estimating relative efficiency of dmu: pareto...
TRANSCRIPT
Estimating Relative Efficiency of DMU: Pareto Principle and MonteCarlo Oriented DEA Approach
Gongbing BiManagement School, University of Science and Technology of China, Jinzhai Road, Hefei,
Anhui province, 230026, China, e-mail: [email protected]
Chenpeng FengManagement School, University of Science and Technology of China, Jinzhai Road, Hefei,
Anhui province, 230026, China, e-mail: [email protected]
Jingjing DingManagement School, University of Science and Technology of China, Jinzhai Road, Hefei,
Anhui province, 230026, China, e-mail: [email protected]
M. Riaz KhanOperations and information system, University of Massachusetts Lowell, Lowell, MA, 01845, USA, e-mail: [email protected]
Abstract—The traditional data envelopment analysis (DEA) models treat a decision making unit(DMU) as a “black box”, which is often criticized for not considering the inner productionmechanism of a production system. The network DEA models developed to overcome thisdeficiency by considering the internal structure of a DMU have recently gained popularity. Theinner data, however, are not generally available for real application purposes. This paper, on onehand, addresses the problem with the traditional DEA for not considering the inner structure and,on the other, with the network models for missing inner data in parallel production settings.Procedures built on managerial information of production processes, as characterized by the Paretoprinciple, are presented that consider the inner production mechanism as well as the dataavailability in a reliable way. Firstly, the production activities of a DMU are classified into a corebusiness unit (CBU) and a non-core business unit (NCBU). Secondly, the internal informationrelated to inputs/outputs is assumed to be available for the DMU under evaluation; whereas forthe other DMUs, this data is generated by using the Pareto principle. In addition, the Monte Carlomethod, also known as the parametric bootstrap, is applied to estimate the efficiency of the DMU.A numerical example illustrates the proposed method.
Keywords Data envelopment analysis (DEA), Pareto principle, Monte Carlo, parallel productionsystem, efficiency.
1. INTRODUCTION
The data envelopment analysis (DEA) is a mathematical pro-
gramming technique that can be used to evaluate the relative
efficiencies of a set of decision making units (DMUs) involving
multiple input/output entities. The technique was originally
introduced by Charnes et al. (1978). In this pioneer paper,
the authors constructed a nonlinear programming model,
referred to as CCR model in literature, to evaluate the efficiency
of an activity conducted by a non-profit organization.
As is known, the CCR model captures both technical and
scale inefficiencies. Banker et al. (1984) proposed a new
approach, which extended the CCR model by separating tech-
nical efficiency and scale efficiency. As a nonparametric tech-
nique, DEA doesn’t require a priori information of production
technology and it has been proven to be an excellent tool for
evaluating DMUs (Zhu, 2000).
Recently, the DEA technique has been widely applied to the
public sector, such as schools and hospitals, and has also been
adopted by a range of business industries. Chiu et al. (2010), forReceived August 2011; Revision November 2011, Accepted
January 2012
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57ISSN 0315-5986 j EISSN 1916-0615
44
instance, used it to evaluate hotels’ performance, Hwang et al.
(2010) used it to formulate stock trading strategies, and Zhou
et al. (2008) used DEA technologies for measuring environ-
mental performance, just to name a few.
The traditional DEA models treat a DMU under evaluation
as a “black box”. Consequently, it is difficult to provide the
manager with specific information concerning the sources of
inefficiency within that DMU. An understanding of the inner
mechanism of a DMU is highly relevant to improving its effi-
ciency. Studies on DMU with parallel structures are geared
towards narrowing this gap. Included among these studies are
Yang et al. (2000), Kao (2009), Castelli et al. (2004).
However, the use of parallel DEA model is often constrained
for not having detailed inner data of other DMUs. This lack
of data poses itself as a bottleneck for the application purposes.
The current study aims at extending the applications of parallel
models to DMUs by incorporating the Pareto principle in the
process.
The Pareto principle, also known as 80/20 rule, is a well
established empirical guideline initially suggested by
Vilfredo Pareto (1971). It says that 80% of the national
wealth is owned by 20% of the population. Ever since its intro-
duction, the principle has been found workable in many other
scenarios and has earned credibility. Koch (2003), for instance,
points out that 80/20 rule is one of the ground rules in commer-
cial fields. He further elaborates that 80% of returns, outputs,
and outcomes, are derived from 20% of inputs, efforts and
reasons. This rule has also been used extensively in the aca-
demic circles. Mizuno et al. (2008) studied the statistical prop-
erties of the expenditure per person in convenience stores. The
results showed that 25% of the major customers accounted for
80% of the overall consumption.
In the current study, we propose to divide the production
activities within a DMU into two subsets or units. The first
unit is termed as the core business unit (CBU), which includes
the main production functions of DMU; the second unit is
referred to as the non-core business unit (NCBU). The Pareto
principle implies that, as a rule of thumb, the CBU produces
80% of total outputs of a DMU, while consumes only 20%
of total inputs. Though Pareto principle has only statistical sig-
nificance, it can provide insight to address the issue of unavail-
ability of data in many real situations. Differentiation between
production functions allows construction of a general model
guided by the Pareto principle. The model is then solved by
using the Monte Carlo method. The proposed methodology
indicates that it is feasible to open the “black box” to estimate
the efficiency of DMU under evaluation, even if the internal
data of other DMUs are missing or cannot be secured.
In what follows, the paper first presents a review of literature
concerning the applications of bootstrap method in DEA in
section 2. Some distinct properties of production possibility
set (PPS) are presented in section 3.1. A special case, which
satisfies 80/20 rule, is discussed in section 3.2 to explain the
model. The model is then extended to a general evaluation
model in section 3.3 by incorporating the Pareto distribution
in which the efficiency can be simulated by Monte Carlo
method. A numerical example is set forth in section 4 to illus-
trate the implementation of the model and demonstrate its
applicability. Finally, the paper ends with conclusions in
section 5.
2. BOOTSTRAP IN DEA
The bootstrap method, first introduced by Efron (1979), is
based on the idea of repeatedly simulating the data generating
process (DGP), usually through re-sampling, and applying
the original estimator to each simulated sample so that resulting
estimates mimic the sampling distribution of the original esti-
mator (Simar and Wilson, 1998). Bootstrap can be classified
into two kinds, namely, parametric bootstrap and nonpara-
metric bootstrap. The parametric bootstrap is also well known
as Monte Carlo. When the distribution of population is
known, the Monte Carlo method can be used to make statistical
inferences by sampling from the population. However, para-
metric bootstrap will not be applicable when the distribution
of population is unknown. In that case, mimicking the sampling
distribution of the original estimator by re-sampling is an
alternative.
In recent years, many research studies involving parametric
or nonparametric bootstrap method have concentrated on DEA
issues. Most of these studies are application-oriented. Yu
(1998), for instance, used a Monte Carlo experiment to
compare parametric and nonparametric approaches. By using
simulated data, Ruggiero (1998) compared different
approaches developed in the context of DEA. Similar works
can be found in Syrj€anen (2004) and Mu~uiz et al. (2006).
In addition, the Monte Carlo studies and experimental
designs related to DEA also extend to measuring the influence
of random noise, number of replications, sensitivity analysis of
the number of DMUs employed, examining the statistical per-
formance of bootstrapping estimator, and so on.
Apart from the Monte Carlo method, recent developments in
the bootstrapping techniques are mainly focused on avoiding
bias in the estimation of efficiency scores and on assessing
the uncertainty surrounding these estimates. Simar and
Wilson (1998) introduced the bootstrap method to analyze
the sensitivity of efficiency scores relative to the sampling vari-
ations of the estimated frontier. They also developed a consist-
ent bootstrap estimation procedure for obtaining confidence
intervals for Malmquist indices and their decompositions
(Simar and Wilson, 1999). In an effort to increase the accuracy
of a frontier, Florens and Simar (2005) proposed a two-stage
approach that provides parametric approximations of nonpara-
metric frontiers by using the bootstrap technique. In addition,
Staat (2002) applied some recently developed bootstrap tech-
niques to derive bias-corrected efficiency scores for a model
representing groups and hierarchies in DEA. Borger et al.
(2008) explored a selection of recently proposed bootstrapping
ESTIMATING RELATIVE EFFICIENCY OF DMU 45
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
techniques to estimate non-parametric convex (DEA) cost fron-
tiers and efficiency scores for transit firms. Essida et al. (2010)
measured the efficiency of high schools in Tunisia. In their
paper, they used a statistical DEA with quasi-fixed inputs in
order to estimate the precision of the measures and to construct
confidence intervals for efficiency measures. Curi et al. (2011)
estimated technical efficiency of each of 18 Italian airports by
means of a bootstrapped DEA model.
3. MODEL
It’s reported that loyal consumers contribute significantly to a
store’s sales in Japan. The statistical law, governing the expen-
diture per person in convenience stores, conforms to Pareto
principle in that country, where the top 2% and 25% of custo-
mers account for, respectively, 25% and 80% of the store’s
sales (Mizuno et al., 2008). The work in this paper is partially
motivated by the evaluation of the efficiencies of aforemen-
tioned stores. Obviously, the inner data of DMUs are unavail-
able or costly for the manager except that of the DMU under
consideration. Consequently, the network DEA models are
inapplicable if we want to evaluate the efficiencies of stores
and estimate their production potential precisely, or, to find
the inefficiency source within the unit. Moreover, the tra-
ditional DEA models tend to overestimate the efficiency of
DMUo for not considering the inner production mechanism.
Inspired by the above arguments, we propose to make full
use of the managerial information in the evaluation model so
as to estimate the production potential of the DMUo precisely
with incomplete information.
3.1. Properties of PPS
Consider an organization conducting n homogeneous decision
making units. Let DMUjð j ¼ 1; . . . ; nÞ be a typical unit, which
uses m inputs Xijði ¼ 1; . . . ;mÞ, to produce s outputs
Yrjðr ¼ 1; . . . ; sÞ. In parallel DEA model, a general hypothesis
is that there are Pj parallel Sub DMUs (SDMUs) within DMUj.
The typical sub unit SDMUj consumes m inputs, as allocated by
the production unit, and it produces s outputs, which make up
for the output of the unit. Thus, we obtain Xj ¼P pj
k¼1 Xkj ,
Yj ¼P pj
k¼1 Ykj .
According to value driver, as proposed earlier, the pro-
duction activities within the DMU are classified into core and
non-core business units, termed respectively as CBU and
NCBU. It is assumed that the proportion of the outputs gener-
ated by CBU is relatively fixed in relation to the total outputs
across all DMUs and is determined by the production activities
in CBU and the homogenous assumption of DMUs. The struc-
ture of a DMU is shown in Figure 1, below.
Based on the classification of sub-units, we assume that
CBU of the system contributes 80% of the total outputs.
Further, in view of the 80/20 principle, it is estimated that
the inputs into CBU account for about 20% of the overall
inputs. Accordingly, NCBU produces 20% of the total
outputs, while consumes 80% of all inputs. There are some pos-
tulates related to the construction of theoretical PPS that need to
be established so that we can evaluate the efficiency of DMUs
and estimate the production potential.
Suppose, SDMUcj denotes CBU and SDMUncj denotes
NCBU within DMUj. Also, let Tc represent the set of theoreti-
cal input/output combination of SDMUcðXc;�YcÞ. Similarly,
Tnc represents the set of SDMUnc and T is the theoretical input/output combination set of DMU. The related postulates are as
follows:
(1) Envelopment: The input/output bundle of SDMUcj
ðXcj;�YcjÞ [ Tc; j ¼ 1; . . . ; n, and that of SDMUncj
belongs to Tnc, i.e., ðXncj;�YncjÞ [ Tnc; j ¼ 1; . . . ; n.
(2) Convexity: If ðXcj;�YcjÞ [ Tc; ðXncj;�YncjÞ[ Tnc; j ¼ 1; . . . ; n, then
Pnj¼1 ljðXcj;�YcjÞ [ Tc andPn
j¼1 ljðXncj;�YncjÞ[ Tnc with lj � 0 andPnj¼1 lj ¼ 1.
(3) Additivity: If SDMUcðXc;�YcÞ and SDMUncðXnc;�YncÞare arbitrarily feasible SDMUs, i.e., ðXc;�YcÞ [ Tc, and
ðXnc;�YncÞ [ Tnc, then ðXc þ Xnc;�Yc � YncÞ [ T .
(4) Free Disposability: If ðX;�YÞ [ T and
ðX�;�Y�Þ ^ ðX;�YÞ, then ðX�;�Y�Þ [ T .
(5) Minimum Extrapolation: T is the intersection set of all T 0
that satisfies all the above postulates.
It should be note that the theoretical PPS above is different from
those associated with Castelli’s single-level hierarchical struc-
ture model and Kao’s parallel model (Castelli et al., 2004; Kao,
2009). Postulate (2’) in these two models, which is distinct from
postulate (2) above, appears as follows:
(2’) Constant Returns to Scale: If ðXcj;�YcjÞ [Tc; ðXncj;�YncjÞ [ Tnc; j ¼ 1; . . . ; n, then
Pnj¼1 lj
ðXcj;�YcjÞ [ Tc andPn
j¼1 ljðXncj;�YncjÞ [ Tnc with
lj � 0.
Obviously, we assume variable returns to scale (VRS) for
SDMU.
Let’s now investigate a more general case. Consider a pro-
duction system that consists of several production lines. Each
production line transforms the same set of inputs into the
same set of outputs using a different technology and process.Figure 1. The Structure of a DMU
BI, FENG, DING, KHAN46
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
The production technology for each production line can be
characterized by a PPS.
Pursuant to the VRS assumption, the PPS’s of T1 and T2 are
mathematically expressed as:
T1 ¼(ðx1
i ; y1r ÞjXn
j¼1
l1j y1
rj � y1r ; r ¼ 1; . . . ; s;
Xn
j¼1
l1j x1
ij � x1i ; i ¼ 1; . . . ;m;
Xn
j¼1
l1j ¼ 1; l1
j � 0
)
T2 ¼(ðx2
i ; y2r ÞjXn
j¼1
l2j y2
rj � y2r ; r ¼ 1; . . . ; s;
Xn
j¼1
l2j x2
ij � x2i ; i ¼ 1; . . . ;m;
Xn
j¼1
l2j ¼ 1; l2
j � 0
)
ð1Þ
where, xij; yrjði ¼ 1; . . . ;m; r ¼ 1; . . . ; sÞ are the ith input and
the rth output of DMUj, respectively. The superscript, 1 or 2,
attached to T indicates production technology. One can notice
thatPn
j¼1 l1j ¼ 1; l1
j � 0;Pn
j¼1 l2j ¼ 1; l2
j � 0 are convex
conditions. Given the technologies, the production technology
of the DMU as a whole, can then be constructed as follows:
T ¼ fðxij; yrjÞjxi � x1i þ x2
i ; i ¼ 1; . . . ;m;
yr � y1r þ y2
r ; r ¼ 1; . . . ; s;
ðx1i ; y
1r Þ [ T1; ðx2
i ; y2r Þ [ T2g ð2Þ
It can be inferred from the definition of T that any input/output combination ðxij; yrjÞði ¼ 1; . . . ;m; r ¼ 1; . . . ; sÞ is feas-
ible if there are two feasible combinations ðx1ij; y
1rjÞ and ðx2
ij; y2rjÞ
belonging to T1 and T2, respectively, such that the sum of the
inputs and outputs dominates ðxij; yrjÞ. To put it differently,
the output bundle ðy1j; . . . ; ysjÞ is producible through DMUj,
given the resource bundle ðx1j; . . . ; xmjÞ, if we are able to
secure the output bundle by properly apportioning the input
resources between technology T1 and T2.
Now, in what follows, we formally derive the properties of
PPS T:
Property 1. T is a convex set.
Proof. Suppose ðX1; Y1Þ and ðX2; Y2Þ belong to T. By defi-
nition, there are sets of multipliers lk1�j ; lk2�
j withPnj¼1 lk1�
j ¼ 1 andPn
j¼1 lk2�j ¼ 1 such that
P2k¼1
Pnj¼1
lk1�j yk
rj � y1r ; r ¼ 1; . . . ; s,
P2k¼1
Pnj¼1
lk1�j xk
ij � x1i ; i ¼ 1; . . . ;m and
P2k¼1
Pnj¼1
lk2�j yk
rj � y2r ; r ¼ 1; . . . ; s,
P2k¼1
Pnj¼1
lk2�j xk
ij � x2i ; i ¼ 1; . . . ;m
For any convex multiplier a;b, we haveP2k¼1
Pnj¼1
ðalk1�j þ blk2�
j Þykrj � ay1
r þ by2r ; r ¼ 1; . . . ; s;
P2k¼1
Pnj¼1
ðalk1�j þ blk2�
j Þxkij � ax1
i þ bx2i ; i ¼ 1; . . . ;m; and
Pnj¼1 ðalk1�
j þ blk2�j Þ ¼ 1; ðk ¼ 1; 2Þ, which ascertains that
aðX1; Y1Þ þ bðX2; Y2Þ [ T . A
Definition 1. Extended DMU set R:
Assume there are n DMUs, each of which consists of two
production lines ðSDMU1j; SDMU2j; j ¼ 1; . . . ; nÞ using pro-
duction technologies T1 and T2, respectively. We define a set
of DMUs as R, which comprises of SDMU1j, SDMU2k, where,
j; k ¼ 1; . . . ; n. Then, it is clear that R has n2 units, which are
denoted as DMUjð j ¼ 1; . . . ; n2Þ.Definition 2. Let ðxij; yrjÞ denote the input and output bundle
associated with set R within DMUj. The convex set T� is then
defined as:
T� ¼(ðxi; yrÞj
Xn2
j¼1
ljxij � xi; i ¼ 1; . . . ;m;Xn2
j¼1
ljyrj � yr;
r ¼ 1; . . . ; s;Xn2
j¼1
lj ¼ 1; lj � 0g ð3Þ
Theorem 1. T ¼ T�.Proof. See Appendix. A
The theorem above serves as a vehicle to study parallel pro-
duction structures that follow the “black box” approach. As
known, relevant properties of DEA models in “black box”
context have been extensively explored in research studies;
and the parallel production counterparts can be easily
deduced through Theorem 1.
3.2. Special case
In this section, we attempt to construct a special DEA model
that will evaluate the efficiency of DMUs based on 80/20
rule. Consider the following formulation:
min
"u�
Xm
i¼1
s�i þXs
r¼1
sþr
!e
#
s.t.Pnj¼1
ðlcj xc
ij þ lncj xnc
ij Þ þ s�i ¼ uoxio i ¼ 1; . . . ;m:
ESTIMATING RELATIVE EFFICIENCY OF DMU 47
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
Xn
j¼1
ðlcj yc
rj þ lncj ync
rj Þ � sþr ¼ yro r ¼ 1; . . . ; s:
Xn
j¼1
lpj ¼ 1 p ¼ c; nc:
8lcj ; l
ncj � 0:
ð4Þ
where lcj and lnc
j are multipliers associated with CBU and
NCBU of DMUj respectively. xcij; y
crj (xnc
ij ; yncrj ) denote, respect-
ively, the ith input and rth output of the CBU (NCBU) of
DMUj. Obviously we have xcij ¼ 0:2xij; x
ncij ¼ 0:8xij;
ycrj ¼ 0:8yrj; y
ncrj ¼ 0:2yrj according to 80/20 rule. Besides,
sþr ; s�i are all slacks and e is a non-Archimedean element
defined to be smaller than any positive real number. It should
be noted that the above model measures the inefficiency of
DMUo and u is efficiency score with a correction term of
ðPm
i¼1 s�i þPs
r¼1 sþr Þe. The production possibility set(PPS)
is obtained by opening the “black box” and combining
SDMUc and SDMUnc. In other words, the PPS consists of ele-
ments(units) generated by adding up virtual
SDMUcðPn
j¼1 lcj Xc
j ;Pn
j¼1 lcj Yc
j Þ and SDMUncðPn
j¼1 lncj Xnc
j ;Pnj¼1 lnc
j Yncj Þ.
By now the manager can estimate the efficiency and pro-
duction potential of DMUo according to the 80/20 rule. We
assume that lc�j ; l
nc�j ; u�; s��; sþ� is the optimal solution to
model (4), and then the production potentials, X� and Y�, are
given by:
X�o ¼Xn
j¼1
ðlc�j Xc
j þ lnc�j Xnc
j Þ ¼ u�Xo � s��
Y�o ¼Xn
j¼1
ðlc�j Yc
j þ lnc�j Ync
j Þ ¼ Yo þ sþ�ð5Þ
Consequently, model (4) can be converted as presented in
model (6), below:
min
"u�
Xm
i¼1
s�i þXs
r¼1
sþr
!e
#
s.t.Xn
j¼1
ð0:2lcj þ 0:8lnc
j Þxij þ s�i ¼ uoxio i ¼ 1; . . . ;m:
Xn
j¼1
ð0:8lcj þ 0:2lnc
j Þyrj � sþr ¼ yro r ¼ 1; . . . ; s:
Xn
j¼1
lpj ¼ 1 p ¼ c; nc:
8lcj ; l
ncj � 0: ð6Þ
We now proceed to use a and b, respectively, to denote the
percentages of inputs consumed and outputs produced by CBU,
when the proportions of inputs and outputs of CBU are identi-
cal across DMUs. Intuitively, the frontier constructed by
NCBUs is dominated by the one characterized by DMUs, and
the frontier of DMUs is dominated in turn by the one character-
ized by CBUs. The productive technologies of CBU and NCBU
will increase and decrease, respectively, as b increases.
As mentioned earlier, the PPS of the proposed model con-
sists of those units determined by adding up virtual SDMUc
and virtual SDMUnc. In order to maximize the productive tech-
nology, the output of SDMUc should account for a higher pro-
portion of the overall output in order to benefit, as far as
possible, from the higher production technology of SDMUc.
This means that, in general, the efficiency of DMUo would
decrease as the value of b is increased due to a positive shift
in the production frontier. We treat this as a property in the
rest of this section as we consider the case of multiple inputs
and single output.
Assume, R denotes Extended DMU set (element of the set is
identified as EDMU) and Q denotes Original DMU set. Then,
EDMUo [ R is constructed by CBUto and NCBUk
o, which are
the CBU of DMUt and the NCBU of DMUk, respectively.
Note that the superscript of CBU or NCBU indicates its
source, i.e., the DMU it belongs to. The subscript indicates
its destination, i.e., the EDMU it goes to.
Lemma 1. If EDMUo is efficient in R, then both DMUt and
DMUk are efficient in Q.
Proof. Assume that DMUt or DMUk is inefficient in Q.
Then, without loss of generality, we can assume that DMUt
is inefficient and the benchmarking point for DMUt is
ðP
l�j Xij;P
l�j YrjÞ. Therefore, ðaXit; bYrtÞ is dominated by
ðP
l�j aXij;P
l�j bYrjÞ, where a and b are the percentages of
DMUs’ input and output, respectively, that are associated
with the input and output of CBU. Note also that ðaXij; bYrjÞis the input-output bundle of CBUj
o .
Expressed differently, if DMUt is inefficient, then EDMUo
is dominated by a combination of EDMU, denoted by
ðNCBUko;PP
l�j CBUjoÞ¼ðNCBUk
o;PðP
l�j aXij;P
l�j bYrjÞÞ.This contradicts the assumption that EDMUo is on the frontier.
Hence, the proof. A
Property 2. The efficiency of DMUo will not increase as b
rises in the multiple inputs and one output case, with the con-
dition that the ratio of inputs and outputs associated with CBU
in each DMU is identical ðb . aÞ.Proof. Let a and b, respectively, denote the percentages of
inputs and outputs of CBU in each DMU. But, according to
Theorem 1, T ¼ T�. This suggests that PPS, by using reference
units from extended set R, is equivalent to PPS constructed by
adding up the PPS’s of SDMU1j and SDMU2j. Thus, we can
view the PPS of a parallel system equivalently from the per-
spective of the PPS of an extended DMU.
Suppose, EDMUo is on the frontier of R. From Lemma 1,
both DMUt and DMUk are efficient in the original PPS. The
inputs and outputs of DMUt and DMUk are denoted by
ðXit; YtÞ and ðXik; YkÞ, where i ¼ 1; . . . ;m indicates the
BI, FENG, DING, KHAN48
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
dimension of inputs, whereas the dimension of output is unity.
Therefore, the EDMUo can be denoted as ðXik þ aðXit � XikÞ;Yk þ bðYk � YtÞÞ.
Obviously, EDMUo is a convex combination of DMUt and
DMUk when a ¼ b, which indicates that EDMUo belongs to
the original PPS constructed by Q in this special case. Since
0 , a , 0:5 and 1 . b . 0:5, i.e., b . a, we can claim that
Yt � Yk. If, on the other hand, Yt , Yk, it will follow that
EDMUo denoted by ðXik þ aðXit � XikÞ; Yk þ bðYt � YkÞÞ ¼ðXik þ aðXit� XikÞ; Yk þ aðYt � YkÞ þ ðb� aÞðYt � YkÞÞ is
dominated by the convex combination ðXik þ aðXit � XikÞ;Yk þ aðYt � YkÞÞ, which belongs to the original PPS. This con-
tradicts the assumption, however, that EDMUo is on the frontier
of R.
Subject to the above analysis, since DbðYt � YkÞ � 0, the
frontier will not get worse as the value of b is increased .
Thus, the efficiency of DMUo will not increase and the prop-
erty is established. A
3.3. General model
In this section we proceed to explore the possibility of incorpor-
ating the Pareto principle into the estimation of production effi-
ciencies of production units with parallel structure, where the
inner production information is not known to DMUo.
Suppose, a and b, respectively, represent the percentages of
inputs and outputs of CBU in DMUo. According to the activi-
ties included in CBU and the experience of manager, the
manager can assume that “b” percent of all outputs produced
by DMUs comes from its CBU. In other words, the percentage
remains constant as it is assumed to be determined mainly by
the production activities included in CBU. Besides, the homo-
geneous assumption vis-a-vis DMUs justifies it. However, the
precise percentage of inputs allocated to CBU in other DMUs
is unknown to the manager of DMUo, as the production effi-
ciencies vary across DMUs. We assume that, based on his
experience, the manager can estimate the probability distri-
bution of inputs consumed by CBU of other DMUs. As
described previously, it can be assumed that the production
information is characterized by Pareto principle. Suppose that
the inputs of CBU X conform to the Pareto distribution function
as follows:
FXðxjX . xmÞ ¼ 1� xm
x
� �bð7Þ
where xm is the minimum of X, as xm � xul. Note, xul is the
maximum of overall inputs. Obviously, the inputs of CBU
can’t exceed the total resource allocation in each DMU.
Thus, we can obtain the following truncated distribution:
FXðxjxm , X � xulÞ ¼1� ðxm=xÞb
1� ðxm=xulÞbð8Þ
Set tm ¼ xm=xul, t ¼ x=xul, where t denotes the percentage of
the inputs of DMUj consumed by CBU in accordance with the
following truncated Pareto distribution:
FTðtjtm , T � 1Þ ¼ 1� ðtm=tÞb
1� tbm
ð9Þ
Here the parameters b and tm are set by the manager based
on his experience. Let tj denote the percentage of inputs of
DMU consumed by CBU. The inputs of NCBU are then
given by ð1� tjÞxul and the general model is presented as
follows:
min
"u�
Xm
i¼1
s�i þXs
r¼1
sþr
!e
#
s.t.Xn
j¼1j=0
ðtjlcj þ ð1� tjÞlnc
j Þxij þ ðalcj þ ð1� aÞlnc
j Þ
� xio þ s�i ¼ uoxio i ¼ 1; . . . ;m:
Xn
j¼1
ðblcj þ ð1� bÞlnc
j Þyrj � sþr ¼ yro r ¼ 1; . . . ; s:
Xn
j¼1
lpj ¼ 1 p ¼ c; nc:
8lcj ; l
ncj � 0:
ð10Þ
where tj denotes the percentage of inputs consumed by CBU in
DMUj. Note that the value of tjð j = oÞ is unknown. For the
sake of simplicity, the meanings of other notations are referred
to the interpretation of model (4). What the decision maker
(DM) knows is its distribution function. However, calculating
the efficiency of DMUo by using traditional method is difficult.
A possible solution can be obtained through parametric
bootstrap.
The parametric bootstrap method, also known as Monte
Carlo, evaluates the mean value and the associated standard
error of the sample efficiencies of DMUo. These statistics
respectively represent the estimate of efficiency and the range
of its reliability. The procedure is described as follows:
(1) N groups of random numbers fUjgNj¼1 are generated auto-
matically, which are uniformly distributed in (0,1).
Provided that there are n DMUs in total, the count of
random numbers in each group is n� 1, as we suppose
that the percentages of inputs and outputs of CBU in
DMUo are known to the manager.
(2) Compute Mj ¼ tmin þ Ujðtmax � tminÞ; j ¼ 1; . . . ;N, where
tmax and tmin are, respectively, the lower and upper
bounds of t. Then, Mj; j ¼ 1; . . . ;N are uniformly distrib-
uted in ðtmin; tmaxÞ.
ESTIMATING RELATIVE EFFICIENCY OF DMU 49
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
(3) Compute Tj ¼ GðMjÞ; j ¼ 1; . . . ;N, where Gð�Þ denotes
the inverse function of the truncated Pareto distribution
given by (9), above. Tjð j ¼ 1; . . . ;N) represents one
sample of t values obtained from this distribution.
(4) We can obtain N sample efficiencies of DMUo through the
model given in (10), above. Now the mean efficiency is
given by u o ¼ 1=NfPN
j¼1 u jog and its standard deviation
can be computed from s ¼ 1=N � 1fPN
j¼1 ðu o�u jo Þ
2g,which indicates the volatility of N efficiencies obtained
by simulation and the reliability about this estimation.
The flow chart of Monte Carlo algorithm is presented in
Figure 2.
Further elaboration on the range of the percentage t of inputs
of CBU in DMUo may be warranted. Suppose that the percen-
tage of outputs of CBU in DMUo is 80%. Then it can be argued
that the corresponding percentage t of inputs can’t reach 1,
because NCBU cannot produce 20% outputs of DMUo
without using any inputs. Furthermore, the productivity of
CBU, by definition, cannot be lower than that of NCBU.
Hence, the upper bound of t cannot exceed 80%.
If the DMU manager can acquire more accurate information
on the productivity of CBU and NCBU, then the upper bound
of t may be further refined. Intuitively, a smaller range of t will
lead to a smaller volatility of efficiencies obtained by simu-
lation. We can expect the outcome to be more precise and
reliable if the range of the percentage t is estimated with
greater precision. It then stands to reason that, in addition to
the values of parameters b and tm, the DM also needs to give
the upper bound on t. Guided by the above reasoning, the trun-
cated distribution of the percentage t in each DMU is adjusted
as follows:
FTðtjtm , T � 0:8Þ ¼ 1� ðtm=tÞb
1� ðtm=0:8Þbð11Þ
While the reliability of the Pareto distribution provided by
DM is of significance, it is beneficial to verify whether this dis-
tribution is statistically acceptable. A plausible method is to use
expert’s opinion in the form of the lower and upper bounds of
DMUo’s efficiency. Based on this expert opinion, the effi-
ciency interval is then compared with the sample efficiency
interval derived from (10), above, by using the Monte Carlo
method. It can then be drawn that the Pareto distribution is stat-
istically acceptable if most of the sample efficiencies fall within
the estimated range; otherwise, the distribution will be rejected,
requiring the DM to modify the parameters until the distri-
bution function passes the test.
4. ILLUSTRATIVE EXAMPLE
In this section, we use simulated data on convenience stores for
illustration. Suppose there are 20 convenience stores in the area
conducting identical business activities. The inputs of each
store are defined below:
(1) Operating Expenses (X1): annual operating expenses of the
store in (thousand) dollars;
(2) Salaries (X2): average annual salaries of all employees in
(thousand) dollar;
(3) Full-time employees (X3): number of full-time employees.
The outputs of each store are defined below:
(1) Profits (Y1): annual profits of each store in (thousand)
dollars;
(2) Sales (Y2): annual sales of each store in (thousand) dollars.
According to the framework outlined above, we divide the cus-
tomer relationship management system of the convenience
store into core consumer unit and non-core consumer unit.
The simulated data of inputs and outputs are shown in
Table 1. It should be mentioned here that, due to limited experi-
ence, the parameters of the truncated distribution tm and b are
assigned arbitrary values in this example.
Suppose that DMU1 is the unit under consideration. The
input values of DMU1 are $299.136, $4691, and 397 employ-
ees, respectively accounting for Operating Expenses, Salaries,
and Labor. The outputs of the unit are $2801 and $8276,
respectively for Profits and Sales.
The efficiency of DMU1 is calculated to be 1 by solving the
BCC model, which suggests that DMU1 is technically efficient.
If the manager wishes to seek the evidence to support a higher
production potential, Pareto principle can be applied. From
model (4), the new efficiency of DMU1 is calculated to beFigure 2. The flow chart of Monte Carlo algorithm
BI, FENG, DING, KHAN50
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
0.9414, which suggests that, based on the 80/20 empirical
assumption, the inputs can be reduced while keeping the
current level of outputs.
Now suppose that, according to his experience, manager’s
estimates of the parameters of the truncated distribution are
b ¼ 1 and tm ¼ 0:1. The parameter t follows the probability
distribution F, shown below, considered to be statistically
correct for simplicity.
FTðtj0:1 , T � 0:8Þ ¼ 8
7� 8
70tð12Þ
The truncated Pareto distribution of (12) is depicted in
Figure 3. This diagram illustrates that the slope of the curve
is monotonically decreasing with a meaning of low probability
with respect to high t value. The probability exceeds 50% when
the value of t is located in the range [0.1, 0.2]. If the range of t is
broaden to [0.1, 0.3], the probability exceeds 70%. The radian
of the curve of truncated Pareto distribution increases as b
increases. Thus, a high b value will lead to a high cumulative
probability ceteris paribus at any specific t.
After a rough estimation of the efficiency of DMUo, using
80/20 principle, we proceed to estimate the efficiency by
applying the parametric bootstrap approach. Following the pro-
cedure outlined in section 3.3, 100 groups of random numbers
are generated according to the distribution function F. Each
group includes 19 random numbers. As mentioned earlier,
the percentages of inputs and outputs of CBU in DMUo are
known. Therefore, 100 sample efficiencies of DMUo can be
obtained by using model (10). The sample efficiencies and
their mean are depicted in Figure 4.
A sample description about the characteristics of sample
efficiencies of DMUo is shown in Figure 5 as histogram. It indi-
cates that the minimum efficiency is 0.7414, while the
maximum can reach 1. The mean efficiency is 0.8944.
Among all samples, it can be observed, 21 achieve an effi-
ciency score of 1, and 56 fall below the mean. The standard
error of the sample mean efficiency of DMUo is 0.0748,
showing high reliability.
Similarly, we calculate the efficiencies of all the DMUs by
applying the two proposed methods (model (4) and model
(10)) and the traditional BCC model. The results are shown
in Table 2. A comparison of the evaluation results, calculated
by different models, shows that the mean efficiency obtained
by model (10) through parametric bootstrap method is not
only not greater than the one obtained by BCC model (uBCC),
it is also not greater than the one delivered by the Pareto prin-
ciple (u80=20). That is to say, uBCC � u80=20 � uMC holds. This
reveals that the proposed method is able to find evidence to
TABLE 1.
Simulated inputs and outputs data of 20 convenience stores
Inputs Outputs
DMU
Operating
expenses Salaries
Full-time
employees Profits Sales
1 299.136 4691 397 2801 8276
2 540.412 4470 533 3034 8393
3 379.552 2940 272 1957 8647
4 200.727 2533 241 1414 7861
5 189.509 1617 135 1558 7678
6 382.864 3706 341 2243 4931
7 266.711 2600 239 1580 5031
8 331.9 4124 356 1058 4309
9 430.954 5135 612 1126 4663
10 278.695 4602 435 1107 3976
11 173.07 2199 138 928 3131
12 361.276 2800 617 892 2914
13 169.564 3802 336 1107 3749
14 337.907 4325 470 995 3056
15 401.7 3234 434 1041 1046
16 155.427 2340 218 1031 2035
17 239.497 3067 164 724 1034
18 159.516 1983 301 740 1310
19 239.631 3332 341 713 1021
20 175.396 2143 419 683 1078
Figure 3. Truncated pareto distribution
ESTIMATING RELATIVE EFFICIENCY OF DMU 51
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
support higher production potential of DMU under evaluation
by considering the internal information of DMUo as well as
manager’s inference to other DMUs with similar production
technologies. In addition, the efficiencies of DMU 2, 3 and 5
all equal one, which reveals that these DMUs significantly
dominate the others. Given the efficiencies of DMU 11, 16 or
18, it is shown that uMC ¼ 1 does not necessarily follows if
uBCC ¼ u80=20 ¼ 1.
It is beneficial to emphasize that the proposed method is a
network approach. Therefore, it helps to alleviate the insuffi-
ciency of “black box” approaches, such as BCC model, for gen-
erating too many units with rating of 1. Furthermore, it helps to
find the inefficiency source within the “black box” which is
beyond the functionality of traditional DEA approach. We
report the performance targets for CBU and NCBU of DMU1
in Table 3. The data under the heading CBU� and NCBU�
denote the performance targets, and the column 2 and 4
(CBU, NCBU) report the real data. Note that the performance
targets associated with uMC ¼ 0:8944 are the average of 100
samples’ projections, and the real data are the observational
data of DMU1, which are assumed to be known by the local
managers. There’s no doubt that the performance targets
reflect the inefficiency source within a DMU and provide an
improved direction which can aid the decision makers to
make decisions.
4.1. Sensitivity analysis
To further show how the selection of the parameters b and b
can influence the result of Model (10), sensitivity analyses
with respect to b and b are performed. It should be noted that
the same sample of t values is used when the analyses associ-
ated with b are reported . This allows holding the effect due
to t constant.
The value of b is moved from 0.6 to 0.9 with a step size of
0.01. The mean efficiency and standard error corresponding to
Figure 5. Distribution of sample efficiencies of DMUo
Figure 4. Simulated DMUs’ efficiency states by Monte Carlo method
BI, FENG, DING, KHAN52
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
each b value are shown in Table 4. Similarly, the value of b is
varied from 0.5 to 1.5 with a step size of 0.05. The correspond-
ing mean efficiency and standard error are reported in Table 5.
Figure 6 and Figure 7 are set forth, respectively, to demonstrate
the trends of mean efficiency and standard error with respect to
changes in b and b.
The left section of Figure 6 demonstrates that the sample
mean efficiency monotonously decreases as the value of b
increases. An inflection point can be found around b ¼ 0:85.
Note that the value of mean efficiency has the largest variation
around the inflection point. In the right section of Figure 6 the
value of standard error first increases and then decreases as b
increases. Jointly with the curve of mean efficiency and stan-
dard error, we can comprehend the variation rule of all
sample efficiencies of DMUo along with the update of b
more intelligently.
Objectively, given a specific sample t, the value of par-
ameter b indicates the difference in productive efficiency
between CBU and NCBU. According to Figure 7, the higher
the efficiency difference between CBU and NCBU, the lower
the efficiency of DMUo. The Figure 7 also illustrates that the
mean efficiency and standard error of DMUo decrease almost
linearly as the value of b increases, which indicates that a
higher b value not only improves the estimation accuracy it
also decreases the estimated efficiency of DMUo.
As mentioned previously, a high b can result in a low possi-
bility of obtaining high sample t values. Similarly, the value of
parameter t also indicates the difference between CBU and
NCBU in terms of productive efficiency when the value of b
is fixed. Consequently, the curve in Figure 7 indicates that
the mean efficiency of DMUo decreases with the value of t.
The changes in mean efficiency illustrated in Figures 6 and 7
are noticeably consistent in the sense that both are sensitive
to the percentages of inputs or outputs of CBU in a DMU.
TABLE 3.
The performance targets of CBU and NCBU of DMU1
Input CBU CBU� NCBU NCBU�
Operating Expenses 59.83 52.37 239.31 201.62
Salaries 938.2 720.45 3752.8 3161.73
Full-time Employees 79.4 79.29 317.6 267.58
TABLE 5.
Mean efficiency and standard error by b value
b
Mean
efficiency
Standard
error b
Mean
efficiency
Standard
error
0.50 0.9121 0.0831 1.05 0.8927 0.0735
0.55 0.9102 0.0825 1.10 0.8910 0.0728
0.60 0.9083 0.0819 1.15 0.8893 0.0721
0.65 0.9065 0.0806 1.20 0.8876 0.0707
0.70 0.9046 0.0800 1.25 0.8859 0.0700
0.75 0.9028 0.0787 1.30 0.8844 0.0686
0.80 0.9010 0.0781 1.35 0.8827 0.0678
0.85 0.8994 0.0775 1.40 0.8811 0.0671
0.90 0.8977 0.0762 1.45 0.8796 0.0656
0.95 0.8960 0.0755 1.50 0.8781 0.0648
1.00 0.8944 0.0748
TABLE 2.
Three efficiency measures of simulated 20 convenience stores
DMU BCC 80/20 Monte Carlo1
1 1.0000 0.9415 0.8944
2 1.0000 1.0000 1.0000
3 1.0000 1.0000 1.0000
4 1.0000 0.9657 0.9192
5 1.0000 1.0000 1.0000
6 0.8595 0.5462 0.5180
7 0.7178 0.6875 0.6563
8 0.5110 0.5079 0.4665
9 0.3975 0.3858 0.3572
10 0.5998 0.5682 0.5221
11 1.0000 1.0000 0.9465
12 0.5775 0.5775 0.5558
13 0.9777 0.9319 0.8573
14 0.4845 0.4833 0.4415
15 0.5000 0.5000 0.4819
16 1.0000 1.0000 0.9405
17 0.8232 0.8232 0.7673
18 1.0000 1.0000 0.9238
19 0.6639 0.6639 0.6065
20 0.9174 0.9174 0.8488
1Suppose that the percentages of inputs and outputs of CBU in
DMUo are known. In particular, let b ¼ 80% and a ¼ 20%.
TABLE 4.
Mean efficiency and standard error by b value
b
Mean
efficiency
Standard
error b
Mean
efficiency
Standard
error
0.60 0.9821 0.0265 0.76 0.9284 0.0600
0.61 0.9805 0.0265 0.77 0.9212 0.0632
0.62 0.9789 0.0283 0.78 0.9132 0.0663
0.63 0.9770 0.0300 0.79 0.9044 0.0707
0.64 0.9750 0.0316 0.80 0.8944 0.0748
0.65 0.9729 0.0332 0.81 0.8829 0.0787
0.66 0.9707 0.0361 0.82 0.8697 0.0825
0.67 0.9681 0.0374 0.83 0.8547 0.0872
0.68 0.9652 0.0400 0.84 0.8375 0.0917
0.69 0.9621 0.0412 0.85 0.8255 0.0927
0.70 0.9587 0.0436 0.86 0.8161 0.0900
0.71 0.9550 0.0458 0.87 0.8078 0.0849
0.72 0.9507 0.0490 0.88 0.7995 0.0800
0.73 0.9459 0.0510 0.89 0.7911 0.0748
0.74 0.9406 0.0539 0.90 0.7824 0.0700
0.75 0.9348 0.0566
ESTIMATING RELATIVE EFFICIENCY OF DMU 53
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
The results of sensitivity analysis show that a DM can lower
the estimated efficiency of DMUo by improving the value of
either b or b, whilst he can reduce the standard error of
sample mean efficiency through enhancing the value of b.
This provides useful guidelines to DM for action when modifi-
cations to parameters are needed.
To close this section, we briefly discuss the practicality
of our approach. The Pareto principle, as the foundation for
estimating the inner data of DMUs, plays an important role in
the formulations of the models. As mentioned earlier, the
principle has been found workable in many other
scenarios and has earned credibility. Therefore, there can be
potential areas for application purposes. From the beginning,
we construct the models in terms of convenience stores. We
have tried to use the real world data in the example.
However, we didn’t find the suitable data with a pity. The
major purpose of the example is to demonstrate the entire appli-
cation process. Finally, we would like to point out that the stan-
dard error of the efficiencies generating by our approach is
really low, this provides evidence to support the reliability of
the approach.
5. CONCLUSIONS
DEA is a widely practiced approach in efficiency evaluation,
especially in the not-for-profit sector of the economy. The tra-
ditional DEA models treat DMUs under evaluation as “black
box” and don’t make a full use of the inner production infor-
mation of a DMU. A Parallel DEA model, on the other hand,
Figure 7. Mean efficiency and standard deviation with change in b value
Figure 6. Mean efficiency and standard deviation with change in b value
BI, FENG, DING, KHAN54
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
takes into account the inner mechanism of DMU with parallel
structure. The inner data, however, are often hard to obtain in
practice, which creates a bottleneck in the application of such
models.
To overcome the insufficiency associated with the “black
box” approach and the bottleneck caused by the missing
inner data in parallel DEA’s application, this paper presents
some DEA based models. The proposed procedure groups the
production activities within a DMU into two units, referred to
as CBU and NCBU, and estimates CBU’s inner input data by
using an empirical Pareto distribution. These models utilize
the input and output data of “black box” approach and the
empirical input of managers based on their experiences to
extract more production information. Relying on this infor-
mation, DEA models can be formulated to estimate the pro-
duction potential and to provide theoretical support to
managers for resource allocation and target setting.
The Monte Carlo method is used in this paper for solving the
proposed models. Sensitivity analysis is performed with respect
to the parameter b, as well as the percentage of outputs of CBU
in DMUs. The results indicate that a decision maker can control
the outcome of a model by adjusting the parameters that
comply with the efficiency interval supplied by the experts.
It’s helpful to explore other ways to estimate the efficiency
of DMU more accurately and reliably when the internal infor-
mation of DMUs is partially unknown. The method discussed
in this study outlines a preliminary approach to handling such
a problem, even though it’s suitable only when a fraction of
CBU in DMUs conforms to a known Pareto distribution.
Further research to extend the current work may follow two
different tracks: one is to expand these models to adequately
account for DMU with outputs obeying some other known stat-
istical distributions; and the other is to develop a methodology
to deal effectively with the problem that exists in the general
Network DEA models.
ACKNOWLEDGEMENTThe authors are grateful to the comments and suggestions
by two anonymous reviewers. This research work is
supported by grants from National Natural Science Funds of
China (No. 70871106; 71171181) and National Natural
Science Funds of China for Innovative Research Groups (No.
70821001).
REFERENCESBanker, R., Charnes, A., and Cooper, W. (1984), “Some models for the
estimation of technical and scale inefficiencies in Data
Envelopment Analysis”, Management Science, 30: 1078–1092.
Borger, B., Kerstens, K., and Staat, M. (2008), “Transit costs and cost
efficiency: bootstrapping non-parametric frontiers”, Research in
Transportation Economics, 23: 53–64.
Castelli, L., Pesenti, R., and Ukovich, W. (2004), “DEA-like models
for the efficiency evaluation of hierarchically structured units”,
European Journal of Operational Research, 154: 465–476.
Charnes, A., Cooper, W. W., and Rhodes, E. (1978), “Measuring the
efficiency of decision making units”, European Journal of
Operational Research, 2: 429–444.
Chiu, Y. H. and Wu, M. F. (2010), “Performance evaluation of
International Tourism Hotels in Taiwan-Application of context-
dependent DEA”, INFOR, 48(3): 155– 170.
Curi, C., Gitto, S., and Mancuso, P. (2011), “New evidence on the effi-
ciency of Italian airports: a bootstrapped DEA analysis”,
Socio-Economic Planning Sciences, 45: 84–93.
Efron, B. (1979), “Bootstrap methods: another look at Jackknife”,
Annals of Statistics, 7: 1–26.
Essida, H., Ouelletteb, P., and Vigeant, S. (2010), “Measuring effi-
ciency of Tunisian schools in the presence of quasi-fixed inputs:
a bootstrap data envelopment analysis approach”, Economics of
Education Review, 29: 589–596.
Florens, J. P. and Simar, L. (2005), “Parametric approximations of
nonparametric frontiers”, Econometrics, 124(1): 91–116.
Hwang, S. H., Chuang, W. C., and Chen, Y. C. (2010), “Formulate
stock trading strategies using DEA: a Taiwanese case”, INFOR,
48(2): 75–81.
Kao, C. (2009), “Efficiency measurement for parallel production
systems”, European Journal of Operational Research, 196:
1107–1112.
Koch, R. (2003), The 80/20 Individual, London: RandomHouse.
Mizuno, K., Toriyama, M., Terano, T., and Takayasu, M. (2008),
“Pareto law of the expenditure of a person in convenience
stores”, Physica A: Statistical mechanics and its applications,
387(15): 3931–3935.
Muniz, M., Paradi, J., Ruggiero, J., and Yang, Z. (2006),
“Evaluating alternative DEA models used to control for non-
discretionary inputs”, Computer & Operations Research, 33:
1173–83.
Pareto, V. (1971), Manual of political economy, New York: A.M. Kelly.
Ruggiero, J. (1998), “Non-discretionary inputs in data envelopment
analysis”, European Journal of Operational Research, 111:
461–469.
Staat, M. (2002), “Bootstrapped efficiency estimates for a model for
groups and hierarchies in DEA”, European Journal of
Operational Research, 138: 1–8.
Simar, L. and Wilson, P. W. (1998), “Sensitivity analysis of efficiency
scores:how to bootstrap in Nonparametric Frontier Models”,
Management Science, 44(1): 49–61.
Simar, L. and Wilson, P. W. (1999), “Estimating and bootstrapping
Malmquist indices”, European Journal of Operational Research,
115: 459–471.
Syrjanen, M. J. (2004), “Non-discretionary and discretionary factors
and scale in data envelopment analysis”, European Journal of
Operational Research, 158: 20–33.
Yang, Y. S., Ma, B. J., and Koike, M. (2000), “Efficiency-measuring
DEA model for production system with k independent
subsystems”, Journal of Operational Research of Japan, 43(3):
343–354.
Yu, C. (1998), “The effects of exogenous variables in efficiency
measurement: a Monte Carlo study”, European Journal of
Operational Research, 105: 569–80.
ESTIMATING RELATIVE EFFICIENCY OF DMU 55
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
Zhou, P., Ang, B. W., and Poh, K. L. (2008), “Measuring environ-
mental performance under different environmental DEA technol-
ogies”, Energy Economics, 30: 1–14.
Zhu, J. (2000), “Further discussion on linear production functions and
DEA”, European Journal of Operational Research, 127: 611–618.
APPENDIX
Define TE; T as follows:
TE ¼ ðX; YÞjXn2
j¼1
ljyrj ¼ yr; r ¼ 1; . . . ; s;
(
Xn2
j¼1
ljxij ¼ xi; i ¼ 1; . . . ;m;
Xn2
j¼1
lj ¼ 1; lj � 0
)
T ¼ ðX; YÞjX2
k¼1
Xn
j¼1
lkj yk
rj ¼ yr; r ¼ 1; . . . ; s;
(
X2
k¼1
Xn
j¼1
lkj xk
ij ¼ xi; i ¼ 1; . . . ;m;
Xn
j¼1
lkj ¼ 1; lk
j � 0
)
ð13Þ
Note that TE; T are PPS’s without assuming inefficient
postulate.
Lemma 2. TE ¼ T
Proof. (1) TE # T
Let DMUj be some DMU in R, and ðx1j; . . . ; xmj; y1j; . . . ; yrjÞbe its input-output bundle. Suppose it is made of SDMU1k, and
SDMU2m, where k;m [ f1; . . . ; ng. Obviously, ðx1j; . . . ; xmj;y1j; . . . ; yrjÞ [ T , since it can be decomposed into input-output
bundle of SDMU1k, and that of SDMU2m. Putting it differently,
if we set multiplier corresponding to SDMU1k and SDMU2m to
1 and other multipliers to zeros, we can see that
ðx1j; . . . ; xmj; y1j; . . . ; yrjÞ satisfies the condition to be an
element of T . Therefore TE # T holds.
(2) TE $ T
For any ðX; YÞ [ T, there exist two set of convex multipliers
ðl11; . . . ; l1
nÞ and ðl21; . . . ; l2
nÞðl1j ; l
2j � 0;
Pnj¼1 l1
j ¼ 1;Pnj¼1 l2
j ¼ 1Þ such that xi ¼Pn
j¼1 l1j x1
ij þPn
j¼1 l2j x2
ij
ði ¼ 1; . . . ;mÞ; yr ¼Pn
j¼1 l1j y1
rj þPn
j¼1 l2j y2
rjðr ¼ 1; . . . ; sÞ.To establish this part, it suffices to show that there always
exists a convex multiplierPn2
j¼1 lj ¼ 1; lj � 0, such that
xi ¼Pn2
j¼1 ljxij; yr ¼Pn2
j¼1 ljyrj, where ðx1j; . . . ; xmj;y1j; . . . ; yrjÞ is the input-output bundle of DMUj in R. In
other words, there is a convex multiplier such that the following
equations hold:
xi ¼Xn
j¼1
ljðx1i1 þ x2
ijÞ þX2n
j¼nþ1
ljðx1i2 þ x2
ið j�nÞÞþ; . . . ;
þXn2
j¼n2�nþ1
ljðx1in þ x2
ið j�n2�nÞÞ
yr ¼Xn
j¼1
ljðy1r1 þ y2
rjÞ þX2n
j¼nþ1
ljðy1r2 þ y1
rð j�nÞÞþ; . . . ;
Xn2
j¼n2�nþ1
ljðy1rn þ y2
rð j�n2�nÞÞð14Þ
where ðx11j; . . . ; x1
mj; y11j; . . . ; y1
sjÞ and ðx21j; . . . ; x2
mj; y21j; . . . ; y2
sjÞð j ¼ 1; . . . ; nÞ are input bundle and output bundle of
SDMU1j, and SDMU2j respectively. That is to say,Pn2
j¼1 lj ¼ 1; lj � 0 must satisfy the following conditions:
l1j ¼
Xð j�1Þnþn
k¼ð j�1Þnþ1
lkð j¼ 1; . . . ;nÞ;
l2j ¼Xn
k¼1
lnð j�1Þþkð j¼ 1; . . . ;nÞ ð15Þ
To further facilitate understanding, we organize the con-
ditions in the following matrix product manner.
l1 lnþ1 . . . ln2�nþ1
l2 lnþ2 . . . ln2�n
. . . . . . . . . . . .ln lnþn . . . ln2
2664
3775
1
1
. . .1
2664
3775 ¼
l21
l22
. . .l2
n
2664
3775 ð16Þ
l1 lnþ1 . . . ln2�nþ1
l2 lnþ2 . . . ln2�n
. . . . . . . . . . . .ln lnþn . . . ln2
2664
3775
T1
1
. . .1
2664
3775 ¼
l11
l12
. . .l1
n
2664
3775 ð17Þ
The above illustration indicates that the row j of the matrix is
summed to l2j , and the column j of the matrix is summed to l1
j .
BI, FENG, DING, KHAN56
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal
Let’s combine the two conditions in the following equations:
Al ¼
11; . . . ; 1zfflfflfflfflffl}|fflfflfflfflffl{n
00; . . . ; 0zfflfflfflfflffl}|fflfflfflfflffl{n
00; . . . ; 0zfflfflfflfflffl}|fflfflfflfflffl{n
. . . 00; . . . ; 0zfflfflfflfflffl}|fflfflfflfflffl{n
00; . . . ; 0 11; . . . ; 1 00; . . . ; 0 . . . 00; . . . ; 0
. . . . . . . . . . . . . . .
00; . . . 0 00; . . . 0 00; . . . 0 . . . 11; . . . ; 1
10; . . . ; 0 10; . . . ; 0 10; . . . ; 0 . . . 10; . . . ; 0
01; . . . ; 0 01; . . . ; 0 01; . . . ; 0 . . . 01; . . . ; 0
. . . . . . . . . . . . . . .
00; . . . ; 1 00; . . . ; 1 00; . . . ; 1 . . . 00; . . . ; 1
26666666666666664
37777777777777775
l1
l2
. . .
ln2
26664
37775 ¼
l11
l12
. . .
l1n
l21
l22
. . .
l2n
266666666666664
377777777777775¼ G ð18Þ
Now we will show that equation (18) always has a nonnega-
tive solution l�1; . . . ; l�n2 . Note thatPn2
j¼1 l�j ¼ 1 automatically
holds, providedPn
j¼1 l1j ¼ 1 and
Pnj¼1 l2
j ¼ 1. Now the
problem reduces to establishing the existence of a nonnegative
solution to equation (18). We claim that a nonnegative solution
always exists by contradiction. Before we proceed, equation
(17) is reduced to (19).
�Al ¼
00; . . . ; 0zfflfflfflfflffl}|fflfflfflfflffl{n
11; . . . ; 1zfflfflfflfflffl}|fflfflfflfflffl{n
00; . . . ; 0zfflfflfflfflffl}|fflfflfflfflffl{n
. . . 00; . . . ; 0zfflfflfflfflffl}|fflfflfflfflffl{n
00; . . . ; 0 00; . . . ; 0 11; . . . ; 1 . . . 00; . . . ; 0
. . . . . . . . . . . . . . .
00; . . . 0 00; . . . 0 00; . . . 0 . . . 11; . . . ; 1
10; . . . ; 0 10; . . . ; 0 10; . . . ; 0 . . . 10; . . . ; 0
01; . . . ; 0 01; . . . ; 0 01; . . . ; 0 . . . 01; . . . ; 0
. . . . . . . . . . . . . . .
00; . . . ; 1 00; . . . ; 1 00; . . . ; 1 . . . 00; . . . ; 1
26666666666666664
37777777777777775
l1
l2
. . .
ln2
26664
37775 ¼
l12
l13
. . .
l1n
l21
l22
. . .
l2n
266666666666664
377777777777775¼ �G
ð19Þ
Note that we eliminate the first row of A and the first element of
G by elementary row operation. Assume that �Al ¼ �G doesn’t
have a nonnegative solution, i.e., �G doesn’t belong to the
conic hull constructed by the column vectors of �A. By Farkas
II lemma, there exists x [ R2n�1, such that
(1) xT �G . 0;
(2) xT �AðiÞ � 0ði ¼ 1; . . . ; n2Þ, �AðiÞ denotes the ith column of�A:
Based on 2, it follows that
(1) xðiÞ � 0 (i ¼ n; . . . ; 2n� 1)(xðiÞ denotes the ith com-
ponent of vector x);
(2) For any k ¼ 1; . . . ; n� 1, we have xðkÞ þ xðiÞ �0ði ¼ n; . . . ; 2n� 1Þ, i.e., xðkÞ � min j¼n;...2n�1�xð jÞðk ¼ 1; . . . ; n� 1Þ
Combining the previous two conditions, we obtain
xT �G ¼Xn�1
k¼1
xðkÞl1kþ1 þ
X2n�1
j¼n
xð jÞl2j
� ð minj¼n;...;2n�1
� xð jÞÞXn�1
k¼1
l1kþ1 þ
X2n�1
j¼n
xð jÞl2j
¼ ð� maxj¼n;...;2n�1
xð jÞÞXn�1
k¼1
l1kþ1 þ
X2n�1
j¼n
xð jÞl2j
� ð� maxj¼n;...;2n�1
xð jÞÞXn�1
k¼1
l1kþ1 þ max
j¼n;...;2n�1xð jÞ
¼ maxj¼n;...;2n�1
xð jÞð1�Xn�1
k¼1
l1kþ1Þ � 0
ð20Þ
This contradicts xT �G . 0. Therefore, �G belongs to the conic
hull constructed by the column vectors of �A, i.e., there is
l ¼ ðl1; l2; . . . ; ln2Þ � 0 such that �Al ¼ �G, which also
means that Al ¼ G . A
Proof of Theorem 1
Proof. Let ðx1j; . . . ; xmj; y1j; . . . ; yrjÞ be an arbitrary point in
TE. We first prove that TE # T . By definition, there exists one
point ð�x1j; . . . ;�xmj;�y1j; . . . ;�yrjÞ in TE such that xij � �xij and
yrj � �yrj. In light of Lemma 2, ð�x1j; . . . ;�xmj;�y1j; . . . ;�yrjÞ also
belongs to T . Therefore ðx1j; . . . ; xmj; y1j; . . . ; yrjÞ [ T , since
there is a point in T such that xij � �xij and yrj � �yrj hold. By
analogy, we can prove TE $ T . Therefore, TE ¼ T holds. A
ESTIMATING RELATIVE EFFICIENCY OF DMU 57
INFOR, Vol. 50, No. 1, February 2012, pp. 44–57 DOI 10.3138/infor.50.1.044ISSN 0315-5986 j EISSN 1916-0615 Copyright # 2012 INFOR Journal