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1 1 Slide © 2008 Thomson South-Western. All Rights Reserved DSCI 3870 Chapter 5 ADVANCED LP APPLICATIONS Additional Reading Material

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2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 5 – Advanced LP Applications Additional Reading Material n Revenue Management n Portfolio Models and Asset Allocation n Game Theory – Part II

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3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Another LP application is revenue management. n Revenue management involves managing the short- term demand for a fixed perishable inventory in order to maximize revenue potential. n 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. n Application areas now include hotels, apartment rentals, car rentals, cruise lines, and golf courses.

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4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management LeapFrog Airways provides passenger service for 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.

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5 5 Slide © 2008 Thomson South-Western. All Rights Reserved LeapFrog uses two fare classes: a discount fare D 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 LeapFrog wants to determine how many seats it should allocate to each ODIF. Revenue Management

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6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management

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7 7 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Decision Variables There are 16 variables, one for each ODIF: IMD = number of seats allocated to Indianapolis-Memphis- Discount class 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

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8 8 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Decision Variables (continued) ITF = number of seats allocated to Indianapolis-Tampa- Full Fare class Full Fare class BMD = number of seats allocated to Baltimore-Memphis- Discount class Discount class BAD = number of seats allocated to Baltimore-Austin- Discount class Discount class BTD = number of seats allocated to Baltimore-Tampa- Discount class Discount class BMF = number of seats allocated to Baltimore-Memphis- Full Fare class Full Fare class BAF = number of seats allocated to Baltimore-Austin- Full Fare class Full Fare class

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9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Decision Variables (continued) BTF = number of seats allocated to Baltimore-Tampa- Full Fare class Full Fare class MAD = number of seats allocated to Memphis-Austin- Discount class Discount class MTD = number of seats allocated to Memphis-Tampa- Discount class Discount class MAF = number of seats allocated to Memphis-Austin- Full Fare class Full Fare class MTF = number of seats allocated to Memphis-Tampa- Full Fare class Full Fare class

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10 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Objective Function Maximize total revenue: Max (fare per seat for each ODIF) Max (fare per seat for each ODIF) x (number of seats allocated to the ODIF) x (number of seats allocated to the ODIF) Max 175IMD + 275IAD + 285ITD + 395IMF + 425IAF + 475ITF + 185BMD + 315BAD + 425IAF + 475ITF + 185BMD + 315BAD + 290BTD + 385BMF + 525BAF + 490BTF + 290BTD + 385BMF + 525BAF + 490BTF + 190MAD + 180MTD + 310MAF + 295MTF + 190MAD + 180MTD + 310MAF + 295MTF

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11 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Constraints There are 4 capacity constraints, one for each flight leg: Indianapolis-Memphis leg Indianapolis-Memphis leg (1) IMD + IAD + ITD + IMF + IAF + ITF < 120 (1) IMD + IAD + ITD + IMF + IAF + ITF < 120 Baltimore-Memphis leg Baltimore-Memphis leg (2) BMD + BAD + BTD + BMF + BAF + BTF < 120 (2) BMD + BAD + BTD + BMF + BAF + BTF < 120 Memphis-Austin leg Memphis-Austin leg (3) IAD + IAF + BAD + BAF + MAD + MAF < 120 (3) IAD + IAF + BAD + BAF + MAD + MAF < 120 Memphis-Tampa leg Memphis-Tampa leg (4) ITD + ITF + BTD + BTF + MTD + MTF < 120 (4) ITD + ITF + BTD + BTF + MTD + MTF < 120

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12 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n Define the Constraints (continued) There are 16 demand constraints, one for each ODIF: (5) IMD < 44(11) BMD < 26(17) MAD < 5 (5) IMD < 44(11) BMD < 26(17) MAD < 5 (6) IAD < 25(12) BAD < 50(18) MTD < 48 (6) IAD < 25(12) BAD < 50(18) MTD < 48 (7) ITD < 40(13) BTD < 42(19) MAF < 14 (7) ITD < 40(13) BTD < 42(19) MAF < 14 (8) IMF < 15(14) BMF < 12(20) MTF < 11 (8) IMF < 15(14) BMF < 12(20) MTF < 11 (9) IAF < 10(15) BAF < 16 (9) IAF < 10(15) BAF < 16 (10) ITF < 8(16) BTF < 9 (10) ITF < 8(16) BTF < 9

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13 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n The Management Scientist Solution Objective Function Value = 94735.000 Variable Value Reduced Cost IMD44.000 0.000 IMD44.000 0.000 IAD 3.000 0.000 IAD 3.000 0.000 ITD 40.000 0.000 ITD 40.000 0.000 IMF 15.000 0.000 IMF 15.000 0.000 IAF 10.000 0.000 IAF 10.000 0.000 ITF 8.000 0.000 ITF 8.000 0.000 BMD 26.000 0.000 BMD 26.000 0.000 BAD 50.000 0.000 BAD 50.000 0.000

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14 Slide © 2008 Thomson South-Western. All Rights Reserved Revenue Management n The Management Scientist Solution (continued) Variable Value Reduced Cost BTD 7.000 0.000 BTD 7.000 0.000 BMF 12.000 0.000 BMF 12.000 0.000 BAF 16.000 0.000 BAF 16.000 0.000 BTF 9.000 0.000 BTF 9.000 0.000 MAD 27.000 0.000 MAD 27.000 0.000 MTD 45.000 0.000 MTD 45.000 0.000 MAF 14.000 0.000 MAF 14.000 0.000 MTF 11.000 0.000 MTF 11.000 0.000

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15 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Models and Asset Management n Asset allocation involves determining how to allocate investment funds across a variety of asset classes such as stocks, bonds, mutual funds, real estate. n Portfolio models are used to determine percentage of funds that should be made in each asset class. n The goal is to create a portfolio that provides the best balance between risk and return.

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16 Slide © 2008 Thomson South-Western. All Rights Reserved John Sweeney is an investment advisor who is 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 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. Portfolio Model

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17 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Exp. Annual Exp. Annual After Tax Liquidity Risk 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 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 Lewisville Transit 12.0% 79 22 Real Estate Realty Partners 22.0% 0 50 First General REIT ( --- See above --- ) First General REIT ( --- See above --- ) Money T-Bill Account 9.6% 80 0 Money Mkt. Fund 10.5% 100 10 Money Mkt. Fund 10.5% 100 10 Saver's Certificate 12.6% 0 0 Saver's Certificate 12.6% 0 0

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18 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Formulate a linear programming problem to 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.)

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19 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Portfolio Limitations Portfolio Limitations 1. The weighted average liquidity factor for the portfolio 1. The weighted average liquidity factor for the portfolio must to be at least 65. must to be at least 65. 2. The weighted average risk factor for the portfolio must 2. The weighted average risk factor for the portfolio must be no greater than 55. be no greater than 55. 3. No more than $60,000 is to be invested in Unidyde 3. No more than $60,000 is to be invested in Unidyde stocks or bonds. stocks or bonds. 4. No more than 40% of the investment can be in any one 4. No more than 40% of the investment can be in any one category except the money category. category except the money category. 5. No more than 20% of the total investment can be in 5. No more than 20% of the total investment can be in any one investment except the money market fund. any one investment except the money market fund.continued

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20 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model Portfolio Limitations (continued) Portfolio Limitations (continued) 6. At least $1,000 must be invested in the Money Market 6. At least $1,000 must be invested in the Money Market fund. fund. 7. The maximum investment in Saver's Certificates is 7. The maximum investment in Saver's Certificates is $15,000. $15,000. 8. The minimum investment desired for debt is $90,000. 8. The minimum investment desired for debt is $90,000. 9. At least $10,000 must be placed in a T-Bill account. 9. At least $10,000 must be placed in a T-Bill account.

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21 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n 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 X10 = $ amount invested in Saver's Certificate

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22 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n 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 +.12X6 +.22X7 +.096X8 +.105X9 +.126X10

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23 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model 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: (2)100X1 + 100X2 + 100X3 + 95X4 + 92X5 + 79X6 + 80X8 + 100X9 > 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) Weighted average risk factor must be no greater than 55: (3)60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 < 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) No more than $60,000 to be invested in Unidyde Corp: (4)X1 + X5 < 60,000 n Define the Constraints

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24 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n 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 (7)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

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25 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n 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 Xj > 0 j = 1,..., 10

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26 Slide © 2008 Thomson South-Western. All Rights Reserved Portfolio Model n 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 X10 = $15,000 invested in Saver's Certificate

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27 Slide © 2008 Thomson South-Western. All Rights Reserved Zero-Sum Games: Dominated Strategies Example RowMinimum -2 0-3 b1b1b1b1 b3b3b3b3 b2b2b2b2 Player B 1 0 3 1 0 3 3 4 -3 3 4 -3 a1a1a2a2a3a3a1a1a2a2a3a3 Player A ColumnMaximum 6 5 3 6 5 3 6 5 -2 6 5 -2 Suppose that the payoff table for a two-person zero- sum game is the following. Here there is no optimal pure strategy. Maximin Minimax

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28 Slide © 2008 Thomson South-Western. All Rights Reserved Dominated Strategies Example b1b1b1b1 b3b3b3b3 b2b2b2b2 Player B 1 0 3 1 0 3 Player A 6 5 -2 6 5 -2 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. 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. 3 4 -3 3 4 -3 a1a1a2a2a3a3a1a1a2a2a3a3 Player A’s Strategy a 3 is dominated by Player A’s Strategy a 3 is dominated by Strategy a 1, so Strategy a 3 can be eliminated.

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29 Slide © 2008 Thomson South-Western. All Rights Reserved Dominated Strategies Example b1b1b1b1 b3b3b3b3 Player B Player A a1a1a2a2a1a1a2a2 Player B’s Strategy b 2 is dominated by Player B’s Strategy b 2 is dominated by Strategy b 1, so Strategy b 2 can be eliminated. b2b2b2b2 1 0 3 1 0 3 6 5 -2 6 5 -2 We continue to look for dominated strategies in order to reduce the size of the game. We continue to look for dominated strategies in order to reduce the size of the game.

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30 Slide © 2008 Thomson South-Western. All Rights Reserved Dominated Strategies Example b1b1b1b1 b3b3b3b3 Player B Player A a1a1a2a2a1a1a2a2 1 3 1 3 6 -2 6 -2 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. 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.

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31 Slide © 2008 Thomson South-Western. All Rights Reserved n Competing for Vehicle Sales Let us continue with the two-dealership game 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

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32 Slide © 2008 Thomson South-Western. All Rights Reserved 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B n Payoff Table: Number of Vehicle Sales Gained Per Week by Dealership A Gained Per Week by Dealership A (or Lost Per Week by Dealership B) (or Lost Per Week by Dealership B) -3 3 -1 3 -2 3 3 -2 3 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Two-Person Zero-Sum Game Example #2

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33 Slide © 2008 Thomson South-Western. All Rights Reserved n The maximin (1) does not equal the minimax (3), so a pure strategy solution does not exist for this problem. n The optimal solution is for both dealerships to adopt a mixed strategy. n There are no dominated strategies, so the problem cannot be reduced to a 2x2 and solved algebraically. n However, the game can be formulated and solved as a linear program. Two-Person Zero-Sum Game Example #2

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34 Slide © 2008 Thomson South-Western. All Rights Reserved n Let us first consider the game from the point of view of Dealership A. n Dealership A will select one of its three strategies based on the following probabilities: PA 1 = the probability that Dealership A selects strategy a 1 PA 2 = the probability that Dealership A selects strategy a 2 PA 3 = the probability that Dealership A selects strategy a 3 Two-Person Zero-Sum Game Example #2

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35 Slide © 2008 Thomson South-Western. All Rights Reserved Two-Person Zero-Sum Game Example #2 n 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 Strategy Expected Gain for Dealership A Dealership B Strategy Expected Gain for Dealership A b 1 EG ( b 1 ) = 2 PA 1 – 3 PA 2 + 3 PA 3 b 1 EG ( b 1 ) = 2 PA 1 – 3 PA 2 + 3 PA 3 b 2 EG ( b 2 ) = 2 PA 1 + 3 PA 2 – 2 PA 3 b 3 EG ( b 3 ) = 1 PA 1 – 1 PA 2 + 3 PA 3 b 3 EG ( b 3 ) = 1 PA 1 – 1 PA 2 + 3 PA 3

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36 Slide © 2008 Thomson South-Western. All Rights Reserved Two-Person Zero-Sum Game Example #2 n Define GAINA to be the optimal expected gain in vehicle sales for Dealership A, which we want to maximize. n Thus, the individual expected gains, EG ( b 1 ), EG ( b 2 ) and EG ( b 3 ) must all be greater than or equal to GAINA. n For example, 2 PA 1 – 3 PA 2 + 3 PA 3 > GAINA n Also, the sum of Dealership A’s mixed strategy probabilities must equal 1. n This results in the LP formulation on the next slide …..

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37 Slide © 2008 Thomson South-Western. All Rights Reserved n Dealership A’s Linear Programming Formulation Max GAINA s.t. 2 PA 1 – 3 PA 2 + 3 PA 3 – GAINA > 0 (Strategy b 1 ) 2 PA 1 + 3 PA 2 – 2 PA 3 – GAINA > 0 (Strategy b 2 ) 1 PA 1 – 1 PA 2 + 0 PA 3 – GAINA > 0 (Strategy b 3 ) 1 PA 1 – 1 PA 2 + 0 PA 3 – GAINA > 0 (Strategy b 3 ) PA 1 + PA 2 + PA 3 = 1 (Prob’s sum to 1) PA 1, PA 2, PA 3, GAINA > 0 (Non-negativity) Two-Person Zero-Sum Game Example #2

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38 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution: Dealership A OBJECTIVE FUNCTION VALUE = 1.333 VARIABLE VALUE REDUCED COSTS VARIABLE VALUE REDUCED COSTS PA1 0.833 0.000 PA1 0.833 0.000 PA2 0.000 1.000 PA2 0.000 1.000 PA3 0.167 0.000 PA3 0.167 0.000 GAINA 1.333 0.000 GAINA 1.333 0.000 Two-Person Zero-Sum Game Example #2

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39 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution: Dealership A CONSTRAINT SLACK/SURPLUS DUAL PRICES 1 0.833 0.000 1 0.833 0.000 2 0.000 -0.333 2 0.000 -0.333 3 0.000 -0.667 3 0.000 -0.667 4 0.000 1.333 4 0.000 1.333 Two-Person Zero-Sum Game Example #2

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40 Slide © 2008 Thomson South-Western. All Rights Reserved n Dealership A’s Optimal Mixed Strategy Offer a cash rebate ( a 1 ) with a probability of 0.833Offer a cash rebate ( a 1 ) with a probability of 0.833 Do not offer free optional equipment ( a 2 )Do not offer free optional equipment ( a 2 ) Offer a 0% loan ( a 3 ) with a probability of 0.167Offer a 0% loan ( a 3 ) 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

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41 Slide © 2008 Thomson South-Western. All Rights Reserved n Let us now consider the game from the point of view of Dealership B. n Dealership B will select one of its three strategies based on the following probabilities: PB 1 = the probability that Dealership B selects strategy b 1 PB 2 = the probability that Dealership B selects strategy b 2 PB 3 = the probability that Dealership B selects strategy b 3 Two-Person Zero-Sum Game Example #2

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42 Slide © 2008 Thomson South-Western. All Rights Reserved n 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 Strategy Expected Loss for Dealership B a 1 EL ( a 1 ) = 2 PB 1 + 2 PB 2 + 1 PB 3 a 1 EL ( a 1 ) = 2 PB 1 + 2 PB 2 + 1 PB 3 a 2 EL ( a 2 ) = -3 PB 1 + 3 PB 2 – 1 PB 3 a 3 EL ( a 3 ) = 3 PB 1 – 2 PB 2 + 3 PB 3 a 3 EL ( a 3 ) = 3 PB 1 – 2 PB 2 + 3 PB 3 Two-Person Zero-Sum Game Example #2

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43 Slide © 2008 Thomson South-Western. All Rights Reserved Two-Person Zero-Sum Game Example #2 n Define LOSSB to be the optimal expected loss in vehicle sales for Dealership B, which we want to minimize. n Thus, the individual expected losses, EL ( a 1 ), EL ( a 2 ) and EL ( a 3 ) must all be less than or equal to LOSSB. n For example, 2 PA 1 + 2 PA 2 + 1 PA 3 < LOSSB n Also, the sum of Dealership B’s mixed strategy probabilities must equal 1. n This results in the LP formulation on the next slide …..

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44 Slide © 2008 Thomson South-Western. All Rights Reserved n Dealership B’s Linear Programming Formulation Min LOSSB s.t. 2 PB 1 + 2 PB 2 + 1 PB 3 – LOSSB < 0 (Strategy a 1 ) 2 PB 1 + 2 PB 2 + 1 PB 3 – LOSSB < 0 (Strategy a 1 ) -3 PB 1 + 3 PB 2 – 1 PB 3 – LOSSB < 0 (Strategy a 2 ) 3 PB 1 – 2 PB 2 + 3 PB 3 – LOSSB < 0 (Strategy a 3 ) 3 PB 1 – 2 PB 2 + 3 PB 3 – LOSSB < 0 (Strategy a 3 ) PB 1 + PB 2 + PB 3 = 1 (Prob’s sum to 1) PB 1 + PB 2 + PB 3 = 1 (Prob’s sum to 1) PB 1, PB 2, PB 3, LOSSB > 0 (Non-negativity) PB 1, PB 2, PB 3, LOSSB > 0 (Non-negativity) Two-Person Zero-Sum Game Example #2

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45 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution: Dealership B OBJECTIVE FUNCTION VALUE = 1.333 VARIABLE VALUE REDUCED COSTS VARIABLE VALUE REDUCED COSTS PB1 0.000 0.833 PB1 0.000 0.833 PB2 0.333 0.000 PB2 0.333 0.000 PB3 0.667 0.000 PB3 0.667 0.000 LOSSB 1.333 0.000 LOSSB 1.333 0.000 Two-Person Zero-Sum Game Example #2

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46 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution: Dealership B CONSTRAINT SLACK/SURPLUS DUAL PRICES 1 0.000 0.833 1 0.000 0.833 2 1.000 0.000 2 1.000 0.000 3 0.000 0.167 3 0.000 0.167 4 0.000 -1.333 4 0.000 -1.333 Two-Person Zero-Sum Game Example #2

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47 Slide © 2008 Thomson South-Western. All Rights Reserved n Dealership B’s Optimal Mixed Strategy Do not offer a cash rebate ( b 1 )Do not offer a cash rebate ( b 1 ) Offer free optional equipment ( b 2 ) with a probability of 0.333Offer free optional equipment ( b 2 ) with a probability of 0.333 Offer a 0% loan ( b 3 ) with a probability of 0.667Offer a 0% loan ( b 3 ) with a probability of 0.667 The expected payoff of this mixed strategy is a loss of 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 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 Zero-Sum Game Example #2

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48 Slide © 2008 Thomson South-Western. All Rights Reserved n Two-Person, Constant-Sum Games (The sum of the payoffs is a constant other than zero.) (The sum of the payoffs is a constant other than zero.) n Variable-Sum Games (The sum of the payoffs is variable.) (The sum of the payoffs is variable.) n n -Person Games (A game involves more than two players.) (A game involves more than two players.) n Cooperative Games (Players are allowed pre-play communications.) (Players are allowed pre-play communications.) n Infinite-Strategies Games (An infinite number of strategies are available for the players.) (An infinite number of strategies are available for the players.) Other Game Theory Models

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49 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 5

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