AUTHOR COPY. DEA cross-efficiency evaluation under variable returns to scale

Transcription

1 Journal of the Operational Research Society (2015) 66, Operational Research Society Ltd. All rights reserved /15 DEA cross-efficiency evaluation under variable returns to scale Sungmook Lim 1 * and Joe Zhu 2 1 Dongguk Business School, Dongguk University, Seoul, South Korea; and 2 School of Business, Worcester Polytechnic Institute, Worcester, USA Cross-efficiency evaluation in data envelopment analysis (DEA) has been developed under the assumption of constant returns to scale (CRS), and no valid attempts have been made to apply the cross-efficiency concept to the variable returns to scale (VRS) condition. This is due to the fact that negative VRS cross-efficiency arises for some decision-making units (DMUs). Since there exist many instances that require the use of the VRS DEA model, it is imperative to develop cross-efficiency measures under VRS. We show that negative VRS cross-efficiency is related to free production of outputs. We offer a geometric interpretation of the relationship between the CRS and VRS DEA models. We show that each DMU, via solving the VRS model, seeks an optimal bundle of weights with which its CRS-efficiency score, measured under a translated Cartesian coordinate system, is maximized. We propose that VRS cross-efficiency evaluation should be done via a series of CRS models under translated Cartesian coordinate systems. The current study offers a valid cross-efficiency approach under the assumption of VRS one of the most common assumptions in performance evaluation done by DEA. Journal of the Operational Research Society (2015) 66(3), doi: /jors Published online 12 March 2014 Keywords: data envelopment analysis; cross-efficiency; variable returns to scale (VRS) 1. Introduction Data envelopment analysis (DEA) is a linear programmingbased technique for assessing relative efficiencies of a homogeneous set of decision-making units (DMUs) that consume multiple inputs to produce multiple outputs. It estimates the efficient frontier based on observed values on inputs and outputs, and thereby it distinguishes efficient units and inefficient ones. In its basic form, DEA allows DMUs to choose their own weights on inputs and outputs in the most favourable way with only a limited set of restrictions. With this self-evaluation mechanism as well as too much weight flexibility allowed, it has frequently been observed in many DEA applications that too many DMUs are determined to be efficient and no further distinction can be made among them. Various remedies have been proposed in the literature to address this drawback of DEA: incorporation of weight restrictions, cross-efficiency evaluation, and so on. Among these, cross-efficiency evaluation is addressed in the current paper. Cross-efficiency evaluation adopts peer appraisal in addition to self-appraisal for evaluating DMUs from a more holistic standpoint (see, eg, Sexton et al, 1986; Doyle and Green, 1994). Two major advantages of the cross-efficiency approach are: (1) it provides for an ordering of the DMUs under the concept of peer appraisal and (2) it eliminates unrealistic weighting *Correspondence: Sungmook Lim, Dongguk Business School, Dongguk University, 30, Pildong-ro 1-gil, Jung-gu, Seoul, , South Korea. sungmook@dongguk.edu schemes without requiring a priori information on weight restrictions from application area experts (Anderson et al, 2002). A number of new cross-efficiency models and applications have been proposed in the literature. For example, Liang et al (2008b) develop a game cross-efficiency approach to address the non-uniqueness of cross-efficiency scores. Ramón et al (2011) present peer-restricted cross-efficiency evaluation to reduce the negative influence of non-unique weights in crossefficiency. Cross-efficiency has also been used in various applications, for example, efficiency evaluations of nursing homes (Sexton et al, 1986), R&D project selection (Oral et al, 1991), preference voting (Green et al, 1996), Olympic Games (Wu et al, 2009), and others. Note that cross-efficiency evaluation has been used for DEA models mostly with the constant returns to scale (CRS) assumption, such as the CRS model or CCR model by Charnes, et al (1978). The literature, to the extent of the authors knowledge, has been almost silent on (and has not properly addressed) the issue that cross-efficiency evaluation for inputoriented DEA models with the variable returns to scale (VRS) assumption, such as the input-oriented VRS model or the BCC model of Banker et al (1984), has the problem of negative cross-efficiency for some units. There are many instances where a change in inputs does not result in the same change in outputs that require the use of the VRS DEA model. If DMUs (eg, bank branches) are of various sizes, then the VRS model is more appropriate to use so that a small-sized DMU is not benchmarked against large-sized DMUs. The VRS DEA model is one

2 Sungmook Lim and Joe Zhu DEA cross-efficiency evaluation under VRS 477 of the basic DEA models that is widely used in various DEA applications. Therefore, it is imperative to develop crossefficiency measures under the condition of VRS. Wu et al (2009) propose to add an additional constraint in the VRS modelsothatcross-efficiencies are non-negative. However, such a modification does not properly address the root cause of the negative VRS cross-efficiency problem. In fact, cross-efficiency evaluation is closely related to the issue of incorporation of weight restrictions in the sense that each DMU is evaluated by weights chosen by other DMUs in addition to its own. It is well known that the incorporation of weight restrictions in DEA models may result in their infeasibility or non-positive efficiency scores of some units. Podinovski and Bouzdine-Chameeva (2013) find that these problems arise when weight restrictions induce free production of outputs (ie, positive outputs with zero inputs) in the underlying technology, which is unacceptable from the production theory point of view. Applying the same concept, we also find that the problem of negative cross-efficiency in the inputoriented VRS DEA model arises when a DMU is crossevaluated by a weight vector associated with an efficient frontier that extends to induce free production of outputs in the underlying technology. We claim that such problematic weights are invalid (or unacceptable) for cross-efficiency evaluation and need to be adjusted. To develop a way of resolving the problem of negative crossefficiency in the input-oriented VRS DEA model, we develop a geometric interpretation of the relationship between the VRS and CRS models. We show that every DMU, via solving the VRS model, seeks a translation of the Cartesian coordinate system and an optimal bundle of weights such that its CRSefficiency score, measured under the chosen coordinate system, is maximized. Therefore, VRS cross-efficiency is related to the CRS cross-efficiency measures. Using the fact that no efficient frontier extends to induce free production of outputs in the CRS model, we propose that cross-efficiency evaluation for the VRS model should be done via a series of CRS models under translated Cartesian coordinate systems. The output-oriented VRS DEA model does not suffer from the problem of negative cross-efficiency since their constraints prevent it. Even for the input-oriented VRS DEA model, optimal bundles of weights chosen by DMUs exhibiting increasing returns to scale (IRS) do not give rise to the problem of negative cross-efficiency since the optimal value of the free variable in the VRS multiplier model is always negative for those DMUs (assuming the VRS multiplier model is stated as model (1) in the next section). However, the problem of invalid weights for cross-efficiency evaluation is hidden and still exists for these cases (see Section 4). Consequently, the proposed approach should be applied to these two cases. The current study is the first to offer a valid cross-efficiency approach under the assumption of VRS one of the most common assumptions in performance evaluation done by DEA. We should note here that the proposed approach is based on coordinate system translation. Therefore, the proposed approach appears to have a similar effect with data translation under the VRS model (Ali and Seiford, 1990; Pastor, 1996). However, data translation for all DMUs does not change the shape of the VRS frontier. As a result, the problem of negative VRS cross-efficiency still exists under translation invariance, and data translation cannot resolve the fundamental problem of weight invalidity related to free production of outputs, which will be detailed in the main body of this paper. The paper is organized as follows. The next section is devoted to an illustration of how the problem of negative cross-efficiency arises in the input-oriented VRS model using a simple one-input and one-output example, and shows how it is related to free production of outputs. A geometric interpretation of the relationship between the VRS and the CRS models is then provided to lay a foundation for our approach of crossefficiency evaluation in the VRS DEA model. The proposed crossefficiency evaluation method is adapted to the output-oriented VRS model, followed by an illustrative application to R&D projection selection. Concluding remarks are given at the end. 2. Negative cross-efficiency and free production We assume that there are n DMUs that consume m inputs to produce s outputs. DMU k (k = 1, 2,, n) uses a vector of inputs x k = (x 1k,, x mk ) T R + m to produce a vector of outputs y k = (y 1k,, y sk ) T R + s. The input-oriented VRS model in multiplier form is as follows: max s:t: X s X m X m u r y r0 - ξ v i x ij - Xs v i x i0 ¼ 1 v i ; u r ε 8i; r; ξ free in sign u r y rj + ξ 0; j ¼ 1; ¼ ; n where ε is a positive non-archimedean infinitesimal. When the above model is solved, an efficiency score of DMU 0 and crossefficiencies of the other DMUs (evaluated by DMU 0 ) are obtained together. Specifically, a cross-efficiency of DMU j is given by P s e 0j ¼ u* r y rj - ξ * P m v* i x (2) ij ð1þ where * denotes an optimal solution to the model. Due to the free variable ξ, cross-efficiency calculated by (2) may be negative when ξ > 0, which results in a problematic situation. Averaging e ij over i,weobtainacross-efficiency score of DMU j. The free variable ξ provides an indication of the type of returns to scale (RTS) that prevail at a particular DMU under evaluation. Specifically, IRS (decreasing returns to scale (DRS)) prevails at (x 0, y 0 )ifandonlyifξ* < 0(ξ* > 0) for all

3 478 Journal of the Operational Research Society Vol. 66, No.3 optimal solutions to the VRS model, whereas CRS prevails at (x 0, y 0 ) if and only if ξ* = 0 in any optimal solution (see Banker et al, 2011). It is worth mentioning that the VRS model itself involves cross-efficiency evaluation in its constraints (and the same is true with the CRS model). Model (1) dictates that each DMU seeks an optimal bundle of weights while making crossefficiencies of the other DMUs not exceeding unity. Also note that these cross-efficiencies are measured (in a linearized form within the constraints of model (1)) no matter which type of RTS prevails at cross-evaluated DMUs. In other words, optimal weights chosen by a DMU exhibiting one type of RTS (say IRS) are used to cross-evaluate the other DMUs exhibiting different types of RTS (say CRS or DRS) within model (1). This interpretation provides a justification of the use of crossefficiency evaluation in DEA as a peer-appraisal approach, particularly under the VRS assumption. We now illustrate how the problem of negative crossefficiency arises in the input-oriented VRS model using a simple one-input and one-output example. Suppose the data set in Table 1 is given, which consists of seven DMUs with a single input and a single output. A VRS efficiency score of each DMU along with its optimal weights are also provided in Table 1. Figure 1 plots the data set and the supporting hyperplane associated with an optimal bundle of weights chosen by DMU Table 1 Data and optimal values DMU Input Output v* u* ξ* Efficiency score A 5/3 1 3/ B 2 2 1/2 1/6 2/3 1 C 4 3 1/4 1/8 1/4 5/8 D 3 4 1/3 1/6 1/3 1 E 6 5 1/6 1/6 1/6 2/3 F 5 6 1/5 1/5 1/5 1 G 7 7 1/7 2/7 1 1 G. Hyperplane H G represents an optimal bundle of weights (v*, u*, ξ*) = (1/7, 2/7, 1) for DMU G, with which DMU G attains an efficiency score of unity. Using an input-oriented radial distance measure, cross-efficiencies of DMUs D, E, and F evaluated by DMU G can be determined with reference to hyperplane H G. For instance, a cross-efficiency of DMU E is E 0 E 1 /E 0 E, which is 3/6 = 1/2, a cross-efficiency of DMU D is 1/3, and a cross-efficiency of DMU F is unity. The same results can also be obtained when we use (2) for calculating cross-efficiency: e GE = (10/7 1)/(6/7) = 1/2, e GD = (8/7 1)/(3/7) = 1/3, and e GF = (12/7 1)/(5/7) = 1. While no problems seem to occur in calculating crossefficiencies of these DMUs, a difficulty will be encountered in determining cross-efficiencies of the other DMUs positioned below the horizontal line (labelled x -axis) that intersects the y-axis at O = (0, ξ*/u*) = (0, 7/2). In fact, model (1) forces cross-efficiencies of DMUs A, B, and C to be determined with reference to the negative-input segment of hyperplane H G.For instance, a cross-efficiency of DMU C is C 0 C 1 /C 0 C, which ( 1)/4 = 1/4, and a cross-efficiency of DMU A is A 0 A 1 /A 0 A, which is ( 5)/(5/3) = 3. The negative sign is due to the position of A 1 and C 1 (left to the y-axis).notethatthesame results can be obtained when we use (2): e GC = (6/7 1)/ (4/7) = 1/4, and e GA = (2/7 1)/(5/3/7) = 3. This is problematic because not only are negative cross-efficiencies obtained, but also e GA can diverge to negative infinity as the slope of H G becomes flatter. We claim that this problem is caused by situations where weights chosen by some DMUs are invalid for cross-evaluating other DMUs; an optimal bundle of weights chosen by DMU G is not valid for determining cross-efficiencies of DMUs A, B, and C. To justify this claim, it is worthwhile to note that the efficient frontier associated with the optimal weights chosen by DMU G extends to induce the point O, which represents a free production of outputs in the underlying technology. Furthermore, model (1) forces DMUs A, B, and C to be Figure 1 Cross-efficiency evaluation by DMU G.

4 Sungmook Lim and Joe Zhu DEA cross-efficiency evaluation under VRS 479 cross-evaluated with reference to the invalid part of the extended efficient frontier that emanates from the unacceptable free production point O and points southwest. This implies that some kind of adjustment is required for those invalid weights to be properly used for cross-efficiency evaluation. We proceed to examine the case of negative values of ξ*. Figure 2 shows the supporting hyperplane associated with an optimal bundle of weights chosen by DMU B. Hyperplane H B represents an optimal bundle of weights (v*, u*, ξ*) = (1/2, 1/6, 2/3) for DMU B, with which DMU B attains an efficiency score of unity. Using an input-oriented radial distance measure, cross-efficiencies of the other DMUs evaluated by DMU B can be determined with reference to hyperplane H B. For instance, a cross-efficiency of DMU E is E 0 E 1 /E 0 E, which is 3/6 = 1/2, and a cross-efficiency of DMU G is G 0 G 1 /G 0 G, which is (11/3)/ 7 = 11/21. Note that the same results can be obtained when we use (2) for calculating cross-efficiency: e BE = (5/6 ( 2/3))/(6/2) = 1/2 and e BG = (7/6 ( 2/3))/(7/2) = 11/21. When the optimal value of the free variable, ξ*, is non-positive, the problem of negative crossefficiency does not seem to occur. In other words, when a DMU is cross-evaluated by another DMU exhibiting IRS, it always attains a positive cross-efficiency. Although seemingly unproblematic for this case, our claim made in the above (using the case of DMU G) still applies. According to Podinovski and Bouzdine-Chameeva (2013), a technology is said to allow free production of outputs when it is possible to produce positive outputs with zero inputs. We Figure 2 Cross-efficiency evaluation by DMU A. extend this definition to include the case of negative outputs with zero inputs such as point O in Figure 2. Positive outputs with zero inputs and negative outputs with zero inputs will be referred to as type I and type II of free production of outputs, respectively. Note that the efficient frontier associated with the optimal weights chosen by DMU B extends to induce the point O (in Figure 2), which represents a type II free production of outputs in the underlying technology. Model (1) forces the other DMUs to be cross-evaluated with reference to the invalid efficient frontier (which extends to induce the unacceptable type II free production point O ) for them. 1 On the basis of the above observations, we conclude that cross-efficiency evaluation via the conventional VRS model is not proper regardless of whether the problem of negative cross-efficiency actually arises or not. Therefore, for the VRS DEA model, some significant change of the framework of cross-efficiency evaluation should be developed. We accomplish this based on a geometric view of the relationship between the VRS and CRS models, which will be presented in the next section. 1 Type II free production of outputs can be interpreted as consumption (opposite to production) of outputs without any inputs, and it can be considered as extended free disposability of outputs, which may not be unacceptable. However, we pursue our development assuming it is unacceptable in this paper, which provides a more general framework. In case type II free production of outputs is acceptable; the developed framework of VRS cross-efficiency can be easily simplified to suit the case.

5 480 Journal of the Operational Research Society Vol. 66, No.3 3. DEA and coordinate systems: a geometric link between the VRS and CRS models To lay a foundation for our cross-efficiency evaluation in the VRS DEA model, we develop a geometric interpretation of the VRS model as a series of CRS models under translated Cartesian coordinate systems. Consider the following theorem. Theorem 1 Given any optimal solution (v*, u*, ξ*) to model (1) chosen by a VRS-efficient DMU (denoted DMU 0 ), a CRS-efficiency score of DMU 0, measured under the translated Cartesian coordinate system defined by an adjusted origin O* = ( β 1 ξ*/v * 1,, β m ξ*/v m *, β m +1 ξ*/u*, 1, β m + s ξ*/u*), s is unity, for any β k R + (k = 1, m +, m + s) such that s k = 1 β k = 1. Proof Under the translated Cartesian coordinate system with origin O*, the input-oriented VRS model (3) with the same setofdmusispresentedasfollows: X s max μ r y r0 - β m + rξ * - κ s:t: X m X m u * r v i x ij + β iξ * - Xs v * i + κ 0; j ¼ 1; ¼ ; n v i x i0 + β iξ * ¼ 1 v * i v i ; μ r 0 8i; r; κ free μ r y rj - β m + rξ * u * r ð3þ Consider a solution (v, μ, κ) = (v*/ Γ, u*/ Γ,0) with Γ = 1+ξ* m s k = 1 β k. Note that Γ > 0 since ξ* 1+ r = 1 u * r y r0 > 1and m k = 1 β k 1. Plugging (v*/ Γ, u*/ Γ,0) into the first set of constraints of model (3), we obtain X m v * i Γ x ij + β iξ * - Xs u * r v * i Γ y rj - β m + rξ * u * r! ¼ 1 X m v * i Γ x ij - Xs u * r y rj + ξ * Xm + s β k X m k¼1 ¼ 1 v * i Γ x ij - Xs u * r y rj + ξ! * 0; 8j where the non-negativity is ensured due to the fact that (v*, u*, ξ*) is a feasible solution to model (1) and Γ > 0. If (v*/ Γ, u*/ Γ, 0) is plugged into the second set of constraints of model (3), we obtain X m v * i Γ x i0 + β iξ *! v * ¼ 1 X m v * i i Γ x i0 + ξ * Xm β k ¼ 1 Γ Xm 1 + ξ* k¼1 k¼1 β k! ¼ 1 using the fact that m i = 1 v i *x i 0 = 1. Therefore, (v*/ Γ, u*/γ, 0) is a feasible solution to model (3). Now examine its objective value, which is X s u * r Γ y r0 - β m + rξ *! u * ¼ 1 X s u * r r Γ y r0 - ξ * Xm + s β k k¼m + 1!! ¼ 1 X s u * r Γ y r0 - ξ * + ξ * 1 - Xm + s β k ¼ 1 Γ Xm 1 + ξ* k¼1 β k! ¼ 1 k¼m + 1 Hence, (v*/ Γ, u*/ Γ, 0) is an optimal solution to model (3) with which DMU 0 attains an efficiency score of unity, implying that DMU 0 is VRS-efficientatwhichCRSprevails since κ = 0 in the optimal solution. This leads to the conclusion that DMU 0 is CRS-efficient under the translated coordinate system. Theorem 1 indicates that each DMU, via solving the VRS model, seeks an optimal bundle of weights and free variable value with which its CRS-efficiency score, measured under a translated Cartesian coordinate system, is maximized. In addition, the theorem shows that the location of the adjusted origin of the translated coordinate system is associated with the chosen optimal bundle of weights and free variable value. It should be noted here that the efficient frontier determined by the CRS model does not extend to induce free production of outputs, and thus the CRS model does not suffer from the problem of negative cross-efficiency. Therefore, we expect that the problem of negative cross-efficiency can be effectively resolved by transforming the VRS model into a series of CRS models. Although Theorem 1 deals with only VRS-efficient DMUs, it can be applied implicitly to VRS-inefficient ones as well since inefficient DMUs can choose the same weights with their reference points (projections) on the efficient frontier to maximize their CRS efficiency. However, it should be pointed out that a VRS efficiency score of an inefficient DMU under the original coordinate system may differ from its CRS efficiency score under a translated coordinate system chosen by Theorem 1. On the basis of Theorem 1, the following corollaries provide a link between VRS optimal weights and free production. Corollary 1 The supporting hyperplane of the efficient frontier associated with an optimal bundle of weights in model (1) chosen by a VRS-efficient DMU exhibiting DRS extends to induce type I free production of outputs in the underlying technology. Proof Suppose there is a VRS-efficient DMU at which DRS prevails. Then an optimal value of ξ* chosen by the unit in model (1) will be positive. According to Theorem 1 and its proof, the supporting hyperplane associated with an optimal bundle of weights (v*, u*, ξ*) chosen by the unit in model (1) under the original coordinate system will have

6 Sungmook Lim and Joe Zhu DEA cross-efficiency evaluation under VRS 481 Figure 3 Cross-efficiency evaluation by DMU G under a translated coordinate system. the same slope with the supporting hyperplane associated with an optimal bundle of weights (v*/ Γ, u*/γ) chosen by the same unit in the corresponding CRS model (or the VRS model with the free variable being 0) under a translated coordinate system chosen by Theorem 1. We now consider a special case of Theorem 1 where only output-associated axes are allowed to be translated. The latter hyperplane (associated with (v*/ Γ, u*/ Γ) will go through the adjusted origin O* = (0,,0,β 1 ξ*/u * 1,, β s ξ*/u * s ), which means that the former hyperplane (associated with (v*, u*, ξ*)) will also go through the point O* under the original coordinate system. The point O* represents positive outputs with zero inputs, which is a type I free production of outputs under the original coordinate system. Corollary 2 The supporting hyperplane of the efficient frontier associated with an optimal bundle of weights in model (1) chosen by a VRS-efficient DMU exhibiting IRS extends to induce type II free production of outputs in the underlying technology. Since the proof of this corollary can be given similarly to that of Corollary 1, it is omitted to save space. If a VRS-efficient DMU exhibits CRS (ie, it is CRS-efficient), the corresponding efficient frontier may or may not extend to induce free production of outputs, depending on the sign of the chosen ξ*. However, an optimal solution to model (1) where ξ* = 0 always exists for a CRS-efficient DMU, and we assume that such solution is always chosen. (In applications, once the CRS condition is identified with a DMU, we can use the CRS model to calculate cross-efficiency scores.) Corollaries 1 and 2 suggest that optimal weights chosen by DMUs at which either DRS or IRS prevails are not valid for cross-evaluating other DMUs, and this provides a good rationale for the development of our cross-efficiency evaluation approach in the VRS DEA model, which will be presented in the next section. Before we proceed, we illustrate Theorem 1 and the related corollaries using the example introduced in the previous section For an illustration, we use DMU G, which is VRS-efficient and exhibits DRS under the original coordinate system with origin O, depicted in Figure 1. The supporting hyperplane H G intersects the y-axis at O = (0, ξ*/u*) = (0, 7/2) and the x-axis at O = ( ξ*/v*, 0) = ( 7, 0). Any point on the line O O can be represented by a convex combination of the two extreme points: O* = ( β 1 ξ*/v*, β 2 ξ*/u*) = ( 7β 1, (7/2)β 2 ) where β 1 + β 2 = 1andβ 1, β 2 R +. Observe that the supporting hyperplane H G of the efficient frontier associated with the optimal weights chosen by DMU G extends to induce the point O (in Figure 1), which represents a type I free production of outputs, as indicated by Corollary 1. Therefore, the optimal weights chosen by DMU G at which DRS prevails are not valid for cross-evaluating the other DMUs; for some DMUs such as A, B, and C, this invalidity results in the actually observable realization of negative cross-efficiency. On the other hand, a VRS efficiency score of DMU G, measured under the translated Cartesian coordinate system with an adjusted origin O*, is unity and CRS prevails at DMU G, meaning that DMU G is CRS-efficient under the translated coordinate system, as indicated by Theorem 1. Considering a special case where β 1 = 1andβ 2 = 0, the Cartesian coordinate system defined by the x-axis and the y -axis with the adjusted origin O* = O = ( 7, 0) renders DMU G CRS-efficient. Note that, under the translated coordinate system, the optimal weights (v*/ Γ, u*/γ) = (v*/(1 + ξ*β 1 ), u*/(1 + ξ*β 1 )) = ((1/7)/2, (2/7)/2) = (1/14, 1/7) chosen by DMU G, represented again by H G, are valid for cross-evaluating the other DMUs since they do not induce free production of outputs. For instance, under the translated coordinate system, the coordinates of DMU A are (26/3, 1), and its cross-efficiency using the weights (1/14, 1/7), is e GA = (1/7)/(13/21) = 3/13. This score can also be determined geometrically under the translated coordinate system in Figure 3: e GA ¼ A 3 A 1 /A 3 A ¼ 2/ð26/3Þ ¼3/13.

7 482 Journal of the Operational Research Society Vol. 66, No.3 Table 2 Raw data of 37 R&D projects on 5 outputs and budget R&D project Indirect economic contribution Direct economic contribution Technical contribution Social contribution Scientific contribution Budget Cross-efficiency evaluation in the VRS model We start with the input-oriented VRS model. Recall that Theorem 1 and the corollaries show that optimal weights chosen by a VRS-efficient DMU exhibiting IRS or DRS are not valid for cross-evaluating other DMUs because the corresponding supporting hyperplane of the efficient frontier extends to induce type I or type II free production of outputs. They also imply a geometric relationship between the VRS and CRS models, which can be stated as follows: the VRS model for any DMU can be casted as the CRS model for the same DMU under a translated Cartesian coordinate system. Using the fact that any supporting hyperplane of the efficient frontier does not extend to induce free production of outputs in the CRS model and is thus always valid for cross-efficiency evaluation, we propose that cross-efficiency evaluation for the VRS model should be done via a series of CRS models under translated Cartesian coordinate systems. For an intuitive exposition, let us first examine DMU G in Figure 1. Solving the VRS model for DMU G is equivalent to solving the CRS model for the same unit under a translated Cartesian coordinate system. While there exist numerous choices of an adjusted origin according to Theorem 1 (ie, any point along O O will do), a convenient choice will be either a point on the x-axis or one on the y-axis. However, if the point on the y-axis, O, is chosen for an adjusted origin, some units (DMUs A, B, and C) will have negative outputs under the translated coordinate system, which is not acceptable. This is not the case with the point on the x-axis, O, and therefore it is selected for an adjusted origin. 2 With such translation of the coordinate system 2 We should note that O is not the only choice for O* along O O that does not give rise to the negative-output problem, and resulting cross-efficiencies depend on the choice of O*. However, the choice of an adjusted origin on the x-axis makes it possible to derive a general formula (that does not depend on coefficients β k ) for VRS cross-efficiency as shown in the subsequent paragraphs.

8 Sungmook Lim and Joe Zhu DEA cross-efficiency evaluation under VRS 483 Table 3 Simple efficiency and cross-efficiency scores under VRS and CRS R&D project Proposed VRS crossefficiency score Conventional VRS crossefficiency score VRS simple efficiency score CRS crossefficiency score CRS simple efficiency score Returns to scale DRS DRS IRS DRS DRS DRS DRS DRS DRS DRS DRS IRS IRS DRS DRS IRS CRS DRS DRS IRS DRS DRS DRS DRS IRS DRS DRS IRS DRS DRS DRS IRS IRS IRS CRS DRS DRS applied, it becomes valid for DMU G to cross-evaluate the other DMUs with reference to its supporting hyperplane H G. A similar reasoning can be applied to DMU A in Figure 2. Solving the VRS model for DMU A is equivalent to solving the CRS model for the same unit under a translated Cartesian coordinate system. Again there exist numerous choices of an adjusted origin along O O according to Theorem 1. While a convenient choice will be either a point on the x-axis or one on the y-axis, the point on the x-axis, O, can be selected to ensure consistency. With such translation of the coordinate system applied, it becomes valid for DMU A to cross-evaluate the other DMUs with reference to its supporting hyperplane H A. We now develop a general formula for cross-efficiency evaluation in the input-oriented VRS model. Suppose that DMU 0 cross-evaluates DMU j using its optimal solution (v*, u*, ξ*) to model (1). The translation of the coordinate system is considered defined by an adjusted origin O* = ( β 1 ξ*/v 1 *,, β m ξ*/v m *, 0,, 0) where 0 repeats s times for the outputassociated coordinates, m k =1 β k = 1 and β k R +. Under this I translated coordinate system, a CRS cross-efficiency e 0j of DMU j is determined using the optimal weights (v*/γ, u*/ Γ) chosen by DMU 0 as follows: e I 0j ¼ P s u * r P m v * r Γ x rj + β rξ * v * r P s u* r y rj Γ y rj ¼ P m v* r x rj + β rξ * P s ¼ u* r y rj P m v* r x rj + ξ * v * r ð4þ We use this formula for a VRS cross-efficiency of DMU j (evaluated by DMU 0 ) under the original coordinate system.

9 484 Journal of the Operational Research Society Vol. 66, No.3 Note that the final formula does not involve coefficients β r and has the same form with the inverse of the conventional crossefficiency formula used in the output-oriented VRS model. According to the following proposition, VRS cross-efficiencies calculated by (4) are positive and less than or equal to unity. Proposition 1 0 < e I 0j 1 Proof Optimal weights used in (4) are obtained by solving model (1). The first set of constraints in model (1) can be rewritten as follows: X m v i x ij + ξ Xs u r y rj >0; j ¼ 1; ¼ ; n Therefore, it follows that 0 < e I 0j 1 for all j. Considering that cross-efficiency evaluation in DEA is a I method of peer evaluation, we note that e 0j implements peer evaluation for the VRS DEA model more fully. Note that the VRS DEA model allows a DMU to choose an optimal value ξ* in addition to (v*, u*), where ξ* indicates the RTS type of the DMU. By Theorem 1, we see that an optimal choice (v*, u*, ξ*) of a DMU determines the normal vector (v*, u*) of the supporting hyperplane associated with the efficient frontier (onto which the DMU is projected), as well as the origin of a new Cartesian coordinate system under which the DMU exhibits CRS (ie, the most productive scale size). The new origin can be determined based on the value of ξ*. Note that the CRS DEA model determines only (v*, u*) fixing ξ* = 0. Therefore, cross-efficiency evaluation in the VRS DEA model should properly take into account the role of ξ*. Recall that the role of ξ* is to determine a new Cartesian coordinate system under which the evaluated DMU attains the most productive scale size (ie, CRS efficiency). Let us define the general concept of peer evaluation in DEA as follows: each DMU cross-evaluates other peer DMUs under its own best evaluation environment. Here the best evaluation environment refers to the weights on the input output factors as well as the new coordinate system that are most favourable to the DMU. Under this best evaluation environment, the DMU itself attains the highest efficiency score as well as the most productive scale size. This concept of peer I evaluation can be implemented by e 0j by allowing each DMU to cross-evaluate other peer DMUs under its own best evaluation environment properly represented by all components of the DMU s optimal weights (v*, u*, ξ*). The problem of negative cross-efficiency cannot be observed in the output-oriented VRS model due to its constraints. However, the same weight invalidity problem related to free production of outputs occurs in cross-efficiency evaluation for the output-oriented VRS DEA model as well. This is the case because the input-oriented and output-oriented VRS models have the same efficient frontier structure. We do not investigate this case in detail here because the same logic can be simply applied. Instead, we just provide a formula for crossefficiency evaluation in the output-oriented VRS model as Table 4 follows: e O 0j ¼ Frequency of negative conventional VRS cross-efficiency R&D project exhibiting DRS Number of negative cross-efficiencies P m v* r x rj P s u* r y rj - ξ * (5) We conclude this section by suggesting a formula for crossefficiency score. As pointed out in Section 3, a VRS crossefficiency of a DMU evaluated by the DMU itself, calculated using (4), differs from its simple VRS efficiency score calculated using (2). This means that one DMU will be given n +1 scores: n cross-efficiencies and one simple efficiency. Therefore, a cross-efficiency score of a DMU can be determined by averaging these n cross-efficiencies. 5. An illustrative application to R&D projection selection We apply our proposed VRS cross-efficiency approach to a set of 37 project proposals relating to the Turkish iron and steel industry studied in Oral et al (1991). Each project is characterized by five output measures: direct economic contribution, indirect economic contribution, technological contribution, scientific contribution, and social contribution. The single input is the budget. Table 2 reports the data. Table 3 shows cross-efficiency and simple efficiency scores determined using alternative formulas and RTS assumptions. VRS cross-efficiency scores calculated using our proposed formula (4) and those by the conventional formula (2) are listed in Columns 2 and 3, respectively, and VRS simple efficiency

10 Sungmook Lim and Joe Zhu DEA cross-efficiency evaluation under VRS 485 Table 5 Project selection results Table 6 Standard deviation of cross-efficiency R&D project Proposed VRS cross-efficiency score CRS crossefficiency score Our selection CRS selection Budget Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Budget sum scoresarereportedincolumn4.inaddition,crscross-and simple efficiency scores and RTS classification are also reported in Columns 5, 6, and 7, respectively. It can be observed that some DMUs have negative values of conventional VRS cross-efficiency scores, which is obviously problematic. Specifically, Table 4 shows the frequency of negative conventional VRS cross-efficiency determined using formula (2) by each DMU (as an evaluator) exhibiting DRS. In fact, all DMUs exhibiting DRS yield at least one negative cross-efficiency, which implies that the problem of negative cross-efficiency in the VRS model is significant and never a rare occasion. The RTS classification is obtained using the approach of Seiford and Zhu (1998) to address the problem of possible multiple optimal solutions in model (1). For CRS DMUs, it is likely that ξ* 0. In our current application, once the CRS R&D project Standard deviation of the proposed VRS cross-efficiency Standard deviation of CRS cross-efficiency DMUs are identified, we use the CRS model to calculate crossefficiency scores. Table 5 shows the project selection results based on the proposed VRS cross-efficiency scores and CRS cross-efficiency scores, in which projects are chosen by the decreasing order of cross-efficiency scores until the budget for the programme (given 1000) is exhausted. There are some ranking differences between the two approaches. For example, project 23 is ranked in 5th place by the proposed VRS cross-efficiency, whereas it is ranked in 10th place by the CRS cross-efficiency. Due to these differences, the two approaches choose different sets of projects; Project 17 is included by one approach but excluded by the other, and Projects 32 and 12 are accepted by only one approach. Although there are some ranking differences and different project selections are made by the two approaches (the

11 486 Journal of the Operational Research Society Vol. 66, No.3 proposed VRS cross-efficiency versus the CRS cross-efficiency), the overall results are broadly similar. However, a striking difference between the two approaches can be found by examining the standard deviation of cross-efficiency, which is reported in Table 6. It shows that the CRS cross-efficiencies have much smaller variation, whereas the VRS crossefficiencies have a wide range of values. This is due to the fact that the efficient frontier under the VRS assumption consists of generally far more faces than that under the CRS assumption. This suggests that VRS cross-efficiency is a more effective approach for switching the self-appraisal model of DEA into a peer-appraisal one. Furthermore, the use of DEA in multicriteria decision-making applications such as portfolio selection may require careful attention to be paid to the variability (variance) of cross-efficiencies in addition to their average (cross-efficiency score) to take into account the risk of selected portfolios (see, eg, Stewart, 1996; Salo and Punkka, 2011; Chen and Zhu, 2011; Lim and Zhu, 2013). In this case, VRS crossefficiency evaluation can provide a more comprehensive way of effectively revealing the variability of cross-efficiencies. 6. Concluding remarks We have investigated the problem of negative cross-efficiency in the VRS DEA model and pointed out that the problem is closely related to free production of outputs. To resolve this issue, we have developed a new and valid method of crossefficiency evaluation in the VRS model based on a novel geometric view of the relationship between the VRS and CRS models. Two other alternative approaches can be applied. One is to s incorporate a non-negativity constraint on r = 1 u r y rj ξ, j in model (1), as is suggested by Wu et al (2009). Although this seems a quick fix, it can be easily observed that this fix distorts the feasible region of multipliers. Specifically, for our example, the non-negativity constraints collectively become ξ/u min y j = 1. Note that, for a feasible solution (v, u, ξ), ξ/u denotes the y-intercept of the associated supporting hyperplane. With this distorted feasible region of multipliers, DMUs F and G cannot attain an efficiency score of unity, even though they are VRSefficient. Therefore, this alternative does not correctly address the issue of negative VRS cross-efficiency score. The other approach is simply not to use problematic (or invalid) optimal weights, such as the one chosen by DMU G in the example, for cross-efficiency calculation. In other words, only optimal weights chosen by DMUs exhibiting CRS are used for cross-efficiency evaluation. While this approach is reasonable in that only valid optimal weights are used in the calculation of cross-efficiency, it may cause an unbalanced (or partial) cross-efficiency evaluation. In other words, it prevents a full range of peer evaluation. It may appear that under VRS cross-efficiency, a small-sized DMU can be benchmarked against a large-sized DMU. However, this particular issue should not be a concern under the concept of cross-efficiency. We point out that the basic idea of cross-efficiency is peer evaluation, namely, applying one DMU s perspective (manifested in its optimal bundle of weights) to others. As recently pointed out by Cook et al (2014), although DEA has a strong link to production theory in economics, the tool is also used for benchmarking in operations management, where a set of measures is selected to benchmark the performance of manufacturing and service operations. In the circumstance of benchmarking, the efficient DMUs, as defined by DEA, may not necessarily form a production frontier, but rather lead to a best-practice frontier. Under this general concept, we believe that size or frontier type should not be an issue of concern, and our proposed approach works. VRS and CRS are just two terms used to characterize the shapes of the DEA best practice frontier. Without the concept of RTS, these two different shapes of DEA frontiers still exist; VRS simply offers a tighter envelopment. Furthermore, under multiple inputs and multiple outputs, it is difficult to define what constitutes a small- or large-sized DMU. The idea of crossefficiency is to benchmark DMUs against each other, regardless of their size, whether under CRS or VRS. Under CRS crossefficiency, a large-sized DMU is also benchmarked against a small-sized DMU. If one uses the concept of RTS to characterize the shape of the DEA frontier and to classify DMUs, one can clearly see the difference between CRS and VRS crossefficiency. In general, a set of DMUs can be classified into three groups: IRS, CRS, and DRS. Under the CRS cross-efficiency, all the CRS-efficient facets are applied to IRS and DRS DMUs, while under the VRS cross-efficiency, IRS, CRS, and DRSefficient facets are applied to all DMUs. Since the general concept of cross-efficiency is to look at the performance of a DMU by using other DMUs weights or facets, it is reasonable to apply IRS, CRS, and DRS facets to all DMUs and to generate VRS cross-efficiency. With the approach developed in the current paper, we can now use the cross-efficiency concept under the VRS assumption.wenotethatnon-uniqueness ofcross-efficiency resulting from multiple optimal multipliers is still an issue with the VRS cross-efficiency approach. As in Doyle and Green (1994), we can add a set of secondary goals in the proposed VRS crossefficiency approach. See also Liang et al (2008a) and Lim (2012). We can also develop a game cross-efficiency approach as in Liang et al (2008b). These are possible future research topics. Acknowledgements The authors are grateful to the two anonymous referees for their constructive comments and suggestions on a previous version of this paper. References Ali AI and Seiford LM (1990). Translation invariance in data envelopment analysis. Operations Research Letters 9(6): Anderson TR, Hollingsworth KB and Inman LB (2002). The fixed weighting nature of a cross-evaluation model. Journal of Productivity Analysis 17(3):

12 Sungmook Lim and Joe Zhu DEA cross-efficiency evaluation under VRS 487 Banker RD, Charnes A and Cooper WW (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30(9): Banker RD, Cooper WW, Seiford LM and Zhu J (2011). Returns to scale in data envelopment analysis. In Cooper WW, Seiford LM and Zhu J (eds). Handbook on Data Envelopment Analysis. Springer: New York. Charnes A, Cooper WW and Rhodes E (1978). Measuring the efficiency of decision making units. European Journal of Operational Research 2(6): Chen C-M and Zhu J (2011). Efficient resource allocation via efficiency bootstraps: An application to R&D project budgeting. Operations Research 59(3): Cook WD, Tone K and Zhu J (2014). Data envelopment analysis: Prior to choosing a model. Omega 44: 1 4. Doyle J and Green R (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society 45(5): Green R, Doyle J and Cook WD (1996). Preference voting and project ranking using DEA and cross-evaluation. European Journal of Operational Research 90(3): Liang L, Wu J, Cook WD and Zhu J (2008a). Alternative secondary goals in DEA cross-efficiency evlauation. International Journal of Production Economics 113(2): Liang L, Wu J, Cook WD and Zhu J (2008b). The DEA game crossefficiency model and its Nash equilibrium. Operations Research 56(5): Lim S (2012). Minimax and maximin formulations of cross-efficiency in DEA. Computers & Industrial Engineering 62(3): Lim S and Zhu J (2013). Use of DEA cross-efficiency evaluation in portfolio selection: An application to Korean stock market. European Journal of Operational Research published online 8 December, doi: /j.ejor Oral M, Kettani O and Lang P (1991). A methodology for collective evaluation and selection of industrial R&D projects. Management Science 37(7): Pastor JT (1996). Translation invariance in data envelopment analysis: A generalization. Annals of Operations Research 66(2): Podinovski VV and Bouzdine-Chameeva T (2013). Weight restrictions and free production in data envelopment analysis. Operations Research 61(2): Ramón N, Ruiz JL and Sirvent I (2011). Reducing differences between profiles of weights: A peer-restricted cross-efficiency evaluation. Omega 39(6): Salo A and Punkka A (2011). Ranking intervals and dominance relations for ratio-based efficiency analysis. Management Science 57(1): Seiford LM and Zhu J (1998). On alternative optimal solutions in the estimation of returns to scale in DEA. European Journal of Operational Research 108(1): Sexton TR, Silkman RH and Hogan AJ (1986). Data envelopment analysis: Critique and extensions. New Directions for Program Evaluation 1986(32): Stewart TJ (1996). Relationships between data envelopment analysis and multicriteria decision analysis. Journal of the Operational Research Society 47(5): Wu J, Liang L and Chen Y (2009). DEA game cross-efficiency approach to Olympic rankings. Omega 37(4): Received 14 February 2013; accepted 3 February 2014 after two revisions