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DSCI 3870 Chapter 5 ADVANCED LP APPLICATIONS Additional Reading Material

DSCI 3870 Chapter 5 ADVANCED LP APPLICATIONS Additional Reading Material. Chapter 5 – Advanced LP Applications Additional Reading Material. Revenue Management Portfolio Models and Asset Allocation Game Theory – Part II. Revenue Management. Another LP application is revenue management.

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DSCI 3870 Chapter 5 ADVANCED LP APPLICATIONS Additional Reading Material

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  1. DSCI 3870 Chapter 5 ADVANCED LP APPLICATIONS Additional Reading Material

  2. Chapter 5 – Advanced LP Applications Additional Reading Material • Revenue Management • Portfolio Models and Asset Allocation • Game Theory – Part II

  3. Revenue Management • 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.

  4. 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 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.

  5. Revenue Management 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.

  6. Revenue Management

  7. 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 class

  8. 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 class BMF = number of seats allocated to Baltimore-Memphis- Full Fare class BAF = number of seats allocated to Baltimore-Austin- Full Fare class

  9. 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 class MTF = number of seats allocated to Memphis-Tampa- Full Fare class

  10. 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

  11. 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

  12. Revenue Management • 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

  13. Revenue Management • 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

  14. Revenue Management • 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

  15. 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.

  16. Portfolio Model 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.

  17. Portfolio Model 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

  18. Portfolio Model 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.)

  19. Portfolio Model 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

  20. Portfolio Model 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.

  21. Portfolio Model • 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

  22. Portfolio Model • 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

  23. Portfolio Model • 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

  24. Portfolio Model • 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

  25. Portfolio Model • 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

  26. Portfolio Model • 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

  27. Zero-Sum Games: Dominated Strategies Example 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

  28. Dominated Strategies Example 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.

  29. Dominated Strategies Example 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.

  30. Dominated Strategies Example 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

  31. 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.)

  32. 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

  33. 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.

  34. 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

  35. 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

  36. 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 …..

  37. 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)

  38. 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

  39. 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

  40. 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.

  41. 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

  42. 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

  43. 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 …..

  44. 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)

  45. 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

  46. 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

  47. 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.)

  48. Other Game Theory Models • 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.)

  49. End of Chapter 5

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