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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 Comparing performance of one branch to the whole Revenue Management Maximize revenue from short-term demand of a fixed-inventory

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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 • Comparing performance of one branch to the whole • Revenue Management • Maximize revenue from short-term demand of a fixed-inventory • Portfolio Models and Asset Allocation • Maximize return from a mix of investments • Game Theory • Compete with another player for a fixed sum (ie market share)

  3. Data Envelopment Analysis • Data envelopment analysis (DEA) • used to determine the relative operating efficiency of units with the same goals and objectives. • Branch of a bank, franchise of a restaurant, etc • DEA creates a hypothetical composite • weighted average (W1, W2,…) of existing units. • Optimal weights determined by the analysis • Goal is to determine E, the efficiency index for one unit • If E < 1, unit is less efficient than the composite and be deemed relatively inefficient. • If E = 1, unit is believed to be efficient compared to the rest.

  4. Data Envelopment Analysis • The DEA Model MIN E s.t.OUTPUTS INPUTS Sum of weights = 1 E, weights > 0

  5. Data Envelopment Analysis • Hospital Administrators at four hospitals want to examine performance • General, University, County, State • Inputs: • Number of FTE nonphysician personnel • Amount spent on supplies • Number of bed-days available • Outputs • Patient days of Medicare Service • Patient days of non-Medicare Service • Number of nurses trained • Number of interns trained

  6. Data Envelopment Analysis • Input • Output

  7. Data Envelopment Analysis • Output

  8. Data Envelopment Analysis • Define the Decision Variables E = Fraction of County's input resources required, compared to the composite hospital wg= Weight applied to General hospital wu = Weight applied to University Hospital wc = Weight applied to County Hospital ws = Weight applied to State Hospital

  9. Data Envelopment Analysis • Define the Objective Function Since our objective is to detect inefficiencies, Minimize the fraction of County’s input resources required by the composite high school: MIN E

  10. Data Envelopment Analysis • Define the Constraints • Sum of the Weights is 1: • wg + wu + wc + ws = 1 • Output Constraints • General form for each output: • output for composite >= output for county • Output for composite = • (Output for general * weight for general) +(output for university * weight for university) + (output for county * weight for county) +(output for state * weight for state)

  11. Data Envelopment Analysis • Output Constraint for Medicare Patient Days • 48.14wg + 34.62wu + 36.72wc + 33.16ws >= 36.72 • Output Constraint for non-Medicare Patient Days • 43.10wg + 27.11wu + 45.98wc + 56.46 ws >= 45.98 • Output constraint for nurses • 253wg + 148wu + 175wc + 160ws >= 175 • Output constraint for Interns • 41wg + 27wu + 23wc + 84ws >= 23

  12. Data Envelopment Analysis • Input Constraints: • General Form • Input for composite <= input for county * E • Input for composite • (input for general * weight for general) +(input for university * weight for university) +(input for county * weight for county) +(input for state * weight for state) Nonnegativity of variables: E, w1, w2, w3> 0

  13. Data Envelopment Analysis • Input Constraint for FTE non-Physicians • 285.2wg + 162.3wu + 275.07wc + 210.4ws <=275.7E • Input constraint for supply expense • 123.8wg + 128.7wu + 348.5wc + 154.1ws <= 348.5E • Input constraint for bed-days • 106.72wg + 64.21wu + 104.1wc + 104.04ws <= 104.1E

  14. Data Envelopment Analysis • Computer 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

  15. Data Envelopment Analysis • Computer Solution (continued) CONSTRAINTSLACK/SURPLUSDUAL VALUES 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

  16. 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.)

  17. 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.

  18. 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.

  19. 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.

  20. Revenue Management Fare Class D D D F F F D D D F F F D D F F ODIF Code IMD IAD ITD IMF IAF ITF BMD BAD BTD BMF BAF BTF MAD MTD MAF MTF Fare 175 275 285 395 425 475 185 315 290 385 525 490 190 180 310 295 Demand 44 25 40 15 10 8 26 50 42 12 16 9 58 48 14 11 ODIF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Origin Indianapolis Indianapolis Indianapolis Indianapolis Indianapolis Indianapolis Baltimore Baltimore Baltimore Baltimore Baltimore Baltimore Memphis Memphis Memphis Memphis Destination Memphis Austin Tampa Memphis Austin Tampa Memphis Austin Tampa Memphis Austin Tampa Austin Tampa Austin Tampa

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. Revenue Management • Computer 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

  28. Revenue Management • Computer 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

  29. 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.

  30. 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. 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.

  31. 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

  32. 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.)

  33. 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

  34. 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.

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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.

  43. 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.

  44. 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.

  45. Two-Person Zero-Sum Game Example • Strategy Choices Strategy 1: Offer a cash rebate on a new vehicle. Strategy 2: Offer free optional equipmenton a new vehicle. Strategy 3: Offer a 0% loan on a new vehicle.

  46. 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

  47. 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).

  48. 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

  49. 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).

  50. 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

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