Fuzzy Multiple Objective Decision Making

83

8

Data Envelopment

Analysis

Traditionally, data envelopment analysis (DEA) and regression-based methods such

as deterministic and stochastic models were widely used to measure the technical

efciency of decision-making units (DMUs). The main difference between DEA and

regression-based methods is that DEA is a non-parametric approach while regression-

based methods are parametric. Several papers compared DEA with regression-based

methods with respect to efciency, exibility, robust, assumptions, and sample sizes

(Cooper and Tone, 1997; Ruggiero, 1998; Chen, 2002).

The abandonment of DEA for measuring technical efciency has been suggested due

to the disadvantages of sensitivity to outliers and failure to reveal measurement errors

(Schmidt, 1985; Greene, 1993). However, the most critical problem of using regres-

sion-based methods is mis-specication (Giannakas et al., 2003; Gonzalez and Castro,

2001). It is necessary to specify a particular production function (e.g., Cobb-Douglas

or translog form) before measuring the frontiers of DMUs, and different production

functions may yield different results. However, it is hard to specify a correct production

function in advance because of the complex relations of input and output variables.

This chapter attempts to provide a exible and robust method for nding the pro-

duction function automatically so that the linear and nonlinear relations between input

and output variables can be considered. Section 8.1 introduces the concepts of DEA

and regression-based frontier models for evaluating the technical efciency of DMUs.

8.1 TRADITIONAL DEA

DEA is a mathematical programming technique that can calculate the relative ef-

ciencies of DMUs according to multiple inputs and outputs. Thus, when facing topics

that involve investigating the efciency of converting multiple inputs into multiple

outputs, DEA can be an appropriate and useful technique. Furthermore, DEA makes

it possible to benchmark the best practice DMU and provides estimates of the poten-

tial improvements for DMUs that are considered inefcient.

DEA has been applied extensively in the managerial and economics elds to solve

multi-criterion problems. Weber (1996) and Liu et al. (2000) applied DEA in evalu-

ation of suppliers for an individual product. DEA has been applied to evaluate the

performance of private sector facilities such as banks and power plants (Berg et al.,

1991; Golany et al., 1994). Instead of investigating the operation efciency of cor-

porations in various industries, DEA is also considered a good technique to provide

the performance indicators when outputs are not dened clearly, for example, the

productivity of public sectors such as universities, hospitals, and government institu-

tions (Bedard, 1985).

84 Fuzzy Multiple Objective Decision Making

The origin of DEA can be traced to the simplex algorithm used to estimate pro-

duction possibility frontiers and access technical efciency proposed by Dantzig

(1951). Farrell (1957) developed a measure of technical efciency calculated from

sample data. Charnes et al. (1978) reintroduced and developed the mathematical

method as DEA. Some comprehensive reviews are provided by Boussoane et al.

(1991), Seiford (1996), and Charnes et al. (1989).

The rst DEA model discussed here is the CCR (Charnes, Cooper and Rhodes)

model introduced in 1978. The input-oriented CCR form of a DEA mathematical

programming model is:

=

∑

∑

=

=

h

uy

vx

max

r

t

rrj

i

m

iij

0

1

1

0

0

(8.1)

∑

∑

≤=

=

=

st

uy

vx

jn

.. 1, 1, ..., ,

r

t

rrj

i

m

iij

1

1

u

r

≥ ε > 0, r = 1,…,t,

vi

m0

,1

,..., ,

i

≥ε

>=

where u

r

is the weight of output r, v

i

is the weight of input i, y

rj

is the output r of DMU

j, x

ij

is the amount of input i of DMU j, t is the number of outputs, m is the number

of inputs, n is the number of DMUs, and ε is a small positive number (in general, ε

is taking ε = 10

–6

.

To maximize the efciency score of a DMU j

0

, the objective function, we choose a

set of weights for all inputs and outputs. The constraint set ensures that the efciency

scores of all DMUs will not exceed 1.0. The last two constraint sets ensure that all

inputs and outputs are included in the model, that is, no weights are set at 0. A score

of 1 represents an efcient DMU j

0

; other values are considered inefcient.

Equation (8.1) is the ratio form of DEA that has an innite number of solutions.

To nd a solution, the formula can be converted into a linear programming problem

by moving the denominator in the rst constraint set in Equation (8.1) and setting

the denominator to 1:

∑

=

=

hu

ymax

rrj

r

t

0

1

0

(8.2)

∑

=

=

s.t. vx 1,

iij

i

t

1

0

∑∑

−≤ =

==

uy vx

jn

0, 1,..., ,

rrjiij

i

m

r

t

11

≥ε>=urt0, 1,..., ,

r

≥ε>=vim0, 1,...,

i