Ann Oper Res (2010) 173: 177–194DOI 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 outputshave 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 thispaper we present a systematic investigation on model building of DEA without transferringundesirable data. We first describe the disposability assumptions and a number of differentperformance measures in the presence of undesirable inputs and outputs, and then discussdifferent combinations of the disposability assumptions and the metrics. This approach leadsto a unified presentation of several classes of DEA models with undesirable inputs and/oroutputs.

Keywords Data Envelopment Analysis · Undesirable inputs and outputs · ExtendedStrong Disposability

1 Introduction

Since the introduction of the DEA in 1978, it has been widely used in efficiency analysis ofmany business and industry applications. Excellent literature surveys can be found in, forinstance, Seiford (1996) and Cooper et al. (2004). The most well-known DEA models arethe CCR model (Charnes et al. 1978) , the BCC model (Banker et al. 1984) , the Additivemodel (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 reviewsand suggestions, which lead to significant improvements of this paper.

W.B. Liu (�)Kent Business School, University of Kent, Canterbury, CT2 7PE, UKe-mail: W.B.Liu@kent.ac.uk

W. MengSchool of Public Administration, East China Normal University, Shanghai, 200062, China

X.X. Li · D.Q. ZhangInstitute 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 existundesirable 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 asfollows:

An intuitive reaction is to apply some transformations. The most acceptable one isf (U) = −U ; the so called the ADD approach suggested by Koopmans (1951). Then theundesirable inputs or outputs will become desirable after this transformation. However, thedata may then become negative, and it is not straightforward to define efficiency scores fornegative 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 onlyranking 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 nonlineartransformation, its behaviors are even more complicated (Scheel 1998). Thus how to prop-erly select a suitable transformation is very much case-dependent. The approaches basedon data-transformation may unexpectedly produce adverse results as discussed in Liu andSharp (1999).

There also exist many approaches that can avoid data transformation. For example, onemay 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 approachis 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 efficientDMUs wish to minimize desirable inputs and undesirable outputs, and to maximize desir-able outputs and undesirable inputs. If one only wishes to investigate operational efficiencyfrom this point of view, there is no need to distinguish between inputs and outputs, but onlyminimum and maximum. We will further extend this approach in this work and discuss itsrelationship with other approaches. Related to ADD, there are several recent papers on DEAmodels 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 closelyrelated to the weighted additive models in order to measure proportional gains from theslacks, which are always positive. It is important to realize that the additive models are ableto handle negative data. After some simple modifications, they can easily handle undesirablecases as well. These will be discussed later on.

Our investigation pays more attention to theoretical aspects of these issues. Our mainidea is to examine them within the general framework proposed in Liu et al. (2006), wheresome 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 anumber of different performance measures or metric functions with attractive properties inthe presence of undesirable inputs and outputs and negative data. We will discuss differentcombinations of the free disposability assumptions and the metrics. This approach leads toa 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 theseinvestigations 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 assumptionsin DEA, which form the foundation of this paper. In Sect. 3 we summarize some metricsused in DEA models following the ideas used in Liu et al. (2006). In Sect. 4, we examineslack 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 negativedata. In Sects. 5–6, we extend the ideas in Liu and Sharp (1999) to formulate general DEAmodels for undesirable data using the radial and Russell efficiency measures. Conclusionsare 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 theinputs 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 specificpurposes. In order to be able to evaluate DMUs, we first have to know our “preference” in theinput-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 manyreal applications where different preferences are indeed useful, see Liu et al. (2006) andZhang et al. (2009) for some examples. Let X = (x1, . . . , xl), Y = (y1, . . . , yl) ∈ Rl . Then inPareto preference, X ≥ Y if and only if xi ≥ yi (i = 1,2, . . . , l). For DMUs with desirableinputs and outputs, DMU1 (X1, Y1) is better than DMU2 (X2, Y2) if X1 ≤ X2, Y1 ≥ Y2 inPareto 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 identifythe “best” DMUs. Most of the preference in business applications is too weak to pick upthe “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 severalassumptions are made on the PPS, such as convexity and no-free-lunch. The most relevantproperty here is the Strong Disposability, which states:

FREE DISPOSAL

The property of free disposal holds if the absorption of any additional amounts of inputswithout 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 cannotbe infinitely large—if so eventually one will not be able to disposal it freely. Assuming thestrong disposal and convexity, then the standard PPS has the following form for desirableinputs 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,∑n

j=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 strongdisposability—Ray (2004: 175) states that, without the assumption of weak disposability,undesirable outputs can be discarded at no cost which undermines the usefulness of theconcept. Thus one has to extend the standard strong disposability to the cases of undesirablevariables. 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 oneabove. Formally this Extended Strong Disposability can be stated as: Let (X,Y ) =(XD,XU,Y D,Y U) ∈ P be desirable and undesirable inputs and outputs respectively,

if WD ≥ XD,WU ≤ XU and ZD ≤ Y D,ZU ≥ Y U, then (WD,WU,ZD,ZU) ∈ P.

There are many practical situations where such free disposability can be observed: takinga 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 goodinputs and few bad inputs. Most electricity generators have pollution control systems, suchas 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 downthese pollution control systems. Similar examples can be found in service sectors where thedesirable and undesirable outputs are numbers of served customers and received complaintsrespectively. If there are plenty of customers, then Extended Strong Disposability holds asit 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 ofExtended Strong Disposability to handle undesirable variables, although this seemingly hasnot been clearly explained in the DEA literature before. Again note that this kind of freedisposability can only hold up to some extent, and often one needs to bound the desirableinputs and undesirable outputs in deriving DEA models.

Then the corresponding PPS with convexity reads:

PPS ={

(XD,XU,Y D,Y U) : XD ≥n∑

i=1

λjXDj ,XU ≤

n∑i=1

λjXUj ,

Y D ≤n∑

i=1

λjYDj ,Y U ≥

n∑i=1

λjYUj ,

n∑i=1

λj = 1, λj ≥ 0

}. (2)

It is very useful to observe that the above PPS can be equivalently formulated via regardingthe undesirable inputs and outputs as desirable outputs and inputs respectively, and thenapplying the standard Strong Disposability. This fact can be summarized in the followingtheorem:

Theorem 2.1 If the construction of the PPS is to avoid data transformations, then treatingundesirable inputs and outputs as desirable outputs and inputs, and assuming the StrongDisposability are equivalent to an assumption of Extended Strong Disposability.

Therefore this theorem provides a theoretical foundation to the approach of exchangingundesirable 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 assumedExtended Strong Disposability. For instance, in many cases assuming Strong Disposabilityfor 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) actuallyassumed Extended Strong Disposability and used the above PPS in the original variables. Inthat model they first used the transformation Y U

j = −Y Uj + W , where Y U

j < W . Then theyassumed the standard Strong Free Disposability and convexity. Thus the PPS with the newvariables reads:{

(X,Y ) : X ≥n∑

j=1

λjXj ,YD ≤

n∑j=1

λjYDj , Y U ≤

n∑j=1

λj YUj ,

n∑j=1

λj = 1, λj ≥ 0

}.

Then back to the original variables via Y Uj = −Y U

j + W , the PPS reads:{(X,Y ) : X ≥

n∑j=1

λjXj ,YD ≤

n∑j=1

λjYDj ,Y U ≥

n∑j=1

λjYUj ,

n∑j=1

λj = 1, λj ≥ 0

},

where Y U 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 equivalenceof 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 butsome of them are negative, then one may first apply ADD to change them into undesirablebut positive variables, and then use Extended Strong Disposability.

It is also possible to derive DEA models to handle undesirable data via replacing StrongDisposability of outputs by the assumptions that the outputs are weakly disposable whileonly the sub-vector of the desirable outputs is strongly disposable (e.g. Färe and Grosskopf2004; Ray 2004). The basic idea is that the undesirable outputs may not be reduced freelyalone but may be so with a proportional reductions of certain desirable outputs. One exampleis formally stated here: undesirable outputs are weakly disposable if

(Y D,Y U) ∈ P (X) and 0 ≤ θ ≤ 1 then (θY D, θY U) ∈ P (X).

In many environment applications, pollution can indeed be freely reduced if some desirableoutputs decrease. Other forms are also possible, see Tone (2004). For instance, the PPS withCRS and desirable inputs reads (Färe and Grosskopf 2004):{

(Y D,Y U ,X) : Y D ≤n∑

j=1

λjYDj ,Y U =

n∑j=1

λjYUj ,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. Whetherone should assume an Extended Strong Disposability or a Weak Disposability in a DEAmodel will much depend on the nature of the applications that it handles. Taking thatthe service example above for instance, if the market has already become very competitivethen it is no longer possible to increase complaints freely. Then one should consider a weakdisposability instead. However in this paper unless otherwise stated, we will always assumeExtended Strong Disposability.

3 Performance metrics

The third building block is performance measurement. In order to find whether or not aparticular 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 )}) interms of their performance. Performance measurement is often given by a merit functionm(·, ·) which is strictly increasing (or decreasing) in the selected preferences, in the sensethat if (W,Z) is better than (X,Y ), then m((W,Z), (X0, Y0)) > m((X,Y ), (X0, Y0)) (orm((W,Z), (X0, Y0)) < m((X,Y ), (X0, Y0))). Assume that (X(λ),Y (λ)) are all desirableand 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 + ε

s∑r=1

s+r , (5)

subject to

Y (λ) − s+ = θY0, λ ∈ S, s+ ≥ 0,

where ε 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)) = maxm∑

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 almostradial merit function for outputs, then the output-oriented performance measurement forDMU (X(λ),Y (λ)) can be defined by

m((X(λ),Y (λ)), (X0, Y0)) = max θ + ε

(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 virtualDMU in the production possibility set P ({(Xj ,Yj )}), and this may be found by solving amathematical 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 decreasingor increasing respectively. This is essentially a DEA model. If the maximum is more thanm((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 otherwiseit is efficient. We will use these ideas in the following sections to formulate some DEAmodels 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 whetherthe input or output data are positive or negative. We first examine the case with desirableinputs and outputs as a starting point. Suppose that DMU0 is to be evaluated. It follows fromthe discussion about GDEA above, we have the following linear programming problem tofind whether DMU0 is efficient or not:

maxm∑

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 extraperformance that the evaluated system should be able to produce if running efficiently. Ifthey 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), thenthe unit is efficient for any set of non-zero weights. Therefore we only consider non-zeroweight sets here. If we treat all the inputs and outputs equally then all the weights should beequal. Equivalently we can take all the weights to be one. Then the above is the well knownAdditive 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 that the inputs and outputof j -th unit can be decomposed into

Xj =(

XDIj

XUIj

), Yj =

(Y DO

j

Y UOj

),

with {DI}, {UI}, {DO}, {UO} being fixed index sets independent of j , such that XDIj ,YDO

j

are desirable inputs and outputs, and XUIj ,YUO

j are undesirable inputs and outputs. For in-stance, DI = {1,2}, UI = {3,4, ..,m}, DO = {1,2,3}, UO = {4,5, . . . , s} so that |DI| = 2,|UI| = m − 2, |DO| = 3, |UO| = s − 3. For example, we assume m = s = 5 below,

X =

⎛⎜⎜⎜⎜⎜⎜⎝

xDI1

xDI2

xUI3

xUI4

xUI5

⎞⎟⎟⎟⎟⎟⎟⎠

, Y =

⎛⎜⎜⎜⎜⎜⎜⎝

yDO1

yDO2

yDO3

yUO4

yUO5

⎞⎟⎟⎟⎟⎟⎟⎠

.

In such a case, let

X =(

XDI

XUI

), Y =

(Y DO

Y UO

).

We will assume Extended Strong Disposability. Note that maximizing desirable outputs (un-desirable inputs) and minimizing undesirable outputs (desirable inputs) can all be achievedby just maximizing the correspondent slack measurements. Then the extra performance ofthe virtual units can be found by solving the following linear programming problem:

maxwDIsDI + wUIs

UI + wDOsDO + wUOsUO, (10)

subject to

n∑j=1

Y DOj λj − sDO = Y DO

0 ,

n∑j=1

Y UOj λj + sUO = Y UO

0 ,

n∑j=1

XDIj λj + sDI = XDI

0 ,

n∑j=1

XUIj λj − sUI = XUI

0 ,

sDI, sUI, sDO, sUO ≥ 0, λ ∈ S,

where wDI,wUI,wDO,wUO are (strictly) positive weight vectors. Then DMU0 is efficient ifand only if the maximum is zero.

However the above DEA model is not units invariant, and cannot produce efficiencyscores directly. For the desirable nonnegative inputs and outputs one can use the Tone (2001)

Ann Oper Res (2010) 173: 177–194 185

formula:

minρ = 1 − 1m

∑m

i=1 s−i /xio

1 + 1s

∑s

r=1 s+r /yro

, (11)

subject to

X0 =n∑

j=1

Xjλj + s−, Y0 =n∑

j=1

Yjλj − s+,

λ ∈ S, s− ≥ 0, s+ ≥ 0.

The model was shown to be units invariant and the scores to be between [0,1]. Later wewill see that the division of X0, Y0 for the slacks defined as above is in fact not the mostreasonable approach, and should be changed. If there are undesirable inputs and outputs,but all the inputs and outputs are nonnegative, then the above DEA model can be readilyextended into the following modified form:

minρ = 1 − 1|DI|+|UO| (

∑sDIi /xDI

i0 + ∑sUOi /yUO

i0 )

1 + 1|DO|+|UI| (

∑sDOr /yDO

r0 + ∑sUIr /xUI

r0 ), (12)

subject ton∑

j=1

Y DOj λj − sDO = Y DO

0 ,

n∑j=1

Y UOj λj + sUO = Y UO

0 ,

n∑j=1

XDIj λj + sDI = XDI

0 ,

n∑j=1

XUIj λj − sUI = XUI

0 ,

sDI, sUI, sDO, sUO ≥ 0, λ ∈ S.

It can be similarly proved that this DEA model is units invariant, and the scores are indeedbetween [0,1]. Although this model is nonlinear, it can be easily transferred into a linear one(Tone 2001). However this model is not translation invariant. For the additive measurement,it follows from Theorem 2.1 that regarding the undesirable inputs and outputs as desirableoutputs and inputs and then applying the Strong Disposability will lead to the same models.In addition, it should be pointed out that one can assume a weak disposability instead, andderive different slack based DEA models as explained in Tone (2004).

The general case where there are negative data for inputs or outputs is non-trivial anddeserves further discussions, as then the standard measure used above can be negative. Al-though technically it should be always possible to transfer such cases and then apply theabove non-negative undesirable models, there are many reasons that people still prefer touse negative data in some applications, see Liu and Sharp (1999) and Sharp et al. (2006) forsome discussions. Below we assume that all the inputs and outputs are desirable but can benegative. In Silva Portela et al. (2004), the SP Range was introduced, and based on it andthe directional distance function approach, the Range Directional Model was formulated as

maxβ, (13)

subject to

n∑j=1

Xjλj ≤ X0 − βP −0 ,

n∑j=1

Yjλj ≥ Y0 + βP +0 , β ≥ 0, λ ∈ S,

186 Ann Oper Res (2010) 173: 177–194

where the SP Range is defined by

P −0 = X0 − Zi, with Zi = min

jxij , P +

0 = Wr − Y0, with Wr = maxj

yrj

to handle negative data. Then 1 − β gives efficiency scores. Although here we do not adoptsuch approaches, the SP Range idea nevertheless seems to be useful in deriving merit func-tions that can be directly applied to negative data. In fact one only needs to replace the X0, Y0

in the Tone’s formula with P −0 ,P +

0 to have the following DEA model:

minρ = 1 − 1m

∑m

i=1 s−i /p−

io

1 + 1s

∑s

r=1 s+r /p+

ro

, (14)

subject to

X0 =n∑

j=1

Xjλj + s−, Y0 =n∑

j=1

Yjλj − s+,

s− ≥ 0, s+ ≥ 0, λ ∈ S.

When p−i0,p

+r0 are zero, the corresponding terms will be dropped from the numera-

tor/denominator, respectively. Let us show that the above measure is in the range [0,1].Note

s− = X0 − Xλ ≤ maxλ

(X0 − Xλ) = X0 − maxλ

(Xλ).

As minλ Xλ ≥ Z, where Zi = minj (xij ), and P −0 = X0 − Z, therefore s− ≤ P −

0 . Also,

s+ = Yλ − Y0 ≤ maxλ

(Yλ − Y0) = maxλ

(Yλ) − Y0.

As maxλ Yλ ≤ M , where Mr = maxj (yrj ), and P +0 = M − Y0, therefore s+ ≤ P +

0 . There-fore, the efficiency measure in Model (14) is in the range [0,1]. It is clear that this DEAmodel is not only units invariant but also translation invariant. The above model is applica-ble to the case where all inputs and outputs are desirable, but may be negative. Followingthe above proof, it is clear that the general Model (12) can be similarly modified so that itcan handle either desirable and undesirable or positive and negative data. We neverthelessomit the details here.

5 DEA models with radial measurement

Now assume that all the components of inputs and outputs are positive.Again we start from the desirable case. Using the output-oriented almost ratio measure-

ment, we then have the output-oriented CCR or BCC models

maxθ,s+,s−,λ

θ + ε

(m∑

i=1

s−i +

s∑r=1

s+r

), (15)

subject to

n∑j=1

Xjλj + s− = X0,

n∑j=1

Yjλj − s+ = θY0, s+ ≥ 0, s− ≥ 0, λ ∈ S.

Ann Oper Res (2010) 173: 177–194 187

For the general case we again decompose the inputs and outputs into desirable and undesir-able parts:

Xj =(

XDIj

XUIj

), Yj =

(Y DO

j

Y UOj

), X =

(XDI

XUI

), Y =

(Y DO

Y UO

).

We will assume Extended Strong Disposability. Unfortunately using the radial based mea-surement, one cannot maximize desirable outputs and minimize undesirable outputs, orminimize desirable inputs and maximize undesirable inputs at the same time. Let us firstexamine two cases where either inputs or outputs are all desirable, which are often met inapplications. In such cases we can apply the almost radial measure to desirable inputs oroutputs, and the additive measure to the mixed variables as it can be used to maximize de-sirable outputs (undesirable inputs) and minimize undesirable outputs (desirable inputs) atthe same time. Then one derives the following DEA models, of which the output-orientedDEA model reads:

max θ + ε(|sDI | + |sUI | + |sDO|), (16)

subject to

n∑j=1

Y DOj λj − sDO = θY DO

0 ,

n∑j=1

XUIj λj − sUI = XUI

0 ,

n∑j=1

XDIj λj + sDI = XDI

0 , λ ∈ S, sDI, sUI, sDO ≥ 0, θ ≥ 1,

and the input-oriented DEA model reads:

min θ − ε(|sDI | + |sDO| + |sUO|), (17)

subject to

n∑j=1

Y DOj λj − sDO = Y DO

0 ,

n∑j=1

Y UOj λj + sUO = Y UO

0 ,

n∑j=1

XDIj λj + sDI = θXDI

0 , λ ∈ S, sDI, sDO, sUO ≥ 0, 0 ≤ θ ≤ 1.

Let us now examine the general case. If we wish to use a single ratio to measure the radialextension or contraction for both desirable and undesirable part of inputs or outputs, thenwe may have to deal with DEA models with objective functions like θ + 1/θ . Thus it is dif-ficult to directly combine Extended Strong Disposability with the standard radial measure inDEA models while keeping the original input-output orientation. Now let us recall that byassuming Extended Strong Disposability we can regard the undesirable inputs as desirableoutputs, and the undesirable outputs as desirable inputs, and then use the standard StrongDisposability. From this point of view, we can derive DEA models of radial type for unde-sirable inputs and outputs, and the extra performance of the virtual units can be found bysolving the following (new)output-oriented DEA model:

max θ + ε(|sDI | + |sUI | + |sDO| + |sUO|), (18)

188 Ann Oper Res (2010) 173: 177–194

subject to

n∑j=1

Y DOj λj − sDO = θY DO

0 ,

n∑j=1

XUIj λj − sUI = θXUI

0 ,

n∑j=1

Y UOj λj + sUO = Y UO

0 ,

n∑j=1

XDIj λj + sDI = XDI

0 ,

λ ∈ S, sDI, sUI, sDO, sUO ≥ 0, θ ≥ 1.

Similarly, we can write down the following (new)input-oriented DEA model with undesir-able inputs and/or outputs.

min θ − ε(|sDI | + |sUI | + |sDO| + |sUO|), (19)

subject to

n∑j=1

Y DOj λj − sDO = Y DO

0 ,

n∑j=1

XUIj λj − sUI = XUI

0 ,

n∑j=1

Y UOj λj + sUO = θY UO

0 ,

n∑j=1

XDIj λj + sDI = θXDI

0 ,

λ ∈ S, sDI, sUI, sDO, sUO ≥ 0, 0 ≤ θ ≤ 1.

Combining the above two models, we have the following non-oriented DEA model withundesirable inputs and/or outputs.

minα/β − ε(|sDI | + |sUI | + |sDO| + |sUO|), (20)

subject to

n∑j=1

Y DOj λj − sDO = βY DO

0 ,

n∑j=1

XUIj λj − sUI = βXUI

0 ,

n∑j=1

Y UOj λj + sUO = αY UO

0 ,

n∑j=1

XDIj λj + sDI = αXDI

0 ,

λ ∈ S, β ≥ 1, 0 ≤ α ≤ 1.

In Model (20), extra performance from both inputs and outputs has been counted incomputing the efficiency score. If this score is unit and all the slacks are zero, then theDMU is efficient. Otherwise, it is inefficient. It is interesting to notice that this model is verysimilar to the Banker MPSS model in studying scale efficiency, see Banker (1984).

We now discuss some relationship between the approaches used in Seiford and Zhu(2002) and this section. In their model, all the inputs are assumed to be desirable. Theyfirst used the output transformation Y U

j = −Y Uj + w, and then the Strong Disposability with

the radial measure to derive the model:

max θ, (21)

subject to

Ann Oper Res (2010) 173: 177–194 189

n∑i=1

λjxij ≤ xi0, i = 1, . . . ,m,

n∑i=1

λj yUrj ≥ θyU

r0, r = 1, . . . , l,

n∑i=1

λjyDrj ≥ θyD

r0, r = l, . . . , s,

n∑i=1

λj = 1, λj ≥ 0, j = 1, .., n.

We know that this model in fact uses Extended Strong Disposability as discussed in Sect. 2.Now suppose that we wish to adopt a desirable output-oriented measure in the above model;that is to say we maximize the performance measure of desirable outputs (like total electric-ity generated) while asking the undesirable ones (like pollution) under control. This seemsto be a sensible strategy as usually it is the desirable outputs that we care most. Then we willarrive at the following DEA model:

max θ,

subject ton∑

i=1

λjxij ≤ xi0, i = 1, . . . ,m,

n∑i=1

λj yUrj ≥ yU

r0, r = 1, . . . , l,

n∑i=1

λjyDrj ≥ θyD

r0, r = l, . . . , s,

n∑i=1

λj = 1, λj ≥ 0, j = 1, .., n.

Back to the original data with the transformation Y Uj = −Y U

j + w, then we have∑n

i=1 λj yUrj = ∑n

i=1 λj (−yUrj + wr) ≥ yr0 = −yU

r0 + wr . Since∑n

i=1 λj = 1, hence it be-comes the following DEA model:

max θ,

subject ton∑

i=1

λjxij ≤ xi0, i = 1, . . . ,m,

n∑i=1

λjyUrj ≤ yU

r0, r = 1, . . . , l,

n∑i=1

λjyDrj ≥ θyD

r0, r = l, . . . , s,

n∑i=1

λj = 1, λj ≥ 0, j = 1, .., n.

190 Ann Oper Res (2010) 173: 177–194

Clearly it is just Model (18) with an empty UI .We end this section by extending discussions to some measures related to radial one. Let

us still assume that all the inputs are desirable for simplicity. Using the directional distanceused in Färe and Grosskopf (2004), it is clear that one can maximize desirable outputs andminimize undesirable outputs, or minimize desirable inputs and maximize undesirable in-puts at the same time. For example assuming Extended Strong Disposability and CRS, onethen has the following DEA model:

max θ, (22)

subject ton∑

j=1

Y DOj λj ≥ Y DO

0 + θGDO,

n∑j=1

Y UOj λj ≤ Y UO

0 − θGUO,

n∑j=1

Xjλj ≤ X0, λ ≥ 0, θ ≥ 0,

where GDO and GUO are selected references. It is interesting to note that if we assume theweak disposability discussed in Sect. 2 and the subsequent PPS instead, then the modelwould become

max θ, (23)

subject ton∑

j=1

Y DOj λj ≥ Y DO

0 + θGDO,

n∑j=1

Y UOj λj = Y UO

0 − θGUO,

n∑j=1

Xjλj ≤ X0, λ ≥ 0, θ ≥ 0,

which replaces the inequality for Y UO in Model (22) with the equality. In applications, extraassumptions like null-joint of desirable and undesirable variables may be needed, see Färeand Grosskopf (2004) for the details.

If one uses θ and 1θ

to measure performance of the desirable and undesirable outputsrespectively and assume Extended Strong Disposability, then one will have the nonlinearmodel in Färe et al. (1989).

The above analysis confirms our observation that most existing DEA models handle un-desirable variables by combing suitable disposability assumptions and performance mea-sures.

6 DEA models with Russell measurement

In the standard CCR and BCC DEA models, the radial measurement is used to measure thechanges of inputs or outputs. Consequently the radial proportional reduction of inputs orextension of outputs is considered to be dominated in these models. This is indeed true inmany economics systems (e.g. Farrell 1957; Shephard 1970). However this hypothesis maybe questionable in studies of input-output relationships in some applications, e.g., scientificresearch. For instance, it is apparent from the data in Meng et al. (2006) that the ratios ofchange in investment and staff of basic research in China are very different during the last

Ann Oper Res (2010) 173: 177–194 191

decade. Therefore, the Enhance Russell Measurement (ERM) DEA can be very useful indetecting changes of components of inputs or outputs in such cases, see Färe and Lovell(1978). The standard output-oriented model with Russell measurement reads:

maxs∑

r=1

θr/s + ε|s−|, (24)

subject to

n∑j=1

yrjλj = θryr0, r = 1, .., s,

n∑j=1

Xjλj + s− = X0, λ ∈ S, s− ≥ 0, θr ≥ 1.

Here we assume that the outputs are positive.Similar to the radial measure, it seems difficult to directly combine Extended Strong

Disposability with the Russell measure in DEA modules while keeping the original input-output orientation. Again for the general case, we can regards undesirable inputs and outputsas desirable outputs and inputs as discussed in Sect. 5. For instance, the output-orientedmodel then reads:

max∑

k∈DO∪UI

θk/(|DO| + |UI|) + ε(|sDI | + |sUO|), (25)

subject to

n∑j=1

yDOrj λj = θry

DOr0 , r ∈ DO,

n∑j=1

xUIij λj = θix

UIi0 , i ∈ UI

n∑j=1

Y UOj λj + sUO = Y UO

0 ,

n∑j=1

XDIj λj + sDI = XDI

0 ,

λ ∈ S, sDI, sUO ≥ 0, θr , θi ≥ 1.

We also have the following non-oriented model:

minh =( ∑

i∈DI∪UO

αi/(|DI| + |UO|))/( ∑

r∈DO∪UI

βr/(|DO| + |UI|))

, (26)

subject to

n∑j=1

yDOrj λj = βry

DOr0 , r ∈ DO,

n∑j=1

xUIrj λj = βrx

UIr0 , r ∈ UI,

n∑j=1

yUOij λj = αiy

UOi0 , i ∈ UO,

n∑j=1

xDIij λj = αix

DIi0 , i ∈ DI,

λ ∈ S, 0 ≤ αi ≤ 1, βr ≥ 1.

Now we state an interesting relationship between the models using the slack measure andRussell measure.

192 Ann Oper Res (2010) 173: 177–194

Proposition 1 Models (12) and (26) are equivalent.

Proof If Model (12) is feasible, then there exist an optimal value ρ∗ and solutionssDI∗i , sDO∗

r , sUO∗i , sUI∗

r , λ∗j ≥ 0. Now define the following variables:

αi = xDIi0 − sDI∗

i

xDIi0

if i ∈ DI, or αi = yUOi0 − sUO∗

i

yUOi0

if i ∈ UO,

and

βr = yDOr0 + sDO∗

r

yDOr0

if r ∈ DO, or βr = xUIr0 + sUI∗

r

xUIr0

if r ∈ UI.

Then it is easy to check that αi ≤ 1, βr ≥ 1 and they are feasible solutions to Model (26).Furthermore it can be checked that the objective value is the same as Model (12). Hence(αi , βr , λ

∗j ) is one feasible solution to Model (25), and h∗ ≤ ρ∗.

If Model (26) is feasible, then there exist an optimal value h∗ and solutions α∗i ≤ 1,

β∗r ≥ 1. Hence define sDI

i , sDOr , sUO

i , sUIr , such that

α∗i = xDI

i0 − sDIi

xDIi0

if i ∈ DI, or α∗i = yUO

i0 − sUOi

yUOi0

if i ∈ UO,

and

β∗r = yDO

r0 + sDOr

yDOr0

if r ∈ DO, or β∗r = xUI

r0 + sUIr

xUIr0

if r ∈ UI.

Then it can be shown that they are feasible solutions to Model (12), and Model (26) has thesame objective function as Model (12). That is, (sDI

i , sDOr , sUO

i , sUIr ) is one feasible solution

of Model (12), and hence h∗ ≥ ρ∗. Therefore we conclude that ρ∗ = h∗. �

Again if using a directional measure like the one used in Seiford and Zhu (2005), thenone can directly combine it with the Extended Strong Disposability or indeed the WeakDisposability to derive the DEA models presented in Seiford and Zhu (2005).

7 Conclusion

In this paper we discuss a general approach of deriving DEA models to handle undesir-able inputs and outputs, which combines the disposability assumptions and the performancemetrics. We first extend the standard Strong Disposability to Extended Strong Disposabilitywhere undesirable variables present. Then we reveal that assuming Extended Strong Dis-posability is equivalent to treating undesirable inputs and outputs as desirable outputs andinputs while assuming the standard Strong Disposability in forming the PPS. By combiningthis approach with different performance measurements, we show that many existing ap-proaches in fact assumed the Extended Strong Disposability, and give a unified presentationof several classes of DEA models with undesirable inputs and/or outputs.

Ann Oper Res (2010) 173: 177–194 193

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