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Chapter 5 Advanced Linear Programming Applications

- Data Envelopment Analysis
- Revenue Management
- Portfolio Models and Asset Allocation
- Game Theory

Data Envelopment Analysis

- Data envelopment analysis (DEA) is an LP application used to determine the relative operating efficiency of units with the same goals and objectives.
- DEA 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.
- If E < 1, unit k is less efficient than the composite unit and be deemed relatively inefficient.
- If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is absolutely efficient.

Data Envelopment Analysis

- The DEA Model

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

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.

The district is evaluating

performances on SAT scores, the

number of seniors finishing high

school, and the number of students

who enter college as a function of the

number of teachers teaching senior

classes, the prorated budget for senior instruction,

and the number of students in the senior class.

Data Envelopment Analysis

- Input

RooseveltLincolnWashington

Senior Faculty 37 25 23

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

Senior Enrollments 850 700 600

Data Envelopment Analysis

- Output

RooseveltLincolnWashington

Average SAT Score 800 830 900

High School Graduates 450 500 400

College Admissions 140 250 370

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 school

Data Envelopment Analysis

- Define the Objective Function

Minimize the fraction of Roosevelt High School's input resources required by the composite high school:

MIN E

Data Envelopment Analysis

- Define the Constraints

Sum of the Weights is 1:

(1) w1 + w2 + w3 = 1

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)

Data Envelopment Analysis

- Define the Constraints (continued)

Input Constraints:

The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are:

(5) 37w1 + 25w2 + 23w3< 37E (Faculty)

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

(7) 850w1 + 700w2 + 600w3< 850E (Seniors)

Nonnegativity of variables:

E, w1, w2, w3> 0

Data Envelopment Analysis

- The Management Scientist Solution

OBJECTIVE FUNCTION VALUE = 0.765

VARIABLEVALUE REDUCED COSTS

E 0.765 0.000

W1 0.000 0.235

W2 0.500 0.000

W3 0.500 0.000

Data Envelopment Analysis

- The Management Scientist Solution (continued)

CONSTRAINTSLACK/SURPLUSDUAL PRICES

1 0.000 -0.235

2 65.000 0.000

3 0.000 -0.001

4 170.000 0.000

5 4.294 0.000

6 0.044 0.000

7 0.000 0.001

Data Envelopment Analysis

- Conclusion

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 college admissions (because of the 0 slack on this constraint (#4)). It is less than 76.5% efficient when using measures of SAT scores and high school graduates (there is positive slack in constraints 2 and 3.)

- Another LP application is revenue management.
- Revenue management involves managing the short-term demand for a fixed perishable inventory in order to maximize revenue potential.
- The methodology was first used to determine how many airline seats to sell at an early-reservation discount fare and many to sell at a full fare.
- Application areas now include hotels, apartment rentals, car rentals, cruise lines, and golf courses.

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.

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.

- 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 class

- 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 class

BMF = number of seats allocated to Baltimore-Memphis-

Full Fare class

BAF = number of seats allocated to Baltimore-Austin-

Full Fare class

- 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 class

MTF = number of seats allocated to Memphis-Tampa-

Full Fare class

- 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

- 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

- Define the Constraints (continued)

There are 16 demand constraints, one for each ODIF:

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

(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

- The Management Scientist Solution

Objective Function Value = 94735.000

VariableValueReduced Cost

IMD 44.000 0.000

IAD 3.000 0.000

ITD 40.000 0.000

IMF 15.000 0.000

IAF 10.000 0.000

ITF 8.000 0.000

BMD 26.000 0.000

BAD 50.000 0.000

- The Management Scientist Solution (continued)

VariableValueReduced Cost

BTD 7.000 0.000

BMF 12.000 0.000

BAF 16.000 0.000

BTF 9.000 0.000

MAD 27.000 0.000

MTD 45.000 0.000

MAF 14.000 0.000

MTF 11.000 0.000

Portfolio Models and Asset Management

- Asset allocation involves determining how to allocate investment funds across a variety of asset classes such as stocks, bonds, mutual funds, real estate.
- Portfolio models are used to determine percentage of funds that should be made in each asset class.
- The goal is to create a portfolio that provides the best balance between risk and return.

John Sweeney is an investment advisor who is

attempting to construct an "optimal portfolio" for a

client who has $400,000 cash to invest. There are ten

different investments, falling into four

broad categories that John and his client

have identified as potential candidate

for this portfolio.

The investments and their important

characteristics are listed in the table on

the next slide. Note that Unidyde Corp.

Corp. under Equities and Unidyde Corp. under Debt

are two separate investments, whereas First General

REIT is a single investment that is considered both an

equities and a real estate investment.

Exp. Annual

After Tax Liquidity Risk

Category Investment Return Factor Factor

Equities Unidyde Corp. 15.0% 100 60

(Stocks) CC’s Restaurants 17.0% 100 70

First General REIT 17.5% 100 75

Debt Metropolis Electric 11.8% 95 20

(Bonds) Unidyde Corp. 12.2% 92 30

Lewisville Transit 12.0% 79 22

Real Estate Realty Partners 22.0% 0 50

First General REIT ( --- See above --- )

Money T-Bill Account 9.6% 80 0

Money Mkt. Fund 10.5% 100 10

Saver's Certificate 12.6% 0 0

Formulate a linear programming problem to

accomplish John's objective as an investment advisor

which is to construct a portfolio that maximizes his

client's total expected after-tax return over the next year, subject to the limitations placed upon him by the client for the portfolio. (Limitations listed on next two slides.)

Portfolio Limitations

1. The weighted average liquidity factor for the portfolio

must to be at least 65.

2. The weighted average risk factor for the portfolio must

be no greater than 55.

3. No more than $60,000 is to be invested in Unidyde

stocks or bonds.

4. No more than 40% of the investment can be in any one

category except the money category.

5. No more than 20% of the total investment can be in

any one investment except the money market fund.

continued

Portfolio Limitations (continued)

6. At least $1,000 must be invested in the Money Market

fund.

7. The maximum investment in Saver's Certificates is

$15,000.

8. The minimum investment desired for debt is $90,000.

9. At least $10,000 must be placed in a T-Bill account.

- Define the Decision Variables

X1 = $ amount invested in Unidyde Corp. (Equities)

X2 = $ amount invested in CC’s Restaurants

X3 = $ amount invested in First General REIT

X4 = $ amount invested in Metropolis Electric

X5 = $ amount invested in Unidyde Corp. (Debt)

X6 = $ amount invested in Lewisville Transit

X7 = $ amount invested in Realty Partners

X8 = $ amount invested in T-Bill Account

X9 = $ amount invested in Money Mkt. Fund

X10 = $ amount invested in Saver's Certificate

- Define the Objective Function

Maximize the total expected after-tax return over the next year:

Max .15X1 + .17X2 + .175X3 + .118X4 + .122X5

+ .12X6 + .22X7 + .096X8 + .105X9 + .126X10

- Define the Constraints

- Total funds invested must not exceed $400,000:
- (1) X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 = 400,000
- Weighted average liquidity factor must to be at least 65:
- 100X1+100X2+100X3+95X4+92X5+79X6+80X8+100X9 >
- 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
- Weighted average risk factor must be no greater than 55:
- 60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 <
- 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)
- No more than $60,000 to be invested in Unidyde Corp:
- X1 + X5 < 60,000

- Define the Constraints (continued)

- No more than 40% of the $400,000 investment can be
- in any one category except the money category:
- (5) X1 + X2 + X3 < 160,000
- (6) X4 + X5 + X6 < 160,000
- X3 + X7 < 160,000
- No more than 20% of the $400,000 investment can be
- in any one investment except the money market fund:
- (8) X2 < 80,000 (12) X7 < 80,000
- (9) X3 < 80,000 (13) X8 < 80,000
- (10) X4 < 80,000 (14) X10 < 80,000
- (11) X6 < 80,000

- Define the Constraints (continued)

At least $1,000 must be invested in the Money Market fund:

(15) X9 > 1,000

The maximum investment in Saver's Certificates is $15,000:

(16) X10 < 15,000

The minimum investment the Debt category is $90,000:

(17) X4 + X5 + X6 > 90,000

At least $10,000 must be placed in a T-Bill account:

(18) X8 > 10,000

Non-negativity of variables:

Xj > 0 j = 1, . . . , 10

- Solution Summary

Total Expected After-Tax Return = $64,355

X1 = $0 invested in Unidyde Corp. (Equities)

X2 = $80,000 invested in CC’s Restaurants

X3 = $80,000 invested in First General REIT

X4 = $0 invested in Metropolis Electric

X5 = $60,000 invested in Unidyde Corp. (Debt)

X6 = $74,000 invested in Lewisville Transit

X7 = $80,000 invested in Realty Partners

X8 = $10,000 invested in T-Bill Account

X9 = $1,000 invested in Money Mkt. Fund

X10 = $15,000 invested in Saver's Certificate

- 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

- 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.

- 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 Example

- 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 Example

- 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 Example

- Payoff Table: Number of Vehicle Sales

Gained Per Week by Dealership A

(or Lost Per Week by Dealership B)

Dealership B

Cash

Rebate

b1

Free

Options

b2

0%

Loan

b3

Dealership A

Cash Rebate a1

Free Options a2

0% Loan a3

2 2 1

-3 3 -1

3 -2 0

- 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 Game Example

- Identifying Maximin and Best Strategy

Dealership B

Cash

Rebate

b1

Free

Options

b2

0%

Loan

b3

Row

Minimum

Dealership A

Cash Rebate a1

Free Options a2

0% Loan a3

1

-3

-2

2 2 1

-3 3 -1

3 -2 0

Best Strategy

For Player A

Maximin

Payoff

- 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 Game Example

- Identifying Minimax and Best Strategy

Dealership B

Best Strategy

For Player B

Cash

Rebate

b1

Free

Options

b2

0%

Loan

b3

Dealership A

Cash Rebate a1

Free Options a2

0% Loan a3

2 2 1

-3 3 -1

Minimax

Payoff

3 -2 0

3 3 1

Column Maximum

- 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

- Saddle Point and Value of the Game

Dealership B

Value of the

game is 1

Cash

Rebate

b1

Free

Options

b2

0%

Loan

b3

Row

Minimum

Dealership A

Cash Rebate a1

Free Options a2

0% Loan a3

1

-3

-2

2 2 1

-3 3 -1

3 -2 0

3 3 1

Column Maximum

Saddle

Point

- 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.

- 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.

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

Player B

Row

Minimum

b1

b2

Player A

Maximin

4

5

a1

a2

4 8

11 5

Column

Maximum

11 8

Minimax

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)

To solve for the optimal probabilities for Player A

we set the two expected values equal and solve for

the value of p.

4p + 11(1 – p) = 8p + 5(1 – p)

4p + 11 – 11p = 8p + 5 – 5p

11 – 7p = 5 + 3p

Hence,

(1 - p) = .4

-10p = -6

p = .6

Player A should select:

Strategy a1 with a .6 probability and

Strategy a2 with a .4 probability.

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)

Expected gain

per game

for Player A

- Value of the Game

For Player A:

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

Expected loss

per game

for Player B

For Player B:

EV = 4q + 8(1 – q) = 4(.3) + 8(.7) = 6.8

Suppose that the payoff table for a two-person zero-

sum game is the following. Here there is no optimal

pure strategy.

Player B

Row

Minimum

b1

b2

b3

Player A

Maximin

a1

a2

a3

-2

0

-3

6 5 -2

1 0 3

3 4 -3

Column

Maximum

6 5 3

Minimax

If a game larger than 2 x 2 has a mixed strategy, we first look for dominated strategies in order to reduce the size of the game.

Player B

b1

b2

b3

Player A

a1

a2

a3

6 5 -2

1 0 3

3 4 -3

Player A’s Strategy a3 is dominated by

Strategy a1, so Strategy a3 can be eliminated.

We continue to look for dominated strategies in order to reduce the size of the game.

Player B

b1

b2

b3

Player A

a1

a2

6 5 -2

1 0 3

Player B’s Strategy b2 is dominated by

Strategy b1, so Strategy b2 can be eliminated.

The 3 x 3 game has been reduced to a 2 x 2. It is now possible to solve algebraically for the optimal mixed-strategy probabilities.

Player B

b1

b3

Player A

a1

a2

6 -2

1 3

Two-Person Zero-Sum Game Example #2

- Competing for Vehicle Sales

Let us continue with the two-dealership game

presented earlier, but with a change to one payoff.

If both Dealership A and Dealership B

choose to offer a 0% loan, the

payoff to Dealership A is now

an increase of 3 vehicle Sales

per week.(The revised payoff

table appears on the next slide.)

Two-Person Zero-Sum Game Example #2

- Payoff Table: Number of Vehicle Sales

Gained Per Week by Dealership A

(or Lost Per Week by Dealership B)

Dealership B

Cash

Rebate

b1

Free

Options

b2

0%

Loan

b3

Dealership A

Cash Rebate a1

Free Options a2

0% Loan a3

2 2 1

-3 3 -1

3 -2 3

Two-Person Zero-Sum Game Example #2

- The maximin (1) does not equal the minimax (3), so a pure strategy solution does not exist for this problem.
- The optimal solution is for both dealerships to adopt a mixed strategy.
- There are no dominated strategies, so the problem cannot be reduced to a 2x2 and solved algebraically.
- However, the game can be formulated and solved as a linear program.

Two-Person Zero-Sum Game Example #2

- Let us first consider the game from the point of view of Dealership A.
- Dealership A will select one of its three strategies based on the following probabilities:

PA1 = the probability that Dealership A selects strategy a1

PA2 = the probability that Dealership A selects strategy a2

PA3 = the probability that Dealership A selects strategy a3

Two-Person Zero-Sum Game Example #2

- Weighting each payoff by its probability and summing provides the expected value of the increase in vehicle sales per week for Dealership A.

Dealership B StrategyExpected Gain for Dealership A

b1EG(b1) = 2PA1 – 3PA2 + 3PA3

b2EG(b2) = 2PA1 + 3PA2 – 2PA3

b3EG(b3) = 1PA1 – 1PA2 + 3PA3

Two-Person Zero-Sum Game Example #2

- Define GAINA to be the optimal expected gain in vehicle sales for Dealership A, which we want to maximize.
- Thus, the individual expected gains, EG(b1), EG(b2) and EG(b3) must all be greater than or equal to GAINA.
- For example,

2PA1 – 3PA2 + 3PA3 >GAINA

- Also, the sum of Dealership A’s mixed strategy probabilities must equal 1.
- This results in the LP formulation on the next slide …..

Two-Person Zero-Sum Game Example #2

- Dealership A’s Linear Programming Formulation

MaxGAINA

s.t.

2PA1 – 3PA2 + 3PA3 – GAINA> 0 (Strategy b1)

2PA1 + 3PA2 – 2PA3 – GAINA> 0 (Strategy b2)

1PA1 – 1PA2 + 0PA3 – GAINA> 0 (Strategy b3)

PA1 + PA2 + PA3 = 1 (Prob’s sum to 1)

PA1, PA2, PA3, GAINA> 0 (Non-negativity)

Two-Person Zero-Sum Game Example #2

- The Management Scientist Solution: Dealership A

OBJECTIVE FUNCTION VALUE = 1.333

VARIABLEVALUE REDUCED COSTS

PA1 0.833 0.000

PA2 0.000 1.000

PA3 0.167 0.000

GAINA 1.333 0.000

Two-Person Zero-Sum Game Example #2

- The Management Scientist Solution: Dealership A

CONSTRAINTSLACK/SURPLUSDUAL PRICES

1 0.833 0.000

2 0.000 -0.333

3 0.000 -0.667

4 0.000 1.333

Two-Person Zero-Sum Game Example #2

- Dealership A’s Optimal Mixed Strategy
- Offer a cash rebate (a1) with a probability of 0.833
- Do not offer free optional equipment (a2)
- Offer a 0% loan (a3) with a probability of 0.167
- The expected value of this mixed strategy is a gain of
- 1.333 vehicle sales per week for Dealership A.

Two-Person Zero-Sum Game Example #2

- Let us now consider the game from the point of view of Dealership B.
- Dealership B will select one of its three strategies based on the following probabilities:

PB1 = the probability that Dealership B selects strategy b1

PB2 = the probability that Dealership B selects strategy b2

PB3 = the probability that Dealership B selects strategy b3

Two-Person Zero-Sum Game Example #2

- Weighting each payoff by its probability and summing provides the expected value of the decrease in vehicle sales per week for Dealership B.

Dealership A StrategyExpected Loss for Dealership B

a1EL(a1) = 2PB1 + 2PB2 + 1PB3

a2EL(a2) = -3PB1 + 3PB2 – 1PB3

a3EL(a3) = 3PB1 – 2PB2 + 3PB3

Two-Person Zero-Sum Game Example #2

- Define LOSSB to be the optimal expected loss in vehicle sales for Dealership B, which we want to minimize.
- Thus, the individual expected losses, EL(a1), EL(a2) and EL(a3) must all be less than or equal to LOSSB.
- For example,

2PA1 + 2PA2 + 1PA3 <LOSSB

- Also, the sum of Dealership B’s mixed strategy probabilities must equal 1.
- This results in the LP formulation on the next slide …..

Two-Person Zero-Sum Game Example #2

- Dealership B’s Linear Programming Formulation

MinLOSSB

s.t.

2PB1 + 2PB2 + 1PB3 – LOSSB< 0 (Strategy a1)

-3PB1 + 3PB2 – 1PB3 – LOSSB< 0 (Strategy a2)

3PB1 – 2PB2 + 3PB3 – LOSSB< 0 (Strategy a3)

PB1 + PB2 + PB3 = 1 (Prob’s sum to 1)

PB1, PB2, PB3, LOSSB> 0 (Non-negativity)

Two-Person Zero-Sum Game Example #2

- The Management Scientist Solution: Dealership B

OBJECTIVE FUNCTION VALUE = 1.333

VARIABLEVALUE REDUCED COSTS

PB1 0.000 0.833

PB2 0.333 0.000

PB3 0.667 0.000

LOSSB 1.333 0.000

Two-Person Zero-Sum Game Example #2

- The Management Scientist Solution: Dealership B

CONSTRAINTSLACK/SURPLUSDUAL PRICES

1 0.000 0.833

2 1.000 0.000

3 0.000 0.167

4 0.000 -1.333

Two-Person Zero-Sum Game Example #2

- Dealership B’s Optimal Mixed Strategy
- Do not offer a cash rebate (b1)
- Offer free optional equipment (b2) with a probability of 0.333
- Offer a 0% loan (b3) with a probability of 0.667
- The expected payoff of this mixed strategy is a loss of
- 1.333 vehicle sales per week for Dealership B.
- Note that expected loss for Dealership B is the same as
- the expected gain for Dealership A. (There is a zero-
- sum for the expected payoffs.)

- Two-Person, Constant-Sum Games

(The sum of the payoffs is a constant other than zero.)

- Variable-Sum Games

(The sum of the payoffs is variable.)

- n-Person Games

(A game involves more than two players.)

- Cooperative Games

(Players are allowed pre-play communications.)

- Infinite-Strategies Games

(An infinite number of strategies are available for the players.)

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