Efficiency aggregation with enhanced Russell measures in data envelopment analysis

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0038-0121/$ - se

doi:10.1016/j.se

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Socio-Economic Planning Sciences 41 (2007) 1–21

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Efficiency aggregation with enhanced Russellmeasures in data envelopment analysis

W.W. Coopera,�, Zhimin Huangb, Susan X. Lib, Barnett R. Parkerc, Jesus T. Pastord

aThe Red McCombs School of Business, University of Texas at Austin, 1 University Station B6500, Austin, TX 78712-1174, USAbSchool of Business, Adelphi University, Garden City, Long Island, NY 11530, USA

cSchool of Business, Pfeiffer University, Misenheimer, NC 28109, USAdCentro de Investigacion Operativa, Universidad Miguel Hernandez, Edificio Torretamarit, Avda. Ferrocarril s/n,

03202-Elche (Alicante), Spain

Available online 23 June 2006

Abstract

In aggregation for data envelopment analysis (DEA), a jointly determined aggregate measure of output and input

efficiency is desired that is consistent with the individual decision making unit measures. An impasse has been reached in

the current state of the literature, however, where only separate measures of input and output efficiency have resulted from

attempts to aggregate technical efficiency with the radial measure models commonly employed in DEA. The latter

measures are ‘‘incomplete’’ in that they omit the non-zero input and output slacks, and thus fail to account for all

inefficiencies that the model can identify. The Russell measure eliminates the latter deficiency but is difficult to solve in

standard formulations. A new approach has become available, however, which utilizes a ratio measure in place of the

standard formulations. Referred to as an enhanced Russell graph measure (ERM), the resulting model is in the form of a

fractional program. Hence, it can be transformed into an ordinary linear programming structure that can generate an

optimal solution for the corresponding ERM model. As shown in this paper, an aggregate ERM can then be formed with

all the properties considered to be desirable in an aggregate measure—including jointly determined input and output

efficiency measures that represent separate estimates of input and output efficiency. Much of this paper is concerned with

technical efficiency in both individual and system-wide efficiency measures. Weighting systems are introduced that extend

to efficiency-based measures of cost, revenue, and profit, as well as derivatives such as rates of return over cost. The

penultimate section shows how the solution to one model also generates optimal solutions to models with other objectives

that include rates of return over cost and total profit. This is accomplished in the form of efficiency-adjusted versions of

these commonly used measures of performance.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Efficiency; Data envelopment analysis; Fractional programming; Individual entity efficiencies; Aggregate efficiencies

e front matter r 2006 Elsevier Ltd. All rights reserved.

ps.2006.03.001

ing author. Fax: +1512 471 0587.

resses: [email protected] (W.W. Cooper), [email protected] (Z. Huang), [email protected] (S.X. Li),

u (B.R. Parker), [email protected] (J.T. Pastor).

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dx.doi.org/10.1016/j.seps.2006.03.001

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ARTICLE IN PRESSW.W. Cooper et al. / Socio-Economic Planning Sciences 41 (2007) 1–212

1. Introduction

Blackorby and Russell [1] examine conditions under which ‘‘technical efficiency’’ measures, as derived indata envelopment analysis (DEA), can be satisfactorily aggregated. See also Russell [2]. The authors of [1]report the following:

Our results are discouraging, indicating that very strong restrictions on the technology and/or the efficiencyindex itself are required to enable consistent aggregation or disaggregation [e.g., in aggregating acrossplants to obtain a company-wide measure] y. Perhaps even more disturbing is the fact that the principalindices proposed by Debreu [3] and Farrell [4], by Fare and Lovell [5] and Zieschang [6] cannot satisfyconditions for any technologies, even linear ones.

These results led Blackorby and Russell to confine their attention to separate ‘‘indices’’ of output and inputefficiency; the search for jointly determined output and input efficiencies has thus been left at an impasse. Suchaggregate measures can be important at the individual firm level, as well as at the economy-wide levelconsidered in traditional economics. For example, they could be of interest to the comptroller of a fast foodchain for use in measuring both system, and individual location, performances. Here, we propose alternateapproaches and measures that differ from those of Blackorby and Russell [1], and also satisfy those criteriagenerally considered decisive in developing such measures.

To begin, we note that the approaches taken by Blackorby and Russell [1], as derived from the classicalDEA literature, use radial measures of efficiency. They encompass only outputs or inputs depending onwhether an ‘‘output oriented’’ or an ‘‘input oriented’’ approach is used. Hence, the results obtained byBlackorby and Russell—as well as others we will cite—are not surprising. However, radial measures are notthe only possibilities. We thus turn to other models, and/or combinations of models and measures, todetermine what might be accomplished.

Before proceeding, we note that others have also studied the aggregation problem. For instance, Fare andZelenyuk [7] used ‘‘value oriented’’ approaches to establish necessary and sufficient conditions for industryefficiency when measured as a sum of the maximal revenues of individual firms. They then showed thatefficiency will be achieved in an aggregate revenue measure if and only if it is achieved for every one of theunderlying decision making units (DMUs). In their approach, efficiency is measured as the ratio of the firm’smaximized revenue to its observed revenue, while industry revenue efficiency is defined as the ratio of industrymaximal to industry observed revenue.

Using Farrell-type decompositions of a cost efficiency index, Fare and Zelenyuk also derived aggregatemeasures of what Farrell [4] referred to as ‘‘technical’’ and ‘‘overall’’ efficiency. Their objective was to preservethe multiplicative form in which Farrell formulated the relations between ‘‘technical,’’ ‘‘allocative,’’ and‘‘overall’’ efficiency. This led to aggregates in terms of weighted geometric means of cost-weighted measures byextending Koopmans [8] theorems on aggregation of profit functions.

Nevertheless, the cost function approach of Fare and Zelenyuk [7] is not combined with the former (revenuefunction) approach—as is done in the present paper; or, then extended to profit and other functions, as is alsodone here. In particular, we show how optimal solutions to one model provide the same to other models, andto measures of efficiency.

2. Russell measures

We now turn to alternative measures of technical efficiency and begin with what is called the‘‘Russell measure.’’ Introduced by Fare and Lovell [5], it was named for R.R. Russell, who subsequentlycontributed to its further development in Russell [2,9]. Originally formulated in input oriented form, it waslater extended in Fare et al. [10, p. 162] in a form that they refer to as the ‘‘Russell graph measure’’—whichsimultaneously minimizes the input efficiency measure and maximizes the output inefficiency measure

ARTICLE IN PRESSW.W. Cooper et al. / Socio-Economic Planning Sciences 41 (2007) 1–21 3

as follows:

min RXm

i¼1

yi;Xs

r¼1

1

,fr

!¼

Pmi¼1yi þ

Psr¼11=fr

mþ s

s.t.

yi0xi0X

Xn

j¼1

xijlj ; i ¼ 1; . . . ;m,

fr0yr0pX

yrjlj ; r ¼ 1; . . . ; s,

0pyip1; 1pfr 8i; r,

0plj ; j ¼ 1; . . . ; n. ð1Þ

Here, xij and yrj represent the coordinates of a point that corresponds, respectively, to the observed values ofinput and output amounts which, from here on, are assumed to be positive for all j ¼ 1; . . . ; n DMUs. The xi0

and yr0 represent the input and output amounts for the DMU0 to be evaluated, where i ¼ 1; . . . ;m andr ¼ 1; . . . ; s. The variables yi0 and fr0 represent measures of input efficiency and output inefficiency,respectively, while the lj are ‘‘structural’’ (also called ‘‘intensity’’) variables.

Remark 1. A referee suggested that an alternative measure is provided by the ‘‘directional distance’’ functionintroduced by Chambers et al. [11] which simultaneously maximizes outputs and minimizes inputs. However,the Russell measure is ‘‘complete’’ in the sense of Cooper et al. [12]. That is, it accounts for all inefficienciesthat the model can identify. This is not the case for the directional distance function as it fails to considerinefficiencies associated with non-zero slacks that may be identified by the model being employed. See theexample in [13, p. 95].

We note that the model in (1) possesses shortcomings. See, for instance, Cooper et al. [12], who note that theobjective in (1) is difficult to compute. They also show that this objective is a weighted average of arithmeticand harmonic means and, hence, is difficult to interpret. We therefore turn to the measure that Pastoret al. [14]1 have termed the ‘‘enhanced Russell graph measure,’’2 which we shorten to ERM ( ¼ enhancedRussell measure) and represent in the following model:

min R ¼

Pmi¼1yi0

�mPs

r¼1fr0=s

s.t. Xn

j¼1

xijljpyi0xi0; i ¼ 1; . . . ;m,

Xn

j¼1

yrjljXfr0yr0; r ¼ 1; . . . ; s,

yip1; frX1; ljX0; i ¼ 1; . . . ;m; r ¼ 1; . . . ; s; j ¼ 1; . . . ; n. ð2Þ

Notice that the yi0 and fr0 are units invariant so that R is also units invariant. To see that this is so, we notethat these inequalities may be formulated as equations without affecting the optimal value of minR ¼ R�.Then, writing yi0 ¼

Pnj¼1 xijlj=xi0; i ¼ 1; . . . ;m and fr0 ¼

Pnj¼1 yrjlj=yr0, r ¼ 1; . . . ;m we see that we can

multiply numerators and denominators by ki; cr40, i ¼ 1; . . . ;m and r ¼ 1; . . . ; s, respectively, withoutaffecting the values of the yi0 and fr0.

1Bardhan et al. [15] develop the same measure but do not provide all the needed detail as is available in Pastor et al. [14]. Tone [16]

formulated a closely related measure, which he refers to as the ‘‘slacks-based measure (SBM)’’. See also the measures referred to as GEMs

(generalized efficiency measures) in Cooper and Pastor [17].2See Fare et al. [10,18] for definitions and discussions of ‘‘graph measures,’’ and their properties.


Finally, R is ‘‘complete’’ and therefore differs from the commonly employed radial measures which (a) failto reflect the non-zero slacks and (b) are either output oriented or input oriented. They thus fail to reflectperformances of the outputs or inputs represented in those constraints not covered by the radial measure.

Model (2) also lends itself to straightforward interpretations. As previously noted, the objective in (2) jointlyminimizes the average of the input efficiencies and maximizes the average of the output inefficiencies, or,alternatively, it jointly minimizes the input and output efficiencies. By this we mean that the denominator ismaximized to give values fr0X1 for any choice of yi0 in the numerator and, simultaneously, the numerator isminimized to give values fr0p1 for any choice of fr0 in the denominator. Thus, the numerator anddenominator are jointly optimized to achieve a minimum value for their ratio. This joint optimization offersvarious advantages. For instance, as shown in Section 8, it becomes possible to use this same ratio form todevelop a profit function to be maximized when revenues are positioned in the numerator and costs in thedenominator in a revenue–cost ratio aggregate.

Now we note that, in the sense just discussed, the numerator in (2) minimizes the average input efficiencywhile the denominator maximizes the average output inefficiency. This being so, the denominator alsominimizes the average output efficiency as measured by its reciprocal, which we represent by

0pXs

r¼1

fr0

,s

!�1p1

with

Xs

r¼1

fr0

,s

! Xs

r¼1

fr0

,s

!�1¼ 1. (2.1)

Hence, R in (2) is the product of average input efficiency and average output efficiency. Following Fare andZelenyuk [19], we define our measure of efficiency as the geometric mean—i.e., the square root—of theproduct of average input efficiency and average output efficiency as represented in the objective of (2).

Remark 2. The constraints in the models explored in this paper omit the convexity condition,Pn

j¼1lj ¼ 1,which is usually introduced to examine returns to scale. The topic of returns to scale is not considered here.Our models, however, evaluate both returns-to-scale and technical efficiency without distinguishing betweenthem. To separate the two, one need merely examine which of the three conditions,

Pnj¼1lj

4o1, occur in the

optimal solutions in order to determine whether returns to scale are decreasing, constant, or increasing.See Banker et al. [20, 49pp].

We now observe that

0pmin R ¼ R�p1

and

0pXm

i¼1

y�i

,mp1; 1p

Xs

r¼1

f�r0

,s, (3)

so that R� provides a suitable measure of efficiency in that it increases with increase in efficiency in thenumerator and denominator.3 It also decomposes into the separate measures of input efficiency and outputinefficiency that are noted in (3), with efficiency attained at R� ¼ 1 if and only if

Pmi¼1 y

�i0=m ¼

Psr¼1f

�r0=s ¼ 1,

where we note that 0py�i0p1 and f�r0X1 conform to properties associated with the input-oriented and output-oriented models used in customary radial measures.

Remark 3. As pointed out to us in a private communication from Hirofumi Fukuyama, the name ‘‘RussellMeasure’’ may be a misnomer. In fact, Russell [9] writes: ‘‘they [Fare and Lovell] mischievously call this indexthe ‘‘Russell Measure’’ but I call it the ‘‘Fare/Lovell’’ efficiency index. . ..’’ However, the former name is now

3Formally, we should useffiffiffiffiffiffiR�p

instead of R� as noted in the discussion leading up to and following (2.1). However, we do not repeat this

extra step in the developments that follow. Instead, we assume that the square root measure is used as needed.


firmly embedded in the literature, so we continue to use it while acknowledging its authorship in Fare andLovell [5].

3. Transformations

The model in (2) is a fractional programming structure. We can therefore utilize the approach in Charnesand Cooper [21] to transform the nonlinear (non-convex) model in (2) into an ordinary linear programmingformulation, which is readily solved by the many computer codes now available. For this purpose, weintroduce a new variable, b, defined by

b ¼Xs

r¼1

fr0

,s

!�1, (4)

so that 0obp1 and bðPs

r¼1fr0=sÞ ¼ 1. See (2.1).We next use b to transform the variables from (2) into new variables defined by

ui ¼ byi; i ¼ 1; . . . ;m,

vr ¼ bfr; r ¼ 1; . . . ; s,

tj ¼ blj ; j ¼ 1; . . . ; n. (5)

We can then multiply the numerator and denominator in the objective of (2) by b without changing its value.Also, since b40, we can multiply both sides of the constraints in (2) by this variable without changing any ofthe orientations. We then write the thus transformed model as follows:

minXm

i¼1

ui=m

s.t. Xs

r¼1

vr ¼ s,

Xn

j¼1

xijtjpuixi0; i ¼ 1; . . . ;m,

Xn

j¼1

yrjtjXvryr0; r ¼ 1; . . . ; s,

uipb; i ¼ 1; . . . ;m,

bpvr; r ¼ 1; . . . ; s,

0ptj ; j ¼ 1; . . . ; n,

0pbp1. ð6Þ

Notice that, in the formulation of (6), we do not need to state explicitly that 0ob.4

We now have an ordinary linear programming formulation that can be readily solved. Suppose, therefore,that we have a solution to (6). If we factor b from both sides of the constraints in (6), we can reproduce theconstraints in (2) by dividing through by b. Factoring b from the objective in (6), and utilizing the definition in(4), we recover the objective in (2). Finally, the last constraint in (6) becomes redundant when b is factored outin this manner. To see why this is so, we need only note that, as in (4), b� ¼ ð

Psr¼1f

�r=sÞ�1 and this, together

with 1pf�r ; 8r, implies that 0ob�p1. Hence, b, the extra variable introduced in going from (2) to (6), is

4It is sufficient to state that 0pb since any feasible solution of (6) necessarily satisfies this inequality strictly; if b ¼ 0, then, via (5), the

input restrictions of (6) show that tj ¼ 0 for all j. Through the output restrictions, however, this implies vr ¼ 0 for all r, which, in turn,

contradicts the first constraint in (6).


omitted when moving from (6) to (2). The objective in (2) is thereby recovered, and the constraints in (2) are allsatisfied without ambiguity.

As proved in Charnes and Cooper [21], an optimum solution for (6) transforms into an optimum solutionfor (2). Indeed, utilizing (4), we have

Xm

i¼1

u�i =m ¼

Pmi¼1y

�i0=mPs

r¼1f�r0=s¼ g�. (7)

To see that this is the case, we note from (6) thatPs

r¼1n�r=s ¼ 1 so that, via (4), we have

Xm

i¼1

m�i

,m ¼

Pmi¼1m

�i =mPs

r¼1n�r=s¼

Pmi¼1bm

�i =mPs

r¼1bn�r=s¼

Pmi¼1y

�i =mPs

r¼1f�r=s

. (7.1)

Thus, all constraints being satisfied optimally, we need only position the y�i0 and f�r0 in the objective of (2).When only the value of g� is sought, it is unnecessary to do more than solve (6). This is the ‘‘ordinary usage’’ inDEA that has been followed since the introduction of these ideas in Charnes et al. [22]. Here, however, interestcenters on (2), so we supplied only needed details on the transformations.

Remark 4. Note that g� ¼ 1, as obtained from (6), establishes efficiency in (2) with y�i ¼ f�r ¼ 1;8i; r whileb� ¼ 1 establishes f�r ¼ 1;8r. See (4).

Remark 5. Blackorby and Russell [1] cite Nataf [23] as demonstrating that aggregates meeting the Klein [24]conditions for ‘‘simultaneous aggregation of inputs and outputs’’ will always be linear and therefore not usefulfor economic analysis. We suggest caution in use of these Klein/Nataf formulations and results, however,because (a) they do not deal with measures of efficiency, and (b) these authors assume that each firm( ¼ DMU) maximizes profit. See Nataf [23, p. 233]. An absence of technical inefficiencies is, however, anecessary condition for maximum profit. Hence, any consideration of technically inefficient production, evenat the level of all individual DMUs, is omitted from the treatments by both Nataf and Klein. Furthermore, wecan regard the formulation in (6), with its linear objective, as only a way of solving the problem in (2), which isnonlinear, and therefore regard this linear programming formulation as only part of the algorithm to beemployed—e.g., in the same sense that the simplex tableaus can be regarded as part of the simplex algorithm.See the discussion of ‘‘algorithmic completion of a model’’ in Charnes and Cooper [25]. Our problems are thusnonlinear, contradicting the finding by Nataf.

4. Aggregation

4.1. Consistency

We now turn to the DEA literature on aggregation where a satisfactory result requires the aggregates andthe individual entity performances to be consistent. In our case, this means that the system-wide, and thecollection of individual entity evaluations, must give the same result. In particular, if system-wide performanceis evaluated as efficient, then all DMUs must also be evaluated as efficient.

Problems arise in the treatment of technical inefficiencies by DEA because of differing assumptions. Forinstance, as noted earlier, the standard assumptions associated with profit maximization (or costminimization) in the economics literature are carried over into the literature on aggregation (see, for instance,the discussions in Varian [26, 9pp] and Samuelson [27, Chapter 4]). This creates problems in DEA treatmentsof technical inefficiencies because assumptions like profit maximization (or cost minimization) imply theabsence of technical inefficiencies.

Problems also arise because the evaluations of any DMU in DEA are relative, and based on comparisonswith other DMUs. A system-wide performance generally lacks a basis for such comparative evaluations withother systems; so, the resulting ‘‘self-evaluations’’ will always be unity, in principle, showing that full efficiencyhas been attained.

To deal with such problems, we proceed as follows. First, we omit the assumption that technicalinefficiencies are absent since it is not suitable for current purposes, viz., the company-wide and economy-wide


aggregates mentioned in our opening paragraph. Next, we avoid the ‘‘self-evaluator’’ possibilities associatedwith the use of relative evaluations in DEA by using the following model in place of (2) to obtain both system-wide and individual entity evaluations:

min

Pmi¼1yiT=mPsr¼1frT=s

s.t.

yiT xiTX

Xn

j¼1

xijlj þ xiTlT ; i ¼ 1; . . . ;m,

frT yrTpXn

j¼1

yrjlj þ yrTlT ; r ¼ 1; . . . ; s,

0pyiTp1; 1pfrT ; 8i; r,

0plj ; j ¼ 1; . . . ; n and 0plT , ð8Þ

where

xiT ¼Xn

j¼1

xij ; i ¼ 1; . . . ;m; yrT ¼Xn

j¼1

yrj ; r ¼ 1; . . . ; s.

That is, xiT and yrT , respectively, represent the system-wide sums of the individual entity (DMU) inputs andoutputs.

This formulation has the character of a mixed integer programming model with lT being a bivalent variable.Recourse to such a mixed integer formulation is not necessary, however, since, as we shall see, l�T ¼ 0 occurswhen system-wide performances are inefficient, while l�T ¼ 1 is a solution, with an alternate optimum, whensuch performances are efficient.

We now introduce:

Theorem 1 (Alternate optima theorem). The system is efficient if and only if the following solutions constitute

alternate optima: (i) l�T ¼ 1 in (8) with all l�j ¼ 0 and (ii) l�j ¼ 1; 8j, and l�T ¼ 0.

Proof. l�T ¼ 1; l�j ¼ 0;8j can occur only if y�iT ¼ f�rT ¼ 1;8i; r in which case we havePm

i¼1y�iT=m=Ps

r¼1f�rT=s ¼ 1, so the system is efficient. This same result occurs if l�j ¼ 1 8j and l�T ¼ 0, since, then,Pn

j¼1xij ¼ xiT ; i ¼ 1; . . . ;m andPn

j¼1yrj ¼ yrT ; r ¼ 1; . . . ; s, so, we must have y�iT ¼ f�rT ¼ 1;8i; r in (8). &

This shows that the properties of these solutions are consistent with the requirements Blackorby and Russell[1, p. 11] cite from Fare and Lovell [5]—viz., system performance will be efficient if and only if theperformance of all entities is efficient.

We also have the following:

Theorem 2. If the system is inefficient, then l�T ¼ 0 in all optimum solutions.

Proof. See the proof of Theorem 7.1 in Cooper et al. [28, p. 209] and Theorem 3.3 in Cooper et al.[29, p. 49]. In other words, only efficient DMUs can enter actively—i.e., with positive l�j —in the DEAevaluations. &

By virtue of these two theorems, we have the desired bivalent property for l�T ; recourse to integerprogramming algorithms is thus not required. We can also omit lT and its coefficients in (8), and thereforerevert to (2) with the insertion of yiT xiT and frT yrT for evaluation on the left-hand side of (2) to determinesystem-wide efficiencies.

4.2. Synthesis

We can go still further and synthesize system-wide solutions from the individual DMU solutions.For this purpose, we again refer to (2), and write the optimal solutions for each of the k ¼ 1; . . . ; n DMUs


as follows:

Xn

j¼1

xijl�j ðkÞ ¼ y�ikxik; i ¼ 1; . . . ;m,

Xn

j¼1

yrjl�j ðkÞ ¼ f�rkyrk; r ¼ 1; . . . ; s. ð9Þ

That is, we here designate DMUk as the DMU0 being evaluated. Summing over these k ¼ 1; . . . ; n relationsgives

Xn

k¼1

Xn

j¼1

xijl�j ðkÞ ¼

Xn

k¼1

y�ikxik ¼ y�i1xi1 þ � � � þ y�inxin ¼ xiTy�iT ; i ¼ 1; . . . ;m,

Xn

k¼1

Xn

j¼1

yrjl�j ðkÞ ¼

Xn

k¼1

f�rkxrk ¼ f�r1xr1 þ � � � þ f�rnyrn ¼ yrTf�rT ; r ¼ 1; . . . ; s. (10)

In this way, the relations on the left determine values of the y�iT , f�rT that enter our proposed measure of system

efficiency. Hence, system-wide performances will be characterized as inefficient if and only if the performancesof some of the DMUj are characterized as inefficient.

An immediate corollary to the above development is that the amount of system inefficiency in inputsi and outputs r is equal to the sum of the inefficiencies of the individual DMUj in the same inputs oroutputs. Moreover, the sources and amounts of the inefficiencies in each of the inputs and outputs ofDMUT are equal to the sums of these amounts in the corresponding inputs and outputs of the inefficientDMUj.

We now synthesize a measure of system efficiency from the individual DMU measures obtained in (10),which we write as

y�iT ¼ y�i1xi1

xiT

þ � � � þ y�inxin

xiT

; i ¼ 1; . . . ;m,

f�rT ¼ f�r1yr1

yrT

þ � � � þ f�rn

yrn

yrT

; r ¼ 1; . . . ; s. (11)

In other words, the y�iT and f�rT represent weighted averages of the y�ik and f�rk with weights given by

0poik ¼xik

xiT

¼p1;Xn

k¼1

oik ¼ 1,

0pork ¼yrk

yrT

¼p1;Xn

k¼1

ork ¼ 1. (12)

From the above development,Pnk¼1ð

Pmi¼1y

�ikoik=m=

Psr¼1f

�rkorr=sÞ

n¼

Pmi¼1y

�iT

m

�Psr¼1f

�rT

s

� �. (13)

Thus, our proposed system measure (as given on the right) is equal to the average of the ratios of the weightedaverages (on the left) that represent the efficiency measure for each of the k ¼ 1; . . . ; n DMUs in the system.

We now show that these choices of y�iT and f�rT constitute an optimal solution to

min

Pmi¼1yiT

m

�Psr¼1frT

s


with

Xn

j¼1

xijljpyiT xiT ; i ¼ 1; . . . ;m,

Xn

j¼1

yrjljXfrT yrT ; r ¼ 1; . . . ; s

and

0pyiTp1; 1pfrT ; ljX0; 8i; r; j. (14)

First, observe that

0pXm

i¼1

y�ikoik ¼ y�iTp1; i ¼ 1; . . . ;m,

1pXm

i¼1

f�rkork ¼ f�rT ; r ¼ 1; . . . ; s, (15)

as defined in (11), so the y�ik, f�rk all satisfy the last mþ s constraints in (2) since the non-negative weights oik

and ork sum to unity for each i and r. We thus have 0py�iTp1 and 1pf�rT ; 8i; r, as required.

Remark 6. A referee has called our attention to the following: The ratios in (12) are referred to as ‘‘price-independent weights’’ in Fare and Zelenyuk [7]—see also Fare et al. [30]—who show that when allocativeinefficiency is present, the relation (technical efficiency)(allocative efficiency) ¼ cost efficiency need not givethe same value of technical efficiency when inefficiency is present as is obtained directly from the inputdistance function they employ. However, as we noted in Remark 1, such distance functions do not allow formix inefficiencies associated with non-zero slacks that are taken into account by cost efficiency. Thus, a non-zero input slack in an input oriented measure of distance would imply that the efficiency score is overstated,while a non-zero output slack in an output oriented measure of distance would imply that its inefficiency scoreis understated. Thus, as shown in Fare et al. [18], when distance measures are used, these possibilities areeliminated if and only if the three efficiencies—Technical, Allocative and Cost—are all equal to unity.

As is noted in Cooper et al. [31], adjustment for mix inefficiencies involves movement to the efficiencyfrontier in order to remove the non-zero slack. This differs from the adjustments for allocative inefficiencieswhich involve movements along efficiency frontiers where no non-zero slacks are involved.5 We do not heredeal further with cost efficiency. Instead, we refer readers to Cooper et al. [12] for discussion of relationsbetween mix and commonly used measures of efficiency.

To complete the argument that we have a solution, we return to (10) and observe that the xij and yrj are free ofthe index k. Collecting the ljðkÞ terms for each such xij and yrj , we define new variables l�j ¼

Pnk¼1l

�j ðkÞ for each

j ¼ 1; . . . ; n. We can then use these variables to satisfy the constraints of (14) in the form of the followingequations6:

Xn

j¼1

xijl�j ¼ y�iT xiT ; i ¼ 1; . . . ;m,

Xn

j¼1

yrjl�j ¼ f�rT yrT ; r ¼ 1; . . . ; s. (16)

The constraints in (14) are thus all satisfied by the choice of y�iT and f�rT given in (10).

5Slack variables may be used, however, to traverse the efficiency frontier as in, for example, Brockett et al. [32] or Cooper et al. [12]. See,

also, Cooper et al. [33] who demonstrate the importance of the slack variables in countering arguments that show the non-existence of

‘‘fully satisfactory’’ DEA measures.6We are assuming that the optimal solutions are unique. More generally, however, one needs to allow for the possible presence of

alternate optima. See the numerical example in the next section of this paper.


We now show that this solution is optimal. If this is not the case, there will exist a solution to (14) for which:

min

Pmi¼1yiT

m

�Psr¼1frT

s¼

Pmi¼1y

0iT

m

�Psr¼1f

0rT

soPm

i¼1y�iT

m

�Psr¼1f

�rT

s. (17)

We then have

y0iT xiT ¼ y0iT ðxi1 þ � � � þ xinÞ ¼ y0iT xi1 þ � � � þ y0iT xin; i ¼ 1; . . . ;m,

y0rT yrT ¼ f0rT ðyr1 þ � � � þ yrnÞ ¼ f0rT yr1 þ � � � þ f0rT yrn; r ¼ 1; . . . ;m, (18)

and, therefore,

y0iT ¼ oi1y0iT þ � � � þ oiny

0iT ; i ¼ 1; . . . ;m.

f0rT ¼ or1f0rT þ � � � þ ornf

0rT ; r ¼ 1; . . . ;m. (19)

This givesPmi¼1y

0iT

m

�Psr¼1f

0rT

s¼

Pmi¼1

Pnj¼1oijy

0iT

m

,Psr¼1

Pnj¼1orjf

0rT

so

Pmi¼1

Pnj¼1oijy

�ij

m

,Psr¼1

Pnj¼1orjf

�rj

s.

We must then have either or both,

Xm

i¼1

Xn

j¼1

oijy0iTo

Xm

i¼1

Xn

j¼1

oijy�ij

or

Xm

i¼1

Xn

j¼1

oijf0rT4

Xs

r¼1

Xn

j¼1

orjf�rj (20)

for some i; r. However, the expressions on the right are optimal in that the ratio form of the fractionalprogramming objectives for each of these j ¼ 1; . . . ; n problems jointly minimize the numerator and maximizethe denominator, so thatPm

i¼1

Pnj¼1oijy

�ij

m

,Psr¼1

Pnj¼1orjf

�rj

sp

Pmi¼1

Pnj¼1oijy

0iT

m

,Psr¼1

Pnj¼1orjf

0rT

s. (21)

We therefore havePmi¼1y

�iT

m

Psr¼1f

�iT

spPm

i¼1y0iT

m

� �Psr¼1f

0rT

s, (22)

contradicting (20). Moreover, we cannot havePmi¼1y

�iT

m

�Psr¼1f

�iT

soPm

i¼1y0iT

m

Psr¼1f

0rT

s

�

since the y�iT ;f�rT values are available for the optimization in (14). Hence, we must havePm

i¼1y�iT

m

Psr¼1f

�iT

s¼

Pmi¼1y

0iT

m

� �Psr¼1f

0rT

s. (23)

5. Numerical example

We illustrate the properties just developed with the simple one input and one output example in Fig. 1,where the points Pj ; j ¼ 1; 2; 3, represent the performances of three DMUs with values ðxij ; yrjÞ representing,respectively, the input amount used, and the output amount produced, by DMUj. The coordinates of P4

ARTICLE IN PRESS

P4(6,10)

P3(3,6)

P2(2,2)P1(1,2)

0

1

2

3

4

5

6

7

8

9

10

11

12

input

output

P5(6,12)

0 1 2 3 4 5 6 7

Fig. 1. Individual and system-wide performances.

Table 1

Solutions for the DMUs

Variable DMU

P1 P2 P3 PT

y 1 1 1 1

f 1 2 1 6=5y=f 1 1/2 1 5/6

l1 1 0 0 0

l2 0 0 0 0

l3 0 2/3 1 2

lT 0 0 0 0

x 1 2 3 6

y 2 2 6 10

W.W. Cooper et al. / Socio-Economic Planning Sciences 41 (2007) 1–21 11

represent the system-wide performance of PT , which is obtained by summing the individual DMUj inputs andoutputs for j ¼ 1; . . . ; 3.

Table 1 lists solutions obtained from these data as described in the preceding section.From the values under PT in the top three rows, we have y�T ¼ 1, f�T ¼

65so y�T=f

�T ¼ 0:833 . . . ¼ 5

6andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0:833 . . .p

¼ 0:91. Hence, the system is inefficient. This same value of 56 is obtained by averaging the y�j =f

�j

values under P1, P2 and P3 to obtain

y�1=f�1 þ y�2=f

�2 þ y�3=f

�3

n¼

1þ 12þ 1

3¼

5

6.

The l�jk values for j ¼ 1; 2; 3 in Table 1 do not sum to the ljT values recorded under PT because

l�11 þ l�12 þ l�13 ¼ 1þ 0þ 0 ¼ 1, whereas l�1T ¼ 0; similarly, l�31 þ l�32 þ l�33 ¼53, while l�3T ¼ 2. As noted in

footnote 6, this suggests the presence of an alternate optimum7which is given by l�11 ¼ 0 and l�31 ¼13with the

7This also implies an alternate optimum under PT , which is given by l�31 ¼ 1, l�32 ¼ 0 and l�33 ¼53so that l11x1 þ l33x3 ¼ xT ¼ 6 and

l11y1 þ l33y3 ¼ yT ¼ 12. This is identical to the solution obtained with l33 ¼ 2.


same minimizing value of y�=f� ¼ 1 recorded under P1. Summing across the row for l3 then produces

l�31 þ l�32 þ l�33 ¼13þ 2

3þ 1 ¼ 2, which is the same as the l�3T ¼ 2 recorded under PT .

Consider, now, the data in the last two rows of Table 1, which we combine with the data in the first tworows, and exhibit the relations in (10) by writing

y�11x11 þ y�12x12 þ y�13x13 ¼ x11 þ x12 þ x13 ¼ 6,

f�11y11 þ f�12y12 þ f�13y13 ¼ y11 þ 2y12 þ y13 ¼ 12.

This yields the same results when the values listed under PT are used since y�T xT ¼ 6 when y�T ¼ 1 and f�T ¼65

gives f�T yT ¼65

yT ¼ 12. Finally, we also see that (11) yields these same values of y�T ¼ 1 and f�T ¼65via

o11y�1 þ o12y

�2 þ o13y

�3 ¼ y�T ¼ 1,

o21f�1 þ o22f

�2 þ o23f

�3 ¼ f�T ¼

65,

where

o11 ¼x11

xT

¼1

6; o12 ¼

x12

xT

¼2

6; o13 ¼

x13

xT

¼3

6,

o21 ¼y21

yT

¼2

10; o22 ¼

y22

yT

¼2

10; o23 ¼

y23

yT

¼6

10.

To interpret these results geometrically, we refer to Fig. 1 and note that the values y�T ¼ 1 and f�T ¼65project

P4 into P5 on the efficiency frontier. This shows a shortfall of two units in the output of P4. Applying theratio y�T=f

�T ¼

56 to P5 moves matters along the frontier until the point with coordinates (5,10) is reached.

The performances of P4 thus exhibit an excess of one unit in its input. Hence, a user of this approach candecide whether to focus on an output shortfall of two units, or an input excess of one unit in the performancesof P4 ¼ PT .

A similar approach to the individual DMUs locates the source of these inefficiencies in the performances ofP2 in the same amounts. As previously noted, it is thus unnecessary to solve (8) since we will already have thesey�ij and f�rj available, along with their associated l�j , from the performances of the individual DMUj with (2).

6. Desiderata

We now turn to the desiderata specified by Blackorby and Russell [1]. First, we note that the objective itselfis homogeneous of degree 1 in the numerator and homogeneous of degree �1 in the denominator. Hence, theobjective in (8) is homogeneous of degree 0, which is one of the desiderata. The components of (8) are alsojointly optimized, as specified by both Blackorby and Russell [1, p. 6] and Fare and Lovell [5], with

0pPm

i¼1y�iT=mPs

r¼1f�rT=s

p1

and

0pXm

i¼1

y�iT=mp1;Xs

r¼1

f�rT=sX1. (24)

Hence, the objective of (8) can be decomposed into an aggregate input measure of input efficiency and anaggregate output measure of output inefficiency. In this respect, (8) conforms to another of the desiderata

specified in [1].Blackorby and Russell [1, p. 7] also note that n disaggregate units can be consistently aggregated (over both

inputs and outputs) into an aggregate index if there exists a continuous and increasing function, F , such that

E0ðy;xÞ ¼ F ðE1ðy1; x1Þ; . . . ;Enðyn; xnÞÞ, (25)

where Ekðyk; xkÞ is the efficiency index for DMUk; k ¼ 1; . . . ; n, with Ekðyk; xkÞ ¼ �kðPs

r¼1 aryrkþPmi¼1bixikÞ; k ¼ 1; . . . ; n, and ar and bi are arbitrary parameters. Hence, yk and xk are linear functions of


yrk and xik for DMUk with, again, ar and bi representing arbitrary parameters—e.g., some (unspecified) set ofweights—and E0ðy;xÞ is the aggregate efficiency indicator with y ¼

Pnk¼1y

k and x ¼Pn

k¼1xk. Blackorby and

Russell [1] also write this as

E0ðy;xÞ ¼ �Xn

k¼1

e�1k ðEkðyk; xkÞÞ

!¼ e

Xs

r¼1

aryr þXm

i¼1

bixi

!, (26)

where the efficiency indexes e and ek; k ¼ 1; . . . ; n, are continuous increasing functions.This � lends itself to use with the measure in (1) but not in (2). To accommodate both types, we replace (26)

with

E0ðy;xÞ ¼ eXs

r¼1

aryr;Xm

i¼1

bixi

!. (27)

This formulation also allows for ratio forms that conform to treatments in the concluding remarks on p. 12 ofBlackorby and Russell [1]— viz., ‘‘we have shown that under reasonable assumptions the efficiency indices,output (input) aggregation is possible only if the efficiency indices are ratios of linear functions of input andoutput quantities. . ..’’ At any rate, these are the kinds of possibilities we are exploring here.

Blackorby and Russell require E0 in (25), above, to be an increasing function of Ekðyk; xkÞ; k ¼ 1; 2; . . . ; n.This is satisfied by (24) in the following manner. The value of this function increases if either the numerator ordenominator measure of efficiency increases so that at least one of the y�i ðkÞ in the numerator increases, or atleast one of the f�r ðkÞ in the denominator decreases—see the discussion of (2.1), above. Consequently, E0 is anincreasing function of Ek, as desired.

The function represented in (8) is also continuous. To see that this is the case, note that the denominator isalways positive and the numerator is bounded above. Moreover, setting lk ¼ l0 and yi, fr ¼ 1; 8i; r, producesa solution. Hence, solutions always exist. Moreover, our function is differentiable at all points and, so, afortiori, it is continuous.

Finally, Blackorby and Russell [1, p. 10] quote Fare and Lovell [5] who suggest the following threeproperties as desirable for efficiency indices: homogeneity, monotonicity and indication of efficient input/output vectors. As already indicated, (8) has the first property. It also has the third property. See (24), andnote that the numerator and denominator are each efficient if and only their component measures are allunity. It is also monotonic in each input and each output. This is a direct consequence of being strictlymonotonic in inputs and outputs (see [14]). In fact, if any of the inputs (outputs) of any of the DMUsincreases, the numerator (denominator) of our efficiency measures decrease (increase), and so does the ratio.See also the development in the following remark.

Remark 7. We can, if we wish, replace the efficiency measure in (2) with a new measure stated directly in termsof the data as is done by Blackorby and Russell [1], for example, in (25) and (26). To accomplish this, we note,as stated earlier, that a necessary condition for optimality in (2) is that the inequalities are all satisfied asequations. Hence, we can restrict attention to solutions that satisfy:

yi ¼

Pnj¼1xijlj

xi0; i ¼ 1; . . . ;m; fr ¼

Pnj¼1yrjlj

yr0

; r ¼ 1; . . . ; s.

Summing, in the manner of the objective in (2), we therefore have

Pmi¼1yi=mPsr¼1fr=s

¼

Pmi¼1

Pnj¼1

xijlj

xi0

�m

Psr¼1

Pnj¼1

yrjlj

yr0

�s

. (28)

This allows us to replace the expression on the left in the objective of (2) with the expression on the right.


To see that the property of degree zero homogeneity is preserved, we note that

kyi ¼kxi1l1 þ � � � þ kximlm

xi0¼

kPn

j¼1xijlj

xi0(29)

and

kfr ¼kyr1l1 þ � � � þ kyrsls

yr0

¼kPn

j¼1yrjlj

yr0

,

for each i ¼ 1; . . . ;m and r ¼ 1; . . . s. Thus,Pmi¼1kyi=mPsr¼1kfr=s

¼kPm

i¼1yi=m

kPs

r¼1fr=s¼

Pmi¼1yi=mPsr¼1fr=s

¼Xm

i¼1

Xn

j¼1

kxijlj

mxi0

,Xs

r¼1

Xn

j¼1

kyrjlj

syr0

¼k

k

Pmi¼1

Pnj¼1

xijlj

xi0

�m

Psr¼1

Pnj¼1

yrjlj

yr0

�s

¼

Pmi¼1

Pnj¼1

xijlj

xi0

�m

Psr¼1

Pnj¼1

yrjlj

yr0

�s

. ð30Þ

Thus, both functions satisfy all homogeneity conditions specified in the discussion leading up to Corollary 1 ofBlackorby and Russell [1, p. 7]. Each consists of a ratio of linear forms, and is homogeneous of degree zero.Finally, we observe that it is also monotonic in each input and output, as cited by Blackorby and Russell [1]from Fare and Lovell [5] and, indeed, is strictly monotonic and hence satisfies this desideratum as shown inCooper et al. [33].

The above characterizations show that all desiderata specified by Blackorby and Russell [1] have beenfulfilled, which is contrary to their p. 7 finding that no aggregation with the above properties is possible. Againon p. 7, Blackorby and Russell confine their attention to measures that are either ‘‘output based’’ or ‘‘inputbased’’, whereas our measures are jointly determined. Furthermore, we supply measures of efficiency for eachinput and output at both the aggregate and individual DMU levels. See the illustrative example in Section 5 ofthis paper.

Remark 8. ERM is not the only model that has all of the properties described above. For example, Pastoret al. [34] show that ERM transforms into a generalized efficiency measure (GEM). This is a modification ofthe additive model, which accords it the property of ‘‘units invariance’’ described immediately after (2).See Cooper and Pastor [17]. See also Bardhan et al. [35], and Cooper et al. [33], where ERM is related to theSBM of Tone [16].

7. Weights

We begin our discussion of weights for inputs and outputs by noting that one possibility is simply to use theset to be aggregated in the proportions ðxi0ðkÞ=

Pnk¼1 xikÞxij and ðyr0ðkÞ=

Pnk¼1 yrkÞyrj to obtain weights for each

input and output. This approach reflects the weighted averages of the input and output values used todetermine y�ij and f�rj in the j ¼ 1; . . . ; n evaluations needed for this purpose. There is a disadvantage, however,in that the resulting aggregates do not reflect the relative importances of the different inputs and outputs in theaggregate measure.

To develop measures that reflect relative importances, we define

y�

i ¼Xn

j¼1

y�ij

,n; f

�

r ¼Xn

j¼1

f�rj

,n. (31)

I.e., y�

i and f�

r represent averages of the weighted averages of the efficiencies in (11) and (12) and thus reflectthe relative importance of each of the j ¼ 1; . . . ; n DMUs in producing each of the i ¼ 1; . . . ;m inputs andr ¼ 1; . . . ; s outputs in the efficiency evaluations. However, this still does not reflect the relative importance ofthe inputs in an overall measure. We thus define new weights that can be used to obtain a measure of relative


cost, adjusted for efficiency, as in (32):

0py�

1o1 þ � � � þ y�

momPsr¼1fr

p1,

where

0poi ¼cixiTPmi¼1cixiT

p1;Xm

i¼1

oi ¼ 1 (32)

and ci represents the unit cost of producing input amount xiT . We then have

Xm

i¼1

y�

i oipXm

i¼1

oi ¼Xm

i¼1

cixiT

,Xm

i¼1

cixiT ¼ 1, (33)

which provides a measure of the relative contribution of each input to total cost, adjusted for efficiency.Turning to outputs, we develop weights in revenue-share form. For this, we observe that f

�

rX1;8r so, forany prices with pr40; yrT40,

f�

r pryrTXpryrT ; r ¼ 1; . . . ; s (34)

and, therefore,

Xs

r¼1

f�

r pryrTX

Xs

r¼1

pryrT .

Hence, we have

Xs

r¼1

f�

rorX1, (35)

where

0por ¼pryrTPsr¼1 pryrT

p1;Xs

r¼1

or ¼ 1.

In combination with (33), this gives

0por ¼

Pmi¼1 y

�

i oiPsr¼1f

�

ror

p1

with

0pXm

i¼1

y�

i oip1 andXs

r¼1

f�

rorX1. (36)

There remain numerous other issues to be dealt with, such as the weights to be used in aggregates such as(36) when unit price or cost data are not available. Similarly, we have not addressed the subjective weights thata manager may wish to employ in evaluating the contributions different plants might make to theaggregates8—e.g., in determining where immediate attention might best be directed. We have, however,shown that suitable weights can be developed in ways that do not appear to yield satisfactory results for usewith radial measure models.

8See, for instance, Thrall [36] who develops what he refers to as ‘‘goal vectors’’ of such weights in terms of modifications of ‘‘dimensional

analysis’’ concepts and definitions used in mathematics and physics.


8. Extensions

8.1. Accomplishment

As previously noted, our measures satisfy all of the numerous desiderata specified in Blackorby andRussell [1]. We have also satisfied other desiderata such as, for instance, those specified in Cooper et al. [12,33]which require the measure to be ‘‘complete’’ so that it reflects all inefficiencies that the model can identify.The Russell measure satisfies this requirement. In its ERM form, as taken from Pastor et al. [14], it isreadily interpreted and easily computed along the lines we have indicated, and thus conforms to thisadditional desideratum in Cooper et al. [12]. Finally, as noted in the sentence leading up to Remark 7,ERM also supplies measures of efficiency for each input and output at both the aggregate and individualDMU levels.

8.2. Further extensions

Of course, this is not the end of the trail opened by Blackorby and Russell [1]. One may therefore turn toextensions of the approaches examined herein, e.g., (2) can be extended to include the use of profit functions.

We now show how our proposed models may be used to obtain optimal solutions to other models andobjectives. For this purpose, we replace (2) with the following maximization structure:

max

Psr¼1 prkfrkyrkPmi¼1 cikyikxik

s.t. Xn

j¼1

cijxijljpcikyikxik; i ¼ 1; . . . ;m,

Xn

j¼1

prjyrjljXprkfrkyrk; r ¼ 1; . . . ; s,

0pyikp1pfrk 8i; r and ljX0 8j, ð37Þ

which is a weighted version of (2). Indeed, the usual assumption that all prices and costs are the same for allDMUs results in their cancellation from both sides of each constraint and reproduces the constraints in (2).Further, we can regard the objective as a weighted version of (2) in which each of the inputs and outputsare differentially weighted, and replace the common weights 1=m and 1=s used in (2). Moreover, thismaximization corresponds to minimizing the reciprocal of the objective which, via joint optimization ofnumerator and denominator, yields the same optimal solutions.

These solutions are not changed by replacing the objective in (37) with

max

Psr¼1 prkfrkyrkPmi¼1 cikyikxik

� 1 ¼

Psr¼1 prkf

�rkyrkPm

i¼1 ciky�ikxik

�

Psr¼1 prkf

�rkyrkPm

i¼1 ciky�ikxik

,

so thatPsr¼1 prkf

�rkyrkPm

i¼1 ciky�ikxik

�

Pmi¼1 ciky

�ikxikPm

i¼1 ciky�ikxik

¼

Psr¼1 prkf

�rkyrk �

Pmi¼1 ciky

�ikxikPm

i¼1 ciky�ikxik

, (38)

which is the ratio of total profit to total cost for each DMUk, k ¼ 1; . . . ; n as derived from (37).This is an efficiency adjusted version of a widely used measure of performance that usually takes the form of

‘‘rate of return over cost.’’ Carrying this over to the evaluation of observed performances, we note thatPsr¼1 prkyrk �

Pmi¼1 cikxikPm

i¼1 cikxik

pPs

r¼1 prkf�rkyrk �

Pmi¼1 ciky

�ikxikPm

i¼1 ciky�ikxik


and, therefore,Psr¼1 prkyrk �

Pmi¼1 cikxikPm

i¼1 cikxik

�Psr¼1 prkf

�rkyrk �

Pmi¼1 ciky

�ikxikPm

i¼1 ciky�ikxik

p1. (39)

This last formulation, (39), provides a measure of performance efficiency with unity achieved if and only if

f�rk ¼ y�ik ¼ 1 8i; r.

Interestingly, if we apply the Charnes–Cooper [21] transformations to this profit-to-cost ratio, the problemis converted to a linear programming model with an ordinary profit function adjusted for efficiency as itsobjective.

Additional extensions can be made by considering other models. One approach would involve the class ofadditive models as they relate to ERM. Such a transformation of ERM can be effected via the relations

yik ¼xik � s�ik

xik

¼ 1�s�ikxik

; i ¼ 1; . . . ;m,

frk ¼yrk þ sþrk

yrk

¼ 1þsþrk

yrk

; r ¼ 1; . . . ; s, (40)

where s�ik; sþrkX0 represent the slacks associated with constraints i ¼ 1; . . . ;m and r ¼ 1; . . . ; s, in the additive

model. See Pastor et al. [14], Cooper and Pastor [17], and Bardhan et al. [15,35]. We thus find that efficiency isachieved in both the additive and ERM models if and only if the slacks are all zero.

Following this route, Pastor et al. [14] provide an alternative to (2) in the form of an additive model that hasmany desirable features. One of these involves ‘‘strong ( ¼ strict) monotonicity’’ in inputs and outputs, whichis a ‘‘property that is very difficult to achieve,’’ as noted in Cooper and Pastor [17].

Additive models have other attractive features. For example, as shown by Scheel and Scholtes [37, p. 154],the measures for such models are ‘‘continuous’’ except for ‘‘data sets of Lebesgue measure zero.’’ Here,‘‘continuous’’ refers to the stronger condition of ‘‘Lipschitz continuity’’ as described in Scheel and Scholtes[37, p. 150]. It also satisfies the properties specified in Theorem 1 of Pastor et al. [14], including the scalingproperties listed as one of the six properties noted there.

Still other extensions are possible. In this paper, we have assumed independence of the individual DMUs;but, a more general treatment would allow for reallocation of resources between DMUs along with returns-to-scale variations that can improve system-wide performance. This, in turn, would require use of ‘‘exact’’( ¼ numerically valued) measures of elasticity as in, for instance, Banker et al. [20]. In such an extension, theefficiency of the individual DMUs would become a necessary but not a sufficient condition of system-wideperformance efficiency. Further improvements in the system might be possible by reallocating resources evenwhen all DMUs perform efficiently.

To explore such possibilities, we would first eliminate inefficiencies by projecting all observations ontopoints on the efficiency frontier. This is accomplished by applying (41) to each DMU in order to define thenew variable indicated by ‘‘4 ’’ in

xi0 ¼ y�i xi0pxi0; i ¼ 1; . . . ;m,

yr0 ¼ f�r yr0Xyr0; r ¼ 1; . . . ; s. (41)

We might then employ dual variables9 and/or the concept of ‘‘super-efficiency’’—as introduced by Andersenand Petersen [41]—to examine when further improvements in system-wide performances might be made bytransferring resources from one efficient DMU to another.

9A duality theory for models with fractional forms is provided in Kydland [38]. For discussions of the state of research in fractional

programming, see Schaible [39] in Gass and Harris [40]. Fractional programming is, however, a dynamically changing field of research

with, as noted by Schaible [39], more than 900 papers appearing since publication of the paper by Charnes and Cooper [21] in 1962.


8.3. Groupings

This form of aggregation analysis can be extended to alternative decompositions of the system into differentgroups and subgroups of DMUs. As is noted in Blackorby and Russell [1, p. 5], the problem is wellsummarized by Cook et al. [42] who conclude their analysis of various groupings of DMUs in DEA with thefollowing statement: ‘‘the necessity arises to combine multiple ratings on a level and to evaluate the groupsthemselves as DMUs. This being the case, an extension of DEA ideas to this more general setting wouldappear to be an important direction for future research.’’

One possibility falls into what Cook et al. [42] refer to as ‘‘natural groupings,’’ which involves simplygrouping the k ¼ 1; . . . ; n DMUs into K disjoint and collectively exhaustive groups. One could then developefficiency indexes for each group, and sum over the groups to obtain the desired ‘‘aggregate efficiency index.’’

Another natural grouping involves nested hierarchies. Here, the standard treatment would proceed asfollows: an efficiency index would first be computed for the group at the lowest level in the hierarchy. Thisgroup would then be combined with the group in the next higher level, and a new index computed. Thisprocess would continue until all groups are covered to yield an overall ‘‘grand aggregate’’ value for theindicator function. See the examples Cooper et al. [28, p. 193] and Cook et al. [42, 183pp].

Other, more complex, problems of grouping are described in Cook et al. [42]. One can add to theirdescription the problem of how to designate best (or optimal) groups. This can raise the problem of‘‘externalities’’ noted in Blackorby and Russell [1, p. 19], as well as issues of returns to scale and other suchpossibilities.

When the sources and amounts of the externalities are known, the approach in Charnes et al. [43] providesone possible mode of treatment. In their case, advertising by competitors was treated as a (non-discretionary)input because of its effects on other DMUs whose own advertising constituted a (discretionary) input. In othercases, externality effects may be present in particular inputs or outputs and the problem becomes one ofidentifying their unknown sources and amounts as well as, possibly, estimating their effects. Research on howthis can be accomplished in operationally implementable forms should be undertaken in this important, butlargely unexplored, area of DEA. See, however, Banker [44, p. 1271] who shows how statistical tests may bedevised to determine whether groups differ significantly in their efficiency scores.

8.4. Stochastic extensions

Finally, we turn to stochastic characterizations where work has already begun by Simar and Zelenyuk [45].They applied bootstrap approaches to the determination of statistical estimates of aggregate measures andtheir distributions. Other approaches might include maximum likelihood estimates like those investigated inBanker [44]. They could also include chance constrained programming, including the ‘‘satisficing’’ objectivereported in Cooper et al. [46]. This has relevance to the relations between DEA, goal programming, andmultiple objective optimization that are reported in Cooper [47].

There are, of course, still other possibilities to be explored, and other problems to be dealt with followingthat line of research detailed in Blackorby and Russell [1]. The menu we have proposed, however, offers anadequate start for those who are interested in exploring such possibilities.

9. Conclusion

In the current paper, we have responded to the findings of Blackorby and Russell [1] and Fare and Lovell[5], as well as others. In particular, we have shown how to meet all conditions these authors have specified forsatisfactory aggregation of DEA performance evaluations. This does not mean that all of what is needed hasnow been accomplished. Numerous additional opportunities and possibilities for research and applicationhave been opened.

We mentioned the need for extending present developments to consider potential resource reallocationsfrom one DMU to another, as well as the need for decomposing overall system performance into subgroupperformance evaluations. The reallocations should cover each input and each output so tradeoff opportunitiesfor reverse reallocations between the various DMUs are also taken into account. Ideally, this should be


accomplished in a manner that determines optimal subgroupings jointly with resource reallocation, evenamong efficient DMUs.

This type of reallocation and regrouping should prove of interest in extending the literatures on supplychain management. It could also open new paths of empirical research in such areas as regional economics.This could, for example, help predict appropriate resource reallocations as the US transits from a service to aknowledge-based economy in the manner described in Kozmetsky and Yue [48].

We noted earlier the possibility of stochastic extensions that could accommodate risk as well as returnevaluations. Such extensions also open possibilities for replacing or augmenting the choice of objectives toinclude the use of ‘‘satisficing’’ instead of ‘‘optimization,’’ as well as combinations of both optimization andsatisficing, by turning to chance constrained programming formulations as found in Cooper et al. [46].

The opportunities described herein would seem to provide a rather useful and important menu for furtherresearch. Pursuit of such topics is likely to open still further possibilities for development that are not presentlyidentifiable, but may be even more important than those currently emerging. See, for example, the historicalcourse of development in goal programming, as described in Cooper [47].

Acknowledgments

The authors are grateful to Valentin Zelenyuk of the National University, Kiev, Ukraine, and HirofumiFukuyama of Fukuoka University, Fukuoka, Japan for comments that helped improve the original version ofthis paper. W.W. Cooper also wishes to express his appreciation to the IC2 Institute of the University of Texasfor support of this research.

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W.W. Cooper is Foster Parker Professor (Emeritus) at the Red McCombs School of Business, The University of Texas at Austin, where he

is also the Nadja Kozmetsky Scott Fellow at the IC2 Institute. The first (founding) Dean at Carnegie Mellon University’s School of Urban

and Public Affairs (now the H.J. Heinz III School of Public Policy and Management), he was also a founding member of the Graduate

School of Industrial Administration at Carnegie Mellon, and occupied the position of University Professor of Public Policy and

Management Science.

Author or coauthor of more than 500 scientific-professional articles and 24 books, Professor Cooper is a member of the

Accounting Hall of Fame and a Fellow of the Econometric Society and the American Association for the Advancement of Science.

He is the recipient of the Lifetime Contributions to Management Accounting of the American Accounting Association, as well as the

Robert W. Hamilton Award for Career Excellence by The University of Texas. Recipient of the John von Neumann Theory Prize,

jointly awarded by the Operations Research Society of America and The Institute of Management Sciences, Professor Cooper

holds Honorary D.Sc. degrees from The Ohio State and Carnegie Mellon Universities in the US, and the degree of Doctorado

Honoris Causa from the University of Alicante in Spain. He was also the first (founding) President of The Institute of Management

Sciences which amalgamated with The Operations Research Society of America to become INFORMS (The Institute for Operations

Research and Management Science).


Zhimin Huang is Professor of Operations Management, School of Business, Adelphi University, Garden City, LI, NY. He received his BS

in industrial engineering from The Beijing University of Aeronautics and Astronautics, China, MS in economics from The Renmin

University of China, and Ph.D. in management science from The University of Texas at Austin. Professor Huang’s research interests

include supply chain management, data envelopment analysis, distribution channels, game theory, chance constrained programming

theory, and multi-criteria decision making analysis. His research has appeared in such journals as Naval Research Logistics, Decision

Sciences, Journal of Operational Research Society, European Journal of Operational Research, Journal of Economic Behavior and

Organization, Optimization, OMEGA, Research in Marketing, Annals of Operations Research, International Journal of Systems Science,

Journal of Productivity Analysis, Journal of Economics, Journal of Mathematical Analysis and Applications, Computers and Operations

Research, Socio-Economic Planning Sciences, and International Journal of Production Economics. Professor Huang serves as Associate

Editor of OMEGA and the Asia-Pacific Journal of Operations Research, and is on the Editorial Board of International Journal of

Information Technology and Decision Making. He is a member of INFORMS, Decision Sciences Institute, and American Marketing

Association.

Susan X. Li is Professor of Management Information Systems, School of Business, Adelphi University, Garden City, LI, NY. She received

her BS in economics from The Renmin University of China and her Ph.D. in management science and information systems from the

Graduate School of Business, The University of Texas at Austin. Professor Li’s research interests include supply chain related information

systems, distribution channels, data envelopment analysis, chance constrained programming theory in investment and insurance portfolio

analysis, and multi-criteria decision making analysis. She has published articles in a variety of journals, including Decision Sciences,

Journal of Operational Research Society, Socio-Economic Planning Sciences, European Journal of Operational Research, The Annals of

Operations Research, OMEGA, Information Systems and Operational Research, Journal of Optimization Theory and Applications, Journal of

Productivity Analysis, Computers and Operations Research, International Journal of Production Economics, and Systems Science and

Mathematical Sciences. Professor Li is a member of INFORMS and Decision Sciences Institute.

Barnett R. Parker is Professor, School of Graduate Studies and School of Business and Economics, Pfeiffer University, Charlotte and

Misenheimer, NC. He earned a B.S.Ch.E. from the University of Massachusetts, Amherst, and an MBA and Ph.D., both from The Simon

School of Business, University of Rochester, NY. Professor Parker’s research interests focus on the application of OR/MS, marketing, and

strategic planning models to problems of service delivery, with an emphasis on developing country settings. His work has taken him to

nearly 50 countries on five continents under funding by such agencies as USAID, WHO, PAHO, and The World Bank. His research has

appeared in a wide variety of refereed journals, including Operations Research, European Journal of Operational Research, OMEGA,

Computers and Operations Research, Socio-Economic Planning Sciences, Journal of Health Administration Education, The International

Journal of Health Planning and Management, Journal of Health Politics, Policy, and Law, American Journal of Preventive Medicine, Journal

of Medical Systems, World Health Forum, and Public Health Nursing. He serves as Editor-in-Chief of Socio-Economic Planning Sciences.

Jesus T. Pastor is Professor of Statistics and Operations Research, Universidad Miguel Hernandez, Elche, Spain. He earned an MBA and

a Ph.D. in mathematical sciences from Valencia University, Spain. Professor Pastor’s research interests include location science, and

banking and efficiency analysis. He has served on the editorial review or advisory boards of more than 20 journals. He has authored or co-

authored nine books in various fields of mathematics, while his research has appeared in a variety of journals such as Operations Research,

OMEGA, Journal of the Operational Research Society, Operations Research Letters, European Journal of Operational Research, Location

Science, Environment and Planning, Economics Letters, Operations Research Letters, Studies in Location Analysis, Information Technology

and Decision Making, Annals of Operations Research, Journal of Productivity Analysis and European Finance Review.

Documents

Efficiency aggregation with enhanced Russell measures in data envelopment analysis