Fuzzy ranking network DEA with general structure

mathematics Article Fuzzy Ranking Network DEA with General Structure Plácido Moreno * and Sebastián Lozano School of Engineering, Universidad de Sevilla, E41092 Seville, Spain; [email protected] *Correspondence: [email protected] Received: 18 July 2020; Accepted: 8 December 2020; Published: 14 December 2020   Abstract: This paper extends two fuzzy ranking data envelopment analysis (DEA) approaches to the case of general networks of processes. The first approach provides an efficiency score for each possibility level which requires solving one linear program for each possibility level. The second approach is even simpler and provides an overall efficiency score solving just one linear program. The proposed approaches are tested on two datasets from the literature and compared with other fuzzy network DEA approaches. The results show that the two methods provide very highly correlated efficiency estimates which are also consistent with those of other fuzzy network DEA approaches. Keywords: efficiency assessment; network DEA; fuzzy data; fuzzy ranking; defuzzification 1. Introduction Data envelopment analysis (DEA) is a well-known non-parametric technique, generally used to assess the relative efficiency of a group decision making units (DMUs). Conventional DEA considers that each DMU consists of just a single process. This process consumes certain amounts of inputs and produces certain amount of outputs. The aim of DEA is to detect inefficiencies, i.e., to check if the same outputs can be obtained with less inputs the so-called input orientation) or the same inputs can produce more outputs (output orientation). Detecting all kinds of inefficiencies is important for both competitiveness and sustainability. There are different DEA models, depending on the orientation, the metric, and the returns to scale (RTS) assumptions considered. Considering certain axioms (like convexity or scalability, for example) and applying the minimum extrapolation principle, DEA can infer from the observed data a so-called production possibility set (PPS), which corresponds to the set of operation points that are considered feasible. The PPS is formed using all linear combinations of the observed DMUs. When variable returns to scale (VRS) are assumed then only convex linear combinations are used. The non-dominated subset of the PPS defines the efficient frontier. The DMUs that fall on the Efficient Frontier (EF) are labelled efficient while those that do not are termed inefficient and are projected onto the EF. The distance of a DMU to the EF is used to compute an efficiency score so that the further from the EF a DMU is (i.e., the larger the input and output improvements that can be achieved) the lower its efficiency score. Accordingly, efficient DMUs have an efficiency score of one since for them no input or output improvements are possible. Different to conventional DEA, network DEA looks at the internal structure of the DMUs, considering the DMU as a network of interrelated processes which consume inputs and produce outputs but also produce and consume intermediate products. As with conventional DEA, many different network DEA approaches have also been proposed (e.g., Kao [ 1 ]; Tone and Tsutsui [ 2 , 3 ]; Cook et al. [ 4 ]; Lozano [ 5 ]). The range of applications of network DEA has also grown accordingly, spanning banking (Lozano [ 6 ]), transportation (e.g., Lozano et al. [ 7 ]), sports (Moreno and Lozano [ 8 ]), supply chain (e.g., Lozano and Adenso-Díaz [9]), etc. Mathematics 2020,8, 2222; doi:10.3390/math8122222 www.mdpi.com/journal/mathematics Mathematics 2020,8, 2222 2 of 18 All the above references have dealt with crisp data, i.e., they assume that there is no uncertainty in the data. This does not mean that DEA cannot work when the data are fuzzy. On the contrary, there are many different fuzzy DEA approaches (e.g., Kao and Liu [ 10 ], Arana-Jimenez et al. [ 11 ], Saati et al. [ 12 ], Le ó n et al. [ 13 ], Lertworasirikul et al. [ 14 ], Wang et al. [ 15 ], Soleimani-damaneh et al. [ 16 ], Arana-Jimenez et al. [ 17 ], etc.). The reader is referred to Hatami-Marbini et al. [ 18 ] and Emrouznejad et al. [19] for a taxonomy and review of fuzzy DEA approaches. Of special relevance are those fuzzy DEA approaches that consider networks of processes. Thus, Kao and Liu [ 20 ] extend Kao and Liu [ 10 ] approach to two-stage systems with fuzzy data. Liu [ 21 ] proposes a methodology to rank fuzzy two-stage efficiencies, while Liu [ 22 ] adds weight restrictions to Kao and Liu [ 20 ] fuzzy two-stage model. Tavana and Khalili-Damghani [ 23 ] also work with Kao and Liu [ 20 ] approach but implementing a leader-follower game theory approach to decompose the two-stage efficiency into single-stage efficiencies. Another variant is presented in Hemmati et al. [ 24 ]. Wang et al. [ 25 ] adopt a bootstrapped truncated-regression model to study the relationship between fuzzy two-stage efficiencies and an exogenous variable. Concerning non-radial network DEA approaches, Shermeh et al. [ 26 ] adapt the Network Slack-Based Measure (SBM) model by Tone and Tsutsui [ 2 ] to be able to deal with fuzzy numbers. Olfat et al. [27] propose a fuzzy extension of the dynamic Network SBM approach by Tone and Tsutsui [3]. Khalili-Damghani and Taghavi-Fard [ 28 ] also study fuzzy two-stage DEA systems. Kao and Lin [ 29 ] consider parallel processes and fuzzy data. Lozano [ 30 , 31 ] compute process efficiencies in parallel and two-stage process DEA systems, respectively. Kao [ 32 ] proposes two approaches, namely the membership grade and the α -cut, for network DEA with fuzzy data. Lozano and Moreno [ 33 ] extend the approaches in Saati et al. [ 12 ], Wang et al. [ 15 ], and Kao and Liu [ 20 ] to general networks of processes. Finally, Mirhedayatian et al. [ 34 ] present a fuzzy network DEA approach with dual-role factors and undesirable outputs to evaluate green supply chains. However, to the best of our knowledge, ranking methods needs further development for network DEA when dealing with fuzzy data and general network of processes. In fact, our work allows any configuration of the stages that form the internal structure of the DMU, which implies an improvement over other models. It could also be argued that previous models seem to be complex enough to prevent some researchers from working with fuzzy data. Therefore, our aim is to develop a general fuzzy network DEA approach without adding unnecessary complexity. The structure of the paper is the following. In Section 2the network DEA is introduced, and a crisp model is formulated. In Section 3, the proposed methods are presented. In Section 4the results of the application of the proposed methods to two datasets from the literature are presented. Finally, Section 5summarizes and concludes. 2. Preliminaries In order to facilitate a better understanding of the proposed fuzzy ranking network DEA approaches, this section presents a brief overview of network DEA models and fuzzy numbers. 2.1. Crisp Network DEA Model It is convenient to formulate first the crisp network DEA model, i.e., the network DEA model without fuzzy data. We will use the same notation that Lozano [ 5 ] and Lozano and Moreno [ 33 ]. This notation helps to formulate the models in a very compact form. Assume the DMUs to assess are all structurally homogeneous, i.e., all of them have the same number and type of processes. Each process may consume a different subset of inputs and may produce a different subset of outputs. Let I(p) be the set of exogenous inputs consumed by process p and, for each i∈I(p) , let xp ij denote the observed amount of exogenous input i used by process p of DMU j. Similarly, let O(p) the set of outputs produced by process p and, for each k∈O(p) , let yp kj denote the amount of output k produced by process p of DMU j. Let PI(i) be the set of processes that Mathematics 2020,8, 2222 3 of 18 consume the input i and xij =P p∈PI(i) xp ij the total amount of input i consumed by DMU j. Let PO(k) be the set of processes that produce the output k and ykj =P p∈PO(k) yp kj the total amount of output k produced by DMU j. A key feature of network DEA is that, in addition to exogenous inputs and outputs, there generally exist intermediate products that are produced by some processes and consumed by others. Let Pout(r) be the set of processes that produce the intermediate product r and, for each p∈Pout(r) , let zp rj the amount of intermediate product r produced by process p of DMU j. Analogously, let Pin(r) be the set of processes that consume the intermediate product r and, for each p∈Pin(r) , let zp rj the amount of intermediate product r used by process p of DMU j. Finally, let us define the sets Rout(p) and Rin(p) that correspond to the intermediate products produced and consumed, respectively, by process p. Once the notation for required data has been introduced let us consider the variables. For the radial, input-oriented model the variables needed are θUniform reduction factor of the input consumption of DMU J λp jIntensity variable of process p of DMU j EJ=Min θ(1) subject to X p∈PI(i)X j λp jxp ij ≤θX p∈PI(i) xp iJ ∀i(2) X p∈PO(k)X j λp jyp kj ≥X p∈PO(k) yp kJ ∀k(3) X p∈Pout(r)X j λp jzp rj −X p∈Pin(r)X j λp jzp rj ≥0∀r(4) X j λp j=1∀p(5) λp j≥0∀j∀pθfree (6) This crisp network DEA model (see Lozano [ 5 ], Lozano and Moreno [ 33 ]) computes the maximum radial reduction of the inputs consumed by a given DMU J. Note that this model has to be solved as many times as DMUs are in the dataset. The idea is to compute a feasible operating point (i.e., within the inferred PPS) that maintains the output level of DMU J but reduces all its inputs as much as possible. The optimal value of the θ variable corresponds to the efficiency score of DMU J. The λp j multipliers, note that there is a specific set of them for each process p, determine the inputs, outputs and intermediate products of each process. Thus, the linear combinations of the observed data define a target feasible operating point that produces at least the same amount of output, as required by Constraint (3), while consuming a fraction θ of the observed inputs, as indicated by Constraints (2). Constraints (4) impose that enough intermediate products are generated internally within the system to satisfy the internal demand of those intermediate products. Therefore, the constraints guarantee that the total amount of outputs produced by DMU J is not reduced and that the intermediate products produced are enough to supply those processes that consume them. Note that although, in the above model, it has been assumed that all processes exhibit variable returns to scale (VRS), other RTS assumptions can be considered in which case Constraints (5) should Mathematics 2020,8, 2222 4 of 18 be modified accordingly (see Lozano [ 5 ]). Moreover, the above formulation corresponds to the input orientation. The corresponding model for the output orientation is shown in Appendix A. 2.2. Fuzzy Numbers For the sake of clarity, an introduction to fuzzy numbers is provided in this section. More detailed information can be found at Dubois and Prade [ 35 ]. A fuzzy number e T is a subset of the real line R with membership function µe T:R→[0, 1]satisfying the following properties: (i) There exists t0∈Rsuch that µe T(t0)=1; (ii) e T is fuzzy convex. In other words, µe T(γt1+(1−γ)t2)≥minnµe T(t1),µe T(t2)o , for any t1 , t2∈R and γ∈[0, 1]; (iii) e Tis upper semicontinuous on R, which means that µ−1 e T([∝, 1]) is closed for all ∝ ∈ [0, 1]; (iv) The support of is µe Tbounded, i.e., the closure of nx∈Rµe T(t)>0ois bounded. In this paper, we deal with LR fuzzy numbers (LRFN) as in Le ó n et al. [ 13 ]. A fuzzy number e T=h(t)L,(t)R,(β)L,(β)RiL,Ris a LRFN if its membership function has the following structure: µe T(t)=                      1if (t)L≤t≤(t)R L(t)L−t (β)Lif (t)L−(β)L≤t≤(t)L Rt−(t)R (β)Rif (t)R≤t≤(t)R+(β)R 0otherwise (7) where L,R:[0, 1]→[0, 1] are non-increasing, continuous shape functions with L(0)=R(0)= 1 and L(1)=R(1)= 0. Furthermore, h(t)L,(t)Ri consists of the real numbers with the highest chance of realization, while (β)Land (β)Rare the left and right spread, respectively. Note that LRFN includes as special cases the commonly used trapezoidal fuzzy numbers (TrFN) and triangular fuzzy numbers (TFN). Thus, TrFN use linear left and right shape functions L(α)=R(α)= 1 −α . TFN use linear shape functions and, in addition, the left and right α= 1 values coincide, i.e., (t)L=(t)R. Finally, the α-cuts of a LRFN e Tare the intervals: e Tα=e TL α,e TU α e TL α=(t)L−L∗(α)·(β)L e TU α=(t)R+R∗(α)·(β)R (8) where the inverse shape functions are defined as L∗(α)=suph:L(h)≥α and R∗(α)= suph:R(h)≥α . In the case of TrFN and TFN these inverse shape functions are simply L∗(α)= R∗(α)=1−α. 3. Proposed Fuzzy Ranking Network DEA Approaches In this section, two different fuzzy ranking methods are presented. Both consider that the input, output and intermediate products are LRFN: e Xp ij =(xp ijL ,xp ijR ,βp ijL ,βp ijR)Li,Ri e Yp kj =(yp kjL ,yp kjR ,ˆ βp kjL ,ˆ βp kjR)ˆ Lk,ˆ Rk e Zp rj =(zp rjL ,zp rjR ,ˆ ˆ βp rjL ,ˆ ˆ βp rjR)ˆ ˆ Lr,ˆ ˆ Rr (9) Mathematics 2020,8, 2222 5 of 18 According to Definition (8), the α-cuts of these LRFN e Xp ij,e Yp kj and e Zp rj are the intervals: e Xp ijα="e Xp ijL α,e Xp ijU α# e Xp ijL α=xp ijL −L∗ i(α)·βp ijL e Xp ijU α=xp ijR +R∗ i(α)·βp ijR (10) e Yp kjα="e Yp kjL α,e Yp kjU α# e Yp kjL α=yp kjL −ˆ L∗ k(α)·ˆ βp kjL e Yp kjU α=yp kjR +ˆ R∗ k(α)·ˆ βp kjR (11) e Zp rjα="e Zp rjL α,e Zp rjU α# e Zp rjL α=zp rjL −ˆ ˆ L ∗ r(α)·ˆ ˆ βp rjL e Zp rjU α=zp rjR +ˆ ˆ R ∗ r(α)·ˆ ˆ βp rjR (12) Note also that it has been assumed that for a given input, output or intermediate product the left shape function is the same for all DMUs and processes and the same occurs with the right shape function. This assumption is commonly made (e.g., Le ó n et al. [ 13 ] or Soleimani-damaneh et al. [ 16 ]) and it is not too restrictive since it only assumes that the data for a certain factor (input, output or intermediate product) are described through LRFN of the same type. This assumption is important because it means that the linear combination of these LRFN, using scalar multipliers, is also a LRFN with the same left and right shape functions. Mathematically, P pP j λp je Xp ij =      P pP j λp jxp ijL ,P pP j λp jxp ijR ,P pP j λp jβp ijL ,P pP j λp jβp ijR      Li,Ri P pP j λp je Yp kj =      P pP j λp jyp kjL ,P pP j λp jyp kjR ,P pP j λp jˆ βp kjL ,P pP j λp jˆ βp kjR      ˆ Lk,ˆ Rk P pP j λp je Zp rj =      P pP j λp jzp rjL ,P pP j λp jzp rjR ,P pP j λp jˆ ˆ βp rjL ,P pP j λp jˆ ˆ βp rjR      ˆ ˆ Lr,ˆ ˆ Rr (13) The α-cuts of these LRFN P pP j λp je Xp ij,P pP j λp je Yp kj and P pP j λp je Yp kj are thus the intervals: P pP j λp je Xp ijα =P pP j λp j(xp ijL −L∗ i(α)βp ijL),P pP j λp j(xp ijR +R∗ i(α)βp ijR) P pP j λp je Yp kjα =P pP j λp j(yp kjL −ˆ L∗ k(α)ˆ βp kjL),P pP j λp j(yp kjR +ˆ R∗ k(α)ˆ βp kjR) P pP j λp je Zp rjα =P pP j λp j(zp rjL −ˆ ˆ L ∗ r(α)ˆ ˆ βp rjL),P pP j λp j(zp rjR +ˆ ˆ R ∗ r(α)ˆ ˆ βp rjR) (14) 3.1. Fuzzy Ranking Method 1 (FRM1) FRM1 uses the ranking method in Tanaka et al. [ 36 ]. This method was proposed in Le ó n et al. [ 13 ] for conventional (i.e., single-process DEA). The starting point is the formulation of the problem using the fuzzy data. The objective is to reduce all inputs as much as possible with respect to the observed Mathematics 2020,8, 2222 6 of 18 values. This corresponds to Equations (15) and (16). Note that the right-hand side of (16) corresponds to the observed input consumption of DMU J (summed for all the processes that consume that input) while the left-hand side is the input consumption of the target operating point (computed as a linear combination of the input consumption of all the DMUs). Similarly, in Constraints (17), the right-hand side represents the observed outputs of DMU J and the left-hand side is the corresponding amounts produced by the target operating computed using a convex linear combination of all the DMUs. Thus, Constraints (17) impose that the observed outputs are not reduced. Finally, Constraints (18) impose that the target operating point must satisfy the intermediate products constraints that guarantee that the internal production of intermediate products is enough to satisfy its internal demand. Min θ(15) subject to X p∈PI(i)X j λp je Xp ij ≤θX p∈PI(i)e Xp iJ ∀i(16) X p∈PO(k)X j λp je Yp kj ≥X p∈PO(k)e Yp kJ ∀k(17) X p∈Pout(r)X j λp je Zp rj −X p∈Pin(r)X j λp je Zp rj ≥0∀r(18) constraints (5)and (6) The Constraints (16)–(18) compare two fuzzy quantities. The key idea in this method is how to interpret the inequality, i.e., when to consider that one fuzzy quantity is larger than or equal to another. Based on Tanaka et al. [ 36 ], Le ó n et al. [ 13 ] proposed the use of the following ranking criterion for two fuzzy numbers e Mand e Nat possibility level α: e M≥αe N⇔ ∀h∈[α, 1]        e ML h≥e NL h e MR h≥e NR h (19) Following this criterion, for each possibility level α∈[0, 1] an input-oriented efficiency score EJ(α) can be computed for each DMU J using the following model: EJ(α)=Min θ(20) subject to X p∈PI(i)X j λp j·xp ijL ≤θ·X p∈PI(i)xp iJL∀i(21) X p∈PI(i)X j λp j·xp ijR ≤θ·X p∈PI(i)xp iJR∀i(22) X p∈PI(i)X j λp j·e Xp ijL α ≤θ·X p∈PI(i)e Xp iJL α ∀i(23) X p∈PI(i)X j λp j·e Xp ijU α ≤θ·X p∈PI(i)e Xp iJU α ∀i(24) X p∈PO(k)X j λp j·yp kjL ≥X p∈PO(k)yp kJL ∀k(25) Mathematics 2020,8, 2222 7 of 18 X p∈PO(k)X j λp j·yp kjR ≥X p∈PO(k)yp kJR ∀k(26) X p∈PO(k)X j λp j·e Yp kjL α ≥X p∈PO(k)e YkJL α∀k(27) X p∈PO(k)X j λp j·e Yp kjU α ≥X p∈PO(k)e YkJU α∀k(28) X p∈Pout(r)X j λp j·zp rjL −X p∈Pin(r)X j λp j·zp rjL ≥0∀r(29) X p∈Pout(r)X j λp j·zp rjR −X p∈Pin(r)X j λp j·zp rjR ≥0∀r(30) X p∈Pout(r)X j λp j·e Zp rjL α −X p∈Pin(r)X j λp j·e Zp rjL α ≥0∀r(31) X p∈Pout(r)X j λp j·e Zp rjU α −X p∈Pin(r)X j λp j·e Zp rjU α ≥0∀r(32) constraints (5)and (6) This fuzzy ranking approach has the drawback, as Soleimani-damaneh et al. [ 16 ] indicates for conventional fuzzy DEA, of having many constraints, more so in the case of fuzzy network DEA (due to the existence of intermediate products). However, the number of constraints can be reduced to almost half (because of many constraints being redundant) in the case of symmetric TFN, in which case the appropriate model to use is the following: EJ(α)=Min θ(33) subject toconstraints (23),(24),(27),(28),(31),(32),(5)and (6) Same as in León et al. [13], the following holds for the efficiency scores computed by FRM1: EJ(α1)≥EJ(α2)∀α1< α2 EJ(α2)=1⇒EJ(α1)=1∀α1< α2 (34) Thus, the efficiency scores EJ(α) represent a monotonous non-increasing function of the possibility level α . In order to rank the efficiency of the different DMUs the following area criterion is proposed. A DMU J ranks above another DMU J’ in terms of efficiency if the area below EJ(α) is greater than that the area below EJ0(α) . The area below EJ(α) is AreaEJ=R1 0EJ(α)dα which in the common case that EJ(α)is computed for several discrete possibility levels αqreduces to AreaEJ=PαqEJαq Pαq1. Finally, let us mention that, same as in Le ó n et al. [ 13 ], the efficiency of DMU J can also be expressed as a fuzzy set whose membership function is µEJ(θ)=supnα:EJ(α)=θo. Mathematics 2020,8, 2222 8 of 18 3.2. Fuzzy Ranking Method 2 (FRM2) FRM2 uses the ranking method in Yao and Wu [ 37 ]. This method was proposed in Soleimani-damaneh et al. [ 16 ] for conventional (i.e., single-process DEA). As before, the key is how to interpret the Inequalities (16)–(18), i.e., when to consider that one fuzzy quantity is larger than or equal to another. Yao and Wu [ 37 ] defined the signed distance between two fuzzy numbers e M and e Nas de M,e N=1 2Z1 0e ML α+e MU α−e NL α−e NR αdα(35) and proposed the following ranking criterion: e M≥e N⇔de M,e N≥0 e M≤e N⇔de M,e N≤0(36) Moreover, Soleimani-damaneh et al. [ 16 ] showed the signed distance between two LRFN e M= n(m)L,(m)R,(β)L,(β)RoL,Rand e N=n(n)L,(n)R,(γ)L,(γ)RoL,Rcan be expressed as de M,e N=(m)L+(m)R−(n)L−(n)R+h(γ)L−(β)LiZ1 0 L∗(α)dα+h(β)R−(γ)RiZ1 0 R∗(α)dα(37) and that the resulting model is equivalent to the corresponding crisp DEA model using appropriately defuzzified factor values. Thus, from the LRFN of each input, output and intermediate product a defuzzified value is computed as xp ij =1 2·(xp ijL +xp ijR +βp ijRR1 0R∗ i(α)dα−βp ijLR1 0L∗ i(α)dα) yp kj =1 2·(yp kjL +yp kjR +ˆ βp kjRR1 0R∗ k(α)dα−ˆ βp kjLR1 0L∗ k(α)dα) zp rj =1 2·(zp rjL +zp rjR +ˆ ˆ βp rjRR1 0R∗ r(α)dα−ˆ βp rjLR1 0L∗ r(α)dα) (38) where L∗ i(α) , L∗ k(α) , L∗ r(α) and R∗ i(α) , R∗ k(α) , R∗ r(α) are all inverse shape functions. In the case of TrFN and TFN L∗(α)=R∗(α)=1−αwhich means that: R1 0L∗ i(α)dα=R1 0L∗ k(α)dα=R1 0L∗ r(α)dα=1 2∀i∀k∀r R1 0R∗ i(α)dα=R1 0R∗ k(α)dα=R1 0R∗ r(α)dα=1 2∀i∀k∀r(39) and therefore xp ij =1 2·        xp ijL +xp ijR +βp ijR −βp ijL 2         yp kj =1 2·        yp kjL +yp kjR +ˆ βp kjR −ˆ βp kjL 2         zp rj =1 2·        zp rjL +zp rjR +ˆ ˆ βp rjR −ˆ ˆ βp rjL 2         (40) Mathematics 2020,8, 2222 9 of 18 Moreover, in the case of symmetric TrFN and TFN, the above expressions reduce to xp ij =xp ijL+xp ijR 2 yp kj =yp kjL+yp kjR 2 zp rj =zp rjL+zp rjR 2 (41) In any case, once the defuzzified values of the input, output and intermediate products have been computed, the following LP is solved: EJ=Min θ(42) subject to X p∈PI(i)X j λp jxp ij ≤θ·X p∈PI(i) xp iJ ∀i(43) X p∈PO(k)X j λp jyp kj ≥X p∈PO(k) yp kJ ∀k(44) X p∈Pout(r)X j λp jzp rj −X p∈Pin(r)X j λp jzp rj ≥0∀r(45) constraints (5)and (6) yielding, for each DMU J, a single efficiency score EJ. Note that the, in the end, above model is just Models (1)–(6) applied to the defuzzified values of the inputs, outputs and intermediate products. Its interpretation is similar, i.e., the model aims at minimizing the amount of inputs consumed by a target feasible operating point that is computed as linear convex combination of all the DMUs. The target operating point must maintain the output level of DMU J and it is also required that the intermediate products produced must be greater than the amount consumed. As mentioned above, the difference between this model and Models (1)–(6) is that in the latter the observed data were crisp while in the above model the crisp input, output, and intermediate product values used for all DMUs (including DMU J) are obtained from the observed fuzzy data through a defuzzification process. As with the previous approach the corresponding output-oriented model is formulated in Appendix A. 4. Illustration of Proposed Approaches In order to illustrate the proposed approach two datasets from the literature will be used. The first one corresponds to a simple two-stage system and was used in Kao and Liu [ 20 ]. This dataset involves 24 DMUs (which represent Taiwanese non-life insurance companies) and considers two inputs (Operating expenses and Insurance expenses), two intermediate products (Direct written premiums and Reinsurance premiums) and two outputs (Underwriting profit and Investment profit). The upper and lower efficiency limits computed by Kao and Liu [ 20 ], which correspond to the upper and lower efficiency estimates for α =0, respectively, are shown in Table 1. In addition, Table 1shows the estimates for α =1. Because in this dataset we are dealing with TFN, the upper and lower efficiencies for α=1 coincide. The efficiency estimates and the corresponding ranking index computed by the proposed FRM1 approach are also shown in Table 1. To compare with the results in Kao and Liu [ 20 ], constant returns to scale (CRS) in all processes and input orientation are considered. Although for FRM1 only three possibility levels are shown in Table 1, calculations were made for eleven levels, i.e., α ranges from 0 to 1 with 0.1 increments. Note that for FRM1 the estimated efficiency score for each possibility level is Mathematics 2020,8, 2222 16 of 18 X p∈PO(k)X j λp j·e Yp kjL α ≥γ·X p∈PO(k)e YkJL α∀k(A12) X p∈PO(k)X j λp j·e Yp kjU α ≥γ·X p∈PO(k)e YkJU α∀k(A13) constraints (5),(29),(30),(31),(32)and (A4) •FRM2 (output orientation) EJ−1=Max γ(A14) subject to X p∈PI(i)X j λp jxp ij ≤X p∈PI(i) xp iJ ∀i(A15) X p∈PO(k)X j λp jyp kj ≥γ·X p∈PO(k) yp kJ ∀k(A16) constraints (5),(45)and (A4) References 1. Kao, C. Efficiency decomposition in network data envelopment analysis: A relational model. Eur. J. Oper. Res. 2009,192, 949–962. [CrossRef] 2. Tone, K.; Tsutsui, M. Network DEA: A slacks-based measure approach. Eur. J. Oper. 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