data envelopment analysis

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University of Wisconsin-MilwaukeeLubar School of Business

Session 3Data Envelopment Analysis:

Comparing Performance of Different Units with Multiple Inputs and Outputs

Alexander KolkerAdjunct Faculty

Alexander Kolker. All rights reserved.

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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


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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-DMUs S1 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 revenueper patient.

• Therefore they are assumed to be efficient, andwould receive a relative efficiency score of 1.

• Surgeon 4, however is relatively inefficient (relatively to thepeers). S4 efficiency score is less than 1.


LOS, days0 1 2 3 4

Surg

ical

kit

s

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 problemThe 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 kSubject 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 datato the interval 0 to 1 by dividing each input and outputdata 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 isthe 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 surgicalquality 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 efficiencylevel of its peers


Extending data envelopment analysis 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 ratiosgive a different picture and it is difficult to combine the entireset of ratios into a single judgment


DEA applicationInputs 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 expenditureOutputs•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 ->

Input 1:

General expenditure,

$ 000

Input 2:

Equipment expenditure, $

000

Output 1:

# of undegrad

s

Output 2:

# of postdocs

Output

3: # of research fellows

Output 4:

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 inputsand outputs, we use our judgment as to what areappropriate constraints to add to the basic DEA model.

For example we might prevent zero weights by addingconstraints such as:

weights>= some small numbers (say, 0.01 rather than 0, as inthe basic model)


On top of that we can argue that, for example, the weights for:• the number of postdocs should 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 ->

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


Data envelopment analysis (DEA) Methodology Summary

• DEA requires the multiple inputs and outputs for each DMUto 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 scorefor a particular DMU by choosing input/output weights thatmaximize the score, thereby presenting the DMU in the bestpossible 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 canalso 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 measurementerrors 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

Healthcare