa,∏a=1nfbaλaei2π∏a=1nρbaλa.
Now we prove fora=n+1. Consider(17)CSFNSfWAϜb1,Ϝb2,…,Ϝbn+1=⊕a=1n+1λaϜba=maxa=1ncab,1−∏a=1n1−tba2λaei2π1−∏a=1n1−ζba2λa,∏a=1nqbaλaei2π∏a=1nϱbaλa,∏a=1nfbaλaei2π∏a=1nρbaλa⊕cnb+1,1−1−tbn+12ei2π1−1−ζbn+12,qbn+1ei2πϱbn+1,fbn+1ei2πρbn+1=maxa=1n+1cab,1−∏a=1n+11−tba2λaei2π1−∏a=1n+11−ζba2λa,∏a=1n+1qbaλaei2π∏a=1n+1ϱbaλa,∏a=1n+1fbaλaei2π∏a=1n+1ρbaλa.
Result holds forn+1; hence, it is held for all natural numbers.Theorem 2 (idempotency property).
LetϜba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρbaa=1,2,…,v be a collection of CSFNSfNs and λ=λ1,λ2,…,λvT be the weight vector of Ϝba, if Ϝba=Ϝ11 for all b,a; then(18)CSFNSfWAϜb1,Ϝb2,…,Ϝbv=⊕a=1vλaϜba=Ϝ11.Theorem 3 (boundedness property).
LetϜba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba be a collection of CSFNSfNs and for all b,a; then(19)Ϝ−≤CSFNSfWA≤Ϝ+,where, Ϝ+=maxacab,maxatbaei2πmaxaζba,minaqbaei2πminaϱba,minafbaei2πminaρba and Ϝ−=minacab,minatbaei2πminaζba,maxaqbaei2πmaxaϱba,maxafbaei2πmaxaρba.Theorem 4 (monotonicity property).
ConsiderϜba=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba and Ϝ¯ba=c¯ab,t¯baei2πζ¯ba,q¯baei2πϱ¯ba,f¯baei2πρ¯ba are two collections of CSFNSfNs along with cab≤c¯ab, tba≤t¯ba, ζba≤ζ¯ba, qba≥q¯ba, ϱba≥ϱ¯ba, fba≥f¯ba, and ρba≥ρ¯ba; then(20)CSFNSfWAϜb1,Ϝb2,…,Ϝbv≤CSFNSfWAϜ¯b1,Ϝ¯b2,…,Ϝ¯bv.


## 3. Complex Spherical FuzzyN-Soft-VIKOR Method

This section delineates the VIKOR method to resolve the MAGDM problems employing the domain ofCSFNSfSs, which is appropriate to deal with two-dimensional data. The proposed method works out to compromise a solution in which an agreement is accomplished by mutual conductance that possesses maximum group utility and minimum individual regret.LetJ=J1,J2,J3,…,Jm be the set of m possible choices whose expertise, possibility, and utility are figured out by k decision-experts Y1,Y2,Y3,…,Yk, with the help of v crucial elements X1,X2,X3,…,Xv, treated as attributes, and let l1,l2,l3,…,lv be considered as the normalized weight vectors of the attributes in accordance with related MAGDM problem. Let λs∈0,1 be the weight of the s-th decision-expert Ys; therefore, the normalized weight vector for the experts is λ=λ1,λ2,λ3,…,λkT. The strategy for complex spherical fuzzy N-soft-VIKOR (CSFNSf-VIKOR) method is described as follows.


### 3.1. Complex Spherical FuzzyN-Soft Performance Matrix

The alternatives are analyzed by decision-experts on the basis of the selected attributes and initially admeasured by numerical labels representing linguistic information relative to the MAGDM, such as 5 for “best,” 4 for “good,” 3 for “bad,” and so. These numerical labels are further epitomized byCSFNSfNs with constraints of grading criteria, which moves us towards the kCSFNSfSs classified by k decision-experts. Furthermore, the kCSFNSfSs regarding k decision-experts are systemized in the form of complex spherical fuzzy N-soft performance matrices CSFNSfPMsZs=Zbasm×v as follows:(21)Zs=c11s,ϕ11s,ψ11s,χ11sc21s,ϕ12s,ψ12s,χ12s…cv1s,ϕ1vs,ψ1vs,χ1vsc12s,ϕ21s,ψ21s,χ21sc22s,ϕ22s,ψ22s,χ22s…cv2s,ϕ2vs,ψ2vs,χ2vs⋮⋮⋱⋮c1ms,ϕm1s,ψm1s,χm1sc2ms,ϕm2s,ψm2s,χm2s…cvms,ϕmvs,ψmvs,χmvs,where Zbas=cabs,ϕbas,ψbas,χbas=cabs,tbasei2πζbas,qbasei2πϱbas,fbasei2πρbas, and b=1,2,3,…,m, a=1,2,3,…,v, and s=1,2,3,…,k stand for the alternatives Jb, attributes Xa, and decision-experts Ys, respectively.


### 3.2. Aggregated Complex Spherical FuzzyN-Soft Performance Matrix

All decision-experts are adequate to manipulate the individual opinion usingCSFNSfWA operator and resulting matrix known as aggregated complex spherical fuzzy N-soft performance matrix is ACSFNSfPM evaluated as follows:(22)Zba=CSFNSfAλZba1,Zba2,…,Zbak=λ1Zba1⊕λ2Zba2⊕⋯⊕λsZbak=maxs=1kcabs,1−∏s=1k1−tbas2λsei2π1−∏s=1k1−ζbas2λs,∏s=1kqbasei2π∏s=1kϱbas,∏s=1kfbasei2π∏s=1kρbas=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba.TheACSFNSfPM is as follows:(23)P=c11,ϕ11,ψ11,χ11c21,ϕ12,ψ12,χ12…ca1,ϕ1v,ψ1v,χ1vc12,ϕ21,ψ21,χ21c22,ϕ22,ψ22,χ22…ca2,ϕ2v,ψ2v,χ2v⋮⋮⋱⋮c1m,ϕm1,ψm1,χm1c2m,ϕm2,ψm2,χm2…cvm,ϕmv,ψmv,χmv.


### 3.3. Selection ofCSFNSf Best Value and CSFNSf Worst Value

TheCSFNSf best value is assessed by the following formula denoted by Z^a:(24)Z^a=maxbZba,ifXa∈ΩB,minbZba,ifXa∈ΩC,where Z^a=c^ab,t^baei2πζ^ba,q^baei2πϱ^ba,f^baei2πρ^ba.TheCSFNSf worst value is assessed by the following formula denoted by Zˇa:(25)Zˇa=maxbZba,ifXa∈ΩC,minbZba,ifXa∈ΩB,where Zˇa=cˇab,tˇbaei2πζˇba,qˇbaei2πϱˇba,fˇbaei2πρˇba.The score valueSZba and accuracy value AZba are utilized to compare the CSFNSfNs in ACSFNSfPM.


### 3.4. EvaluatingSb and ℝb

The normalized Euclidean distance is utilized to evaluate the group utility measureSb and individual regret measure ℝb with the provision of normalized weights of attributes as follows:(26)Sb=∑a=1vladZ^a,ZbadZˇa,Zba,(27)ℝb=max1<a<vladZ^a,ZbadZˇa,Zba.The normalized Euclidean distancedZ^a,Zba and dZˇa,Zba are calculated as follows:(28)dZ^a,Zba=14c^abN−12−cabN−122+t^ba2−tba22+q^ba2−qba22+f^ba2−fba22+ζ^ba2−ζba22+ϱ^ba2−ϱba22+ρ^ba2−ρba221/2,(29)dZˇa,Zba=14cˇabN−12−cabN−122+tˇba2−tba22+qˇba2−qba22+fˇba2−fba22+ζˇba2−ζba22+ϱˇba2−ϱba22+ρˇba2−ρba221/2.The optimal values ofSb and ℝb are as follows:(30)S^=minbSb,Sˇ=maxbSb,ℝ^=minbℝb,ℝˇ=maxbℝb.TheS^ and ℝ^ correspond to a maximum majority rule index and a minimum regret of opponent strategy, respectively.


### 3.5. Compromise Measureℚb

The configurations of utility measure and regret measure of feasible choiceJb link up for the compromise measure ℚb as follows:(31)ℚ=ξSb−S^bSˇb−S^b+1−ξℝb−ℝ^bℝˇb−ℝ^b,where ξ∈0,1 is the coefficient strategy of the majority of the attributes and in most cases taken as 0.5, usually for the sake of equal weightage of both the configurations, ξ=0.5. Moreover, ξ=1, representing that the compromise solution is biased towards the maximum group utility. On the other hand, ξ=0 shows the biasness towards minimum individual regret.


### 3.6. Ranking of Alternatives

The values of ranking measuresSb, ℝb, and ℚb, corresponding to each alternative, are arranged in ascending order so that we get three ranking lists that further play an important role in finding a compromise solution. Moreover, the alternative with minimum value regarding three ranking lists is considered as the best feasible option.


### 3.7. Compromise Solution

For compromise solution consisting of alternativeJ1 with a minimum value of ranking measure ℚ, the following conditions should hold, which are described as follows:ℂ1: acceptable advantage:(32)ℚJ2−ℚJ1≥1m−1,whereJ1 and J2 are the alternatives with the initial and subsequent position in the ranking list and m representing the number of alternatives.ℂ2: acceptable stability: J1 should be ranked first with respect to S or ℝ. The compromise solution within the proposed method can be assumed stable in all possible situations of “voting by majority rule ξ>0.5,” “by consensus ξ=0.5,” or “by veto ξ<0.5.”Moreover, if the conditionℂ1 is not satisfied, the compromise solution set contains the alternatives satisfying the following inequality:(33)ℚJb−ℚJ1<1m−1,∀1≤b≤m.But, if conditionℂ2 is not fulfilled, both alternatives J1 and J2 are known as compromise solutions of the MAGDM problem.The flowchart of the proposedCSFNSf-VIKOR method is given in Figure 1.Figure 1
Flowchart of theCSFNSf-VIKOR method.


## 3.1. Complex Spherical FuzzyN-Soft Performance Matrix

The alternatives are analyzed by decision-experts on the basis of the selected attributes and initially admeasured by numerical labels representing linguistic information relative to the MAGDM, such as 5 for “best,” 4 for “good,” 3 for “bad,” and so. These numerical labels are further epitomized byCSFNSfNs with constraints of grading criteria, which moves us towards the kCSFNSfSs classified by k decision-experts. Furthermore, the kCSFNSfSs regarding k decision-experts are systemized in the form of complex spherical fuzzy N-soft performance matrices CSFNSfPMsZs=Zbasm×v as follows:(21)Zs=c11s,ϕ11s,ψ11s,χ11sc21s,ϕ12s,ψ12s,χ12s…cv1s,ϕ1vs,ψ1vs,χ1vsc12s,ϕ21s,ψ21s,χ21sc22s,ϕ22s,ψ22s,χ22s…cv2s,ϕ2vs,ψ2vs,χ2vs⋮⋮⋱⋮c1ms,ϕm1s,ψm1s,χm1sc2ms,ϕm2s,ψm2s,χm2s…cvms,ϕmvs,ψmvs,χmvs,where Zbas=cabs,ϕbas,ψbas,χbas=cabs,tbasei2πζbas,qbasei2πϱbas,fbasei2πρbas, and b=1,2,3,…,m, a=1,2,3,…,v, and s=1,2,3,…,k stand for the alternatives Jb, attributes Xa, and decision-experts Ys, respectively.


## 3.2. Aggregated Complex Spherical FuzzyN-Soft Performance Matrix

All decision-experts are adequate to manipulate the individual opinion usingCSFNSfWA operator and resulting matrix known as aggregated complex spherical fuzzy N-soft performance matrix is ACSFNSfPM evaluated as follows:(22)Zba=CSFNSfAλZba1,Zba2,…,Zbak=λ1Zba1⊕λ2Zba2⊕⋯⊕λsZbak=maxs=1kcabs,1−∏s=1k1−tbas2λsei2π1−∏s=1k1−ζbas2λs,∏s=1kqbasei2π∏s=1kϱbas,∏s=1kfbasei2π∏s=1kρbas=cab,tbaei2πζba,qbaei2πϱba,fbaei2πρba.TheACSFNSfPM is as follows:(23)P=c11,ϕ11,ψ11,χ11c21,ϕ12,ψ12,χ12…ca1,ϕ1v,ψ1v,χ1vc12,ϕ21,ψ21,χ21c22,ϕ22,ψ22,χ22…ca2,ϕ2v,ψ2v,χ2v⋮⋮⋱⋮c1m,ϕm1,ψm1,χm1c2m,ϕm2,ψm2,χm2…cvm,ϕmv,ψmv,χmv.


## 3.3. Selection ofCSFNSf Best Value and CSFNSf Worst Value

TheCSFNSf best value is assessed by the following formula denoted by Z^a:(24)Z^a=maxbZba,ifXa∈ΩB,minbZba,ifXa∈ΩC,where Z^a=c^ab,t^baei2πζ^ba,q^baei2πϱ^ba,f^baei2πρ^ba.TheCSFNSf worst value is assessed by the following formula denoted by Zˇa:(25)Zˇa=maxbZba,ifXa∈ΩC,minbZba,ifXa∈ΩB,where Zˇa=cˇab,tˇbaei2πζˇba,qˇbaei2πϱˇba,fˇbaei2πρˇba.The score valueSZba and accuracy value AZba are utilized to compare the CSFNSfNs in ACSFNSfPM.


## 3.4. EvaluatingSb and ℝb

The normalized Euclidean distance is utilized to evaluate the group utility measureSb and individual regret measure ℝb with the provision of normalized weights of attributes as follows:(26)Sb=∑a=1vladZ^a,ZbadZˇa,Zba,(27)ℝb=max1<a<vladZ^a,ZbadZˇa,Zba.The normalized Euclidean distancedZ^a,Zba and dZˇa,Zba are calculated as follows:(28)dZ^a,Zba=14c^abN−12−cabN−122+t^ba2−tba22+q^ba2−qba22+f^ba2−fba22+ζ^ba2−ζba22+ϱ^ba2−ϱba22+ρ^ba2−ρba221/2,(29)dZˇa,Zba=14cˇabN−12−cabN−122+tˇba2−tba22+qˇba2−qba22+fˇba2−fba22+ζˇba2−ζba22+ϱˇba2−ϱba22+ρˇba2−ρba221/2.The optimal values ofSb and ℝb are as follows:(30)S^=minbSb,Sˇ=maxbSb,ℝ^=minbℝb,ℝˇ=maxbℝb.TheS^ and ℝ^ correspond to a maximum majority rule index and a minimum regret of opponent strategy, respectively.


## 3.5. Compromise Measureℚb

The configurations of utility measure and regret measure of feasible choiceJb link up for the compromise measure ℚb as follows:(31)ℚ=ξSb−S^bSˇb−S^b+1−ξℝb−ℝ^bℝˇb−ℝ^b,where ξ∈0,1 is the coefficient strategy of the majority of the attributes and in most cases taken as 0.5, usually for the sake of equal weightage of both the configurations, ξ=0.5. Moreover, ξ=1, representing that the compromise solution is biased towards the maximum group utility. On the other hand, ξ=0 shows the biasness towards minimum individual regret.


## 3.6. Ranking of Alternatives

The values of ranking measuresSb, ℝb, and ℚb, corresponding to each alternative, are arranged in ascending order so that we get three ranking lists that further play an important role in finding a compromise solution. Moreover, the alternative with minimum value regarding three ranking lists is considered as the best feasible option.


## 3.7. Compromise Solution

For compromise solution consisting of alternativeJ1 with a minimum value of ranking measure ℚ, the following conditions should hold, which are described as follows:ℂ1: acceptable advantage:(32)ℚJ2−ℚJ1≥1m−1,whereJ1 and J2 are the alternatives with the initial and subsequent position in the ranking list and m representing the number of alternatives.ℂ2: acceptable stability: J1 should be ranked first with respect to S or ℝ. The compromise solution within the proposed method can be assumed stable in all possible situations of “voting by majority rule ξ>0.5,” “by consensus ξ=0.5,” or “by veto ξ<0.5.”Moreover, if the conditionℂ1 is not satisfied, the compromise solution set contains the alternatives satisfying the following inequality:(33)ℚJb−ℚJ1<1m−1,∀1≤b≤m.But, if conditionℂ2 is not fulfilled, both alternatives J1 and J2 are known as compromise solutions of the MAGDM problem.The flowchart of the proposedCSFNSf-VIKOR method is given in Figure 1.Figure 1
Flowchart of theCSFNSf-VIKOR method.


## 4. Application to Group Decision-Making

In this section, the proposedCSFNSf-VIKOR method is carried out for the MAGDM problem in the domain of business administration. Precisely, the proposed CSFNSf-VIKOR method is implemented for the selection of firms for the Saudi oil refinery project in Pakistan.


### 4.1. Selection of Firm for the Saudi Oil Refinery Project in Pakistan

China-Pakistan Economic Corridor (CPEC) is a prudent economic project that upgrades the financial activities in Pakistan and China, serving as a gateway for the Middle East, Europe, and Africa. Specifically, there are advantageous aspects for the people of Gwadar as this project has industrious and everlasting benefits. In a sign of its enormous appliances, Saudi Arabia made a decision to establish a mega oil city in Pakistan at the spot of Gwadar and get involved in business and investment under the flagship of CPEC. A firm or company will be hired for the formation of the mega oil city, as the oil will be imported from Gulf and will be stored at the proposed Gwadar oil city, within the time period of one year. For this purpose, a team of three decision-makersY1, Y2, and Y3 are chosen, and the selected concise list of five firms is as follows:J1: Rabigh Refining and Petrochemical firmJ2: Petromin CompanyJ3: Saudi Aramco Total Refining and Petrochemical firmJ4: Alkhorayef Lubricants CompanyJ5: Yanbu Aramco Sinopec Refining Company (YASREF) Ltd.According to the proficiency and expertise of the MAGDM problem, weights for the decision-makersY1, Y2, and Y3 are 0.34, 0.31, and 0.35, respectively.Decision-makers thoroughly study the five firms according to the requirements of the project and captured data from the websiteshttp://www.dnb.com (Dun and Bradstreet Data Cloud) and http://www.investopedia.com, on the basis of precise characteristics or parameters discussed in Table 1.Table 1
Study of attributes for the MAGDM problem.
X1Strong risk ManagementIt is imperative to select a business firm that is financially strong and fully endorsed.Benefit typeX2Modern technology:It includes the latest apparatus and machinery, up-to-date software, and other high-tech innovations.Benefit typeX3Cost priceThis attribute includes initial cost, maintenance cost, and production cost. The firm with the lowest cost is preferable.Cost typeX4Plan and visionIt is necessary to choose a firm whose strategy related to the project is impactful and profitable.Benefit typeX5ExperienceIt includes business-related experience, knowledge, and expertise with fluctuating track record and success.Benefit typeDecision-makers assigned weight vector0.26,0.24,0.05,0.15,0.30T to the attributes taking into account the excellence and type of the attributes. For the selection of the best firm, we adopt the CSFNSf-VIKOR method as follows:Step 1: decision-makers modeled six-soft set, arranged in Table2, related to attributes of the MAGDM problem, where five asterisks mean “excellent performance,” four asterisks mean “great performance,” three asterisks mean “good performance,” two asterisks mean “average performance,” one asterisk means “bad performance,” and big dot means “very bad performance.”The criteria for the association of theCSFNs to 6-soft sets of decision-makers is given in Table 3, so that the opinions of the decision-makers in the form of CSFNSfPMsZ1, Z2, and Z3 are arranged in Tables 4, 5, and 6, simultaneously.Step 2: the judgments of all performance matrices of decision-makers amalgamated intoACSFNSfPM using CSFNSfAλ operator are summarized in Table 7.Step 3: for the evaluation of the worst and the best values, the score values of the entries ofACSFNSfPM are calculated, keeping in view the benefit and cost type criteria where all the attributes are benefit type except X3. The worst and best values are calculated from equations (24) and (25) that are shown in Table 8.Step 4: the normalized Euclidean distance is formulated by equations (28) and (29), which is utilized to measure the separation of best value from each entry of CSFNSfPM and also from worst value simultaneously, organized in Table 9. These distances are further employed in equations (26) and (27) to investigate the group utility measure S and individual regret measure ℝ, calculated in Table 10.Step 5: the compromise measureℚb related to each alternative or firm, with ξ=0.50, is estimated through equation (31) and pinned up in Table 10.Step 6: furthermore, the alternatives are ranked on the basis of the group utility, regret measure, and compromise measure that is illuminated in Table11.Table 2
Decision-makers opinions regarding attributes.
AttributesAlternativesY1Y2Y3X1J1∗=1∗∗=2⋅=0J2∗=1⋅=0∗=1J3∗∗∗∗∗=5∗∗∗∗=4∗∗∗∗∗=5J4∗∗∗∗=4∗∗∗=3∗∗∗∗=4J5∗∗∗=3∗∗=2∗∗∗=3X2J1⋅=0∗=1⋅=0J2∗=1∗=1∗=1J3∗∗∗∗=4∗∗∗∗∗=5∗∗∗∗∗=5J4∗∗∗=3∗∗∗∗=4∗∗∗∗=4J5∗∗=2∗∗∗=3∗∗∗=3X3J1⋅=0⋅=0⋅=0J2⋅=0∗=1∗=1J3∗∗∗∗=4∗∗∗∗∗=5∗∗∗∗∗=5J4∗∗∗∗=4∗∗∗∗∗=5∗∗∗∗=4J5∗∗∗=3∗∗∗=3∗∗∗=3X4J1⋅=0⋅=0∗=1J2∗∗=2∗=1∗∗=2J3∗∗∗∗∗=5∗∗∗∗∗=5∗∗∗∗=4J4∗∗∗∗=4∗∗∗=3∗∗∗∗∗=5J5∗∗∗=3∗∗∗=3∗∗=2X5J1∗=1⋅=0⋅=0J2∗=1∗=1∗=1J3∗∗∗∗∗=5∗∗∗∗∗=5∗∗∗∗∗=5J4∗∗∗∗=4∗∗∗∗=4∗∗∗∗=4J5∗∗∗=3∗∗∗=3∗∗∗=3Table 3
Grading criteria forCSF6SS.
hzw/JDegree of yesNeutral membershipDegree of noGradestba2πζbaqba2πϱbafba2πρbacab=00,0.30π0,0.01700,0.0340π0.90,1.001.80π,2.00πcab=10.15,0.300.30π,0.60π0.0170,0.08190.0340π,0.1638π0.75,0.901.50π,1.80πcab=20.30,0.500.60π,1.00π0,0.01700,0.0340π)0.50,0.751.00π,1.50πcab=30.50,0.751.00π,1.50π0.0170,0.08190.0340π,0.1638π0.30,0.500.60π,1.00πcab=40.75,0.901.50π,1.80π0,0.01700,0.0340π)0.15,0.300.30π,0.60πcab=50.90,1.001.80π,2.00π0.0170,0.08190.0340π,0.1638π0,0.150,0.30πTable 4
CSFNSfPM of decision-maker Y1.
X1X2X3X4X5J11,0.160ei0.340π,0.021ei0.044π,0.092ei1.860π0,0.070ei0.120π,0.015ei0.028π,0.985ei1.972π0,0.08ei0.140π,0.016ei0.030π,0.986ei1.974π0,0.090ei0.160π,0.017ei0.032π,0.987e1.976iπ1,0.170ei0.360π,0.022ei0.026π,0.930e1.880iπJ21,0.180ei0.380π,0.230ei0.480π,0.940ei1.900π1,0.190ei0.400π,0.025ei0.052π,0.950ei1.920π0,0.130ei0.240π,0.012ei0.026π,0.994ei1.990π2,0.310ei0.640π,0.011ei0.024π,0.076e1.54iπ1,0.190ei0.400π,0.024ei0.050π,0.950e1.920iπJ35,0.900ei1.820π,0.020ei0.060π,0.010ei0.040π4,0.760ei1.540π,0.010ei0.022π,0.200ei0.420π4,0.770ei1.560π,0.011ei0.024π,0.220ei0.420π5,0.910ei1.840π,0.018ei0.038π,0.010e0.040iπ5,0.920ei1.860π,0.019ei0.040π,0.020e0.060iπJ43,0.510ei1.04π,0.028ei0.058π,0.450ei0.920π2,0.320ei0.660π,0.013ei0.028π,0.770ei1.560π3,0.520ei1.060π,0.020ei0.060π,0.46ei0.940π3,0.530ei1.080π,0.031ei0.064π,0.470e0.960iπ3,0.540ei1.100π,0.032ei0.066π,0.480e0.980iπJ51,0.200ei0.42π,0.026ei0.054π,0.960ei1.940π2,0.350ei0.680π,0.014ei0.030π,0.780ei1.580π1,0.210ei0.420π,0.027ei0.056π,0.970ei1.920π2,0.340ei0.700π,0.015ei0.032π,0.790e1.600iπ2,0.350ei0.720π,0.016ei0.034π,0.810e1.640iπTable 5
CSFNSfPM of decision-maker Y2.
X1X2X3X4X5J12,0.360ei0.740π,0.012ei0.022π,0.820ei1.660π1,0.220ei0.460π,0.028ei0.058π,0.960ei1.900π0,0.100ei0.180π,0.013ei0.028π,0.989ei1.980π0,0.110ei0.200π,0.014ei0.030π,0.991e1.984iπ0,0.120ei0.220π,0.015ei0.032π,0.993e1.988iπJ20,0.110ei0.180π,0.013ei0.028π,0.990ei1.978π1,0.230ei0.480π,0.030ei0.062π,0.950ei1.880π1,0.240ei0.500π,0.031ei0.064π,0.940ei1.86π1,0.250ei0.520π,0.033ei0.068π,0.930e1.84iπ1,0.260ei0.540π,0.034ei0.070π,0.920e1.860iπJ34,0.820ei1.660π,0.017ei0.032π,0.250ei0.420π5,0.930ei1.88π,0.020ei0.420π,0.030ei0.080π5,0.940ei1.900π,0.021ei0.040π,0.040ei0.100π5,0.950ei1.920π,0.023ei0.048π,0.050e0.120iπ5,0.960ei1.940π,0.024ei0.050π,0.060e0.140iπJ42,0.390ei0.800π,0.014ei0.026π,0.840ei1.700π3,0.570ei1.160π,0.036ei0.074π,0.530ei1.040π3,0.580ei1.180π,0.037ei0.076π,0.530ei1.040π3,0.590ei1.200π,0.038ei0.078π,0.530e1.080iπ2,0.380ei0.780π,0.013ei0.024π,0.830e1.680iπJ51,0.270ei0.560π,0.036ei0.074π,0.940ei1.860π1,0.280ei0.580π,0.037ei0.076π,0.950ei1.88π2,0.410ei0.840π,0.015ei0.028π,0.850ei1.720π2,0.420ei0.860π,0.016ei0.034π,0.860e1.94iπ2,0.430ei0.880π,0.017ei0.032π,0.870e1.760iπTable 6
CSFNSfPM of decision-maker Y3.
X1X2X3X4X5J10,0.030ei0.040π,0.011ei0.060π,0.981ei1.964π0,0.040ei0.060π,0.012ei0.026π,0.982ei1.966π0,0.05ei0.080π,0.013ei0.028π,0.983ei0.968π1,0.270ei0.520π,0.040ei0.082π,0.960e1.900iπ0,0.060ei0.100π,0.014ei0.030π,0.984e1.970iπJ21,0.290ei0.480π,0.038ei0.078π,0.940ei1.86π1,0.280ei0.540π,0.039ei0.080π,0.950ei1.88π1,0.270ei0.520π,0.041ei0.084π,0.960e1.900π2,0.440ei0.900π,0.016ei0.030π,0.880e1.780iπ1,0.260ei0.500π,0.042ei0.086π,0.970e1.920iπJ35,0.990ei1.960π,0.071ei0.144π,0.100ei0.220π5,0.980ei1.940π,0.072ei0.146π,0.110ei0.240π5,0.970ei1.920π,0.073ei0.148π,0.120ei0.260π4,0.840ei1.700π,0.014ei0.026π,0.270e0.460iπ5,0.960ei1.900π,0.074ei0.150π,0.130e0.280iπJ43,0.630ei1.280π,0.043ei0.084π,0.560ei1.140π3,0.640ei1.300π,0.044ei0.086π,0.570ei1.160π3,0.650ei1.320π,0.043ei0.088π,0.580ei1.180π2,0.480ei0.980π,0.012ei0.022π,0.860e1.700iπ3,0.620ei1.260π,0.042ei0.082π,0.310e0.640iπJ52,0.450ei0.920π,0.015ei0.028π,0.890ei1.760π2,0.460ei0.940π,0.014ei0.026π,0.880ei1.740π2,0.470ei0.960π,0.013ei0.024π,0.870ei1.720π3,0.600ei1.220π,0.039ei0.082π,0.540e1.100iπ2,0.490ei0.940π,0.011ei0.020π,0.850e1.720iπTable 7
Aggregated complex spherical fuzzyN-soft performance matrix.
X1X2X3X4X5J12,0.2254ei0.4656π,0.0133ei0.0280π,0.09079ei1.8300π1,0.1321ei0.2698π,0.0168ei0.0340π,0.9761ei1.9472π0,0.0784ei0.1376π,0.0139ei0.0286π,0.9858ei1.9736π1,0.1802ei0.3426π,0.0216ei0.0436π,0.9786e1.9516iπ1,0.1249ei0.2506π,0.0166ei0.0352π,0.9680e1.9444iπJ21,0.2114ei0.4130π,0.0229ei0.0480π,0.9552ei1.9094π1,0.2372ei0.4776π,0.0309ei0.0638π,0.9500ei1.8934π1,0.2222ei0.4392π,0.0247ei0.0518π,0.9651ei1.9174π2,0.3487ei0.7170π,0.0176ei0.0358π,0.8516e1.7118iπ1,0.2387ei0.4822π,0.0325ei0.0670π,0.9475e1.9012iπJ35,0.9475ei1.8722π,0.0296ei0.0678π,0.0607ei0.1608π5,0.9329ei1.8540π,0.0247ei0.0520π,0.0901ei0.2064π5,0.9270ei1.8492π,0.0260ei0.0546π,0.1049ei0.2274π5,0.9088ei1.8176π,0.0178ei0.0356π,0.0522e1.4160iπ5,0.9494ei1.9044π,0.0328ei0.0680π,0.0541e1.3360iπJ43,0.5305ei1.0812π,0.0262ei0.0514π,0.5895ei1.1996π3,0.5288ei1.0976π,0.0273ei0.0560π,0.6172ei1.2402π3,0.5892ei1.1984π,0.0358ei0.0738π,0.5182ei1.0564π3,0.5344ei1.0890π,0.0236ei0.0468π,0.6027e1.2054iπ3,0.5337ei1.0876π,0.0266ei0.0520π,0.5966e1.2136iπJ52,0.3316ei0.6826π,0.0237ei0.0472π,0.9288ei1.8505π2,0.3766ei0.7608π,0.0189ei0.0380π,0.8649ei1.7248π2,0.3842ei0.7878π,0.0174ei0.0336π,0.8963ei1.7854π3,0.4769ei0.9738π,0.0214ei0.0426π,0.7099e1.4402iπ2,0.4294ei0.8583π,0.0142ei0.0276π,0.8422e1.7044iπTable 8
CSFNSf best and worst values related to the attributes.
AttributeBest valueWorst valueX15,0.9475ei1.8722π,0.0296ei0.0678π,0.0607ei0.1608π1,0.2114ei0.4130π,0.0229ei0.0480π,0.9552ei1.9094πX25,0.9329ei1.8540π,0.0247ei0.0520π,0.0901ei0.2064π1,0.1321ei0.2698π,0.0168ei0.0340π,0.9761ei1.9472πX30,0.0784ei0.1376π,0.0139ei0.0286π,0.9858ei1.9736π5,0.9270ei1.8492π,0.0260ei0.0546π,0.1049ei0.2274πX45,0.9088ei1.8176π,0.0178ei0.0356π,0.0522e1.4160iπ1,0.1802ei0.3426π,0.0216ei0.0436π,0.9786e1.9516iπX55,0.9494ei1.9044π,0.0328ei0.0680π,0.0541e1.3360iπ1,0.1249ei0.2506π,0.0166ei0.0352π,0.9680e1.9444iπTable 9
Normalized Euclidean distance.
AlternativedZ^1,Zb1dZ^2,Zb2dZ^3,Zb3dZ^4,Zb4dZ^5,Zb5J10.93040.96600.000491.00351.0316J20.99860.91240.04970.83190.9954J3001.036200J40.58600.55960.58060.56280.5957J50.91690.78900.17850.64300.8308Table 10
The values ofSb, ℝb, and ℚb.
AlternativeSbℝbℚbJ10.93220.24220.9014J20.90290.28940.9833J30.050.050J40.57690.17320.5559J50.78100.24160.7418Table 11
Ranking of firms.
AlternativesJ1J2J3J4J5Ranking order ofS54123Ranking order ofℝ54123Ranking order ofℚ54123The firmJ3 has rank order 1 on the basis of the compromise measure ℚ, and also, the remaining two conditions of CSFNSf-VIKOR method are satisfied and calculated as follows:(1)
ℚJ4−ℚJ3=0.5559−0=0.5559≥1/6−1=0.2(2)
J3 is the best firm regarding S and ℝ, and ranking orders are J3≻J4≻J5≻J1≻J2Thus, the firmJ3 is best for the establishment of the Saudi oil refinery project in Pakistan.


## 4.1. Selection of Firm for the Saudi Oil Refinery Project in Pakistan

China-Pakistan Economic Corridor (CPEC) is a prudent economic project that upgrades the financial activities in Pakistan and China, serving as a gateway for the Middle East, Europe, and Africa. Specifically, there are advantageous aspects for the people of Gwadar as this project has industrious and everlasting benefits. In a sign of its enormous appliances, Saudi Arabia made a decision to establish a mega oil city in Pakistan at the spot of Gwadar and get involved in business and investment under the flagship of CPEC. A firm or company will be hired for the formation of the mega oil city, as the oil will be imported from Gulf and will be stored at the proposed Gwadar oil city, within the time period of one year. For this purpose, a team of three decision-makersY1, Y2, and Y3 are chosen, and the selected concise list of five firms is as follows:J1: Rabigh Refining and Petrochemical firmJ2: Petromin CompanyJ3: Saudi Aramco Total Refining and Petrochemical firmJ4: Alkhorayef Lubricants CompanyJ5: Yanbu Aramco Sinopec Refining Company (YASREF) Ltd.According to the proficiency and expertise of the MAGDM problem, weights for the decision-makersY1, Y2, and Y3 are 0.34, 0.31, and 0.35, respectively.Decision-makers thoroughly study the five firms according to the requirements of the project and captured data from the websiteshttp://www.dnb.com (Dun and Bradstreet Data Cloud) and http://www.investopedia.com, on the basis of precise characteristics or parameters discussed in Table 1.Table 1
Study of attributes for the MAGDM problem.
X1Strong risk ManagementIt is imperative to select a business firm that is financially strong and fully endorsed.Benefit typeX2Modern technology:It includes the latest apparatus and machinery, up-to-date software, and other high-tech innovations.Benefit typeX3Cost priceThis attribute includes initial cost, maintenance cost, and production cost. The firm with the lowest cost is preferable.Cost typeX4Plan and visionIt is necessary to choose a firm whose strategy related to the project is impactful and profitable.Benefit typeX5ExperienceIt includes business-related experience, knowledge, and expertise with fluctuating track record and success.Benefit typeDecision-makers assigned weight vector0.26,0.24,0.05,0.15,0.30T to the attributes taking into account the excellence and type of the attributes. For the selection of the best firm, we adopt the CSFNSf-VIKOR method as follows:Step 1: decision-makers modeled six-soft set, arranged in Table2, related to attributes of the MAGDM problem, where five asterisks mean “excellent performance,” four asterisks mean “great performance,” three asterisks mean “good performance,” two asterisks mean “average performance,” one asterisk means “bad performance,” and big dot means “very bad performance.”The criteria for the association of theCSFNs to 6-soft sets of decision-makers is given in Table 3, so that the opinions of the decision-makers in the form of CSFNSfPMsZ1, Z2, and Z3 are arranged in Tables 4, 5, and 6, simultaneously.Step 2: the judgments of all performance matrices of decision-makers amalgamated intoACSFNSfPM using CSFNSfAλ operator are summarized in Table 7.Step 3: for the evaluation of the worst and the best values, the score values of the entries ofACSFNSfPM are calculated, keeping in view the benefit and cost type criteria where all the attributes are benefit type except X3. The worst and best values are calculated from equations (24) and (25) that are shown in Table 8.Step 4: the normalized Euclidean distance is formulated by equations (28) and (29), which is utilized to measure the separation of best value from each entry of CSFNSfPM and also from worst value simultaneously, organized in Table 9. These distances are further employed in equations (26) and (27) to investigate the group utility measure S and individual regret measure ℝ, calculated in Table 10.Step 5: the compromise measureℚb related to each alternative or firm, with ξ=0.50, is estimated through equation (31) and pinned up in Table 10.Step 6: furthermore, the alternatives are ranked on the basis of the group utility, regret measure, and compromise measure that is illuminated in Table11.Table 2
Decision-makers opinions regarding attributes.
AttributesAlternativesY1Y2Y3X1J1∗=1∗∗=2⋅=0J2∗=1⋅=0∗=1J3∗∗∗∗∗=5∗∗∗∗=4∗∗∗∗∗=5J4∗∗∗∗=4∗∗∗=3∗∗∗∗=4J5∗∗∗=3∗∗=2∗∗∗=3X2J1⋅=0∗=1⋅=0J2∗=1∗=1∗=1J3∗∗∗∗=4∗∗∗∗∗=5∗∗∗∗∗=5J4∗∗∗=3∗∗∗∗=4∗∗∗∗=4J5∗∗=2∗∗∗=3∗∗∗=3X3J1⋅=0⋅=0⋅=0J2⋅=0∗=1∗=1J3∗∗∗∗=4∗∗∗∗∗=5∗∗∗∗∗=5J4∗∗∗∗=4∗∗∗∗∗=5∗∗∗∗=4J5∗∗∗=3∗∗∗=3∗∗∗=3X4J1⋅=0⋅=0∗=1J2∗∗=2∗=1∗∗=2J3∗∗∗∗∗=5∗∗∗∗∗=5∗∗∗∗=4J4∗∗∗∗=4∗∗∗=3∗∗∗∗∗=5J5∗∗∗=3∗∗∗=3∗∗=2X5J1∗=1⋅=0⋅=0J2∗=1∗=1∗=1J3∗∗∗∗∗=5∗∗∗∗∗=5∗∗∗∗∗=5J4∗∗∗∗=4∗∗∗∗=4∗∗∗∗=4J5∗∗∗=3∗∗∗=3∗∗∗=3Table 3
Grading criteria forCSF6SS.
hzw/JDegree of yesNeutral membershipDegree of noGradestba2πζbaqba2πϱbafba2πρbacab=00,0.30π0,0.01700,0.0340π0.90,1.001.80π,2.00πcab=10.15,0.300.30π,0.60π0.0170,0.08190.0340π,0.1638π0.75,0.901.50π,1.80πcab=20.30,0.500.60π,1.00π0,0.01700,0.0340π)0.50,0.751.00π,1.50πcab=30.50,0.751.00π,1.50π0.0170,0.08190.0340π,0.1638π0.30,0.500.60π,1.00πcab=40.75,0.901.50π,1.80π0,0.01700,0.0340π)0.15,0.300.30π,0.60πcab=50.90,1.001.80π,2.00π0.0170,0.08190.0340π,0.1638π0,0.150,0.30πTable 4
CSFNSfPM of decision-maker Y1.
X1X2X3X4X5J11,0.160ei0.340π,0.021ei0.044π,0.092ei1.860π0,0.070ei0.120π,0.015ei0.028π,0.985ei1.972π0,0.08ei0.140π,0.016ei0.030π,0.986ei1.974π0,0.090ei0.160π,0.017ei0.032π,0.987e1.976iπ1,0.170ei0.360π,0.022ei0.026π,0.930e1.880iπJ21,0.180ei0.380π,0.230ei0.480π,0.940ei1.900π1,0.190ei0.400π,0.025ei0.052π,0.950ei1.920π0,0.130ei0.240π,0.012ei0.026π,0.994ei1.990π2,0.310ei0.640π,0.011ei0.024π,0.076e1.54iπ1,0.190ei0.400π,0.024ei0.050π,0.950e1.920iπJ35,0.900ei1.820π,0.020ei0.060π,0.010ei0.040π4,0.760ei1.540π,0.010ei0.022π,0.200ei0.420π4,0.770ei1.560π,0.011ei0.024π,0.220ei0.420π5,0.910ei1.840π,0.018ei0.038π,0.010e0.040iπ5,0.920ei1.860π,0.019ei0.040π,0.020e0.060iπJ43,0.510ei1.04π,0.028ei0.058π,0.450ei0.920π2,0.320ei0.660π,0.013ei0.028π,0.770ei1.560π3,0.520ei1.060π,0.020ei0.060π,0.46ei0.940π3,0.530ei1.080π,0.031ei0.064π,0.470e0.960iπ3,0.540ei1.100π,0.032ei0.066π,0.480e0.980iπJ51,0.200ei0.42π,0.026ei0.054π,0.960ei1.940π2,0.350ei0.680π,0.014ei0.030π,0.780ei1.580π1,0.210ei0.420π,0.027ei0.056π,0.970ei1.920π2,0.340ei0.700π,0.015ei0.032π,0.790e1.600iπ2,0.350ei0.720π,0.016ei0.034π,0.810e1.640iπTable 5
CSFNSfPM of decision-maker Y2.
X1X2X3X4X5J12,0.360ei0.740π,0.012ei0.022π,0.820ei1.660π1,0.220ei0.460π,0.028ei0.058π,0.960ei1.900π0,0.100ei0.180π,0.013ei0.028π,0.989ei1.980π0,0.110ei0.200π,0.014ei0.030π,0.991e1.984iπ0,0.120ei0.220π,0.015ei0.032π,0.993e1.988iπJ20,0.110ei0.180π,0.013ei0.028π,0.990ei1.978π1,0.230ei0.480π,0.030ei0.062π,0.950ei1.880π1,0.240ei0.500π,0.031ei0.064π,0.940ei1.86π1,0.250ei0.520π,0.033ei0.068π,0.930e1.84iπ1,0.260ei0.540π,0.034ei0.070π,0.920e1.860iπJ34,0.820ei1.660π,0.017ei0.032π,0.250ei0.420π5,0.930ei1.88π,0.020ei0.420π,0.030ei0.080π5,0.940ei1.900π,0.021ei0.040π,0.040ei0.100π5,0.950ei1.920π,0.023ei0.048π,0.050e0.120iπ5,0.960ei1.940π,0.024ei0.050π,0.060e0.140iπJ42,0.390ei0.800π,0.014ei0.026π,0.840ei1.700π3,0.570ei1.160π,0.036ei0.074π,0.530ei1.040π3,0.580ei1.180π,0.037ei0.076π,0.530ei1.040π3,0.590ei1.200π,0.038ei0.078π,0.530e1.080iπ2,0.380ei0.780π,0.013ei0.024π,0.830e1.680iπJ51,0.270ei0.560π,0.036ei0.074π,0.940ei1.860π1,0.280ei0.580π,0.037ei0.076π,0.950ei1.88π2,0.410ei0.840π,0.015ei0.028π,0.850ei1.720π2,0.420ei0.860π,0.016ei0.034π,0.860e1.94iπ2,0.430ei0.880π,0.017ei0.032π,0.870e1.760iπTable 6
CSFNSfPM of decision-maker Y3.
X1X2X3X4X5J10,0.030ei0.040π,0.011ei0.060π,0.981ei1.964π0,0.040ei0.060π,0.012ei0.026π,0.982ei1.966π0,0.05ei0.080π,0.013ei0.028π,0.983ei0.968π1,0.270ei0.520π,0.040ei0.082π,0.960e1.900iπ0,0.060ei0.100π,0.014ei0.030π,0.984e1.970iπJ21,0.290ei0.480π,0.038ei0.078π,0.940ei1.86π1,0.280ei0.540π,0.039ei0.080π,0.950ei1.88π1,0.270ei0.520π,0.041ei0.084π,0.960e1.900π2,0.440ei0.900π,0.016ei0.030π,0.880e1.780iπ1,0.260ei0.500π,0.042ei0.086π,0.970e1.920iπJ35,0.990ei1.960π,0.071ei0.144π,0.100ei0.220π5,0.980ei1.940π,0.072ei0.146π,0.110ei0.240π5,0.970ei1.920π,0.073ei0.148π,0.120ei0.260π4,0.840ei1.700π,0.014ei0.026π,0.270e0.460iπ5,0.960ei1.900π,0.074ei0.150π,0.130e0.280iπJ43,0.630ei1.280π,0.043ei0.084π,0.560ei1.140π3,0.640ei1.300π,0.044ei0.086π,0.570ei1.160π3,0.650ei1.320π,0.043ei0.088π,0.580ei1.180π2,0.480ei0.980π,0.012ei0.022π,0.860e1.700iπ3,0.620ei1.260π,0.042ei0.082π,0.310e0.640iπJ52,0.450ei0.920π,0.015ei0.028π,0.890ei1.760π2,0.460ei0.940π,0.014ei0.026π,0.880ei1.740π2,0.470ei0.960π,0.013ei0.024π,0.870ei1.720π3,0.600ei1.220π,0.039ei0.082π,0.540e1.100iπ2,0.490ei0.940π,0.011ei0.020π,0.850e1.720iπTable 7
Aggregated complex spherical fuzzyN-soft performance matrix.
X1X2X3X4X5J12,0.2254ei0.4656π,0.0133ei0.0280π,0.09079ei1.8300π1,0.1321ei0.2698π,0.0168ei0.0340π,0.9761ei1.9472π0,0.0784ei0.1376π,0.0139ei0.0286π,0.9858ei1.9736π1,0.1802ei0.3426π,0.0216ei0.0436π,0.9786e1.9516iπ1,0.1249ei0.2506π,0.0166ei0.0352π,0.9680e1.9444iπJ21,0.2114ei0.4130π,0.0229ei0.0480π,0.9552ei1.9094π1,0.2372ei0.4776π,0.0309ei0.0638π,0.9500ei1.8934π1,0.2222ei0.4392π,0.0247ei0.0518π,0.9651ei1.9174π2,0.3487ei0.7170π,0.0176ei0.0358π,0.8516e1.7118iπ1,0.2387ei0.4822π,0.0325ei0.0670π,0.9475e1.9012iπJ35,0.9475ei1.8722π,0.0296ei0.0678π,0.0607ei0.1608π5,0.9329ei1.8540π,0.0247ei0.0520π,0.0901ei0.2064π5,0.9270ei1.8492π,0.0260ei0.0546π,0.1049ei0.2274π5,0.9088ei1.8176π,0.0178ei0.0356π,0.0522e1.4160iπ5,0.9494ei1.9044π,0.0328ei0.0680π,0.0541e1.3360iπJ43,0.5305ei1.0812π,0.0262ei0.0514π,0.5895ei1.1996π3,0.5288ei1.0976π,0.0273ei0.0560π,0.6172ei1.2402π3,0.5892ei1.1984π,0.0358ei0.0738π,0.5182ei1.0564π3,0.5344ei1.0890π,0.0236ei0.0468π,0.6027e1.2054iπ3,0.5337ei1.0876π,0.0266ei0.0520π,0.5966e1.2136iπJ52,0.3316ei0.6826π,0.0237ei0.0472π,0.9288ei1.8505π2,0.3766ei0.7608π,0.0189ei0.0380π,0.8649ei1.7248π2,0.3842ei0.7878π,0.0174ei0.0336π,0.8963ei1.7854π3,0.4769ei0.9738π,0.0214ei0.0426π,0.7099e1.4402iπ2,0.4294ei0.8583π,0.0142ei0.0276π,0.8422e1.7044iπTable 8
CSFNSf best and worst values related to the attributes.
AttributeBest valueWorst valueX15,0.9475ei1.8722π,0.0296ei0.0678π,0.0607ei0.1608π1,0.2114ei0.4130π,0.0229ei0.0480π,0.9552ei1.9094πX25,0.9329ei1.8540π,0.0247ei0.0520π,0.0901ei0.2064π1,0.1321ei0.2698π,0.0168ei0.0340π,0.9761ei1.9472πX30,0.0784ei0.1376π,0.0139ei0.0286π,0.9858ei1.9736π5,0.9270ei1.8492π,0.0260ei0.0546π,0.1049ei0.2274πX45,0.9088ei1.8176π,0.0178ei0.0356π,0.0522e1.4160iπ1,0.1802ei0.3426π,0.0216ei0.0436π,0.9786e1.9516iπX55,0.9494ei1.9044π,0.0328ei0.0680π,0.0541e1.3360iπ1,0.1249ei0.2506π,0.0166ei0.0352π,0.9680e1.9444iπTable 9
Normalized Euclidean distance.
AlternativedZ^1,Zb1dZ^2,Zb2dZ^3,Zb3dZ^4,Zb4dZ^5,Zb5J10.93040.96600.000491.00351.0316J20.99860.91240.04970.83190.9954J3001.036200J40.58600.55960.58060.56280.5957J50.91690.78900.17850.64300.8308Table 10
The values ofSb, ℝb, and ℚb.
AlternativeSbℝbℚbJ10.93220.24220.9014J20.90290.28940.9833J30.050.050J40.57690.17320.5559J50.78100.24160.7418Table 11
Ranking of firms.
AlternativesJ1J2J3J4J5Ranking order ofS54123Ranking order ofℝ54123Ranking order ofℚ54123The firmJ3 has rank order 1 on the basis of the compromise measure ℚ, and also, the remaining two conditions of CSFNSf-VIKOR method are satisfied and calculated as follows:(1)
ℚJ4−ℚJ3=0.5559−0=0.5559≥1/6−1=0.2(2)
J3 is the best firm regarding S and ℝ, and ranking orders are J3≻J4≻J5≻J1≻J2Thus, the firmJ3 is best for the establishment of the Saudi oil refinery project in Pakistan.


## 5. Sensitivity Test

In the sensitivity test, the role of coefficient strategyξ playing as the weight of group utility measure and 1−ξ as the weight of individual regret measure is investigated to demonstrate the ability of the proposed CSFNSf-VIKOR model and the stability of the solution, which is evaluated by applying the presented methodology.ξ is assigned different values from the unit interval [0, 1], and then we analyze the strength and potency of the computed decisions from the proposed methodology. In this case, it is assumed that the variation in the coefficient strategy weight occurs because the MAGDM panel can prioritize both (group utility or individual regret) measures, according to the future necessities of the proposed case study. However, ξ∈0.6,1 illustrates that the compromise solution is subjective to maximum group utility, and ξ∈0,0.4 represents the biasness towards minimum individual regret.Figure2 illustrate that the variation in ξ from 0 to 1 affects the value of the compromise measure ℚ of alternatives, but the ranking order of the alternatives and the best solution (i.e., J3) are static in all cases.Figure 2
Sensitivity test.


## 6. Comparative Analysis

This section elaborates the superiority and caliber of the proposedCSFNSf-VIKOR method by solving the proposed MAGDM problem “Selection of firm for Saudi oil refinery project in Pakistan” by complex spherical fuzzy-VIKOR (CSF-VIKOR) method, proposed by Akram et al. [33]. For the selection of the best firm, we adopt the CSF-VIKOR method as follows:Step 1: the initial judgments of the decision-makers are in the form of grade level as shown in Table2 that further transformed into CSFNs, following the grading criteria defined in Table 3. The personal assessment of the decision-makers related to the parameters are denoted by CSFNSfPMs (Z1, Z2, and Z3) and arranged in Tables 12, 13, and 14 respectively.Step 2: the aggregated complex spherical fuzzy performance matrixACSFPM securing the collective opinion of the decision-makers through a complex spherical fuzzy weighted average CSFWA operator is employing as follows:(34)Zba=1−∏s=1k1−tbas2λsei2π1−∏s=1k1−ζbas2λs,∏s=1kqbasei2π∏s=1kϱbas,∏s=1kfbasei2π∏s=1kρbas.TheACSFPM shown in Table 15.Step 3: furthermore, for the evaluation of worst and best attributes inACSFPM, equations (24) and (25) are brought into play, despite that the comparison of complex spherical fuzzy numbers in ACSFPM is computed by exploiting the score function of each entry as follows:(35)ScZba=tba2−qba2−fba2+ζba2−ϱba2−ρba2.CSF’s best and worst values related to the attributes are given in Table16.Step 4: the group utility measureS and individual regret measure ℝ of each alternative Jb are valuated through equations (26) and (27) in the light of normalized distance measures, indicated in the following equations and worthy attributes.(36)dZ^a,Zba=13t^ba2−tba22+q^ba2−qba22+f^ba2−fba22+ζ^ba2−ζba22+ϱ^ba2−ϱba22+ρ^ba2−ρba22,(37)dZˇa,Zba=13tˇba2−tba22+qˇba2−qba22+fˇba2−fba22+ζˇba2−ζba22+ϱˇba2−ϱba22+ρˇba2−ρba22.The normalized distance in equations (36) and (37) is symmetric as well as it gives positive values corresponding to each entry in ACSFPM, listed in Table 17. In addition to that, the group utility measure S, individual regret measure ℝ and compromise ranking ℚ (around with equation (31) and ξ=0.50) are calculated in Table 18.Step 5: following the values ofS, ℝ, and ℚ from Table 1 in ascending order, the priority order of alternatives is presented in Table 19.Step 6: the firmJ3 has rank order one on the basis of the compromise measure ℚ, and also, the remaining two conditions of the CSF-VIKOR method are satisfied and calculated as follows:(1)
ℚJ4−ℚJ3=0.5262−0=0.5262≥1/6−1=0.2(2)
J3 is best firm regarding S and ℝTable 12
Complex spherical fuzzy performance matrix of decision-makerY1.
X1X2X3X4X5J10.160ei0.340π,0.021ei0.044π,0.092ei1.860π0.070ei0.120π,0.015ei0.028π,0.985ei1.972π0.08ei0.140π,0.016ei0.030π,0.986ei1.974π0.09ei0.160π,0.017ei0.032π,0.987e1.976iπ0.170ei0.360π,0.022ei0.026π,0.930e1.88iπJ20.180ei0.380π,0.230ei0.480π,0.940ei1.900π0.190ei0.400π,0.025ei0.052π,0.950ei1.920π0.130ei0.240π,0.012ei0.026π,0.994ei1.990π0.310ei0.640π,0.011ei0.024π,0.076e1.54iπ0.190ei0.400π,0.024ei0.050π,0.950e1.920iπJ30.900ei1.820π,0.02ei0.060π,0.010ei0.040π0.760ei1.540π,0.010ei0.022π,0.200ei0.420π0.770ei1.560π,0.011ei0.024π,0.220ei0.420π0.910ei1.840π,0.018ei0.038π,0.010e0.040iπ0.920ei1.860π,0.019ei0.040π,0.020e0.060iπJ40.510ei1.04π,0.028ei0.058π,0.450ei0.920π0.320ei0.660π,0.013ei0.028π,0.770ei1.560π0.520ei1.060π,0.020ei0.060π,0.46ei0.940π0.530ei1.080π,0.031ei0.064π,0.470e0.960iπ0.540ei1.100π,0.032ei0.066π,0.480e0.980iπJ50.200ei0.42π,0.026ei0.054π,0.960ei1.94π0.350ei0.680π,0.014ei0.030π,0.780ei1.58π0.210ei0.420π,0.027ei0.056π,0.970ei1.92π0.340ei0.700π,0.015ei0.032π,0.790e1.600iπ0.350ei0.720π,0.016ei0.034π,0.810e1.64iπTable 13
Complex spherical fuzzy performance matrix of decision-makerY2.
X1X2X3X4X5J10.360ei0.740π,0.012ei0.022π,0.820ei1.660π0.220ei0.460π,0.028ei0.058π,0.960ei1.900π0.100ei0.180π,0.013ei0.028π,0.989ei1.980π0.110ei0.200π,0.014ei0.030π,0.991e1.984iπ0.120ei0.220π,0.015ei0.032π,0.993e1.988iπJ20.110ei0.180π,0.013ei0.028π,0.990ei1.978π0.230ei0.480π,0.030ei0.062π,0.950ei1.88π0.240ei0.500π,0.031ei0.064π,0.940ei1.86π0.250ei0.520π,0.033ei0.068π,0.930e1.84iπ0.260ei0.540π,0.034ei0.070π,0.920e1.860iπJ30.820ei1.660π,0.017ei0.032π,0.250ei0.420π0.930ei1.88π,0.020ei0.420π,0.030ei0.080π0.940ei1.900π,0.021ei0.040π,0.040ei0.100π0.950ei1.920π,0.023ei0.048π,0.050e0.120iπ0.960ei1.940π,0.024ei0.050π,0.060e0.140iπJ40.390ei0.800π,0.014ei0.026π,0.840ei1.700π0.570ei1.16π,0.036ei0.074π,0.530ei1.04π0.580ei1.18π,0.037ei0.076π,0.530ei1.04π0.590ei1.200π,0.038ei0.078π,0.530e1.08iπ0.380ei0.78π,0.013ei0.024π,0.830e1.680iπJ50.27ei0.560π,0.036ei0.074π,0.940ei1.860π0.280ei0.580π,0.037ei0.076π,0.950ei1.88π0.410ei0.840π,0.015ei0.028π,0.850ei1.72π0.420ei0.860π,0.016ei0.034π,0.860e1.94iπ0.430ei0.880π,0.017ei0.032π,0.870e1.76iπTable 14
Complex spherical fuzzy performance matrix of decision-makerY3.
X1X2X3X4X5J10.030ei0.040π,0.011ei0.060π,0.981ei1.964π0.040ei0.060π,0.012ei0.026π,0.982ei1.966π0.05ei0.080π,0.013ei0.028π,0.983ei0.968π0.270ei0.520π,0.040ei0.082π,0.960e1.900iπ0.060ei0.100π,0.014ei0.030π,0.984e1.970iπJ20.290ei0.480π,0.038ei0.078π,0.940ei1.86π0.280ei0.540π,0.039ei0.080π,0.950ei1.88π0.270ei0.52π,0.041ei0.084π,0.960ei1.900π0.440ei0.900π,0.016ei0.030π,0.880e1.78iπ0.260ei0.500π,0.042ei0.086π,0.970e1.92iπJ30.990ei1.960π,0.071ei0.144π,0.100ei0.220π0.980ei1.940π,0.072ei0.146π,0.110ei0.240π0.970ei1.920π,0.073ei0.148π,0.120ei0.260π0.840ei1.700π,0.014ei0.026π,0.270e0.460iπ0.960ei1.900π,0.074ei0.150π,0.130e0.280iπJ40.630ei1.280π,0.043ei0.084π,0.560ei1.140π0.640ei1.300π,0.044ei0.086π,0.570ei1.160π0.650ei1.320π,0.043ei0.088π,0.580ei1.180π0.480ei0.980π,0.012ei0.022π,0.860e1.700iπ0.620ei1.260π,0.042ei0.082π,0.310e0.640iπJ50.450ei0.920π,0.015ei0.028π,0.890ei1.760π0.460ei0.940π,0.014ei0.026π,0.880ei1.740π0.470ei0.960π,0.013ei0.024π,0.870ei1.720π0.600ei1.220π,0.039ei0.082π,0.540e1.100iπ0.490ei0.940π,0.011ei0.020π,0.850e1.720iπTable 15
Aggregated complex spherical fuzzy performance matrix.
X1X2X3X4X5J10.2254ei0.4656π,0.0133ei0.0280π,0.09079ei1.8300π0.1321ei0.2698π,0.0168ei0.0340π,0.9761ei1.9472π0.0784ei0.1376π,0.0139ei0.0286π,0.9858ei1.9736π0.1802ei0.3426π,0.0216ei0.0436π,0.9786e1.9516iπ0.1249ei0.2506π,0.0166ei0.0352π,0.9680e1.9444iπJ20.2114ei0.4130π,0.0229ei0.0480π,0.9552ei1.9094π0.2372ei0.4776π,0.0309ei0.0638π,0.9500ei1.8934π0.2222ei0.4392π,0.0247ei0.0518π,0.9651ei1.9174π0.3487ei0.7170π,0.0176ei0.0358π,0.8516e1.7118iπ0.2387ei0.4822π,0.0325ei0.0670π,0.9475e1.9012iπJ30.9475ei1.8722π,0.0296ei0.0678π,0.0607ei0.1608π0.9329ei1.8540π,0.0247ei0.0520π,0.0901ei0.2064π0.9270ei1.8492π,0.0260ei0.0546π,0.1049ei0.2274π0.9088ei1.8176π,0.0178ei0.0356π,0.0522e1.4160iπ0.9494ei1.9044π,0.0328ei0.0680π,0.0541e1.3360iπJ40.5305ei1.0812π,0.0262ei0.0514π,0.5895ei1.1996π0.5288ei1.0976π,0.0273ei0.0560π,0.6172ei1.2402π0.5892ei1.1984π,0.0358ei0.0738π,0.5182ei1.0564π0.5344ei1.0890π,0.0236ei0.0468π,0.6027e1.2054iπ0.5337ei1.0876π,0.0266ei0.0520π,0.5966e1.2136iπJ50.3316ei0.6826π,0.0237ei0.0472π,0.9288ei1.8505π0.3766ei0.7608π,0.0189ei0.0380π,0.8649ei1.7248π0.3842ei0.7878π,0.0174ei0.0336π,0.8963ei1.7854π0.4769ei0.9738π,0.0214ei0.0426π,0.7099e1.4402iπ0.4294ei0.8583π,0.0142ei0.0276π,0.8422e1.7044iπTable 16
CSF best and worst values related to the attributes.
AttributeBest valueWorst valueX10.9475ei1.8722π,0.0296ei0.0678π,0.0607ei0.1608π0.2114ei0.4130π,0.0229ei0.0480π,0.9552ei1.9094πX20.9329ei1.8540π,0.0247ei0.0520π,0.0901ei0.2064π0.1321ei0.2698π,0.0168ei0.0340π,0.9761ei1.9472πX30.0784ei0.1376π,0.0139ei0.0286π,0.9858ei1.9736π0.9270ei1.8492π,0.0260ei0.0546π,0.1049ei0.2274πX40.9088ei1.8176π,0.0178ei0.0356π,0.0522e1.4160iπ0.1802ei0.3426π,0.0216ei0.0436π,0.9786e1.9516iπX50.9494ei1.9044π,0.0328ei0.0680π,0.0541e1.3360iπ0.1249ei0.2506π,0.0166ei0.0352π,0.9680e1.9444iπTable 17
Differences of alternatives from best values ofACSFPM.
AlternativedZ^1,Zb1dZ^2,Zb2dZ^3,Zb3dZ^4,Zb4dZ^5,Zb5J10.95850.96800.000051.01751.0544J21.01110.89600.05260.82921.0070J3001.048000J40.56680.53020.63710.53460.5803J50.94120.77140.18440.64410.8277Table 18
The values ofSb, ℝb, and ℚb.
J10.93640.24640.9152J20.989340.28650.9757J30.050.050J40.55150.16510.5262J50.77250.24200.81346Table 19
Ranking of firms with respect toCSF-VIKOR.
AlternativesJ1J2J3J4J5Ranking order ofS54123Ranking order ofℝ54123Ranking order ofℚ54123Thus, the firmJ3 is also the best feasible choice for the establishment of the Saudi oil refinery project in Pakistan with respect to CSF-VIKOR method.


### 6.1. Discussion

The comparison provides the following results:(1)
The compromise solution assessed by theCSFNSf-VIKOR and CSF-VIKOR methods is the same exact, as indicated in Table 19, which implies that J3 is the best firm for the development of the Saudi oil refinery at the place of Gwadar. Together with the same results, the rankings of the alternatives in both methods are unvarying. So, we can claim that the outcomes of the proposed CSFNSf-VIKOR method are equivalent to the CSF-VIKOR method. The correlation between CSFNSf-VIKOR and CSF-VIKOR methods is illustrated in Table 20, and their graphical representations are put forward in Figure 3.(2)
We also implement the techniques ofSF-VIKOR [27] and fuzzy-VIKOR [8] on the proposed MAGDM problem. The transparency and accuracy in the outcomes improve and glorify the level of trust and confidence about the proposed method. The comparison results are pinned up in Table 21.(3)
The comparative study of theCSFNSf-VIKOR method with the existing techniques demonstrates and certifies the superiority of the proposed method as fuzzy-VIKOR, SF-VIKOR, and CSF-VIKOR methods have some imperfections and limitations that they all are impotent to understand the grades level with parameterized figures of such contemporary MAGDM problems.Figure 3
Comparative analysis.Table 20
Comparison withCSF-VIKOR.
MethodRanking of firmsBest firmCSFNSf-VIKOR (proposed)J3≻J4≻J5≻J1≻J2J3CSF-VIKOR [33]J3≻J4≻J5≻J1≻J2J3Table 21
Comparison withSF-VIKOR and fuzzy-VIKOR.
MethodRanking of firmsBest firmCSFNSf-VIKOR (proposed)J3≻J4≻J5≻J1≻J2J3SF-VIKOR [27]J3≻J4≻J5≻J1≻J2J3Fuzzy-VIKOR [8]J3≻J4≻J5≻J1≻J2J3


## 6.1. Discussion

The comparison provides the following results:(1)
The compromise solution assessed by theCSFNSf-VIKOR and CSF-VIKOR methods is the same exact, as indicated in Table 19, which implies that J3 is the best firm for the development of the Saudi oil refinery at the place of Gwadar. Together with the same results, the rankings of the alternatives in both methods are unvarying. So, we can claim that the outcomes of the proposed CSFNSf-VIKOR method are equivalent to the CSF-VIKOR method. The correlation between CSFNSf-VIKOR and CSF-VIKOR methods is illustrated in Table 20, and their graphical representations are put forward in Figure 3.(2)
We also implement the techniques ofSF-VIKOR [27] and fuzzy-VIKOR [8] on the proposed MAGDM problem. The transparency and accuracy in the outcomes improve and glorify the level of trust and confidence about the proposed method. The comparison results are pinned up in Table 21.(3)
The comparative study of theCSFNSf-VIKOR method with the existing techniques demonstrates and certifies the superiority of the proposed method as fuzzy-VIKOR, SF-VIKOR, and CSF-VIKOR methods have some imperfections and limitations that they all are impotent to understand the grades level with parameterized figures of such contemporary MAGDM problems.Figure 3
Comparative analysis.Table 20
Comparison withCSF-VIKOR.
MethodRanking of firmsBest firmCSFNSf-VIKOR (proposed)J3≻J4≻J5≻J1≻J2J3CSF-VIKOR [33]J3≻J4≻J5≻J1≻J2J3Table 21
Comparison withSF-VIKOR and fuzzy-VIKOR.
MethodRanking of firmsBest firmCSFNSf-VIKOR (proposed)J3≻J4≻J5≻J1≻J2J3SF-VIKOR [27]J3≻J4≻J5≻J1≻J2J3Fuzzy-VIKOR [8]J3≻J4≻J5≻J1≻J2J3


## 7. Advantages of Proposed Method

(i)
The principle of the VIKOR method relies on group utility measure and individual regret measure of each feasible choice, obtained using the normalized weighted vector of attributes within the normalized Euclidean distances, compromise the effects of grades level along with the two-dimensional frame.(ii)
Decision-making using the VIKOR method provides compromise solution in close proximity of the ideal solution in conjunction with a two-dimensional fusion ofCSFNSf model that enjoys four opinions of true, neutral, false, and level of the alternatives in reference to the soft information. Moreover, by keeping the level of the alternatives zero, two-dimensional information of four perspectives diversified into the CSF data that can be readily managed by the presented model.(iii)
The proposed strategic model even accommodates one-dimensional MAGDM problems within the ambit of spherical fuzzy information and picture fuzzy information by exerting periodic terms and ordered grades zero and raises the assurance of the proposed method via the accuracy of the results.(iv)
Additionally, the proficiency about the neutral part elongates the presented model as the abstraction of the fuzzyN-soft models including intuitionistic fuzzy N-soft, Pythagorean fuzzy N-soft, and complex Pythagorean fuzzy N-soft environments.


## 8. Conclusion

Many real MAGDM problems have a complex pattern. A hybrid decision-making model has been introduced in this paper, which is based on the VIKOR method but allows us to solve problems posed in the elaborate form of complex spherical fuzzyN-soft sets. This model was developed by Akram et al. [34]. An advantage of decision-making within the two-dimensional frame of CSFNSfSs is that the experts are free to use four conjectures of true, neutral, false, and level of the alternatives in reference to the soft information. The proposed CSFNSf-VIKOR method has been in discussion corresponding to the conflicting criteria under different methodologies. The linear normalized weights of the attributes and normalized Euclidean distances have been interpolated together, for the sake of two main features of the acclaimed VIKOR methodology, known as maximum group utility and minimum individual regret. Moreover, the coefficient of weight strategy pertaining to majority opinions and minimum regret of opponents have been exhibited for the compromise measure (ranking function). Furthermore, by keeping the level of the alternatives at zero, two-dimensional information comprising four conjectures becomes CSF knowledge that can therefore be embedded in the proposed model.The applicability and adequacy of the proposed method have been exemplified through a MAGDM problem whereupon a feasible firm is required for the development of the Saudi oil refinery project in Pakistan. For the implementation of the proposedCSFNSf-VIKOR method, experts’ inputs have been supported in the form of a nonbinary evaluation system. The CSFNSfWA operator has been deployed for the construction of ACSFNSfPM, and score degree has been established for the comparison of two complex entities in ACSFNSfPM. In the end, a comparison has been executed, which strengthens the presented method through the results’ transparency. The one-dimensional MAGDM problems under the auspices of spherical fuzzy information and picture fuzzy information can be conciliated with this structure by setting phase terms and ordered grades at zero. This speaks for the strength and validity of the presented approach. The proposed model, in accordance with neutral information, constitutes a generalization of fuzzy N-soft knowledge including intuitionistic fuzzy N-soft, Pythagorean fuzzy N-soft, and complex Pythagorean fuzzy N-soft data.For future direction, theCSFNSf-VIKOR method can be employed for selections in other types of MAGDM problems, including solar panel selection, sites for hotel business selection, in the field of medical and sustainable suppliers. Moreover, the limitations of the proposed model invite us to formulate new paradigms. For instance, the CSFNSf-VIKOR method is deficient to settle conditions in which the sum of squares of truth membership, neutral membership, and falsity membership of amplitude terms (or phase terms) exceeds 1. Therefore, we can work for establishing a complex T-spherical fuzzy N-soft-VIKOR methodology to extend the current boundary constraints. Moreover, a computer program can be created to handle the difficulties that appear in the presence of large numbers of alternatives and attributes.



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*Source: 1490807-2021-08-31.xml*