A.11 Other Advanced Applications

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Readings
Readings
Chapter 5
Advanced Linear Programming Applications
BA 452 Lesson A.11 Other Advanced Applications
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Overview
Overview
BA 452 Lesson A.11 Other Advanced Applications
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Overview
Data Envelopment Analysis measures the relative efficiency of operating units
with the same goals and objectives, and the same types of resources. Applies
to hospitals, banks, courts, …
Revenue Management Problems are Resource Allocation Problems when
inputs are fixed. Revenue Management Problems thus help airlines determine
how many seats to sell at a discount.
BA 452 Lesson A.11 Other Advanced Applications
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Tool Summary
Tool Summary
 Do not make integer restrictions, and maybe the solution are integers.
• Second Example: IMD = number of Indianapolis-Memphis-Discount
seats
 Use compound variables:
• Second Example: IMD = number of Indianapolis-Memphis-Discount
seats
 Constrain a weighted average with a linear constraint:
• Third Example: Constrain the weighted average risk factor to be no
greater than 55: 60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 +
10X9 < 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
 Interpret the unrealistic assumptions needed for a linear formulation:
• First Example: Assume the set of alternative inputs and outputs for a
high school is convex. Thus if 100 teaching hours gets 20 more
students admitted to college, then 10 teaching hours gets at least 2
admitted.
• Second Example: Assume demand has only two values.
• Third Example: Assume risk is a linear function of investment shares.
BA 452 Lesson A.11 Other Advanced Applications
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Tool Summary
Tool Summary
 Critique the unrealistic or approximate variables used because data on
better variables may not be available:
• First Example: High school output is measured by only three variables.
• Average SAT Scores, even though maximizing those scores is not
the same as maximizing learning.
• The number of High School Graduates, even though there is no
accounting for learning beyond a minimal level.
• The number of College Admissions, even though there is no
accounting for the quality or selectivity of the colleges.
• First Example: High school input is measured by only three variables.
• Senior Faculty
• Budget ($100,000's)
• Senior Enrollments, even though there is no measure for the quality
of those students before their senior year.
BA 452 Lesson A.11 Other Advanced Applications
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Data Envelopment Analysis
Data Envelopment Analysis
BA 452 Lesson A.11 Other Advanced Applications
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Data Envelopment Analysis
Overview
Data Envelopment Analysis measures the relative
efficiency of operating units with the same goals and
objectives, and the same types of resources. Data
Envelopment Analysis applies to fast-food outlets within the
same chain, to hospitals, banks, courts, schools, and so
on.
BA 452 Lesson A.11 Other Advanced Applications
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Data Envelopment Analysis
Overview




Data Envelopment Analysis creates a fictitious composite unit made
up of an optimal weighted average (w1, w2,…) of existing units.
An individual unit, k, can be compared by determining E, the fraction
of unit k’s input resources required by the optimal composite unit to
achieve k’s goals and objectives.
If E < 1, unit k is less efficient than the composite unit, and is
deemed relatively inefficient.
If E = 1, there is no evidence that unit k is inefficient, but one cannot
conclude that k is best without further information (such as the value
of 1 point higher Average SAT Score compared with 1 more College
Admission in the following example).
BA 452 Lesson A.11 Other Advanced Applications
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Data Envelopment Analysis
Overview
Min E
s.t.
Weighted outputs > Unit k’s output
(for each measured output)
Weighted inputs < E [Unit k’s input]
(for each measured input)
Sum of weights = 1
E, weights > 0
BA 452 Lesson A.11 Other Advanced Applications
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Data Envelopment Analysis
Question: The Langley County School District is trying to determine the
relative efficiency of its three high schools. In particular, it wants to
evaluate Roosevelt High.
The district is evaluating performances on SAT scores, the number of
seniors finishing high school, and the number of students admitted to
college [outputs] as a function of the number of teachers teaching
senior classes, the prorated budget for senior instruction, and the
number of seniors enrolled [inputs].


Input
Senior Faculty
Budget ($100,000's)
Senior Enrollments
Roosevelt Lincoln Washington
37
25
23
6.4
5.0
4.7
850
700
600
Output
Average SAT Score
High School Graduates
College Admissions
Roosevelt Lincoln Washington
800
830
900
450
500
400
140
250
370
BA 452 Lesson A.11 Other Advanced Applications
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Data Envelopment Analysis
Answer:
 Define the decision variables
E = Fraction of Roosevelt's input resources required by the composite high
school
w1 = Weight applied to Roosevelt's input/output resources by the composite
high school
w2 = Weight applied to Lincoln’s input/output resources by the composite
high school
w3 = Weight applied to Washington's input/output resources by the
composite high school
 Define the objective function. Minimize the fraction of Roosevelt
High School's input resources required by
the composite high school: Min E
BA 452 Lesson A.11 Other Advanced Applications
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Data Envelopment Analysis



Constrain the sum of the weights to one: (1) w1 + w2 + w3 = 1
Constrain each output of the composite school to be at least Roosevelt’s:
(2) 800w1 + 830w2 + 900w3 > 800 (SAT Scores)
(3) 450w1 + 500w2 + 400w3 > 450 (Graduates)
(4) 140w1 + 250w2 + 370w3 > 140 (College Admissions)
Constrain the inputs used by the composite high school to be no more than
the multiple, E, of the inputs available to Roosevelt:
(5) 37w1 + 25w2 + 23w3 < 37E (Faculty)
(6) 6.4w1 + 5.0w2 + 4.7w3 < 6.4E (Budget)
(7) 850w1 + 700w2 + 600w3 < 850E (Seniors)
BA 452 Lesson A.11 Other Advanced Applications
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Data Envelopment Analysis
Interpretation: The output shows
that the composite school is
made up of equal weights of
Lincoln and Washington.
Roosevelt is 76.5% efficient
compared to this composite
school when measured by High
School Graduates (because of
the 0 slack on this constraint
(#3)). It is less than 76.5%
efficient if output were only
measured by SAT Scores and
College Admissions (there is
positive slack in constraints 2
and 4.)
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management
Revenue Management
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management
Overview
Revenue Management Problems are Resource Allocation
Problems when inputs are fixed. Revenue Management
Problems thus help airlines determine how many seats to
sell at an early-reservation discount fare and many to sell
at a full fare. Other applications include hotels, apartment
rentals, car rentals, cruise lines, and golf courses.
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management
Answer: LeapFrog Airways provides passenger service for Indianapolis,
Baltimore, Memphis, Austin, and Tampa. LeapFrog has two WB828
airplanes, one based in Indianapolis and the other in Baltimore. Each
morning the Indianapolis based plane flies to Austin with a stopover in
Memphis. The Baltimore based plane flies to Tampa with a stopover in
Memphis. Both planes have a coach section with a 120-seat capacity,
and they arrive in Memphis at the same time.
LeapFrog uses two fare classes: a discount fare D class and a full fare
F class. Leapfrog’s products, each referred to as an origin destination
itinerary fare (ODIF), are listed on the next slide with their fares and
forecasted demand.
LeapFrog wants to determine how many seats it should allocate to
each ODIF.
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management
ODIF
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Origin
Indianapolis
Indianapolis
Indianapolis
Indianapolis
Indianapolis
Indianapolis
Baltimore
Baltimore
Baltimore
Baltimore
Baltimore
Baltimore
Memphis
Memphis
Memphis
Memphis
Destination
Memphis
Austin
Tampa
Memphis
Austin
Tampa
Memphis
Austin
Tampa
Memphis
Austin
Tampa
Austin
Tampa
Austin
Tampa
Fare
Class
D
D
D
F
F
F
D
D
D
F
F
F
D
D
F
F
ODIF
Code
IMD
IAD
ITD
IMF
IAF
ITF
BMD
BAD
BTD
BMF
BAF
BTF
MAD
MTD
MAF
MTF
Fare
175
275
285
395
425
475
185
315
290
385
525
490
190
180
310
295
Demand
44
25
40
15
10
8
26
50
42
12
16
9
58
48
14
11
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management
Answer: Define 16 decision variables, one for each ODIF. For
example, IMD = number of seats allocated to Indianapolis-MemphisDiscount class.
Fare
ODIF
ODIF
Origin
Destination Class Code
Fare Demand
1
Indianapolis
Memphis
D
IMD
175
44
2
Indianapolis
Austin
D
IAD
275
25
3
Indianapolis
Tampa
D
ITD
285
40
4
Indianapolis
Memphis
F
IMF
395
15
5
Indianapolis
Austin
F
IAF
425
10
6
Indianapolis
Tampa
F
ITF
475
8
7
Baltimore
Memphis
D
BMD
185
26
8
Baltimore
Austin
D
BAD
315
50
9
Baltimore
Tampa
D
BTD
290
42
10
Baltimore
Memphis
F
BMF
385
12
11
Baltimore
Austin
F
BAF
525
16
12
Baltimore
Tampa
F
BTF
490
9
13
Memphis
Austin
D
MAD
190
58
14
Memphis
Tampa
D
MTD
180
48
15
Memphis
Austin
F
MAF
310
14
16
Memphis
Tampa
F
MTF
295
11
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management
Simplification: Although the Revenue Management
problem is an Integer Linear Programming problem, it has
a special form that allows it to be formulated without integer
constraints, and the solutions turn out to be integers.
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management

Define the objective function. Maximize total revenue:
Max (fare per seat for each ODIF)
x (number of seats allocated to the ODIF)
Max 175IMD + 275IAD + 285ITD + 395IMF
+ 425IAF + 475ITF + 185BMD + 315BAD
+ 290BTD + 385BMF + 525BAF + 490BTF
+ 190MAD + 180MTD + 310MAF + 295MTF
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management
Define the 4 capacity constraints, one for each flight leg:
Indianapolis-Memphis leg (1) IMD + IAD + ITD + IMF + IAF + ITF < 120
Baltimore-Memphis leg (2) BMD + BAD + BTD + BMF + BAF + BTF < 120
Memphis-Austin leg (3) IAD + IAF + BAD + BAF + MAD + MAF < 120
Memphis-Tampa leg (4) ITD + ITF + BTD + BTF + MTD + MTF < 120

BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management

Define the first 8 demand constraints, one for each ODIF:
(5) IMD < 44
(6) IAD < 25
(7) ITD < 40
(8) IMF < 15
(9) IAF < 10
(10) ITF < 8
(11) BMD < 26 (12) BAD < 50
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management

Define the remaining 8 demand constraints, one for each ODIF:
(13) BTD < 42
(14) BMF < 12
(15) BAF < 16
(16) BTF < 9
(17) MAD < 58 (18) MTD < 48
(19) MAF < 14 (20) MTF < 11
BA 452 Lesson A.11 Other Advanced Applications
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Revenue Management
Interpretation: Total revenue = $94,735.00,
with the specified seat allocation.
ODIF
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Origin
Indianapolis
Indianapolis
Indianapolis
Indianapolis
Indianapolis
Indianapolis
Baltimore
Baltimore
Baltimore
Baltimore
Baltimore
Baltimore
Memphis
Memphis
Memphis
Memphis
Destination
Memphis
Austin
Tampa
Memphis
Austin
Tampa
Memphis
Austin
Tampa
Memphis
Austin
Tampa
Austin
Tampa
Austin
Tampa
Fare
Class
D
D
D
F
F
F
D
D
D
F
F
F
D
D
F
F
BA 452 Lesson A.11 Other Advanced Applications
ODIF
Code
IMD
IAD
ITD
IMF
IAF
ITF
BMD
BAD
BTD
BMF
BAF
BTF
MAD
MTD
MAF
MTF
Fare
175
275
285
395
425
475
185
315
290
385
525
490
190
180
310
295
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BA 452
Quantitative Analysis
End of Lesson A.11
BA 452 Lesson A.11 Other Advanced Applications
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