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

240

);

TH1 YV('Y1I') YV('Y2I') YV('Y1F') YV('Y2F') YV('Y3F') MODELST

A

FM1 13.20.8 12.85.4 5 1

FM2 17.8 1 1.5 3.8 6.5 7.5 1

FM3 21.4 1.2 1.75 4.8 7.85 8.75 1

FIG4.RES

Chapter 12

Chapter 14

FLEXIBLE MEASURES

-

CLASSIFYING INPUTS

AND OUTPUTS

Wade D. Cook

1

and Joe Zhu

2

1

Schulich School of Business, York University, Toronto, Ontario, Canada,

M

3J 1P3,

wcook@shulich.yorku.ca

2

Department of Management, Worcester Polytechnic Institute, Worcester, MA 01609,

j

zhu@wpi.edu

Abstract: In standard data envelopment analysis (DEA), it is assumed that the input

versus output status of each of the chosen analysis measures is known. In some

situations, however, certain measures can play either input or output roles.

Consider using the number of interns on staff in a study of hospital efficiency.

Such a factor clearly constitutes an output measure for a hospital, being one

form of training provided by the organization, but at the same time is an

important component of the hospital’s total staff complement, hence is an

input. This chapter presents DEA models to accommodate such flexible

measures. Both an individual DMU model and an aggregate model are

suggested as methodologies for deriving the most appropriate designations for

flexible measures.

Key words: Data Envelopment Analysis (DEA), Flexible, Inputs, Outputs

1. INTRODUCTION

In data envelopment analysis (DEA) (Charnes, Cooper and Rhodes 1978;

CCR), efficiency of peer decision making units (DMUs) is defined as the

ratio of weighted outputs to weighted inputs. It is assumed that the input

versus output status of each performance measure related to the DMUs is

known priori to the application of DEA model. For example, in a study of

power plant efficiency by Cook et. al. (1998, 2005), one of the outputs is a

function of what is termed ‘outages’. This measure is designed to represent

the percentage of time that the plant is available to be in operation. On the

262

other hand, this variable can legitimately be considered as an environmental

input that represents the ability of the plant to perform as it should.

Similar arguments to those above can be made regarding the evaluation

of research productivity by universities, such as described in Beasley (1990,

1995). There, “research income” is treated as both an output and input. A

related problem setting is that in which research - granting agencies (e.g.,

NSERC in Canada and NSF in the USA) wish to allocate funds to those

researchers and universities such as to have the greatest impact. In this

environment, graduate students can play the role of either an input (a

resource available to the faculty member, effecting his/her productivity), or

as an output (trained personnel, hence a benefit resulting from research

funding). Medical interns have a similar interpretation in the evaluation of

hospital efficiency. In a very different environment, Cook et al (1992), in

evaluating robotics installations, use the measure “uptime” as an output that

represents the percentage of production time available. At the same time, one

might make the argument that this measure is, as well, an input that clearly

influences the overall operation of the technology.

In many problem situations such as those described, the input versus

output status of certain measures can be deemed as flexible. Cook and Zhu

(2007) develop DEA models for classifying a measure into an input or

output. Cook, Green and Zhu (2006) presents a methodology for dealing

with performance evaluation settings where factors can simultaneously play

both input and output roles.

This chapter presents the approach in Cook and Zhu (2007) in an effort to

facilitate the derivation of the input/output status of variables when

flexibility is an issue. The approach is illustrated with an example.

2. IDENTIFYING THE INPUT OUTPUT STATUS

Suppose we have n peer DMUs, {

j

DMU : j = 1, 2, …, n}, and each

j

DMU

produces multiple outputs y

rj

, (r = 1, 2, ..., s), by utilizing multiple

inputs x

ij

, (i = 1, 2, ..., m). When a

o

DMU is under evaluation by the CCR

model, we have (Charnes, Cooper and Rhodes, 1978)

OF FLEXIBLE MEASURES

Chapter 14

263

1

max

s.t. 1 , 1 2

, 0, ,

s

rro

r

m

iio

i

s

rrj

r

m

iij

i

ri

y

x

y

j

= , ,...,n

x

ri

μ

ν

μ

ν

μν

=

≤

≥∀

∑

(1)

In Cook and Zhu (2007), it is assumed that there are L “flexible

measures”, whose input/output status we wish to determine. The values

assumed by these measures are denoted as

lj

w for

j

DMU (l=1,..,L). For

each measure l, binary variables d

l

∈{0,1} are introduced, where d

l

= 1

designates that factor l is an output, and d

l

= 0 designates it as an input.

Let

l

γ

be the weight for each measure l. The following programming

problem is established to let

o

DMU to choose the best status (input or

output) for each measure l

1

.

11

max

(1 )

s.t. 1 1 2

(1 )

{0,1}, , , 0, , ,

sL

rro ll lo

rl

mL

iio l l lo

il

sL

rrj lllj

rl

mL

iij l l lj

il

l ril

ydw

xdw

ydw

j

= , ,...,n

xdw

dl ril

μγ

νγ

μγ

νγ

μνγ

==

+

+−

+

≤

+−

∈∀ ≥ ∀

∑∑

(2)

For any fixed value d

l

(whether 0 or 1), model (2) becomes the following

linear program

1

Note that since either d

l

= 0 or d

l

= 1, then measure l will end up either as an output or an

input.

Cook & Zhu, Flexible Measures

-

Classifying Inputs and Outputs

264

11

s

r=1 1 i=1 1

11

max

s.t. (1 ) 0 1,..., ;

(1 ) 1

{0,1}, , , 0, , ,

sL

rro ll lo

rl

LsL

rrj lllj iij l l lj

ll

mL

iio l l lo

il

lril

ydw

uy d w vx d w j n

vx d w

dvril

μγ

γγ

γ

μγ

==

+

+−−−≤=

+− =

∈≥∀

∑∑

∑∑ ∑∑

∑∑

(3)

Model (3) is clearly nonlinear. It can, however, be linearized by letting

l

δ

=

ll

d

γ

, and, for each l, imposing the constraints

(1 ),

ll

ll l

Md

M

d

δ

δγ

γδ

≤

≤+ −

(4)

where M is a large positive number. These linear integer restrictions capture

the nonlinear expression d

l

γ

=

l

δ

, without actually having to directly

specify it in the optimization model. Note that if d

l

= 1 then

l

γ

=

l

δ

, and if d

l

= 0, then

l

δ

= 0. We, therefore, have the following mixed integer linear

program

11

s

r=1 1 i=1 1

11 1

max

s.t. 2 0 1,..., ;

1

sL

rro l lo

rl

LsL

rrj l lj iij l lj

ll

mL L

iio l lo l lo

il l

ll

ll l

yw

y

wvx w j n

vx w w

Md

M( - d )

μδ

μδ γ

γδ

δ

δγ

γδ

==

== =

+

+−−≤=

+−=

≤

≤+

∑∑

∑∑∑∑

∑∑ ∑

{0,1}, , 0, , 0, ,

lllri

dri

δγ μν

∈≥≥∀

(5)

We point out that when only one flexible measure w is present, model (3)

becomes

Chapter 14

265

1

s

r=1 i=1

1

max

s.t. (1 ) 0 1,..., ;

(1 ) 1

0, , 0, , {0,1}.

s

rro o

r

s

rrj j iij j

m

iio o

i

ri

ydw

y

dw vx d w j n

vx d w

vrid

μγ

μγ γ

γ

γμ

=

+

+− −− ≤ =

+− =

≥≥∀∈

∑

∑∑

∑

(6)

Replacing d

γ

=

δ

by a single constraint set as given by (4) for L=1,

model (6) reduces to the integer linear program

1

s

r=1 i=1

1

max

s.t. 2 0 1,..., ;

1

{0,1}, , 0, , 0, ,

s

rro o

r

s

rrj j iij j

m

iio o o

i

ri

yw

ywvxw j n

vx w w

Md

M( -d)

dri

μδ

μδ γ

γδ

δ

δγ

γδ

γδ μ ν

=

+

+− −≤ =

+−=

≤

≤+

∈≥≥∀

∑

∑∑

∑

(7)

If d = 1, then

γ

=

δ

and the factor w becomes an output. Model (7) then

becomes

1

s

r=1 i=1

1

max

s.t. 0 1,..., ;

1

0, , 0, ,

s

rro o

r

s

rrj j iij

m

iio

i

ri

yw

y

wvx j n

vx

ri

μδ

δμν

=

+

+− ≤ =

=

≥≥∀

∑

∑∑

∑

If d = 0, then

δ

= 0 and the factor w becomes an input, and model (7)

becomes

Cook & Zhu, Flexible Measures

-

Classifying Inputs and Outputs

266

1

s

r=1 i=1

1

max

s.t. 0 1,..., ;

1

0, , 0, ,

s

rro

r

s

rrj iij j

m

iio o

i

ri

y

vx w j n

vx w

ri

μ

μγ

γ

γμν

=

−−≤=

+=

≥≥∀

∑

∑∑

∑

In the above development, we solve model (5), and obtain a set of optimal

*

l

d for each DMU. One criterion for deciding the input versus output status

of the measure, would be to base it on the majority choice among the DMUs.

An alternative approach would be to use the designation (input or output),

that renders the aggregate efficiency of the collection of DMUs as large as

possible. Such a model would be helpful if ties are encountered using model

(5)

Cook and Zhu (2007), therefore, assume that the optimal input/output will

be the one created by the model which optimizes the aggregate or average

ratio of outputs to inputs, namely

11 1 1

11

()

max

()(1)

s.t. 1 1 2

(1 )

{0,1}, , , 0, ,

sn L n

rrj lllj

rj l j

mn L n

iij lllj

ij l j

sL

rrj lllj

rl

mL

iij l l lj

il

lril

ydw

xdw

ydw

j

= , ,...,n

xdw

dr

μγ

νγ

μγ

νγ

μνγ

== = =

==

+

+−

+

≤

+−

∈≥∀

∑∑ ∑ ∑

∑∑

,.il

(8)

For notational convenience, let

∑

=

n

j

rjr

yy

1

~

,

∑

=

n

j

iji

xx

1

~

, and

∑

=

n

j

ljl

ww

1

~

. Using (4), the aggregate problem is then equivalent to the

following integer linear programming problem

Chapter 14

267

11

s

r=1 1 i=1 1

11 1

max

s.t. 2 0, 1,...,

1

{

sL

rr l l

rl

LsL

rrj l lj iij l lj

ll

mL L

ii ll ll

il l

ll

ll l

l

yw

y

wvx w j n

vx w w

Md

M( - d )

d

μδ

μδ γ

γδ

δ

δγ

γδ

==

== =

+

+−−≤=

+−=

≤

≤+

∈

∑∑

∑∑∑∑

∑∑ ∑

%%

%% %

0,1}, , , 0, , ,

ril

μνγ

≥∀

(9)

3. APPLICATION

In this section, we apply the models to the data set in Beasley (1990).

Table 14-1 reports the data set which consists of two inputs, General

Expenditure and Equipment Expenditure and three outputs, consisting of

three types of students. The “flexible measure” here is the Research Income.

Table 14-2 reports the results from model (7), where the second column

shows the optimal d and the third column, the optimal value to model (7). In

this case, 20 out of the 50 universities treat the research income measure as

an output, i.e., the majority of 30 treat it as an input.

When we apply our aggregate model (9), the optimal d = 1, with the

optimal objective function value being 0.69329. This indicates that from an

aggregate efficiency perspective, research income is treated as an output.

Part of the explanation for the different results between the two approaches

may be that for the 30 cases where the input was the preferred status using

model (7), that preference may not have been as strong as was the preference

of output versus input in the other 20 cases. Hence, the majority concept may

be flawed by failing to take account of the difference in efficiency for each

DMU, when the flexible measure is used as an output versus an input.

Cook & Zhu, Flexible Measures

-

Classifying Inputs and Outputs

268

Table 14-1. University data

DMU

General

expenditure

Equipment

expenditure

UG

Students

PG

Teaching

PG

Research

Income

Universit

y

1 528 64 145 0 26 254

Universit

y

2 2605 301 381 16 54 1485

Universit

y

3 304 23 44 3 3 45

Universit

y

4 1620 485 287 0 48 940

Universit

y

5 490 90 91 8 22 106

Universit

y

6 2675 767 352 4 166 2967

Universit

y

7 422 0 70 12 19 298

Universit

y

8 986 126 203 0 32 776

Universit

y

9 523 32 60 0 17 39

Universit

y

10 585 87 80 17 27 353

Universit

y

11 931 161 191 0 20 293

Universit

y

12 1060 91 139 0 37 781

Universit

y

13 500 109 104 0 19 215

Universit

y

14 714 77 132 0 24 269

Universit

y

15 923 121 135 10 31 392

Universit

y

16 1267 128 169 0 31 546

Universit

y

17 891 116 125 0 24 925

Universit

y

18 1395 571 176 14 27 764

Universit

y

19 990 83 28 36 57 615

Universit

y

20 3512 267 511 23 153 3182

Universit

y

21 1451 226 198 0 53 791

Universit

y

22 1018 81 161 5 29 741

Universit

y

23 1115 450 148 4 32 347

Universit

y

24 2055 112 207 1 47 2945

Universit

y

25 440 74 115 0 9 453

Universit

y

26 3897 841 353 28 65 2331

Universit

y

27 836 81 129 0 37 695

Universit

y

28 1007 50 174 7 23 98

Universit

y

29 1188 170 253 0 38 879

Universit

y

30 4630 628 544 0 217 4838

Universit

y

31 977 77 94 26 26 490

Universit

y

32 829 61 128 17 25 291

Universit

y

33 898 39 190 1 18 327

Universit

y

34 901 131 168 9 50 956

Universit

y

35 924 119 119 37 48 512

Universit

y

36 1251 62 193 13 43 563

Universit

y

37 1011 235 217 0 36 714

Universit

y

38 732 94 151 3 23 297

Universit

y

39 444 46 49 2 19 277

Universit

y

40 308 28 57 0 7 154

Universit

y

41 483 40 117 0 23 531

Universit

y

42 515 68 79 7 23 305

Universit

y

43 593 82 101 1 9 85

Universit

y

44 570 26 71 20 11 130

Universit

y

45 1317 123 293 1 39 1043

Universit

y

46 2013 149 403 2 51 1523

Universit

y

47 992 89 161 1 30 743

Universit

y

48 1038 82 151 13 47 513

Universit

y

49 206 1 16 0 6 72

Universit

y

50 1193 95 240 0 32 485

Chapter 14

269

Table 14-2. Results from model (7)

DMU D Efficiency DMU D Efficiency

University1 1 1 University26 0 1

University2 0 1 University27 1 0.855471

University3 0 0.837244 University28 0 1

University4 1 0.685697 University29 1 0.824968

University5 0 1 University30 0 1

University6 0 1 University31 1 0.775853

University7 1 1 University32 0 0.896402

University8 1 0.811941 University33 1 1

University9 0 1 University34 0 1

University10 1 0.906595 University35 1 1

University11 0 0.890126 University36 0 0.8369

University12 1 0.709313 University37 1 0.830789

University13 0 0.803249 University38 0 0.833414

University14 0 0.767744 University39 0 0.791219

University15 0 0.704214 University40 1 0.741404

University16 0 0.54274 University41 1 1

University17 1 0.819451 University42 0 0.847172

University18 1 0.627824 University43 0 0.920638

University19 1 1 University44 0 1

University20 0 1 University45 0 1

University21 0 0.699625 University46 0 1

University22 1 0.716738 University47 1 0.688445

University23 0 0.617112 University48 0 0.938878

University24 0 1 University49 0 1

University25 1 1 University50 0 0.841683

4. CONCLUSIONS

Conventional DEA analyses require that each variable or measure be

assigned an explicit designation specifying whether it is an input or output.

In various settings, however, it remains that there are variables whose status

is flexible. In cases where a resource can as well represents a tangible

product of the organization (medical interns, graduate students, research

funding, etc.), this flexibility is present.

The current chapter presents the Cook and Zhu’ (2007) modification to

the standard CCR model that permits the inclusion of flexible measures in

the DEA analysis. The approach assigns an optimal designation, whether

input or output to each such variable. Two models are given for

accomplishing this, namely an individual DMU model, and one that

optimizes the aggregate efficiency of the collection of DMUs.

Cook & Zhu, Flexible Measures

-

Classifying Inputs and Outputs

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

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