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251

normalized between 0 and 1, and we have that if, and only if,

1ρ = , then

it is efficient, because

1ρ = implies that all slacks are zero and the DMU

locates on the efficient frontier. The slack-based score is units invariant and

has monotonic property.

Base upon (4), we propose a stratification procedure in the same manner

to the original context-dependent DEA as

1lll

JJE

= −

, where the set of

efficient DMUs

E is determined from the slack-based efficiency score.

That is, the set of

can be found as the DMUs with optimal value

of 1

to the following programming problem:

Minimize

subject to

0, 0, 0

jij i ik

jrj r rk

xs x

ys y

−

∈

− +

−

− =

≥≥≥

∑

(11)

The intensity vector

in (11) shows the reference set ()

SBM

Rl of

DMU

k in the efficiency level

l based on the context

E , where

ll< .

{

}

( ) 0 in (11)

SBM l

Rl jJλ= ∈ > (12)

For context-dependent DEA by radial efficiency measure, the reference set

()

CCR

Rl of DMU k in the efficiency level

l based on the context

E is

similarly given by

{

}

() 0 in (2)

CCR l

Rl jJλ= ∈ > (13)

Note that the reference set based on the context

is the same to the

reference set of slack-based DEA model. The DMUs in the reference set can

be used as benchmark targets for inefficient DMU. The context-dependent

DEA provides several benchmark targets by setting evaluation context. To

reduce the inefficiency, it is effective to improve the performance step by

step. So the benchmark targets should be provided according to the

efficiency level that the DMU exists. Therefore, the sequence of reference

sets

(),

Rl ( 1),..., (1)

Rl R− is used as the step by step benchmark

targets.

Morita & Zhu, Context-dependent DEA

252

DMU 5

DMU 1

DMU 2

DMU 3

DMU 4

DMU 6

DMU 7

DMU 8

DMU 9

DMU 10

Input 1

Input 2

Figure 13-2. Stratification by Slack-based Measure

Now, the attractiveness based on the evaluation context

is measured

with respect to the DMUs in the subset

J . For example, the attractiveness

for DMU

k based on the evaluation context

E is obtained from the

following programming problem.

maximize

subject to

,,0

xxyy

∈

≥

≤

≥≤≥

∑

(14)

Note that this fractional programming problem can be transformed into a

linear programming problem using the Charnes-Cooper transformation as

shown in Charnes, Cooper and Rhodes (1978).

The slack-based attractiveness and slack-based super-efficiency score

(Tone, 2002a) share some similarity. We consider the subset

−

of the

production possibility set excluding DMU

k .

()

{}

,,,0

jj jj

jk jk

Pxyx xy y

xy x x y y

λλλ

−

≠≠

⎧⎫

⎪⎪

= ≥≤≥

⎨⎬

⎪⎪

⎩⎭

≥≤

∑∑

∩

(15)

Chapter 13

253

The

L distance between

(

)

xy and

−

is used as the index of slack-

based super-efficiency, which is obtained from the following programming

problem.

Maximize

subject to , 1,...,

, 1,...,

,,0

ijij

rjrj

iikrrk

xxim

yyrs

xxyy

≠

≥ =

≤ =

≥≤≥

∑

(16)

Note that in slack-based super-efficiency, the evaluation context

(reference set) changes for each efficient. Our slack-based context-dependent

DEA uses the same evaluation context.

4. APPLICATION

A company in Osaka, Japan has 14 sales branches, and each sales branch

does business independently in its covered district. When a sales branch

having poor performance tried to improve its performance by benchmarking

the best sales branches, it was very difficult to adopt their performance,

because there was a huge gap between the best practices and the

underperforming sales branches. This indicates that it is necessary to provide

an attainable benchmark target via a step-wise improvement.

This company measures the performance of sales branches in terms of

four indices, namely Management, Mobility, Planning and Presentation.

Table 13-2 shows the data of 14 sales branches. The purpose of the analysis

is to find the good performer, and to provide an appropriate benchmark

target for poor performers.

Each sales branch is viewed as a DMU and the four indices are used as

four outputs with one input of unity. By calculating (1) for

1l = , we obtain

θ = for k = H, J and L. The first-level efficient frontier is

E = {H, J,

L} (the original DEA frontier). Next, we exclude the DMUs in set

E from

and obtain

J =

{A, B, C, D, E, F, G, I, K, M, N}. We have

θ =

for

k = A, F, G, I and N. Therefore, the efficient frontier of

E =

{A, F,

G, I, N} (second level efficient frontier). By repeating this process, we

finally obtain

E =

{C} (the fifth-level efficient frontier) and 5L = . Next,

by applying the slack-based context dependent DEA, we have the efficiency

Morita & Zhu, Context-dependent DEA

254

scores and the stratification of sales branches. Table 13-3 reports the radial

and slack-based efficiency scores for each sales branch.

Table 13-2. Index Values for 14 Branches

Branch Management Mobility Planning Presentation

A 14.1 25.4 4.9 4.4

B 13.2 20.5 6.4 4.7

C 11.5 16.5 4.0 4.4

D 15.6 23.6 6.4 7.4

E 15.2 18.8 6.4 0.8

F 15.2 22.4 9.0 6.0

G 14.7 23.9 9.1 8.3

H 18.0 29.0 11.3 10.2

I 16.4 23.5 6.3 7.9

J 18.1 26.4 13.0 10.0

K 10.4 20.0 4.8 8.6

L 15.1 25.6 8.8 10.2

M 12.9 16.9 7.9 7.4

N 12.8 20.4 8.8 8.7

Table 13-3. Efficiency Scores and Levels

Radial measure Slack-based measure

Branch

θ Level ρ Level

A 0.758 2 0.352 3

B 0.465 4 0.423 4

C 0.402 5 0.162 5

D 0.728 3 0.558 3

E 0.681 4 0.101 4

F 0.681 2 0.626 2

G 0.804 2 0.698 2

H 1.000 1 1.000 1

I 0.813 2 0.568 2

J 1.000 1 1.000 1

K 0.835 3 0.247 4

L 1.000 1 0.868 2

M 0.711 3 0.375 4

N 0.845 2 0.543 3

Table 13-4 and Table 13-5 show the stratification of efficient frontier by

radial measure and slack-based measure, respectively, which indicate five

levels for both measures. C has the least efficiency and form the last level of

efficient frontier. Note that the efficiency level does not necessarily follow

Chapter 13

255

the order of the efficiency scores. For example, D on level 3 has a larger

efficiency score than F on level 2 by radial measure.

Table 13-4. Levels by CRS Radial Measure

Levels Frontier branches CRS efficiency range

Level 1 H, J, L 1

Level 2 A, F, G, I, N 0.681 – 0.845

Level 3 D, K, M 0.711 – 0.835

Level 4 B, E 0.465 – 0.681

Level 5 C 0.402

Table 13-5. Levels by Slack-based Measure

Levels Frontier branches Slack-based efficiency range

Level 1 H, J 1

Level 2 F, G, I, L 0.568 – 0.868

Level 3 A, D, N 0.352 – 0.558

Level 4 B, E, K, M 0.101 – 0.423

Level 5 C 0.162

The difference between radial and slack-based measures is found in the

sales branches with non-zero slack value. For example, L has a radial

efficiency score of one, however, it is not efficient because of the non-zero

slacks in indices of management, mobility and planning. The slack-based

efficiency score reflects the amount of non-zero slack, and the efficiency

score is less than one. L is categorized into level 2 by slack-based context-

dependent DEA.

Table 13-6. Reference Sets as Benchmark Targets by Radial Measure

Branch Level

(1)

CCR

R (2)

CCR

R (3)

CCR

R (4)

CCR

A 2 H

B 4 H, J G, I, A D, M

C 5 J I, A D B

D 3 J, H I, G, A

E 4 J I, F D, M

F 2 J, H

G 2 H

H 1

I 2 J

J 1

K 3 H N

L 1

M 3 H G, N

N 2 H

Morita & Zhu, Context-dependent DEA

256

The reference sets and their diagram for benchmark targets

()

CCR

Rl by

radial measure are shown in Table 13-6 and Table 13-3. Table 13-6 tells us

that, for example, the benchmark targets of C are found as J on level 1, I and

A on level 2, D on level 3 and B on level 4. J may not be an appropriate

benchmark target for C, because J on level 1 is far from C on level 5. At the

first step, C should benchmark B on level 4. Note that although L is in level

1 by the radial measure, it is not benchmarked by other branches because of

the non-zero slacks. However, the performance of L is excellent and should

be used as a benchmark for inefficient branches. Table 13-7 and Table 13-5

show the reference sets and their benchmark targets

()

SBM

Rl by slack-based

measure. Actually, by using the slack-based measure, L is benchmarked

from branches on level 3 based on the second level evaluation context.

Level 1 Level 2 Level 3 Level 4 Level 5

Figure 13-3. Diagram of Benchmark Targets by Radial Measure

Table 13-7. Reference Sets as Benchmark Targets by Slack-based Measure

Branch Level

(1)

SBM

R (2)

SBM

R (3)

SBM

R (4)

SBM

A 3 J L

B 4 J L D, N

C 5 J L N M

D 3 J L, I

E 4 H L, I D, N

F 2 J

G 2 J

H 1

I 2 J

J 1

K 4 J L N

L 2 H

M 4 J L N

N 3 J L

Chapter 13

257

Level 1 Level 2 Level 3 Level 4 Level 5

Figure 13-4. Diagram of Benchmark Targets by Slack-based Measure

Table 13-8. Attractiveness Scores by Radial Measure for the First and Second Levels

Branches

Evaluation context

Level Branch

H 1.242 1.523

J 1.429 1.646

L 1.190 1.319

F 1 1.200

G 1 1.237

I 1 1.062

A 1 1.076

N 1 1.162

Table 13-9. Attractiveness Scores by Slack-based Measure for the First and Second Levels

Branches

Evaluation context

Level Branch

H 1.143 1.313

J 1.149 1.322

L 1 1.144

F 1 1.061

G 1 1.078

I 1 1.029

We now turn to the attractiveness scores for the branches in the first and

second levels. Table 13-8 and Table 13-9 report the attractiveness scores by

radial measure and slack-based measure, respectively. On both of the

evaluation contexts

and

, J is the best branch because it has the

largest attractiveness score.

Morita & Zhu, Context-dependent DEA

258

Consider the sales branches on the first and second levels based on the

evaluation context

E . We can rank them in the order of J, H, L, G, F and I

by slack-based measure. L is ranked in the third position by radial as well as

slack-based context-dependent DEA, which indicate that the performance of

L is good. It is reasonable that L is used as a benchmark by less efficient

branches, which can be realized by the slack-based measure, not by the

radial measure.

5. CONCLUDING REMARKS

This chapter presented a relative new DEA approach--context-dependent

DEA. Interested read should refer to Seiford and Zhu (1999a), Seiford and

Zhu (2003), Zhu (2003) and Morita, Hirokawa Zhu (2005) for detailed

applications. Chen, Morita and Zhu (2005) also provided an illustrative

application to measuring the performance of Tokyo public libraries. This

chapter also show that non-zero slacks can be incorporated into the context-

dependent DEA.

REFERENCES

1. Andersen, P. and N. C. Petersen, (1993), “A procedure for ranking

efficient units in data envelopment analysis”, Management Sciences, vol.

39, no. 10, 1261-1264.

2. Charnes, A., W. W. Cooper and E. Rhodes, (1978), “Measuring the

efficiency of decision making units”, European Journal of Operational

Research, vol. 2, 429-444.

3. Chen, Y., H. Morita and J. Zhu, (2005), “Context-dependent DEA with

an application to Tokyo public libraries”, International Journal of

Information Technology and Decision Making, Vol. 4, No. 3, 385-394.

4. Morita, H., K. Hirokawa and J. Zhu, (2005), “A slack-based measure of

efficiency in context-dependent data envelopment analysis”, Omega,

vol. 33, 357-362.

5. Seiford, L.M. and J. Zhu, (1999a), “Profitability and marketability of the

top 55 US commercial banks”, Management Science, Vol. 45, No. 9,

1270-1288.

6. Seiford, L.M. and J. Zhu, (1999b), “Infeasibility of super-efficiency

DEA models”, INFOR, vol. 37, no. 2, 174-187.

7. Seiford, L.M. and J. Zhu, (2003), “Context-dependent data envelopment

analysis: measuring attractiveness and progress”, Omega, vol. 31, no. 5,

397-408.

Chapter 13

259

8. Tone, K. (2002a), “A slack-based measure of super-efficiency in data

envelopment analysis”, European Journal of Operational Research, vol.

143, no. 1, 32-41.

9. Tone, K. (2002b), “A slack-based measure of efficiency in data

envelopment analysis”, European Journal of Operational Research, vol.

143, no. 3, 498-509.

10. Tversky, A. and I. Simonson, (1993), “Context-dependent preferences”,

Management Sciences, vol. 39, 1179-1189.

11. Zhu, J. (1996), Data envelopment analysis with preference structure.

Journal of Operational Research Society, Vol. 47, No.1, 136-150

12. Zhu, J. (2003), “Quantitative models for performance evaluation and

benchmarking –Data envelopment analysis with spreadsheets and DEA

Excel solver”, Kluwer, Boston.

Part of the material in this chapter is based upon OMEGA, vol. 31, no. 5, L. M.

Seiford and J. Zhu, Context-dependent data envelopment analysis: measuring

attractiveness and progress, pp. 397-408 (2003), and OMEGA, vol. 33, H. Morita, K.

Hirokawa and J. Zhu, A slack-based measure of efficiency in context-dependent

data envelopment analysis, pp. 357-362 (2005), with permission from Elsevier

Science.

Morita & Zhu, Context-dependent DEA

Chapter 12

NETWORK DEA

Rolf Färe

, Shawna Grosskopf

and Gerald Whittaker

Department of Economics and Department of Agricultural Economics, Oregon State

University, Corvallis, OR 97331 USA

Department of Economics, Oregon State University, Corvallis, OR 97331 USA,

shawna.grosskopf@orst.edu

National Forage Seed Production Research Center, Agricultural Research Sevice, USDA,

Corvallis, OR 97331 USA

Abstract: This chapter describes network DEA models, where a network consists of sub-

technologies. A DEA model typically describes a technology to a level of

abstraction necessary for the analyst’s purpose, but leaves out a description of

the sub-technologies that make up the internal functions of the technology.

These sub-technologies are usually treated as a “black box”, i.e., there is no

information about what happens inside them. The specification of the sub-

technologies enables the explicit examination of input allocation and

intermediate products that together form the production process. The

combination of sub-technologies into networks provides a method of

analyzing problems that the traditional DEA models cannot address. We apply

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.

Data Envelopment Analysis (DEA), Network, Intermediate Products, Dynamic

Production, Technology Adoption.

Key words: