DEA models with undesirable inputs and outputs

Ann Oper Res (2010) 173: 177–194 DOI 10.1007/s10479-009-0587-3 DEA models with undesirable inputs and outputs W.B. Liu · W. Meng · X.X. Li · D.Q. Zhang Published online: 15 July 2009 © Springer Science+Business Media, LLC 2009 Abstract Data Envelopment Analysis (DEA) models with undesirable inputs and outputs have been frequently discussed in DEA literature, e.g., via data transformation. These stud- ies were scatted in the literature, and often confined to some particular applications. In this paper we present a systematic investigation on model building of DEA without transferring undesirable data. We first describe the disposability assumptions and a number of different performance measures in the presence of undesirable inputs and outputs, and then discuss different combinations of the disposability assumptions and the metrics. This approach leads to a unified presentation of several classes of DEA models with undesirable inputs and/or outputs. Keywords Data Envelopment Analysis · Undesirable inputs and outputs · Extended Strong Disposability 1 Introduction Since the introduction of the DEA in 1978, it has been widely used in efficiency analysis of many business and industry applications. Excellent literature surveys can be found in, for instance, Seiford (1996) and Cooper et al. (2004). The most well-known DEA models are the CCR model (Charnes et al. 1978) , the BCC model (Banker et al. 1984) , the Additive model (Charnes et al. 1985), and the Cone Ratio model (Charnes et al. 1989). These DEA The authors wish to express their sincere thanks to the referees for his or her constructive reviews and suggestions, which lead to significant improvements of this paper. W.B. Liu (�) Kent Business School, University of Kent, Canterbury, CT2 7PE, UK e-mail: W.B.Liu@kent.ac.uk W. Meng School of Public Administration, East China Normal University, Shanghai, 200062, China X.X. Li · D.Q. Zhang Institute of Policy and Management, Chinese Academy of Sciences, Beijing, 100080, China 178 Ann Oper Res (2010) 173: 177–194 models were all formulated for desirable inputs and outputs. However, there frequently exist undesirable inputs and/or outputs in real applications. In DEA literature, there already existed much research concerning applications with un- desirable inputs and/or outputs. Some of the existing approaches are briefly summarized as follows: An intuitive reaction is to apply some transformations. The most acceptable one is f (U) = −U ; the so called the ADD approach suggested by Koopmans (1951). Then the undesirable inputs or outputs will become desirable after this transformation. However, the data may then become negative, and it is not straightforward to define efficiency scores for negative data. Scheel (2001) attempted this task but his definitions seemed to be compli- cated. Translation f (U) = −U + β is another widely used one (e.g. Ali and Seiford 1990; Pastor 1996; Scheel 2001; Seiford and Zhu 2002). However it is well-known that not only ranking but also classification may depend on β . Another transformation is the multiplica- tive inverse: f (U) = 1/U (e.g. Golany and Roll 1989; Lovell et al. 1995). Being a nonlinear transformation, its behaviors are even more complicated (Scheel 1998). Thus how to prop- erly select a suitable transformation is very much case-dependent. The approaches based on data-transformation may unexpectedly produce adverse results as discussed in Liu and Sharp (1999). There also exist many approaches that can avoid data transformation. For example, one may regard undesirable inputs as desirable outputs, or undesirable outputs as desirable in- puts, see Liu and Sharp (1999) for an initial attempt to formulate this method. This approach is an attractive method in studying operational efficiency due to its simplicity and elegance, although it changes the physical input-output relationship. Its starting point is that efficient DMUs wish to minimize desirable inputs and undesirable outputs, and to maximize desir- able outputs and undesirable inputs. If one only wishes to investigate operational efficiency from this point of view, there is no need to distinguish between inputs and outputs, but only minimum and maximum. We will further extend this approach in this work and discuss its relationship with other approaches. Related to ADD, there are several recent papers on DEA models handling negative data (but desirable) with directional distance functions, such as, Färe and Grosskopf (2004) (where a weak free disposability was used), Silva Portela et al. (2004), and Yu (2004). As shown in Seiford and Zhu (2005), such approaches are closely related to the weighted additive models in order to measure proportional gains from the slacks, which are always positive. It is important to realize that the additive models are able to handle negative data. After some simple modifications, they can easily handle undesirable cases as well. These will be discussed later on. Our investigation pays more attention to theoretical aspects of these issues. Our main idea is to examine them within the general framework proposed in Liu et al. (2006), where some essential building blocks of a DEA model were identified and illustrated. The princi- pal objective of this paper is to discuss the free disposability assumptions and to describe a number of different performance measures or metric functions with attractive properties in the presence of undesirable inputs and outputs and negative data. We will discuss different combinations of the free disposability assumptions and the metrics. This approach leads to a unified presentation for some existing DEA models with undesirable inputs and/or out- puts. We also discuss some relationship among the existing approaches. We undertake these investigations by utilizing some of the ideas used in Liu et al. (2006). The plan of this paper is as follows: In Sect. 2 we discuss disposability assumptions in DEA, which form the foundation of this paper. In Sect. 3 we summarize some metrics used in DEA models following the ideas used in Liu et al. (2006). In Sect. 4, we examine slack based models. We combine our ideas with the works in Tone (2001) and Silva Portela Ann Oper Res (2010) 173: 177–194 179 et al. (2004) to formulate additive models, which are applicable to undesirable or negative data. In Sects. 5–6, we extend the ideas in Liu and Sharp (1999) to formulate general DEA models for undesirable data using the radial and Russell efficiency measures. Conclusions are summarized in Sect. 7. 2 Disposability assumptions in DEA Assumed that there are n decision-making units to be evaluated. Let Xj and Yj denote the inputs and outputs of DMUj with j = 1,2, . . . , n. The first building block of any DEA mod- els is preference. It is clear that decision making units are built or operated for some specific purposes. In order to be able to evaluate DMUs, we first have to know our “preference” in the input-output space (X,Y ). To set up certain order relationship among the input-output pos- sibilities, preference aims to clarify the precise meaning of our fuzzy desires like “higher”, “lower”; “better”, or “worse”. In the standard DEA models like CCR and BCC, the clas- sic Pareto preference is assumed, as in this work. However, we emphasize there are many real applications where different preferences are indeed useful, see Liu et al. (2006) and Zhang et al. (2009) for some examples. Let X = (x1, . . . , xl), Y = (y1, . . . , yl) ∈ Rl . Then in Pareto preference, X ≥ Y if and only if xi ≥ yi (i = 1,2, . . . , l). For DMUs with desirable inputs and outputs, DMU1 (X1, Y1) is better than DMU2 (X2, Y2) if X1 ≤ X2, Y1 ≥ Y2 in Pareto preference. If, e.g., the outputs are all undesirable, then DMU1 is better than DMU2 if X1 ≤ X2, Y1 ≤ Y2 in Pareto preference, and so on. The second building block is the Production Possibility Set (PPS). DEA works by per- forming multiple comparisons and by so doing, implies a measure of preference. However, we need a sufficient number of peers so that the comparisons can meaningfully identify the “best” DMUs. Most of the preference in business applications is too weak to pick up the “best” among finite set of DMUs since there are not enough peers for the comparisons. The PPS P ({(Xj ,Yj )}) contains all the realizable DMUs associated with the preference, al- though some of them may not in fact exist. In DEA, they are referred to as “virtual” DMUs, and are also included in the comparisons. If a real DMU (Xj ,Yj ) is found to be the “best” in P ({(Xj ,Yj )}), then it is considered to be efficient. In the standard DEA models several assumptions are made on the PPS, such as convexity and no-free-lunch. The most relevant property here is the Strong Disposability, which states: FREE DISPOSAL The property of free disposal holds if the absorption of any additional amounts of inputs without any reduction in outputs is always possible. Let P be Production Possibility Set: if (X,Y ) ∈ P and W ≥ X,Z ≤ Y then (W,Z) ∈ P. Let us note that such free disposal can only hold up to some extent in practice as W cannot be infinitely large—if so eventually one will not be able to disposal it freely. Assuming the strong disposal and convexity, then the standard PPS has the following form for desirable inputs and outputs P ({(Xj ,Yj )}) = { X ≥ X(λ) = n∑ j=1 λjXj ,Y ≤ Y (λ) = n∑ j=1 λjYj , λ ∈ S } , (1) where either S = {λj ≥ 0, j = 1,2, . . . , n} or S = {λj ≥ 0,∑nj=1 λj = 1} in the DEA litera- ture. 180 Ann Oper Res (2010) 173: 177–194 To handle undesirable inputs or outputs satisfactorily, one needs to modify the strong disposability—Ray (2004: 175) states that, without the assumption of weak disposability, undesirable outputs can be discarded at no cost which undermines the usefulness of the concept. Thus one has to extend the standard strong disposability to the cases of undesirable variables. There seems to exist several possible ways: One can directly extend the above strong disposability via using the same state- ment but with the preferences adopted for undesirable cases as explained in block one above. Formally this Extended Strong Disposability can be stated as: Let (X,Y ) = (XD,XU,YD,YU) ∈ P be desirable and undesirable inputs and outputs respectively, if WD ≥ XD,WU ≤ XU and ZD ≤ YD,ZU ≥ YU, then (WD,WU,ZD,ZU) ∈ P. There are many practical situations where such free disposability can be observed: taking a post-office for instance, letters with correct addresses are good inputs but those with in- correct addresses are bad ones. Therefore one can produce a given output with more good inputs and few bad inputs. Most electricity generators have pollution control systems, such as equipments to reduce sulfur dioxide in their production processes. Thus undesirable out- puts like sulfur dioxide can be “freely” increased, at least to some extent, by shutting down these pollution control systems. Similar examples can be found in service sectors where the desirable and undesirable outputs are numbers of served customers and received complaints respectively. If there are plenty of customers, then Extended Strong Disposability holds as it is possible to freely increase complaints without reducing numbers of serviced customers. Furthermore it will be seen later that many existing DEA models in fact use this type of Extended Strong Disposability to handle undesirable variables, although this seemingly has not been clearly explained in the DEA literature before. Again note that this kind of free disposability can only hold up to some extent, and often one needs to bound the desirable inputs and undesirable outputs in deriving DEA models. Then the corresponding PPS with convexity reads: PPS = { (XD,XU,YD,YU) : XD ≥ n∑ i=1 λjX D j ,X U ≤ n∑ i=1 λjX U j , YD ≤ n∑ i=1 λjY D j ,Y U ≥ n∑ i=1 λjY U j , n∑ i=1 λj = 1, λj ≥ 0 } . (2) It is very useful to observe that the above PPS can be equivalently formulated via regarding the undesirable inputs and outputs as desirable outputs and inputs respectively, and then applying the standard Strong Disposability. This fact can be summarized in the following theorem: Theorem 2.1 If the construction of the PPS is to avoid data transformations, then treating undesirable inputs and outputs as desirable outputs and inputs, and assuming the Strong Disposability are equivalent to an assumption of Extended Strong Disposability. Therefore this theorem provides a theoretical foundation to the approach of exchanging undesirable variables with desirable ones, which will be examined in more detail later on. In the next sections, we will show that many existing DEA models have in fact assumed Extended Strong Disposability. For instance, in many cases assuming Strong Disposability for transferred variables is just to assume the Extended Strong Disposability for the original Ann Oper Res (2010) 173: 177–194 181 variables. Let us further illustrate here that the model in Seiford and Zhu (2002) actually assumed Extended Strong Disposability and used the above PPS in the original variables. In that model they first used the transformation Y¯ Uj = −YUj + W , where YUj < W . Then they assumed the standard Strong Free Disposability and convexity. Thus the PPS with the new variables reads:{ (X,Y ) : X ≥ n∑ j=1 λjXj ,Y D ≤ n∑ j=1 λjY D j , Y¯ U ≤ n∑ j=1 λj Y¯ U j , n∑ j=1 λj = 1, λj ≥ 0 } . Then back to the original variables via Y¯ Uj = −YUj + W , the PPS reads:{ (X,Y ) : X ≥ n∑ j=1 λjXj ,Y D ≤ n∑ j=1 λjY D j ,Y U ≥ n∑ j=1 λjY U j , n∑ j=1 λj = 1, λj ≥ 0 } , where YU is bounded up by W . Thus one may say that this model is in fact based on Ex- tended Strong Disposability. Here the convexity plays a key role in deriving the equivalence of PPS if W �= 0. Therefore it follows that the above equivalence holds for ADD transfor- mation with convexity, CRS, IRS or DRS. On the other hand, if outputs are desirable but some of them are negative, then one may first apply ADD to change them into undesirable but positive variables, and then use Extended Strong Disposability. It is also possible to derive DEA models to handle undesirable data via replacing Strong Disposability of outputs by the assumptions that the outputs are weakly disposable while only the sub-vector of the desirable outputs is strongly disposable (e.g. Färe and Grosskopf 2004; Ray 2004). The basic idea is that the undesirable outputs may not be reduced freely alone but may be so with a proportional reductions of certain desirable outputs. One example is formally stated here: undesirable outputs are weakly disposable if (YD,YU) ∈ P (X) and 0 ≤ θ ≤ 1 then (θYD, θYU) ∈ P (X). In many environment applications, pollution can indeed be freely reduced if some desirable outputs decrease. Other forms are also possible, see Tone (2004). For instance, the PPS with CRS and desirable inputs reads (Färe and Grosskopf 2004):{ (YD,YU ,X) : YD ≤ n∑ j=1 λjY D j ,Y U = n∑ j=1 λjY U j ,X ≥ n∑ j=1 λjXj ,λj ≥ 0 } . Then one may derive suitable DEA models to handle environment applications. The ex- tended strong disposability and the weak disposability are in fact independent. Whether one should assume an Extended Strong Disposability or a Weak Disposability in a DEA model will much depend on the nature of the applications that it handles. Taking that the service example above for instance, if the market has already become very competitive then it is no longer possible to increase complaints freely. Then one should consider a weak disposability instead. However in this paper unless otherwise stated, we will always assume Extended Strong Disposability. 3 Performance metrics The third building block is performance measurement. In order to find whether or not a particular DMU (X0, Y0) is the “best” in P ({(Xj ,Yj )}), we need to use some performance 182 Ann Oper Res (2010) 173: 177–194 measurement to measure how much better of (X0, Y0) than its peers in P ({(Xj ,Yj )}) in terms of their performance. Performance measurement is often given by a merit function m(·, ·) which is strictly increasing (or decreasing) in the selected preferences, in the sense that if (W,Z) is better than (X,Y ), then m((W,Z), (X0, Y0)) > m((X,Y ), (X0, Y0)) (or m((W,Z), (X0, Y0)) < m((X,Y ), (X0, Y0))). Assume that (X(λ),Y (λ)) are all desirable and better than (X0, Y0), that is X(λ) = n∑ j=1 λjXj ≤ X0, Y (λ) = n∑ j=1 λjYj ≥ Y0, λ ∈ S. Then an additive merit function for both inputs and outputs is defined by m(X(λ),X0) = ∑ wis − i , (3) subject to n∑ j=1 xijλj + s−i = xi0, λ ∈ S, s−i ≥ 0, i = 1, ..,m, and m(Y(λ),Y0) = ∑ urs + r , (4) subject to n∑ j=1 yrjλj − s+r = yr0, λ ∈ S, s+r ≥ 0, r = 1, .., s respectively, where wi,ur are assigned positive weights. An almost radial merit function can be defined by using slacks as follows. Assume that Y0 > 0. Define m(Y(λ),Y0) = max θ + � s∑ r=1 s+r , (5) subject to Y (λ) − s+ = θY0, λ ∈ S, s+ ≥ 0, where � is a very small positive number. Then combining (3) and (4) together, the additive performance measurement (non- oriented) for DMU (X(λ),Y (λ)) is defined by m((X(λ),Y (λ)), (X0, Y0)) = max m∑ i=1 wis − i + s∑ r=1 urs + r , (6) subject to n∑ j=1 Xjλj + s− = X0, n∑ j=1 Yjλj − s+ = Y0, s+ ≥ 0, s− ≥ 0, λ ∈ S. Ann Oper Res (2010) 173: 177–194 183 If we use the additive merit function with the same weights for inputs, and almost radial merit function for outputs, then the output-oriented performance measurement for DMU (X(λ),Y (λ)) can be defined by m((X(λ),Y (λ)), (X0, Y0)) = max θ + � ( m∑ i=1 s−i + s∑ r=1 s+r ) , (7) subject to n∑ j=1 Xjλj + s− = X0, n∑ j=1 Yjλj − s+ = θY0, s+ ≥ 0, s− ≥ 0, λ ∈ S. Then whether or not DMU (X0, Y0) is efficient depends on if one can find a better virtual DMU in the production possibility set P ({(Xj ,Yj )}), and this may be found by solving a mathematical programming problem like (GDEA): (min)maxm((X,Y ), (X0, Y0)), (8) subject to (X,Y ) ∈ P ({(Xj ,Yj )}), where min or max will be used depending whether or not the merit function is decreasing or increasing respectively. This is essentially a DEA model. If the maximum is more than m((X0, Y0), (X0, Y0)) then there exists an λ0 ∈ S such that the virtual DMU (X(λ0), Y (λ0)) is better than (X0, Y0) in the defined preference. Thus (X0, Y0) is not efficient, and otherwise it is efficient. We will use these ideas in the following sections to formulate some DEA models with undesirable inputs and/or outputs. 4 Slack based DEA models In this section we use the additive merit functions. We make no assumption as to whether the input or output data are positive or negative. We first examine the case with desirable inputs and outputs as a starting point. Suppose that DMU0 is to be evaluated. It follows from the discussion about GDEA above, we have the following linear programming problem to find whether DMU0 is efficient or not: max m∑ i=1 wis − i + s∑ r=1 urs + r , (9) subject to n∑ j=1 Xjλj + s− = X0, n∑ j=1 Yjλj − s+ = Y0, λj ≥ 0, s−, s+ ≥ 0, λ ∈ S. where wi,ur are the positive weights. The values s−i and s+r identify the amounts of extra performance that the evaluated system should be able to produce if running efficiently. If they are zero, DMU0 is classified to be efficient by this DEA model. It follows that if a unit 184 Ann Oper Res (2010) 173: 177–194 is efficient for a particular set of non-zero weights (i.e. none of the weights is zero), then the unit is efficient for any set of non-zero weights. Therefore we only consider non-zero weight sets here. If we treat all the inputs and outputs equally then all the weights should be equal. Equivalently we can take all the weights to be one. Then the above is the well known Additive Model of DEA. It is clear that the standard Strong Disposability is assumed in the above standard model. For the case with undesirable inputs and outputs, we will assume tha