A fuzzy DEA slacks-based approach

A Fuzzy DEA Slacks-Based Approach Manuel Arana-Jim´ eneza, M. Carmen S´ anchez-Gil a,1, Sebasti´ an Lozanob aDepartment of Statistics and Operational Research, University of C´adiz, Spain bDepartment of Industrial Management University of Seville, Spain Abstract This paper deals with the problem of efficiency assessment using Data Envelopment Analysis (DEA) when the input and output data are given as fuzzy sets. In particular, a fuzzy extension of the measure of inefficiency proportions, a well-known slacks-based additive inefficiency measure, is considered. The proposed approach also provides fuzzy input and output targets. Computational experiences and comparison with other fuzzy DEA approaches are reported. Keywords: Relative efficiency; fuzzy data; DEA; measure of inefficiency proportions; fuzzy targets 2018 MSC: 1. Introduction In order to assess the relative efficiency of a set of homogeneous Decision Making Units (DMUs) a non-parametric methodology, namely Data envelopment analysis (DEA), can be used. DEA only requires data about the inputs consumed and the outputs produced by the DMUs and, from that, using some5 optimization models, an efficiency score and an efficient target can be computed for each DMU ([49], [12]). There are di↵erent ways of carrying out the ⇤[email protected] Email addresses: [email protected] (Manuel Arana-Jim´ enez ), [email protected] (M. Carmen S´ anchez-Gil ), [email protected] (Sebasti´ an Lozano) Preprint submitted to Journal of L A T EX Templates August 26, 2020 Manuscript Click here to view linked References projection onto the efficient frontier and computing the corresponding efficiency scores: radial approaches (e.g. [29]), hyperbolic approaches (e.g. [18]), directional distance function approaches (e.g. [32]), slacks-based approaches10 (e.g. [43]), multidirectional efficiency approaches (e.g. [33]), efficiency potential approaches (e.g. [40]), etc. One of the challenges in DEA applications is the difficulty in quantifying the exact value of some of the input and output data in real-world problems, where the observed values are often undetermined or incomplete. One way15 to handle this uncertainty in DEA data is using fuzzy sets. Thus, imprecise or vague data in DEA can be represented by linguistic terms characterized by fuzzy numbers.[23]. There are di↵erent fuzzy DEA (FDEA) approaches that can be used when the data are fuzzy. The best reference on the subject is [17], which extended20 a previous review by [22] and proposed a taxonomy that classifies the FDEA methods into ↵-level set approaches (e.g. [27], [38]), fuzzy ranking approaches (e.g. [21], [20]), possibility approaches (e.g. [44]), fuzzy arithmetic approaches (e.g. [45]) and fuzzy random/type-2 fuzzy sets (e.g. [42]). [48] presents a more recent survey of the FDEA literature. A summary of existing FDEA approaches25 is shown in Table 1. This list does not pretend to be exhaustive but it includes a number of representative approaches. For comparison, the proposed approach has also been included. It can be seen in the Table 1 that the ↵-level set, the fuzzy ranking and the possibility approaches are the most common. Most FDEA approaches involve30 radial, input-oriented, multiplier formulations. There are some non-radial approaches, mostly based on the Enhanced Russell Graph Measure (ERGM) and Slacks-Based Measure (SBM), as well as a few additive approaches, using both multiplier and envelopment formulations. Some approaches compute fuzzy efficiency scores while others compute a crisp efficiency or an efficiency35 score for each possibility level. Also some FDEA approaches allow ranking the DMUs. What most approaches neglect (or, more exactly, are unable to compute) are efficient targets. Only a few FDEA approaches can compute targets, crisp 2 Table 1: Summary of existing fuzzy DEA approaches. Notes: CRS =Constant Returns to Scale; VRS =Variable Returns to Scale; SBM =Slacks-Based Measure; ERGM =Enhanced Russell Graph Measure Method Reference Main features ↵level Kao and Liu (2000) Ratio, multiplier formulation; CRS; fractional parametric programming; fuzzy efficiency scores Saati et al. (2002) Radial, input-oriented, multiplier formulation; CRS; interval linear programming; ↵level efficiency scores Saati and Memariani (2009) Non-oriented SBM, multiplier formulation; CRS; interval linear programming; ↵level efficiency scores Hsiao et al. (2011) Non-oriented SBM, envelopment formulation; VRS; fuzzy efficiency and super-efficiency scores Hatami-Marbini et al. (2012) Additive, non-oriented, envelopment formulation; VRS; interval linear programming; ↵level efficiency scores Puri and Yadav (2013) Radial and non-radial, input-oriented, envelopment formulations; CRS; fuzzy efficiency scores Wu et al. (2015) Non-radial, non-oriented ERGM envelopment formulation; CRS; undesirable outputs; fuzzy efficiency scores Fuzzy ranking Guo and Tanaka (2001) Radial, input-oriented, multiplier formulation; CRS; fuzzy linear programming; fuzzy efficiency scores Le´ on et al. (2003) Radial, input-oriented, envelopment formulation; VRS; fuzzy linear programming; possibility-level efficiency scores Soleimani-damaneh et al. (2006) Radial, input-oriented, envelopment formulation; VRS; crisp efficiency scores; crisp targets Ghasemi et al. (2015) Multiplier formulation; generalized DEA; VRS; Fuzzy expected value; DMU ranking Khaleghi et al. (2015) Radial, input-oriented, multiplier formulation; CRS; fully fuzzy; level-sum method; fuzzy efficiency scores, DMU ranking Kordrostami et al. (2016) Radial, input-oriented, envelopment formulation; CRS; integer data; crisp efficiency scores; crisp targets Arana-Jim´ enez et al. (2020) Radial, input-oriented, envelopment formulation; CRS; fully fuzzy; fuzzy partial order; lexicographic weighted Tchebyche↵method; fuzzy efficiency scores, fuzzy targets Proposed approach Normalized additive, non-oriented, envelopment formulation; VRS; fuzzy partial order; crisp efficiency scores; fuzzy targets Possibility approach Lertworasirikul et al. (2003) Radial, input-oriented, multiplier and envelopment formulations; VRS; possibility-level efficiency scores Wang and Chin (2011) Radial, input-oriented, multiplier formulation; CRS; Fuzzy expected value; double frontier analysis; crisp efficiency scores; DMU ranking Ruiz and Sirvent (2017) Radial, input-oriented, multiplier formulation; CRS; benevolent and aggressive possibility-level crossefficiency scores Izadikhah and Khoshroo (2018) Non-radial, non-oriented modified ERGM envelopment formulation; CRS; undesirable outputs; possibilitylevel super-efficiency scores Perykani et al. (2019) Radial, input-oriented, multiplier formulation; CRS; adjustable pessimistic-optimistic parameter; possibilitylevel efficiency scores Fuzzy arithmetic Wang et al. (2009) Radial, input-oriented, multiplier formulation; CRS; fuzzy efficiency scores; DMU ranking Azar et al. (2016) Additive, non-oriented, multiplier formulation; CRS; common set of weights; fuzzy efficiency scores; DMU ranking 3 in some cases and fuzzy in others. As shown in Table 1, the proposed fuzzy ranking approach is based on40 the normalized additive envelopment formulation. Unlike the conventional additive DEA model considered in other FDEA approaches and which does not normalize the input and output slacks, the proposed approach is units invariant. Moreover, it makes use of a fuzzy partial order, which has proved to be a flexible FDEA modelling tool ( e.g. [4]). This combination of features leads45 to a simple and elegant linear programming (LP) formulation that provides crisp efficiency scores as well as fuzzy targets. This target computing feature and its focus on quantifying the potential input and output improvements are the main advantages of the proposed approach with respect to other more complex approaches that do not provide efficient targets. We believe that50 such information is essential for managers to take action and orientate their improvement e↵orts. The structure of the paper is as follows. In Section 2, some notions related to the Fuzzy sets and DEA methodologies are introduced. In Section 3, uncertainty on inputs and outputs is addressed with trapezoidal fuzzy numbers. The55 corresponding fuzzy DEA technology is defined, and a fuzzy DEA model is proposed to compute an inefficiency measure as well as a fuzzy target for each DMU. Section 4 illustrates the proposed approach on a dataset from the literature, comparing it with some existing fuzzy DEA approaches. Finally, Section 5 summarizes and concludes.60 2. Preliminaries 2.1. Fuzzy sets Theory A fuzzy set on Rnis a mapping u:Rn![0,1]. For each fuzzy set uand for any ↵2(0,1] , let us define the ↵-level set as [u]↵={x2Rn|u(x)↵}. Let us denote the support of uby supp(u) where supp(u)={x2Rn|u(x)>0}. The65 closure of supp(u) defines the 0-level set of u, .i.e. [u]0=cl(supp(u)) where cl(M) 4 means the closure of the subset M⇢Rn. A fuzzy number is a type of fuzzy set (see [14, 15]) defined as follows. Definition 1. A fuzzy set u on Ris said to be a fuzzy number if: 1. u is normal, i.e. there exists x02Rsuch that u(x0)=1;70 2. u is an upper semi-continuous function; 3. u(x+(1 )y)min{u(x),u(y)},x,y2R,2[0,1]; 4. [u]0is compact. Definition 2. A fuzzy number ˜ u=(u1,u2,u3,u4)is said to be a trapezoidal fuzzy number if its membership function is given by ˜ u(x)= 8 > > > > > > > > > < > > > > > > > > > : xu1 u2u1,if u1x<u2, 1,if u2xu3, u4x u4u3,if u3<xu4, 0,otherwise. (1) Its corresponding ↵-levels are determined by [ ˜ u]↵=[u1+↵(u2u1),u4 ↵(u4u3)].We denote the set of all trapezoidal fuzzy numbers as TrFN. The75 subset of non-negative TrFNs is denoted by TrFN+. A TrFN ˜ uis a triangular fuzzy number if and only if u2=u3. Figure 1 shows illustrative examples of a trapezoidal and a triangular fuzzy number. Definition 3. Given two trapezoidal fuzzy numbers ˜ a=(a1,a2,a3,a4)2TrFN and ˜ b=(b1,b2,b3,b4)2TrFN, the following arithmetical operations are defined:80 (i) Addition, ˜ a+˜ b=(a1+b1,a2+b2,a3+b3,a4+b4) (2) (ii) Multiplication by a scalar 2R, ˜ a=8 > > > < > > > : (a1,a2,a3,a4)if 0; (a4,a3,a2,a1)if <0. (3) (iii) Multiplication of two TrFN, ˜ a˜ b=˜ c=(c1,c2,c3,c4), where c1=min{a1b1,a1b4,a4b1,a4b4}c4=max{a1b1,a1b4,a4b1,a4b4} c2=min{a2b2,a2b3,a3b2,a3b3}c3=max{a2b2,a2b3,a3b2,a3b3} (4) 5 Figure 1: Two trapezoidal fuzzy numbers, their sum and their product, as defined in 3. In cyan, the particular case of a triangular fuzzy number ˜ a=(1,2,3) =(1,2,2,3). In blue, a trapezoidal fuzzy number (3.5,4,5,5.5). The [˜ a]↵and [˜ b]↵levels, for ↵=0.2,0.4,0.6,0.8, are represented with dashed lines. In the particular case of two non-negative TrFN ˜ aand ˜ b, their multiplication is just ˜ a˜ b=(a1b1,a2b2,a3b3,a4b4). Examples of the addition and of the multiplication of a trapezoidal and a triangular fuzzy number are shown in Figure 1. In the Fuzzy DEA problem addressed in this research, we will consider that the input and output variables (and some model variables) are TrFN+. The arithmetic operations between them are those established in Definition 3. Besides, it is necessary to provide a partial order relationship between two trapezoidal fuzzy numbers. To this aim, we will use LU-fuzzy partial orders, which are well known in the literature (see, e.g., [46, 41]): µ⌧()⌫if and only if µ↵()⌫↵and µ↵()⌫↵,for all ↵2[0,1]. 6 In the particular case of TrFNs, and following the characterization result given in [3] for triangular fuzzy numbers, we can say that given two trapezoidal fuzzy numbers ˜ u=(u1,u2,u3,u4) and ˜ v=(v1,v2,v3,v4), ˜ u⌧()˜ vif and only if ui()vi,for all i=1,2,3,4.(5) 2.2. DEA methodology85 Consider a set of NDMUs. For j2{1,...,N}, each DMUjhas Minputs Xj=(x1j,...,xMj)2RM, and produces Soutputs Yj=(y1j,...,ySj)2RS. DEA aims to measure the relative efficiency of these DMUs. A DMU is inefficient if it can reduce its inputs without reducing its outputs or it can increase its outputs without increasing its inputs. The first step in the DEA methodology90 is establishing the production possibility set (PPS) (a.k.a. DEA technology), which contains all the operating points that are deemed feasible. The usual axioms for a basic DEA technology, called T, are the following: (A1) Envelopment: (Xj,Yj)2T, for all j2{1,...,N}. (A2) Free disposability: (x,y)2T,(x0,y0)2RM+S,x0=x,y05y)(x0,y0)2T.95 (A3) Constant Returns to Scale (CRS): (x,y)2T)(x,y)2T, for all 2R+. (A4) Convexity: (x,y),(x0,y0)2T, then (x,y)+(1 )(x0,y0)2T, for all 2[0,1]. Let us recall that the technology which verifies (A1)-(A4) is known as Constant Returns to Scale (VRS), and noted as TCRS. Following the minimum extrapo-100 lation principle, the DEA PPS is the intersection of all the sets that satisfy the above axioms and can be mathematically formulated as ([9]) TCRS =8 > > > < > > > : (x,y)2RM+S +:x= N X j=1 jXj,y5 N X j=1 jYj, j09 > > > = > > > ; There is another common DEA technology labelled Variable Returns to Scale (VRS) which corresponds to dropping Axiom (A3) ([7]) 7 TVRS =8 > > > < > > > : (x,y)2RM+S +:x= N X j=1 jXj,y5 N X j=1 jYj, N X j=1 j=1,j09 > > > = > > > ; Note that the CRS and VRS DEA technologies di↵er basically in that the105 latter imposes the convexity constraint PN j=1j=1. Recall that, given a technology T(TCRS,TVRS, or any other under consideration), a DMU pis said to be efficient if and only if for any (x,y)2Tsuch that x5Xpand y=Yp, then (x,y)=(Xp,Yp). Radial efficiency measures can be computed by: a) reducing all the inputs equi-proportionally without decreasing the outputs (input-oriented model): E(x,y)=min{✓|(✓x,y)2T} b) expanding all the outputs equi-proportionally without increasing the inputs (output-oriented model): E1(x,y)=max{|(x,y)2T} Instead of a radial measure of efficiency, an additive DEA model can be110 used considering the input and output slacks. i.e. the amount that each input can be reduced, and each output can be increased. This leads to the following additive measure of inefficiency ([10]) Max M X i=1 s i+ S X r=1 s+ r(6) s.t. N X j=1 jxij +s i=xip,i=1,...,M N X j=1 jyrj =yrp +s+ r,r=1,...,S, s i0,i=1,...,M s+ r0,r=1,...,S j0,j=1,...,N, 8 Note that a certain DMUp is efficient if the optimal solution of the above model has an objective function equal to zero. This means that all input slacks115 s i, and all output slacks s+ rmust be zero at the optimum ([13]). A drawback of the above model is that its objective function sums input and output slacks that may be measured in di↵erent units. One way of solving this is to normalize the slacks, i.e. the input and output improvements, and express them in relative terms. Then, for a given DMUp, the corresponding Measure120 of Inefficiency Proportions (MIP) ([11]) results from changing the objective function of the above model to Max PM i=1s i/xip +PS r=1s+ r/yrp (7) Another relevant inefficiency measure is the directional distance function (DDF), which uses a directional vector g=(gx,gy)=(gx 1,...,gx M,gy 1,...,gy S),0, and leads to the following DEA model ([8])125 p=Max (8) s.t. N X j=1 jxij xip gx i,i=1,...,M N X j=1 jyrj yrp +gy r,r=1,...,S, N X j=1 j=1 j,0,j=1,...,N, Finally, another slacks-based measure of inefficiency was proposed by [19] as a generalization of the above optimization problem considering the sum of directional distance functions based on the unit vectors ex i,i=1,...,Mand ey r, r=1,...,S(instead of a single directional vector g=(gx,gy)) and di↵erent variables iand r, instead of a single variable for all inputs and outputs.130 These iand rdimensionless variables can be interpreted as the number of 9 0 2 4 6 8 10 12 02468 Input Output A B C D E F G H Figure 2: Observed DMUs and their corresponding targets marked with blue triangles. The thick, gray line joining DMUs A and C is a crisp approximation of the true fuzzy efficient frontier. inefficiency measure is a necessary but not a sufficient condition for establishing the efficiency of a DMU. This simple example also allows us to illustrate the proposed FDEA approach graphically as regards the efficiency status classification. Thus, Figure 2240 shows the observed DMUs, labeled A to H. The plotted horizontal and vertical error bars represent the 0level (i.e., the closure of the support) of the corresponding trapezoidal fuzzy number, for both the input and output dimensions. The targets computed by the proposed approach are marked with blue triangles and its corresponding horizontal and vertical error bars. Note that DMUs245 A and C are efficient, and hence, they are projected onto themselves. The efficiency status of DMU G is undetermined, its inefficiency score ˜ I(˜ xG,˜ yG) is zero although it is not projected onto itself but onto DMU C. Inefficient DMUs D, E, F, and H are projected onto DMU C while the remaining inefficient DMU (namely DMU B) is projected onto a target that is a linear convex combination250 16 Table 3: Comparison with other FDEA approaches Method ↵-level A B C D E F G H Le´ on et al. (2003) [30] 0 1 1 1 0.7500 0.6429 0.6050 1 0.6923 (Radial efficiency) 0.5 1 0.9412 1 0.6623 0.6172 0.5227 1 0.6400 0.75 1 0.8675 1 0.6144 0.6010 0.4776 1 0.5854 1 1 0.7500 1 0.5385 0.5714 0.4062 0.45 0.5000 Saati et al. (2002) [38] 0 1 1 1 1 1 0.640 0.867 0.764 (Radial efficiency) 0.5 1 0.764 1 0.706 0.835 0.462 0.628 0.457 0.75 1 0.602 1 0.574 0.668 0.390 0.533 0.343 1 0.750 0.469 1 0.462 0.536 0.328 0.450 0.250 Hatami-Marbini 0 6 7 2.333 8.25 8.5 9.5 9.5 10 et al. (2012) [24] 0.5 0 5 0 6.125 6 7.75 7.5 7.75 (Sum of slacks ine↵.) 0.75 0 4 0 5.063 4.75 6.875 6.5 6.625 1 0 2.5 0 4 3.5 6 5.5 5.5 I(xp,yp) – 0 1.071 0 1.429 1.584 3.013 0 3.231 of DMUs A and C. The thick grey line joining these DMUs is an approximate crisp representation of the true fuzzy efficient frontier. Figures 3 and 4 show the solutions, for each of the DMUs, computed by the proposed approach (FDEA). For each DMU, the left panel shows, in blue color, the fuzzy inefficiency measure ˜ I(˜ x,˜ y)=˜ +˜ . The middle panel shows255 the input data (in black color) and the input target (in magenta color and dash line). The right panel, shows the output data (in black color) and the output target (in magenta color and dash line). When a DMU pis inefficient not only ˜ I(˜ xp,˜ xe),˜ 0 but also its input target is shifted to the left (i.e. the original input is reduced) whereas its output target is shifted to the right (i.e. the original260 output is increased). Finally, in Table 3, the results of the proposed approach are compared with those of other fuzzy DEA approaches, namely [30], [38] and [24]. The last row of the table considers the crisp inefficiency measure I(xp,yp) computed by model (FDEA3). For the inefficient DMUs, with I(xp,yp)>0, their efficiency265 measurements from [30] and [38] are less than one for nearly all the ↵levels. Equivalently in the case of [24], the corresponding sum of slacks is greater than 17 zero for all ↵levels. On the other hand, for the three DMUs with I(xp,yp)=0, and therefore possibly efficient, their efficiency scores from [30] and [38] are equal to one for all ↵levels in the case of DMUs A and C and for some alpha270 levels in the case of DMU G. The sum of slacks computed by [24] are zero for DMUs A and C for most ↵levels and greater than zero for DMU C for all ↵levels. 5. Conclusions In this paper, a new fuzzy DEA slacks-based measure of inefficiency has275 been proposed. It requires solving a crisp linear optimization model and allows computing corresponding input and output targets. The normalized character of additive metric used in the objective function implies that it is units invariant. Computational experiments have been presented to validate the proposed approach and compare it with existing fuzzy DEA methods.280 One of the advantages of the proposed approach with respect to other more complex approaches is that, apart from computing efficiency scores, it focuses on the input and output improvements that can be achieved, called slacks in DEA parlance. These improvements are also expressed in the form of efficient targets and provide managers with useful information on the variables on285 which the improvement e↵ort should be concentrated and the extent of those e↵orts. Most existing FDEA approaches, particularly but not only those that use multiplier formulations, do not provide efficient targets and thus their managerial usefulness is limited. A limitation of the proposed approach is that a null value of the computed290 inefficiency score is a necessary nut not sufficient condition for efficiency, i.e. it cannot discriminate between efficient and weakly efficient DMUs. This issue does not happen in the crisp DEA case, in which maximizing the sum of input and output slacks guarantees efficiency. This highlights the di↵erences between the crisp and fuzzy data scenarios. Devising an appropriate phase295 II, as in radial DEA approaches, is a topic for further research. Extending the 18 −1.0 −0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU A β ~+γ ~ 1.0 2.0 3.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU A Inputs 2.0 2.5 3.0 3.5 4.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU A Outputs 0.354 0.357 0.360 0.0 0.2 0.4 0.6 0.8 1.0 DMU B β ~+γ ~ 1.5 2.5 3.5 4.5 0.0 0.2 0.4 0.6 0.8 1.0 DMU B Inputs 1.5 2.5 3.5 4.5 0.0 0.2 0.4 0.6 0.8 1.0 DMU B Outputs −1.0 −0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU C β ~+γ ~ 3.0 4.0 5.0 6.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU C Inputs 5.0 5.5 6.0 6.5 7.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU C Outputs 0.472 0.476 0.480 0.0 0.2 0.4 0.6 0.8 1.0 DMU D β ~+γ ~ 34567 0.0 0.2 0.4 0.6 0.8 1.0 DMU D Inputs 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 DMU D Outputs Figure 3: Solutions of proposed FDEA approach for DMUs A to D. proposed to include undesirable outputs is also a worthy endeavour. 19 0.45 0.50 0.55 0.60 0.0 0.2 0.4 0.6 0.8 1.0 DMU E β ~+γ ~ 3456789 0.0 0.2 0.4 0.6 0.8 1.0 DMU E Inputs 4.5 5.0 5.5 6.0 6.5 7.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU E Outputs 0.96 0.98 1.00 1.02 1.04 0.0 0.2 0.4 0.6 0.8 1.0 DMU F β ~+γ ~ 3 4 5 6 7 8 0.0 0.2 0.4 0.6 0.8 1.0 DMU F Inputs 34567 0.0 0.2 0.4 0.6 0.8 1.0 DMU F Outputs −1.0 −0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU G β ~+γ ~ 9.0 9.5 10.0 10.5 11.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU G Inputs 5.6 5.8 6.0 6.2 6.4 0.0 0.2 0.4 0.6 0.8 1.0 DMU G Outputs 1.070 1.080 0.0 0.2 0.4 0.6 0.8 1.0 DMU H β ~+γ ~ 3.0 4.0 5.0 6.0 0.0 0.2 0.4 0.6 0.8 1.0 DMU H Inputs 1234567 0.0 0.2 0.4 0.6 0.8 1.0 DMU H Outputs Figure 4: Solutions of proposed FDEA approach for DMUs E to H. 20 6. Acknowledgements The first author was partially supported by the research project MTM2017- 89577-P (MINECO, Spain). The second author was partially supported by300 the Spanish Ministry of Economy and Competitiveness, grant AYA2016-75931- C2-1-P and from the Consejer´ ıa de Educaci´ on y Ciencia (Junta de Andaluc´ ıa, reference TIC-101). The third author acknowledges the financial support of the Spanish Ministry of Science, Innovation and Universities, grant PGC2018- 095786-B-I00.305 7. References [1] Arana-Jim´ enez M (Ed.). Optimiality conditions in vector optimization. Bentham Science Publishers, Ltd.: Bussum, 2010. [2] Arana-Jim´ enez, M., Antczak, T. The minimal criterion for the equivalence between local and global optimal solutions in nondi↵erentiable optimiza-310 tion problem. Mathematical Methods in the Applied Sciences, 40 (2017) 6556- 6564. [3] Arana-Jim´ enez, M. Nondominated solutions in a fully fuzzy linear programming problem. Mathematical Methods in the Applied Sciences, 41 (2018) 7421-7430.315 [4] Arana-Jim´ enez, M., S´ anchez-Gil, M.C., Lozano, S. Efficiencya Assessment and Target Setting Using a Fully Fuzzy DEA Approach. International Journal of Fuzzy Systems, 22, 4 (2020) 1056-1072. [5] Arya, A., Yadav, S.P. A Fuzzy Dual SBM Model with Fuzzy Weights: An Application to the Health Sector. Advances in Intelligent Systems and320 Computing, 546 (2017) 230-238 [6] Azar, A., Zarei Mahmoudabadi, M., Emrouznejad, A. A New Fuzzy Additive Model for Determining the Common Set of Weights in Data Envelopment Analysis, Journal of Intelligent & Fuzzy Systems, 30, 1 (2016) 61-69. 21 [7] Banker, R.J., Charnes, A., Cooper, W.W. Some models for estimating tech-325 nical and scale inefficiencies in data envelopment analysis. Management Science, 30, 9 (1984) 1078-1092 [8] Chambers, R.G., Chung, Y., F¨ are, R. Profit directional distance functions and Nerlovian efficiency. Journal of Optimization Theory and Applications, 95 (1998) 351-354.330 [9] Charnes, A., Cooper, W.W., Rhodes, E. Measuringthe efficiencies of DMUs. European Journal of Operational Research, 2, 6 (1978) 429-444. [10] Charnes, A., Cooper, W.W., Golany, B., Seiford, L., Stutz, J. Foundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functions. Journal of Econometrics, 30 (1985) 91-107.335 [11] Cooper, W.W., Park, K.S., Pastor, J.T. RAM: A Range Adjusted Measure of Inefficiency for Use with Additive Models, and Relations to Other Models and Measures in DEA. Journal of Productivity Analysis, 11 (1999) 5-42. [12] Cooper, W.W., Seiford, L.M., Zhu, J. Handbook on Data Envelopment Analysis. Springer, New York, (2004)340 [13] Cooper, W.W., Seiford, L.M., Tone, K. Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA- Solver Software, 2nd edition, Springer, New York, 2006 [14] Dubois, D., Prade, H. Operations on fuzzy numbers.Ins. J. Systems Sci.,9 (1978)613-626.345 [15] Dubois, D., Prade, H. Fuzzy Sets and Systems: Theory and Applications. Academic Press: New York, (1980). [16] Ebrahimnejad, A., Nasseri, S.H., Lotfi, F.H., Soltanifar, M.. A primal-dual method for linear programming problems with fuzzy variables.Eur. J. Ind. Eng., 4 (2010) 189-209.350 22 [17] Emrouznejad, A., Tavana, M. Hatami-Marbini, A. The state of the art in fuzzy data envelopment analysis. Studies in Fuzziness and Soft Computing, 309 (2014) 1-45. [18] F¨ are, R., Margaritis, D., Rouse, P., Roshdi, I. Estimating the hyperbolic distance function: A directional distance function approach. European Journal355 of Operational Research, 254, (2016) 312-319. [19] F¨ are, R., Grosskopf, S. Directional distance functions and slacks-based measures of efficiency. European Journal of Operational Research, 200 (2010) 320-322 . [20] Ghasemi, M.R., Ignatius, J., Lozano, S., Emrouznejad, A., Hatami-Marbini,360 A. A fuzzy expected value approach under generalized data envelopment analysis. Knowledge-Based Systems, 89 (2015) 148-159. [21] Guo, P., Tanaka, H. Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets and Systems, 119 (2001) 149-160. [22] Hatami-Marbini, A., Emrouznejad, A., Tavana, M. A taxonomy and review365 of the fuzzy data envelopment analysis literature: Two decades in the making. European Journal of Operational Research, 214 (2011a) 457-472. [23] Hatami-Marbini, A., Tavana, M., Ebrahimi, A. A fully fuzzified data envelopment analysis model. International Journal of Information and Decision Sciences, 3, 3 (2011) 252-264.370 [24] Hatami-Marbini, A., Tavana, M., Emrouznejad, A., Saati, S. Efficiency measurement in fuzzy additive data envelopment analysis. International Journal of Industrial and Systems Engineering, 10, 1 (2012) 1-20. [25] Hsiao, B., Chern, C.C., Chiu, Y.H., Chiu, C.R. Using fuzzy super-efficiency slack-based measure data envelopment analysis to evaluate Taiwan’s com-375 mercial bank efficiency. Expert Systems with Applications, 38 (2011) 9147- 9156. 23 [26] Izadikhah, M., Khoshroo, A. Energy management in crop production using a novel fuzzy data envelopment analysis model. RAIRO-Operations Research, 52 (2018) 595-617.380 [27] Kao, C., Liu, S.T. Fuzzy efficiency measures in data envelopment analysis, Fuzzy Sets and Systems, 113 (2000) 427-437. [28] Khaleghi, S., Noura, A, Hosseinzadeh Lotfi, F. Measuring Efficiency and Ranking Fully Fuzzy DEA. Indian Journal of Science and Technology, 8, 30 (2015) 1-6.385 [29] Korhonen, P.J., Dehnokhalaji, A., Nasrabadi, N. A lexicographicradial projection onto the efficient frontier in Data Envelopment Analysis. European Journal of Operational Research, 265 (2018) 1005-1012. [30] Le´ on, T., Liern, V., Ruiz, J.L., Sirvent, I. A fuzzy mathematical programming approach to the assessment of efficiency with DEA models. Fuzzy390 Sets and Systems, 139 (2003) 407-419 [31] Lertworasirikul, S., Fang, S.C., Nuttle, H.L.W., Joines, J.A. Fuzzy BCC model for data envelopment analysis. Fuzzy Optimization and Decision Making, 2, 4 (2003) 337-358. [32] Lozano, S., Soltani, N. DEA target setting using lexicographic and en-395 dogenous Directional Distance Function approaches. Journal of Productivity Analysis, 50, 1-2 (2018) 55-70. [33] Lozano, S., Soltani, N. Efficiency assessment using a multidirectional DDF approach. International Transactions in Operational Research, 27 (2020) 2064- 2080.400 [34] Peykani, O., Mohammadi, E., Emrouznejad, A., Pishvae, M.S., Rostamy- Malkhalifeh, M. Fuzzy data envelopment analysis: An adjustable approach. Expert Systems With Applications, 36 (2019) 439-452. 24 [35] Puri, J., Yadav, S.P. A concept of fuzzy input mix-efficiency in fuzzy DEA and its application in banking sector. Expert Systems with Applications, 40,405 5 (2013) 1437-1450. [36] Ruiz, J.L., Sirvent, I. Fuzzy cross-efficiency evaluation: a possibility approach. Fuzzy Optimization and Decision Making, 16 (2017) 111-126. [37] Saati, S., Memariani, A. SBM model with fuzzy input-output levels in DEA. Australian Journal of Basic and Applied Sciences, 3, 2 (2009) 352-357.410 [38] Saati, S., Memariani, A., Jahanshahloo, G.R. Efficiency Analysis and Ranking of DMUs with Fuzzy Data. Fuzzy Optimization and Decision Making,1 (2002) 255-267. [39] Soleimani-damaneh, M., Jahanshahloo, G.R., Abbasbandy, S. Computational and theoretical pitfalls in some current performance measurement415 techniques; and a new approach. Applied Mathematics and Computation, 181 (2006) 1199-1207. [40] Soltani, N., Lozano, S. Potential-Based Efficiency Assessment and Target Setting. Computers and Industrial Engineering, 126 (2018) 611-624. [41] Stefanini, L, Arana-Jim´ enez, M. Karush-Kuhn—Tucker conditions for in-420 terval and fuzzy optimization in several variables under total and directional generalized di↵erentiability. Fuzzy Sets and Systems, 362 (2019) 1-34. [42] Tavana, M., Khanjani Shiraz, R., Hatami-Marbini, A., Agrell, P.J., Paryab, K. Chance-constrained DEA models with random fuzzy inputs and outputs. Knowledge Based Systems, 52 (2013) 32-52.425 [43] Tone, K. A slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research, 130, 3 (2001) 498-509. [44] Wang, Y.M., Chin, K.S. Fuzzy data envelopment analysis: a fuzzy expected value approach. Expert Systems with Applications, 38 (2011) 11678-11685. 25