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Slides by. . . . . . . . . . . . . John Loucks St. Edward’s Univ. Chapter 12 Advanced Optimization Applications. Data Envelopment Analysis Revenue Management Portfolio Models and Asset Allocation Nonlinear Optimization Constructing an Index Fund.

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  1. Slides by . . . . . . . . . . . . John Loucks St. Edward’s Univ.

  2. Chapter 12 Advanced Optimization Applications • Data Envelopment Analysis • Revenue Management • Portfolio Models and Asset Allocation • Nonlinear Optimization • Constructing an Index Fund

  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 • Computer Solution Objective Function Value = 0.765 VariableValue Reduced Cost E 0.765 0.000 W1 0.000 0.235 W2 0.500 0.000 W3 0.500 0.000

  13. 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.)

  14. 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.

  15. 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.

  16. 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.

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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.

  26. 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.

  27. 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

  28. 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.)

  29. 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

  30. 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.

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. Conservative Portfolio A portfolio manager has been asked to develop a portfolio for the firm’s conservative clients who express a strong aversion to risk. The manager’s task is to determine the proportion of the portfolio to invest in each of six mutual funds so that the portfolio provides the best return possible with a minimum risk. The annual return (%) for five 1-year periods for the six mutual funds are shown on the next slide. The portfolio manager thinks that the returns for the five years shown in the table are scenarios that can be used to represent the possibilities for the next year.

  38. Conservative Portfolio

  39. Conservative Portfolio • Define the Decision Variables FS = proportion invested in foreign stock mutual fund IB = proportion invested in intermediate-term bond fund LG = proportion invested in large-cap growth fund LV = proportion invested in large-cap value fund SG = proportion invested in small-cap growth fund SV = proportion invested in small-cap value fund

  40. Conservative Portfolio • Constraints Minimum returns for five scenarios: – M + 10.06FS + 17.64IB + 32.41LG + 32.36LV + 33.44SG + 24.56SV ≥ 0 – M + 13.12FS + 3.25IB + 18.71LG + 20.61LV + 19.40SG + 25.32SV ≥ 0 – M + 13.47FS + 7.51IB + 33.28LG + 12.93LV + 3.85SG – 6.70SV ≥ 0 – M + 45.42FS – 1.33IB + 41.46LG + 7.06LV + 58.68SG + 5.43SV ≥ 0 – M – 21.93FS + 7.36IB – 23.26LG – 5.37LV – 9.02SG + 17.31SV ≥ 0 Sum of the proportions must equal 1: FS + IB + LG + LV + SG + SV = 1 Non-negativity M, FS, IB, LG, LV, SG, SV ≥ 0 • Objective Function Maximize the minimum return for the portfolio: Max M

  41. Conservative Portfolio • Optimal Solution The optimal value of the objective function is 6.445. (The optimal portfolio will earn 6.445% in the worst-case scenario.) 55.4% of the portfolio should be invested in the intermediate-term bond fund. 13.2% of the portfolio should be invested in the large-cap growth fund. 31.4% of the portfolio should be invested in the small-cap value fund.

  42. Moderate Portfolio A portfolio manager would like to construct a portfolio for clients who are willing to accept a moderate amount of risk in order to attempt to achieve better returns. Suppose that clients in this risk category are willing to accept some risk, but do not want the annual return for the portfolio to drop below 2%. The annual return (%) for five 1-year periods for the six mutual funds are shown on the next slide. The portfolio manager thinks that the returns for the five years shown in the table are scenarios that can be used to represent the possibilities for the next year.

  43. Moderate Portfolio • Constraints Minimum returns for five scenarios: – M + 10.06FS + 17.64IB + 32.41LG + 32.36LV + 33.44SG + 24.56SV ≥ 2 – M + 13.12FS + 3.25IB + 18.71LG + 20.61LV + 19.40SG + 25.32SV ≥ 2 – M + 13.47FS + 7.51IB + 33.28LG + 12.93LV + 3.85SG – 6.70SV ≥ 2 – M + 45.42FS – 1.33IB + 41.46LG + 7.06LV + 58.68SG + 5.43SV ≥ 2 – M – 21.93FS + 7.36IB – 23.26LG – 5.37LV – 9.02SG + 17.31SV ≥ 2 Sum of the proportions must equal 1: FS + IB + LG + LV + SG + SV = 1 Non-negativity M, FS, IB, LG, LV, SG, SV ≥ 0

  44. Moderate Portfolio • Objective Function The coefficient of FS in the objective function is given by: 0.2(10.06) + 0.2(13.12) + 0.2(13.47) + 0.2(45.42) + 0.2( – 21.93) + 12.03 The coefficient of IB is given by: 0.2(17.64) + 0.2(3.25) + 0.2(7.51) = 0.2( – 1.33) = 0.2(7.36) = 6.89 … and so on. Thus, the objective function is: Maximize the minimum return for the portfolio: Max 12.03FS + 6.89IB + 20.52LG + 13.52LV + 21.27SG + 13.18SV

  45. Moderate Portfolio • Optimal Solution Invest 10.8% of the portfolio in a large-cap growth mutual fund. Invest 41.5% in a small-cap growth mutual fund. Invest 47.7% in a small-cap value mutual fund. This allocation provides a maximum expected return of 17.33%. The portfolio return will only be 2% if scenarios 3 or 5 occur (constraints 3 and 5 are binding). The portfolio return will be 29.093% if scenario 1 occurs, 22.149% if scenario 2 occurs, and 31.417% if scenario 4 occurs.

  46. Nonlinear Optimization • Many business processes behave in a nonlinear manner. • The price of a bond is a nonlinear function of interest rates. • The price of a stock option is a nonlinear function of the price of the underlying stock. • The marginal cost of production often decreases with the quantity produced. • The quantity demanded for a product is often a nonlinear function of the price.

  47. A nonlinear optimization problem is any optimization problem in which at least one term in the objective function or a constraint is nonlinear. • Nonlinear terms include • The nonlinear optimization problems presented on the upcoming slides can be solved using computer software such as LINGO and Excel Solver. Nonlinear Optimization

  48. Nonlinear Optimization • Armstrong Bike Co. Armstrong Bike Co. produces two new lightweight bicycle frames, the Flyer and the Razor, that are made from special aluminum and steel alloys. The cost to produce a Flyer frame is $100, and the cost to produce a Razor frame is $120. We can not assume that Armstrong will sell all the frames it can produce. As the selling price of each frame model – Flyer and Razor - increases, the quantity demanded for each model goes down.

  49. Nonlinear Optimization • Assume that the demand for Flyer frames F and the demand for Razor frames R are given by: F = 750 – 5PF R = 400 – 2PR where PF = the price of a Flyer frame PR = the price of a Razor frame. • The profit contributions (revenue – cost) are: PF F- 100F for Flyer frames PR R- 120R for Razor frames

  50. Nonlinear Optimization • Profit Contribution as a Function of Demand • Solving F = 750 - 5PF for PF we get: PF = 150 -1/5F Substituting 150 -1/5F for PF in PF F- 100F we get: PF F- 100F = F(150 -1/5F) - 100F = 50F-1/5F 2 • Solving R = 400 - 2PR for PR we get: PR = 200 -1/2R Substituting 200 -1/2R for PR in PR R- 120R we get: PR R- 120R = R(200 -1/2R) - 120R = 80R-1/2R2

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