DEA introduction

1. 國立臺北護理健康大學 NTUHS
Data Envelopment Analysis (DEA)
Orozco Hsu
2021-06-10
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2. About me
• Education
• NCU (MIS)、NCCU (CS)
• Work Experience
• Telecom big data Innovation
• AI projects
• Retail marketing technology
• User Group
• TW Spark User Group
• TW Hadoop User Group
• Taiwan Data Engineer Association Director
• Research
• Big Data/ ML/ AIOT/ AI Columnist
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「How can you not get romantic about baseball ? 」

3. Tutorial
Content
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Linear Programming (Maximizing Profit)
DEA introduction
Homework
Operation Research

4. Code
• Download code
• https://github.com/orozcohsu/ntunhs_2020/tree/master/alg_20210610
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5. Quiz 1
• 1. Operations research is the application of ________methods to
arrive at the optimal Solutions to the problems.
• A. economical
• B. scientific
• C. artistic
• D. a and b both
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6. Quiz 2
• 2. Operations research is based upon collected information,
knowledge and advanced study of various factors impacting a
particular operation. This leads to more informed ________.
• A. Management processes
• B. Decision making
• C. Procedures
• D. Machine learning
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7. Quiz 3
• 3. What is the objective function in linear programming problems?
• A. A constraint for available resource
• B. An objective for research and development of a company
• C. A linear function in an optimization problem
• D. A set of non-negativity conditions
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8. Quiz 4
• About the Data Envelopment Analysis which is TRUE?
• A. We can also use regression line to analysis inefficiency DMUs
• B. In CCR model of input oriented, that means we want to maximum the
outputs
• C. Compare with CRS and VRS, CSR has more benchmarks (efficiency score=1)
• D. Data envelopment analysis may generate inaccurate efficiency scores,
especially when the sample size is small relative to the number of inputs and
outputs
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9. Operation Research
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10. Operation Research
• The field of operations research
provides QUANTITATIVE methods to
solve problems including MAXIMIZE
profits or MINIMIAZE losses,
investigating the outcomes under
fluctuating market conditions, and to
facilitate decision making.
• A new field which started in the late
1930’s and has grown and expanded
tremendously in the last 30 years.
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11. Operation Research
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(Simulation)

12. Operation Research
• Operations Research in one word
• OPTIMIAZTION
• Operations Research in one sentence
• DO THE THINGS BEST UNDER CONSTRANITS.
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https://en.wikipedia.org/wiki/Convex_optimization
Convex optimization is a subfield of mathematical optimization
that studies the problem of minimizing convex functions over
convex sets. Many classes of convex optimization problems
admit polynomial-time algorithms, whereas mathematical
optimization is in general NP-hard

13. Operation Research
• Three main problem classes in OR
• Mathematical programming
• Linear Programming with objective
function and constraints.
• Numerical optimization
• Convex Optimization: Gradient-based
• Non-gradient: Bayesian, Genetic
Algorithms…etc.
• Simulation
• Used to repeated random sampling and
obtains numerical result to approximate a
probability distribution , like MCMC
(Model Carlo Markov Chain)
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https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo
Newton method
Tangent line

14. Operation Research
• Linear Programming
• Translate into a model.
• A simple technique where we depict
complex relationships through linear
functions and then find the optimum points.
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15. Operation Research
• Nonlinear Programming
• The process of solving an optimization problem where
some of the constraints or the objective function are
nonlinear.
• Objective function
• CONCAVE (maximization problem)
• CONVEX (minimization problem)
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Maximize: f(x) = x1 + x2
Subject to:
x1 ≥ 0
x2 ≥ 0
x1
2 + x2
2 ≥ 1
x1
2 + x2
2 ≤ 2

16. Operation Research
• The gradient decent optimization
• Gradient descent is A FIRST-ORDER iterative
optimization algorithm for finding a LOCAL
minimum of a differentiable function.
• The idea is to take repeated steps in the OPPOSITE
DIRECTION of the gradient of the function at the
current point, because this is the direction of
steepest descent.
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For bigdata consideration, we used to use mini-batch for gradient decent (SGD).
https://towardsdatascience.com/batch-mini-batch-stochastic-gradient-descent-7a62ecba642a

17. Operation Research
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• Linear Programming
• Object function is linear
• Non-Linear Programming
• Object function is non-linear (convex)
• Convex
• Global optimization
• Non-Convex
• Local optimization
• Time complexity
• Polynomial time
• NP complete

18. Operation Research
• Monte Carlo: Simulation to draw
quantities of interest from the distribution.
• Markov Chain: Stochastic process in which
future states are independent of past state
given the present state.
• MCMC: A class of method in which we can
simulate draw that are slightly dependent
and are approximately from posterior
distribution.
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https://www.youtube.com/watch?v=R9NQY2Hyl14

19. Linear Programming (Maximizing Profit)
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20. Linear Programming (Maximizing Profit)
• Linear Programming (LP) and the Simple algorithm has been around
for decades.
• It was first introduced in the U.S. Air Force for helping with strategical
planning back in the 40s.
• Ever since then, many industries are taking advantage of it to
maximize profit and minimize cost, among other things.
• Often used in process of data-driven decision making.
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https://arxiv.org/ftp/cs/papers/0611/0611008.pdf
*Why LP cannot solve large instances of NP-complete problems in polynomial time?
(convex optimization)

21. Linear Programming (Maximizing Profit)
• Objective function
• Maximize or Minimize
• Constrains
• linear inequalities
Maximize: 3x + 2y
Subject to: y < 20 – 6x
y < 12 – 2x
y < 10 -x
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22. Linear Programming (Maximizing Profit)
• 2D Process:
• (K: Capital, L: Labor)
• Constrains:
• GH
• Best solution:
• At point E (OJN)
• Feasible region reaches the isoquant
for 200Q (the highest possible)
• Take action:
• 200 units of output with process 2
by using 8L and 8K
K/L =2
K/L =1
K/L =0.5
200Q = D(6L, 12K), E(8L, 8K), F(12L, 6K)
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23. Linear Programming (Maximizing Profit)
• Mix of constrains
• Find out the maximize point
2D line intersects
graphic_solution.ipynb
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24. Linear Programming (Maximizing Profit)
• This demonstrates why we don't try to
GRAPH the FEASIBLE REGION when
there are more than two decision
variables.
• Three dimensional graphs aren't that
easy to draw and you can forget about
making the sketch when there are four
or more decision variables.
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25. Linear Programming (Maximizing Profit)
• Problem we need to solve
• The company produces four furniture items: chairs, tables, desks,
and bookcases.
• By improving the operations of the firm and its resources allocation, we can
potentially maximize the profit
5 board 10 man-hours 3 ounces 4 square fee of leather
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26. Linear Programming (Maximizing Profit)
• The bottom neck is that all these material have the following total
quantities available, every week
• Our task is to decide how to better allocate these resources together
in order to make the most profit.
Material item quantities
Board 20,000
Man-hours 4,000
Ounces of glue 2,000
Square feet of leather 3,000
Square feet of glass 500
Product Profit($)
Chair 45
Table 80
Desk 110
Bookcase 55
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27. Linear Programming (Maximizing Profit)
• This is the Maximization LP (Linear Programming) problem
• c for chair, t for table, d for desk, and b for bookcase
constraints
resources available 27

28. Linear Programming (Maximizing Profit)
• Optimal profit
•$20,857
• Chair
•57
• Table
•228
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Activity-Analysis-Problem.ipynb

29. DEA introduction
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30. DEA introduction
• Question:
• How to evaluate the efficiency of those 3
hospitals?
• Answer:
• You may refer to many indicators such like
Occupancy Rate, Average Length of Stay, Days of
Stay…etc.
• Each hospital has different indicators, but how
to evaluate?
• Use scatter plot that can help, but only works
with maximum 3 indicators in visualization.
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Decision Making Unit (DMU): It represents the basic unit of analysis.
In this case, 10 hospitals equals to 10 DMU.

31. DEA introduction
• Data Envelopment Analysis (DEA)
• Consider multiple efficiency indicators, that can analyze hospital efficiency
(input) and hospital service quality (output)
• Use efficiency score as a key indicator to evaluation
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DMU X1 (Labor) X2 (Medical Devices) Y1 (Operation) Y2 (Occupancy Rate)
Hospital A 50 60 40 30
Hospital B 75 95 55 65
Hospital C 100 120 150 130
Input Output
variable
Input/ Output variables are no limit. If there are more variables, the number of DMUs should be more
only then can the analysis result be valid. DMU ≥ Max{a*b ; 3*(a+b)} => a: input variables; b: output variables

32. DEA introduction
• Solve the equations (u1, u2, v1, v2)
• Hospital A: (u1*40+u2*30)/(v1*50+v2*60) ≤ 1
• Hospital B: (u1*55+u2*65)/(v1*75+v2*95) ≤ 1
• Hospital C: (u1*150+u2*130)/(v1*100+v2*120) ≤ 1
• u1, u2, v1, v2 >=0
• Calculating Hospital A efficiency score
• Put u1, u2, v1, v2 into (u1*40+u2*30)/(v1*50+v2*60)
• There will be at least one (or more than one) hospital
efficiency score= 1
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33. DEA introduction
• DEA has two main oriented method
• Constant return to scale (CRS): When
adding one point of resource input will
bring one point of output, then the
relationship between input and output
variables is a fixed return.
• Variable return to scale (VRS): If one
point of investment produces more than
one point of results, or one point of
investment is less than one point of
results.
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DMU Efficiency score
Hospital A 0.93
Hospital B 1.00
Hospital C 1.00
Hospital D 0.92
A
B
C
D (inefficient DMU)
CSR Frontier

34. DEA introduction
• DEA
• It only gives you relative efficiencies -
efficiencies relative to the data
considered. It does not, and cannot,
give you absolute efficiencies.
• DEA with supervise learning
• Without any hypothesis (ex. data from
normal distribution)
• Without any handling with outlier data
• Without any feature selection process
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35. DEA introduction
• CCR model:
• The model was first introduced in 1978, developed a linear programming
• It is based on the assumption that constant return to scale exists at the
efficient frontiers.
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36. DEA introduction
• BBC model:
• The model is based on the VRS technology
assumption and it measures the pure technical
efficiency.
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https://www.researchgate.net/publication/333403650_Use_of_data_envelop
ment_analysis_to_benchmark_environmental_product_declarations-
a_suggested_framework

37. DEA introduction
• We will start DEA from this table. (We use CCR assumption)
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Price or Cost
(Non Beneficial)
Input
Storage Space
(Beneficial)
Output
Camera Quality
(Beneficial)
Output
Looks
(Beneficial)
Output
Mobile1 DMU1 250/527.97=0.4735 16/50.6=0.3162 12/22.98=0.5222 4/8.79=0.4551
Mobile2 DMU2 225/527.97=0.4262 16/50.6=0.3162 8/22.98=0.3482 5/8.79=0.5689
Mobile3 DMU3 300/527.9=-0.5682 32/50.6=0.6325 16/22/98=0.6963 4.5/8.79=0.512
Mobile4 DMU4 275/527.97=0.5209 32/50.6=0.6325 8/22.98=0.3482 4/8.79=0.4551
527.97 50.6 22.98
8.79
෍
𝑗=0
𝑛
𝑋𝑡𝑗2

38. DEA introduction
• Formulating the problem
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n: number of DMU (4)
m: number of input criteria (1)
S: number of output criteria (3)
Xik and yrk denote the values of ith input criterion and rth output criterion for
kth DMU
ur and vi are the non-negative variable weights to be determined by the
solution of the minimization proble
4
1 3

39. DEA introduction
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• Formulating the problem
g1 = min(0.4735V1)
Subject to
-0.3162u1 – 0.5222u2 – 0.4551u3 + 0.4735V1 >= 0
-0.3162u1 – 0.3482u2 – 0.5689u3 + 0.4262V1 >= 0
-0.6325u1 – 0.6963u2 – 0.5120u3 + 0.5682V1 >= 0
-0.6325u1 – 0.3482u2 – 0.4551u3 + 0.5209V1 >= 0
0.3162u1 + 0.5222u2 + 0.4551u3 = 1
u1, u2, u3 >=0
We need to calculate g1, g2, g3, g4

40. DEA introduction
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• Formulating the problem
g2 = min(0.4262V1)
Subject to
-0.3162u1 – 0.5222u2 – 0.4551u3 + 0.4735V1 >= 0
-0.3162u1 – 0.3482u2 – 0.5689u3 + 0.4262V1 >= 0
-0.6325u1 – 0.6963u2 – 0.5120u3 + 0.5682V1 >= 0
-0.6325u1 – 0.3482u2 – 0.4551u3 + 0.5209V1 >= 0
0.3162u1 + 0.3483u2 + 0.5689u3 = 1
u1, u2, u3 >=0
We need to calculate g1, g2, g3, g4

41. DEA introduction
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• Formulating the problem
g3 = min(0.5682V1)
Subject to
-0.3162u1 – 0.5222u2 – 0.4551u3 + 0.4735V1 >= 0
-0.3162u1 – 0.3482u2 – 0.5689u3 + 0.4262V1 >= 0
-0.6325u1 – 0.6963u2 – 0.5120u3 + 0.5682V1 >= 0
-0.6325u1 – 0.3482u2 – 0.4551u3 + 0.5209V1 >= 0
0.6325u1 + 0.6963u2 + 0.512u3 = 1
u1, u2, u3 >=0
We need to calculate g1, g2, g3, g4

42. DEA introduction
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• Formulating the problem
g4 = min(0.5209V1)
Subject to
-0.3162u1 – 0.5222u2 – 0.4551u3 + 0.4735V1 >= 0
-0.3162u1 – 0.3482u2 – 0.5689u3 + 0.4262V1 >= 0
-0.6325u1 – 0.6963u2 – 0.5120u3 + 0.5682V1 >= 0
-0.6325u1 – 0.3482u2 – 0.4551u3 + 0.5209V1 >= 0
0.6325u1 + 0.3482u2 + 0.4551u3 = 1
u1, u2, u3 >=0
We need to calculate g1, g2, g3, g4

43. DEA introduction
• Practice DEA
• Go to the pyDEA folder
• Type python -m
pyDEA.main_gui to RUN
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44. DEA introduction
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Using dea_example.xlsx

45. DEA introduction
DMU Efficiency Score
Mobile1 DMU1 0.9682
Mobile2 DMU2 1
Mobile3 DMU3 1
Mobile4 DMU4 1
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46. DEA introduction
• How to improve the performance of
inefficiency DMU? (CCR model)
• Input oriented: Focuses on minimizing
the level of inputs or efficiency with an
assumption of fixed level of outputs.
• Output oriented: Focuses on efficient
outputs with an assumption of fixed level
of inputs.
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Projection to frontier

47. DEA introduction
• DEA and regression
• On regression line DMUs have
average efficiency.
• Regression can
accommodate Multiple
inputs or outputs but not
both.
• Regression provides only
average relationships not
best practice.
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48. DEA introduction
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49. DEA introduction
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50. Homework
• Analyze a simple DEA problem, here is our data set
• Graphic analysis
• What are the efficiencies of DMUs?
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51. Homework
• Hint
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52. Reference
• http://d1c25a6gwz7q5e.cloudfront.net/papers/403.pdf
• https://github.com/jzuccollo/pyDEA
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