Content uploaded by Peter Bogetoft

Author content

All content in this area was uploaded by Peter Bogetoft on Jul 22, 2017

Content may be subject to copyright.

1

Measuring potential sub-unit efficiency to counter the

aggregation bias in benchmarking

Heinz Ahna, Peter Bogetoftb and Ana Lopesc

a Corresponding Author. Technische Universität Braunschweig, Institute of Management Control and Business Ac-

counting, Fallersleber-Tor-Wall 23, D-83100 Braunschweig, Germany. Phone: +49 531 391-3610. Email:

hw.ahn@tu-bs.de.

b Copenhagen Business School, Department of Economics, Porcelaenshaven 16 A, DK-2000 Frederiksberg,

Denmark. Phone: +45 23326495. Email: pb.eco@cbs.dk.

c Universidade Federal de Minas Gerais, Av. Antonio Carlos, 6627, 31270-901, Pampulha – Belo Horizonte, Minas

Gerais, Brazil. Phone: +55 31 99212-7972. Email: analopes.ufmg@gmail.com.

Abstract: The paper deals with benchmarking cases where highly aggregated decision making units

are part of the data set. It is shown that these units – consisting of sub-units which are not further

known by the evaluator – are likely to receive an unjustifiable harsh evaluation, here referred to as

aggregation bias. To counter this bias, we present an approach which allows to calculate the poten-

tial sub-unit efficiency of a decision making unit by taking into account the possible impact of its

sub-units’ aggregation without having disaggregated sub-unit data. Based on data envelopment

analysis, the approach is operationalized in several ways. Finally, we apply our method to the

benchmarking model actually used by the Brazilian Electricity Regulator to measure the cost effi-

ciency of the Brazilian distribution system operators. For this case, our results reveal that the po-

tential effect of the aggregation bias on the operators’ efficiency scores is enormous.

Keywords: Benchmarking; Data envelopment analysis; DEA; Aggregation bias; Potential sub-unit

efficiency; Regulation

2

1. Introduction

In efficiency analysis, it is customary to work with aggregated production data. Not only non-

homogenous inputs and outputs are aggregated into a few input and output categories, but also

spatially distributed production entities as well as temporally consecutive production processes are

aggregated, often into only one overall, consolidated unit to be evaluated. In the terminology of

data envelopment analysis (DEA) which is addressed in the paper, this means that rows (variables)

as well as columns (individual production entities/processes) are aggregated before the actual eval-

uation of the consolidated unit. While the latter is usually referred to as decision making unit

(DMU), its individual production entities and production processes, respectively, are hereinafter

called sub-units.

The above-mentioned aggregations affect the evaluation. We can usually identify more ineffi-

ciency the more we aggregate. However, especially the aspect of sub-unit aggregations has not

received much attention in the efficiency analysis literature. Little is known about the magnitude

of the aggregation impact, and little is known about how this impact can be measured. The present

paper wants to shed some light on this issue, stressing particularly the possible bias resulting from

the evaluation of consolidated DMUs.

This bias occurs under the usual DEA assumption of convexity of the production possibility set.

Here, a DMU can improve its overall profit by adjusting production to variations in prices over

space and time, while its technical efficiency based on physical production data will deteriorate

because a convex technology favors producing the average output using the average input. In other

words, if a consolidated DMU operates several sub-units that serve different areas with different

needs and characteristics, the DMU may easily appear inefficient even though it is in fact operating

3

optimally in the different areas. Such a shortcoming, subsequently referred to as aggregation bias,

analogously applies for different time spans of a period which are to be served differently by a

DMU.

A series of plausible examples can be found which are prone to the aggregation bias, ranging

• from special area-related cases like the one of German savings banks ,whose local branches

have to cover an assigned region, no matter how diverse the areas of this region may be (Ahn

and Le, 2015),

• to the general case of companies which are present on diverse markets, e.g. worldwide with

corresponding international production sites, as well as

• from special time-related cases like the one of Taiwanese design consultancies, for which it has

been shown that it is advantageous to match their the time-based business strategy with the

actual conditions of the market environment (Sung et al., 2010),

• to the general case of companies which have to adjust their production to seasonally fluctuations

in demand, e.g. in the travel industry.

As a further example where the aggregation bias matters, we will emphasize the regulatory context

in the paper. On one site, firms are regulated because they benefit from a regional monopoly. On

the other site, these firms have to deal with the specifics of their particular market region. Such a

region may be characterized by a significant heterogeneity of its individual service areas, e.g., due

to variations of the production environment and customer structure. Consequently, a firm affected

by this scenario (as DMU) will develop different strategies for its different service areas (as sub-

units) to maximize its overall profit. Due to the described aggregation bias, however, such a DMU

is likely to receive an unjustifiable harsh evaluation by the regulatory authority which does not take

into account the sub-unit structure.

4

In addition to the negative impact of the aggregation bias on mere efficiency evaluation, at least

two further issues arise. First, aggregated evaluations are difficult to connect with more detailed

evaluations of other evaluators of the same production system: economists using aggregate data

and engineers using detailed data main obtain conflicting results; marketing analysts can be im-

pressed by a firm’s adaptability to changing market conditions, and yet, an aggregate evaluation

may show deficient allocative efficiency because of the underlying convex model. This impedes

the communication and trust between the respective stakeholders.

Second, the aggregation bias induces adverse incentive effects. A contract with a DMU designed

to improve technical and allocative efficiency (cf., e.g., Bogetoft and Otto, 2011; Bogetoft, 2012)

may diminish responsiveness to variations in the market condition because adjustments to inputs

and outputs will manifest as increased technical inefficiency in the aggregated evaluation. Hence,

contracting on technical efficiency may lead to lower profitability and, more generally, poorer goal

fulfillment. It is important that incentive schemes take this problem into account. For example,

leading regulators of electrical networks have made corresponding amendments to their revenue

cap regulation (cf., e.g., the merger approach used by the Norwegian regulator as discussed in

Bogetoft, 2012).

Against this background, the paper suggests an approach to counter the aggregation bias by evalu-

ating the potential impact of sub-unit aggregations on a DMU’s DEA efficiency without actually

having disaggregated sub-unit data. The approach involves hypothetical disaggregations to inves-

tigate whether the conclusions derived from the aggregate information could be significantly al-

tered by more detailed information.

5

The impact our approach can have on efficiency analysis is verified on the basis of data from the

Brazilian Electricity Regulator used in 2015 to measure the cost efficiency of the Brazilian distri-

bution system operators (DSOs). These DSOs are confronted with a – in particular geographically

– heterogeneous business environment that is susceptible to the aggregation bias. Accordingly, our

findings impressively indicate the relevance the aggregation bias may have. In comparison to the

results of the Brazilian DSO model, the number of DSOs identified as efficient clearly raises when

using our approach. Furthermore, the extent by which the particular efficiency scores increase is

noteworthy.

This paper is organized as follows. After an overview of the related literature in Section 2, Section

3 provides a conceptual introduction to the idea of distinguishing between disaggregated sub-unit

efficiency and aggregated, organizational efficiency. We provide simple examples of the aggrega-

tion bias, explain how this bias arises, and outline the measure of potential sub-unit efficiency (PSE)

which serves to correct a biased evaluation of organizational efficiency. As starting point to for-

malize the PSE concept, Section 4 discusses the condition under which the aggregation bias does

not appear. On this basis, Section 5 focuses on the opposite case. After proposing a general ap-

proach to measure PSE referring to Farrell efficiency, we elaborate an operational PSE measure

for the case of missing sub-unit information and also simplify the resulting mixed-integer program.

In Section 6, our model is applied to the case of the Brazilian DSO regulation to investigate to what

extent these DSOs may be affected by the aggregation bias. Final conclusions are provided in Sec-

tion 7.

6

2. Literature review

We are not aware of papers that are directly related to the present study. There are, however, at

least four lines of research with links to the issues we address.

One line of studies relates to sensitivity analysis. To determine a simple but sufficiently detailed

model, one may examine how sensitive the results (efficiency scores etc.) are to aggregations over

space and time, respectively. This issue is of course only one of several that can be addressed by

sensitivity analysis. For other applications in DEA models, see for example Charnes and Neralic

(1990), Ray and Sahu (1992) as well as Charnes et al. (1996). However, our approach does not

calculate the consequences of given aggregations, but we seek for the disaggregation that has the

maximal effect on the results. We construct worst-case scenarios to put the evaluated DMUs in the

best possible light.

Another line of research involves investigating under which conditions it is possible to make exact

aggregations. Many studies address the relationship between the aggregation of variables and the

separability of the DMU into several sub-units. Although aggregating sub-units and variables is

not the same, the former can be considered an instance of the latter if we interpret inputs and outputs

from different sub-units as different inputs and outputs. Then the question is when these can be

aggregated over space and time, respectively, i.e., when it is safe to aggregate the variables de-

scribing the same type of input or output in the different sub-units. In this context, there are a few

studies explicitly linking aggregation and separability issues to the efficiency measurement prob-

lem (cf. Färe and Lovell, 1988). We deviate here by examining specific production entities instead

of general structures. Furthermore, we are mainly concerned with the impact of inexact aggrega-

tions rather than with the circumstances under which exact aggregations are possible. However, in

7

Section 4, we discuss a fundamental condition, sub-unit alignment, that allows for exact aggrega-

tions with respect to efficiency evaluations.

A third line of research is to combine the methods from efficiency analysis, including the estima-

tion of production models using observed productions, with those reported in the production plan-

ning literature, including the use of linear programming to model networks of production processes.

Based on the seminal paper of Färe and Grosskopf (2000a) about network DEA, a series of con-

ceptual variations (cf., e.g., Lewis and Sexton, 2004) as well as application-oriented contributions

(cf., e.g., Mirhedayatian et al., 2014) were provided. We deviate here from this literature by two

aspects. First, we do not address the sequential interrelation of production processes of a DMU.

Instead, we address sub-units of a DMU which are assumed as independent production instances.

Second, we assume that no detailed information about these sub-units are available and investigate

how this limitation could affect the DEA efficiency scores.

A final line of inquiry is that of DEA-based merger analysis. In a series of contributions, bench-

marking models were used to make ex ante predictions of the likely gains from mergers, cf., e.g.,

Bogetoft et al., 2003 as well as Bogetoft and Wang, 2005; the approach suggested in the latter

paper has meanwhile been adopted by regulators in the Netherlands and Norway to guide decision

making and incentive regulation in the context of mergers in the hospital and the energy sector (cf.

Bogetoft, 2012). Furthermore, Andersen and Bogetoft (2007) as well as Bogetoft et al. (2007) ex-

amined the effect of allowing more general reallocations of some of the resources and services

within a large number of market participants. While those studies progressed from less aggregated

to more aggregated production units, the present paper goes in the opposite direction, from aggre-

gated to disaggregated production units. We ask what bias may occur when DMUs are evaluated

as if there are synergies, but when such synergies in fact do not exist because of geographic or

8

temporal barriers. Hence, whereas the merger literature studies hypothetical mergers, the present

paper studies hypothetical disaggregations of consolidated DMUs.

To sum up, we seem to add a new aspect to the vast DEA literature. Yet, from a broader perspective,

our paper can be assigned to the research stream of DEA under the condition of a so-called cen-

tralized management (cf., e.g., Lozano and Villa, 2004, and most recently, Afsharian et al., 2017).

This notion covers contributions in which a central evaluator not merely puts every DMU in its

best possible light, but incorporates management control mechanisms into the efficiency measure-

ment. These mechanisms incentivize the DMUs (as agents) to make decisions in accordance with

the goals of the regulator (as principal). In line with this, our approach applies such a mechanism

which counters the aggregation bias, preventing large companies to reorganize into smaller units

only because this leads to higher efficiency scores. The underlying idea of distinguishing between

disaggregated sub-unit efficiency and aggregated organizational efficiency is illustrated in the fol-

lowing, using two motivational examples.

9

3. Exemplary description of the aggregation bias and its quantification

Most companies comprise individual sub-units operated in parallel as well as over time. When

these sub-units are described in aggregate terms, the adaptation to variations in supply and demand

over space and time will be suppressed. As already outlined, actually favorable sub-unit adaptions

may lead to unfavorable aggregate evaluations of the organization as a whole. In such a setting, we

can distinguish between two efficiency notions:

• Sub-unit efficiency: Individual sub-units cannot be improved.

• Organizational efficiency: The aggregate of the sub-units cannot be improved.

The issue we investigate is that an organization may have fully efficient sub-units but still appears

inefficient on an aggregate level – we call this the aggregation bias.

A simple example is the so-called Fox paradox (cf. Fox, 1999, 2004). One version of the phenom-

enon is illustrated in Table 1, which describes the case of two companies, e.g. electrical network

firms, both serving rural and urban costumers. DMU A serves 2 rural and 4 urban costumers and

spends 1 on the rural and 1 on the urban costumers. The unit costs (UC) of the two types of con-

sumers are therefore 1/2 and 1/4, respectively. In total, DMU A has spent 2 to serve 6 costumers,

and the aggregate UC are 2/6. The other numbers can be interpreted analogously.

DMU

Rural UC

Urban UC

Aggregate UC

A

1/2

1/4

2/6 = 0.33

B

1,5/2

21/80

22,5/82 = 0.27

Table 1 Fox paradox as example for the aggregation problem

The interesting observation is now that DMU A has

10

• lower rural unit costs (cost rural / rural costumer) and

• lower urban unit costs (cost urban / urban costumers) but

• higher overall unit costs (cost / costumers).

That is, sub-unit by sub-unit, DMU A is more efficient than DMU B, and yet, at the aggregate

level, DMU B is the more efficient unit. The explanation is simple: The relatively more efficient

activity of serving urban costumers plays a larger part in DMU B than in DMU A.

In some cases, we can consider this situation a resolvable allocation problem: DMU A allocates an

excessively large share of its consumers to the least efficient sub-unit. If costumers are freely trans-

ferable, DMU A is indeed responsible for this misallocation, and the aggregate appraisal might be

fair. The important point, however, is that in cases such as the electrical network industry (and

many others, as outlined in Section 1), companies cannot be held responsible for all allocation

problems. For example, they cannot freely reallocate the costumers between rural and urban areas.

In such a situation, the aggregate evaluation becomes biased. DMU A is blamed for performance

aspects it cannot control, one of the most obvious mistakes in adequate performance evaluations.

Indeed, DMU A should appear efficient because for no common composition of costumers can

DMU B outperform DMU A.

Färe and Grosskopf (2000b) describe how to conceptually avoid the Fox paradox by applying

solely additive efficiency measures. However, this way to define efficiency is not compatible with

the classical DEA approach addressed in this paper which applies a ratio efficiency measure.

Hence, the aggregation bias remains a possible pitfall of DEA, as is discussed in the following

example.

Imagine that there are 32 DMUs, each of which has used two inputs to produce the same amount

11

of one output. Fig. 1 illustrates these DMUs as numbered, filled points, and Table 2 lists the re-

spective data set. We observe that all DMUs except for DMU 32 are fully efficient.

Now, assume that we also consider all consolidated versions of the efficient DMUs, i.e., all sums

of two efficient DMUs including the sum of one DMU with itself. When we also allow for

downscaling with 1/2 to scale down the inputs of the consolidated units to give an output of 1, we

obtain the other, unfilled points in Fig. 1.

Fig. 1. Simple DEA example to describe the aggregation problem

Input 1

Input 2

12

!

DMU

Input 1

Input 2

Output

PSE* score

DMU

Input 1

Input 2

Output

PSE* score

1

2.00

0.00

1.00

1.00

17

0.47

0.60

1.00

1.75

2

1.90

0.03

1.00

1.00

18

0.43

0.70

1.00

1.57

3

1.80

0.07

1.00

1.00

19

0.40

0.80

1.00

1.42

4

1.70

0.10

1.00

1.00

20

0.37

0.90

1.00

1.30

5

1.60

0.13

1.00

1.00

21

0.33

1.00

1.00

1.20

6

1.50

0.17

1.00

1.00

22

0.30

1.10

1.00

1.11

7

1.40

0.20

1.00

1.00

23

0.27

1.20

1.00

1.03

8

1.30

0.23

1.00

1.00

24

0.23

1.30

1.00

1.00

9

1.20

0.27

1.00

1.03

25

0.20

1.40

1.00

1.00

10

1.10

0.30

1.00

1.11

26

0.17

1.50

1.00

1.00

11

1.00

0.33

1.00

1.20

27

0.13

1.60

1.00

1.00

12

0.90

0.37

1.00

1.30

28

0.10

1.70

1.00

1.00

13

0.80

0.40

1.00

1.42

29

0.07

1.80

1.00

1.00

14

0.70

0.43

1.00

1.57

30

0.03

1.90

1.00

1.00

15

0.60

0.47

1.00

1.75

31

0.00

2.00

1.00

1.00

16

0.50

0.50

1.00

2.00

32

1.00

1.00

1.00

1.00

* PSE: potential sub-unit efficiency

Table 2 Data set of the simple DEA example and PSE scores

Let us look at one example of these consolidated units: If we add 0.5*DMU 1 and 0.5*DMU 31,

we obtain the input combination (1,1), producing an output of 1. These are precisely the same

inputs and outputs that characterize DMU 32. If, therefore, DMU 32 is really a consolidated DMU,

it might actually be sub-unit efficient even though it is inefficient at the aggregated level. In fact,

if we generalize this example, we can conclude that all unfilled points are potentially sub-unit ef-

ficient. Therewith, our concept of potential sub-unit efficiency (PSE) is basically outlined.1

This concept also takes super-efficiency into account. Concerning the numbered DMUs, e.g., the

ones from 9 through 23 may represent super-efficient performance. Table 1 shows their respective

PSE scores, which are above 1. Thus, these DMUs could have increased their inputs and still could

have been the result of running two efficient sub-units. A PSE score of 1.20, for example, means

1 The model to calculate the PSE scores listed in Table 1 and referred to in the next paragraph will be presented in

Section 5.2.

13

that the corresponding DMU could have expanded the use of both inputs by 20 % and still would

have been potentially sub-unit efficient, i.e., it still could be considered an aggregation of efficient

sub-units.

As an interesting observation, it can be derived that centrally located DMUs have larger aggrega-

tion corrections. The reason is that they are more likely to be the results of a consolidation of sub-

units with rather different input-output profiles. The more extreme DMUs have a smaller correc-

tion, i.e., when evaluated as consolidated units, their aggregation bias is smaller, because they are

less likely to be the result of aggregating sub-units with very different input-output profiles.

In the following, we step by step formalize our PSE concept. As starting point, the next section

formulates and discusses the condition of sub-unit alignment under which the aggregation bias does

not occur.

14

4. Efficiency under the condition of sub-unit alignment

Let us consider a DMU that uses m inputs ! " #$

%&'()*to produce n outputs + " #$

,*&'(). Let T

be the production possibility set, with - " #$

%$, . We assume that T is closed, convex and freely

disposable. (x,y) "*T is weakly efficient (or strictly non-dominated) if

! . /0 + 1 2 3 -*456*788* /0 2 " #$

%$, *0 /0 2 9 ( (1)

where (a,b) >> 0 means that all coordinates of (a,b) are strictly positive. Let the set of weakly

efficient productions in T be denoted W (T). We note that weak efficiency is weaker than the clas-

sical economic notion of efficiency which characterizes a production as efficient when we cannot

increase any output (decrease any input) without decreasing (increasing) another output (input). A

production that can be improved in some but not all dimensions is weakly efficient.

We shall say also that (x,y) "*T is allocatively efficient with respect to a price vector :;0 <= "

*#$

%$, &'() if and only if

!0 + " 76> ?7@

:AB0CB="D <+B . ;!B. (2)

In our theoretical analysis, it is advantageous to use the weak efficiency notion. First, the notion

simplifies several of the results presented below for which we would have to assume unique solu-

tions or solutions with strictly positive prices if we would work with ordinary efficiency to avoid

picking up points on horizontal and vertical segments of the production possibility frontier (cf.,

e.g., Bogetoft and Pruzan, 1991, Appendix A). Second, with convex sets, the weakly efficient pro-

ductions are all those that result from optimal economic behavior (profit maximization) for some

non-negative and non-zero price vector. Finally, in the efficiency analysis literature, it is common

to work with notions of efficiency that are weaker than the classical economic one.

15

To take up again our assumptions, by disposability, W (T) constitutes the ”North-East” boundary

points of T. Moreover, since T is closed and convex, it is the intersection of all halfspaces {(x´,y´ ) |

vy – ux ≤ r} containing T (cf. Rockafellar, 1970, Theorem 11.5). Now, note that:

Lemma 1. Any halfspace containing T has a normal (–u,v), where (u,v) is non-negative.

Proof. Assume that v has a negative component vi. Let ei = (0,...,0,1,0,...,0), and denote the i’th unit

vector in output space. Thus, for any (x,y) " T and any r, there exists a positive scalar E such that

(y – Eei)v – xu*F r, violating the free disposability of T. If instead u has a negative component, we

can increase vy – ux by increasing the corresponding input coordinate, which again would violate

free disposability. □

This finding implies that all supporting hyperplanes {(x´,y´) | vy´ – ux´ = r} through a boundary

point have (u,v) ≥ 0. In efficiency terms, this expression may alternatively be written as follows:

Lemma 2. A production (x,y) " T is weakly technically efficient if and only if it is allocatively

efficient with respect to some price vector (u,v)*" #$

%$, &'().

Hence, the weakly technically efficient productions are allocatively efficient under some (non-

negative and non-zero) price vector and vice versa.

Now, to investigate the aggregation problem, let us assume that the production of a DMU can be

split into |H| sub-units, h*"*H = {1,…,|H|}. For any h, let (xh,yh) " #$

%$, be the production vector,

let T h be the set of possible productions for h, and when applicable, let (uh,vh) " #$

%$, \{0} be the

non-negative prices of the inputs and outputs of h. For simplicity, we assume that inputs and out-

puts are freely disposable, i.e., for any T h we assume that !B0 +B " -G0 !BB H !B0 +BB I +B*

* !BB0 +BB " -G. Furthermore, we assume that T h, h*"*H, are closed and convex.

16

We can now provide a necessary and sufficient condition for aggregations to create no biases in

the evaluations: This necessary and sufficient condition for efficiency at the aggregate level is al-

locative efficiency at the sub-unit level with regard to the same set of prices – we call this the

alignment condition.

Proposition 1. If T h is convex for all h " H, the aggregate production :!G0 +G=

G"J is weakly

technical efficient at the aggregate level

:!G

G"J 0 +G= " K: -G=

G"J (3)

if and only if the sub-unit productions (xh,yh), h "*H, are allocatively efficient at the sub-unit level

with the same rate of substitutions in all sub-units:

L ;0 < " #$

%$, & ( M !G0 +G* " *76>* *?7@

A0C "DN*<+ .;!**ON " PQ (4)

Proof: When -G is convex for all h " H, so is -G

G"J . Therefore, when the aggregate output is

efficient, :!G

G"J 0 +G= " K: -G

G"J ), we know from Lemma 2 that it is allocativly efficient with

respect to some price vector

L ;0 < " #$

%$, & ( M !G0 +G

G"J * " *76>* *?7@

A0C " DR

R"S

*<+ .;! (5)

In turn, this relationship implies

L ;0 < " #$

%$, & ( M !G0 +G" *76>* *?7@

A0C "DN*<+ .;!**ON " P (6)

because if we use the same price vector as in (5) and :!G0 +G= does not solve ?7@ A0C "DR<+ .;!

for some h, we could have found a better solution in (5) by substituting the solution into

17

?7@ A0C "DR<+ .;! for the old :!G0 +G= in (5). This finding shows that allocative efficiency at the

sub-unit level with the same rate of substitution in all sub-units is a necessary (only if) condition

for aggregate efficiency.

The sufficiency also follows from a contradiction. Let :!G0 +G= * " * 76> *T/! A0C "DR<+ .;!0 N "

P be a solution to (6). Now, if !G0 +G

G"J does not solve (5), there exists an alternative solution

!G0 +G

G"J such that < +G. ; !GU < +G. ; !G

G"JG"JG"JG"J . Thus, for at least one

h, we have <VG. ;!GU<+G.;!G, which contradicts the fact that we had a solution (6) to begin

with. Hence, a solution to (6) yields a solution to (5), and by Lemma 2, the result is weak technical

efficiency. This proves the sufficiency (if) part of the proposition. □

Proposition 1 shows that to attain technical efficiency at the aggregate level, it is not enough to be

technical and allocatively efficient at the sub-unit level. Optimal profit-maximizing behavior does

not ensure aggregate technical efficiency. We also need for the sub-units to be aligned by a com-

mon price vector in the sense that the rates of substitution are the same in all sub-units. In more

organizational terms, we can conclude that it is not sufficient to have efficient sub-units; we need

goal concordance among them as well.

Note that goal concordance in this case is considered with respect to a common weighting of the

inputs and outputs, which it is not necessarily advantageous. If prices differ, concordance regarding

the same weights is sub-optimal. In such a case, the alignment condition in Proposition 1, which

implies that scale and scope effects do not come into play, causes a discrepancy between sub-unit

performance and aggregated, organizational performance. In other words: in addition to optimal

behavior in the different sub-units, we need price proportionality in the sub-units to aggregate in-

formation without obscuring the evaluations.

18

One instance of Proposition 1 is particularly clear. If all sub-units are the same, T 1 = T 2 =…= T |H|, a

necessary condition for technical efficiency at the aggregate level is that they all operate if not at

the same point, then at least at the same facet. This follows immediately from the alignment con-

dition in Proposition 1. Furthermore, if the technologies are strictly convex – at least in the relevant

(efficient) part – we conclude that the aggregate production is efficient if and only if all sub-units

are using exactly the same (efficient) production.

This situation illustrates how desirable adjustments to spatial/temporal variations in prices at the

sub-unit level will tend to appear as technical inefficiencies at the aggregate level. Optimal eco-

nomic behavior in the sub-units leads to aggregate inefficiency if prices over space/time are not

accidentally proportional. It is fair to say, therefore, that aggregate efficiency is quite unlikely and

certainly not always desirable; such efficiency may come at the cost of inadequate adaptions to

local variations in prices or, more generally, to variations in demand and supply.

We note that these statements regard aggregate efficiency in an absolute sense, i.e., when compared

with a theoretical production model. In practice, a DMU’s efficiency is usually measured relative

to other DMUs, and because they may be ”handicapped” by variations in the prices on the dis-

aggregated level as well, a DMU may be more likely to be (relatively) as efficient as an aggregate

unit.

In the next section, we propose an approach to make more fair evaluations when DMUs cannot

freely allocate production between different sub-units, i.e., when the condition of sub-unit align-

ment is not valid.

19

5. Potential sub-unit efficiency

5.1. A PSE approach based on Farrell efficiency

In practice, it is uncertain if inefficiency at the aggregate level is the result of slack in the sub-unit

productions or the result of an aggregation of non-aligned but efficient sub-units. We may, how-

ever, ask if a DMU’s observed inefficiency is sufficiently small to be explained by unobserved but

potentially desirable adjustments of its sub-unit productions to local or temporal variations in

prices. Taking this perspective a bit further, we may ask how much of aggregate inefficiency we

can explain by the aggregation of sub-units alone.

If the observed aggregate production can be expressed as an aggregation of efficient sub-unit pro-

ductions, we say that the aggregate production is characterized by potential sub-unit efficiency

(PSE). More precisely, we say that (x,y) " T is potentially :-G0 N " P= sub-unit efficient (PS effi-

cient) if and only if

L !G0 +G"WXX -G0 N " PM ! I !G0 + H +G

G"JG"J (7)

where WXX -G is the efficient production in T h for some notion of efficiency. Hence, a PS efficient

aggregate production of a DMU is one that results from aggregating efficient – but not necessarily

aligned – sub-unit productions. To test such an aggregate production, we let the sub-units be hy-

pothetical and construct sub-unit productions that put a DMU in the best possible light. On this

basis, we can evaluate whether a production (x,y) is possibly sub-unit efficient, and also measure

the corresponding PSE score.

As always, efficiency can be operationalized in different ways. We here refer to Farrell’s idea of

20

determining input efficiency (E) and output efficiency (F) by means of proportional input and out-

put adjustments, respectively. Based on this well-known concept, we can measure the PSE of (x,y)

on the input side, PSEI (x,y), as

YZW[!0 + \ ?7@'W]W! I !G0 + H +G0 :!G0 +G= " WXX:-G

G"J =0 N " P)

G"J Q (8)

The interpretation of this program is straightforward: We seek the largest possible expansion of the

inputs used in (x,y) such that the resulting production still uses no more inputs to produce at least

the same output as some combination of efficient sub-units. A score of PSEI (x,y) = 1 denotes the

case that (x,y) is PS efficient. If PSEI (x,y) exceeds 1, (x,y) is in fact super-PS efficient. If PSEI (x,y)

is less than 1, even when we take limitations of an alignment between the sub-units into account,

there are some savings to be made in the production (x,y).

Likewise, we can measure PSE on the output side by

YZW^!0 + \ ?_`'a]! I !G0a+ H +G0 :! G0 +G= " WXX:-G

G"J =0 N " P)Q

G"J (9)

The interpretation of this program is again simple: We seek the largest possible contraction of the

outputs produced in (x,y) such that the resulting production still uses no more inputs to produce at

least the same output as some combination of efficient sub-units. A score of PSEO (x,y) = 1 denotes

the case that (x,y) is PS efficient. If PSEO (x,y) is less than 1, (x,y) is super-PS efficient because

there exists a combination of efficient sub-units that produce less than y using at least the same

inputs x. If PSEO (x,y) is greater than 1, even when we allow for limitations of an alignment between

the sub-units, there is some potential to expand the outputs in (x,y).

5.2. Operational PSE measures in case of missing sub-unit information

21

In many evaluation settings, detailed information about the sub-units is missing. We do not know

the set of sub-units H or the production possibilities in these, Th. We shall now discuss how to

measure PSE when such sub-unit information is missing.

Taking a usual benchmarking study as starting point, we assume that each of K DMUs,

DMU1,...,DMUK have used inputs !b" #$

% to produce outputs +b" #$

,. Let T be the underlying

production possibility set, and let T* be an estimate of T based on ':!b0 +b=]c " d). If, for exam-

ple, we assume convexity, free disposability and weakly increasing (non-decreasing) returns to

scale, the minimal extrapolation estimate of T using DEA would be

-e\ ':!0 += " #$

%$, ]! H Eb!b0 + I Eb+b0 EbH f)

b"gb"gb"g (10)

To cope with missing information about the sub-unit production possibilities T h, we propose using

the basic idea of non-parametric benchmarking, namely to rely on observed best practices. One

simple approach is to project the original observations on the efficient frontier of T* and to only

use combinations of these observations in the evaluation of PSE. To this end, let the Farrell input

efficiency of DMUk be

Ek = min{E | (Ex,y) " T h} (11)

and consider

Eff (T h) = {(Ekxk,yk), k " K} ON " P. (12)

That is, we let the efficient outcomes of sub-unit h be the efficient versions of the original DMU

observations. Thereby, an observation (xk,yk) is PS efficient if it is the sum of efficient versions of

the actual observations. This direct aggregation approach leads to a PSE measure which is based

22

on a minimum of speculations regarding what is feasible and which is easy to interpret. The greater

Eff (T k), the more PS efficient the DMUs will appear, i.e., we will obtain larger PSEI and smaller

PSEO scores. We can therefore say that the approach of relying solely on the efficient versions of

the original observation is cautious in the sense that it does not lead to excessively good PSE eval-

uations.

Using the described approximation of the efficient sub-unit outcomes, the calculation of PSEI can

be outlined in four steps:

Step 1: Compute the Farrell input efficiencies Ek of each of the original observations (xk,yk), k*"*K.

Step 2: Determine the efficient sub-unit outcomes as

Eff (T h) = {(Ekxk,yk), k " K} ON " P. (13)

Step 3: Determine the PSE reference technology as all possible additions of efficient observations

- \ ': !G

G"J 0 +G

G"J =]:!G0 +G= " WXX -b*ON " P). (14)

Step 4: Determine the PS input efficiency of the original observations as

YZW[!0 + \ ?7@'W]W! I !0 + H +0 :!0 += " -). (15)

This formulation of the direct aggregation approach can be simplified. The last three steeps essen-

tially correspond to the solution of the mixed-integer program

YZW[!0 + \ ?7@

i0jk0l0jmW

nQ oQ****W! I EbWb!p

b0*****q \ f0 l 0 T

g

brs

23

+ H Eb+t

b0*************u \ f0 l 0 v

g

brs

Eb" (0f0w0x0 l 0*********c0 l 0 d

Eb

g

brs I PQ********* (16)

In this program, we look for a combination of efficient productions (Ekxk,yk), k = 1,..., K such that

the combination uses more inputs to produce less outputs than (Ex,y).

Of course, the procedure outlined above can also be executed using projections in the output direc-

tion in Steps 2 and 4:

Step 2*: Determine the efficient sub-unit outcomes as

Eff (T h) = {(xk, Fkyk), k " K} ON " P. (17)

Step 4*: Determine the PS output efficiency of the original observations as

YZWy:!0 += \ ?_`*'a]! I !0 a+ H +0 :!0 += " -). (18)

The mixed-integer program solving PSEO becomes

24

YZWy!0 + \ ?_`

z0jk0l0jma

nQ oQ****! I Ebab!p

b0*********q \ f0 l 0 T*

g

brs

a+ H Eb+t

b0**********u \ f0 l 0 v

g

brs

Eb" (0f0w0x0 l 0*********c0 l 0 d

Eb

g

brs I PQ********* (19)

In applications, the number of underlying sub-units H to consider may be uncertain. Then, we have

different options. One possibility is to choose the most restrictive option and only allow DMUs to

be hypothetically decomposed into two sub-units, i.e., H = 2. The option leads to cautious results

in the sense that it yields the smallest set of PS efficient outcomes and therefore the smallest scores

of the PSEI measure and the largest scores of the PSEO measure. The DEA example in Section 3

was explained using this version of the direct aggregation approach.

Another possibility is to choose the most flexible option and say that we allow DMUs to be hypo-

thetically decomposed into any number of sub-units, i.e., H = any natural number. This option

yields the largest set of PS efficient outcomes and therefore the largest scores of the PSEI measure

and the smallest scores of the PSEO measure.

5.3. A simplified (relaxed) approach

The aggregation approach is conceptually simple because it directly constructs possible PS effi-

cient outcomes by adding together efficient versions of actual outcomes. Still, computationally, the

approach may seem complicated because it involves mixed-integer programming. It may therefore

be interesting to study the following simplified (relaxed) version of the mixed-integer program:

25

YZW[!0 + \ ?7@

i0jk0l0jmW

nQ oQ****W! I EbWb!p

b0*****q \ f0 l 0 T

g

brs

+ H Eb+t

b0*************u \ f0 l 0 v

g

brs

Eb" #{0*************************c0 l 0 d

Eb

g

brs H fQ********** (20)

This problem is relaxed by allowing the E values to be real numbers as opposed to integers, and by

removing the upper constraint on the sum of these values.2

The program is again straightforward to interpret. It involves performing an output-oriented DEA

efficiency analysis of (x,y) in which we assume convexity, increasing (non-decreasing) returns to

scale (IRS), and free disposability. Furthermore the inputs and outputs are reversed, i.e., the inputs

x are treated as outputs and outputs y are treated as inputs.

Because the relaxation leads to slightly larger scores of PSEI, the efficiency measures calculated in

this manner may exceed the scores calculated using the direct aggregation approach. Hence, the

DMUs will tend to appear more efficient. As we will observe, however, the relaxation may be

modest. In fact, in the example presented in Section 6, the relaxed problem leads to the same scores

yielded by the direct aggregation approach with H = 2.

2 Forcing the sum of E values to be at least one is also effectively a relaxation because we could have introduced such

a constraint in the mixed-integer formulations above without affecting the scores of PSE(x,y).

26

Instead of considering the relaxed formulation an approximation, it is possible to motivate the for-

mulation in its own right. One approach could be to assume that the underlying technology is an

IRS technology. The weighted sum of efficient sub-units can now be rewritten as follows:

: EbWb!b0 Eb+b= \ : |b}bWb!b0 |b}b+b

g

brs =

g

brs

g

brs

g

brs (21)

Where }b\ ceiling Eb" f0w0x0 l and 0 ≤ αk = Eb~}b ≤ 1 for all k=1,…,K. Here, the ceiling

function ceiling(z) is the smallest integer not less than z. Hence, the reference unit used to evaluate

the PSE can be interpreted as the result of two operations:

• Downscaling: The efficient versions of the original observations can be downscaled, making

them possibly super-efficient by the increasing return to scale assumption.3

• Aggregation: The reference unit can be any direct aggregation of a finite number of efficient

and possibly super-efficient sub-units.

Hence, if we accept the IRS assumption (like in the following example), the simplified approach

is conceptually easy to promote in its own right.

3 Note that if (x,y) is feasible in an IRS DEA technology, any downscaled version k(x,y) for k < 1 is either

efficient or super-efficient.

27

6. Application to the Brazilian DSO model

6.1. Motivation

The Brazilian distribution system operators (DSOs) are regulated on the basis of a DEA model

with weight restrictions to determine efficient cost levels (cf. ANEEL, 2015). This example does

not only serve to merely illustrate our approach based on real-world data, but also sheds some light

on actual issues of benchmarking in the Brazilian energy distribution sector.

First, the size of the Brazilian DSOs entails a quite heterogeneous business environment for deliv-

ering their services. Facing in particular considerably different geographical conditions, it is likely

that many of the DSOs should in fact be regarded as consolidations of diverse sub-DSOs that have

limited possibilities to create synergies. If this is the case, the evaluations based on the Brazilian

DSO model may be affected by the aggregation bias.

Second, the fact that weight restrictions are used in the Brazilian DSO model may mitigate the

heterogeneity problem because the resulting isoquants attain a lower curvature. Therefore, although

the aggregation bias is a reasonable presumption, its importance can only be evaluated by a numeric

analysis based on our new approach.

6.2. The Brazilian DSO model

The Brazilian DSO regulation is in many ways in line with the international literature on regulatory

benchmarking. Corresponding models typically use a series of indicators of the capacity provided,

the transport work undertaken and the customer services delivered as cost drivers (cf. Bogetoft,

2012). The respective input and outputs used in the Brazilian DSO model are shown in Table 3,

which also indicates the tasks covered by the different cost drivers.

28

Model variables

Task

INPUT

x_OPEX_adjusted

OUTPUT

y_Underground_all_tension_levels

Physical assets

y_Air_distribution_network

Physical assets

y_High_network

Physical assets

y_Averaged_market

Transport service

y_Consumers_number

Customer service

z_Neg_non_technical_losses_adjusted

Quality

z_Neg_interruption_adjusted

Quality

Table 3 Brazilian DSO model variables

We note that the use of physical assets to capture capacity provision is quite common in regulatory

practice. It is also noteworthy that the model does not contain direct information about the charac-

teristics of service areas, such as precipitation and vegetation, even though these conditions vary

considerably from DSO to DSO as well as across the areas serviced by the individual DSOs. An-

other remarkable feature of the model is the direct inclusion of quality indicators as negative out-

puts. Quality is usually considered a property of the basic services and is typically handled by

second-stage corrections or add-on regulatory instruments (cf. Bogetoft, 2012).

The Brazilian DSO model also di44ers from common regulatory benchmarking models via the use

of restrictions on the dual weights. In total, seven such restrictions are used, as shown in Table 4.

It can be observed that two of the restrictions limit the possible rate of substitution between outputs,

whereas the remaining five restrict the output costs for individual outputs, compared to the input

OPEX (operational expenditure). The first two constraints are so-called Type I assurance regions,

whereas the latter five are Type II assurance regions.

29

Constraint

Lower limit

Ratio

Upper limit

A

1

<

y_Underground_all_tension_levels/y_Air_distribution_network

<

2

B

0.58

<

y_Air_distribution_network/x_OPEX_adjusted

<

2.2

C

0.4

<

y_High_network/y_Air_distribution_network

<

1

D

0.001

<

y_Averaged_market/x_OPEX_adjusted

<

0.06

E

0.03

<

y_Consumers_number/x_OPEX_adjusted

<

0.145

F

0.01

<

z_Neg_Non_technical_losses_adjusted/x_OPEX_adjusted

<

0.15

G

0

<

z_Neg_interruption_adjusted/x_OPEX_adjusted

<

0.002

Table 4 Weight restrictions used in the Brazilian DSO model

Weight restrictions can be considered either an expression of preferences or an expression of partial

information about rates of substitutions. For example, the last restriction listed in Table 4 can be

an expression that the value of avoiding the loss of an hour cannot exceed 0.002 kBRL, i.e., that

the value of an hour of lost electricity cannot exceed 2 BRL. Alternatively, the restrictions can be

an expression that the actual cost of cutting down on the hours of interruption is never higher than

2 BRL per hour. It is not known whether the restrictions are actually expressions of regulatory

preferences or of specific knowledge of cost effects (cf. Bogetoft and Lopes, 2015).

6.3. Findings

The use of weight restrictions is interesting regarding the aggregation bias because they lead to

more linear isoquants, which one would expect to limit the bias. In that respect, it can be determined

that the constraints have a non-trivial impact on the Brazilian DSO model results, i.e. the constraints

actually matter. For the 61 DSOs of our data set, this is illustrated in Fig. 2. Here, the model results

obtained using weight restrictions (the monotonically increasing black points) are compared with

the pure IRS scores obtained without weight restrictions (the upper series of grey points).

30

Fig. 2. Impact of weight restrictions in the Brazilian DSO model

Next, we have calculated the PSEI scores of the Brazilian DSOs using our simplified (relaxed)

approach. The results are shown in Fig. 3. Here, the DSOs are sorted from the smallest PSEI score

to the largest one. As explained above, this sorting illustrates the expansion of costs that is possible

assuming efficient sub-units. We observe that a large share of the DSOs can in fact be considered

as PS efficient. Only 13 of the 61 DSOs remain inefficient, with a PSEI score less than 1, which

means that 48 DSOs are classified as fully PS efficient (and many of them are super-efficient). By

means of the Brazilian DSO model, only 8 DSOs were classified as efficient. We also observe that

nearly half of the DSOs have PSEI scores greater than 1.5, suggesting that they could in fact have

increased their OPEX by 50% and it would still be possible to consider them as sub-unit efficient.

Brazilian DSOs

Efficiencies without and with weight restrictions

black points:

Brazilian DSO model with weight restrictions

grey points:

model without weight restrictions

31

Fig. 3. PSEI scores

The effects of applying our approach are dramatic. Most DSOs obtain significantly better scores

when we consider them as consolidated units and investigate whether they could in fact be decom-

posed into fully efficient sub-units. This finding is illustrated in Fig. 4, in which we compare the

Brazilian DSO model efficiencies (the monotonic series of black points) with the PSEI efficiencies

derived from the simplified approach (the upper series of non-monotonic grey points).

It is obvious that the derived PS efficiency scores are very lenient on the DSOs. This suggests that

one should consider restricting the number of sub-units H in which the PSE analysis is allowed to

hypothetically disaggregate the DSOs. However, our goal was to demonstrate that the aggregation

of data at the DSO level can have a huge impact on the results, i.e. that the potential aggregation

bias can be enormous.

Brazilian DSOs

PS efficiency of the Brazilian DSOs

32

Fig. 4. Comparison of efficiency scores: the Brazilian DSO model versus the PSE model

Brazilian DSOs

efficiency scores

black points: efficiency scores of Brazilian DSO model

grey points: PSEIscores

33

7. Conclusions

In this paper, we have argued that the occurrence of highly aggregated organizational units in a

benchmarking study may skew the results. Such aggregated DMUs are likely to receive excessively

harsh evaluations. We have illustrated this aggregation bias and reflected upon the condition under

which the bias does not occur, namely the alignment condition. Only with aligned productions of

a DMU’s sub-units, an aggregation of these productions does not affect the efficiency analysis of

the DMU. Basically, price proportionality with respect to the sub-units is needed to allow for an

exact aggregation of their productions without obscuring the evaluation on the DMU level.

As explained in the paper, this condition may be problematic in a many real-world cases, where

DMUs have to manage sub-units with different business environments resulting from, e.g., differ-

ent locations or periods. For such cases, we propose a DEA-based approach for compensating for

the possible aggregation bias by calculating a DMU’s potential sub-unit efficiency – PSE. This

concept allows to measuring the extent to which the respective DMU can be viewed as an aggre-

gation of efficient sub-units. A PSE score less than one indicates that – even accounting for given

limitations of an alignment between the sub-units – activities are not performed efficiently. To this

effect, we elaborated how to determine PSE scores under different assumptions.

As example, we applied the PSE concept to the DEA model used by the Brazilian Electricity Reg-

ulator in 2015 to measure the cost efficiency of the Brazilian distribution system operators (DSOs).

Because of the size of these DSOs and the heterogeneity of their service areas, it is highly likely

that many of the DSOs are in fact subject to biased evaluations. Our numerical results showed that

the biases may be considerable. In comparison to the results of the Brazilian DSO model, the num-

ber of DSOs classified as efficient significantly raised, along with a substantial increase of many

34

of the efficiency scores.

The implications of our findings are twofold. From the perspective of a central evaluator, e.g. a

regulator, it is important to be aware of a possible aggregation bias. It seems necessary to investi-

gate whether there are good reasons that the DMUs to be analyzed operate sub-units in different

business environments which require different strategies for performing optimally. In this case,

incorporating the PSE concept into the particular efficiency analysis is a helpful control mechanism

to take the issue into account, inducing fairer and broader accepted evaluations.

Our findings can also be of great relevance from the perspective of particular DMUs under evalu-

ation, since the impact of the aggregation bias on efficiency scores was shown to be potentially

huge. On the one side, it might be in the interest of affected companies to prove that a benchmarking

analysis without considering the bias would be flawed. On the other side, companies may also react

from a strategic point of view, since our findings imply that “playing the regulation” by reorganiz-

ing into smaller sub-units may have a considerable payoff.

As one possibility for further research, our findings could be aligned to bootstrapping in DEA (cf.

the seminal paper of Simar and Wilson, 1999, and for a more recent application, e.g., Gitto and

Mancuso, 2012). We speculate that uncertainty as estimated by bootstrapping is largely inversely

related to the extent of the consolidation bias. While we outlined in Section 3 that the aggregation

bias tends to increase from more to less extreme types of DMUs, the bias correction that can be

derived from a bootstrapping exercise has the opposite tendency. This finding indicates that DMUs

which we are more uncertain about in a typical efficiency analysis are also those that are less likely

to have a large aggregation bias. Vice versa, the firms that we are more certain about in the technical

evaluations will also be those that are more likely to have a large aggregation bias. A thorough

35

investigation of this topic might be fruitful.

36

Acknowledgments

The first author gratefully acknowledges the financial support from the Deutsche Forschungs-ge-

meinschaft (DFG) in the context of the research fund AH 90/5-1.

The third author appreciates the financial support from the Foundation for the Research Support in

Minas Gerais (FAPEMIG) and Companhia Energética de Minas Gerais (CEMIG), Coordination for

the Improvement of Higher Education Personnel (CAPES) and National Council for Scientific and

Technological Development (CNPq) in the context of the research funds APQ-03165-11,

999999.000003/2015-08 and 444375/2015-5, respectively.

37

References

Afsharian, M., Ahn, H., Thanassoulis, E., 2017. A DEA-based incentives system for centrally man-

aged multi-unit organisations. European Journal of Operational Research 259 (2), 587–598.

Ahn, H., Le, M.H., 2015. DEA efficiency of German savings banks: evidence from a goal-oriented

perspective. Journal of Business Economics 85 (5), 953–975.

Andersen, J., Bogetoft, P., 2007. Gains from quota trade: Theoretical models and an application to

the Danish fishery. European Review of Agricultural Economics 34 (1), 105–127.

ANEEL, 2015. Operational costs methodology (translated from Portuguese). Tech. rep., Agẽncia

Nacional De Energia Elétrica ANEEL, Technical Note 66/2015, Brasilia.

Bogetoft, P., 2012. Performance Benchmarking: Measuring and Managing Performance. Springer,

New York.

Bogetoft, P., Boye, K., Neergaard-Petersen, H., Nielsen, K., 2007. Reallocating sugar beet con-

tracts: can sugar production survive in Denmark? European Review of Agricultural Economics 34

(1), 1–20.

Bogetoft, P., Lopes, A., 2015. Comments on the Brazilian benchmarking model for energy distri-

bution regulation. Fourth cycle of tariff review – technical note 407/2014. Working paper,

nespufmg.com.br/content/upload/p/d3d9446802a44259755d38e6d163e820.pdf (accessed April

2017).

Bogetoft, P., Otto, L., 2011. Benchmarking with DEA, SFA, and R. Springer, New York.

38

Bogetoft, P., Otto, L., 2015. Benchmarking package. Tech. rep., R Project.

Bogetoft, P., Pruzan, P., 1991. Planning with Multiple Criteria. North-Holland, Amsterdam.

Bogetoft, P., Strange, N., Thorsen, B.J., 2003. Efficiency and merger gains in the Danish forestry

extension service. Forest Science 49 (4), 585–595.

Bogetoft, P., Wang, D., 2005. Estimating the potential gains from mergers. Journal of Productivity

Analysis 23 (2), 145–171.

Charnes, A., Rousseau, J., Semple, J., 1996. Sensitivity and stability of efficiency classifications

in data envelopment analysis. Journal of Productivity Analysis 7 (1), 5–18.

Charnes, A., Neralic, L., 1990. Sensitivity analysis of the additive model in data envelopment anal-

ysis. European Journal of Operational Research 48 (3), 332–341.

Färe, R., Grosskopf, S., 2000a. Network DEA. Socio-Economic Planning Sciences 34 (1), 35–49.

Färe, R., Grosskopf, S., 2000b. Outfoxing a paradox. Economics Letters 69 (2), 159–163.

Färe, R., Lovell, C.K., 1988. Aggregation and efficiency. In: Eichhorn, W. (Ed), Measurement in

Economics: Theory and Applications of Economic Indices. Physica-Verlag, Heidelberg, pp. 639–

647.

Farrell, M.J., 1957. The measurement of productive efficiency. Journal of the Royal Statistical

Society 120 (3), 253–281.

Fox, K.J., 1999. Efficiency at different levels of aggregation: public vs. private sector firms. Eco-

nomics Letters 65 (2), 173–176.

39

Fox, K.J., 2004. Problems with (dis)aggregating productivity, and another productivity paradox.

Working Paper, University of New South Wales.

Gitto, S., Manusco, P., 2012. Bootstrapping the Malmquist indexes for Italian airports. Interna-

tional Journal of Production Economics 135 (1), 403–411.

Lewis, H.F., Sexton, T.R., 2004. Network DEA: efficiency analysis of organizations with complex

internal structure. Computers & Operations Research 31 (9), 1365–1410.

Lozano, S., Villa, G., 2004. Centralized resource allocation using data envelopment analysis. Jour-

nal of Productivity Analysis 22 (1-2), 143–161.

Mirhedayatian, S.M., Azadi, M., Farzipoor Saen, R., 2014. A novel network data envelopment

analysis model for evaluating green supply chain management. International Journal of Production

Economics 147 (2), 544–554.

Ray, P., Sahu, S., 1992. Productivity measurement in multi-product manufacturing firms evalua-

tion and control through sensitivity analysis. International Journal of Production Economics 28 (1),

71–84.

Rockafellar, R.T., 1970. Convex Analysis. Princeton University Press: Princeton, New Jersey.

Simar, L., Wilson, P.W., 1999. Estimating and bootstrapping Malmquist indices. European Journal

of Operational Research 115 (3), 459–471.

Sung, T.-J., Lu, Y.-T., Ho, S.-S., 2010. Time-Based Strategy and Business Performance under En-

vironmental Uncertainty: An Empirical Study of Design Firms in Taiwan. International Journal of

Design 4 (3), 29–42.

40

Appendix

Implementing the abovementioned measures of PSE is very simple. Using the R package ”Bench-

marking” (Bogetoft and Otto, 2015), we can implement the aggregation approach with two sub-

units, H = 2, as follows:

#Initialize

EFF <- Efficiencies of DMUs in T* model, K x 1 vector

mx <- Inputs of original DMUs, K x m matrix

my <- Outputs of original DMUs, K x n matrix

# Calculate the projection on the efficient frontier

mx_eff <- EFF * mx

# Construct all PSE mergers

merge_matrix <- (K(K+1)/2) x K matrix in which each row contains at least

one and at the most two ones and the rest are 0 entries

X_merge <- merge_matrix % * % mx_eff

Y_merge <- merge_matrix % * % my

# Evaluate original DMUs against PSE mergers

PSE_INPUT_BASED_AGGREGATION <-

dea(X=my,Y=mx,XREF=Y_merge,YREF=X_merge,RTS="FDH", ORIENTA-

TION="out")$eff

Furthermore, we can implement the aggregation approach in the general case with arbitrary H by

using this code:

# Initialize

EFF <- Efficiencies of DMUs in T* model, K x 1 vector

mx <- Inputs of original DMUs, K x m matrix

my <- Outputs of original DMUs, K x n matrix

# Calculate the projection on the efficient frontier

41

mx_eff <- EFF * mx

# Evaluate original DMUs against PSE mergers

PSE_INPUT_BASED_AGGREGATION <-

dea(X=my,Y=mx,XREF=my,YREF=mx_eff,RTS="ADD", ORIENTATION="out")$eff

Similarly, we would implement the simplified (relaxed) approach as follows:

# Initialize

EFF <- Efficiencies of DMUs in T* model, K x 1 vector

mx <- Inputs of original DMUs, K x m matrix

my <- Outputs of original DMUs, K x n matrix

# Calculate the projection on the efficient frontier

mx_eff <- EFF * mx

# Calculate the PSE by the aggregation approach

PSE_INPUT_BASED_SIMPLE <-

dea(X=my,Y=mx,XREF=my,YREF=mx_eff,RTS="IRS", ORIENTATION="out")$eff

Hence, it is easy to calculate the PSE of the DMUs and thereby to understand the possible bias

resulting from working with excessively aggregated DMUs.

- A preview of this full-text is provided by Springer Nature.
- Learn more

Preview content only

Content available from Journal of Business Economics

This content is subject to copyright. Terms and conditions apply.