ata definitions & notation nonlinear programming model...

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DEA Introductin 11/13/2007 page 1 of 82 ata nvelopment nalysis Dennis L. Bricker College of Engineering American University of Armenia 23 March 2006 DEA Introductin 11/13/2007 page 2 of 82 Definitions & Notation Nonlinear Programming Model The DEA LP Model DEA Dual LP Model Examples Further References Summary of Strengths & Weaknesses of DEA DEA Introductin 11/13/2007 page 3 of 82 Given: a set of n “decision-making units” (DMU) for DMU #j, inputs X ij , i=1, …m outputs Y rj , r=1,…s Estimate: The relative efficiencies of the DMUs DEA Introductin 11/13/2007 page 4 of 82 If the DMUs are typical “firms” or “industries” and the inputs (resources) and outputs (goods & services) all have market values, then the efficiency of DMU #j could be computed by: total market value of outputs produced total market value of inputs consumed j E = Suppose, however, that the DMUs are public entities or not-for-profit organizations which use resources and produce outputs which have no easily-determined market values.

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Page 1: ata Definitions & Notation Nonlinear Programming Model ...user.engineering.uiowa.edu/~dbricker/slides_handouts/DEA_intro.pdf · • Examples • Further ... rrj rrj rr mm iij iij

DEA Introductin 11/13/2007 page 1 of 82

ata nvelopment nalysis

Dennis L. Bricker College of Engineering American University of Armenia 23 March 2006

DEA Introductin 11/13/2007 page 2 of 82

• Definitions & Notation

• Nonlinear Programming Model

• The DEA LP Model

• DEA Dual LP Model

• Examples

• Further References

• Summary of Strengths & Weaknesses of DEA

DEA Introductin 11/13/2007 page 3 of 82

Given:

• a set of n “decision-making units” (DMU)

• for DMU #j, inputs Xij, i=1, …m

outputs Yrj, r=1,…s

Estimate:

• The relative efficiencies of the DMUs

DEA Introductin 11/13/2007 page 4 of 82

If the DMUs are typical “firms” or “industries” and the

inputs (resources) and outputs (goods & services) all have

market values,

then the efficiency of DMU #j could be computed by:

total market value of outputs producedtotal market value of inputs consumedjE =

Suppose, however, that

the DMUs are public entities or not-for-profit organizations

which use resources and produce outputs which have no

easily-determined market values.

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DEA Introductin 11/13/2007 page 5 of 82

The state wishes to evaluate different public schools,

whose inputs might include not only funds provided by the state

(which of course have market value!), but also less “tangible” inputs

such as:

• student-teacher ratio • socio-economic factors (median income of the population,

proportion of single-parent families, etc.) • physical facilities • parent participation in school activities

Likewise, the outputs may have intangible value, e.g.,

• standardized test scores of students • psychological tests of student attitudes • fraction of graduates seeking higher education

How can a state governing agency evaluate the performance of the schools in any kind of objective way?

DEA Introductin 11/13/2007 page 6 of 82

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That is, DMU #k wishes to solve the

optimization problem:

1

1

s

r rkrm

i iki

v YMaximize

u X

=

=

subject to: ??? any other constraints to be imposed ???

0, 1,...0 , 1,...

r

i

u r sv j m

≥ =≥ =

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DEA Introductin 11/13/2007 page 9 of 82

Restrictions on values u & v: The “prices” selected by DMU #k for the evaluation of its efficiency

cannot yield an efficiency greater than 100%,

not only for itself, but for every other DMU!

That is, the prices u & v selected by DMU #k must satisfy the

inequalities

1

1

1

s

r rjrm

i iji

v Y

u X

=

=

≤∑

∑ for every DMU j (including k )

DEA Introductin 11/13/2007 page 10 of 82

1

,

1

s

r rkr

k mu v

i iki

v YE Maximum

u X

=

=

=∑

subject to

1

1

1 for all 1,2,

s

r rjrm

i iji

v Yj n

u X

=

=

≤ =∑

∑…

0, 1,...0 , 1,...

r

i

u r sv j m

≥ =≥ =

DEA Introductin 11/13/2007 page 11 of 82

Of course, we must give each of the other

DMUs the same opportunity to choose the

values of its inputs and outputs by which their

efficiencies are to be evaluated!

DEA Introductin 11/13/2007 page 12 of 82

1

,

1

s

r rkr

k mu v

i iki

v YE Maximum

u X

=

=

=∑

subject to

1

1

1 for all 1, 2,

s

r rjrm

i iji

v Yj n

u X

=

=

≤ =∑

∑…

0, 1,...0 , 1,...

r

i

u r sv j m

≥ =≥ =

kE

Page 4: ata Definitions & Notation Nonlinear Programming Model ...user.engineering.uiowa.edu/~dbricker/slides_handouts/DEA_intro.pdf · • Examples • Further ... rrj rrj rr mm iij iij

DEA Introductin 11/13/2007 page 13 of 82

Unfortunately, these optimization problems as stated have

nonlinear objective function and nonlinear constraints...

The constraints are easily linearized by multiplying both

sides by the denominator, which is nonnegative.

1

1

1 1

1 for all 1, 2,

for all 1, 2,

s

r rjrm

i iji

s m

r rj i ijr i

v Yj n

u X

v Y u X j n

=

=

= =

≤ =

⇒ ≤ =

∑ ∑

DEA Introductin 11/13/2007 page 14 of 82

In order to linearize the objective function, we note that the objective & constraint functions are unchanged if the prices u & v were to be multiplied by any positive scalar, that is, for any α > 0,

( )

( )1 1

1 1

s s

r rj r rjr rm m

i ij i iji i

v Y v Y

u X u X

α

α

= =

= =

=∑ ∑

∑ ∑

It should be valid, then, to arbitrarily restrict the total value of the inputs of DMU #k to be equal to 1.00:

1

1m

i iki

u X=

=∑

DEA Introductin 11/13/2007 page 15 of 82

The problem

is equivalent to

which has a linear objective!

DEA Introductin 11/13/2007 page 16 of 82

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DEA- Port Arthur Police Dept 9/28/2004 page 1 of 6

You have been assigned to evaluate the efficiency of the eight precincts of the Port

Charles Police Department. The data available for each precinct are as follows:

Inputs:

Number of policemen

Number of vehicles used

Outputs:

Number of responses to service requests by patrol units (thousands/year)

Number of convictions obtained each year (hundreds/year)

The following data has been collected:

Precinct # policemen # vehicles # responses # convictions

A 200 60 6 8

B 250 65 5.5 9

C 300 90 8 9.5

D 400 120 10 11

E 350 100 9.5 9

F 300 80 5 7.5

G 275 85 9 8

H 325 75 4.5 10

The city wishes to use this information to determine which precincts, if any, are inefficient.

DEA- Port Arthur Police Dept 9/28/2004 page 2 of 6

The LP model which can be used to compute the efficiency of precinct C.

1 2Maximize 8 9.5v v

subject to

1 2300 90 1u u

1 2 1 26 8 200 60 0v v u u

1 2 1 25.5 9 250 65 0v v u u

1 2 1 28 9.5 300 90 0v v u u

1 2 1 210 11 400 120 0v v u u

1 2 1 24.5 10 325 75 0v v u u

1 2 1 29.5 9 350 100 0v v u u

1 2 1 25 7.5 300 80 0v v u u

1 2 1 29 8 275 85 0v v u u

1 2 1 20, 0, 0, 0u u v v

The total number of LP problems which need to be solved in order to compute the

efficiencies of the eight precincts is _________

DEA- Port Arthur Police Dept 9/28/2004 page 3 of 6

The output below was computed by the APL workspace “DEA” which can be downloaded from

the website at URL http://www.engineering.uiowa.edu/~dbricker/APL_software.html

i ID Efficiency Freq R 1 A 1 4 1

2 B 1 2 3

3 C 0.8727 0 6

4 D 0.809 0 7

5 E 0.9042 0 5

6 F 0.6875 0 8

7 G 1 3 2

8 H 0.963 0 4

Freq = frequency of appearance in reference sets of inefficient DMUs

R = rank based upon (Efficiency + Freq)

Slack inputs/outputs i/o C D E F H

responses 0 0 0 0 1.61111convictions 0 0 0 0 0policemen 2.43056 4.86111 22.7083 6.25 35.1852vehicles 0 0 0 0 0

DEA- Port Arthur Police Dept 9/28/2004 page 4 of 6

Prices

i ID responses convictions policemen vehicles 1 A 0.125 0.03125 0.005 0

2 B 0 0.111111 0.00111111 0.0111111

3 C 0.0925926 0.0138889 0 0.0111111

4 D 0.0694444 0.0104167 0 0.00833333

5 E 0.0833333 0.0125 0 0.01

6 F 0.025 0.075 0 0.0125

7 G 0.0909091 0.0227273 0.00363636 0

8 H 0 0.0962963 0 0.0133333

Reference Sets

For each DMU, the DMUs in its reference set are listed:

3 C 4 D 5 E 6 F 8 H 1 A 1 A 7 G 2 B 2 B

7 G 7 G 1 A 1 A

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DEA- Port Arthur Police Dept 9/28/2004 page 5 of 6

a. In order to make itself look as “efficient” as possible, what “prices” would be

assigned by precinct C to the outputs (# responses & # convictions) and to the

inputs (# policemen & # vehicles)?

Solution:

Variable Price

# responses __________/thousand responses = _______/response

# convictions __________/hundred convictions = _______/conviction

# policemen __________/policeman

# vehicles __________/vehicle

DEA- Port Arthur Police Dept 9/28/2004 page 6 of 6

b. Using these prices for precinct C, compute the ratio of the total value of the

output variables responses and convictions to the total value of input variables

policemen and vehicles.

Solution:0.092596 8 0.0138889 9.5 0.87271255

87.3%0 300 0.011111 90 1

which is in agreement with the efficiency computed for precinct C.

c. Using these same prices which would be assigned by precinct C, which precincts

would be judged to be 100% efficient?

Precincts ____ and ____.

d. By how much should precinct C cut its number of policemen in order to become

“efficient” (assuming that they could maintain their current output levels)? ____