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### 專題討論

Topic：ANOTHER VIEW OF EFFICIENCY

IMPROVEMENT IN DATA

ENVELOPMENT ANALYSIS

Source：Journal of the Chinese Institute of Industrial

Engineers, Vol. 26, No. 2 (2009)

Authors：Tien-Hui Chen

Chiao-Pin Bao

Shiow-Yun Chang

• Abstract

• Introduction

• DEA Methodology ( Include decomposition)

• Alternative targets of factors

• An illustration

• Treatment of the exogenous inputs

• Conclusions

• My review of the Paper

• Q&A

This study modifies the original DEA model by

decomposing the normalizing equation in order

to for it to be associated with different dual

variables.

As a consequence, to improve efficiency the

adjustment proportion of each input or output

factor can be different.

In essence, the proposed approach can not only

set targets of factors for inefficient decision

making units to eliminate inefficiency, but can

also deal with the exogenous variables in a DEA

context.

1、Data envelopment analysis (DEA) is a systematic

programming approach for measuring relative efficiencies

within a group of decision making units (DMUs), which

utilize several inputs to produce a set of outputs.

2、If the efficiency score of a DMU is equal to one, then the

DMU is classified as efficient; otherwise it is inefficient.

3、This study decomposes the normalizing equation in the

original DEA model inorder for it to be associated with

different dual variables to obtain improved inputs and outputs

targets for inefficient DMUs. The proposed approach can not

only set targets of factors for inefficient DMUs to achieve

Pareto efficiency, but can also deal with the exogenous variables in a DEA context.

Basic DEA model-(model 1)

Dual model-(model 2)

and all slacks are of zero in the DMUj DEA run

efficient

if

inefficient

if

Decomposition of the normalizing equation-(model 3)

Decompose the normalizing equation(2)

Dual model-(model 4)

efficient

if

inefficient

if

Dual model-(model 4)

1、However, implies that DMUj is inefficient because evidence (from

the efficiencies of efficient DMUs) shows that DMUj could reduce

its input r in the proportion without worsening any output,

r = 1,2,…,m .

2、Therefore, an improvement possibility for the inefficient DMUj to

eliminate input inefficiency is to decrease input r with the amount

of .

3、Thus, it is not necessary to reduce all of the inputs in the same

proportion as in the traditional model.

Solve model 4

Specify the values of for

To continue

1、The contribution of input r to the current efficiency of .

2、The proposed procedure can ensure that the best score of

by model (4) is the same as that of the traditional DEA model.

If is relatively inefficient, the reference

coordinates of on the frontier are

,

According to constraints (11) and (12)

We have

So for the inefficient to achieve Pareto

efficiency it must

Decrease input r

increase output i

In equation (13), the larger the value of , the

smaller the adjustment proportion of input r for

the inefficient .

The targets to eliminate inefficiency for

using model (2) are

For the outputs, as shown in equations (14) and (16)

is the unit revenue of output i for the

Using equations (13) and (15) are stated as

and , respectively.

is the unit cost of input r for the

1、Compare the total reduction in cost for performance improvement.

2、For an inefficient the total reduction costs in eliminating input inefficiency.

Set the reduced targets for each input to eliminate the input wastage

The reduced targets of input r is

Otherwise

Notably

The improvement targets are the same.

=

For example

1

2

6

1、Using LINGO 6.0.

2、

= .

Set of weights and the values of and

1、Using CCR model.

Improvement DMUE and DMUF

=0.600*5=3

=0.385*5=1.925

Based on equations (13) and (15), the improvement input targets for

and are presented in Table 3.

Total cost of DMUs

If , we find

If the unit cost ratio fo input X1 to input X2 of

DMUE is greater than 0.5, i.e.

Then the proposed approach offers a larger reduction

in cost with regard to eliminating input inefficiency;

otherwise, the traditional method provides a better

choice.

：input X1 to input X2 is greater than

Provide a better targets to eliminate input inefficiency

In this illustration, there are only two inputs under the

same respective output target so that the decision maker

can determine which targets of inputs should be applied

based on the cost ratio.

For cases with more than two inputs, decision makers can

apply Equations (17) and (18) to obtain and compare the

total reduction costs for efficiency improvement in order

to arrive at a better decision.

In many realistic situations, a few of the input factors are within

the DMU’s control, i.e. they can be varied at the discretion of

decision makers; however, some are uncontrollable.

BM model

A dual of BM model

with m inputs and in

which input m is an

exogenousvariable

is shown as follow.

Because the exogenous variables are not possible to vary them at

the discretion of management.

1、

2、

3、Then the dual variable is eliminated in the objective

function.

BM model

Therefore, the proposed method can deal with the exogenous variables

as the BM model.

1、The characteristic of a DEA model is that it allows DMUs to

select the best weights in calculating their efficiencies.

2、This study modifies the CCR model by decomposing the

normalizing equation to provide another choice for efficiency improvement.

3、If the efficiency score by the CCR model is one, then the DMUs being evaluated are already Pareto efficiency.

4、This paper provides another choice of efficiency

Q：何謂柏拉圖效率( Pareto efficient )？

A：經濟學定義：是指在不損及他人的情

。而完全競爭市場的均衡就是柏拉圖

Q：此篇文章的目的最主要在探討何種

A：此篇文章最主要在探討，為了改善