Slides by John Loucks St. Edward’s University

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© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Slides by

JohnLoucks

St. Edward’sUniversity

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Agenda

Some Review from Last Class Data Envelopment Analysis Revenue Management Game Theory Concepts

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Chapter 5 Advanced Linear Programming

Applications Data Envelopment Analysis

• Compares one unit to similar others• Ie branch of a bank, franchise of a chain

Revenue Management• Maximize revenue with a fixed inventory

Portfolio Models and Asset Allocation• Determine best portfolio composition

Game Theory• Competition with a zero sum

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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 Index

• Allows 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.

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Data Envelopment Analysis The DEA Model

MIN Es.t. OUTPUTS

INPUTSSum of weights = 1E, weights > 0

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Data Envelopment Analysis

The Langley County School District is trying todetermine the relative efficiency of its three highschools. In particular, it wants to evaluate RooseveltHigh.

Outputs:performances on SAT scores, the number of seniors finishing high school the number of students who enter college

Inputs number of teachers teaching senior classesthe prorated budget for senior instructionnumber of students in the senior class.

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Data Envelopment Analysis Input

Roosevelt1 Lincoln2 Washington3 Senior Faculty 37 25 23Budget ($100,000's) 6.4 5.0 4.7Senior Enrollments 850 700 600

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Data Envelopment Analysis Output

Roosevelt1 Lincoln2 Washington3 Average SAT Score 800 830 900

High School Graduates 450 500 400

College Admissions 140 250 370

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Data Envelopment Analysis Define the Decision Variables

E = Fraction of Roosevelt's input resources required by the composite high schoolw1 = Weight applied to Roosevelt's input/output

resources by the composite high schoolw2 = Weight applied to Lincoln’s input/output

resources by the composite high schoolw3 = Weight applied to Washington's input/output resources by the composite high school

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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 E

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Data Envelopment Analysis Define the Constraints

Sum of the Weights is 1: (1) w1 + w2 + w3 = 1 Output 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 ) +

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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 + 830w2 + 900w3 > 800 (SAT Scores)

(3) 450w1 + 500w2 + 400w3 > 450 (Graduates)

(4) 140w1 + 250w2 + 370w3 > 140 (College Admissions)

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Data Envelopment Analysis Input Constraints

• General Form• Input for composite <= input for Roosevelt

* E• Input for composite =

• (Input for Roosevelt * Input for Roosevelt ) +(Input for Lincoln * Input for Lincoln ) + (Input for Washington * Input for Washington )

(5) 37w1 + 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 > 0

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MIN E ST (1) w1 + w2 + w3 = 1

(2) 800w1 + 830w2 + 900w3 > 800 (SAT Scores)

(3) 450w1 + 500w2 + 400w3 > 450 (Graduates)

(4) 140w1 + 250w2 + 370w3 > 140 (College Admissions)(5) 37w1 + 25w2 + 23w3 < 37E (Faculty)

(6) 6.4w1 + 5.0w2 + 4.7w3 < 6.4E (Budget) (7) 850w1 + 700w2 + 600w3 < 850E

(Seniors) (8) E, w1, w2, w3 > 0

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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

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 0.001

Zero 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

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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.

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Revenue Management

General Form MAX (revenue per unit * units allocated) ST

• CAPACITY • DEMAND • NONNEGATIVE

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Revenue Management

LeapFrog Airways provides passenger service forIndianapolis, 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 toTampa with a stopover in Memphis. Both planes have a coach section with a 120-seat capacity.

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LeapFrog uses two fare classes: a discount fare Dclass and a full fare F class. Leapfrog’s products, eachreferred to as an origin destination itinerary fare (ODIF), are listed on the next slide with their fares andforecasted demand. LeapFrog wants to determine how many seats it should allocate to each ODIF.

Revenue Management

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IND BAL

MEM

AUS TAM

Each day a planeLeaves both IND And BAL for AUS and TAMRespectively.

Both flights lay overIn MEM

No return flights (for simplicity)

Each plane holds 120

Leg 1 Leg 2

Leg 3 Leg 4

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Orig DestIND MEMIND AUSIND TAMBAL MEMBAL AUSBAL TAMMEM AUSMEM TAM

8 different origin-destination combinations

Plus two different fare classes: Discount and Full Fare

8 Orig-Desination combinations * 2 fare classes = 16 combinations

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ODIF123456789

10111213141516

OriginIndianapoli

sIndianapoli

sIndianapoli

sIndianapoli

sIndianapoli

sIndianapoli

sBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreMemphisMemphisMemphisMemphis

DestinationMemphis

AustinTampa

MemphisAustinTampa

MemphisAustinTampa

MemphisAustinTampaAustinTampa AustinTampa

FareClass

DDDFFFDDDFFFDDFF

ODIFCodeIMDIADITDIMFIAFITF

BMDBADBTDBMFBAFBTFMADMTDMAFMTF

Fare175275285395425475185315290385525490190180310295

Demand

44254015108

26504212169

58481411

Revenue Management

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Revenue Management Define the Decision VariablesThere are 16 variables, one for each ODIF:IMD = number of seats allocated to Indianapolis-Memphis-

Discount classIAD = number of seats allocated to Indianapolis-Austin- Discount classITD = number of seats allocated to Indianapolis-Tampa- Discount classIMF = number of seats allocated to Indianapolis-Memphis- Full Fare classIAF = number of seats allocated to Indianapolis-Austin-Full Fare class

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Revenue Management Define the Decision Variables (continued)ITF = number of seats allocated to Indianapolis-Tampa- Full Fare classBMD = number of seats allocated to Baltimore-Memphis- Discount classBAD = number of seats allocated to Baltimore-Austin- Discount classBTD = number of seats allocated to Baltimore-Tampa- Discount classBMF = number of seats allocated to Baltimore-Memphis- Full Fare classBAF = number of seats allocated to Baltimore-Austin- Full Fare class

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Revenue Management Define the Decision Variables (continued)BTF = number of seats allocated to Baltimore-Tampa- Full Fare classMAD = number of seats allocated to Memphis-Austin- Discount classMTD = number of seats allocated to Memphis-Tampa- Discount classMAF = number of seats allocated to Memphis-Austin- Full Fare classMTF = number of seats allocated to Memphis-Tampa- Full Fare class

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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

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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 <

120

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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 < 9

Revenue Management

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

315BAD + 290BTD + 385BMF + 525BAF +

490BTF + 190MAD + 180MTD + 310MAF +

295MTFST: IMD + IAD + ITD + IMF + IAF + ITF < 120

BMD + BAD + BTD + BMF + BAF + BTF < 120IAD + IAF + BAD + BAF + MAD + MAF < 120ITD + ITF + BTD + BTF + MTD + MTF < 120

IMD < 44, BMD < 26, MAD < 58, IAD < 25, BAD < 50MTD < 48, ITD < 40, BTD < 42, MAF < 14, IMF < 15BMF < 12, MTF < 11, IAF < 10, BAF < 16, ITF < 8BTF < 9

IMD, IAD, ITD, IMF, IAF, ITF, BMD, BAD, BTD, BMF, BAF, BTF, MAD, MTD, MAF, MTF > 0

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Revenue Management Computer Solution

Revenue Contribution is $96265

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Revenue Management Computer Solution

(continued) - IMD dual value is 90

- IMF dual value is 310

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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).

continue

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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.

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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 Game

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Competing for Vehicle SalesSuppose that there are only two vehicle

dealer-ships in a small city. Each dealership is consideringthree strategies that are designed to take sales of new vehicles from the other dealership over afour-month period. The strategies, assumed to be the same for both dealerships, are on the next slide.

Two-Person Zero-Sum Game Example

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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.


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2 2 1

CashRebate

b1

0%Loan

b3

FreeOptions

b2

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 a1

Free Options a2

0% Loan a3

Dealership A


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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).


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Identifying Maximin and Best Strategy

RowMinimum

1-3-2

2 2 1

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

-3 3 -1 3 -2 0

Cash Rebate a1

Free Options a2

0% Loan a3

Dealership A

Best Strategy

For Player AMaximinPayoff


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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).


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Identifying Minimax and Best Strategy

2 2 1

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

-3 3 -1 3 -2 0

Cash Rebate a1

Free Options a2

0% Loan a3

Dealership A

Column Maximum 3 3 1

Best Strategy

For Player B

MinimaxPayoff


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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 strategy

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RowMinimum

1-3-2

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

-3 3 -1 3 -2 0

Cash Rebate a1

Free Options a2

0% Loan a3

Dealership A

Column Maximum 3 3 1

Pure Strategy Example

Saddle Point and Value of the Game

2 2 1

SaddlePoint

Value of thegame is 1

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Pure Strategy Example

Pure Strategy Summary Player A should choose Strategy a1 (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.

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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.

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Mixed Strategy Example

b1 b2

Player B

11 5

a1

a2

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.

ColumnMaximum 11

8

RowMinimum

4 5

Maximin

Minimax

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p = the probability Player A selects strategy a1(1 - p) = the probability Player A selects strategy a2

If Player B selects b1:EV = 4p + 11(1 – p)

If Player B selects b2:EV = 8p + 5(1 – p)

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4p + 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.

4p + 11 – 11p = 8p + 5 – 5p11 – 7p = 5 + 3p

-10p = -6p = .6Player A should select:

Strategy a1 with a .6 probability and Strategy a2 with a .4 probability.

Hence,(1 - p) = .4

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q = the probability Player B selects strategy b1(1 - q) = the probability Player B selects strategy b2

If Player A selects a1:EV = 4q + 8(1 – q)

If Player A selects a2:EV = 11q + 5(1 – q)

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Value of the GameFor Player A:

EV = 4p + 11(1 – p) = 4(.6) + 11(.4) = 6.8

For Player B:EV = 4q + 8(1 – q) = 4(.3) + 8(.7) = 6.8

Expected gain

per gamefor Player A

Expected lossper game

for Player B

Slides by John Loucks St. Edward’s University

Documents

composite unit

cengage learning

accessible website

composite high schoolw2

composite high schoolw3

composite high schoolw1

roosevelt output

high school graduates