Zhu J., Cook W.D. (Eds.) Modeling Data Irregularities and Structural Complexities in Data Envelopment Analysis

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1. INTRODUCTION

Many production systems (technologies) may be conceptualized as the

joint, interacting action of a finite number of production sub-

technologies called Activities. (Shephard and Färe, 1975 p. 43)

In this chapter we will apply the idea of sub-technologies to enter into

the “black box” of DEA. We will discuss three “black boxes”: i) a static

production technology ii) a dynamic production technology and iii)

technology adoption, i.e. embodied technical change.

Production of electricity by a coal-fired plant can be seen as a static

network model. One sub-technology produces electricity and sulfur dioxide

(

), where electricity is a final output and sulfur dioxide is a by-product

which may then be an input into a second technology, pollution abatement.

We can think of the optimization problem of this power plant that uses the

sub-technologies of electricity production and pollution abatement as a

network model. This facilitates the explicit analysis of allocation across sub-

technologies of both inputs (e.g., labor) and intermediate products (

SO )

that also serve as inputs.

The discrete (Ramsey, 1928) model can be formulated as a dynamic

network model. Using potato production as an example application, a

network is constructed where one year of potato production may be either

consumed as a final output or saved to become an input for the next growing

season. The inter-temporal dependence can be used to study optimal

savings.

Our third network model, technology adoption, can be formulated on the

basis of two machines, each with different technologies. These sub-

technologies are modeled as a network where inputs are allocated among the

machines, and hence determine which technology will be adopted. This is

more generally known in economics as embodied technical change.

In the chapter, we will discuss the theoretical content of each of the three

“black boxes” described above, and present simple empirical examples in a

DEA or Activity Analysis framework. We apply the network DEA methods

to three examples; a static production technology with intermediate products,

a dynamic production technology, and technology adoption (or embodied

technological change). The data and GAMS code for two examples of

network DEA models are listed in appendices.

Chapter 12

211

2. STATIC NETWORK MODEL

Here we introduce the static network model. It consists of a finite set of

sub-technologies (or activities) that are connected to form a network. This

network model enables us to study the processes that usually remain hidden

within the “black box” of DEA. The static model is particularly useful for

analyzing the allocation of intermediate products. This model also provides

the basic structure for the dynamic model and the technology adoption

model discussed in the following sections. We end the section by analyzing

property rights specified using a network model.

We restrict our presentation to three sub-technologies

P ,

P , and

P .

These three sub-technologies are connected by the directed network shown

in Figure 12-1.

Figure 12-1. Representation of sub-technologies

To complete a network of these three sub-technologies, we add a

distribution process and a sink, or collection of final outputs

. Inputs are

denoted by

(,)

xx x

=∈ℜK , the network exogenous vector, i.e., total

availability is attached to the three sub-technologies,

,1,2,3...

xi= where 0

denotes source and i denotes use (Figure 12-2). For example,

is the

input vector from source 0 used in activity 1. The total amounts used in the

three activities cannot exceed the total amount available, i.e.,

≥

∑

In Figure 12-2 activity

P uses

as an exogenous input and produces

yy+

This model is adopted from Färe and Grosskopf (1996).

Färe, Grosskopf & Whittaker, Network DEA

212

as outputs. The

y is an input into activity 3, while

y is the final output

from

P . Activity

P uses

as an exogenous input and

y ,

y as

intermediate inputs. The final product from activity

P is output vector

y .

The total network output is the sum of the final output of the three activities,

444

123

.yyy++

Note that even if an activity does not produce one of the listed outputs, that

output is set at zero and the network structure remains the same. It is also

noteworthy that some outputs may be both final and intermediate outputs,

e.g. spare parts.

Based on the description above, a generic network model takes the form

()

{

()()

()

}

444

123

43 11

11 0

43 22

22 0

43333

3012

123

000

() :

Px y y y

yyPx

yP xy y

xxxx

=++

+∈

∈+

++≤

(12.1)

where 1,2,3

Pi= are output sets, i.e., ( ) { : can produce }Px y x y

. Thus

the network model

()Px is formed by the individual sub-technologies.

Figure 12-2. A network technology

A variety of estimations can be accomplished using ( )Px as the

reference technology, including

()

•

Chapter 12

213

profit maximization: let

∈ℜ be output prices and

∈

ℜ be

input prices. Then profit maximization is

max . . ( )

ywxstyPx−∈

ii)

revenue maximization:

max . . ( )

ystyPx∈

iii)

directional output distance function estimation: let , 0

∈

ℜ≠ be

a directional vector. Then the directional output distance function is

estimated as

(

)

max . . ( )

ty g Px

ββ

+∈

iv)

output distance function estimation:

min . . ( )

ty Px

∈

The DEA network model is obtained by formulating a DEA or Activity

Analysis model for each activity.

Recall that if there are 1, ,kK= K observations of inputs and outputs

(, )

y , then a DEA output set is

{

}

() : , 1, ,

,1,,

0, 1, , .

kkm m

kkn n

Px y zy y m M

zy x n N

zkK

=≥=

∑

≤=

∑

≥=

(12.2)

The

z are the intensity variables, here non-negative, making ( )Px

homogeneous of degree one, i.e., it satisfies constant returns to scale

(CRS), ( ) ( ), 0Px Px

λλ

=>. By restricting the intensity variables as

≤

∑

the model will exhibit non-increasing returns to scale (NIRS),

() ()Px Px

⊆ . If the sum is constrained to equal one, variable returns to

scale (VRS) are obtained.

Färe, Grosskopf & Whittaker, Network DEA

214

The following restrictions on the data are imposed

() 0 1, ,

iymM

ii y k K

iii x n N

iv x k K

∑

(12.3)

Output condition (i) states that each output is produced by some k (DMU),

and (ii) requires that each activity produce some output. The input

conditions (iii) and (iv) say that each input is used by some k and that each k

uses at least one input. When the DEA model in (12.2) satisfies the

conditions in (12.3), one can prove that it has the following properties

A.1 (0) {0}; no free lunch.

A.2 ( ) ; scarcity.

A.3 ( ) ( ); free disposability of inputs.

A.4 , ( ) ( ); free disposability of outputs

A.5 ( ) .

A.6 ( ) (

P x is bounded for each x

x x Px Px

yyy Px yPx

P x is a closed set for each x

Px P

λλ

∈ℜ

≥⇒ ⊇

≤∈⇒∈

∈ℜ

= ); constant returns to scale.x

If each of the sub-technologies satisfies A.1 - A.6 one can show that the

network technology ( )Px also satisfies these conditions

. Clearly the

network model allows for a number of sub-technologies, and is not limited to

those described above.

Next we give a numerical example how the output distance function (iv)

can be estimated in the network model (12.1). The data for the three sub-

technologies

,,PPand

P are given in Tables 12-1, 12-2 and 12-3.

The sub-technology

P consists of three observations. These observations of

the sub-technology are referred to as sub-DMU’s in this chapter. As usual,

these willl form the technology

P .

These conditions are due to Kemeny, Morgenstern and Thompson Kenmeny, J. G.,

Morgenstern, O., and Thompson, G. L. (1956). “A generalization of the von Neumann

model of an expanding economy.” Econometrica, 24, 115-135.

See Shephard (1970)

Färe and Grosskopf (1996)

Chapter 12

215

Table 12-1. Data for sub-technology

P .

Sub-DMU

Input 1

Input 2

Output

1 1 2 4

2 2 1 3

3 2 2 5

Observations for the second example sub-technology are set out in Table

12-2.

Table 12-2. Data for sub-technology

Sub-DMU

Input 1

Input 2

Output

yy+

1 2 3 4

2 3 2 3

3 3 3 5

Table 12-3. Data for sub-technology

Sub-DMU

Input 1

Input 2

Intermediate

Input

Intermediate

Input

Output

1 3 2 2 2 2

2 4 2 1 2 3

3 4 3 3 2 2

Table 12-4. Data for the network technology

Sub-DMU

Input 1

Input 2

Output 1

Output 2

Output 3

1 6 7 2 2 2

2 9 5 2 1 3

3 9 8 2 3 4

As shown in Figure 12-2, the sub-technology

P uses outputs from the

sub-technologies

P and

P as inputs.

Färe, Grosskopf & Whittaker, Network DEA

216

In addition to the three sub-technologies we have the network technology

data (Figure 12-4).

To understand how this table relates the data of the three sub-

technologies, consider DMU 1. Input 1 for DMU 1 is the sum

123

01 01 01

()(123)6xxx++ =++=. Output 1 for DMU 1 is

43 3

111 1

(( ) ) (4 2) 2yyyy=+−=−=, and so on for the other elements of the

network.

Next let us specify the DEA model that solves for the output distance

function for DMU 1. This problem is

(/ ) max

31 1 1

111 2 3

11 11

12 301

1111

12302

111

123

Sub technology 1

2 yz4z3z5

z1z2z2 x

z2z1z2 x

z0z0z0

λγ

•+ ≤ •+ •+ •

•+ •+ •≤

≥≥≥

32 2 2

121 2 3

2222

12301

2222

12302

222

123

Sub technology 2

2 yz4z3z5

z2z3z3 x

z3z2z3 x

z0z0z0

•+ ≤•+•+•

•+ •+ •≤

≥≥≥

333

11 23

3333

12301

3333

12302

3333

1231

3333

1232

123

01 01 01

123

02 02 02

Sub technology 3

4z 2z 3z 4

z3z4z4 x

z2z2z3 x

z2z1z3 y

z2z2z2 y

Total Input

xxx6

xxx7

•≤ •+ •+ •

•+ •+ •≤

++≤

Chapter 12

217

In Appendix I, we provide GAMS programs, including the data sets and

results, that show how this type of problem can be solved.

A DEA network model less general than that presented here was used by

Lewis and Sexton (2004) to study major league baseball. There they

“demonstrate the advantages of the Network DEA Model over the standard

DEA Model. Specifically, the Network DEA Model can detect inefficiencies

that the standard DEA Model misses” by allowing examination of the

individual stages of the production process. Färe and Whittaker (1996) use

the Network DEA model to analyze the efficiency of cattle producers with

permits for grazing on public land. The network in that study had sub-

technologies for grain and cattle production, and for different time periods.

Before we provide a simple empirical example, let us look at how this

model was used by Färe, Grosskopf and Lee (2004) to analyze property

rights. We assume there are two firms modeled as

P and

P . The upstream

firm of

P produces two outputs using input

. One output is a marketed

commodity

y (paper, for example) and the other an undesirable externality

u (waste water, for example). The downstream firm

P produces a

marketed output

y using

and

u as inputs. This makes the downstream

firm’s production dependent on the upstream firm. This model is illustrated

in Figure 12-3.

Figure 12-3. The Network model with externalities

Some additional assumptions need to be introduced to complete the model.

First we assume that

y and

u are nulljoint, i.e.,

2222 2 2

() (,) 0 0.if y P x u and u then y∈==

In words this condition requires that if no externality is produced, then no

marketed output can be produced. Or, conversely, to produce

y , some

externality

u (also called a bad) also is produced. The externality

u is

assumed to be a pollutant that affects the downstream firm; i.e.,

Färe, Grosskopf & Whittaker, Network DEA

218

(

)

(

)

22 222 222

11 1 1

,,.uuPxu Pxu

′′

≥⇒ ⊆

To evaluate the effects of assigning the property rights and the effect of

internalizing the externality we can compute the profit under different

arrangements. Assume that the upstream firm has the property right, then

this firm maximizes its profit thus

;

112

11 11 12 1 1

(,, )

.. ( , ) ( )

max

xy u

ywx st puPx−∈

(12.4)

Equation (12.4) gives us a solution vector including the output of pollutant

u . The downstream firm must use

u as an input, and so solves the

problem

22 22 2 2 22*

(,)

.. ( , )

max

ywx styPxu−∈

(12.5)

If instead the downstream firm is assigned the property right, then it solves

the problem

222

22 22 2 2 22

(,,)

.. ( , )

max

xy u

ywx styPxu−∈

. (12.6)

Now

u may take the value

2**

u , which the upstream has to take as its

parameter in solving

11 11 12** 1 1

(,)

.. ( , ) ( )

max

ywx st pu Px−∈

. (12.7)

The solutions of (12.4)-(12.7) may be compared to the case of

internalizing the cost of disposing of the pollutant. To accomplish this we

solve

112 22

11 11 2 2 22

(,, , , )

12 1 1

2222

max

.. ( , ) ( )

(, ).

xyx y u

ywx pywx

st p u P x

yPxu

−+ −

∈

(12.8)

We cannot use the CRS model here, since it only yields zero profit. Thus the NIRS or the

VRS model should be used.

Chapter 12

219

By comparing the five problems (12.4)-(12.8), one may evaluate the

assignment of property rights versus internalization of the externality

. Next

we use sub-technologies to specify a dynamic model.

3. DYNAMIC NETWORK MODEL

Let us start by formulating a dynamic model as a network. Assume there

are three time periods t-1, t, t+1, and that each has its production

technology

,1,,1Pttt

=− +. A dynamic model has the property that a

decision in one time period impacts on later time periods. For example, if I

save now, then my possible consumption may increase later. Therefore we

introduce intermediate products, i.e., those products that are held over

between time periods,

∈ℜ . If 1

, then

is the intermediate

vector of outputs produced at time t and entering the production process at

t+1, i.e., an input at

. Figure 12-4 illustrates this setup.

inal Outputs

−1 t

−1

Intermediate

Inputs

t 1−

t 1+

Figure 12-4. Dynamic sub-technologies

Each of the production sub-technologies P

uses exogenous inputs

produce the final output

−

and intermediate inputs

−

. To complete the

network model (see Figure 12-5) we add initial conditions (distribution

process in the static model) and transversality conditions (sink in the static

model).

Of course, this simple model can be extended by the addition of more variables, firms, and

sub-technologies.

Färe, Grosskopf & Whittaker, Network DEA