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Slides by John Loucks St. Edward’s Slides by John Loucks St. Edward’s

Slides by John Loucks St. Edward’s - PowerPoint Presentation

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Slides by John Loucks St. Edward’s - PPT Presentation

University Agenda Some Review from Last Class Data Envelopment Analysis Revenue Management Game Theory Concepts Chapter 5 Advanced Linear Programming Applications Data Envelopment Analysis ID: 757953

strategy player number game player strategy game number revenue seats fare allocated class composite management envelopment data analysis memphis

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Slide1

Slides by

John

Loucks

St. Edward’s

UniversitySlide2

Agenda

Some Review from Last ClassData Envelopment AnalysisRevenue ManagementGame Theory ConceptsSlide3

Chapter 5 Advanced Linear Programming Applications

Data Envelopment Analysis

Compares one unit to similar othersIe branch of a bank, franchise of a chainRevenue Management

Maximize revenue with a fixed inventoryPortfolio Models and Asset Allocation

Determine best portfolio compositionGame Theory

Competition with a zero sumSlide4

Data Envelopment Analysis

Data envelopment analysis (DEA): used to determine the relative operating efficiency of units with the same goals and objectives.DEA creates a

hypothetical composite optimal weighted average (W1,

W2,…) of existing units

.E – Efficiency IndexAllows comparison between composite and unit“what the outputs of the composite would be, given the units inputs”

If E

< 1, unit is less efficient than the composite unit If E = 1, there is no evidence that unit k is inefficient.Slide5

Data Envelopment Analysis

The DEA Model

MIN

E

s.t. OUTPUTS INPUTS

Sum of weights = 1 E, weights > 0 Slide6

Data Envelopment Analysis

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.

Outputs:

performances on SAT scores, the number of seniors finishing high school

the number of students who enter collegeInputs number of teachers teaching senior classes the prorated budget for senior instruction number of students in the senior class. Slide7

Data Envelopment Analysis

Input

Roosevelt1 Lincoln2

Washington3 Senior Faculty 37 25 23 Budget ($100,000's) 6.4 5.0 4.7 Senior Enrollments 850 700 600

Slide8

Data Envelopment Analysis

Output

Roosevelt1

Lincoln2 Washington3 Average SAT Score 800 830 900 High School Graduates 450 500 400 College Admissions 140 250 370Slide9

Data Envelopment Analysis

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 schoolSlide10

Data Envelopment Analysis

Define the Objective Function

Since our objective is to DETECT INEFFICIENCIES

, we want to minimize the fraction of Roosevelt High School's input resources required by the composite high school:

MIN ESlide11

Data Envelopment Analysis

Define the Constraints

Sum of the Weights is 1: (1) w1 +

w2 + w

3 = 1Output Constraints

General form for each output: output for composite >= output for Roosevelt

Output for composite = (Output for Roosevelt * weight for Roosevelt ) +(output for Lincoln * weight for Lincoln )

+ (output for Washington * weight for Washington ) + Slide12

Data Envelopment Analysis

Output Constraints: Since w1

= 1 is possible, each output of the composite school must be at least as great as that of Roosevelt: (2) 800w1 + 830w

2 + 900w3

> 800 (SAT Scores) (3) 450w1

+ 500w2 + 400

w3 > 450 (Graduates) (4) 140w1 + 250w2 + 370

w3 > 140 (College Admissions)Slide13

Data Envelopment Analysis

Input ConstraintsGeneral Form

Input for composite <= input for Roosevelt * EInput for composite =

(Input for

Roosevelt * Input for Roosevelt ) +(Input

for Lincoln * Input for Lincoln ) +

(Input for Washington * Input for Washington ) (5) 37w

1 + 25w2 + 23w3 < 37E (Faculty) (6) 6.4w1 + 5.0w2 + 4.7w3 < 6.4E (Budget) (7) 850w1

+ 700w2

+ 600w3 < 850E (Seniors) Nonnegativity : E, w1, w2

, w3 > 0Slide14

Data Envelopment Analysis

MIN EST (1) w1 +

w2 + w3 = 1

(2) 800w1

+ 830w2 + 900w

3 > 800

(SAT Scores) (3) 450w1 + 500w2 + 400w

3 > 450 (Graduates) (4) 140w1 + 250w2 + 370w3 > 140 (College Admissions) (5) 37w1 + 25w2 + 23w

3 < 37

E (Faculty) (6) 6.4w1 + 5.0w2 + 4.7w3 < 6.4E (Budget) (

7) 850w1 + 700w2 + 600w

3 < 850E (Seniors)

(8)

E

,

w

1

,

w

2

,

w

3

>

0Slide15

Data Envelopment Analysis

Computer

Solution

OBJECTIVE FUNCTION VALUE = 0.765

VARIABLE VALUE REDUCED COSTS E 0.765 0.000

W1 (R) 0.000 0.235 W2 (L) 0.500 0.000 W3 (W) 0.500 0.000*Composite is 50% Lincoln, 50% Washington*Roosevelt is no more than 76.5% efficient as composite Slide16

Data Envelopment Analysis

Computer Solution

(continued

)

CONSTRAINT

SLACK/SURPLUS DUAL VALUES 1 0.000

-0.235 2 (SAT) 65.000 0.000 3 (grads) 0.000 -0.001 4 (college) 170.000 0.000 5 (fac

) 4.294 0.000

6 (budget) 0.044 0.000 7 (seniors) 0.000 0.001Zero Slack – Roosevelt is 76.5% efficient in this area (ie grads)

Positive slack – Roosevelt is LESS THAN 76.5% efficient (ie

SAT) ie SAT scores are 65 points higher in the composite school

Slide17

Revenue Management

Another LP application is revenue management.

Revenue management

managing

the short-term demand for a fixed perishable inventory in order to

maximize revenue potential.

first used to determine how many airline seats to sell at an early-reservation discount fare and many to sell at a full fare.Slide18

Revenue Management

General FormMAX (revenue per unit * units allocated)STCAPACITY

DEMAND NONNEGATIVESlide19

Revenue Management

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 morningthe 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. Slide20

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 ManagementSlide21

IND

BAL

MEM

AUS

TAM

Each day a plane

Leaves both IND

And BAL for

AUS and TAM

Respectively.

Both flights lay over

In MEM

No return flights

(for simplicity)

Each plane holds 120

Leg 1

Leg 2

Leg 3

Leg 4Slide22

Orig

Dest

INDMEM

INDAUS

INDTAM

BAL

MEMBALAUSBALTAM

MEMAUSMEMTAM8 different origin-destination combinationsPlus two different fare classes: Discount and Full Fare8 Orig-Desination combinations * 2 fare classes = 16 combinations Slide23

ODIF

1234

5678

910

111213

1415

16OriginIndianapolisIndianapolis

IndianapolisIndianapolisIndianapolisIndianapolisBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreMemphisMemphisMemphisMemphis

Destination

MemphisAustinTampaMemphisAustinTampaMemphisAustinTampaMemphisAustin

TampaAustinTampa AustinTampa

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

1411

Revenue ManagementSlide24

Revenue Management

Define the Decision Variables

There are 16 variables, one for each ODIF:

IMD = number of seats allocated to Indianapolis-Memphis-

Discount class

IAD = number of seats allocated to Indianapolis-Austin- Discount class

ITD = number of seats allocated to Indianapolis-Tampa- Discount class

IMF = number of seats allocated to Indianapolis-Memphis- Full Fare class

IAF = number of seats allocated to Indianapolis-Austin-Full Fare classSlide25

Revenue Management

Define the Decision Variables (continued)

ITF = number of seats allocated to Indianapolis-Tampa-

Full Fare class

BMD = number of seats allocated to Baltimore-Memphis-

Discount class

BAD = number of seats allocated to Baltimore-Austin-

Discount class

BTD = number of seats allocated to Baltimore-Tampa-

Discount classBMF = number of seats allocated to Baltimore-Memphis- Full Fare class

BAF = number of seats allocated to Baltimore-Austin-

Full Fare classSlide26

Revenue Management

Define the Decision Variables (continued)

BTF = number of seats allocated to Baltimore-Tampa-

Full Fare class

MAD = number of seats allocated to Memphis-Austin-

Discount class

MTD = number of seats allocated to Memphis-Tampa-

Discount class

MAF = number of seats allocated to Memphis-Austin-

Full Fare classMTF = number of seats allocated to Memphis-Tampa- Full Fare classSlide27

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 + 295MTFSlide28

Revenue Management

Define the Constraints

There are 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

<

120Slide29

Revenue Management

Define the Constraints (continued)

Demand Constraints Limit the amount of seats for each ODIF

There

are 16 demand constraints, one for each ODIF:

(5) IMD < 44 (11) BMD < 26 (17) MAD <

58

(6) IAD < 25 (12) BAD <

50 (18) MTD < 48

(7) ITD

<

40 (13) BTD

<

42 (19) MAF

<

14

(8) IMF

<

15 (14) BMF

<

12 (20) MTF

<

11

(9) IAF

<

10 (15) BAF

<

16

(10) ITF

<

8 (16) BTF

<

9Slide30

Revenue Management

Max 175IMD + 275IAD + 285ITD + 395IMF + 425IAF + 475ITF + 185BMD + 315BAD

+ 290BTD + 385BMF + 525BAF + 490BTF + 190MAD + 180MTD + 310MAF + 295MTF

ST: IMD + IAD + ITD + IMF + IAF + ITF

< 120

BMD + BAD + BTD + BMF + BAF + BTF

< 120IAD + IAF + BAD + BAF + MAD + MAF < 120

ITD + ITF + BTD + BTF + MTD + MTF < 120IMD < 44, BMD < 26, MAD < 58, IAD < 25, BAD

< 50

MTD < 48, ITD < 40, BTD < 42, MAF < 14, IMF

< 15BMF <

12, MTF < 11, IAF

<

10, BAF

<

16, ITF

<

8

BTF

<

9

IMD, IAD, ITD, IMF, IAF, ITF, BMD, BAD, BTD, BMF, BAF, BTF, MAD, MTD, MAF, MTF

> 0Slide31

Revenue Management

Computer

Solution

Revenue Contribution is $96265

Slide32

Revenue Management

Computer Solution

(continued

)

IMD dual value is

90

IMF dual value is

310Slide33

Introduction to Game Theory

In

decision analysis

, a single decision maker seeks to select an optimal alternative.

In

game theory

, there are two or more decision makers, called players, who compete as adversaries against each other.

It is assumed that each player has the same information and will select the strategy that provides the best possible outcome from his point of view.Each player selects a strategy independently without knowing in advance the strategy of the other player(s). continueSlide34

Introduction to Game Theory

The combination of the competing strategies provides the

value of the game

to the players.

Examples of competing players are teams, armies, companies, political candidates, and contract bidders.Slide35

Two-person

means there are two competing players in the game.

Zero-sum

means the gain (or loss) for one player is equal to the corresponding loss (or gain) for the other player.

The gain and loss balance out so that there is a zero-sum for the game.

What one player wins, the other player loses.

Two-Person Zero-Sum GameSlide36

Competing for Vehicle Sales

Suppose that there are only two vehicle dealer-ships in a small city. Each dealership is considering

three strategies that are designed

to take

sales of

new

vehicles

from the other dealership over a four-month period. The strategies, assumed to be

the same

for both dealerships, are on the next slide.

Two-Person Zero-Sum Game ExampleSlide37

Strategy Choices

Strategy

1: Offer a

cash rebate

on a new vehicle. Strategy 2: Offer free optional

equipment on a

new vehicle.

Strategy

3: Offer a

0% loan

on

a new vehicle.

Two-Person Zero-Sum Game ExampleSlide38

2 2 1

Cash

Rebate

b

1

0%

Loan

b

3

FreeOptionsb2

Dealership B

Payoff Table: Number of Vehicle Sales

Gained Per Week by Dealership A

(or Lost Per Week by Dealership B)

-3 3 -1

3 -2 0

Cash Rebate

a

1

Free Options

a

2

0% Loan

a

3

Dealership A

Two-Person Zero-Sum Game ExampleSlide39

Step 1:

Identify the minimum payoff for each

row (for Player A).

Step 2:

For Player A, select the strategy that provides

the maximum of the row minimums (called

the

maximin).Two-Person Zero-Sum GameSlide40

Identifying Maximin and Best Strategy

Row

Minimum

1

-3

-2

2 2 1

Cash

Rebate

b

1

0%

Loan

b

3

Free

Options

b

2

Dealership B

-3 3 -1

3 -2 0

Cash Rebate

a

1

Free Options

a

2

0% Loan

a

3

Dealership A

Best Strategy

For Player A

Maximin

Payoff

Two-Person Zero-Sum Game ExampleSlide41

Step 3:

Identify the maximum payoff for each column

(for Player B).

Step 4:

For Player B, select the strategy that provides

the minimum of the column maximums

(called the

minimax).Two-Person Zero-Sum GameSlide42

Identifying Minimax and Best Strategy

2 2 1

Cash

Rebate

b

1

0%

Loan

b

3

FreeOptions

b2

Dealership B

-3 3 -1

3 -2 0

Cash Rebate

a

1

Free Options

a

2

0% Loan

a

3

Dealership A

Column Maximum

3 3 1

Best Strategy

For Player B

Minimax

Payoff

Two-Person Zero-Sum Game ExampleSlide43

Pure Strategy

Whenever an optimal

pure strategy

exists:

the maximum of the row minimums equals the minimum of the column maximums (Player A’s

maximin

equals Player B’s minimax

)the game is said to have a saddle point (the intersection of the optimal strategies)the value of the saddle point is the value of the game

neither player can improve his/her outcome by changing strategies even if he/she learns in advance the opponent’s strategySlide44

Row

Minimum

1

-3

-2

Cash

Rebate

b

1

0%

Loan

b

3

Free

Options

b

2

Dealership B

-3 3 -1

3 -2 0

Cash Rebate

a

1

Free Options

a

2

0% Loan

a

3

Dealership A

Column Maximum

3 3 1

Pure Strategy Example

Saddle Point and Value of the Game

2 2 1

Saddle

Point

Value of the

game is 1Slide45

Pure Strategy Example

Pure Strategy Summary

Player A should choose Strategy

a

1

(offer a cash rebate).

Player A can expect a

gain of at least 1 vehicle sale per week.Player B should choose Strategy b3 (offer a 0% loan).

Player B can expect a

loss of no more than 1 vehicle sale per week.Slide46

Mixed Strategy

If the maximin value for Player A does not equal the minimax value for Player B, then a pure strategy is not optimal for the game.

In this case, a

mixed strategy

is best.

With a mixed strategy, each player employs more than one strategy.

Each player should use one strategy some of the time and other strategies the rest of the time.

The optimal solution is the relative frequencies with which each player should use his possible strategies.Slide47

Mixed Strategy Example

b

1

b

2

Player B

11 5

a

1

a

2

Player A

4 8

Consider the following two-person zero-sum game. The maximin does not equal the minimax. There is not an optimal pure strategy.

Column

Maximum

11 8

Row

Minimum

4

5

Maximin

MinimaxSlide48

Mixed Strategy Example

p

= the probability Player A selects strategy

a

1

(1

-

p) = the probability Player A selects strategy a2

If Player B selects

b1:EV = 4

p + 11(1 –

p)

If Player B selects

b

2

:

EV = 8

p

+ 5(1 –

p

)Slide49

Mixed Strategy Example

4

p

+ 11(1 –

p

) = 8p

+ 5(1 – p)

To solve for the optimal probabilities for Player Awe set the two expected values equal and solve forthe value of p.

4

p + 11 – 11p = 8p + 5 – 5p

11 – 7

p = 5 + 3

p

-10

p

= -6

p

= .6

Player A should select:

Strategy

a

1

with a .6 probability and

Strategy

a

2

with a .4 probability.

Hence,

(1

-

p

) = .4Slide50

Mixed Strategy Example

q

= the probability Player B selects strategy

b

1

(1

-

q) = the probability Player B selects strategy b2

If Player A selects

a1:EV = 4

q + 8(1 –

q)

If Player A selects

a

2

:

EV = 11

q

+ 5(1 –

q

)Slide51

Mixed Strategy Example

Value of the Game

For Player A:

EV = 4

p

+ 11(1 –

p

) = 4(.6) + 11(.4) = 6.8

For Player B:

EV = 4

q

+ 8(1 –

q

) = 4(.3) + 8(.7) = 6.8

Expected gain

per game

for Player A

Expected loss

per game

for Player B