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Readings - PPT Presentation

Readings Chapter 5 Advanced Linear Programming Applications Overview Overview 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 ID: 412621

school revenue high management revenue school management high analysis envelopment data composite input memphis output number unit roosevelt resources

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Slide1

Readings

Chapter 5

Advanced Linear Programming ApplicationsSlide2

Overview

OverviewSlide3

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.Slide4

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.Tool SummarySlide5

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.

Tool SummarySlide6

Data Envelopment Analysis

Data Envelopment AnalysisSlide7

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.

Data Envelopment AnalysisSlide8

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

’s

goals and objectives.

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).Data Envelopment AnalysisSlide9

Overview

Min E s.t. Weighted outputs >

Unit

’s

output

(for each measured output)

Weighted inputs < E [Unit k’s input] (for each measured input) Sum of weights = 1 E, weights > 0 Data Envelopment AnalysisSlide10

Input

Roosevelt

Lincoln

Washington

Senior Faculty 37 25 23

Budget ($100,000's) 6.4 5.0 4.7

Senior Enrollments 850 700 600

Output

Roosevelt

Lincoln

Washington

Average SAT Score 800 830 900

High School Graduates 450 500 400

College Admissions 140 250 370

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].

Data Envelopment AnalysisSlide11

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

= Weight applied to Washington's

input/output

resources by the composite high schoolDefine the objective function. Minimize the fraction of Roosevelt High School's input resources required by the composite high school: Min E

Data Envelopment AnalysisSlide12

onstrain the sum of the weights to one: (1) w1 +

w2 + w3 = 1Constrain each output of the composite school to be at least Roosevelt’s: (2) 800w1 + 830w

+ 900

> 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) 850w

+ 700

+ 600

850

(Seniors)

Data Envelopment AnalysisSlide13

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.)

Data Envelopment AnalysisSlide14

Revenue Management

Revenue ManagementSlide15

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.Slide16

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.

Revenue ManagementSlide17

Revenue ManagementSlide18

Answer: Define 16 decision

variables,

one

for each

ODIF.

For example, IMD

= number of seats allocated to Indianapolis-Memphis-Discount class.Revenue ManagementSlide19

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.

Revenue ManagementSlide20

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

Revenue ManagementSlide21

Define

the 4

capacity constraints, one for each flight leg: