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Readings

Readings. Readings. Chapter 5 Advanced Linear Programming Applications. Overview. Overview. Overview. Tool Summary. Tool Summary Do not make integer restrictions , and maybe the solution are integers. Second Example: IMD = number of Indianapolis-Memphis-Discount seats

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Readings

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  1. Readings • Readings • Chapter 5 • Advanced Linear Programming Applications

  2. Overview • Overview

  3. Overview

  4. Tool Summary Tool Summary • Do not make integer restrictions, and maybe the solution are integers. • Second Example: IMD = number of Indianapolis-Memphis-Discount seats • Use compound variables: • Second Example: IMD = number of Indianapolis-Memphis-Discount seats • Constrain a weighted average with a linear constraint: • Third Example: Constrain the weighted average risk factor to be no greater than 55: 60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 < 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • Interpret the unrealistic assumptions needed for a linear formulation: • First Example: Assume the set of alternative inputs and outputs for a high school is convex. Thus if 100 teaching hours gets 20 more students admitted to college, then 10 teaching hours gets at least 2 admitted. • Second Example: Assume demand has only two values. • Third Example: Assume risk is a linear function of investment shares.

  5. Tool Summary Tool Summary • Critique the unrealistic or approximate variables used because data on better variables may not be available: • First Example: High school output is measured by only three variables. • Average SAT Scores, even though maximizing those scores is not the same as maximizing learning. • The number of High School Graduates, even though there is no accounting for learning beyond a minimal level. • The number of College Admissions, even though there is no accounting for the quality or selectivity of the colleges. • First Example: High school input is measured by only three variables. • Senior Faculty • Budget ($100,000's) • Senior Enrollments, even though there is no measure for the quality of those students before their senior year.

  6. Portfolio Selection with Asset Classification • Portfolio Selection with Asset Classification

  7. Portfolio Selection with Asset Classification Overview Portfolio Selection Problems with Asset Classification help financial managers restrict the maximum investment in any single class (equities, debt, real estate, …) to ensure diversification, and so to reduce risk.

  8. Portfolio Selection with Asset Classification Question: 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 candidates for this portfolio. The investments and their important characteristics are listed in the table on the next slide. Note that UnidydeCorp. 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.

  9. Portfolio Selection with Asset Classification 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

  10. Portfolio Selection with Asset Classification Construct a portfolio that maximizes John's client's total expected after-tax return over the next year, subject to the following constraints placed upon him by the client for the portfolio. 1. The weighted average liquidity factor for the portfolio must be at least 65. 2. The weighted average risk factor for the portfolio must be no greater than 55. (Using a weighted average or any linear function is an inaccurate risk measure.) 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. 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.

  11. Portfolio Selection with Asset Classification • Answer: • Define the decision variables X1 = $ amount invested in Unidyde Corp. (Equities) X2 = $ amount invested in CC’s Restaurants (Equities) X3 = $ amount invested in First General REIT (Equities, Real Estate) X4 = $ amount invested in Metropolis Electric (Debt) X5 = $ amount invested in Unidyde Corp. (Debt) X6 = $ amount invested in Lewisville Transit (Debt) X7 = $ amount invested in Realty Partners (Real Estate) X8 = $ amount invested in T-Bill Account X9 = $ amount invested in Money Mkt. Fund X10 = $ amount invested in Saver's Certificate

  12. Portfolio Selection with Asset Classification • 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 • Define theconstraint that total investment is at most $400,000: • X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 < 400,000

  13. Portfolio Selection with Asset Classification • Constrain the weighted-average-liquidity factor to be at least 65: • 100X1 + 100X2 + 100X3 + 95X4 + 92X5 + 79X6 + 80X8 + 100X9 > • 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • Constrain the weighted average risk factor to be no greater than 55: • 60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 < • 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • Constrain that no more than $60,000 be invested in Unidyde Corp: • X1 + X5 < 60,000 • Constrain that no more than 40% of the total investment be in any one category except the money category: • X1 + X2 + X3 < 0.40 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • X4 + X5 + X6 < 0.40 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • X3 + X7 < 0.40 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)

  14. Portfolio Selection with Asset Classification • Constrain that no more than 20% of the investment can be in any one investment, except the money market: • X1 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • X2 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • X3 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • (11) X4 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • (12) X5 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • (13) X6 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • (14) X7 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • (15) X8 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • (16) X10 < 0.20 (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10)

  15. Portfolio Selection with Asset Classification • Constrain that at least $1,000 be in the Money Market: • (17) X9 > 1,000 • Constrain that the maximum investment in Saver's Certificates is $15,000: • (18) X10 < 15,000 • Constrain that the minimum investment in the Debt category is $90,000: • (19) X4 + X5 + X6 > 90,000 • Constrain that at least $10,000 be in a T-Bill account: • (20) X8 > 10,000

  16. Portfolio Selection with Asset Classification Interpretation: 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

  17. Portfolio Selection with Planning Scenarios • Portfolio Selection with Planning Scenarios

  18. Portfolio Selection with Planning Scenarios Overview Portfolio Selection Problems with Planning Scenarios help financial managers predict returns based on past performance. From predicted returns, managers can compute minimum returns for risk-averse investors, and expected returns for risk-taking investors.

  19. Portfolio Selection with Planning Scenarios Question: Portfolio manager Max Gaines needs to develop an investment portfolio for his clients who are willing to accept a moderate amount of risk. His task is to determine the proportion of the portfolio to invest in each of the five mutual funds listed below so that the portfolio provides an annual return of no less than 3%, and it maximizes the average annual return assuming that each planning scenario is equally likely.

  20. Portfolio Selection with Planning Scenarios Formulate and solve an appropriate linear program based on the following past performance.

  21. Portfolio Selection with Planning Scenarios Answer: Max R1 + R2 + R3 + R4 R1 = 22 IS + 14 LC + 19 MC + 13 SC + 7 IB R1 > 3 (Scenario 1) R2 = 26 IS + 18 LC + 18 MC + 11 SC + 8 IB R2 > 3 (Scenario 2) R3 = 6 IS + 10 LC + 5 MC + 2 SC + 9 IB R3 > 3 (Scenario 3) R4 = -3 IS + 5 LC + 1 MC + 6 SC – 3 IB R4 > 3 (Scenario 4) IS + LC + MC + SC + IB = 1 (Total proportion must equal 1) IS, LC, MC, SC, IB > 0 (Non-negativity)

  22. Portfolio Selection with Planning Scenarios

  23. Portfolio Selection with Maximin • Portfolio Selection with Maximin

  24. Portfolio Selection with Maximin Overview Portfolio Selection Problems with a Maximin (Maximize a Minimum) Objective help financial managers compute minimum returns for risk-averse investors. Risk-averse investors are pessimists, and want to maximize their investment return in the worst-case (minimum return) scenario.

  25. Portfolio Selection with Maximin Question: Portfolio manager Max Gaines needs to develop an investment portfolio for his conservative clients. His task is to determine the proportion of the portfolio to invest in each of the four mutual funds listed below so that the portfolio provides the best return possible with a minimum risk.

  26. Portfolio Selection with Maximin Answer: Formulating this problem depends on the definition of “minimum risk”. Risk is different from variability. Risk is only concerned with the potential for loss. Hence, the best return possible with a minimum risk means to maximize investment return in the worst-case (minimum return) scenario.

  27. Portfolio Selection with Maximin Let IS = proportion of portfolio invested in International Stock, and so on. Let M = guaranteed minimum return (with zero risk). Max M s.t. - M + 12 IS + 14 LC + 9 MC + 3 SC > 0 (Scenario 1) - M + 16 IS + 18 LC + 18 MC + 11 SC > 0 (Scenario 2) - M + 6 IS + 10 LC + 5 MC - 2 SC > 0 (Scenario 3) IS + LC + MC + SC = 1 (Total proportion must equal 1) M, IS, LC, MC, SC >0 (Non-negativity) (If the optimal minimal return M turns out to be negative, then the problem above is infeasible. In that case, methods are needed that are more advanced than we cover in this class.)

  28. Portfolio Selection with Maximin Solution:

  29. Portfolio Selection with Maximin Solution: The optimum is remarkable because the rate of return in planning scenarios 2 and 3 exceed the guaranteed minimum return.

  30. Production Scheduling with Adjustment Costs Production Scheduling with Adjustment Costs

  31. Production Scheduling with Adjustment Costs Overview Production Scheduling Problems with Adjustment Costs help managers find an efficient low-cost production schedule for one or more products over several periods in the future (weeks or months) when there are costs of increasing or decreasing production or storage from one period to the next.

  32. Production Scheduling with Adjustment Costs Question: Anderson makes replacement windows. In January, the company produced 15,000 windows and ended the month with 9,000 windows in inventory. Anderson’s management team would like to develop a production schedule for the next two months. A smooth production schedule is desirable because it maintains the current workforce. However, given sales forecasts, the management team does not think a smooth schedule is possible.

  33. Production Scheduling with Adjustment Costs Anderson’s accounting department estimates that increasing production from one month to the next will cost $4 per unit increase, and decreasing production from one month to the next will cost $2 per unit decrease. Assume there is no cost of changing storage from one month to the next.

  34. Production Scheduling with Adjustment Costs Formulate a linear programming model to minimize the cost of adjusting production so that you meet your sales forecasts in each month.

  35. Production Scheduling with Adjustment Costs Answer: The essential data are: A) Increasing production from one month to the next will cost $4 per unit increase, and decreasing production from one month to the next will cost $2 per unit decrease. B) January production and storage is 15,000 and 9,000. C) Data for February and March are:

  36. Production Scheduling with Adjustment Costs • Let • F = number of windows manufactured in February • M = number of windows manufactured in March • Im = increase in production level necessary during month m • Dm = decrease in production level necessary during month m • sm = ending inventory in month m • Min 4I1 + 4I2 + 2D1 + 2D2 • s.t. • 9000 + F - s1 = 15,000 • or • (1)F- s1 = 6000 February Demand • (2) s1 + M - s2 = 16,000 March Demand

  37. Production Scheduling with Adjustment Costs • F - 15,000 = I1 - D1 • or • (3) F - I1 + D1 = 15,000 Change in February Production • M - F = I2 - D2 • or • (4) M - F - I2 + D2 = 0 Change in March Production • (5) F 14,000 February Production Capacity • (6) M 15,000 March Production Capacity • (7) s1  6,000 February Storage Capacity • (8) s2  7,000 March Storage Capacity

  38. Production Scheduling with Adjustment Costs

  39. Production Scheduling with Adjustment Costs Solution: The third and fourth constraints F - 15,000 = I1 - D1 M- F = I2 - D2 for the changes in production (on the left-hand side of those equations) decompose those changes into either an increase or a decrease but not both. For example, the February change in production is a decrease of D1 = 3000.

  40. BA 452 Quantitative Analysis End of Lesson A.10

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