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Slides by JOHN LOUCKS St. Edward’s University

Slides by JOHN LOUCKS St. Edward’s University. Chapter 5 Advanced Linear Programming Applications. Data Envelopment Analysis Revenue Management Portfolio Models and Asset Allocation Game Theory. Data Envelopment Analysis.

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Slides by JOHN LOUCKS St. Edward’s University

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  1. Slides by JOHN LOUCKS St. Edward’s University

  2. Chapter 5 Advanced Linear Programming Applications • Data Envelopment Analysis • Revenue Management • Portfolio Models and Asset Allocation • Game Theory

  3. 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.

  4. 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

  5. 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.

  6. 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

  7. Data Envelopment Analysis • Output RooseveltLincolnWashington Average SAT Score 800 830 900 High School Graduates 450 500 400 College Admissions 140 250 370

  8. 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

  9. 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

  10. 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)

  11. 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

  12. 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

  13. 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

  14. 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.)

  15. 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.

  16. 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.

  17. 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.

  18. Revenue Management

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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.

  28. 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.

  29. 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

  30. 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.)

  31. 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

  32. 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.

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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.

  41. Two-Person Zero-Sum Game • 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.

  42. 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.

  43. 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.

  44. 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

  45. Two-Person Zero-Sum Game • 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).

  46. 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

  47. Two-Person Zero-Sum Game • 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).

  48. 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

  49. 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

  50. Pure Strategy Example • 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

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