Data Envelopment Analysis

1
University of Wisconsin-Milwaukee
Lubar School of Business
Session 3
Data Envelopment Analysis:
Comparing Performance of Different Units
with Multiple Inputs and Outputs
Alexander Kolker
Adjunct Faculty
Alexander Kolker. All rights reserved.

2
OUTLINE
•Data Envelopment Analysis:
•What is it ?
•What is it for?
• A concept of the efficiency frontier and a scoring function
•DEA as a linear optimization problem
• Examples of DEA using Excel Add-in Solver
• Extending DEA using Value judgment

3
DEA: The Main Concept Points
• Data envelopment analysis (DEA) is a technique that can
be used to measure the multiple dimensions of
performance (efficiency) of producing units
• These producing units are called Decision-Making Units
(DMU)
• DEA allows multiple inputs and outputs to be used that
develop a single efficiency score

• DEA can be used to measure comparative efficiency
(performance) of hospitals, physicians, group practices,
or any other producing unit- DMU using a so-called
scoring function (defined below)
(Cont.)
Alexander Kolker. All rights reserved. 4
• In the heart of the DEA is finding the "best" producer
(DMU) among many other producers (comparative
DMUs).

(Cont.)
• If one producer (DMU1) is better than another producer
(DMU2) by either making more output with the same input
or making the same output with less input then this
producer (DMU1) is more efficient than another one
(DMU2).
• The procedure of comparing the producers aimed at finding
efficient ones vs. less efficient and by how much can be
formulated as a linear optimization problem.
• Analyzing the efficiency of N producers is then a set of N
linear optimization problems.

DEA vs. Statistical Approach
•A typical statistical approach is characterized as a central
tendency approach and it evaluates producers relative to an
average producer
•In contrast, DEA is an extreme point method and compares
each producer with only the "best" producers. Extreme point
method is not always the right tool for a problem but it is an
appropriate approach in many cases
•A fundamental assumption behind an extreme point method
is that if a given producer is capable of producing Y units of
output with X units of inputs, then other producers should
also be able to do the same if they were to operate efficiently
Note: This assumption is not always true. Not everybody can become an Olympics
champion, no matter how much one is trained

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DEA-- A Simple Example
Surgeons-DMUsS1 to S4
Inputs: S1 S2 S3 S4
Length of stay-
LOS
2.5 1 3 4
Surgical kits,
units
1 3 2 3
Output: Net
Revenue per
patient, $
$4000 $4000 $4000 $3000

•Surgeons S1, S2, and S3 use different combinations of
resources: LOS and surgical kits
• However they produce the same output: the net revenue
per patient.
• Therefore they are assumed to be efficient, and
would receive a relative efficiency score of 1.
• Surgeon 4, however is relatively inefficient (relatively to the
peers). S4 efficiency score is less than 1.

LOS, days
0 1 2 3 4
Surgicalkits
4
3
2
1
S2
S1
S3
S4Inefficiency
• S4 must reduce either the medically necessary LOS, or the
use of surgical kits, or both, to become as efficient as
his/her peers.
• The amount of the reduction necessary is called
inefficiency

• Thus, DEA allows calculating how much the
input/output mix must be changed for inefficient
DMUs to reach efficiency relatively to their peers

DEA as a linear optimization problem
The score of a DMU is defined as the ratio of the weighted
outputs and the weighted inputs, i.e.
Score=(weighted sum of outputs)/(weighted sum of inputs)
Now,
For each DMU k:
Maximize the defined above score of unit k
Subject to (s.t.):
For every unit j (including k):
Score(j)<=1

• Thus, unit k may choose a scoring function that makes it
look as good as possible (maximized) subject to no other
unit getting a score >1 using the same scoring function.
• If unit k gets a score of 1, it means that there is no other
unit strictly dominating k
• Now, let’s translate this problem into a more formal Linear
Optimization problem:

The above optimization problem is a nonlinear problem.
However, it can be converted into a linear optimization
problem.
To do this:
• an additional constraint is introduced setting the
denominator of the objective function equal to 1
(technically this can be done as the above nonlinear problem
has one degree of freedom - multiplying all the weights by a
(positive) scale factor would leave the solution value
unchanged).

Thus,

Exercise 1.
There are 4 performance metrics collected for 6 hospital’s
units (DMUs)
• The first 2 metrics are treated as inputs:
cost per patient (1) and % of Medicaid patients for each DMU (2);
• The last 2 metrics are treated as outputs:
Surgical quality score (1) and the length of medically necessary
patient stay-LOS (2)
Unit / Input/Output ->
Input 1:
Cost/patient
Input 2:
% Medicaid/Medicare
Output 1:
Surgical Quality
score
Output 2:
LOS days
unit 1 $8,939 55 25.2 6
unit 2 $8,625 49 28.2 5
unit 3 $10,813 58 29.4 8
unit 4 $10,638 51 26.4 11
unit 5 $6,240 51 27.2 7
unit 6 $4,719 41 25.5 12

• Which units (DMU) can be considered efficient, and
which ones are less efficient?
• If a unit is less efficient, how much more output does it
need to produce in order to become as efficient as the
best units (increase the surgical quality score and/or
decrease the LOS) ?
Use DEA excel template: file DEA-6 DMU 2 In 2 Out
and Add-in Solver

DEA set up Excel template

Explanation of the template and Excel Solver set-up:
• In each Tab named unit 1, unit 2, ….., unit 6 (k=6)
4 decision variables-weights are in cells b22:e22
(2 input weights and 2 output weights)
• Objective function is in cell i13;
it is set up as =sumproduct(d15:e15, d$22:e$22), i.e.
score= v1*output1+v2*output2
• Constraint 1, i.e. weighted input=1 is in cell h14
=sumproduct(b15:c15, b$22:c$22)=1
• Constraints sum of weighted outputs<=sum of weighted
inputs for all k=6 units are in cells f15:f20 and h15:h20,
respectively. They set up as corresponding ‘=sumproducts’

Solver set-up panel

• Raw input data are provided in different measures and
scales of different orders of magnitude
• Therefore decision variables-weights are calculated in
Solver in corresponding inverse measures and could have
different orders of magnitude
• In order to minimize rounding errors the Solver box “Use
Automatic Scaling” can be checked on
• However, a more reliable way is to normalize input data
to the interval 0 to 1 by dividing each input and output
data point by the maximal value for this variable

DEA Results and Discussion
Unit #
Unit
efficiency
score Input 1-Cost/patient
Input 2-
%
Medicaid/Me
dicare
Output 1-Surgical
Quality score Output 2-LOS,
days
6 1.000 $4,719 41% 25.5 12
2 1.000 $8,625 49% 28.2 5
5 1.000 $6,240 51% 27.2 7
4 0.8318 $10,638 51% 26.4 11
3 0.8295 $10,813 58% 29.4 8
1 0.8085 $8,939 55% 25.2 6

Key Points:
•Units 6, 2 and 5 are efficient. They have the score=1
•Units 4, 3 and 1 are less efficient than their peers. Unit 1 is
the least efficient with the lowest score=0.808
• What can unit 1 do to improve its efficiency score ?
It can, for example, decrease LOS, or improve the surgical
quality score, or do both.
But by how much?

• If we run the DEA model testing different reduced LOS for
unit 1, we can identify that LOS=4.8 days makes the
efficiency score for this unit equal to 1, i.e. its efficiency
much improved to the level of its peers
• Unit 1 can also simultaneously decrease the LOS and
increase the surgical quality score.
For example, LOS=5 days (instead of 6 days) and the
surgical quality score 32.2 (instead of 25.2) make the
efficiency score equal to 1, i.e. return this unit to efficiency
level of its peers

Extending data envelopmentanalysis using value
judgment
We will construct a DEA model for comparing university
departments concerned with the same discipline.
Let’s consider two business schools using the following data:

School 1 School 2
Student numbers
Undergraduates 161 190
Postgraduates 111 90
Research Fellows 32 12
Total number 304 292
Expenditure ($'000)
General expenditure 970 600
Equipment expenditure 64 55
Total expenditure 1034 655
Other data
Academic staff 35 27
Research budget ($’000) 220 120
Research rating 3 3

How can we compare these two departments using this data?
A traditionally used method is ratios, for example:
School 1 School 2
Expenditure per:
student 3.2 2.05
staff member 29.5 26.7
Research budget per:
$ of expenditure 0.21 0.12
Students per:
staff member 8.7 17
(the staff/student ratio)
Equipment expenditure per:
student 0.21 0.08
A problem with comparison via ratios is that different ratios
give a different picture and it is difficult to combine the entire
set of ratios into a single judgment

DEA application
Inputs and outputs
What do we have to choose as our inputs and outputs?
The answer is not as obvious as it might seem. For illustration
the following inputs and outputs are chosen:
Inputs
•General expenditure
•Equipment expenditure
Outputs
•Number of undergraduates
•Number of postgraduates
•Number of research fellows
• Research budget

The input and output information is summarized
below:
Unit /
Input/Output ->
Input 1:
General
expenditure,
$ 000
Input 2:
Equipment
expenditure,
$ 000
Output 1:
# of
undegrads
Output 2:
# of
postdocs
Output 3:
# of
research
fellows
Output 4:
Research
budget,
$000
school 1 $970 $64 161 111 22 $220
school 2 $600 $55 190 90 12 $120

Results (use the file DEA-2 schools.xlsx)
School 1:
Solution:
w(i)-
weights 1.000 0.000 0.000 0.276 0.724 0.000
School 2:
Solution:
w(i)-
weights 0.000 1.164 1.000 0.000 0.000 0.000
School 2
Objective
Function: Output
score-> max
1.000
School 1
Objective
Function:
Output score->
max
1.000
These weights and the same efficiency scores =1 seem
unrealistic.
Many input/ output factors are ignored: they are equal to 0
Unit /
Input/Output ->
Input1:
General
expenditure,
$ 000
Input2:
Equipment
expenditure,$
000
Output1:
# of
undegrad
s
Output2:
# of
postdocs
Output
3: # of
research
fellows
Output4:
Research
budget,
$000

How can the basic model be improved ?
In order to improve the model we introduce more constraints.
This addition of constraints involves value judgments.
Just as we exercised our judgment in choosing the inputs
and outputs, we use our judgment as to what are
appropriate constraints to add to the basic DEA model.
For example we might prevent zero weights by adding
constraints such as:
weights>= some small numbers (say, 0.01 rather than 0, as in
the basic model)

On top of that we can argue that, for example, the
weights for:
• the number of postdocsshould be >= the number of
undegrads
• the number of research fellows >= the number of
postdocs
• the number of research fellows >= twice the number
of undergrads
• research budget >= general expenditure
Other appropriate constraints can be added to the
basic model using modified weights constraints

Results for DEA model with additional constraints:
School 1:
Solution:
w(i)-
weights 0.010 0.990 0.204 0.408 0.408 0.010
School 2:
Solution:
w(i)-
weights 0.010 1.156 0.239 0.477 0.477 0.010
School 2
Objective
Function: Output
score-> max
0.891
School 1
Objective
Function:
Output score->
max
1.000
These weights and the efficiency scores seem more realistic.
We conclude that school 2 is less efficient than school 1
Unit /
Input/Output ->
Input1:
General
expenditure,
$ 000
Input2:
Equipment
expenditure,
$ 000
Output1:
# of
undegrads
Output2:
# of
postdocs
Output
3: # of
research
fellows
Output4:
Research
budget,
$000

Data envelopment analysis (DEA) Methodology
Summary
• DEA requires the multiple inputs and outputs for each DMU
to be specified
• DEA defines efficiency score for each DMU as a weighted
sum of outputs [total output] divided by a weighted sum of
inputs [total input]
• DEA restricts all efficiency scores to the range 0 to 1
• DEA calculates the numerical value of the efficiency score
for a particular DMU by choosing input/output weights that
maximize the score, thereby presenting the DMU in the best
possible light

Strengths and Limitations of DEA
Strengths of DEA
DEA can be a powerful tool when used wisely. A few of the
characteristics that make it powerful are:
• DEA can handle multiple input and multiple output models
• It doesn't require an assumption of a functional form
relating inputs to outputs.
• DMUs are directly compared against a peer or combination
of peers
• Inputs and outputs can have very different units. For
example, X1 could be in units of lives saved and X2 could be
in units of dollars without requiring an a priori tradeoff
between the two.

Limitations of DEA
The same characteristics that make DEA a powerful tool can
also create problems. These limitations should be kept in
mind when choosing whether or not to use DEA:
• Since DEA is an extreme point technique, noise (even
symmetrical noise with zero mean) such as measurement
errors can cause problems
• DEA is good at estimating "relative" efficiency of a DMU but
it converges very slowly to "absolute" efficiency.
• DEA can tell how well you are doing compared to your
peers but not compared to a "theoretical maximum."
• Since a standard formulation of DEA creates a separate
linear program for each DMU, large problems can be
computationally intensive.

QUESTIONS ?

Data Envelopment Analysis

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