# Spherical Linear Diophantine Fuzzy TOPSIS Algorithm for Green Supply Chain Management System

Authors: Ibtesam Alshammari; Mani Parimala; Cenap Ozel; Muhammad Riaz
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Enhancing competitive pressure is one of the most significant roles of supply chain management. The competitive environment and customer perception have shifted in favour of an ecological mentality. As a result, green supplier selection (GSS) has emerged as a critical problem. The challenge of green supplier selection striving for agility, durability, ecological sensitivity, leanness, and sustainability is tackled in this paper. In terms of recycling applications, environmental applications, carbon footprint, and water consumption, the environmental parameters evaluated in GSS and traditional supplier selection differ. Because of the form of the problem, a resolution is defined, which comprises an algorithm entrenched in the spherical linear Diophantine fuzzy sets (SLDFSs) Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) technique. Before discussing the approach of the SLDF model, some background information on SLDF sets is provided. To assure the uniqueness of this robust extension, different operations on SLDFSs are described, along with some concise interpretations to help the reader comprehend these ideas. A robust TOPSIS approach has been utilized in the issue of GSS by taking into consideration the multicriteria decision-making (MCDM) technique particularly useful in several areas, like analyzing and choosing traditional and environmental conventionalities. Due to linguistic criteria and the inability to assess all criteria, the fuzzy technique must be used with the TOPSIS method to lessen the consequences of instability and ambiguity. The spherical linear Diophantine fuzzy TOPSIS approach is employed, as it simplifies the evaluation of decision-makers and criteria. The hybrid technique resulting from integrating the SLDFS and TOPSIS is extremely successful in selecting which provider is more suited among the alternatives established on the criteria set by the order of significance, and this method may also be incorporated into similar issues.


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## Body

## 1. Introduction

One of the main archetypes in the supply chain (SC) management system is the green notion, which may be thought of as an organizational philosophy. Because of environmental legislation and consumer demands on sustainability, the notion of green supply chain management (GSCM) has gained traction as explained by Govindan et al. [1]. Mishra et al. [2] in 2019 explained that GSCM is an administrative approach that incorporates the environmental thinking structure in at-most all supply chain operations such as material selection, purchasing, product design, and production process which helps the companies to achieve more acquisitions while improving their sustainability impact by downsizing the adverse accouterments of potential consequences.The GSCM must begin from the start of the supply chain, with raw material purchase, and continue through every stage, by incorporating disposal or recycling of the product. It is not adequate to spotlight just greenness in infiltrating SC operations for sustainability strategies and goals; firms must also address the environmental burdens of retiring operations among stakeholders or partners in order to improve supplier achievements as analyzed by Banaeian et al. [3] in 2018. As a result, suppliers play an important role in helping businesses enhance their environmental achievement as explained by Mathiyazhagan et al. [4] in 2018. As a result, businesses have focused on the green supplier selection (GSS) issue when building GSCM.MCDM techniques may be an appropriate way of dealing with the GSS situation and comparing vendors. Though there are many published studies on GSS in both fuzzy and crisp contexts, further investigation is necessary with diverse criteria, skill, and linguistic characteristics taken into consideration.MCDM techniques may be an appropriate way of dealing with the GSS situation and comparing vendors. Further investigations are required with various skills, measures, linguistic characteristics, and even many studies published on GSS in both crisp and fuzzy contexts.Intuitionistic fuzzy sets (IFSs) [5, 6], Pythagorean fuzzy sets (PyFS) [7–9], and q-rung orthopair fuzzy sets (q-ROPFS) [10, 11] are basic ideas in computational intelligence with numerous applications in fuzzy set (FS) [12, 13] system modeling and decision-making under uncertainty. Nonetheless, all of these ideas are subject to stringent limitations on the membership grade (MG) and nonmembership grade (NMG). Moreover, by including the neutral-membership grade in the abovementioned grades, we get the picture fuzzy sets (PFSs) [14], spherical fuzzy sets (SFSs) [15, 16], m-polar spherical fuzzy sets (m-PSFSs) [12], and q-rung ortho picture fuzzy sets (q-RPFS) [17]. All these notions have some strict restrictions over the grade functions imposed on them. To alleviate these constraints, we suggest a novel extension of fuzzy sets known as SLDFSs [18, 19], which takes into account reference parameters as well. Spherical linear Diophantine fuzzy sets can naturally classify issues and offer acceptable solutions by picking alternative pairings of reference parameters. Some fundamental procedures for spherical linear Diophantine fuzzy sets are provided.Some famous academics have created many nondeterministic extensions of the TOPSIS approach in recent years, including fuzzy sets [20], IFSs [21], PyFSs [22, 23], and other hybrid structures. Based on intuitionistic fuzzy numbers (IFNs), Xu et al. [24–26] proposed weighted average operators, geometric operators, and induced generalised operators. In the framework of IFNs, Jose [27] explored aggregation operators associated with the score function for MCDM. The TOPSIS technique in recent years in extensions of fuzzy sets with abstinence membership values such as PFSs [28], SFSs [29], and q-RPFSs [30] has been developed and utilized in many different fields widely in medical diagnosis problems, enterprise resource planning systems, dry bulk carrier selection, green supply selection, etc. Many researchers have studied the TOPSIS method for decision-making problems, including Adeel et al. [31], Akram and Arshad [32], Xu and Zhang [33], Selvachandran and Peng [34], Kumar and Garg [35], Biswas and Sarkar [22], Boran et al. [21], Li and Nan [36], and Eraslan and Karaaslan [37].Although PFSs, SFSs, q-RPFSs, and m-PSFSs have extensive applications in diverse domains of real life, these ideas have limits connected to membership, abstinence, and nonmembership grades. To overcome these constraints, we provide the innovative notion of spherical linear Diophantine fuzzy set (SLDFS) with reference parameters. Because of the usage of reference parameters, the suggested SLDFS model is more efficient and adaptable than existing techniques. SLDFS classifies data in MADM situations by varying the physical sense of reference parameters. With the use of reference parameters, this set covers the spaces of existing buildings and enlarges the space for MG, AG, and NMG. The reason for the suggested paradigm is presented step by step throughout the manuscript. Now, we will go through some of the paper’s main goals.(1)
In some real-world scenarios, the total of the triplets MG, abstinence grade (AG), and NMG to which an option meeting an attribute specified by a decision-maker (DM) is assigned may be greater than one (ex.0.9+0.3+0.5>1), and their sum of squares may also be greater than one (ex. 0.92+0.32+0.52>1). In such cases, PFS and SFS fail. To address these shortcomings, the constraints on MG, AG, and NMGs are changed to tq+aq+mq in the case of q-RPFS. We are capable of dealing with MG, AG, and NMGs independently to some extent even for extremely high values of “q.” In some practical issues, where MG t, AG a, and NMG f are equal to 1 (i.e., t=a=f=1), we get 1q+1q+1q≥1, which defines the q-RPFS condition. In such cases, MADM with q-RPFS may not be employed(2)
Our initial goal is to close this research gap using the unique idea of SLDFS. We are capable of dealing with the picture, spherical, and q-rung picture character of alternatives under the influence of reference factors using this approach. (For example, for (0.8+0.4+0.6>1), we can provide reference parameters like 0.80.3+0.40.4+0.60.2≤1, where 0.3,0.4,0.2 are the reference parameters for MG, AG, and NMGs, respectively.) Because the suggested technique resembles the well-defined linear equation of three variables ax+by+cz=d in number theory, SLDFS is the best name to describe the suggested model(3)
Our next and second purpose is to introduce reference parameter functions in SLDFS, which cannot cope with parameterizations like PFSs, SFSs, and q-RPFSs. The suggested approach improves on previous techniques, and the decision-maker (DM) has complete freedom in selecting grades. This structure additionally categorizes the problem by modifying the tactile feel of reference parameters(4)
The final goal is to build a solid link betwixt the suggested technique and the MCDM challenges. Using this innovative viewpoint, we construct an enhanced TOPSIS technique that employs SLDF weighted geometric aggregation (SLDFWG) operators. We show an application of the recently published TOPSIS approach based on SLDFSs to an MCDM problem involving the selection of green supply chain systems(5)
After reading this paper, the reader can understand the concept of SLDFG and its operations. Also, the reader can apply real-life problems using the proposed algorithm(6)
If the reader looks for limitations, the calculation part is slightly complicatedThe following is how the manuscript is structured. Review of the literature is presented in Section2. Use of the spherical linear Diophantine fuzzy TOPSIS approach to analyze the green supplier carrier selection process is explained in Section 3. Section 4 walks you through the application and the results step by step. In Section 5, we presented the implications and perspectives. Finally, Section 6 leads the paper to an end.


## 2. Preliminaries

We will review and give some fundamental definitions of the SLDFSs.Definition 1 (see [18]).
SLDFSSDis an element on the nonvoid setQof the form(1)SD=ζ,mDζ,aDζ,nDζ,αDζ,βDζ,ηDζ:ζ∈Q,where the following holds:
(1)
mDζ,aDζ,nDζ,rDζ are MG, AG, NMG, and the refusal grade (RG)(2)
αζ,βζ,ηζ,γζ∈0,1 are the reference parameters of MG, AG, NMG, and RG, respectively(3)
The condition0≤αDζmDζ+βDζaDζ+ηDζnDζ≤1∀ζ∈Q and with 0≤αζ+βζ+ηζ≤1(4)
γDζrDζ=1−αDζmDζ+βDζaDζ+ηDζnDζ(5)
SD=mD,aD,nD,αD,βD,ηD is the spherical linear Diophantine fuzzy number (SLDFN), with the constraints 0≤α+β+η≤1 and 0≤αDmD+βDaD+ηDnD≤1Definition 2.
A SLDFS onQis said to be(i)
absolute SLDFS, ifSD1=ζ,1,0,0,1,,0,0:ζ∈Q(ii)
null SLDFS, ifSD0=ζ,0,0,1,0,0,1:ζ∈QDefinition 3.
LetSD=mD,aD,nD,αD,βD,ηDbe a SLDFN, then(1)
the score function (SF) is displayed bySSD and is depicted as
(2)SSD=12mD−aD−nD+αD−βD−ηD,where S:SDQ⟶−1,1(2)
the accuracy function (AF) is displayed byASD and is depicted as
(3)ASD=12mD+aD+nD3+αD+βD+ηD,where A:SDQ⟶0,1 and SDQ is the assortment of all SLDFNs on QDefinition 4.
Two LDFNsSD1 and SD2 can be comparable using SF and AF. It is defined as follows:
(i)
SD1>SD2 if SSD1>SSD2(ii)
SD1<SD2 if SSD1<SSD2(iii)
IfSSD1=SSD2, then
(a)
SD1>SD2 if ASD1>ASD2(b)
SD1<SD2 if ASD1<ASD2(c)
SD1=SD2 if ASD1=ASD2Definition 5.
LetSDi=mDi,aDi,nDi,αDi,βDi,ηDi for i∈Δ is a convene of SLDFNs on Q and X>0 then
(i)
SD1c=nD1,1−aD1,mD1,ηD1,βD1,αD1(ii)
SD1=SD2⇔mD1=mD2,aD1=aD2,nD1=nD2,αD1=αD2,βD1=βD2,ηD1=ηD2(iii)
SD1⊆SD2⇔mD1≤mD2,aD1≥aD2,nD1≥nD2,αD1≤αD2,βD1≥βD2,ηD1≥ηD2(iv)
SD1⊕SD2=mD1+mD2−mD1mD2,aD1aD2,nD1nD2,αD1+αD2−αD1αD2,βD1βD2,ηD1ηD2(v)
SD1⊗SD2=mD1mD2,aD1+aD2−aD1aD2,nD1+nD2−nD1nD2,αD1αD2,βD1+βD2−βD1βD2,ηD1+ηD2−ηD1ηD2(vi)
SD1∪SD2=mD1∨mD2,aD1∧aD2,nD1∧nD2,αD1∨αD2,βD1∧βD2,ηD1∧ηD2(vii)
SD1∩SD2=mD1∧mD2,aD1∨aD2,nD1∨nD2,αD1∧αD2,βD1∨βD2,ηD1∨ηD2(viii)
XSD1=1−1−mD1X,aD1X,nD1X,1−1−αD1X,βD1X,ηD1X(ix)
SD1X=mD1X,1−1−aD1X,1−1−nD1X,αD1X,1−1−βD1X,1−1−nD1XExample 1.
In medicine, the combination of medications is for better therapy. Medicines are chemicals or substances that are used to treat, stop, or prevent disease, alleviate symptoms, or aid in the identification of disorders. Medical advancements have enabled doctors to heal numerous ailments and save lives. A combination drug, often known as a fixed-dose combination (FDC), is a medication that contains two or more active components in a single dosage form. Aspirin/paracetamol/caffeine, for example, is a combination medication used to relieve pain, particularly tension headaches and migraines. LetG=g1,g2,g3,g4 represent the combination of two life-saving medications. Two or more medications can be mixed in the formulation of medicine to increase its impact. If the reference or control parameter is taken into account, α = excellent impact against infection produced during surgeries; β = not high impact against infection produced during surgeries (uneffected or neutral); γ = no high impact against infection produced during surgeries.The SLDF information can be taken as Table1 for these reference or control parameters.Table 1
SLDFSs.
SDmD,aD,nD,αD,βD,ηDg10.92,0.24,0.41,0.54,0.12,0.11g20.51,0.46,0.81,0.38,0.32,0.11g30.73,0.45,0.53,0.34,0.23,0.21g40.72,0.26,0.74,0.29,0.34,0.21All cited mathematicians claim that these operations produce SLDFNs, and their constructed aggregation operators produce results for SLDFNs. Now, we check these operations for arbitrary SLDFNs. LetSD1=0.92,0.56,0.45,0.64,0.26,0.09 and SD2=0.85,0.65,0.58,0.43,0.32,0.21 be two SLDFNs, then
(i)
SD1c=0.45,0.44,0.92,0.09,0.74,0.64(ii)
SD2⊆SD1 by Definition 5 (iii)(iii)
SD1⊕SD2=0.988,0.364,0.261,0.7948,0.0832,0.0189(iv)
SD1⊗SD2=0.782,0.846,0.769,0.2752,0.4968,0.2811(v)
SD1∪SD2=0.92,0.56,0.45,0.64,0.26,0.09=SD1(vi)
SD1∩SD2=0.85,0.65,0.58,0.43,0.32,0.21=SD2IfX=0.3, then we have the following:
(vii)
XSD1=0.5313,0.8403,0.7870,0.2640,0.6676,0.4856(viii)
SD1X=0.9753,0.2183,0.1642,0.8747,0.0864,0.0279


## 3. Methodology: SLDF-TOPSIS Algorithms

The selection of environmentally friendly vendors is seen as an MCDM problem. As an example, in order to solve the issue of TOPSIS, an MCDM approach is proposed. This section examines and summarizes the SLDFSs and the TOPSIS approach.Algorithm 1 (SLDF-TOPSIS).
There are several criteria in decision-making challenges. Certain of them are critical and required light of the topic at hand, while others are less so. So, one must choose the appropriate and significant criteria, which is done with the support of a professional viewpoint or according to the problem’s requirements.
Step 1. Pinpoint the problem:E=ek, the group of decision-makers/experts, the assemblage of alternatives/attributes is A=ai and C=cj is the family of parameters/criteria, where i,j,k∈N and i=1,2,3,⋯,p, j=1,2,3,⋯,q, and k=1,2,3,⋯,r.
The decision matrix is represented as(4)D=aijp×q=a11a12⋯a1qa21a22⋯a2q⋮⋮⋱⋮ap1ap2⋯apq.
Step 2. The rating of each alternative based on the associated criterion is created using spherical linear Diophantine fuzzy information for MADM and is presented in the decision matrix as(5)DP=aijp×q=mij,aij,nij,αij,βij,ηijp×q=m11,a11,n11,α11,β11,η11m12,a12,n12,α12,β12,η12⋯m1q,a1q,n1q,α1q,β1q,η1qm21,a21,n21,α21,β21,η21m22,a22,n22,α22,β22,η22⋯m2q,a2q,n2q,α2q,β2q,η2q⋮⋮⋱⋮mp1,ap1,np1,αp1,βp1,ηp1mp2,ap2,np2,αp2,βp2,ηp2⋯mpq,apq,npq,αpq,βpq,ηpq.
In matrixDP, the entries mij,aij,nij,αij,βij,ηij represent membership, abstinence, nonmembership grades, and their reference parameters, respectively, where i=1,2,3,⋯,p, j=1,2,3,⋯,q. These grades satisfy the following properties under spherical linear Diophantine fuzzy environment:
(1)
0≤mij,aij,nij,αij,βij,ηij≤1(2)
0≤αij+βij+ηij≤1(3)
0≤αijmij+βijaij+ηijnij≤1, where i=1,2,3,⋯,p, and j=1,2,3,⋯,q
Ifwij denotes the weight allocated by Ek to Cj keeping into consideration the linguistic variables (LVs) (Table 2), construct weighted parameter matrix
(6)P=wijp×q=w11w12⋯w1qw21w22⋯w2q⋮⋮⋱⋮wp1wp2⋯wpq.
Step 3. Assume we have a group of"r" decision-makers, each with their own decision weights. As a result, each expert’s choice and importance differ from one another. Assume that the SLD-LV takes into account the significance of each expert. Let ek=mk,ak,nk,αk,βk,ηk, k=1,2,3,⋯,r be SLDFNs for the assessment of “r” decision-makers, and the weights of kth decision-makers are given as follows:
(7)ωk=1−1−mkζ2+akζ2+nkζ2+1−αkζ2+βkζ2+ηkζ2/3∑k=1r1−1−mkζ2+akζ2+nkζ2+1−αkζ2+βkζ2+ηkζ2/3,where k=1,2,3,⋯,r and ∑k=1rωk=1.
Step 4. Construct SLDF decision matrixZi=zjkip×q, where zjki is a SLDFS object, for i expert. The aggregating matrix is A=Z1+Z2+⋯+Zn/n=żjkp×q, where
(8)żij=SLDFWGzij1,zij2,⋯,zijk=ω1zij1⊗ω2zij2⊗⋯⊗ωkzijk=Πk=1rmijkωk,1−Πk=1r1−mijkωk,1−Πk=1r1−nijkωk,Πk=1rαijkωk,1−Πk=1r1−βijkωk,1−Πk=1r1−ηijkωk.
Hence, we get aggregated decision matrix fori=1,2,3,⋯,p, j=1,2,3,⋯,q and k=1,2,3,⋯,r.
Step 5. Acquire the weighted SLDF decision matrix(9)J=z˘jkl×q=z˘11z˘12⋯z˘1qz˘21z˘22⋯z˘2q⋮⋮⋱⋮z˘p1z˘p2⋯z˘pq,where z˘k=ξk×żjk(10)z˘k=SLDFWGzij1,zij2,⋯,zijk=ξ1zij1⊗ξ2zij2⊗⋯⊗ξkzijk=Πk=1rmijkξk,1−Πk=1r1−mijkξk,1−Πk=1r1−nijkξk,Πk=1rαijkξk,1−Πk=1r1−βijkξk,1−Πk=1r1−ηijkξk.
Step 6. The weights of the criterion and the aggregated decision matrix are utilized to create the aggregated weighted SLDF decision matrix in this phase. This matrix may be produced and assessed as follows:(11)z˘k⊗żij=z˘kżij=Zi=zjkip×q.
Here, each entry is SLDFN andi=1,2,⋯,p, j=1,2,⋯,q.
Step 7. Locate SLDF-valued positive ideal solution (SLDFV-PIS) and SLDF-valued negative ideal solution (SLDFV-NIS), confined in a form(12)sj+=ρ¨1+,ρ¨2+,⋯,ρ¨q+=∨imij,∧iaij,∧inij,∨iαij,∧iβij,∧iγij,(13)sj−=ρ¨1−,ρ¨2−,⋯,ρ¨q−=∧imij,∨iaij,∨inij,∧iαij,∨iβij,∨iγij.
Step 8. The normalized Euclidean distance (NED) of each alternative and its SLDFNIS can be defined as(14)DEN+=16n∑j=1qimij−imj+2+iaij−iaj+2+inij−inj+2+iαij−iαj+2+iβij−iβj+2+iγij−iγj+2.
The normalized Euclidean distance (NED) of each alternative and its SLDFPIS can be defined as(15)DEN−=16n∑j=1qimij−imj−2+iaij−iaj−2+inij−inj−2+iαij−iαj−2+iβij−iβj−2+iγij−iγj−2.
Step 9. Compute the SLDF relative closeness with the formula(16)Cj+=DEN−DEN++DEN−.
Step 10. Finally, the most advantageous ranking procedure of the choices is chosen. The ideal scenario is the one with the greatest updated coefficient value.Table 2
LTs for significant criteria weights.
LVsSLDFNCertainly low importance (CLI)0.00,0.00,0.94,0.05,0.02,0.85Very low importance (VLI)0.10,0.20,0.90,0.12,0.14,0.73Low importance (LI)0.35,0.31,0.80,0.11,0.23,0.65Below average importance (BAI)0.35,0.45,0.75,0.17,0.25,0.57Average importance (AI)0.43,0.56,0.48,0.19,0.28,0.51Above average importance (AAI)0.55,0.65,0.35,0.31,0.33,0.36High importance (HI)0.65,0.80,0.20,0.20,0.35,0.35Very high importance (VHI)0.80,0.90,0.10,0.40,0.37,0.21Certainly high importance (CHI)0.90,1.00,0.03,0.52,0.39,0.05Exactly equal (EE)0.96,0.95,0.19,0.79,0.10,0.02The proposed SLDF-TOPSIS is expressed as a flow diagram in Figure1.Figure 1
Flow diagram of the proposed algorithm.


## 4. Numerical Example

### 4.1. Case Study

Engenderment systems have advanced to incipient heights in tandem with advances in information technology. As a result of ecumenical competition, businesses have transmuted their core capabilities, ameliorated their subsisting environment, and developed incipient business models for themselves and their stakeholders. In this component, the recommended GSS technique is implemented by the management of an agricultural implement company (represented as ABC) in Turkey. The managers’ goal is to assess supplier performance by investigating the priority ranking of the GSS criteria from the industrial expert platform. The company that primarily manufactures lawnmowers expands its product line without jeopardizing its reputation for excellence and perpetuated innovation, and it wishes to implement its environmental management system throughout the supply chain, which includes cooperation activities with all suppliers in accordance with industrial practices. GSS has been apperceived as a required decision-making activity for the ABC firm in light of this circumstance.


### 4.2. Problem Description

ABC Corporation wants to select the finest green supplier in light of industrial fundamentals. A committee of 4 experts was formed to evaluate vendors. Face-to-face interviews were used to obtain data. Four decision-makers/experts from various areas of the firm have been requested to offer input on the suggested method, symbolised byE=e1,e2,e3,e4. It has been concluded that the needed part can be obtained from one of five suppliers defined by A=a1,a2,a3,a4,a5. To pick green suppliers and construct a decision-making tool in the evaluation, the required criteria must first be appropriately identified. The six most significant and prevalent environmental factors are delivery performance, pollution control, production, quality service, environmental representation, and technology capability. The choice of criteria C=c1,c2,c3,c4,c5,c6, where the criteria are c1=delivery performance, c2=pollution control, c3=production, c4=quality service, c5=environmental representation, and c6=technology capability.


### 4.3. Application of SLDF-TOPSIS to GSS

We apply Algorithm1 (SLDF-TOPSIS) in this example as follows.Step 1.
LetE=ek:k=1,2,3,4 be the team comprising experts, A=ai:i=1,2,3,⋯,5 the set of alternatives covered by the study, and C=cj:j=1,2,⋯,6 the family of criteria. The importance of each decision-maker is expressed by LT, and each LT is an SLDFN shown in Table 2, where we have different types of level of importance as low importance (LI), average importance (AI), and high importance (VI).Step 2.
The decision power of experts is considered “linguistic terms (LTs).” Each expert has different opinions about LTs. First, we define some LTs for rating criteria and experts as shown in Table3. For example, for the LT “very good/very important (VG/VI),” we have numeric value 0,96,0.15,0.12,0.65,0.10,0.21 as a SLDFN. If an expert assign this LT to a criteria for a decision-making problem, then it means that criteria have 90% satisfaction value, 15% indeterminacy value, and 12% of dissatisfaction value. For the assigned LT, the reference parameters have 65% satisfaction value and 10% indeterminacy value and 21% dissatisfaction value corresponding to the three grades, and these reference parameters control the behavior of membership, abstinence, and nonmembership grades.Step 3.
Consider we have four decision-makers or experts in the selection committee, and all experts have different opinions. The weight of each decision maker is calculated by utilizing equation (7) as follows: ω1=1−0.2151/3/4−0.2151/3−0.3497/3−0.5758/3−0.9749/3=0.3073, and similarly, ω2=0.2764, ω3=0.2358, ω4=0.1804. Hence, the weight vector of four experts is ω=0.3073,0.2764,0.2358,0.1804T with the constraint ∑k=14ωk=1.Step 4.
Table2 contains the LT as well as the SLDFNs. These LTs are used to rate the alternatives based on the supplied criteria. The assessment values of each alternative Ai: i=1,2,3,⋯,p=5 according to each criteria Cj: j=1,2,3,⋯,q=6 are provided by four experts given in Table 4, and it is based on SLDFNs. For alternative Ai, i=1,2,3,4,5, and criteria Cj, j=1,2,3,4,5,6, we have the following LTs according to these four decision-makers. For the alternative Ai and criteria Cj, we have four LTs for four decision-makers: HI, VHI, CHI, and EE, respectively. We have six SLDFNs corresponding to six criteria. We now apply equation (8) to do further input data computations. We apply equation (8) to Table 4 to obtain the aggregated SLDF decision table shown in Table 5. For the alternative a1 and for criteria c1, we will use the LTs “MG, G, MG, and F” from Table 3. We use the weight vector ω=0.3073,0.2764,0.2358,0.1804T calculated in Step 3 and calculate the aggregated value by using equation (8) given in Table 4.Step 5.
We compute the aggregated weights for the six criteria listed in Table6 using equation (10). Because equation (10) is the same aggregated formula as equation (8), we use it to aggregate weights for six criteria, and all of the computations are the same as in Step 4.Step 6.
We now assess the aggregated SLDF decision table using aggregated weights and alternative ratings. Using equation (11), we get Table 7. Table 7 is obtained by multiplying Table 5 with Table 6. This multiplication is done according to the definition of multiplication of two SLDFNs. The first SLDFN for option a1 and criteria c1 is 0.7451,1.0000,0.2175,0.4345,0.3000,0.2030 shown in Table 5, while the second SLDFN for criteria c1 is 0.9138,0.1885,0.2939,0.6270,0.1230,0.2176 shown in Table 6. We acquire an SLDFN as an aggregated decision SLDFN 0.6809,1.0000,0.4475,0.2724,0.3861,0.3764 for option a1 and criteria c1 by multiplying both SLDFNs. The same approach may be used to compute all other values.Step 7.
Now, using equations (12) and (13) in Table 7, we determine Table 8 SLDF relative PIS and NIS for the calculated aggregated weight. The mathematical methods for selecting MG, AG, and NMG from the aggregated decision Table 7 are briefly described in equations (12) and (13).Step 8.
Let us calculate the distance measure by using the proposed algorithm from relative PIS and relative NIS provided in Tables7 and 8. For positive ideal solution, we find the normalized Euclidean distance between six criteria of the alternative a1 given in Table 7 and Sj+ given in Table 8. The value DEN+ for alternative a1 is calculated as follows: DEN+=1/6×60.6809−0.68092+1.0000−0.58932+0.4475−0.44752+0.2724−0.27242+0.3861−0.34372+0.3764−0.37642+⋯+0.6294−0.68572+0.9035−0.87702+0.4644−0.37802+0.2116−0.24312+0.3682−0.31822+0.4405−0.40532=0.0191. On the same pattern, we can calculate all other values for PIS and NIS corresponding to all the alternatives. The distance measure of each alternative is given in Table 9.Step 9.
Calculating the SLDF relative closeness asCj+=DEN−/DEN++DEN−=0.0117/0.0191+0.0117=0.3796 and its ranking is presented in Table 10.Step 10.
The priority structure of the green supplier as seen in Table10 (see Figure 2) is a2>a5>a4>a1>a3; thus, a2 is the best green supplier.Table 3
Expert’s importance in terms of SLDFNs.
ExpertLinguistic variablesSLDF numberse1Very good (VG)0.96,0.15,0.12,0.65,0.10,0.21e2Good (G)0.90,0.20,0.30,0.62,0.13,0.22e3Medium good (MG)0.85,0.33,0.41,0.57,0.17,0.25e4Fair (F)0.72,0.45,0.53,0.47,0.19,0.31Table 4
Judgments/assessments given by four experts in terms of LTs.
AlternativesExpertsCriteriac1c2c3c4c5c6a1e1CHIAAIHIBAILIEEe2AIVHIEECHIAAIHIe3EEAIAAILIHIVHIe4CHIEEVHIAIHIAIa2e1LIHICHIVHIEEAIe2AAIHICHIVHIHIEEe3LIVHIBAIHIAIAAIe4AAIVHIEECHIVHIEEa3e1CHIAAIEEAILIVHIe2EEBAICHIHIAAIVHIe3LIAIAAIHIEEHIe4LIEEVHIHIAAICHIa4e1EEAAICHIAIVHIAAIe2AIEEHIHILIVHIe3HIEEBAICHIAAIEEe4CHILIVHIAAIEEAIa5e1LIAAIVHIEEHICHIe2HIAAIAIVHIEEHIe3EELIHICHIAAIVHIe4CHIAAIHIVHIEEAIWeightse1GFMGGVGVGe2GVGVGFGGe3VGFGGVGFe4GGVGFGVGTable 5
Aggregated SLDF decision table.
AlternativesCriteriaSLDFNsa1c10.7451,1.0000,0.2175,0.4345,0.3000,0.2030c20.6365,0.8160,0.2979,0.3509,0.2934,0.3121c30.7226,0.8627,0.2192,0.3674,0.2877,0.2485c40.4716,1.0000,0.6062,0.2132,0.2924,0.4778c50.5132,0.6584,0.5066,0.1879,0.3095,0.4649c60.7143,0.8721,0.2360,0.3560,0.2736,0.2662a2c10.4303,0.4939,0.6573,0.1766,0.2774,0.5388c20.7087,0.8501,0.1598,0.2669,0.3584,0.2950c30.7288,1.0000,0.3180,0.4308,0.3130,0.2075c40.7782,1.0000,0.1127,0.3562,0.3690,0.2200c50.6901,0.8612,0.2589,0.3416,0.2682,0.2854c60.6577,0.8456,0.3289,0.4089,0.2161,0.2837a3c10.6184,1.0000,0.5217,0.3058,0.2516,0.3676c20.5065,0.7053,0.5073,0.2770,0.2585,0.4186c30.8002,1.0000,0.1761,0.4993,0.2930,0.1548c40.5725,0.7451,0.2992,0.1969,0.3292,0.4040c50.5459,0.7275,0.5234,0.2811,0.2503,0.4121c60.7782,1.0000,0.1127,0.3562,0.3690,0.2200a4c10.6932,1.0000,0.2619,0.3574,0.2694,0.2696c20.6743,0.8540,0.4118,0.4153,0.2008,0.2860c30.6445,1.0000,0.3409,0.2926,0.3444,0.3136c40.5998,1.0000,0.2936,0.2670,0.3356,0.3501c50.6023,0.7977,0.4603,0.2981,0.2794,0.3759c60.6655,0.8369,0.2805,0.3797,0.2846,0.2852a5c10.6248,1.0000,0.4574,0.2734,0.2690,0.3660c20.4944,0.5892,0.5077,0.2428,0.3076,0.4449c30.6181,0.7990,0.2636,0.2440,0.3377,0.3617c40.8699,1.0000,0.1131,0.5246,0.3023,0.1184c50.7468,0.8788,0.2339,0.4154,0.2404,0.2187c60.7003,1.0000,0.1925,0.3130,0.3554,0.2732Table 6
Aggregated SLDF criteria weight table.
AlternativesCriteriaSLDFNswc10.9138,0.1885,0.2939,0.6270,0.1230,0.2176c20.8116,0.3363,0.4259,0.5404,0.1552,0.2677c30.9108,0.2211,0.2745,0.6174,0.1291,0.2248c40.8019,0.3534,0.4907,0.5356,0.1672,0.2693c50.9321,0.1732,0.2424,0.6361,0.1138,0.2146c60.8812,0.2457,0.2990,0.5944,0.1303,0.2375Table 7
Aggregated weighted SLDF decision table.
AlternativesCriteriaSLDFNsa1c10.6809,1.0000,0.4475,0.2724,0.3861,0.3764c20.5166,0.8779,0.5969,0.1896,0.4031,0.4963c30.6581,0.8931,0.4335,0.2268,0.3797,0.4174c40.3782,1.0000,0.7994,0.1142,0.4107,0.6184c50.4784,0.7176,0.6262,0.1195,0.3881,0.5797c60.6294,0.9035,0.4644,0.2116,0.3682,0.4405a2c10.3932,0.5893,0.7580,0.1107,0.3663,0.6392c20.5752,0.9005,0.5176,0.1442,0.4580,0.4837c30.6638,1.0000,0.5052,0.2660,0.4017,0.3857c40.6240,1.0000,0.5481,0.1908,0.4745,0.4301c50.6432,0.8852,0.4385,0.2173,0.3515,0.4388c60.5796,0.8835,0.5296,0.2431,0.3182,0.4538a3c10.5651,1.0000,0.6623,0.1917,0.3437,0.5052c20.4111,0.8044,0.7171,0.1497,0.3736,0.5742c30.7288,1.0000,0.4023,0.3083,0.3843,0.3448c40.4591,0.8352,0.6431,0.1055,0.4414,0.5645c50.5088,0.7747,0.6389,0.1788,0.3356,0.5383c60.6857,1.0000,0.3780,0.2117,0.4512,0.4053a4c10.6334,1.0000,0.4788,0.2241,0.3593,0.4285c20.5473,0.9031,0.6623,0.2244,0.3248,0.4771c30.5870,1.0000,0.5218,0.1807,0.4290,0.4679c40.4810,1.0000,0.6402,0.1430,0.4467,0.5251c50.5614,0.8327,0.5911,0.1896,0.3614,0.5098c60.5864,0.8770,0.4956,0.2257,0.3778,0.4550a5c10.5709,1.0000,0.6169,0.1714,0.3589,0.5040c20.4013,0.7274,0.7174,0.1312,0.4151,0.5935c30.5630,0.8434,0.4657,0.1506,0.4232,0.5052c40.6976,1.0000,0.5483,0.2810,0.4190,0.3558c50.6961,0.8998,0.4196,0.2642,0.3268,0.3864c60.6171,1.0000,0.4339,0.1860,0.4394,0.4458Table 8
Relative SLDF: PIS and NIS.
AlternativesCriteriaSLDFNsSj+c10.6809,0.5893,0.4475,0.2724,0.3437,0.3764c20.5752,0.7274,0.5176,0.2244,0.3248,0.4771c30.7288,0.8434,0.4023,0.3083,0.3797,0.3448c40.6976,0.8352,0.5481,0.2810,0.4107,0.3558c50.6961,0.7176,0.4196,0.2642,0.3268,0.3864c60.6857,0.8770,0.3780,0.2431,0.3182,0.4053Sj−c10.3932,1.0000,0.7580,0.1107,0.3861,0.6392c20.4013,0.9031,0.7174,0.1312,0.4580,0.5935c30.5630,1.0000,0.5218,0.1506,0.4290,0.5052c40.3782,1.0000,0.7994,0.1055,0.4745,0.6184c50.4784,0.8998,0.6389,0.1195,0.3881,0.5797c60.5796,1.0000,0.5296,0.1860,0.4512,0.4550Table 9
Distance measure of each alternative.
AlternativesDEN+DEN−a10.01910.0117a20.01380.0161a30.01830.0091a40.01580.0097a50.01450.0170Table 10
SLDF closeness coefficient of each alternative.
AlternativesCj+Ranka10.37964a20.53911a30.33305a40.38043a50.53852Figure 2
Priority order of the green supplier.


## 4.1. Case Study

Engenderment systems have advanced to incipient heights in tandem with advances in information technology. As a result of ecumenical competition, businesses have transmuted their core capabilities, ameliorated their subsisting environment, and developed incipient business models for themselves and their stakeholders. In this component, the recommended GSS technique is implemented by the management of an agricultural implement company (represented as ABC) in Turkey. The managers’ goal is to assess supplier performance by investigating the priority ranking of the GSS criteria from the industrial expert platform. The company that primarily manufactures lawnmowers expands its product line without jeopardizing its reputation for excellence and perpetuated innovation, and it wishes to implement its environmental management system throughout the supply chain, which includes cooperation activities with all suppliers in accordance with industrial practices. GSS has been apperceived as a required decision-making activity for the ABC firm in light of this circumstance.


## 4.2. Problem Description

ABC Corporation wants to select the finest green supplier in light of industrial fundamentals. A committee of 4 experts was formed to evaluate vendors. Face-to-face interviews were used to obtain data. Four decision-makers/experts from various areas of the firm have been requested to offer input on the suggested method, symbolised byE=e1,e2,e3,e4. It has been concluded that the needed part can be obtained from one of five suppliers defined by A=a1,a2,a3,a4,a5. To pick green suppliers and construct a decision-making tool in the evaluation, the required criteria must first be appropriately identified. The six most significant and prevalent environmental factors are delivery performance, pollution control, production, quality service, environmental representation, and technology capability. The choice of criteria C=c1,c2,c3,c4,c5,c6, where the criteria are c1=delivery performance, c2=pollution control, c3=production, c4=quality service, c5=environmental representation, and c6=technology capability.


## 4.3. Application of SLDF-TOPSIS to GSS

We apply Algorithm1 (SLDF-TOPSIS) in this example as follows.Step 1.
LetE=ek:k=1,2,3,4 be the team comprising experts, A=ai:i=1,2,3,⋯,5 the set of alternatives covered by the study, and C=cj:j=1,2,⋯,6 the family of criteria. The importance of each decision-maker is expressed by LT, and each LT is an SLDFN shown in Table 2, where we have different types of level of importance as low importance (LI), average importance (AI), and high importance (VI).Step 2.
The decision power of experts is considered “linguistic terms (LTs).” Each expert has different opinions about LTs. First, we define some LTs for rating criteria and experts as shown in Table3. For example, for the LT “very good/very important (VG/VI),” we have numeric value 0,96,0.15,0.12,0.65,0.10,0.21 as a SLDFN. If an expert assign this LT to a criteria for a decision-making problem, then it means that criteria have 90% satisfaction value, 15% indeterminacy value, and 12% of dissatisfaction value. For the assigned LT, the reference parameters have 65% satisfaction value and 10% indeterminacy value and 21% dissatisfaction value corresponding to the three grades, and these reference parameters control the behavior of membership, abstinence, and nonmembership grades.Step 3.
Consider we have four decision-makers or experts in the selection committee, and all experts have different opinions. The weight of each decision maker is calculated by utilizing equation (7) as follows: ω1=1−0.2151/3/4−0.2151/3−0.3497/3−0.5758/3−0.9749/3=0.3073, and similarly, ω2=0.2764, ω3=0.2358, ω4=0.1804. Hence, the weight vector of four experts is ω=0.3073,0.2764,0.2358,0.1804T with the constraint ∑k=14ωk=1.Step 4.
Table2 contains the LT as well as the SLDFNs. These LTs are used to rate the alternatives based on the supplied criteria. The assessment values of each alternative Ai: i=1,2,3,⋯,p=5 according to each criteria Cj: j=1,2,3,⋯,q=6 are provided by four experts given in Table 4, and it is based on SLDFNs. For alternative Ai, i=1,2,3,4,5, and criteria Cj, j=1,2,3,4,5,6, we have the following LTs according to these four decision-makers. For the alternative Ai and criteria Cj, we have four LTs for four decision-makers: HI, VHI, CHI, and EE, respectively. We have six SLDFNs corresponding to six criteria. We now apply equation (8) to do further input data computations. We apply equation (8) to Table 4 to obtain the aggregated SLDF decision table shown in Table 5. For the alternative a1 and for criteria c1, we will use the LTs “MG, G, MG, and F” from Table 3. We use the weight vector ω=0.3073,0.2764,0.2358,0.1804T calculated in Step 3 and calculate the aggregated value by using equation (8) given in Table 4.Step 5.
We compute the aggregated weights for the six criteria listed in Table6 using equation (10). Because equation (10) is the same aggregated formula as equation (8), we use it to aggregate weights for six criteria, and all of the computations are the same as in Step 4.Step 6.
We now assess the aggregated SLDF decision table using aggregated weights and alternative ratings. Using equation (11), we get Table 7. Table 7 is obtained by multiplying Table 5 with Table 6. This multiplication is done according to the definition of multiplication of two SLDFNs. The first SLDFN for option a1 and criteria c1 is 0.7451,1.0000,0.2175,0.4345,0.3000,0.2030 shown in Table 5, while the second SLDFN for criteria c1 is 0.9138,0.1885,0.2939,0.6270,0.1230,0.2176 shown in Table 6. We acquire an SLDFN as an aggregated decision SLDFN 0.6809,1.0000,0.4475,0.2724,0.3861,0.3764 for option a1 and criteria c1 by multiplying both SLDFNs. The same approach may be used to compute all other values.Step 7.
Now, using equations (12) and (13) in Table 7, we determine Table 8 SLDF relative PIS and NIS for the calculated aggregated weight. The mathematical methods for selecting MG, AG, and NMG from the aggregated decision Table 7 are briefly described in equations (12) and (13).Step 8.
Let us calculate the distance measure by using the proposed algorithm from relative PIS and relative NIS provided in Tables7 and 8. For positive ideal solution, we find the normalized Euclidean distance between six criteria of the alternative a1 given in Table 7 and Sj+ given in Table 8. The value DEN+ for alternative a1 is calculated as follows: DEN+=1/6×60.6809−0.68092+1.0000−0.58932+0.4475−0.44752+0.2724−0.27242+0.3861−0.34372+0.3764−0.37642+⋯+0.6294−0.68572+0.9035−0.87702+0.4644−0.37802+0.2116−0.24312+0.3682−0.31822+0.4405−0.40532=0.0191. On the same pattern, we can calculate all other values for PIS and NIS corresponding to all the alternatives. The distance measure of each alternative is given in Table 9.Step 9.
Calculating the SLDF relative closeness asCj+=DEN−/DEN++DEN−=0.0117/0.0191+0.0117=0.3796 and its ranking is presented in Table 10.Step 10.
The priority structure of the green supplier as seen in Table10 (see Figure 2) is a2>a5>a4>a1>a3; thus, a2 is the best green supplier.Table 3
Expert’s importance in terms of SLDFNs.
ExpertLinguistic variablesSLDF numberse1Very good (VG)0.96,0.15,0.12,0.65,0.10,0.21e2Good (G)0.90,0.20,0.30,0.62,0.13,0.22e3Medium good (MG)0.85,0.33,0.41,0.57,0.17,0.25e4Fair (F)0.72,0.45,0.53,0.47,0.19,0.31Table 4
Judgments/assessments given by four experts in terms of LTs.
AlternativesExpertsCriteriac1c2c3c4c5c6a1e1CHIAAIHIBAILIEEe2AIVHIEECHIAAIHIe3EEAIAAILIHIVHIe4CHIEEVHIAIHIAIa2e1LIHICHIVHIEEAIe2AAIHICHIVHIHIEEe3LIVHIBAIHIAIAAIe4AAIVHIEECHIVHIEEa3e1CHIAAIEEAILIVHIe2EEBAICHIHIAAIVHIe3LIAIAAIHIEEHIe4LIEEVHIHIAAICHIa4e1EEAAICHIAIVHIAAIe2AIEEHIHILIVHIe3HIEEBAICHIAAIEEe4CHILIVHIAAIEEAIa5e1LIAAIVHIEEHICHIe2HIAAIAIVHIEEHIe3EELIHICHIAAIVHIe4CHIAAIHIVHIEEAIWeightse1GFMGGVGVGe2GVGVGFGGe3VGFGGVGFe4GGVGFGVGTable 5
Aggregated SLDF decision table.
AlternativesCriteriaSLDFNsa1c10.7451,1.0000,0.2175,0.4345,0.3000,0.2030c20.6365,0.8160,0.2979,0.3509,0.2934,0.3121c30.7226,0.8627,0.2192,0.3674,0.2877,0.2485c40.4716,1.0000,0.6062,0.2132,0.2924,0.4778c50.5132,0.6584,0.5066,0.1879,0.3095,0.4649c60.7143,0.8721,0.2360,0.3560,0.2736,0.2662a2c10.4303,0.4939,0.6573,0.1766,0.2774,0.5388c20.7087,0.8501,0.1598,0.2669,0.3584,0.2950c30.7288,1.0000,0.3180,0.4308,0.3130,0.2075c40.7782,1.0000,0.1127,0.3562,0.3690,0.2200c50.6901,0.8612,0.2589,0.3416,0.2682,0.2854c60.6577,0.8456,0.3289,0.4089,0.2161,0.2837a3c10.6184,1.0000,0.5217,0.3058,0.2516,0.3676c20.5065,0.7053,0.5073,0.2770,0.2585,0.4186c30.8002,1.0000,0.1761,0.4993,0.2930,0.1548c40.5725,0.7451,0.2992,0.1969,0.3292,0.4040c50.5459,0.7275,0.5234,0.2811,0.2503,0.4121c60.7782,1.0000,0.1127,0.3562,0.3690,0.2200a4c10.6932,1.0000,0.2619,0.3574,0.2694,0.2696c20.6743,0.8540,0.4118,0.4153,0.2008,0.2860c30.6445,1.0000,0.3409,0.2926,0.3444,0.3136c40.5998,1.0000,0.2936,0.2670,0.3356,0.3501c50.6023,0.7977,0.4603,0.2981,0.2794,0.3759c60.6655,0.8369,0.2805,0.3797,0.2846,0.2852a5c10.6248,1.0000,0.4574,0.2734,0.2690,0.3660c20.4944,0.5892,0.5077,0.2428,0.3076,0.4449c30.6181,0.7990,0.2636,0.2440,0.3377,0.3617c40.8699,1.0000,0.1131,0.5246,0.3023,0.1184c50.7468,0.8788,0.2339,0.4154,0.2404,0.2187c60.7003,1.0000,0.1925,0.3130,0.3554,0.2732Table 6
Aggregated SLDF criteria weight table.
AlternativesCriteriaSLDFNswc10.9138,0.1885,0.2939,0.6270,0.1230,0.2176c20.8116,0.3363,0.4259,0.5404,0.1552,0.2677c30.9108,0.2211,0.2745,0.6174,0.1291,0.2248c40.8019,0.3534,0.4907,0.5356,0.1672,0.2693c50.9321,0.1732,0.2424,0.6361,0.1138,0.2146c60.8812,0.2457,0.2990,0.5944,0.1303,0.2375Table 7
Aggregated weighted SLDF decision table.
AlternativesCriteriaSLDFNsa1c10.6809,1.0000,0.4475,0.2724,0.3861,0.3764c20.5166,0.8779,0.5969,0.1896,0.4031,0.4963c30.6581,0.8931,0.4335,0.2268,0.3797,0.4174c40.3782,1.0000,0.7994,0.1142,0.4107,0.6184c50.4784,0.7176,0.6262,0.1195,0.3881,0.5797c60.6294,0.9035,0.4644,0.2116,0.3682,0.4405a2c10.3932,0.5893,0.7580,0.1107,0.3663,0.6392c20.5752,0.9005,0.5176,0.1442,0.4580,0.4837c30.6638,1.0000,0.5052,0.2660,0.4017,0.3857c40.6240,1.0000,0.5481,0.1908,0.4745,0.4301c50.6432,0.8852,0.4385,0.2173,0.3515,0.4388c60.5796,0.8835,0.5296,0.2431,0.3182,0.4538a3c10.5651,1.0000,0.6623,0.1917,0.3437,0.5052c20.4111,0.8044,0.7171,0.1497,0.3736,0.5742c30.7288,1.0000,0.4023,0.3083,0.3843,0.3448c40.4591,0.8352,0.6431,0.1055,0.4414,0.5645c50.5088,0.7747,0.6389,0.1788,0.3356,0.5383c60.6857,1.0000,0.3780,0.2117,0.4512,0.4053a4c10.6334,1.0000,0.4788,0.2241,0.3593,0.4285c20.5473,0.9031,0.6623,0.2244,0.3248,0.4771c30.5870,1.0000,0.5218,0.1807,0.4290,0.4679c40.4810,1.0000,0.6402,0.1430,0.4467,0.5251c50.5614,0.8327,0.5911,0.1896,0.3614,0.5098c60.5864,0.8770,0.4956,0.2257,0.3778,0.4550a5c10.5709,1.0000,0.6169,0.1714,0.3589,0.5040c20.4013,0.7274,0.7174,0.1312,0.4151,0.5935c30.5630,0.8434,0.4657,0.1506,0.4232,0.5052c40.6976,1.0000,0.5483,0.2810,0.4190,0.3558c50.6961,0.8998,0.4196,0.2642,0.3268,0.3864c60.6171,1.0000,0.4339,0.1860,0.4394,0.4458Table 8
Relative SLDF: PIS and NIS.
AlternativesCriteriaSLDFNsSj+c10.6809,0.5893,0.4475,0.2724,0.3437,0.3764c20.5752,0.7274,0.5176,0.2244,0.3248,0.4771c30.7288,0.8434,0.4023,0.3083,0.3797,0.3448c40.6976,0.8352,0.5481,0.2810,0.4107,0.3558c50.6961,0.7176,0.4196,0.2642,0.3268,0.3864c60.6857,0.8770,0.3780,0.2431,0.3182,0.4053Sj−c10.3932,1.0000,0.7580,0.1107,0.3861,0.6392c20.4013,0.9031,0.7174,0.1312,0.4580,0.5935c30.5630,1.0000,0.5218,0.1506,0.4290,0.5052c40.3782,1.0000,0.7994,0.1055,0.4745,0.6184c50.4784,0.8998,0.6389,0.1195,0.3881,0.5797c60.5796,1.0000,0.5296,0.1860,0.4512,0.4550Table 9
Distance measure of each alternative.
AlternativesDEN+DEN−a10.01910.0117a20.01380.0161a30.01830.0091a40.01580.0097a50.01450.0170Table 10
SLDF closeness coefficient of each alternative.
AlternativesCj+Ranka10.37964a20.53911a30.33305a40.38043a50.53852Figure 2
Priority order of the green supplier.


## 5. Comparison, Limitations, and Advantages

(i)
In this manuscript, we present the perception of SLDFS and therefore a TOPSIS-based algorithm. The model’s most notable feature is that it envelopes the assessment spaces of FSs, IFSs, q-ROFSs, PFSs, SFSs, and q-RPFSs. It also has the reference parameters, which is an additional point, and the comparison analysis is given in Table11(ii)
Table12 shows the ranking results of five options using known methodologies and a novel notion utilizing the TOPSIS method. Comparison in Table 12 illustrates that the optimal selection produced by the suggested method is more like the current approaches, which is expressive in and of itself and validates the new method’s dependability and validity. Now comes the question of why we need to describe a novel algorithm based on this unique structure. There are several grounds, as described above, why the suggested methodology is superior to other previous techniques(iii)
As a limitation, SLDFS does not provide information about the roughness of data and cannot handle multivalued parameterizations(iv)
This framework can also be used to define and solve MCDM problems in a broader context. The developed TOPSIS technique based on SLDFSs outperforms several current methods. There is a little disparity betwixt the suggested approach’s ranking results and those found in the literature. In reality, depending on their ordering tactics, various equations can allow somewhat different effects. Because of the addition of reference parameters in SLDFSs, more comprehensive information may be extracted from data input. So, when comparing to other methodologies, our SLDFS-based technique provides more precise findingsTable 11
Comparison between some existing sets and SLDFS.
SetAdvantagesLimitationsFS [12, 13]It can handle imprecise informationIt cannot handle nonmembership valuesIFS [5, 6, 25]It contains the membership and nonmembership valuesIt will not work for the caset+f>1PyFS [7–9, 23]It can be used when the sum of membership and nonmembership grades exceeds 1It will not work for the caset2+f2>1q-ROFS [10, 11]It is more efficient than PyFs. We can handle the sum ofq-power of membership and nonmembership values less than 1It cannot be used fort=f=1 or if “q” is small with tq+fq>1PFS [14]It is efficient than IFS. This theory is used when we express the sum of membership, neutral, and nonmembership less than 1It will not work for the caset+a+f>1SFS [15]It is better than PFS and PyFS. It works for the caset2+a2+f2<1It will not work for the caset2+a2+f2>1q-RPFS [17]It is better than PFS. It could handle when the sum ofq-power of membership, neutral, and nonmembership is less than 1It cannot be used fort=a=f=1 or if the value of “q” is smaller with tq+aq+fq>1SLDF (proposed)(1) It can handle all circumstances where FS, IFS, PyFS, q-ROFS, PFS, SFS, and q-RPFS cannot be usedCalculation is complicated(2) It takes a parameterization approach and operates under the effect of reference parameters(3) Membership, abstinence, and nonmembership grades may be freely selected from [0, 1]Table 12
Comparison with existing approaches.
TechniquesApproachRanking alternativesWang et al. [28]PFSa2>a1>a5>a4>a3Kahraman et al. [29]SFSa2>a4>a2>a1>a3Akram and Shumaiza [30]q-RPFSa2>a5>a4>a3>a1Proposed algorithmSLDFSa2>a5>a4>a1>a3


## 6. Conclusion

We looked into several fuzzy set extensions, such as IFSs, PFSs, and q-ROFSs. SLDFSs are a new fuzzy set extension that we started. This concept, when combined with the usage of two extra reference factors, can contribute to a more efficient and flexible structure for the modeling of fuzzy systems and DM under ambivalence circumstances. From a geometric standpoint, we compared SLDFSs to various current extensions of fuzzy sets. We introduced several fundamental operations for SLDFS and created the SLDFS-based TOPSIS approach by utilizing spherical linear Diophantine fuzzy geometric aggregation operators. Furthermore, we successfully applied the suggested strategy to a multiattribute decision-making issue involving the selection of green supply chain management systems. When compared to other current methodologies, numerical findings suggest that the SLDFS-based methodology is more realistic and versatile in real-world situations.



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*Source: 3136462-2022-07-29.xml*