Estimating Relative Efﬁciency of DMU: Pareto …bs.ustc.edu.cn/UserFiles/Editor/file/20130930/...2013/09/30 · The Pareto principle, also known as 80/20 rule, is a well established

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,


Jingjing DingManagement School, University of Science and Technology of China, Jinzhai Road, Hefei,


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


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:



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



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.



(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



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



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



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



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



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).

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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 .



Let’s combine the two conditions in the following equations:

Al ¼

11; . . . ; 1zfflfflfflfflffl}|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

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



Documents

Estimating Relative Efﬁciency of DMU: Pareto …bs.ustc.edu.cn/UserFiles/Editor/file/20130930/...2013/09/30 · The Pareto principle, also known as 80/20 rule, is a well established