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A multiplier bound approach to assess relative efficiency in DEA without slacks
European Journal of Operational Research 203 (2010) 261–269
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A multiplier bound approach to assess relative efficiency in DEA without slacks Nuria Ramón, José L. Ruiz *, Inmaculada Sirvent Centro de Investigación Operativa, Universidad Miguel Hernández, Avd. de la Universidad, s/n. 03202-Elche (Alicante), Spain
a r t i c l e
i n f o
Article history: Received 29 July 2008 Accepted 13 July 2009 Available online 18 July 2009 Keywords: DEA AR Non-zero weights Zero-slacks
a b s t r a c t In this paper, we propose a new approach to deal with the non-zero slacks in data envelopment analysis (DEA) assessments that is based on restricting the multipliers in the dual multiplier formulation of the used DEA model. It guarantees strictly positive weights, which ensures reference points on the Paretoefficient frontier, and consequently, zero slacks. We follow a two-step procedure which, after specifying some weight bounds, results in an ‘‘Assurance Region”-type model that will be used in the assessment of the efficiency. The specification of these bounds is based on a selection criterion among the optimal solutions for the multipliers of the unbounded DEA models that tries to avoid the extreme dissimilarity between the weights that is often found in DEA applications. The models developed do not have infeasibility problems and we do not have problems with the alternate optima in the choice of weights that is made. To use our multiplier bound approach we do not need a priori information about substitutions between inputs and outputs, and it is not required the existence of full dimensional efficient facets on the frontier either, as is the case of other existing approaches that address this problem. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction In most cases in practice, the DEA models assess the efficiency of the inefficient units by using reference points on the frontier of the production possibility set (PPS) that are not Pareto-efficient. This happens as a result of the fact that these models usually yield zero weights for the optimal multipliers, or equivalently (by duality), strictly positive values for the optimal slacks, which means that the efficiency scores obtained for these units do not account for all sources of inefficiency. Bessent et al. (1988) deal with the so-called ‘‘not naturally enveloped inefficient units”, which are defined as those that have a mix of inputs and/or outputs which is different from that of any other point on the efficient frontier. These authors report the results corresponding to several studies that reveal the high frequency of the not naturally enveloped inefficiency units in practice. These units are actually those in F [ NF according to the classification of the decision making units (DMUs) in Charnes et al. (1991) (the DMUs in F are on the weak efficient frontier whereas those in NF are projected onto points in F). It has been paid much attention in the literature to this type of DMUs, where we can find a wide variety of approaches intended to provide efficiency scores for them trying to avoid the problems with the non-zero slacks. In this paper we propose a new approach to address this problem that deals with the dual multiplier formulation of the DEA models. This approach is based on imposing restrictions on the * Corresponding author. E-mail address: [email protected] (J.L. Ruiz). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.07.009
weights, which have been frequently used to obtaining non-zero weights, and also allow incorporating value judgements into the analysis (see Allen et al. (1997) and Thanassoulis et al. (2004) for a general discussion). The key issue when using weight restrictions is the setting of the bounds to be located in these constraints. We determine some weight bounds as the result of using an ancillary criterion of selection that makes it possible a specific choice of non-zero weights among the optimal solutions of the unbounded DEA models. We guarantee that the resulting DEA formulations provide efficiency scores that reflect the comparison of the unit under assessment with reference points on the Pareto-efficient frontier, which ensures zero slacks. We can find in the literature different approaches to setting bounds in weight restrictions of DEA models. Weight bounds can be obtained by resorting to the opinion of some experts involved in the underlying production process, as in Beasley (1990) or in Takamura and Tone (2003). In this latter case, the authors process the information from experts by using additionally AHP (Analytic Hierarchy Process). We can also use the information on prices and/or costs as in Thompson et al. (1995, 1996). Or we can even combine expert opinion and price information as in Thompson et al. (1990, 1992). Other authors have proposed to use the optimal weights of some units considered as model DMUs in order to specify the bounds in the weight restrictions (see Charnes et al. (1990) and Brockett et al. (1997)). These approaches have been mainly used with either Cone Ratio (CR) models (Charnes et al. (1990)) or Assurance Regions (AR) models (Thompson et al. (1986)), which were originally developed with the purpose of incorporating value judgements into the analysis, i.e., prior information and/or
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accepted beliefs or preferences concerning the underlying process of assessing efficiency and, in addition, they often lead to non-zero weights. However, it often happens in practice that we do not have available either expert opinion or prices/costs information. In that case, there are different methods that can be of help for the estimation of weight bounds, and these obviously rely on the information provided by the data. Most of them are based on somehow handling the optimal weights of the unbounded DEA models. For example, once the unbounded DEA model is solved in a first stage and we have compiled a weight matrix for all the variables, Roll et al. (1991) and Roll and Golany (1993) claim that we can set the bounds: (1) after eliminating the outliers and the extreme weights, (2) by imposing that a certain percentage of weights falls within the bounds or (3) at an acceptable ratio of variation for each weight within the range of the unbounded weights. As acknowledged by these authors, one of the main difficulties with these techniques is that the bounds obtained may vary depending on the chosen solution among the alternate optima for the weights. There are also some approaches that are based on imposing some specifically developed lower bounds on the multipliers in the dual formulation of the used DEA model. The idea behind these approaches is obviously to have strictly positive weights and, consequently, by duality, zero slacks. In fact, the use of the well-known non-archimedean e in this formulation leads to non-zero weights, but this model does not produce efficiency scores that can be readily used. To avoid this, Chang and Guh (1991) propose to replace this e with a specific lower bound that is obtained from the smallest non-zero value of the multipliers for all the variables of the unbounded DEA model. The main difficulty with this approach is that the resulting model may become infeasible (Chen et al. (2003) show how to modify this bound to avoid the infeasibility problems), together with the fact that the bounds are obtained following a procedure that does not consider the possible existence of alternate optima for the weights. Thus, different optimal solutions may lead to different bounds and these may lead to different efficiency scores for the units under assessment. Chen et al. (2003) develop an alternative multiplier bound approach in which the lower bounds are determined by strong complementary slackness condition (SCSC) solution pairs for extreme efficient DMUs. However, as acknowledged by the authors, the results obtained may vary depending on the SCSC solution that is chosen. We also include here the approach in Dyson and Thanassoulis (1988) which, in the case of having a single input (or a single output), provides lower bounds for the output (input) weights by imposing the condition that it cannot be used less than some percentage of the average input level the DMU being assessed uses per unit of output, which is estimated by means of a regression analysis. The existing work on facet models, which relates to the extension of the facets of the frontier, is in particular intended to address the problems with zero weights, and consequently with the slacks. Some of these approaches are based on the extension of full dimensional efficient facets (FDEFs) of the frontier. See Green et al. (1996), Olesen and Petersen (1996) and Portela and Thanassoulis (2006). In the two former, the authors define some new technologies from the FDEFs of the original frontier, whereas in the latter the proposed approach uses an AR model in which the bounds of the AR constraints are estimated from the marginal rates of substitution of these FDEFs. Obviously, these approaches guarantee nonzero weights, but they require the existence of such FDEFs on the frontier to be used, which does not happen very often in practice. See also Bessent et al. (1988), with the ‘‘constrained facet analysis” (CFA), for an algorithm along this line that proposes to project each inefficient unit onto an extended facet of the frontier that is suitably selected. CFA, however, guarantees neither that the facet wanted exists nor that the algorithm implemented finds this facet even if this exists. Lang et al. (1995) follow a similar idea and mod-
ify the previous approach by proposing to extend a facet of the PPS of a previously specified dimension, calling their approach ‘‘controlled envelopment analysis” (CEA). We note that CEA avoids the zero weights by imposing a non-archimedean e used as lower bound for the multipliers in the CCR model without specifying how this value is to be implemented for use in practice. Finally, it should be noted that the problem addressed here has also been approached in a different manner by dealing with the primal envelopment formulation of the DEA models. That is the case of the so-called ‘‘generalized efficiency measures” (GEMs) (see, e.g., Cooper et al. (1999), Pastor et al. (1999) and Tone (2001)), which are efficiency measures especially designed to account for both radial and non-radial inefficiencies, and so, avoiding the problems with the slacks. We propose here another multiplier bound approach intended to provide efficiency scores for the inefficient units that account for all sources of inefficiency. It can be used when we do not have available either information on prices/costs or expert opinion reflecting value judgements related to the involved inputs and outputs, and also if FDEFs on the frontier do not exist. We develop a two-step procedure which, in a first step, is aimed at specifying some weight bounds, which are then used in the second step in the weight restrictions that are incorporated into the dual multiplier formulation of the used DEA model. The resulting models will provide us with the efficiency scores that are wanted. We will show that these models can be equivalently formulated as AR models. As for the specification of the weight bounds, these are determined on the basis of a choice among the optimal solutions for the multipliers of the extreme efficient DMUs in the unbounded DEA models that is made by using a selection criterion. In absence of information on the relative importance of the involved variables, this criterion seeks to avoid the extreme dissimilarity between the weights of the optimal DEA solutions that is often found in practice. Thus, we look for non-zero weights while at the same time we try to avoid large differences between the multipliers as much as possible. We would like also to stress both that the procedures proposed to setting weight bounds are not affected by the possible existence of alternate optima in the unbounded DEA models and that the models we develop do not have infeasibility problems. The paper unfolds as follows: In Section 2 we develop the twostep procedure we propose to the setting of the bounds to be located in the weight restrictions of the DEA models used for the assessment of the inefficient units. Section 3 includes an illustrative example. In Section 4 we outline two possible extensions of the proposed approach. Section 5 concludes.
2. A two-step procedure to assess DMUs in F [ NF without slacks The high frequency of non-zero slacks (or, equivalently, zero weights) when assessing inefficient units in practice is mainly due to the total weight flexibility implicitly used in DEA. As has been widely explained in the literature, this total weight flexibility often leads to unreasonable results, since the weights assigned to the variables considered are frequently inconsistent with the prior knowledge or accepted views on the relative value of the involved inputs and outputs. In particular, the DEA models, trying to show the units under assessment in their best possible light, usually exploit this total weight flexibility to the point of evaluating the DMUs by putting the weight solely on a few set of variables and ignoring the rest of the originally considered inputs and outputs by assigning them a zero weight. This also brings with it that we have strictly positive values for the optimal slacks when assessing the efficiency of the inefficient units, since these are projected onto the facets of the weak efficient frontier, which means that these units are evaluated with reference to points that are not
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Pareto-efficient. Thus, the efficiency scores obtained do not account for all sources of inefficiency. As said before, we propose here a new multiplier bound approach to deal with this problem that is based on the two-step procedure we describe below. Throughout the paper we assume that we have n DMUs, each DMUj using m inputs ðxij ; i ¼ 1; . . . ; mÞ and producing s outputs ðyrj ; r ¼ 1; . . . ; sÞ. Assume also that the relative efficiency of these DMUs is assessed with the CCR model (Charnes et al., 1978) input-oriented (the developments here can be straightforwardly adapted for the output-oriented case), which consists of the following pair of dual problems
Min h n P kj xij 6 hxi0 ; s:t:
i ¼ 1; . . . ; m;
j¼1 n P
ð1Þ
kj yrj P yr0 ;
r ¼ 1; . . . ; s;
j¼1
kj P 0;
j ¼ 1; . . . ; n;
force all these multipliers to vary in between the bounds z and h. As a result, the ratios between each couple of multipliers are all greater than or equal to z=h: Since we are maximizing z=h in (3), with this problem we seek to maximize the minimum of the ratios between multipliers and, in this sense, we look for the least dissimilar optimal weights that allow DMUj0 to be rated as efficient. Thus, if, for example, uj0 ¼ 0:3; this would mean that it is not possible to assess DMUj0 as efficient with a set of weights in which the lowest multiplier is higher than 30% of the highest one.1 As said before, in Step I we solve (3) for each DMUj in E, which are those that support the efficient frontier and then are fundamental in the sense that they are used to evaluate all the points that represent the performances of the DMUs that are to be assessed (see Appendix A for the aspects concerned with solving this problem). Then, we aggregate all the information provided by the extreme efficient units by means of the following scalar
u ¼ min uj ;
ð4Þ
j2E
and
Max s:t:
s P r¼1 m P
lr yr0 mi xi0 ¼ 1;
i¼1 m P
mi xij þ
s P r¼1
i¼1
ð2Þ
lr yrj 6 0; j ¼ 1; . . . ; n;
mi P 0; i ¼ 1; . . . ; m; lr P 0; r ¼ 1; . . . ; s: Based on this DEA model we can partition the set of DMUs into the 0 0 classes E, E , F, NE, NE and NF as in Charnes et al. (1991). The DMUs 0 in E and E are Pareto-efficient. E consists of the extreme efficient 0 units, whereas those in E are Pareto-efficient units that can be expressed as linear combinations of DMUs in E. F is the set of weakly 0 efficient units. Finally, the DMUs in NE, NE and NF are inefficient 0 and are projected onto points that are in E, E and F respectively. We develop a two step procedure in two different situations: Firstly, we address the situation in which we do not distinguish between inputs and outputs in the implicit choice of weights that is made. This case mainly has a methodological interest. And next, we extend the idea previously developed to the more interesting case in practice of restricting the weights of the inputs and the weights of the outputs separately. 2.1. Restricting input multipliers and output multipliers indistinctly Step I: For each extreme efficient unit DMUj0 (see Thrall (1996) for the identification of DMUs in E), in the first step we propose to solve the following problem
Max uj0 ¼ hz m P s:t: mi xij0 ¼ 1; i¼1 s P
r¼1
ð3:1Þ
lr yrj0 ¼ 1;
m P
mi xij þ
i¼1
ð3:2Þ
s P r¼1
ð3Þ
lr yrj 6 0; j 2 E; ð3:3Þ
z 6 mi 6 h;
i ¼ 1; . . . ; m;
ð3:4Þ
z 6 lr 6 h;
r ¼ 1; . . . ; s;
ð3:5Þ
mi ; lr ; z; h P 0: The idea behind this problem is the following: With (3.1–3.3) we allow for all the optimal multipliers in the unbounded CCR model for DMUj0 (which is rated as efficient). The constraints (3.4) and (3.5)
which is the minimum of the uj0 0 s across the DMUs in E. Thus, u quantifies in a specific manner the disequilibrium between the optimal multipliers in a CCR model needed to assess the DMUs in E as efficient. In other words, u provides a lower bound for the ratios between the optimal multipliers of the DMUs in E (u will be always lower than or equal to 1) so that the optimal weights satisfying this condition guarantee that the extreme efficient units that support the efficient frontier remain as such. Proposition 1 is very important in order to guarantee that the inefficient units are assessed with reference to points on the Pareto-efficient frontier and so avoiding the non-zero slacks: Proposition 1. u > 0. Proof. See Appendix B.
h
Step II: Let DMU0 be an inefficient unit in F [ NF. Then, to assess its relative efficiency, we propose to use as an efficiency score the optimal value of the following LP problem
Max s:t:
s P r¼1 m P
lr yr0 m
i xi0 i¼1 m P
¼ 1;
mi xij þ
i¼1
ð5:1Þ
s P r¼1
lr yrj 6 0; j 2 E; ð5:2Þ
z 6 mi 6 h;
i ¼ 1; . . . ; m;
z 6 lr 6 h;
r ¼ 1; . . . ; s;
z h
P u ;
ð5Þ
ð5:3Þ ð5:4Þ ð5:5Þ
mi ; lr ; z; h P 0: We can see that (5) is the result of incorporating (5.3–5.5) into the dual formulation of the classical CCR model. As a consequence, with this model we force the DMU0 under evaluation to be assessed with reference to a set of weights that cannot be more dissimilar than those of the DMU in E that needs to unbalance more its weights (as measured by u ) in order to be rated as efficient. We would like to emphasize the following features of our approach: Firstly, Proposition 2 establishes that we do not have infeasibility problems with model (5): 1 In order to make the readership easier, our developments in this paper are in terms of absolute weights, which implicitly means to assume that the involved variables are measured in units such that the comparisons between multipliers are meaningful. If this is not so, as it often happens in practice, the absolute weights should be replaced with the corresponding virtuals, which can be done straightforwardly in this approach, as explained with more detail in Section 4. Alternatively, a referee suggested to scale all inputs and outputs into the interval [0,1].
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Proposition 2. Model (5) is feasible. Proof. See Appendix B.
h
Furthermore, as a result of Proposition 1 we also have that with (5) DMU0 is assessed with reference to a set of strictly positive weights. Consequently, by duality, the slacks in the efficiency assessment will be all zero, and so, the efficiency score of the inefficient units (the optimal value of (5)) will be reflecting the result of their comparison with a Pareto-efficient reference point. From a geometrical point of view, the effect of the choice of weights made with (5) is to introduce some ‘‘hypothetical efficient facets” (as termed in Chen et al. (2003)) where the inefficient DMUs are projected on. This is shown with more detail in Section 3 which includes a graphical illustration. It is also important to make clear that ours is not a usual absolute weight bound approach, since the absolute weight restrictions in (5) have not the usual form that forces each multiplier to lie in between two previously specified bounds, as z and h are not constants but they are actually variables in this problem. Thus, in (5) z and h may vary, with the only condition that the minimum ratio between weights, which is that associated with the couple of more dissimilar weights, be not lower than u . In fact, (5) is actually equivalent to an AR model, as we show next2:
Proposition 3. If ðmi ; lr ; z ; h Þ is an optimal solution of (5) then ðmi ; lr Þ is an optimal solution of the following LP problem
Max s:t:
s P r¼1 m P
lr yr0 m
i xi0 i¼1 m P
¼ 1;
mi xij þ
i¼1
u 6
s P r¼1
mi 1 mi0 6 u lr 1 lr0 6 u mi 1 lr 6 u
ð6:1Þ
lr yrj 6 0; j 2 E; 0
i; i ¼ 1; . . . ; m;
ð6:2Þ 0
i
ð6Þ
ð6:3Þ
u 6 r; r 0 ¼ 1; . . . ; s; r < r0 ; ð6:4Þ u 6 i ¼ 1; . . . ; m; r ¼ 1; . . . ; s; ð6:5Þ mi ; lr P 0 i ¼ 1; . . . ; m; r ¼ 1; . . . ; s: And conversely, if ðmi ; lr Þ is an optimal solution of (6) then ðmi ; lr ; z ; h Þ is an optimal solution of (5) for some z ; h 2 R. In addition, (5) and (6) have the same optimal value. Proof. See Appendix C.
cient frontier that is used as reference in the assessment of DMU0 . See, e.g., Cooper et al. (2007b) for developments concerning the AR-models. To sum up, we have developed a multiplier bound approach that makes it possible to assess the efficiency of the inefficient units with reference to a set of non-zero weights by using an AR model. A specific weight bound to be used in the DEA formulations has been proposed as the result of using a selection criterion among the optimal weights of the unbounded DEA models. The presence of alternate optima for the weights of the extreme efficient units in the unbounded DEA models provides opportunities for ancillary criteria of selection among them.4 When we have available neither information on prices/costs nor expert opinion we have to rely on the data as the only source of information to set the corresponding bounds. In particular, in absence of information on the relative importance of the involved variables, the proposed AR model makes a choice of weights that seeks to avoid the problems previously mentioned with the total weight flexibility by impeding the inefficient DMUs to be assessed with reference to sets of extremely dissimilar weights. It is the empirical efficient frontier which establishes the limit for the allowable similarities among weights, since we do not permit the extreme efficient units to become inefficient. Cook and Seiford (2008) state that ‘‘the AR concept was developed to prohibit large differences in the values of multipliers”, which goes on about the idea of having less dissimilar weights we are using in this paper. Thus, we depart from the classical DEA, which allows large differences between the weights of the DMU under assessment in its objective of maximizing the resulting efficiency score, and move in an opposite direction by looking for non-zero weights while at the same time we try to avoid as much as possible situations of extreme dissimilarity between the multipliers.5 Finally, it is also worth mentioning other benefits of our approach: (1) it does not require the existence of FDEFs, (2) the models developed do not have infeasibility problems and (3) we do not have problems with the alternate optima for the weights in the unbounded DEA models.
h
On the basis of Proposition 3, to proceed with the second step of our two-step procedure, we might equivalently solve this AR-model to assess the efficiency of DMU0 , which includes both type I and type II AR constraints since (6.3–6.5) are restrictions on the ratios between the weights of the inputs, the outputs and both the inputs and the outputs.3 Therefore, we can use conventional DEA software including AR-models in order to carry out the second step of our procedure. In particular, this will provide us with the efficiency scores b i ¼ P k xij ; i ¼ 1; . . . ; m, and and the associated targets, X j2E j P 0 b Y r ¼ j2E kj yrj ; r ¼ 1; . . . ; s, the kj s being the optimal solutions of variables associated with the constraints (6.2) in the model dual to (6). These targets are the coordinates of the point on the Pareto-effi2 It should be mentioned the difficulties with the use of absolute weight bounds with the possible infeasibility of the corresponding models and other side effects, as those discussed in Podinovski (1999, 2001), claiming that the models with absolute weigh bounds may not show the unit under assessment in the best possible light relative to other DMUs. 3 As a result of the equivalence to (3), (6) is also feasible, in spite of including AR-II constraints.
2.2. Restricting input multipliers and output multipliers separately In the previous subsection we have proposed a multiplier bound approach to assess the inefficient units eliminating the slacks without making any distinction between the multipliers corresponding to the inputs and those corresponding to the outputs. In this sense, that approach could be seen as an alternative to both the use of the non-archimedean e and the approach in Chang and Guh (1991), since these two approaches are based on imposing a common lower bound to all the multipliers of the inputs and outputs. In this section we extend this idea to address the more interesting case in practice of restricting the input multipliers and the output multipliers separately. That is the case of, for instance, the Cone Ratio models (Charnes et al. (1990)) in its more frequent use in practice, the type I AR models (Thompson et al. (1986)) and the model proposed in Bal et al. (2008), which minimizes separately the coefficient of variation of the input weights and that of 4 Some authors have pointed out the need to deal with the alternate optima for the weights in the DEA models. See Cooper et al. (2007a) which discusses this issue and proposes a specific choice of optimal weights based on an additional criterion of selection. See also Liang et al. (2008) which propose to use alternative secondary goals to deal with the non-uniqueness of the DEA optimal weights in the context of the cross-efficiency evaluation. 5 We can find in the literature several approaches that implement procedures that avoid extreme weights with different purposes. See, for instance, Li and Reeves (1999) which develop a multi-criteria DEA (MCDEA) model that avoids extreme weights and at the same time improves discrimination power among the efficient DMUs. See also Bal et al. (2008) which propose to minimize the coefficient of variation of the input and output weights with the same purpose.
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the output weights in an attempt to have more homogeneous weights. To adapt the two-step procedure in 2.1 to this new situation it should be modified as follows: Step I: For each DMUj0 in E solve the following problem
Max uj0 s:t:
mi xi0 ¼ 1;
lr yr0 ¼ 1;
m P
mi xij þ
s P r¼1
zI 6 mi 6 hI ; zO 6 lr 6 hO ;
m
i xi0 i¼1 m P
¼ 1;
mi xij þ
r¼1
lr lr0
ð7Þ
ð7:6Þ
zO hO
P uj0 ;
ð7:7Þ
lr yr0 m
¼ 1;
mi xij þ
s P r¼1
i¼1
ð9:3Þ
0
And conversely, if ðmi ; lr Þ is an optimal solution of (9) then ðmi ; lr ; zI ; hI ; zO ; hO Þ is an optimal solution of (8) for some zI ; hI ; zO ; hO 2 R. In addition, (8) and (9) have the same optimal value. Proof. See Appendix C. h
The idea of (7) is similar to that of (3) with the exception that z=h is now maximized separately for the inputs and the outputs by maximizing the minimum of the corresponding two ratios. Therefore, in solving (7) the bounds provided are obtained as the result of restricting the input multipliers on one hand and the output multipliers on the other trying to avoid as much as possible extreme dissimilarities between the input weights and the output weights independently. It should be noted that (7) is now a non-linear problem, which has a global optimum that can be found by using any conventional software of global optimization such as LINGO. As in the previous section, the optimal values of (7) will be also used to define the scalar u ¼ minj2E uj to be used in the second step. Step II: Let DMU0 an inefficient unit in F [ NF. To assess its relative efficiency, we propose to solve the following LP problem
i xi0 i¼1 m P
0
ð7:5Þ
P uj0 ;
r¼1 m P
1
ð9Þ
ð9:2Þ
u 6 6 u ; r; r ¼ 1; . . . ; s; r < r ; ð9:4Þ mi ; lr P 0; i ¼ 1; . . . ; m; r ¼ 1; . . . ; s:
zI hI
s P
lr yrj 6 0; j 2 E;
ð7:2Þ
ð7:4Þ
r ¼ 1; . . . ; s;
s P
u 6 mmii0 6 u1 ; i; i0 ¼ 1; . . . ; m; i < i0 ;
lr yrj 6 0; j 2 E; ð7:3Þ i ¼ 1; . . . ; m;
ð9:1Þ
ð7:1Þ
mi ; lr ; zI ; hI ; zO ; hO ; uj0 P 0:
s:t:
lr yr0
i¼1
Max
r¼1 m P
i¼1
i¼1
r¼1
s:t:
s P
m P
s P
Max
ð8:1Þ
lr yrj 6 0; j 2 E; ð8:2Þ
We can see that now in solving (8) we will be implicitly solving a type I AR model since the constraints used only involve ratios between input weights and ratios between output weights, as could be expected. To illustrate graphically the idea of the proposed procedure consider the following small example depicted in Fig. 1 which consists of 4 DMUs, with two inputs and one constant output, whose relative efficiency is to be assessed: A(1,8;1), B(2,3;1), C(5,2;1) and D(10,2.5;1). The CCR model reveals that A, B and C are extreme efficient points whereas D is inefficient. Its efficiency score is 4/5 = 0.8, 0 which is the result of projecting this DMU onto the point D (8,2;1) on the weak efficient frontier, and so, it is detected a slack s 1D ¼ 3 in the first input, as compared to DMU C (by duality, the optimal multiplier associated with x1 is 0). Thus, this efficiency score is not accounting for all the inefficiency of DMU D. To avoid this, we propose to alternatively evaluate the efficiency of DMU D by using our two-step procedure. In the first step, we have the following results by solving (7): for DMU A, uA ¼ 1=5 ¼ 0:2, which is associated with the optimal weights m1 ¼ 5=13; m2 ¼ 1=13 and l ¼ 1; for DMU B, uB ¼ 1, for m1 ¼ m2 ¼ 1=5 and l ¼ 1 and, for DMU C, uC ¼ 1=3 ¼ 0:33, for m1 ¼ 1=11; m2 ¼ 3=11 and l ¼ 1. Each of these optimal solutions represents the coefficients of a supporting hyperplane for the PPS 5 1 x1 13 x2 þ y ¼ 0; at DMU A, at the corresponding unit: HA 13 1 3 x1 11 x2 þ HB 15 x1 15 x2 þ y ¼ 0, at DMU B, and HC 11 y ¼ 0, at DMU C. For each of these DMUs, these coefficients are
ð8Þ
zI 6 mi 6 hI ; i ¼ 1; . . . ; m; zO 6 lr 6 hO ; r ¼ 1; . . . ; s;
ð8:3Þ ð8:4Þ
zI P hI zO P hO
ð8:5Þ
u ; u ;
x2
HA
ð8:6Þ
mi ; lr ; zI ; hI ; zO ; hO P 0:
A
As in the previous section, model (8) assesses the efficiency of DMU0 by using a formulation in which the weights are restricted from the information provided by the data in the first step. To be specific, DMU0 is now assessed with reference to a set of weights in which both the input weights among themselves and the output weights among themselves cannot be more dissimilar than u . Moreover, the discussion in Section 2.1 on the features and advantages of the proposed procedure also applies in this case of dealing with the input and output weights separately. The following proposition is the equivalent to Proposition 3 for this case, and also allows us to obtain the optimal solution of (8) by using an equivalent AR formulation:
y=1 HB
B C
D’’
D’
HW H*
HC x1
Proposition 4. If ðmi ; lr ; zI ; hI ; zO ; hO Þ is an optimal solution of (8) then ðmi ; lr Þ is an optimal solution of the following LP problem
D
Fig. 1. Graphical illustration.
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the least dissimilar weights in the sense previously explained. For instance, uC ¼ 1=3 ¼ 0:33 means that it is impossible to find a set of positive optimal weights for DMU C such that the magnitude of the multiplier of the variable with minimum weight is higher than 33.3% of that of the variable with maximum weight (without making it become inefficient). For the assessment of the inefficient DMU D in the second step we use the minimum of these values, u ¼ minfuA ; uB ; uC g ¼ 1=5 ¼ 0:2; since this bound guarantees the existence of optimal weights for the 3 efficient DMUs that make it possible to maintain them as such. This u ¼ 1=5 ¼ 0:2 will force DMU D to be assessed with reference to a set of weights that cannot be more dissimilar than those of DMU A, which is the unit that needs to unbalance more its weights in order to be rated as efficient. This minimum also represents a lower bound for the ratios between the coefficients of the hyperplane where DMU D is projected on. In particular, when assessing DMU D with (8), its efficiency score is now 2/ 00 3 = 0.66 (and not 4/5 = 0.8), as the result of projecting it onto D , 4 2 2 which is on the hyperplane H 90 x1 9 x2 þ 3 y ¼ 0 (which cor1 x1 þ 13 x2 ¼ 1 at the level y = 1 in Fig. 1). Obviresponds to the line 15 00 ously, the coordinates of the point D are not the efficient targets 00 for DMU D, since the projection point D is outside the PPS, but the targets for DMU D are the coordinates of DMU C, which are actually those associated with the optimal solution provided by the AR model (as explained before) and is also on H . DMU C is in addition on the Pareto-efficient frontier of the PPS, which consists of the segments AB and BC, and this allows us to eliminate the slacks in the assessment of DMU D. The assessment of DMU D depicted in Fig. 1 gives us a geometrical idea of the procedure we have proposed. We depart from the hyperplane HW ¼ 12 x2 ¼ 1, which contains a part of the weak frontier of the PPS, and move by looking for more similar weights up to hyperplane H , where the constraint hzII P 0:2 is binding. Going further than H would mean to assess DMU D with a lower efficiency score. To be precise, the supporting hyperplanes at DMU C in between H and HC are associated with feasible solutions of (8) when DMU D is evaluated, but projecting it onto some of them would yield a lower efficiency score for this unit. We can see that with our approach we avoid the weighting schemes with extremely dissimilar values for the multipliers. However, we do not look for close multipliers, but our aim is to find the least dissimilar weights which, in addition, allow us to maintain all the DMUs in E as efficient.
It is to be noted that in this particular example the PPS does have two facets of full dimension at the frontier, so we could also use the approaches based on the projection onto the extension of these FDEFs to assess the inefficient DMU D. To be specific, following Green et al. (1996) and Olesen and Petersen (1996), DMU D would be projected onto HC , which extends the facet determined by DMUs B and C. The efficiency score of DMU D would be 0.6285 in that case. The hyperplane HA , which is the extension of AB, is associated with some feasible solution of the models used in these approaches, but projecting DMU D onto HA would yield a lower efficiency score for this unit. CFA would lead to these same results. To use CEA, we should select the facet of maximum dimension to be extended which also guarantees non-zero weights. In this example, this would be the segment BC, and so DMU D would be again projected onto HC with the same results as before. Portela and Thanassoulis (2006) determine a lower and an upper bound for the ratios between each couple of multipliers as the minimum and maximum of the corresponding marginal rates of substitution estimated with all the FDEFs of the frontier. The resulting AR-model for the assessment of DMU D would have as feasible solutions those associated with all the supporting hyperplanes for the PPS at DMU B which are in between HA and HC . Obviously, DMU D would be projected again onto HC since this yields the higher efficiency score. We cannot provide a direct comparison with the approaches based on imposing lower bounds on the multipliers, as we explain next. The approach in Chen et al. (2003) would select a hyperplane in between HW and HC which would depend on the choice of the SCSC solution that were made. In Chang and Guh (1991) this specification also depends on the choice of an optimal solution for the multipliers in the unbounded DEA model. And this would also happen with the absolute weight bound procedures listed in Roll et al. (1991) and Roll and Golany (1993).
3. Illustrative example In this section we illustrate the use of the proposed procedure with a new example with a larger data set in a higher dimensional space. Consider the sample consisting of 20 DMUs, each one using 4 inputs to produce 2 outputs. The data together with the corresponding CCR efficiency scores and the associated optimal slacks are recorded in Table 1. We can see that the DMUs 1, 4, 5, 6, 8
Table 1 Data and CCR efficiency analysis results. DMU
x1
x2
x3
x4
y1
y2
CCR (score)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1.7 5.9 5.6 2.9 2.6 1.7 3 1.6 5.3 3.3 4 8 8 4.3 1.9 3 5.9 2.8 1.7 3.3
1.7 3.3 5 2 3.3 2.5 5 4 2.5 2 2.5 5 4 3.2 2.5 5.5 4.2 4.6 2.2 1.7
2.3 3.3 3.3 1.7 3.3 1.4 2.5 2 3.3 2 5 1.7 3.5 3.3 3.5 2.4 4.9 2.3 1.4 3.3
1 1.3 1.4 2 1.7 2 2 1.5 2 2 3.3 1.1 1.7 2 1.7 1.3 2 1.9 1.3 3.3
2 1 1 2 1 1.5 2 2 1 2 2 2 2 1.5 2 1 2 1 1 2
1 2 2 2 2 1.5 1 1 2 1 2 3 2 1 1 1.5 2 2 1 1
1 0.8794 0.7749 1 1 1 0.7720 1 0.8914 0.9500 0.8000 1 0.7547 0.4764 0.8831 0.8505 0.6973 0.9956 0.8079 1
Slacks x1
x2
x3
x4
y1
y2
0.64 0
0 0.55
1.59 0.64
0 0
0.55 0.21
0 0
0
0.08
0
0
0
0.31
1.41 0.64 0.3
0 0 0
1.34 0 2.3
0 0.23 0.64
0.89 0 0
0 0.67 0
1.67 0.17 0 0 0 0 0 1.6
0 0 0 2.24 0 2.1 0.45 0
0.72 0 0.86 0 1.4 0 0 1
0 0 0.39 0 0 0 0 2.3
0 0 0 0 0 0.63 0 0
0 0.06 0 0 0 0 0 0
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and 12 are efficient (they are actually extreme efficient points) whereas the remaining fourteen DMUs are inefficient. To be precise, these inefficient DMUs belong to F [ NF, since they all have some non-zero slacks. This suggests the need for a procedure such as the one proposed here to alternatively assess their efficiency. It is to be noted that there is no FDEF at the frontier of the PPS defined by this data set, in spite of having enough extreme efficient points, the maximum dimension of the efficient facets being 4. Therefore, none of the approaches based on the extension of FDEFs could be used for this efficiency assessment. To proceed with our procedure in the first step, we solve (7) for each of the 6 extreme efficient DMUs. Table 2 records the optimal values uj together with an associated optimal solution for the multipliers. As expected, all these multipliers are strictly positive. The minimum of these uj 0 s is 0.11345, which will be used in the second step as the bound u for the allowable dissimilarity between the resulting multipliers. To be precise, the inefficient units will be evaluated with reference to a set of weights in which the ratios of the input weights and those of the output weights cannot be lower than 0.11345.
Table 3 records the scores for the inefficient DMUs and an optimal solution for the multipliers obtained with both the classical CCR model and our two-step procedure. It can be seen that, unlike the optimal solutions provided by the CCR model, our two-step procedure yields non-zero weights, as a consequence of avoiding these extreme dissimilarities between the multipliers. Therefore, our approach guarantees that the associated efficiency scores are the result of the comparison of the DMU under assessment with a Pareto-efficient point on the frontier of the PPS. Obviously, these efficiency scores are lower than those provided by the CCR model as a result of eliminating the slacks and, consequently, accounting for all sources of inefficiency. We highlight the case of DMU 20, which is on the weak efficient frontier, and whose efficiency score is eventually 0.8031 when compared with a Pareto-efficient referent. Finally, it is important to point out that we cannot make comparisons with some of the existing multiplier approaches since these cannot be used with these data. As said before, there is no FDEF at the frontier of the PPS, so we cannot use the models based on their extension. Perhaps, CEA could be used, since it does not necessarily requires FDEFs, but its formulations depend on a nonarchimedean e to guarantee non-zero weights and the authors do not provide any guideline for its specification. As for the approaches based on imposing lower bounds on the multipliers, we have the same difficulties already mentioned in the discussion of the small example in Section 2.
Table 2 Step I: Specification of the weight bounds. DMU_1
DMU_4
DMU_5
DMU_6
DMU_8
DMU_12
uj0
0.89171
1
0.11345
0.29231
0.14286
0.35908
1 2 3 4 1 2
0.15957 0.14229 0.14229 0.15957 0.34582 0.30837
0.11628 0.11628 0.11628 0.11628 0.25000 0.25000
0.19807 0.02247 0.02247 0.19807 0.05368 0.47316
0.29200 0.08535 0.08535 0.08535 0.25651 0.41015
0.22801 0.03257 0.22801 0.03257 0.43105 0.13789
0.04808 0.04808 0.13390 0.13390 0.09657 0.26895
m m m m l l
4. Extensions (1) It is well-known that the results of the DEA models with weight restrictions based on absolute weights, both those with absolute weights bounds and the AR models, are dependent on the units of measurement of the inputs and
Table 3 Step II: Efficiency scores. Score
m1
m2
m3
m4
l1
l2
DMU_2
CCR Two-step
0.8794 0.8011
0 0.1596
0.2152 0.1423
0 0.1423
0.2231 0.1596
0 0.3458
0.5 0.3084
DMU_3
CCR Two-step
0.7749 0.7317
0.1054 0.0909
0 0.0238
0 0.0238
0.2929 0.2097
0 0.0393
0.5 0.3462
DMU_7
CCR Two-step
0.7720 0.7171
0.02 0.0991
0 0.0233
0.284 0.2056
0.115 0.0359
0.5 0.3393
0 0.0385
DMU_9
CCR Two-step
0.8914 0.7878
0 0.0241
0.216 0.147018
0 0.0241
0.23 0.2125
0 0.0423
0.5 0.3727
DMU_10
CCR Two-step
0.9500 0.8923
0 0.0297
0.335 0.2616
0.165 0.1598
0 0.0297
0.5 0.4222
0 0.0479
DMU_11
CCR Two-step
0.8 0.7149
0 0.1344
0.4 0.1344
0 0.0152
0 0.0152
0.14 0.1498
0.36 0.2077
DMU_13
CCR Two-step
0.7547 0.7014
0 0.0224
0.155 0.1016
0 0.0224
0.2235 0.1975
0.145 0.1092
0.355 0.2414
DMU_14
CCR Two-step
0.4764 0.4697
0 0.0204
0.0344 0.0204
0.2030 0.1795
0.11 0.1274
0.6667 0.2796
0 0.0503
DMU_15
CCR Two-step
0.8831 0.8175
0.5 0.3606
0.02 0.0409
0 0.0409
0 0.0409
0.45 0.2725
0.1 0.2725
DMU_16
CCR Two-step
0.8505 0.7619
0.16 0.1240
0 0.0324
0.0375 0.0324
0.3385 0.2860
0.11 0.0536
0.5933 0.4722
DMU_17
CCR Two-step
0.6973 0.6655
0.0763 0.0733
0.0333 0.0212
0 0.0212
0.205 0.1873
0.06 0.0640
0.44 0.2688
DMU_18
CCR Two-step
0.9956 0.9208
0.1893 0.1824
0 0.0207
0.0652 0.0207
0.1632 0.1824
0 0.0494
0.5 0.4357
DMU_19
CCR Two-step
0.8079 0.7819
0.3294 0.3321
0 0.0377
0.0357 0.0377
0.3 0.2307
0.16 0.1640
0.84 0.6179
DMU_20
CCR Two-step
1 0.8031
0 0.0402
0.4 0.3542
0 0.0402
0 0.0402
0.415 0.3163
0.17 0.1706
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outputs. In the particular case of our approach, this means that the bounds u obtained might vary depending on the scaling of the variables and, consequently, this would also happen with the results of the corresponding models. However, the virtual inputs and outputs are dimensionless, and so, the use of virtuals instead of absolute weights would lead to the same results regardless of the units of measurement of the variables. In addition, absolute weights restrictions are frequently difficult to interpret whereas with virtual weights restrictions the decision maker has a clear idea of being restricting the relative importance of the factors (the relative contribution of each input and output to its efficiency score). See Sarrico and Dyson (2004) for a discussion about this. The approach in this paper can be easily adapted to be used with virtuals instead of absolute weights, if it is desired to avoid the mentioned difficulties with the use of absolute weights. To do it, we just need to replace the absolute weights in all the formulations with the corresponding virtuals. To be precise, in the particular case of restricting the input weights and the output weights separately, we should replace both the constraints (7.4) and (7.5), in the first step, and the constraints (8.3) and (8.4) in the second step with
zI 6 mi xi0 6 hI ; zO 6 lr yr0 6 hO ;
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s:
ð10Þ
(2) The approach developed here is based on a specific choice of weights among the multipliers of the extreme efficient units. We could similarly use our two-step procedure if the minimum u were calculated across a different set of DMUs instead of E, say a group of model DMUs. The idea of using optimal weights of model DMUs was initially proposed in Charnes et al. (1990) and in Brockett et al. (1997). These authors show how to define cones that favour certain (model) DMUs, and so favouring desired patterns of input usage and output production in the efficiency assessment by using optimal multiplier vectors of the model DMUs. To be specific, this approach was used in the analysis of efficiency of commercial banks by means of Cone Ratio models where the weights of several banks considered as excellent were used as benchmarks for the rest of banks under assessment (note, however, that this ignores the possibility of having alternate optima for the multipliers for the model DMUs, which would be avoided with our approach). It should also be noted that in that case the efficient units not considered as model DMUs could become inefficient.
ence to some new defined production possibility sets, such as the so-called extended facet PPS, TEXFA , and the full dimensional efficient facet PPS, TFDEF , which are built from the FDEFs of the original frontier. Obviously, this approach depends on the existence of such facets, which rarely happens in practice. In our proposal here, we have decided to maintain the original technology and have opted to restrict the total weight flexibility, as a mean to avoid the weak efficient frontier. In absence of any information from experts, this obliges us to use an ancillary criterion of selection among weights, which, in our case, has been based on the idea of avoiding the extreme dissimilarity between weights that is often found in DEA analyses. Thus, with our approach we go one step further than those also based on imposing bounds on the multipliers and are only concerned with providing a set of non-zero weights. In summary, the benefits of our multiplier bound approach to the assessment of efficiency without slacks (with non-zero weights) over other related approaches can be listed as follows: (1) it does not need any type of prior information to be used, (2) it does not require the existence of FDEFs on the frontier, (3) the models proposed do not have infeasibility problems, (4) we do not have problems with the alternate optima in the choice of weights that is made and (5) we not only guarantee non-zero weights but also we avoid as much as possible weighting schemes with extremely dissimilar values. In any case, we still believe that the problem addressed here should be investigated more deeply in the near future. Acknowledgments We are very grateful to Ministerio de Ciencia e Innovación (MTM2009-10479) for its financial support. Appendix A. Computational aspects Formulation (3) is a program with a fractional objective and linear constraints. Its optimal value can be easily obtained by using the following change of variables (see Charnes and Cooper, 1962)
b ¼ 1h ; m~i ¼ bmi ;
ðA1:1Þ
l~ r ¼ blr ; ~z ¼ bz;
which leads to the following LP problem whose optimal value coincide with that of (3).
~ j0 ¼ ~z Max u m P s:t: m~i xi0 ¼ b; i¼1
5. Conclusions In the DEA analyses, the inefficient units are almost always assessed by using reference points on the weak efficient frontier of the production possibility set, which means that the associated efficiency scores do not account for all sources of inefficiency. The fact that this happens so frequently in practice makes this issue become a problem of crucial importance in efficiency analyses, and, in our opinion, it has been insufficiently discussed in the literature. In the core of the matter, the key issue is the fact that DEA cannot provide an estimation of the frontier of full dimension (i.e., with the same dimension of the input-output space), as a result of the insufficient variation in the data (see Olesen and Petersen (1996) for a detailed discussion about this including developments). Some authors have opted to approach this problem by modifying the DEA technology, as is the case of Olesen and Petersen (1996) who propose to assess the efficiency with refer-
s P
l~ r yr0 ¼ b;
r¼1
m P i¼1
m~i xij þ
s P
ðA1:2Þ
l~ r yrj 6 0; j 2 E;
r¼1
~z 6 m ~i 6 1; i ¼ 1; . . . ; m; ~z 6 l ~ r 6 1; r ¼ 1; . . . ; s; m~ ; l~ r ; ~z; h~ P 0: i
Appendix B B.1. Proof of Proposition 1 Since each DMUj0 in E is, in particular, a Pareto-efficient point on the frontier, there exists at least one optimal solution of (2) for this unit consisting of strictly positive weights. This guarantees
N. Ramón et al. / European Journal of Operational Research 203 (2010) 261–269
that uj0 > 0, since we are maximizing in (3), and so, that u > 0 too. h B.2. Proof of Proposition 2 We can easily prove this statement by simply constructing a feasible solution of problem (5). For instance, if ðmi ; lr ; z ; h Þ is an optimal solution of (3) for a given DMUj0 in E, then it can be ~ ~ r ; ~z; hÞ, ~i ¼ mi =M; l ~r ¼ ~i ; l where m immediately shown that ðm P lr =M; ~z ¼ z =M and h~ ¼ h =M, M being mi¼1 mi xi0 , is a feasible solution of (5). h Appendix C. Equivalences to AR-models C.1. Proof of Proposition 3 Let us denote by v (5) and v (6) the optimal values of (5) and (6) respectively. As a result of Let ðmi ; lr ; z ; h Þ be an optimal solution of (5). m (5.3) and (5.5) we have that both mi hz u and mi m0 i
0
0
hz u ; i; i ¼ 1; . . . ; m; i – i .
i0
m
u mi0 u1 ; i; i0 ¼
Thus,
i
0
1; . . . ; m; i – i , and (6.3) holds. We can similarly show that both (6.4) and (6.5) also hold. Therefore, we can conclude that ðmi ; lr Þ is a feasible solution of (6), and so, v ð5Þ 6 v ð6Þ. Conversely, let ðmi ; lr Þ be an optimal solution of (6). Define z :¼ minfmi ; ur g and h :¼ maxfmi ; ur g. Then, (5.3) and (5.4) hold. In addition, since both z and h must coincide with some of either 0 0 the mi s or the lr s, say z ¼ mi0 and h ¼ mi0 , and we have that either z h
mi
0
¼ m0 P u (if i0 < i0 ) or i0 0
h z
0
mi0
0
¼ m0 6 u1 (if i0 > i0 Þ, it holds i0
z h
P u in
any case. Therefore, ðmi ; lr ; z ; h Þ is a feasible solution of (5), and so, v ð5Þ P v ð6Þ. Then, we have that v ð5Þ ¼ v ð6Þ, and since ðmi ; lr Þ is a feasible solution of (6) with optimal value equals v (6), then it is an opti mal solution of (6). Analogously, we can assert that ðmi ; lr ; z ; h Þ is an optimal solution of (5). h C.2. Proof of Proposition 4 The proof of this proposition is similar to that of Proposition 1. In this case, the quantities zI ; hI ; zO ; hO 2 R in the sufficient condition must be defined as follows: zI :¼ minfmi g; hI :¼ maxfmi g; zO :¼ minfur g and hO :¼ maxfur g. h References Allen, R., Athanassopoulos, A., Dyson, R.G., Thanassoulis, E., 1997. Weights restrictions and value judgements in data envelopment analysis: Evolution, development and future directions. Annals of Operations Research 73, 13–34. Bal, H., Örkcü, H.H., Çelebioglu, S., 2008. A new method based on the dispersion of weights in data envelopment analysis. Computers and Industrial Engineering 54 (3), 502–512. Beasley, J.E., 1990. Comparing university departments. Omega 18 (2), 171–183. Bessent, A., Bessent, W., Elam, J., Clark, T., 1988. Efficiency frontier determination by constrained facet analysis. Operations Research 36 (5), 785–796. Brockett, P.L., Charnes, A., Cooper, W.W., Huang, Z.M., Sun, D.B., 1997. Data transformation in DEA cone ratio envelopment approaches for monitoring bank performances. European Journal of Operational Research 98, 250–268. Chang, K.P., Guh, Y.Y., 1991. Linear production functions and data envelopment analysis. European Journal of Operational Research 52, 215–223. Charnes, A., Cooper, W.W., 1962. Programming with linear fractional functionals. Naval Research Logistics Quarterly 9, 181–186.
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Interfaces with Other Disciplines
A multiplier bound approach to assess relative efficiency in DEA without slacks Nuria Ramón, José L. Ruiz *, Inmaculada Sirvent Centro de Investigación Operativa, Universidad Miguel Hernández, Avd. de la Universidad, s/n. 03202-Elche (Alicante), Spain
a r t i c l e
i n f o
Article history: Received 29 July 2008 Accepted 13 July 2009 Available online 18 July 2009 Keywords: DEA AR Non-zero weights Zero-slacks
a b s t r a c t In this paper, we propose a new approach to deal with the non-zero slacks in data envelopment analysis (DEA) assessments that is based on restricting the multipliers in the dual multiplier formulation of the used DEA model. It guarantees strictly positive weights, which ensures reference points on the Paretoefficient frontier, and consequently, zero slacks. We follow a two-step procedure which, after specifying some weight bounds, results in an ‘‘Assurance Region”-type model that will be used in the assessment of the efficiency. The specification of these bounds is based on a selection criterion among the optimal solutions for the multipliers of the unbounded DEA models that tries to avoid the extreme dissimilarity between the weights that is often found in DEA applications. The models developed do not have infeasibility problems and we do not have problems with the alternate optima in the choice of weights that is made. To use our multiplier bound approach we do not need a priori information about substitutions between inputs and outputs, and it is not required the existence of full dimensional efficient facets on the frontier either, as is the case of other existing approaches that address this problem. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction In most cases in practice, the DEA models assess the efficiency of the inefficient units by using reference points on the frontier of the production possibility set (PPS) that are not Pareto-efficient. This happens as a result of the fact that these models usually yield zero weights for the optimal multipliers, or equivalently (by duality), strictly positive values for the optimal slacks, which means that the efficiency scores obtained for these units do not account for all sources of inefficiency. Bessent et al. (1988) deal with the so-called ‘‘not naturally enveloped inefficient units”, which are defined as those that have a mix of inputs and/or outputs which is different from that of any other point on the efficient frontier. These authors report the results corresponding to several studies that reveal the high frequency of the not naturally enveloped inefficiency units in practice. These units are actually those in F [ NF according to the classification of the decision making units (DMUs) in Charnes et al. (1991) (the DMUs in F are on the weak efficient frontier whereas those in NF are projected onto points in F). It has been paid much attention in the literature to this type of DMUs, where we can find a wide variety of approaches intended to provide efficiency scores for them trying to avoid the problems with the non-zero slacks. In this paper we propose a new approach to address this problem that deals with the dual multiplier formulation of the DEA models. This approach is based on imposing restrictions on the * Corresponding author. E-mail address: [email protected] (J.L. Ruiz). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.07.009
weights, which have been frequently used to obtaining non-zero weights, and also allow incorporating value judgements into the analysis (see Allen et al. (1997) and Thanassoulis et al. (2004) for a general discussion). The key issue when using weight restrictions is the setting of the bounds to be located in these constraints. We determine some weight bounds as the result of using an ancillary criterion of selection that makes it possible a specific choice of non-zero weights among the optimal solutions of the unbounded DEA models. We guarantee that the resulting DEA formulations provide efficiency scores that reflect the comparison of the unit under assessment with reference points on the Pareto-efficient frontier, which ensures zero slacks. We can find in the literature different approaches to setting bounds in weight restrictions of DEA models. Weight bounds can be obtained by resorting to the opinion of some experts involved in the underlying production process, as in Beasley (1990) or in Takamura and Tone (2003). In this latter case, the authors process the information from experts by using additionally AHP (Analytic Hierarchy Process). We can also use the information on prices and/or costs as in Thompson et al. (1995, 1996). Or we can even combine expert opinion and price information as in Thompson et al. (1990, 1992). Other authors have proposed to use the optimal weights of some units considered as model DMUs in order to specify the bounds in the weight restrictions (see Charnes et al. (1990) and Brockett et al. (1997)). These approaches have been mainly used with either Cone Ratio (CR) models (Charnes et al. (1990)) or Assurance Regions (AR) models (Thompson et al. (1986)), which were originally developed with the purpose of incorporating value judgements into the analysis, i.e., prior information and/or
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accepted beliefs or preferences concerning the underlying process of assessing efficiency and, in addition, they often lead to non-zero weights. However, it often happens in practice that we do not have available either expert opinion or prices/costs information. In that case, there are different methods that can be of help for the estimation of weight bounds, and these obviously rely on the information provided by the data. Most of them are based on somehow handling the optimal weights of the unbounded DEA models. For example, once the unbounded DEA model is solved in a first stage and we have compiled a weight matrix for all the variables, Roll et al. (1991) and Roll and Golany (1993) claim that we can set the bounds: (1) after eliminating the outliers and the extreme weights, (2) by imposing that a certain percentage of weights falls within the bounds or (3) at an acceptable ratio of variation for each weight within the range of the unbounded weights. As acknowledged by these authors, one of the main difficulties with these techniques is that the bounds obtained may vary depending on the chosen solution among the alternate optima for the weights. There are also some approaches that are based on imposing some specifically developed lower bounds on the multipliers in the dual formulation of the used DEA model. The idea behind these approaches is obviously to have strictly positive weights and, consequently, by duality, zero slacks. In fact, the use of the well-known non-archimedean e in this formulation leads to non-zero weights, but this model does not produce efficiency scores that can be readily used. To avoid this, Chang and Guh (1991) propose to replace this e with a specific lower bound that is obtained from the smallest non-zero value of the multipliers for all the variables of the unbounded DEA model. The main difficulty with this approach is that the resulting model may become infeasible (Chen et al. (2003) show how to modify this bound to avoid the infeasibility problems), together with the fact that the bounds are obtained following a procedure that does not consider the possible existence of alternate optima for the weights. Thus, different optimal solutions may lead to different bounds and these may lead to different efficiency scores for the units under assessment. Chen et al. (2003) develop an alternative multiplier bound approach in which the lower bounds are determined by strong complementary slackness condition (SCSC) solution pairs for extreme efficient DMUs. However, as acknowledged by the authors, the results obtained may vary depending on the SCSC solution that is chosen. We also include here the approach in Dyson and Thanassoulis (1988) which, in the case of having a single input (or a single output), provides lower bounds for the output (input) weights by imposing the condition that it cannot be used less than some percentage of the average input level the DMU being assessed uses per unit of output, which is estimated by means of a regression analysis. The existing work on facet models, which relates to the extension of the facets of the frontier, is in particular intended to address the problems with zero weights, and consequently with the slacks. Some of these approaches are based on the extension of full dimensional efficient facets (FDEFs) of the frontier. See Green et al. (1996), Olesen and Petersen (1996) and Portela and Thanassoulis (2006). In the two former, the authors define some new technologies from the FDEFs of the original frontier, whereas in the latter the proposed approach uses an AR model in which the bounds of the AR constraints are estimated from the marginal rates of substitution of these FDEFs. Obviously, these approaches guarantee nonzero weights, but they require the existence of such FDEFs on the frontier to be used, which does not happen very often in practice. See also Bessent et al. (1988), with the ‘‘constrained facet analysis” (CFA), for an algorithm along this line that proposes to project each inefficient unit onto an extended facet of the frontier that is suitably selected. CFA, however, guarantees neither that the facet wanted exists nor that the algorithm implemented finds this facet even if this exists. Lang et al. (1995) follow a similar idea and mod-
ify the previous approach by proposing to extend a facet of the PPS of a previously specified dimension, calling their approach ‘‘controlled envelopment analysis” (CEA). We note that CEA avoids the zero weights by imposing a non-archimedean e used as lower bound for the multipliers in the CCR model without specifying how this value is to be implemented for use in practice. Finally, it should be noted that the problem addressed here has also been approached in a different manner by dealing with the primal envelopment formulation of the DEA models. That is the case of the so-called ‘‘generalized efficiency measures” (GEMs) (see, e.g., Cooper et al. (1999), Pastor et al. (1999) and Tone (2001)), which are efficiency measures especially designed to account for both radial and non-radial inefficiencies, and so, avoiding the problems with the slacks. We propose here another multiplier bound approach intended to provide efficiency scores for the inefficient units that account for all sources of inefficiency. It can be used when we do not have available either information on prices/costs or expert opinion reflecting value judgements related to the involved inputs and outputs, and also if FDEFs on the frontier do not exist. We develop a two-step procedure which, in a first step, is aimed at specifying some weight bounds, which are then used in the second step in the weight restrictions that are incorporated into the dual multiplier formulation of the used DEA model. The resulting models will provide us with the efficiency scores that are wanted. We will show that these models can be equivalently formulated as AR models. As for the specification of the weight bounds, these are determined on the basis of a choice among the optimal solutions for the multipliers of the extreme efficient DMUs in the unbounded DEA models that is made by using a selection criterion. In absence of information on the relative importance of the involved variables, this criterion seeks to avoid the extreme dissimilarity between the weights of the optimal DEA solutions that is often found in practice. Thus, we look for non-zero weights while at the same time we try to avoid large differences between the multipliers as much as possible. We would like also to stress both that the procedures proposed to setting weight bounds are not affected by the possible existence of alternate optima in the unbounded DEA models and that the models we develop do not have infeasibility problems. The paper unfolds as follows: In Section 2 we develop the twostep procedure we propose to the setting of the bounds to be located in the weight restrictions of the DEA models used for the assessment of the inefficient units. Section 3 includes an illustrative example. In Section 4 we outline two possible extensions of the proposed approach. Section 5 concludes.
2. A two-step procedure to assess DMUs in F [ NF without slacks The high frequency of non-zero slacks (or, equivalently, zero weights) when assessing inefficient units in practice is mainly due to the total weight flexibility implicitly used in DEA. As has been widely explained in the literature, this total weight flexibility often leads to unreasonable results, since the weights assigned to the variables considered are frequently inconsistent with the prior knowledge or accepted views on the relative value of the involved inputs and outputs. In particular, the DEA models, trying to show the units under assessment in their best possible light, usually exploit this total weight flexibility to the point of evaluating the DMUs by putting the weight solely on a few set of variables and ignoring the rest of the originally considered inputs and outputs by assigning them a zero weight. This also brings with it that we have strictly positive values for the optimal slacks when assessing the efficiency of the inefficient units, since these are projected onto the facets of the weak efficient frontier, which means that these units are evaluated with reference to points that are not
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Pareto-efficient. Thus, the efficiency scores obtained do not account for all sources of inefficiency. As said before, we propose here a new multiplier bound approach to deal with this problem that is based on the two-step procedure we describe below. Throughout the paper we assume that we have n DMUs, each DMUj using m inputs ðxij ; i ¼ 1; . . . ; mÞ and producing s outputs ðyrj ; r ¼ 1; . . . ; sÞ. Assume also that the relative efficiency of these DMUs is assessed with the CCR model (Charnes et al., 1978) input-oriented (the developments here can be straightforwardly adapted for the output-oriented case), which consists of the following pair of dual problems
Min h n P kj xij 6 hxi0 ; s:t:
i ¼ 1; . . . ; m;
j¼1 n P
ð1Þ
kj yrj P yr0 ;
r ¼ 1; . . . ; s;
j¼1
kj P 0;
j ¼ 1; . . . ; n;
force all these multipliers to vary in between the bounds z and h. As a result, the ratios between each couple of multipliers are all greater than or equal to z=h: Since we are maximizing z=h in (3), with this problem we seek to maximize the minimum of the ratios between multipliers and, in this sense, we look for the least dissimilar optimal weights that allow DMUj0 to be rated as efficient. Thus, if, for example, uj0 ¼ 0:3; this would mean that it is not possible to assess DMUj0 as efficient with a set of weights in which the lowest multiplier is higher than 30% of the highest one.1 As said before, in Step I we solve (3) for each DMUj in E, which are those that support the efficient frontier and then are fundamental in the sense that they are used to evaluate all the points that represent the performances of the DMUs that are to be assessed (see Appendix A for the aspects concerned with solving this problem). Then, we aggregate all the information provided by the extreme efficient units by means of the following scalar
u ¼ min uj ;
ð4Þ
j2E
and
Max s:t:
s P r¼1 m P
lr yr0 mi xi0 ¼ 1;
i¼1 m P
mi xij þ
s P r¼1
i¼1
ð2Þ
lr yrj 6 0; j ¼ 1; . . . ; n;
mi P 0; i ¼ 1; . . . ; m; lr P 0; r ¼ 1; . . . ; s: Based on this DEA model we can partition the set of DMUs into the 0 0 classes E, E , F, NE, NE and NF as in Charnes et al. (1991). The DMUs 0 in E and E are Pareto-efficient. E consists of the extreme efficient 0 units, whereas those in E are Pareto-efficient units that can be expressed as linear combinations of DMUs in E. F is the set of weakly 0 efficient units. Finally, the DMUs in NE, NE and NF are inefficient 0 and are projected onto points that are in E, E and F respectively. We develop a two step procedure in two different situations: Firstly, we address the situation in which we do not distinguish between inputs and outputs in the implicit choice of weights that is made. This case mainly has a methodological interest. And next, we extend the idea previously developed to the more interesting case in practice of restricting the weights of the inputs and the weights of the outputs separately. 2.1. Restricting input multipliers and output multipliers indistinctly Step I: For each extreme efficient unit DMUj0 (see Thrall (1996) for the identification of DMUs in E), in the first step we propose to solve the following problem
Max uj0 ¼ hz m P s:t: mi xij0 ¼ 1; i¼1 s P
r¼1
ð3:1Þ
lr yrj0 ¼ 1;
m P
mi xij þ
i¼1
ð3:2Þ
s P r¼1
ð3Þ
lr yrj 6 0; j 2 E; ð3:3Þ
z 6 mi 6 h;
i ¼ 1; . . . ; m;
ð3:4Þ
z 6 lr 6 h;
r ¼ 1; . . . ; s;
ð3:5Þ
mi ; lr ; z; h P 0: The idea behind this problem is the following: With (3.1–3.3) we allow for all the optimal multipliers in the unbounded CCR model for DMUj0 (which is rated as efficient). The constraints (3.4) and (3.5)
which is the minimum of the uj0 0 s across the DMUs in E. Thus, u quantifies in a specific manner the disequilibrium between the optimal multipliers in a CCR model needed to assess the DMUs in E as efficient. In other words, u provides a lower bound for the ratios between the optimal multipliers of the DMUs in E (u will be always lower than or equal to 1) so that the optimal weights satisfying this condition guarantee that the extreme efficient units that support the efficient frontier remain as such. Proposition 1 is very important in order to guarantee that the inefficient units are assessed with reference to points on the Pareto-efficient frontier and so avoiding the non-zero slacks: Proposition 1. u > 0. Proof. See Appendix B.
h
Step II: Let DMU0 be an inefficient unit in F [ NF. Then, to assess its relative efficiency, we propose to use as an efficiency score the optimal value of the following LP problem
Max s:t:
s P r¼1 m P
lr yr0 m
i xi0 i¼1 m P
¼ 1;
mi xij þ
i¼1
ð5:1Þ
s P r¼1
lr yrj 6 0; j 2 E; ð5:2Þ
z 6 mi 6 h;
i ¼ 1; . . . ; m;
z 6 lr 6 h;
r ¼ 1; . . . ; s;
z h
P u ;
ð5Þ
ð5:3Þ ð5:4Þ ð5:5Þ
mi ; lr ; z; h P 0: We can see that (5) is the result of incorporating (5.3–5.5) into the dual formulation of the classical CCR model. As a consequence, with this model we force the DMU0 under evaluation to be assessed with reference to a set of weights that cannot be more dissimilar than those of the DMU in E that needs to unbalance more its weights (as measured by u ) in order to be rated as efficient. We would like to emphasize the following features of our approach: Firstly, Proposition 2 establishes that we do not have infeasibility problems with model (5): 1 In order to make the readership easier, our developments in this paper are in terms of absolute weights, which implicitly means to assume that the involved variables are measured in units such that the comparisons between multipliers are meaningful. If this is not so, as it often happens in practice, the absolute weights should be replaced with the corresponding virtuals, which can be done straightforwardly in this approach, as explained with more detail in Section 4. Alternatively, a referee suggested to scale all inputs and outputs into the interval [0,1].
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Proposition 2. Model (5) is feasible. Proof. See Appendix B.
h
Furthermore, as a result of Proposition 1 we also have that with (5) DMU0 is assessed with reference to a set of strictly positive weights. Consequently, by duality, the slacks in the efficiency assessment will be all zero, and so, the efficiency score of the inefficient units (the optimal value of (5)) will be reflecting the result of their comparison with a Pareto-efficient reference point. From a geometrical point of view, the effect of the choice of weights made with (5) is to introduce some ‘‘hypothetical efficient facets” (as termed in Chen et al. (2003)) where the inefficient DMUs are projected on. This is shown with more detail in Section 3 which includes a graphical illustration. It is also important to make clear that ours is not a usual absolute weight bound approach, since the absolute weight restrictions in (5) have not the usual form that forces each multiplier to lie in between two previously specified bounds, as z and h are not constants but they are actually variables in this problem. Thus, in (5) z and h may vary, with the only condition that the minimum ratio between weights, which is that associated with the couple of more dissimilar weights, be not lower than u . In fact, (5) is actually equivalent to an AR model, as we show next2:
Proposition 3. If ðmi ; lr ; z ; h Þ is an optimal solution of (5) then ðmi ; lr Þ is an optimal solution of the following LP problem
Max s:t:
s P r¼1 m P
lr yr0 m
i xi0 i¼1 m P
¼ 1;
mi xij þ
i¼1
u 6
s P r¼1
mi 1 mi0 6 u lr 1 lr0 6 u mi 1 lr 6 u
ð6:1Þ
lr yrj 6 0; j 2 E; 0
i; i ¼ 1; . . . ; m;
ð6:2Þ 0
i
ð6Þ
ð6:3Þ
u 6 r; r 0 ¼ 1; . . . ; s; r < r0 ; ð6:4Þ u 6 i ¼ 1; . . . ; m; r ¼ 1; . . . ; s; ð6:5Þ mi ; lr P 0 i ¼ 1; . . . ; m; r ¼ 1; . . . ; s: And conversely, if ðmi ; lr Þ is an optimal solution of (6) then ðmi ; lr ; z ; h Þ is an optimal solution of (5) for some z ; h 2 R. In addition, (5) and (6) have the same optimal value. Proof. See Appendix C.
cient frontier that is used as reference in the assessment of DMU0 . See, e.g., Cooper et al. (2007b) for developments concerning the AR-models. To sum up, we have developed a multiplier bound approach that makes it possible to assess the efficiency of the inefficient units with reference to a set of non-zero weights by using an AR model. A specific weight bound to be used in the DEA formulations has been proposed as the result of using a selection criterion among the optimal weights of the unbounded DEA models. The presence of alternate optima for the weights of the extreme efficient units in the unbounded DEA models provides opportunities for ancillary criteria of selection among them.4 When we have available neither information on prices/costs nor expert opinion we have to rely on the data as the only source of information to set the corresponding bounds. In particular, in absence of information on the relative importance of the involved variables, the proposed AR model makes a choice of weights that seeks to avoid the problems previously mentioned with the total weight flexibility by impeding the inefficient DMUs to be assessed with reference to sets of extremely dissimilar weights. It is the empirical efficient frontier which establishes the limit for the allowable similarities among weights, since we do not permit the extreme efficient units to become inefficient. Cook and Seiford (2008) state that ‘‘the AR concept was developed to prohibit large differences in the values of multipliers”, which goes on about the idea of having less dissimilar weights we are using in this paper. Thus, we depart from the classical DEA, which allows large differences between the weights of the DMU under assessment in its objective of maximizing the resulting efficiency score, and move in an opposite direction by looking for non-zero weights while at the same time we try to avoid as much as possible situations of extreme dissimilarity between the multipliers.5 Finally, it is also worth mentioning other benefits of our approach: (1) it does not require the existence of FDEFs, (2) the models developed do not have infeasibility problems and (3) we do not have problems with the alternate optima for the weights in the unbounded DEA models.
h
On the basis of Proposition 3, to proceed with the second step of our two-step procedure, we might equivalently solve this AR-model to assess the efficiency of DMU0 , which includes both type I and type II AR constraints since (6.3–6.5) are restrictions on the ratios between the weights of the inputs, the outputs and both the inputs and the outputs.3 Therefore, we can use conventional DEA software including AR-models in order to carry out the second step of our procedure. In particular, this will provide us with the efficiency scores b i ¼ P k xij ; i ¼ 1; . . . ; m, and and the associated targets, X j2E j P 0 b Y r ¼ j2E kj yrj ; r ¼ 1; . . . ; s, the kj s being the optimal solutions of variables associated with the constraints (6.2) in the model dual to (6). These targets are the coordinates of the point on the Pareto-effi2 It should be mentioned the difficulties with the use of absolute weight bounds with the possible infeasibility of the corresponding models and other side effects, as those discussed in Podinovski (1999, 2001), claiming that the models with absolute weigh bounds may not show the unit under assessment in the best possible light relative to other DMUs. 3 As a result of the equivalence to (3), (6) is also feasible, in spite of including AR-II constraints.
2.2. Restricting input multipliers and output multipliers separately In the previous subsection we have proposed a multiplier bound approach to assess the inefficient units eliminating the slacks without making any distinction between the multipliers corresponding to the inputs and those corresponding to the outputs. In this sense, that approach could be seen as an alternative to both the use of the non-archimedean e and the approach in Chang and Guh (1991), since these two approaches are based on imposing a common lower bound to all the multipliers of the inputs and outputs. In this section we extend this idea to address the more interesting case in practice of restricting the input multipliers and the output multipliers separately. That is the case of, for instance, the Cone Ratio models (Charnes et al. (1990)) in its more frequent use in practice, the type I AR models (Thompson et al. (1986)) and the model proposed in Bal et al. (2008), which minimizes separately the coefficient of variation of the input weights and that of 4 Some authors have pointed out the need to deal with the alternate optima for the weights in the DEA models. See Cooper et al. (2007a) which discusses this issue and proposes a specific choice of optimal weights based on an additional criterion of selection. See also Liang et al. (2008) which propose to use alternative secondary goals to deal with the non-uniqueness of the DEA optimal weights in the context of the cross-efficiency evaluation. 5 We can find in the literature several approaches that implement procedures that avoid extreme weights with different purposes. See, for instance, Li and Reeves (1999) which develop a multi-criteria DEA (MCDEA) model that avoids extreme weights and at the same time improves discrimination power among the efficient DMUs. See also Bal et al. (2008) which propose to minimize the coefficient of variation of the input and output weights with the same purpose.
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the output weights in an attempt to have more homogeneous weights. To adapt the two-step procedure in 2.1 to this new situation it should be modified as follows: Step I: For each DMUj0 in E solve the following problem
Max uj0 s:t:
mi xi0 ¼ 1;
lr yr0 ¼ 1;
m P
mi xij þ
s P r¼1
zI 6 mi 6 hI ; zO 6 lr 6 hO ;
m
i xi0 i¼1 m P
¼ 1;
mi xij þ
r¼1
lr lr0
ð7Þ
ð7:6Þ
zO hO
P uj0 ;
ð7:7Þ
lr yr0 m
¼ 1;
mi xij þ
s P r¼1
i¼1
ð9:3Þ
0
And conversely, if ðmi ; lr Þ is an optimal solution of (9) then ðmi ; lr ; zI ; hI ; zO ; hO Þ is an optimal solution of (8) for some zI ; hI ; zO ; hO 2 R. In addition, (8) and (9) have the same optimal value. Proof. See Appendix C. h
The idea of (7) is similar to that of (3) with the exception that z=h is now maximized separately for the inputs and the outputs by maximizing the minimum of the corresponding two ratios. Therefore, in solving (7) the bounds provided are obtained as the result of restricting the input multipliers on one hand and the output multipliers on the other trying to avoid as much as possible extreme dissimilarities between the input weights and the output weights independently. It should be noted that (7) is now a non-linear problem, which has a global optimum that can be found by using any conventional software of global optimization such as LINGO. As in the previous section, the optimal values of (7) will be also used to define the scalar u ¼ minj2E uj to be used in the second step. Step II: Let DMU0 an inefficient unit in F [ NF. To assess its relative efficiency, we propose to solve the following LP problem
i xi0 i¼1 m P
0
ð7:5Þ
P uj0 ;
r¼1 m P
1
ð9Þ
ð9:2Þ
u 6 6 u ; r; r ¼ 1; . . . ; s; r < r ; ð9:4Þ mi ; lr P 0; i ¼ 1; . . . ; m; r ¼ 1; . . . ; s:
zI hI
s P
lr yrj 6 0; j 2 E;
ð7:2Þ
ð7:4Þ
r ¼ 1; . . . ; s;
s P
u 6 mmii0 6 u1 ; i; i0 ¼ 1; . . . ; m; i < i0 ;
lr yrj 6 0; j 2 E; ð7:3Þ i ¼ 1; . . . ; m;
ð9:1Þ
ð7:1Þ
mi ; lr ; zI ; hI ; zO ; hO ; uj0 P 0:
s:t:
lr yr0
i¼1
Max
r¼1 m P
i¼1
i¼1
r¼1
s:t:
s P
m P
s P
Max
ð8:1Þ
lr yrj 6 0; j 2 E; ð8:2Þ
We can see that now in solving (8) we will be implicitly solving a type I AR model since the constraints used only involve ratios between input weights and ratios between output weights, as could be expected. To illustrate graphically the idea of the proposed procedure consider the following small example depicted in Fig. 1 which consists of 4 DMUs, with two inputs and one constant output, whose relative efficiency is to be assessed: A(1,8;1), B(2,3;1), C(5,2;1) and D(10,2.5;1). The CCR model reveals that A, B and C are extreme efficient points whereas D is inefficient. Its efficiency score is 4/5 = 0.8, 0 which is the result of projecting this DMU onto the point D (8,2;1) on the weak efficient frontier, and so, it is detected a slack s 1D ¼ 3 in the first input, as compared to DMU C (by duality, the optimal multiplier associated with x1 is 0). Thus, this efficiency score is not accounting for all the inefficiency of DMU D. To avoid this, we propose to alternatively evaluate the efficiency of DMU D by using our two-step procedure. In the first step, we have the following results by solving (7): for DMU A, uA ¼ 1=5 ¼ 0:2, which is associated with the optimal weights m1 ¼ 5=13; m2 ¼ 1=13 and l ¼ 1; for DMU B, uB ¼ 1, for m1 ¼ m2 ¼ 1=5 and l ¼ 1 and, for DMU C, uC ¼ 1=3 ¼ 0:33, for m1 ¼ 1=11; m2 ¼ 3=11 and l ¼ 1. Each of these optimal solutions represents the coefficients of a supporting hyperplane for the PPS 5 1 x1 13 x2 þ y ¼ 0; at DMU A, at the corresponding unit: HA 13 1 3 x1 11 x2 þ HB 15 x1 15 x2 þ y ¼ 0, at DMU B, and HC 11 y ¼ 0, at DMU C. For each of these DMUs, these coefficients are
ð8Þ
zI 6 mi 6 hI ; i ¼ 1; . . . ; m; zO 6 lr 6 hO ; r ¼ 1; . . . ; s;
ð8:3Þ ð8:4Þ
zI P hI zO P hO
ð8:5Þ
u ; u ;
x2
HA
ð8:6Þ
mi ; lr ; zI ; hI ; zO ; hO P 0:
A
As in the previous section, model (8) assesses the efficiency of DMU0 by using a formulation in which the weights are restricted from the information provided by the data in the first step. To be specific, DMU0 is now assessed with reference to a set of weights in which both the input weights among themselves and the output weights among themselves cannot be more dissimilar than u . Moreover, the discussion in Section 2.1 on the features and advantages of the proposed procedure also applies in this case of dealing with the input and output weights separately. The following proposition is the equivalent to Proposition 3 for this case, and also allows us to obtain the optimal solution of (8) by using an equivalent AR formulation:
y=1 HB
B C
D’’
D’
HW H*
HC x1
Proposition 4. If ðmi ; lr ; zI ; hI ; zO ; hO Þ is an optimal solution of (8) then ðmi ; lr Þ is an optimal solution of the following LP problem
D
Fig. 1. Graphical illustration.
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the least dissimilar weights in the sense previously explained. For instance, uC ¼ 1=3 ¼ 0:33 means that it is impossible to find a set of positive optimal weights for DMU C such that the magnitude of the multiplier of the variable with minimum weight is higher than 33.3% of that of the variable with maximum weight (without making it become inefficient). For the assessment of the inefficient DMU D in the second step we use the minimum of these values, u ¼ minfuA ; uB ; uC g ¼ 1=5 ¼ 0:2; since this bound guarantees the existence of optimal weights for the 3 efficient DMUs that make it possible to maintain them as such. This u ¼ 1=5 ¼ 0:2 will force DMU D to be assessed with reference to a set of weights that cannot be more dissimilar than those of DMU A, which is the unit that needs to unbalance more its weights in order to be rated as efficient. This minimum also represents a lower bound for the ratios between the coefficients of the hyperplane where DMU D is projected on. In particular, when assessing DMU D with (8), its efficiency score is now 2/ 00 3 = 0.66 (and not 4/5 = 0.8), as the result of projecting it onto D , 4 2 2 which is on the hyperplane H 90 x1 9 x2 þ 3 y ¼ 0 (which cor1 x1 þ 13 x2 ¼ 1 at the level y = 1 in Fig. 1). Obviresponds to the line 15 00 ously, the coordinates of the point D are not the efficient targets 00 for DMU D, since the projection point D is outside the PPS, but the targets for DMU D are the coordinates of DMU C, which are actually those associated with the optimal solution provided by the AR model (as explained before) and is also on H . DMU C is in addition on the Pareto-efficient frontier of the PPS, which consists of the segments AB and BC, and this allows us to eliminate the slacks in the assessment of DMU D. The assessment of DMU D depicted in Fig. 1 gives us a geometrical idea of the procedure we have proposed. We depart from the hyperplane HW ¼ 12 x2 ¼ 1, which contains a part of the weak frontier of the PPS, and move by looking for more similar weights up to hyperplane H , where the constraint hzII P 0:2 is binding. Going further than H would mean to assess DMU D with a lower efficiency score. To be precise, the supporting hyperplanes at DMU C in between H and HC are associated with feasible solutions of (8) when DMU D is evaluated, but projecting it onto some of them would yield a lower efficiency score for this unit. We can see that with our approach we avoid the weighting schemes with extremely dissimilar values for the multipliers. However, we do not look for close multipliers, but our aim is to find the least dissimilar weights which, in addition, allow us to maintain all the DMUs in E as efficient.
It is to be noted that in this particular example the PPS does have two facets of full dimension at the frontier, so we could also use the approaches based on the projection onto the extension of these FDEFs to assess the inefficient DMU D. To be specific, following Green et al. (1996) and Olesen and Petersen (1996), DMU D would be projected onto HC , which extends the facet determined by DMUs B and C. The efficiency score of DMU D would be 0.6285 in that case. The hyperplane HA , which is the extension of AB, is associated with some feasible solution of the models used in these approaches, but projecting DMU D onto HA would yield a lower efficiency score for this unit. CFA would lead to these same results. To use CEA, we should select the facet of maximum dimension to be extended which also guarantees non-zero weights. In this example, this would be the segment BC, and so DMU D would be again projected onto HC with the same results as before. Portela and Thanassoulis (2006) determine a lower and an upper bound for the ratios between each couple of multipliers as the minimum and maximum of the corresponding marginal rates of substitution estimated with all the FDEFs of the frontier. The resulting AR-model for the assessment of DMU D would have as feasible solutions those associated with all the supporting hyperplanes for the PPS at DMU B which are in between HA and HC . Obviously, DMU D would be projected again onto HC since this yields the higher efficiency score. We cannot provide a direct comparison with the approaches based on imposing lower bounds on the multipliers, as we explain next. The approach in Chen et al. (2003) would select a hyperplane in between HW and HC which would depend on the choice of the SCSC solution that were made. In Chang and Guh (1991) this specification also depends on the choice of an optimal solution for the multipliers in the unbounded DEA model. And this would also happen with the absolute weight bound procedures listed in Roll et al. (1991) and Roll and Golany (1993).
3. Illustrative example In this section we illustrate the use of the proposed procedure with a new example with a larger data set in a higher dimensional space. Consider the sample consisting of 20 DMUs, each one using 4 inputs to produce 2 outputs. The data together with the corresponding CCR efficiency scores and the associated optimal slacks are recorded in Table 1. We can see that the DMUs 1, 4, 5, 6, 8
Table 1 Data and CCR efficiency analysis results. DMU
x1
x2
x3
x4
y1
y2
CCR (score)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1.7 5.9 5.6 2.9 2.6 1.7 3 1.6 5.3 3.3 4 8 8 4.3 1.9 3 5.9 2.8 1.7 3.3
1.7 3.3 5 2 3.3 2.5 5 4 2.5 2 2.5 5 4 3.2 2.5 5.5 4.2 4.6 2.2 1.7
2.3 3.3 3.3 1.7 3.3 1.4 2.5 2 3.3 2 5 1.7 3.5 3.3 3.5 2.4 4.9 2.3 1.4 3.3
1 1.3 1.4 2 1.7 2 2 1.5 2 2 3.3 1.1 1.7 2 1.7 1.3 2 1.9 1.3 3.3
2 1 1 2 1 1.5 2 2 1 2 2 2 2 1.5 2 1 2 1 1 2
1 2 2 2 2 1.5 1 1 2 1 2 3 2 1 1 1.5 2 2 1 1
1 0.8794 0.7749 1 1 1 0.7720 1 0.8914 0.9500 0.8000 1 0.7547 0.4764 0.8831 0.8505 0.6973 0.9956 0.8079 1
Slacks x1
x2
x3
x4
y1
y2
0.64 0
0 0.55
1.59 0.64
0 0
0.55 0.21
0 0
0
0.08
0
0
0
0.31
1.41 0.64 0.3
0 0 0
1.34 0 2.3
0 0.23 0.64
0.89 0 0
0 0.67 0
1.67 0.17 0 0 0 0 0 1.6
0 0 0 2.24 0 2.1 0.45 0
0.72 0 0.86 0 1.4 0 0 1
0 0 0.39 0 0 0 0 2.3
0 0 0 0 0 0.63 0 0
0 0.06 0 0 0 0 0 0
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and 12 are efficient (they are actually extreme efficient points) whereas the remaining fourteen DMUs are inefficient. To be precise, these inefficient DMUs belong to F [ NF, since they all have some non-zero slacks. This suggests the need for a procedure such as the one proposed here to alternatively assess their efficiency. It is to be noted that there is no FDEF at the frontier of the PPS defined by this data set, in spite of having enough extreme efficient points, the maximum dimension of the efficient facets being 4. Therefore, none of the approaches based on the extension of FDEFs could be used for this efficiency assessment. To proceed with our procedure in the first step, we solve (7) for each of the 6 extreme efficient DMUs. Table 2 records the optimal values uj together with an associated optimal solution for the multipliers. As expected, all these multipliers are strictly positive. The minimum of these uj 0 s is 0.11345, which will be used in the second step as the bound u for the allowable dissimilarity between the resulting multipliers. To be precise, the inefficient units will be evaluated with reference to a set of weights in which the ratios of the input weights and those of the output weights cannot be lower than 0.11345.
Table 3 records the scores for the inefficient DMUs and an optimal solution for the multipliers obtained with both the classical CCR model and our two-step procedure. It can be seen that, unlike the optimal solutions provided by the CCR model, our two-step procedure yields non-zero weights, as a consequence of avoiding these extreme dissimilarities between the multipliers. Therefore, our approach guarantees that the associated efficiency scores are the result of the comparison of the DMU under assessment with a Pareto-efficient point on the frontier of the PPS. Obviously, these efficiency scores are lower than those provided by the CCR model as a result of eliminating the slacks and, consequently, accounting for all sources of inefficiency. We highlight the case of DMU 20, which is on the weak efficient frontier, and whose efficiency score is eventually 0.8031 when compared with a Pareto-efficient referent. Finally, it is important to point out that we cannot make comparisons with some of the existing multiplier approaches since these cannot be used with these data. As said before, there is no FDEF at the frontier of the PPS, so we cannot use the models based on their extension. Perhaps, CEA could be used, since it does not necessarily requires FDEFs, but its formulations depend on a nonarchimedean e to guarantee non-zero weights and the authors do not provide any guideline for its specification. As for the approaches based on imposing lower bounds on the multipliers, we have the same difficulties already mentioned in the discussion of the small example in Section 2.
Table 2 Step I: Specification of the weight bounds. DMU_1
DMU_4
DMU_5
DMU_6
DMU_8
DMU_12
uj0
0.89171
1
0.11345
0.29231
0.14286
0.35908
1 2 3 4 1 2
0.15957 0.14229 0.14229 0.15957 0.34582 0.30837
0.11628 0.11628 0.11628 0.11628 0.25000 0.25000
0.19807 0.02247 0.02247 0.19807 0.05368 0.47316
0.29200 0.08535 0.08535 0.08535 0.25651 0.41015
0.22801 0.03257 0.22801 0.03257 0.43105 0.13789
0.04808 0.04808 0.13390 0.13390 0.09657 0.26895
m m m m l l
4. Extensions (1) It is well-known that the results of the DEA models with weight restrictions based on absolute weights, both those with absolute weights bounds and the AR models, are dependent on the units of measurement of the inputs and
Table 3 Step II: Efficiency scores. Score
m1
m2
m3
m4
l1
l2
DMU_2
CCR Two-step
0.8794 0.8011
0 0.1596
0.2152 0.1423
0 0.1423
0.2231 0.1596
0 0.3458
0.5 0.3084
DMU_3
CCR Two-step
0.7749 0.7317
0.1054 0.0909
0 0.0238
0 0.0238
0.2929 0.2097
0 0.0393
0.5 0.3462
DMU_7
CCR Two-step
0.7720 0.7171
0.02 0.0991
0 0.0233
0.284 0.2056
0.115 0.0359
0.5 0.3393
0 0.0385
DMU_9
CCR Two-step
0.8914 0.7878
0 0.0241
0.216 0.147018
0 0.0241
0.23 0.2125
0 0.0423
0.5 0.3727
DMU_10
CCR Two-step
0.9500 0.8923
0 0.0297
0.335 0.2616
0.165 0.1598
0 0.0297
0.5 0.4222
0 0.0479
DMU_11
CCR Two-step
0.8 0.7149
0 0.1344
0.4 0.1344
0 0.0152
0 0.0152
0.14 0.1498
0.36 0.2077
DMU_13
CCR Two-step
0.7547 0.7014
0 0.0224
0.155 0.1016
0 0.0224
0.2235 0.1975
0.145 0.1092
0.355 0.2414
DMU_14
CCR Two-step
0.4764 0.4697
0 0.0204
0.0344 0.0204
0.2030 0.1795
0.11 0.1274
0.6667 0.2796
0 0.0503
DMU_15
CCR Two-step
0.8831 0.8175
0.5 0.3606
0.02 0.0409
0 0.0409
0 0.0409
0.45 0.2725
0.1 0.2725
DMU_16
CCR Two-step
0.8505 0.7619
0.16 0.1240
0 0.0324
0.0375 0.0324
0.3385 0.2860
0.11 0.0536
0.5933 0.4722
DMU_17
CCR Two-step
0.6973 0.6655
0.0763 0.0733
0.0333 0.0212
0 0.0212
0.205 0.1873
0.06 0.0640
0.44 0.2688
DMU_18
CCR Two-step
0.9956 0.9208
0.1893 0.1824
0 0.0207
0.0652 0.0207
0.1632 0.1824
0 0.0494
0.5 0.4357
DMU_19
CCR Two-step
0.8079 0.7819
0.3294 0.3321
0 0.0377
0.0357 0.0377
0.3 0.2307
0.16 0.1640
0.84 0.6179
DMU_20
CCR Two-step
1 0.8031
0 0.0402
0.4 0.3542
0 0.0402
0 0.0402
0.415 0.3163
0.17 0.1706
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outputs. In the particular case of our approach, this means that the bounds u obtained might vary depending on the scaling of the variables and, consequently, this would also happen with the results of the corresponding models. However, the virtual inputs and outputs are dimensionless, and so, the use of virtuals instead of absolute weights would lead to the same results regardless of the units of measurement of the variables. In addition, absolute weights restrictions are frequently difficult to interpret whereas with virtual weights restrictions the decision maker has a clear idea of being restricting the relative importance of the factors (the relative contribution of each input and output to its efficiency score). See Sarrico and Dyson (2004) for a discussion about this. The approach in this paper can be easily adapted to be used with virtuals instead of absolute weights, if it is desired to avoid the mentioned difficulties with the use of absolute weights. To do it, we just need to replace the absolute weights in all the formulations with the corresponding virtuals. To be precise, in the particular case of restricting the input weights and the output weights separately, we should replace both the constraints (7.4) and (7.5), in the first step, and the constraints (8.3) and (8.4) in the second step with
zI 6 mi xi0 6 hI ; zO 6 lr yr0 6 hO ;
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s:
ð10Þ
(2) The approach developed here is based on a specific choice of weights among the multipliers of the extreme efficient units. We could similarly use our two-step procedure if the minimum u were calculated across a different set of DMUs instead of E, say a group of model DMUs. The idea of using optimal weights of model DMUs was initially proposed in Charnes et al. (1990) and in Brockett et al. (1997). These authors show how to define cones that favour certain (model) DMUs, and so favouring desired patterns of input usage and output production in the efficiency assessment by using optimal multiplier vectors of the model DMUs. To be specific, this approach was used in the analysis of efficiency of commercial banks by means of Cone Ratio models where the weights of several banks considered as excellent were used as benchmarks for the rest of banks under assessment (note, however, that this ignores the possibility of having alternate optima for the multipliers for the model DMUs, which would be avoided with our approach). It should also be noted that in that case the efficient units not considered as model DMUs could become inefficient.
ence to some new defined production possibility sets, such as the so-called extended facet PPS, TEXFA , and the full dimensional efficient facet PPS, TFDEF , which are built from the FDEFs of the original frontier. Obviously, this approach depends on the existence of such facets, which rarely happens in practice. In our proposal here, we have decided to maintain the original technology and have opted to restrict the total weight flexibility, as a mean to avoid the weak efficient frontier. In absence of any information from experts, this obliges us to use an ancillary criterion of selection among weights, which, in our case, has been based on the idea of avoiding the extreme dissimilarity between weights that is often found in DEA analyses. Thus, with our approach we go one step further than those also based on imposing bounds on the multipliers and are only concerned with providing a set of non-zero weights. In summary, the benefits of our multiplier bound approach to the assessment of efficiency without slacks (with non-zero weights) over other related approaches can be listed as follows: (1) it does not need any type of prior information to be used, (2) it does not require the existence of FDEFs on the frontier, (3) the models proposed do not have infeasibility problems, (4) we do not have problems with the alternate optima in the choice of weights that is made and (5) we not only guarantee non-zero weights but also we avoid as much as possible weighting schemes with extremely dissimilar values. In any case, we still believe that the problem addressed here should be investigated more deeply in the near future. Acknowledgments We are very grateful to Ministerio de Ciencia e Innovación (MTM2009-10479) for its financial support. Appendix A. Computational aspects Formulation (3) is a program with a fractional objective and linear constraints. Its optimal value can be easily obtained by using the following change of variables (see Charnes and Cooper, 1962)
b ¼ 1h ; m~i ¼ bmi ;
ðA1:1Þ
l~ r ¼ blr ; ~z ¼ bz;
which leads to the following LP problem whose optimal value coincide with that of (3).
~ j0 ¼ ~z Max u m P s:t: m~i xi0 ¼ b; i¼1
5. Conclusions In the DEA analyses, the inefficient units are almost always assessed by using reference points on the weak efficient frontier of the production possibility set, which means that the associated efficiency scores do not account for all sources of inefficiency. The fact that this happens so frequently in practice makes this issue become a problem of crucial importance in efficiency analyses, and, in our opinion, it has been insufficiently discussed in the literature. In the core of the matter, the key issue is the fact that DEA cannot provide an estimation of the frontier of full dimension (i.e., with the same dimension of the input-output space), as a result of the insufficient variation in the data (see Olesen and Petersen (1996) for a detailed discussion about this including developments). Some authors have opted to approach this problem by modifying the DEA technology, as is the case of Olesen and Petersen (1996) who propose to assess the efficiency with refer-
s P
l~ r yr0 ¼ b;
r¼1
m P i¼1
m~i xij þ
s P
ðA1:2Þ
l~ r yrj 6 0; j 2 E;
r¼1
~z 6 m ~i 6 1; i ¼ 1; . . . ; m; ~z 6 l ~ r 6 1; r ¼ 1; . . . ; s; m~ ; l~ r ; ~z; h~ P 0: i
Appendix B B.1. Proof of Proposition 1 Since each DMUj0 in E is, in particular, a Pareto-efficient point on the frontier, there exists at least one optimal solution of (2) for this unit consisting of strictly positive weights. This guarantees
N. Ramón et al. / European Journal of Operational Research 203 (2010) 261–269
that uj0 > 0, since we are maximizing in (3), and so, that u > 0 too. h B.2. Proof of Proposition 2 We can easily prove this statement by simply constructing a feasible solution of problem (5). For instance, if ðmi ; lr ; z ; h Þ is an optimal solution of (3) for a given DMUj0 in E, then it can be ~ ~ r ; ~z; hÞ, ~i ¼ mi =M; l ~r ¼ ~i ; l where m immediately shown that ðm P lr =M; ~z ¼ z =M and h~ ¼ h =M, M being mi¼1 mi xi0 , is a feasible solution of (5). h Appendix C. Equivalences to AR-models C.1. Proof of Proposition 3 Let us denote by v (5) and v (6) the optimal values of (5) and (6) respectively. As a result of Let ðmi ; lr ; z ; h Þ be an optimal solution of (5). m (5.3) and (5.5) we have that both mi hz u and mi m0 i
0
0
hz u ; i; i ¼ 1; . . . ; m; i – i .
i0
m
u mi0 u1 ; i; i0 ¼
Thus,
i
0
1; . . . ; m; i – i , and (6.3) holds. We can similarly show that both (6.4) and (6.5) also hold. Therefore, we can conclude that ðmi ; lr Þ is a feasible solution of (6), and so, v ð5Þ 6 v ð6Þ. Conversely, let ðmi ; lr Þ be an optimal solution of (6). Define z :¼ minfmi ; ur g and h :¼ maxfmi ; ur g. Then, (5.3) and (5.4) hold. In addition, since both z and h must coincide with some of either 0 0 the mi s or the lr s, say z ¼ mi0 and h ¼ mi0 , and we have that either z h
mi
0
¼ m0 P u (if i0 < i0 ) or i0 0
h z
0
mi0
0
¼ m0 6 u1 (if i0 > i0 Þ, it holds i0
z h
P u in
any case. Therefore, ðmi ; lr ; z ; h Þ is a feasible solution of (5), and so, v ð5Þ P v ð6Þ. Then, we have that v ð5Þ ¼ v ð6Þ, and since ðmi ; lr Þ is a feasible solution of (6) with optimal value equals v (6), then it is an opti mal solution of (6). Analogously, we can assert that ðmi ; lr ; z ; h Þ is an optimal solution of (5). h C.2. Proof of Proposition 4 The proof of this proposition is similar to that of Proposition 1. In this case, the quantities zI ; hI ; zO ; hO 2 R in the sufficient condition must be defined as follows: zI :¼ minfmi g; hI :¼ maxfmi g; zO :¼ minfur g and hO :¼ maxfur g. h References Allen, R., Athanassopoulos, A., Dyson, R.G., Thanassoulis, E., 1997. Weights restrictions and value judgements in data envelopment analysis: Evolution, development and future directions. Annals of Operations Research 73, 13–34. Bal, H., Örkcü, H.H., Çelebioglu, S., 2008. A new method based on the dispersion of weights in data envelopment analysis. Computers and Industrial Engineering 54 (3), 502–512. Beasley, J.E., 1990. Comparing university departments. Omega 18 (2), 171–183. Bessent, A., Bessent, W., Elam, J., Clark, T., 1988. Efficiency frontier determination by constrained facet analysis. Operations Research 36 (5), 785–796. Brockett, P.L., Charnes, A., Cooper, W.W., Huang, Z.M., Sun, D.B., 1997. Data transformation in DEA cone ratio envelopment approaches for monitoring bank performances. European Journal of Operational Research 98, 250–268. Chang, K.P., Guh, Y.Y., 1991. Linear production functions and data envelopment analysis. European Journal of Operational Research 52, 215–223. Charnes, A., Cooper, W.W., 1962. Programming with linear fractional functionals. Naval Research Logistics Quarterly 9, 181–186.
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