EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1158-1179 ISSN 1307-5543 – ejpam.com Published by New York Business Global Data envelopment analysis in the context of spherical fuzzy inputs and outputs Kshitish Kumar Mohanta1, Deena Sunil Sharanappa1, Devika Dabke2, Lakshmi Narayan Mishra3, Vishnu Narayan Mishra1,∗ 1 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, 484 887, Madhya Pradesh, India 2 Department of Mathematics, Central University of Karnataka, Kalaburgi, 585 367, Karnataka, India 3 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632 014, Tamil Nadu, India Abstract. In this study, Data Envelopment Analysis (DEA) models are improved by employing spherical fuzzy sets (SFSs), which is an extension of generalized fuzzy sets. SFSs were recently in- troduced as a novel type of fuzzy set that allows decision-makers to express their level of uncertainty directly. As a result, SFSs provide a more preferred domain for decision-makers. Fundamental Charnes-Cooper-Rhodes (CCR) model is discussed on the context of spherical trapezoidal fuzzy numbers (STrFNs), which consider each data value’s truth, indeterminacy, and falsehood degrees, and a unique solution technique is implemented. This method converts a spherical fuzzy DEA (SF-DEA) model into three pair of crisp DEA model, which may then be solved using one of many existing approaches. The largest optimal interval is determined for each DMU such that the efficiency score lies inside that interval. Furthermore, an example demonstrates this novel method and clearly explains the DMUs’ ranking technique. 2020 Mathematics Subject Classifications: 90C90, 90C70, 90C08 Key Words and Phrases: Efficiency Analysis, Data Envelopment Analysis, Spherical Trape- zoidal Fuzzy Number, CCR Model 1. Introduction The idea of fuzzy set was established by Zadeh [41] in 1965, and fuzzy set theory has been widely employed in practical applications of uncertainty modeling. Many academics have been interested in fuzzy set theory as a result of its expansion and application. The membership degree of set elements of a fuzzy set was defined by the characteristic ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4391 Email addresses: kshitishkumar.math@gmail.com (K. K. Mohanta), deena.sunil@igntu.ac.in (D. S. Sharanappa), devikash131@gmail.com (D. Dabke), vishnunarayanmishra@gmail.com (V. N. Mishra), lakshminarayanmishra04@gmail.com (L. N. Mishra) https://www.ejpam.com 1158 © 2022 EJPAM All rights reserved. V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1159 function on the unit interval [0, 1] in the study of fuzzy set (FS) theory. The fuzzy set’s non-membership degree is calculated by subtracting the membership degree from 1. Atanassov [7] in 1986 expanded Zadeh’s fuzzy set notion to intuitionistic fuzzy set (IFS), and its membership and non-membership degrees are defined separately. However, the sum of IFSs’ membership and non-membership degrees must fall within the range [0, 1]. Smarandache [38] in 1999 proposed neutrosophic logic and neutrosophic sets (NSs) as an extension of intuitionistic fuzzy sets. A neutrosophic set is one in which each element of the universe contains different degrees of truthiness, indeterminacy, and falsehood. They can be calculated individually, and their sum can range between 0 and 3. NSs used in solving many optimization technique and MCDM problem [26, 33]. Yager [39] in 2013 developed Pythagorean fuzzy sets which have a membership degree and a non-membership degree that satisfy the condition that the square sum of membership and non-membership degrees is at most equal to one, and are a generalisation of Intuitionistic Fuzzy Sets (IFS). Cuong and Kreinovich [10] in 2014 invented picture fuzzy sets Picture fuzzy sets-based models may be appropriate in circumstances requiring additional varieties of human opinions, such as yes, abstain, no, and rejection. Kahraman and Gundogdu [24] in 2018 proposed spherical fuzzy sets (SFS) as an extension of Pythagorean, neutrosophic, and picture fuzzy sets. SFS allows decision makers to generalise additional extensions of fuzzy sets by constructing a membership function on a spherical surface and separately assigning the parameters of that membership function to a broader domain. SFS have been applied to many multicriteria decision-making methods [4, 6, 25, 32, 36]. The difference between Intuitionistic fuzzy set, Pythagorean fuzzy set, Neutrosophic fuzzy set, and spherical fuzzy set are shown in Figure (1) where TA, FA and IA represents the truth, falsity and indeterminate membership grade for the fuzzy set A. Figure 1: Representation of the fuzzy set and it’s extension in geometrically 7 3d drawing ⟨0, 0, 0⟩ ⟨1, 0, 0⟩ ⟨0, 1, 0⟩ ⟨0, 0, 1⟩ TA FA IA Spherical Fuzzy Set Neutrosophic Fuzzy Set Intuitionistic Fuzzy Set Pythogorean Fuzzy Set 4 V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1160 For development, growth, and sustainability, all public or commercial organizations re- quire an accurate performance evaluation. In today’s competitive market, these businesses are under pressure to turn inputs into outputs at the lowest possible cost. This pressure motivates them to be more efficient. To be more specific, one of the major functions of government in the public sector, when the traditional disciplines of a competitive market are missing, is to offer public goods and services. As a result, identifying efficient providers can improve efficiency by acknowledging and disseminating best practices. Farrell [17] in 1957 developed the mathematical model for evaluating the efficiency of the DMUs, which was extended by Charnes et al. [9] in 1978 and developed a linear mathematical pro- gramming (LP) model to measure the comparative efficiency of the DMUs is called the Charnes-Cooper-Rhodes (CCR) model under the assumption of constant returns to scale (CRS). Banker et al. [8] in 1984 extended the pioneering work [9] and proposed a model conventionally called the Banker-Chames-Cooper (BCC) model to measure the relative efficiency under the assumption of variable returns to scale (VRS). The data envelop- ment analysis (DEA) is a non-parametric linear programming technique that considers the weighted sum of outputs to the weighted sum of inputs when evaluating the relative efficiency of a set of homogeneous DMUs. In the usual efficiency evaluation, it converts a single input/output ratio to a multiple input/output ratio. This approach is regarded as an effective multicriteria decision procedure and has been widely applied in various disciplines. In recent years, there has been a widespread use of DEA in a variety of in- dustries, including banking institutions [29], the insurance business [23], financial services [30], education [35], supply chain management [20], crisis management [34], sustainability [3], energy [18] and health-care services [28]. Sengupta [37] in 1992 used fuzzy sets in DEA for the first time. The DEA techniques employing fuzzy theory may be grouped into four basic groups, according to Hatami- Marbini et al. [22]: parametric approaches, possibility approaches, ranking approaches, defuzzification approaches, and many additional approaches have been brought to fuzzy DEA advancement. Emrouznejad et al. [16] in 2014 categorized the fuzzy DEA approaches in six types: the tolerance technique, the α-level based approach, the fuzzy ranking ap- proach, the possible approach, the fuzzy arithmetic, and the fuzzy random/type-2 fuzzy set and reviewed the literature during the last 30 years. Zhou and Xu [42] in 2020 provides a summary of the fuzzy data envelopment analysis research and its successful implemen- tations. Several ways to dealing with inaccurate, ambiguous, partial, and/or missing data in DEA have been proposed. To detect inaccurate input and output data, stochastic approaches [12] and interval DEA models are widely utilised. There have also been var- ious research articles published in DEA that make use of intuitionistic fuzzy sets [5, 21]. First time, Edalatpanah [13] in 2018 extended the DEA model in the context of single value Neutrosophic number. For offering a solution to the efficiency of private institu- tions. Kahraman et al. [27] in 2019 presented a hybrid algorithm based on a neutrosophic analytic hierarchy process (AHP) and DEA. Abdelfattah [1] in 2019 proposed a suitable approach for solving the DEA model in which all inputs and outputs are neutrosophic number. Following that, several approaches to solving DEA models utilising neutrosophic fuzzy sets are utilised [14, 15]. Mao et al. [31] in 2020 used single- valued neutrosophic sets V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1161 (SVNSs) in DEA with undesirable output. Yang et al. [40] in 2020 used triangular single valued neutrosophic number for measuring the hospital efficiency base on data envelop- ment analysis. Abdelfattah [2] in 2021 developed the parametric approach in neutrosophic data envelopment analysis and measured the efficiency of the regional hospitals in Tunisia using parametric neutrosophic data envelopment analysis. We noticed a few research gaps in this exciting field. SFSs are a generalisation and ex- tension of Picture Fuzzy Sets that define a membership function on a spherical surface and assign the parameters of that membership function independently over a larger domain, which has not been utilised in DEA with trapezoidal inputs and outputs. Trapezoidal fuzzy numbers are the most acceptable form of a fuzzy number because it covers more ambiguity than other fuzzy numbers. The acceptability area of SFTrNs provides better information assessment flexibility as a consequence of combining the benefits of SFSs with trapezoidal fuzzy numbers. Also, for SFSs, the situation of uncertain decision-making evaluations has not been considered. In this research, a novel efficient solution strategy is provided for solving Spherical Fuzzy DEA models in which all inputs and outputs are spherical trapezoidal fuzzy numbers (STrFNs) and the reference set or peer group for in- efficient DMUs is defined. We offered an example to show the method’s applicability and validity. Section (2) discusses some advanced knowledge, concepts, and arithmetic operations on SFs and STrFNs. In section (3), we create the previously proposed DEA model in spherical fuzzy environment. In section (4), offer a strategy for solving it. Section (5) presented a numerical illustration for the proposed model. Section (6) concludes with findings and future directions. 2. Preliminary Definition 1 ([19]). Let U be a universe. A spherical fuzzy set X̂ over U is defined by X̂ = {⟨x;ϕx, φx, ψx⟩ : x ∈ U}, (1) where ϕx, φx and ψx are called membership function, non-membership function and hesi- tancy function, respectively. They are respectively defined by ϕx, φx, ψx : U → [0, 1], such that 0 ≤ ϕ2x + φ2 x + ψ2 x ≤ 1. Definition 2 ([11]). A Spherical Trapezoidal Fuzzy Numbers (STrFNs) is denoted by X̂ = ⟨xL, xM1 , xM2 , xU ;ϕx, φx, ψx⟩, where the three membership functions for the truth, falsity, and indeterminacy of x can be defined as follows: τ(x) =  x− xL xM1 − xL ϕx, if x ∈ [xL, xM1 ], ϕx, if x ∈ [xM1 , xM2 ] xU − x xU − xM2 ϕx, if x ∈ [xM2 , xU ], 0, otherwise, (2) V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1162 ι(x) =  xM1 − x+ (x− xL)φx xM1 − xL , if x ∈ [xL, xM1 ] φx, if x ∈ [xM1 , xM2 ], xU − x+ (x− xM2)φx xU − xM2 , if x ∈ [xM2 , xU ], 1, otherwise, (3) ν(x) =  xM1 − x+ (x− xL)ψx xM1 − xL , if x ∈ [xL, xM1 ], ψx, if x ∈ [xM1 , xM2 ] xU − x+ (x− xM2)ψx xU − xM2 , if x ∈ [xM2 , xU ], 1, otherwise, (4) where 0 ≤ τ(x)2 + ι(x)2 + ν(x)2 ≤ 1, ∀ x ∈ X̂. Definition 3 ([11]). Suppose X̂i = ⟨xLi , x M1 i , xM2 i , xUi ;ϕxi , φxi , ψxi⟩, for i = 1, 2, · · · , n are n STrFNs. Then the arithmetic relations are defined as (i) X̂1⊕X̂2 = ⟨xL1+xL2 , x M1 1 +xM1 2 , xM2 1 +xM2 2 , xU1 +x U 2 ; (ϕ 2 x1 +ϕ2x2 −ϕ2x1 ϕ2x2 ) 1 2 , φx1φx2 , [(1− ϕ2x2 )ψ2 x1 + (1− ϕ2x1 )ψ2 x2 − ψ2 x1 ψ2 x2 ] 1 2 ⟩. (ii) X̂1−X̂1 = ⟨xL1−xU2 , x M1 1 −xM2 2 , xM2 1 −xM1 2 , xU1 −xL2 ; (ϕ2x1 +ϕ2x2 −ϕ2x1 ϕ2x2 ) 1 2 , φx1φx2 , [(1− ϕ2x2 )ψ2 x1 + (1− ϕ2x1 )ψ2 x2 − ψ2 x1 ψ2 x2 ] 1 2 ⟩. (iii) X̂1 ⊗ X̂1 = ⟨xL1 xL2 , x M1 1 xM1 2 , xM2 1 xM2 2 , xU1 x U 2 ;ϕx1ϕx2 , (φ 2 x1 + φ2 x2 − φ2 x1 φ2 x2 ) 1 2 , [(1 − φ2 x2 )ψ2 x1 + (1− φ2 x1 )ψ2 x2 − ψ2 x1 ψ2 x2 ] 1 2 ⟩. (iv) λX̂1 = { ⟨λxL1 , λx M1 1 , λxM2 1 , λxU1 ; (1− (1− ϕ2x1 )λ) 1 2 , φλ x1 , [(1− ϕ2x1 )λ − (1− ϕ2x1 − ψ2 x1 )λ] 1 2 ⟩, λ > 0. ⟨λxU1 , λx M2 1 , λxM1 1 , λxL1 ; (1− (1− ϕ2x1 )λ) 1 2 , φλ x1 , [(1− ϕ2x1 )λ − (1− ϕ2x1 − ψ2 x1 )λ] 1 2 ⟩, λ < 0. (v) ∑n i=1 λiX̂i = ⟨ ∑n i=1 λix L i , ∑n i=1 λix M1 i , ∑n i=1 λix M2 i , ∑n i=1 λix U i ; ( 1− ∏n i=1(1−ϕ2xi )λi )1/2 ,∏n i=1 φ λi xi , (∏n i=1(1− ϕ2xi )λi − ∏n i=1(1− ϕ2xi − ψ2 xi )λi )1/2 ⟩, ∀λi ≥ 0. Definition 4. The α−cut, β−cut and γ−cut for a STrFN X̂ = ⟨xL, xM1 , xM2 , xU ;ϕx, φx, ψx⟩, can be defined as X̂(α,β,γ) = {x : ϕx ≥ α,φ ≤ β, ψ ≤ γ}, (5) where 0 ≤ α ≤ ϕx, φx ≤ β ≤ 1 and ψx ≤ γ ≤ 1. Using definition (3) and equation (5) , the lower limits L(α), L(β) and L(γ), and upper limits U(α), U(β) and U(γ) of α, β and γ-level cut for STrFN are defined as X̂α = [L X̂ (α), U X̂ (α)] = [ xL + α( xM1 − xL ϕx ), xU − α( xU − xM2 ϕx ) ] , V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1163 X̂β = [L X̂ (β), U X̂ (β)] = [(β − φx)x L + (1− β)xM1 1− φx , (β − φx)x U + (1− β)xM2 1− φx ] , X̂γ = [L X̂ (γ), U X̂ (γ)] = [(γ − ψx)x L + (1− γ)xM1 1− ψx , (γ − ψx)x U + (1− γ)xM2 1− ψx ] , then X̂(α,β,γ) = (X̂α, X̂β, X̂γ). (6) Definition 5. Let X̂ and Ŷ are two STrFNs. The arithmatic relation for (α, β, γ)-cut of the STrFNs can be defined as (i) X̂p + Ŷp = [ L X̂ (p), U X̂ (p) ] + [ L Ŷ (p), U Ŷ (p) ] = [ L X̂ (p) + L Ŷ (p), U X̂ (p) + U Ŷ (p) ] , (ii) X̂p − Ŷp = [ L X̂ (p), U X̂ (p) ] − [ L Ŷ (p), U Ŷ (p) ] = [ L X̂ (p)− U Ŷ (p), U X̂ (p)− L Ŷ (p) ] , (iii) λX̂p =  [λL X̂ (p), λU X̂ (p)], λ > 0, 0, λ = 0, [λU X̂ (p), λL X̂ (p)], λ < 0, (iv) X̂p Ŷp = [L X̂ (p), U X̂ (p)] [L Ŷ (p), U Ŷ (p)] = [L X̂ (p) U Ŷ (p) , U X̂ (p) L Ŷ (p) ] , where p = α or β or γ. Remark 1. Any real number a ∈ R may be written as a spherical triangular fuzzy number a = ⟨a, a, a, a; 1, 0, 0⟩. 3. Spherical Fuzzy Data Envelopment Analysis (SF-DEA) Suppose that there are n decision making units (DMUs) each having m inputs and r outputs as represented by the vectors x ∈ Rm and y ∈ Rr, respectively. We define the input matrix X as X = [x1, · · · , xm] ∈ Rm×n, and the output matrix Y as Y = [yl, · · · , yr] ∈ Rr×n, xi ∈ Rm, ∀ i = 1, 2, · · · ,m, yk ∈ Rr, ∀ k = 1, 2, 3, · · · , r and assume that X > 0 and Y > 0. Charnes et al. [9] developed this model for measuring the efficiency of DMUo, (o = 1, 2, 3, · · · , n), that is, max uk,vi θ = ∑r k=1 ukyko∑m i=1 vixio , subject to ∑r k=1 ukykj∑m i=1 vixij ≤ 1, j = 1, 2, · · ·n, (7) uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m, V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1164 which is equivalent to the linear programming(LPo) problem, i.e, max uk,vi θ = r∑ k=1 ukyko, subject to m∑ i=1 vixio = 1, (8) r∑ k=1 ukykj ≤ m∑ i=1 vixij , j = 1, 2, · · ·n, uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m, which is called CCR model. If any of the observed data for inputs and/or outputs in this model are inaccurate, unclear, or ambiguous, then the efficiency score of the DMUo will be inaccurate. Let us assume that inputs and outputs are STrFNs while the variables uk and vi are real numbers; thus, (α, β, γ)−cut approach of the CCR model will be written as follows: max uk,vi θ(α,β,γ) = r∑ k=1 ukŷko (α,β,γ), subject to m∑ i=1 vix̂io (α,β,γ) = 1̂(α,β,γ), (9) r∑ k=1 ukŷkj (α,β,γ) ≤ m∑ i=1 vix̂ij (α,β,γ), j = 1, 2, · · ·n uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m, where x̂ij = ⟨xLij , x M1 ij , xM2 ij , xUij , ϕxij , φxij , ψxij ⟩ and ŷkj = ⟨yLkj , y M1 kj , y M2 kj , y U kj , ϕykj , φykj , ψykj ⟩ for i = 1, 2, 3, · · · , n, j = 1, 2, 3, · · · ,m, k = 1, 2, 3, · · · , r, and 1̂ = ⟨1, 1, 1, 1; 1, 0, 0⟩ are the STrFNs and the efficiency score is lies between 0 and 1. That implies max uk,vi θ(α,β,γ) = r∑ k=1 uk ([ Lŷko(α), Uŷko(α) ] , [ Lŷko(β), Uŷko(β) ] (10) [ Lŷko(γ), Uŷko(γ) ]) , s.t. m∑ i=1 vi ([ Lx̂io (α), Ux̂io (α) ] , [ Lx̂io (β), Ux̂io (β) ] , [ Lx̂io (γ), Ux̂io (γ) ]) V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1165 = ([ L1̂(α), U1̂(α) ] , [ L1̂(β), U1̂(β) ] , [ L1̂(γ), U1̂(γ) ]) , r∑ k=1 uk ( [Lŷkj (α), Uŷkj (α) ] , [ Lŷkj (β), Uŷkj (β) ] , [ Lŷkj (γ), Uŷkj (γ) ]) − m∑ i=1 vi ([ Lx̂ij (α), Ux̂ij (α) ] , [ Lx̂ij (β), Ux̂ij (β) ] , [ Lx̂ij (γ), Ux̂ij (γ) ]) ≤ 0, j = 1, 2, · · ·n, and uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m. Using definition (5), we have max uk,vi θ(α,β,γ) = ([ r∑ k=1 ukLŷko(α), r∑ k=1 ukUŷko(α) ] r∑ k=1 uk [ Lŷko(β), r∑ k=1 ukUŷko(β) ] , r∑ k=1 uk [ Lŷko(γ), r∑ k=1 ukUŷko(γ) ]) , s.t. ([ m∑ i=1 viLx̂io (α), m∑ i=1 viUx̂io (α) ] , [ m∑ i=1 viLx̂io (β), m∑ i=1 viUx̂io (β) ] (11) [ m∑ i=1 viLx̂io (γ), m∑ i=1 viUx̂io (γ) ]) = ( [1, 1], [1, 1], [1, 1] ) , ([ r∑ k=1 ukLŷkj (α), r∑ k=1 ukUŷkj (α) ] , [ r∑ k=1 ukLŷkj (β), r∑ k=1 ukUŷkj (β) ] , [ r∑ k=1 ukLŷkj (γ), r∑ k=1 ukUŷkj (γ) ]) − ([ m∑ i=1 viLx̂ij (α), m∑ i=1 viUx̂ij (α) ] , [ m∑ i=1 viLx̂ij (β), m∑ i=1 viUx̂ij (β) ] , [ m∑ i=1 viLx̂ij (γ), m∑ i=1 viUx̂ij (γ) ]) ≤ 0, j = 1, 2, · · ·n, and uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m, which is the spherical fuzzy DEA model with (α, β, γ)−cut approach. The SF-DEA model converted into three pair of DEA models to evaluate the lower and upper bounds of the efficiency score in (α, β, γ)− cut approach. The mathematical model for α− cut approach V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1166 is defined as θαL ∗ = inf α∈[0,t1]  θαL = maxuk,vi ∑r k=1 ukLŷko(α), s.t ∑m i=1 viUx̂io (α) = 1,∑r k=1 ukLŷkj (α)− ∑m i=1 viUx̂ij (α) ≤ 0,∑r k=1 ukUŷkj (α)− ∑m i=1 viLx̂ij (α) ≤ 0, j = 1, 2, · · ·n, uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m, (12) θαU ∗ = sup α∈[0,t1]  θαU = maxuk,vi ∑r k=1 ukUŷko(α), s.t ∑m i=1 viLx̂io (α) = 1,∑r k=1 ukLŷkj (α)− ∑m i=1 viUx̂ij (α) ≤ 0,∑r k=1 ukUŷkj (α)− ∑m i=1 viLx̂ij (α) ≤ 0, j = 1, 2, · · ·n, uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m, (13) where t1 = inf(ϕxij , ϕykj ), ∀ i, j, k. Similarly, The mathematical model for β−cut and γ-cut approach are defined as fol- lows. θβL ∗ = inf β∈[t2,1]  θβL = maxuk,vi ∑r k=1 ukLŷko(β), s.t ∑m i=1 viUx̂io (β) = 1,∑r k=1 ukLŷkj (β)− ∑m i=1 viUx̂ij (β) ≤ 0∑r k=1 ukUŷkj (β)− ∑m i=1 viLx̂ij (β) ≤ 0, j = 1, 2, · · ·n uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m, (14) θβU ∗ = sup β∈[t2,1]  θβU = maxuk,vi ∑r k=1 ukUŷko(β), s.t ∑m i=1 viLx̂io (β) = 1,∑r k=1 ukLŷkj (β)− ∑m i=1 viUx̂ij (β) ≤ 0,∑r k=1 ukUŷkj (β)− ∑m i=1 viLx̂ij (β) ≤ 0, j = 1, 2, · · ·n, uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m. (15) V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1167 and θγL ∗ = inf γ∈[t3,1]  θγL = maxuk,vi ∑r k=1 ukLŷko(γ), s.t ∑m i=1 viUx̂io (γ) = 1,∑r k=1 ukLŷkj (γ)− ∑m i=1 viUx̂ij (γ) ≤ 0,∑r k=1 ukUŷkj (γ)− ∑m i=1 viLx̂ij (γ) ≤ 0, j = 1, 2, · · ·n, uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m. (16) θγU ∗ = sup γ∈[t3,1]  θγU = maxuk,vi ∑r k=1 ukUŷko(γ), s.t ∑m i=1 viLx̂io (γ) = 1,∑r k=1 ukLŷkj (γ)− ∑m i=1 viUx̂ij (γ) ≤ 0,∑r k=1 ukUŷkj (γ)− ∑m i=1 viLx̂ij (γ) ≤ 0, j = 1, 2, · · ·n, uk ≥ 0, k = 1, 2, · · · , r, vi ≥ 0, i = 1, 2, · · · ,m, (17) where t2 = sup(φxij , φykj ), ∀i, j, k and t3 = sup(ψxij , ψykj ), ∀ i, j, k. The efficiency score in α−cut, β−cut and γ−cut approach must be lies in the optimal interval [θαL ∗, θαU ∗], [θβL ∗ , θβU ∗ ] and [θγL ∗ , θγU ∗ ] respectively. Theorem 1. The lower bound of the optimal interval in (α, β, γ)- cut are equal, that is θαL ∗ = θβL ∗ = θγL ∗ . (18) Proof. Since the (α, β, γ)- cut for a SFTrN X̂ = ⟨xL, xM1 , xM2 , xU ;ϕx, φx, ψx⟩ is defined in equation (6), we have lim (α,β,γ)→(0,1,1) X̂(α,β,γ) = ([ L X̂ (0), U X̂ (0) ] , [ L X̂ (1), U X̂ (1) ] , [ L X̂ (1), U X̂ (1) ]) , = ( [xL, xU ], [xL, xU ], [xL, xU ] ) . It follows that lim α→0 θαL = lim β→1 θβL = lim γ→1 θγL. θαL ∗ = θβL ∗ = θγL ∗ . V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1168 4. Method for Solving SF-DEA model Let us consider the inputs and outputs of the DMUs are the STrFNs . The following steps can be used to calculate the efficiency score of the DMUs. Step 2: Transform the DEA model into the SF-DEA model using the (α, β, γ)-cut technique, as shown in equation (9) & (10). Step 2: Convert three pairs of crisp DEA models as shown in the equation (12) & (13), equation (14) & (15), and equation (16) & (17). Step 3: Solve this crisp DEA model and find the optimal interval [θαL ∗, θαU ∗], [θβL ∗ , θβU ∗ ] and [θγL ∗ , θγU ∗ ] for α-cut, β−cut and γ-cut respectively. Step 4: The largest optimal interval [θ∗L, θ ∗ U ] for each DMU was computed by taking the union of the optimal intervals and ranking all DMUs based on the mean efficiency score of each DMUs. That is [θ∗L, θ ∗ U ] = [θαL ∗, θαU ∗] ∪ [θβL ∗ , θβU ∗ ] ∪ [θγL ∗ , θγU ∗ ]. (19) Mean efficiency(θ) = θ∗L + θ∗U 2 . (20) The solution method for the SF-DEA model is depicted in the flow chart shown in Figure (2). Figure 2: Method of Solution for SF-DEA model Plotting Technique In LATEX 5 Collect inputs and outputs Fuzzifier STrFNs Inputs and Outputs SF-DEA model Convert crisp DEA model Optimal interval for (α, β, γ)− cut largest opti- mal interval Mean Efficiency (α, β, γ)−approach Solving Union V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1169 5. Numerical Example Let us consider 12 DMUs with inputs and outputs are STrFNs, shown in Table (1) and Table (2). The efficiency score for each DMU was evaluated by proceeding with the above technique given in Section (4). Table 1: Spherical fuzzy Inputs data DMU Input 1 Input 2 Input 2 D1 ⟨11, 15, 18, 25; 0.8, 0.3, 0.4⟩ ⟨31, 35, 40, 50; 0.5, 0.4, 0.6⟩ ⟨12, 14, 19, 21; 0.7, 0.1, 0.3⟩ D2 ⟨6, 7, 11, 13; 0.7, 0.5, 0.5⟩ ⟨12, 18, 25, 30; 0.5, 0.2, 0.4⟩ ⟨10, 14, 16, 19; 0.4, 0.3, 0.6⟩ D3 ⟨17, 21, 24, 28; 0.9, 0.2, 0.3⟩ ⟨48, 50, 55, 60; 0.8, 0.5, 0.3⟩ ⟨30, 35, 38, 42; 0.5, 0.2, 0.4⟩ D4 ⟨11, 15, 17, 24; 0.5, 0.2, 0.5⟩ ⟨22, 25, 27, 35; 0.8, 0.2, 0.4⟩ ⟨8, 14, 18, 23; 0.7, 0.5, 0.3⟩ D5 ⟨22, 25, 27, 31; 0.8, 0.4, 0.1⟩ ⟨34, 38, 41, 46; 0.6, 0.1, 0.5⟩ ⟨24, 30, 32, 38; 0.4, 0.5, 0.2⟩ D6 ⟨13, 19, 24, 28; 0.6, 0.2, 0.5⟩ ⟨36, 41, 47, 51; 0.4, 0.5, 0.5⟩ ⟨12, 13, 17, 22; 0.9, 0.1, 0.2⟩ D7 ⟨20, 24, 27, 32; 0.4, 0.3, 0.4⟩ ⟨41, 44, 50, 52; 0.6, 0.4, 0.3⟩ ⟨3, 7, 11, 15; 0.8, 0.2, 0.4⟩ D8 ⟨11, 12, 15, 18; 0.7, 0.4, 0.6⟩ ⟨32, 34, 37, 40; 0.7, 0.4, 0.4⟩ ⟨16, 19, 21, 24; 0.7, 0.4, 0.1⟩ D9 ⟨21, 24, 31, 35; 0.9, 0.3, 0.3⟩ ⟨41, 45, 47, 55; 0.5, 0.4, 0.4⟩ ⟨4, 7, 11, 17; 0.4, 0.5, 0.3⟩ D10 ⟨17, 18, 21, 24; 0.9, 0.3, 0.1⟩ ⟨51, 58, 61, 65; 0.9, 0.3, 0.2⟩ ⟨21, 24, 26, 30; 0.8, 0.2, 0.4⟩ D11 ⟨9, 12, 18, 22; 0.6, 0.3, 0.5⟩ ⟨13, 18, 23, 27; 0.7, 0.3, 0.3⟩ ⟨31, 34, 37, 45; 0.7, 0.3, 0.3⟩ D12 ⟨18, 24, 27, 32; 0.4, 0.4, 0.2⟩ ⟨51, 54, 58, 63, 0.6, 0.2, 0.5⟩ ⟨32, 36, 39, 41; 0.5, 0.1, 0.4⟩ Table 2: Spherical fuzzy Outputs data DMU Output 1 Output 2 D1 ⟨118, 123, 125, 135; 0.7, 0.1, 0.4⟩ ⟨134, 137, 141, 148; 0.9, 0.2, 0.3⟩ D2 ⟨134, 138, 140, 144; 0.4, 0.3, 0.3⟩ ⟨182, 186, 189, 192; 0.5, 0.3, 0.2⟩ D3 ⟨205, 209, 215, 220; 0.6, 0.1, 0.3⟩ ⟨141, 145, 147, 150; 0.7, 0.5, 0.4⟩ D4 ⟨123, 127, 132, 134; 0.8, 0.1, 0.2⟩ ⟨128, 131, 134, 138; 0.4, 0.7, 0.3⟩ D5 ⟨194, 196, 200, 215; 0.6, 0.2, 0.5⟩ ⟨184, 186, 190, 203; 0.8, 0.5, 0.2⟩ D6 ⟨140, 145, 147, 152; 0.5, 0.2, 0.5⟩ ⟨94, 106, 111, 115; 0.6, 0.4, 0.1⟩ D7 ⟨112, 118, 126, 131; 0.7, 0.4, 0.5⟩ ⟨170, 176, 181, 185; 0.4, 0.7, 0.2⟩ D8 ⟨141, 146, 153, 155; 0.8, 0.5, 0.3⟩ ⟨129, 136, 141, 144; 0.7, 0.2, 0.4⟩ D9 ⟨67, 78, 82, 88; 0.6, 0.5, 0.2⟩ ⟨211, 218, 222, 225; 0.5, 0.3, 0.6⟩ D10 ⟨161, 167, 178, 181; 0.4, 0.6, 0.5⟩ ⟨141, 148, 152, 155; 0.7, 0.5, 0.1⟩ D11 ⟨117, 126, 129, 137; 0.8, 0.5, 0.3⟩ ⟨125, 128, 134, 138; 0.6, 0.1, 0.3⟩ D12 ⟨136, 139, 143, 147; 0.7, 0.6, 0.2⟩ ⟨185, 188, 194, 198; 0.4, 0.6, 0.6⟩ The SF-DEA model for DMU D1 can be written as max u,v θ = ⟨118, 123, 125, 135; 0.7, 0.1, 0.4⟩u1 + ⟨134, 137, 141, 148; 0.9, 0.2, 0.3⟩u2, s.t ⟨11, 15, 18, 25; 0.8, 0.3, 0.4⟩v1 + ⟨31, 35, 40, 50; 0.5, 0.4, 0.6⟩v2 + ⟨12, 14, 19, 21; 0.7, 0.1, 0.3⟩v3 = 1, V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1170 ⟨118, 123, 125, 135; 0.7, 0.1, 0.4⟩u1 + ⟨134, 137, 141, 148; 0.29, 0.2, 0.3⟩u2 ≤ ⟨11, 15, 18, 25; 0.8, 0.3, 0.4⟩v1 + ⟨31, 35, 40, 50; 0.5, 0.4, 0.6⟩v2 + ⟨12, 14, 19, 21; 0.7, 0.1, 0.3⟩v3, ⟨134, 138, 140, 144; 0.4, 0.3, 0.3⟩u1 + ⟨182, 186, 189, 192; 0.5, 0.3, 0.2⟩u2 ≤ ⟨6, 7, 11, 13; 0.7, 0.5, 0.5⟩v1 + ⟨12, 18, 25, 30; 0.5, 0.2, 0.4⟩v2 + ⟨10, 14, 16, 19; 0.4, 0.3, 0.6⟩v3, ... ⟨136, 139, 143, 147; 0.7, 0.6, 0.2⟩u1 + ⟨185, 188, 194, 198; 0.4, 0.6, 0.6⟩u2 ≤ ⟨18, 24, 27, 32; 0.4, 0.4, 0.2⟩v1 + ⟨51, 54, 58, 63; 0.6, 0.2, 0.5⟩v2 + ⟨32, 36, 39, 41; 0.5, 0.1, 0.4⟩v3, and u1, u2, v1, v2, v3 ≥ 0. Step 1 Using equation (11), the (α, β, γ)−cut approach of the SF-DEA model for the DMU D1 can be written as follows: max u,v θα,β,γ = ([( 118 + 2α 0.7 ) u1 + ( 134 + 3α 0.9 ) u2, ( 135− 10α 0.7 ) u1 + ( 148− 7α 0.9 ) u2 ] , [(β − 0.1)118 + (1− β)123 0.9 u1 + (β − 0.2)134 + (1− β)137 0.8 u2, (β − 0.1)135 + (1− β)125 0.9 u1 + (β − 0.2)148 + (1− β)141 0.8 u2 ] , [(γ − 0.4)118 + (1− γ)123 0.6 u1 + (γ − 0.3)134 + (1− γ)137 0.7 u2, (γ − 0.4)135 + (1− γ)125 0.6 u1 + (γ − 0.3)148 + (1− γ)141 0.7 u2 ]) , s.t ([( 11 + 4α 0.8 ) v1 + ( 31 + 4α 0.5 ) v2 + ( 12 + 2α 0.7 ) v3, ( 25− 7α 0.8 ) v1 + ( 50− 10α 0.5 ) v2 + ( 21− 2α 0.7 ) v3 ] , [(β − 0.3)11 + (1− β)15 0.7 v1 + (β − 0.4)31 + (1− β)35 0.6 v2 + (β − 0.1)12 + (1− β)14 0.9 v3, (β − 0.3)25 + (1− β)18 0.7 v1 + (β − 0.4)50 + (1− β)40 0.6 v2 + (β − 0.1)21 + (1− β)19 0.9 v3 ] , [(β − 0.4)11 + (1− β)15 0.6 v1 + (β − 0.6)31 + (1− β)35 0.4 v2 + (β − 0.3)12 + (1− β)14 0.7 v3, V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1171 (β − 0.4)25 + (1− β)18 0.6 v1 + (β − 0.6)50 + (1− β)40 0.4 v2 + (β − 0.3)21 + (1− β)19 0.7 v3 ]) = 1,([( 118 + 2α 0.7 ) u1 + ( 134 + 3α 0.9 ) u2, ( 135− 10α 0.7 ) u1 + ( 148− 7α 0.9 ) u2 ] , [(β − 0.1)118 + (1− β)123 0.9 u1 + (β − 0.2)134 + (1− β)137 0.8 u2, (β − 0.1)135 + (1− β)125 0.9 u1 + (β − 0.2)148 + (1− β)141 0.8 u2 ] ,[(γ − 0.4)118 + (1− γ)123 0.6 u1 + (γ − 0.3)134 + (1− γ)137 0.7 u2, (γ − 0.4)135 + (1− γ)125 0.6 u1 + (γ − 0.3)148 + (1− γ)141 0.7 u2 ]) ≤ ([( 11 + 4α 0.8 ) v1 + ( 31 + 4α 0.5 ) v2 + ( 12 + 2α 0.7 ) v3, ( 25− 7α 0.8 ) v1 + ( 50− 10α 0.5 ) v2 + ( 21− 2α 0.7 ) v3 ] , [(β − 0.3)11 + (1− β)15 0.7 v1 + (β − 0.4)31 + (1− β)35 0.6 v2 + (β − 0.1)12 + (1− β)14 0.9 v3, (β − 0.3)25 + (1− β)18 0.7 v1 + (β − 0.4)50 + (1− β)40 0.6 v2 + (β − 0.1)21 + (1− β)19 0.9 v3 ] , [(γ − 0.4)11 + (1− γ)15 0.6 v1 + (γ − 0.6)31 + (1− γ)35 0.4 v2 + (γ − 0.3)12 + (1− γ)14 0.7 v3, (γ − 0.4)25 + (1− γ)18 0.6 v1 + (γ − 0.6)50 + (1− γ)40 0.4 v2 + (γ − 0.3)21 + (1− γ)19 0.7 v3 ]) , ...([( 136 + 3α 0.7 ) u1 + ( 185 + 3α 0.4 ) u2, ( 147− 4α 0.7 ) u1 + ( 198− 4α 0.4 ) u2 ] , [(β − 0.6)136 + (1− β)139 0.4 u1 + (β − 0.6)185 + (1− β)188 0.4 u2, (β − 0.6)147 + (1− β)143 0.4 u1 + (β − 0.6)198 + (1− β)194 0.4 u2 ] , V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1172[(γ − 0.2)136 + (1− γ)139 0.8 u1 + (γ − 0.6)185 + (1− γ)188 0.4 u2, (γ − 0.2)147 + (1− γ)143 0.8 u1 + (γ − 0.6)198 + (1− γ)194 0.4 u2 ]) ≤ ([( 18 + 6α 0.4 ) v1 + ( 51 + 3α 0.6 ) v2 + ( 32 + 4α 0.5 ) v3, ( 32− 5α 0.4 ) v1 + ( 63− 5α 0.6 ) v2 + ( 41− α 0.5 ) v3 ] , [(β − 0.4)18 + (1− β)24 0.6 v1 + (β − 0.2)51 + (1− β)54 0.8 v2 + (β − 0.1)32 + (1− β)36 0.9 v3, (β − 0.4)32 + (1− β)27 0.6 v1 + (β − 0.2)63 + (1− β)58 0.8 v2 + (β − 0.1)41 + (1− β)39 0.9 v3 ] , [(γ − 0.2)18 + (1− γ)24 0.8 v1 + (γ − 0.5)51 + (1− γ)54 0.5 v2 + (γ − 0.4)32 + (1− γ)14 0.6 v3, (γ − 0.2)32 + (1− γ)27 0.8 v1 + (γ − 0.5)63 + (1− γ)58 0.5 v2 + (γ − 0.4)41 + (1− γ)39 0.6 v3 ]) , and u1, u2, v1, v2, v3 ≥ 0, where α ∈ [0, t1], β ∈ [t2, 1] and γ ∈ [t3, 1], t1 = inf(ϕxij , ϕykj ), t2 = sup(φxij , φykj ), t3 = sup(ψxij , ψykj ), ∀ i, j, k. V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1173 Step 2 The above model was transformed into three pair of crisp DEA models. θαL ∗ = inf α∈[0,0.4]  θαL = maxuk,vi ( 118 + 2α 0.7 ) u1 + ( 134 + 3α 0.9 ) u2 s.t ( 25− 7α 0.8 ) v1 + ( 50− 10α 0.5 ) v2 + ( 21− 2α 0.7 ) v3 = 1,( 118 + 2α 0.7 ) u1 + ( 134 + 3α 0.9 ) u2 − ( 25− 7α 0.8 ) v1 − ( 50 −10α 0.5 ) v2 − ( 21− 2α 0.7 ) v3 ≤ 0, ...( 136 + 3α 0.7 ) u1 + ( 185 + 3α 0.4 ) u2 − ( 32− 5α 0.4 ) v1 − ( 63 − 5α 0.6 ) v2 − ( 41− α 0.5 ) v3 ≤ 0,( 135− 10α 0.7 ) u1 + ( 148− 7α 0.9 ) u2 − ( 11 + 4α 0.8 ) v1 − ( 31 + 4α 0.5 ) v2 − ( 12 + 2α 0.7 ) v3 ≤ 0, ...( 147− 4α 0.7 ) u1 + ( 198− 4α 0.4 ) u2 − ( 18 + 6α 0.4 ) v1 − ( 51 + 3α 0.6 ) v2 − ( 32 + 4α 0.5 ) v3 ≤ 0, and u1, u2, v1, v2, v3 ≥ 0, θαU ∗ = sup α∈[0,0.4]  θαU = maxuk,vi ( 135− 10α 0.7 ) u1 + ( 148− 7α 0.9 ) u2, s.t ( 11 + 4α 0.8 ) v1 + ( 31 + 4α 0.5 ) v2 + ( 12 + 2α 0.7 ) v3 = 1,( 118 + 2α 0.7 ) u1 + ( 134 + 3α 0.9 ) u2 − ( 25− 7α 0.8 ) v1 − ( 50 −10α 0.5 ) v2 − ( 21− 2α 0.7 ) v3 ≤ 0, ...( 136 + 3α 0.7 ) u1 + ( 185 + 3α 0.4 ) u2 − ( 32− 5α 0.4 ) v1 − ( 63 − 5α 0.6 ) v2 − ( 41− α 0.5 ) v3 ≤ 0,( 135− 10α 0.7 ) u1 + ( 148− 7α 0.9 ) u2 − ( 11 + 4α 0.8 ) v1 − ( 31 + 4α 0.5 ) v2 − ( 12 + 2α 0.7 ) v3 ≤ 0, ...( 147− 4α 0.7 ) u1 + ( 198− 4α 0.4 ) u2 − ( 18 + 6α 0.4 ) v1 − ( 51 + 3α 0.6 ) v2 − ( 32 + 4α 0.5 ) v3 ≤ 0, and u1, u2, v1, v2, v3 ≥ 0, Similarly, other two pair of DEA models for β-cut and γ-cut determined using equa- tion (14) & (15) and equation (16) & (17). V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1174 Step 3 Solving the above three pair of DEA model using the value of α ∈ [0, 0.4], β ∈ [0.7, 1] and γ ∈ [0.6, 1] and obtain the optimal interval for efficiency score of the DMU D1. Similarly, the optimal interval for all DMUs were calculated as shown in Table (3). Table 3: Efficiency Score in α, β, γ−cut DMU [θαL ∗, θαU ∗] [θ β L ∗ , θ β U ∗ ] [θ γ L ∗, θγ U ∗] D1 [0.327213951, 0.544156783] [0.327213951, 0.458144777 ] [ 0.327213951, 0.487868248] D2 [ 0.482615392, 0.802766814] [ 0.482615392, 0.636611817] [ 0.482615392, 0.682741675] D3 [ 0.331550137, 0.526182357] [ 0.331550137, 0.430341206] [ 0.331550137, 0.458495452] D4 [ 0.352868079, 0.606499314] [ 0.352868079, 0.516346142] [ 0.352868079, 0.568938001] D5 [0.353924801, 0.622102844] [ 0.353924801, 0.495217527] [ 0.353924801, 0.533268136] D6 [0.373781071, 0.59695553] [ 0.373781071, 0.509640297] [ 0.373781071, 0.547609621] D7 [0.410372628, 0.636753029 ] [ 0.410372628, 0.561771929 ] [ 0.410372628, 0.600530298] D8 [0.388465032, 0.63146065 ] [ 0.388465032, 0.532694159 ] [ 0.388465032, 0.571762135] D9 [ 0.450806804, 0.808444737 ] [ 0.450806804, 0.652290118] [ 0.450806804, 0.721735487] D10 [0.349282411, 0.566211624] [ 0.349282411, 0.479628115 ] [ 0.349282411, 0.510998311] D11 [0.36111111, 0.643058285] [ 0.36111111, 0.523612246] [ 0.36111111, 0.559335959] D12 [0.220904311, 0.344251309] [ 0.220904311, 0.281483055] [ 0.220904311, 0.29655302] Step 4 The largest optimal interval for DMU D1 is obtained by using equation (19), that is, union of the optimal interval for (α, β, γ)−cut. Similarly, the largest optimal interval for other DMUs are obtained as shown in Table (4). The DMUs were compared by taking mean of the largest optimal interval as show in Table (4) and Figure (3). The DMUs have been ranked in the following order D2 > D9 > D7 > D8 > D11 > D5 > D6 > D4 > D10 > D1 > D3 > D12. The DMU D2 is highly efficient then other DMUs and D12 is the least efficient. Table 4: Optimal Interval, Mean efficiency score and Ranking DMU [θtL ∗ , θtU ∗ ] Mean Ranking D1 [0.327213951, 0.544156783] 0.435685367020560 10 D2 [ 0.482615392, 0.802766814] 0.642691102824787 1 D3 [ 0.331550137, 0.526182357] 0.428866246964638 11 D4 [ 0.352868079, 0.606499314] 0.479683696432378 8 D5 [0.353924801, 0.622102844] 0.488013822795279 6 D6 [0.373781071, 0.59695553] 0.485368300723181 7 D7 [0.410372628, 0.636753029 ] 0.523562828586664 3 D8 [0.388465032, 0.63146065 ] 0.509962840919284 4 D9 [ 0.450806804, 0.808444737 ] 0.629625770540691 2 D10 [0.349282411, 0.566211624] 0.457747017566756 9 D11 [0.36111111, 0.643058285] 0.502084697510003 5 D12 [0.220904311, 0.344251309] 0.282577809805037 12 6. Conclusion Spherical fuzzy sets (SFSs) are a relatively new academic topic rapidly growing in pop- ularity and being used to a wide range of decision-making concerns, particularly mathe- matical programming problems. This study focuses on DEA models with spherical fuzzy V. N. Mishra et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 1158-1179 1175 Figure 3: Mean Efficiency score in SF-DEA 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 DMUs E ff ic ie n c y S c o re Mean Efficiency inputs and outputs. We introduce the Spherical fuzzy DEA (SF-DEA) models and offer a unique approach to solve them.The SF-DEA model transformed three pairs of crisp DEA models to determine the optimal interval in which the (α, β, γ)-cut efficiency lies. The DMUs were ranked based on the mean efficiency score, with the largest optimal interval determined by combining the (α, β, γ)-cut optimal intervals. Finally, we offer an example to demonstrate the method’s applicability and validity. Future research should use this innovative technique to solve additional DEA mod- els, including the BCC and SBM models, and provides encouraging results. 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