1. 1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2. 2 2 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 8 Nonlinear Optimization Models n A Production Application n Constructing an Index Fund n Markowitz Portfolio Model n Blending: The Pooling Problem n Forecasting Adoption of a New Product

3. 3 3 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction n Many business processes behave in a nonlinear manner. The price of a bond is a nonlinear function of interest rates. 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 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 marginal cost of production often decreases with the quantity produced. The quantity demanded for a product is often a nonlinear function of the price. The quantity demanded for a product is often a nonlinear function of the price.

4. 4 4 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction n A nonlinear optimization problem is any optimization problem in which at least one term in the objective function or a constraint is nonlinear. n Nonlinear terms include n The nonlinear optimization problems presented on the upcoming slides can be solved using computer software such as LINGO and Excel Solver.

5. 5 5 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. demanded for each model goes down. Example: Production Application n Armstrong Bike Co.

6. 6 6 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Assume that the demand for Flyer frames F and the demand for Razor frames R are given by: F = 750 – 5 P F F = 750 – 5 P F R = 400 – 2 P R R = 400 – 2 P R where P F = the price of a Flyer frame P R = the price of a Razor frame. P R = the price of a Razor frame. n The profit contributions (revenue – cost) are: P F F  100 F for Flyer frames P F F  100 F for Flyer frames P R R  120 R for Razor frames P R R  120 R for Razor frames Example: Production Application

7. 7 7 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Profit Contribution as a Function of Demand Solving F = 750  5 P F for P F we get:Solving F = 750  5 P F for P F we get: P F = 150   F P F = 150   F Substituting 150   F for P F in P F F  100 F we get: Substituting 150   F for P F in P F F  100 F we get: P F F  100 F = F (150   F )  100 F = 50 F   F 2 P F F  100 F = F (150   F )  100 F = 50 F   F 2 Solving R = 400  2 P R for P R we get:Solving R = 400  2 P R for P R we get: P R = 200   R P R = 200   R Substituting 200   R for P R in P R R  120 R we get: Substituting 200   R for P R in P R R  120 R we get: P R R  120 R = R (200   R )  120 R = 80 R   R 2 P R R  120 R = R (200   R )  120 R = 80 R   R 2 Example: Production Application

8. 8 8 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Total Profit Contribution Total Profit Contribution = 50 F –  F 2 + 80 R –  R 2 Total Profit Contribution = 50 F –  F 2 + 80 R –  R 2 This function is an example of a quadratic function This function is an example of a quadratic function because the nonlinear terms have a power of 2. because the nonlinear terms have a power of 2. Example: Production Application

9. 9 9 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A supplier can deliver a maximum of 500 A supplier can deliver a maximum of 500 pounds of the aluminum alloy and 420 pounds of the steel alloy weekly. The number of pounds of each alloy needed per frame is summarized below. Aluminum Alloy Steel Alloy Aluminum Alloy Steel Alloy Flyer 2 3 Flyer 2 3 Razor 4 2 Razor 4 2 How many Flyer and Razor frames should Armstrong produce each week? Example: Production Application

10. 10 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Max 50 F –  F 2 + 80 R –  R 2 (Total Weekly Profit) s.t. 2 F + 4 R < 500 (Aluminum Available) 3 F + 2 R < 420 (Steel Available) 3 F + 2 R < 420 (Steel Available) F, R > 0 (Non-negativity) F, R > 0 (Non-negativity) Example: Production Application n Problem Formulation

11. 11 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Total Profit Contribution First, we will solve the unconstrained version of First, we will solve the unconstrained version of this nonlinear program to find the values of F and R that maximize the above total profit contribution function (with the production constraints ignored). Example: Production Application

12. 12 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Production Application n Optimal Solution for Unconstrained Problem 3 F + 2 R < 420 2F + 4 R < 500 UnconstrainedOptimum (125, 80) Profit = $6,325.00 x1x1x1x1 x2x2x2x2 25020015010050 50 100 150 200 250 300 FeasibleRegion

13. 13 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Total Profit Contribution Now we will solve the constrained version of this Now we will solve the constrained version of this nonlinear program to find the values of F and R that maximize the total profit contribution function with the production constraints enforced. Example: Production Application

14. 14 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Production Application n Objective Function Contour Lines $6,325.00 x2x2x2x2 25020015010050 50 100 150 200 250 300 $6,200.00Contour $6,075.47Contour $5,500.00Contour x1x1x1x1

15. 15 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Production Application n Optimal Solution for Constrained Problem x1x1x1x1 x2x2x2x2 25020015010050 50 100 150 200 250 300 $6,075.47Contour ConstrainedOptimum (92.45, 71.32) Profit = $6,075.47

16. 16 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Production Application n Optimal Solution Produce 92.45 Flyer frames per week.Produce 92.45 Flyer frames per week. Produce 71.32 Razor frames per week.Produce 71.32 Razor frames per week. Profit per week is $6,075.47.Profit per week is $6,075.47. Use 470.2 pounds of aluminum alloy per week (of the 500 pounds available per week).Use 470.2 pounds of aluminum alloy per week (of the 500 pounds available per week). Use the entire 420 pounds of steel alloy available per week.Use the entire 420 pounds of steel alloy available per week.

17. 17 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Local and Global Optima n A feasible solution is a local optimum if there are no other feasible solutions with a better objective function value in the immediate neighborhood. For a maximization problem the local optimum corresponds to a local maximum. For a maximization problem the local optimum corresponds to a local maximum. For a minimization problem the local optimum corresponds to a local minimum. For a minimization problem the local optimum corresponds to a local minimum.  A feasible solution is a global optimum if there are no other feasible points with a better objective function value in the feasible region.  Obviously, a global optimum is also a local optimum.

18. 18 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Local Optima n Nonlinear optimization problems can have multiple local optimal solutions, in which case we want to find the best local optimum. n Nonlinear problems with multiple local optima are difficult to solve and pose a serious challenge for optimization software. n In these cases, the software can get “stuck” and terminate at a local optimum. n There can be a severe penalty for finding a local optimum that is not a global optimum. n Developing algorithms capable of finding the global optimum is currently a very active research area.

19. 19 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Local Optima n Consider the function n The shape of this function is shown on the next slide. n The hills and valleys in the graph show that this function has several local maximums and local minimums. n There are two local minimums, one of which is the the global minimum. n There are three local maximums, one of which is the global maximum.

20. 20 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Local Optima

21. 21 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Single Local Optimum n Consider the function n The shape of this function is shown on the next slide. n A function that is bowl-shaped down is called a concave function. n The maximum value for this particular function is 0 and the point (0, 0) gives the optimal value of 0. n Functions such as this one have a single local maximum that is also a global maximum. n This type of nonlinear problem is relatively easy to maximize.

22. 22 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Single Local Optimum

23. 23 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Single Local Optimum n Consider the function n The shape of this function is shown on the next slide. n A function that is bowl-shaped up is called a convex function. n The minimum value for this particular function is 0 and the point (0, 0) gives the optimal value of 0. n Functions such as this one have a single local minimum that is also a global minimum. n This type of nonlinear problem is relatively easy to minimize.

24. 24 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Single Local Optimum Y X Z 2 4 0 -2 -4 -2 0 2 4 40 20

25. 25 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Dual Values n Recall that the dual value is the change in the value of the optimal solution per unit increase in the right- hand side of the constraint. n The interpretation of the dual value for nonlinear models is exactly the same as it is for LPs. n However, for nonlinear problems the allowable increase and decrease are not usually reported. n This is because for typical nonlinear problems the allowable increase and decrease are zero. n That is, if you change the right-hand side by even a small amount, the dual value changes.

26. 26 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Constructing an Index Fund n Index funds are a very popular investment vehicle in the mutual fund industry. n Vanguard 500 Index Fund is the largest mutual fund in the U.S. with over $70 billion in net assets in 2005. n An index fund is an example of passive asset management. n The key idea behind an index fund is to construct a portfolio of stocks, mutual funds, or other securities that closely matches the performance of a broad market index such as the S&P 500. n Behind the popularity of index funds is research that basically says “you can’t beat the market.”

27. 27 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Lymann Brothers Investments Lymann Brothers has a substantial number of clients who wish to own a mutual fund portfolio that closely matches the performance of the S&P 500 stock index. A manager at Lymann Brothers has selected five mutual funds (shown on the next slide) that will be considered for inclusion in the portfolio. The manager must decide what percentage of the portfolio should be invested in each mutual fund.

28. 28 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund Annual Returns (Planning Scenarios) Annual Returns (Planning Scenarios) Mutual Fund Year 1Year 2Year 3Year 4 International Stock 25.64 27.62 5.80 -3.13 Large-Cap Blend 15.31 18.77 11.06 4.75 Mid-Cap Blend 18.74 18.43 6.28 -1.04 Small-Cap Blend 14.19 12.37 -1.92 7.32 Intermediate Bond 7.88 9.45 10.56 3.31 S&P 500 13.00 12.00 7.00 2.00 n Mutual Fund Performance in 4 Selected Years

29. 29 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Define the 9 Decision Variables IS = proportion of portfolio invested in international stock LC = proportion of portfolio invested in large-cap blend MC = proportion of portfolio invested in mid-cap blend SC = proportion of portfolio invested in small-cap blend IB = proportion of portfolio invested in intermediate bond R 1 = portfolio return for scenario 1 (year 1) R 2 = portfolio return for scenario 2 (year 2) R 3 = portfolio return for scenario 3 (year 3) R 4 = portfolio return for scenario 4 (year 4)

30. 30 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Define the Objective Function Min ( R 1 – 13) 2 + ( R 2 – 12) 2 + ( R 3 – 7) 2 + ( R 4 – 2) 2 n Define the 6 Constraints (including non-negativity) 25.64 IS + 15.31 LC + 18.74 MC + 14.19 SC + 7.88 IB = R 1 27.62 IS + 18.77 LC + 18.43 MC + 12.37 SC + 9.45 IB = R 2 5.80 IS + 11.06 LC + 6.28 MC  1.92 SC + 10.56 IB = R 3 5.80 IS + 11.06 LC + 6.28 MC  1.92 SC + 10.56 IB = R 3  3.13 IS + 4.75 LC  1.04 MC + 7.32 SC + 3.31 IB = R 4  3.13 IS + 4.75 LC  1.04 MC + 7.32 SC + 3.31 IB = R 4 IS + LC + MC + SC + IB = 1 IS, LC, MC, SC, IB > 0

31. 31 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Optimal Solution for Lymann Brothers Example R 1 = 12.51(12.51% portfolio return for scenario 1) R 1 = 12.51(12.51% portfolio return for scenario 1) R 2 = 12.90 (12.90% portfolio return for scenario 2) R 2 = 12.90 (12.90% portfolio return for scenario 2) R 3 = 7.13 ( 7.13% portfolio return for scenario 3) R 3 = 7.13 ( 7.13% portfolio return for scenario 3) R 4 = 2.51 ( 2.51% portfolio return for scenario 4) R 4 = 2.51 ( 2.51% portfolio return for scenario 4) IS = 0 ( 0.0% of portfolio in international stock) IS = 0 ( 0.0% of portfolio in international stock) LC = 0 ( 0.0% of portfolio in large-cap blend) LC = 0 ( 0.0% of portfolio in large-cap blend) MC =.332 (33.2% of portfolio in mid-cap blend) MC =.332 (33.2% of portfolio in mid-cap blend) SC =.161 (16.1% of portfolio in small-cap blend) SC =.161 (16.1% of portfolio in small-cap blend) IB =.507 (50.7% of portfolio in intermediate bond) IB =.507 (50.7% of portfolio in intermediate bond) 100.0% of portfolio

32. 32 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Lymann Brothers Portfolio Return vs. S&P 500 Return Scenario Portfolio ReturnS&P 500 Return 112.5113.00 112.5113.00 2 12.9012.00 2 12.9012.00 3 7.13 7.00 3 7.13 7.00 4 2.51 2.00 4 2.51 2.00

33. 33 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Markowitz Portfolio Model n There is a key tradeoff in most portfolio optimization models between risk and return. n The index fund model (Lymann Brothers example) presented earlier managed the tradeoff passively. n The Markowitz mean-variance portfolio model provides a very convenient way for an investor to actively trade-off risk versus return. n We will now demonstrate the Markowitz portfolio model by extending the Lymann Brothers example.

34. 34 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Markowitz Portfolio Model n In the Lymann Brothers example there were four scenarios and the return under each scenario was defined by the variables R 1, R 2, R 3, and R 4. n If p s is the probability of scenario s, and there are n scenarios, then the expected return for the portfolio R is n If we assume that the four scenarios in the Lymann Brothers model are equally likely, then

35. 35 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Markowitz Portfolio Model n The measure of risk most often associated with the Markowitz model is the variance of the portfolio. n For our example, the portfolio variance is n For our example, the four planning scenarios are equally likely. Thus,

36. 36 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Markowitz Portfolio Model n The portfolio variance is the average of the sum of the squares of the deviations from the mean value under each scenario. n The larger the variance value, the more widely dispersed the scenario returns are about the average return value. n If the portfolio variance were equal to zero, then every scenario return R i would be equal.

37. 37 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n There are two basic ways to formulate the Markowitz model: (1) Minimize the variance of the portfolio subject to constraints on the expected return, and(1) Minimize the variance of the portfolio subject to constraints on the expected return, and (2) Maximize the expected return of the portfolio subject to a constraint on risk.(2) Maximize the expected return of the portfolio subject to a constraint on risk. We will now demonstrate the first (1) formulation, assuming that Lymann Brothers’ client requires the expected portfolio return to be at least 9 percent. We will now demonstrate the first (1) formulation, assuming that Lymann Brothers’ client requires the expected portfolio return to be at least 9 percent. Example: Markowitz Portfolio Model

38. 38 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Define the Objective Function Minimize the portfolio variance: Minimize the portfolio variance: n Define the Constraints Define the return for each scenario: 25.64 IS + 15.31 LC + 18.74 MC + 14.19 SC + 7.88 IB = R 1 27.62 IS + 18.77 LC + 18.43 MC + 12.37 SC + 9.45 IB = R 2 5.80 IS + 11.06 LC + 6.28 MC  1.92 SC + 10.56 IB = R 3 5.80 IS + 11.06 LC + 6.28 MC  1.92 SC + 10.56 IB = R 3  3.13 IS + 4.75 LC  1.04 MC + 7.32 SC + 3.31 IB = R 4  3.13 IS + 4.75 LC  1.04 MC + 7.32 SC + 3.31 IB = R 4 Example: Markowitz Portfolio Model

39. 39 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Define the Constraints (continued) All the money must be invested in the portfolio: IS + LC + MC + SC + IB = 1 IS + LC + MC + SC + IB = 1 Define the expected return for the portfolio: The portfolio return must be at least 9 percent: Non-negativity: IS, LC, MC, SC, IB > 0 Example: Markowitz Portfolio Model

40. 40 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Markowitz Portfolio Model n Optimal Solution R 1 = 10.63(10.63% portfolio return for scenario 1) R 1 = 10.63(10.63% portfolio return for scenario 1) R 2 = 12.20 (12.20% portfolio return for scenario 2) R 2 = 12.20 (12.20% portfolio return for scenario 2) R 3 = 8.93 ( 8.93% portfolio return for scenario 3) R 3 = 8.93 ( 8.93% portfolio return for scenario 3) R 4 = 4.24 ( 4.24% portfolio return for scenario 4) R 4 = 4.24 ( 4.24% portfolio return for scenario 4) R bar = 9.00( 9.00% expected portfolio return)R bar = 9.00( 9.00% expected portfolio return) IS = 0 ( 0.0% of portfolio in international stock) IS = 0 ( 0.0% of portfolio in international stock) LC =.251 (25.1% of portfolio in large-cap blend) LC =.251 (25.1% of portfolio in large-cap blend) MC = 0 ( 0.0% of portfolio in mid-cap blend) MC = 0 ( 0.0% of portfolio in mid-cap blend) SC =.141 (14.1% of portfolio in small-cap blend) SC =.141 (14.1% of portfolio in small-cap blend) IB =.608 (60.8% of portfolio in intermediate bond) IB =.608 (60.8% of portfolio in intermediate bond) 100.0% of portfolio

41. 41 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Blending: The Pooling Problem n Blending problems arise when a manager must decide how to blend two or more components (resources) to produce one or more products. n It is often the case that while transporting or storing the blending components, the components must share a pipeline or storage tank. n In this case, the components are called pooled components.

42. 42 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Blending: The Pooling Problem n Two types of decisions arise: n What should the proportions be for the components that are to be pooled? n How much of the pooled and non-pooled components will be used to make each of the final products?

43. 43 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem Grand Strand refinery wants to refine three petroleum components into regular and premium gasoline in order to maximize total profit contribution. Components 1 and 2 are pooled in a single storage tank. Component 3 has its own storage tank. The maximum number of gallons available for the three components are 5000, 10,000, and 10,000, respectively. The three components cost $2.50, $2.60, and $2.84, respectively. Regular gasoline sells for $2.90 and premium sells for $3.00. At least 10,000 gallons of regular gasoline must be produced. The product specifications for regular and premium gasoline are shown on the next slide.

44. 44 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem Product Specifications Regular gasolineAt most 30% component 1Regular gasolineAt most 30% component 1 At least 40% component 2 At most 20% component 3 Premium gasolineAt least 25% component 1Premium gasolineAt least 25% component 1 At most 45% component 2 At least 30% component 3

45. 45 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem n Define the 6 Decision Variables y 1 = gallons of component 1 in the pooling tank y 1 = gallons of component 1 in the pooling tank y 2 = gallons of component 2 in the pooling tank y 2 = gallons of component 2 in the pooling tank x pr = gallons of pooled components 1 and 2 in regular gas x pp = gallons of pooled components 1 and 2 in premium gas x 3 r = gallons of component 3 in regular gasoline x 3 p = gallons of component 3 in premium gasoline

46. 46 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem n Define the Objective Function Maximize the total contribution to profit (which is revenue from selling regular and premium gasolines minus cost of buying components 1, 2, and 3): Max 2.90( x pr + x 3 r ) + 3.00( x pp + x 3 p ) – 2.50 y 1 – 2.60 y 2 – 2.84( x 3 r + x 3 p ) – 2.50 y 1 – 2.60 y 2 – 2.84( x 3 r + x 3 p ) (Note: x pr + x 3 r = gallons of regular gasoline sold, x pp + x 3 p = gallons of premium gasoline sold, x pp + x 3 p = gallons of premium gasoline sold, x 3 r + x 3 p = gallons of component 3 consumed) x 3 r + x 3 p = gallons of component 3 consumed)

47. 47 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem n Define the 11 Constraints Conservation equation: 1) y 1 + y 2 = x pr + x pp Component availability: 2)y 1 < 5,000 3) y 2 < 10,000 4) x 3 r + x 3 p < 10,000 4) x 3 r + x 3 p < 10,000 Minimum production of regular gasoline: 5) x pr + x 3 r > 10,000

48. 48 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem n Define the 11 Constraints (continued) Regular gasoline specifications: 6)7)8)

49. 49 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem n Define the 11 Constraints (continued) Premium gasoline specifications: 9) 9)10)11) Non-negativity: x pr, x pp, x 3 r, x 3 p, y 1, y 2 > 0

50. 50 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem n Computer Output Objective Function Value = 1045000.000 Objective Function Value = 1045000.000 Variable Value Reduced Cost Variable Value Reduced Cost x pr 8000.000 0.000 x pr 8000.000 0.000 x 3 r 2000.000 0.000 x 3 r 2000.000 0.000 x pp 6000.000 0.000 x pp 6000.000 0.000 x 3 p 2571.429 0.000 x 3 p 2571.429 0.000 y 1 5000.000 0.000 y 1 5000.000 0.000 y 2 9000.000 0.000 y 2 9000.000 0.000

51. 51 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem n Computer Output (continued) Constraint Slack/Surplus Dual Value Constraint Slack/Surplus Dual Value 1 0.000 1.412 1 0.000 1.412 2 1000.000 0.000 2 1000.000 0.000 3 5428.571 0.000 3 5428.571 0.000 4 0.000 -3.061 4 0.000 -3.061 5 142.857 0.000 5 142.857 0.000 6 1142.857 0.000 6 1142.857 0.000 7 0.000 0.229 7 0.000 0.229 8 0.000 -2.197 8 0.000 -2.197 9 0.000 0.865 9 0.000 0.865 10 0.000 0.000 10 0.000 0.000 11 0.000 -0.123 11 0.000 -0.123

52. 52 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Blending - The Pooling Problem Regular Gasoline Regular Gasoline Premium Gasoline Premium Gasoline Component 1 Component 1 2,857.143 (28.57%) 2,857.143 (28.57%) 2,142.857 (25.00%) 2,142.857 (25.00%) Component 2 Component 2 5,142.857 (51.43%) 5,142.857 (51.43%) 3,857.143 (45.00%) 3,857.143 (45.00%) Component 3 Component 3 2,000.000 (20.00%) 2,000.000 (20.00%) 2,571.429 (30.00%) 2,571.429 (30.00%) Total Total 10,000.000 (100.00%) 8,571.429 (100.00%) 8,571.429 (100.00%) n Solution

53. 53 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecasting Adoption of a New Product n Forecasting new adoptions (purchases) after a product introduction is an important marketing problem. n We introduce here a forecasting model developed by Frank Bass. n Nonlinear programming is used to estimate the parameters of the Bass forecasting model.

54. 54 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecasting Adoption of a New Product n The Bass model has three parameters that must be estimated. m is the number of people estimated to eventually adopt a new product m is the number of people estimated to eventually adopt a new product q is the coefficient of imitation which measures the likelihood of adoption due to a potential adopter influenced by someone who has already adopted the product q is the coefficient of imitation which measures the likelihood of adoption due to a potential adopter influenced by someone who has already adopted the product p is the coefficient of imitation which measures the likelihood of adoption assuming no influence from someone who has already adopted the product. p is the coefficient of imitation which measures the likelihood of adoption assuming no influence from someone who has already adopted the product.

55. 55 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecasting Adoption of a New Product n Developing the Forecasting Model F t, the forecast of the number of new adopters during time period t, is F t, the forecast of the number of new adopters during time period t, is F t = (likelihood of a new adoption in time period t ) F t = (likelihood of a new adoption in time period t ) x (number of potential adopters remaining at x (number of potential adopters remaining at the end of time period t – 1) the end of time period t – 1)

56. 56 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Developing the Forecasting Model Essentially, the likelihood of a new adoption is the likelihood of adoption due to innovation plus the likelihood of adoption due to imitation.Essentially, the likelihood of a new adoption is the likelihood of adoption due to innovation plus the likelihood of adoption due to imitation. Let C t  1 denote the number of people who have adopted the product up to time t  1.Let C t  1 denote the number of people who have adopted the product up to time t  1. Hence, C t  1 / m is the fraction of the number of people estimated to adopt the product by time t – 1.Hence, C t  1 / m is the fraction of the number of people estimated to adopt the product by time t – 1. The likelihood of adoption due to imitation is q ( C t  1 / m ).The likelihood of adoption due to imitation is q ( C t  1 / m ). The likelihood of adoption due to innovation and imitation is p + q ( C t  1 / m ).The likelihood of adoption due to innovation and imitation is p + q ( C t  1 / m ). Forecasting Adoption of a New Product

57. 57 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Developing the Forecasting Model The number of potential adopters remaining at the end of time period t – 1 is m  C t  1.The number of potential adopters remaining at the end of time period t – 1 is m  C t  1. Hence, the complete forecast model is given byHence, the complete forecast model is given by F t = ( p + q ( C t  1 / m )) ( m  C t  1 ) F t = ( p + q ( C t  1 / m )) ( m  C t  1 ) Forecasting Adoption of a New Product

58. 58 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Nonlinear Optimization Problem Formulation F t = ( p + q ( C t  1 / m )) ( m  C t  1 ), t = 1, …., N F t = ( p + q ( C t  1 / m )) ( m  C t  1 ), t = 1, …., N E t = F t  S t, t = 1, …., N E t = F t  S t, t = 1, …., N where N = number of time periods of data available E t = forecast error for time period t E t = forecast error for time period t S t = actual number of adopters (or a multiple of S t = actual number of adopters (or a multiple of that number such as sales) in time period t that number such as sales) in time period t Forecasting Adoption of a New Product

59. 59 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Forecasting New-Product Adoption n Maid For You Maid For You is a residential cleaning service firm that has been quite successful developing a client base in the Chicago area. The firm plans to expand to other major metropolitan areas during the next few years. Maid For You would like to use its Chicago subscription data (on the next slide) to develop a model for forecasting service subscriptions in regions where it might expand. The first step is to estimate values for p (coefficient of innovation) and q (coefficient of imitation).

60. 60 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Forecasting New-Product Adoption n Subscribers and Cumulative Subscribers (1000s) Month Subscribers S t Cum. Subscribers C t Month Subscribers S t Cum. Subscribers C t 1 0.53 0.53 1 0.53 0.53 2 2.94 3.47 2 2.94 3.47 3 3.60 7.07 3 3.60 7.07 4 4.8511.92 4 4.8511.92 5 3.4415.36 5 3.4415.36 6 2.7618.12 6 2.7618.12 7 1.8219.94 7 1.8219.94 8 0.9320.87 8 0.9320.87 9 0.6121.48 9 0.6121.48

61. 61 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Define the Objective Function Minimize the sum of the squared forecast errors: Minimize the sum of the squared forecast errors: Example: Forecasting New-Product Adoption

62. 62 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Forecasting New-Product Adoption n Define the Constraints Define the forecast for each time period: 1) F 1 = pm 2)F 2 = ( p + q ( 0.53/ m )) ( m – 0.53) 3) F 3 = ( p + q ( 3.47/ m )) ( m – 3.47) 4) F 4 = ( p + q ( 7.07/ m )) ( m – 7.07) 4) F 4 = ( p + q ( 7.07/ m )) ( m – 7.07) 5) F 5 = ( p + q (11.92/ m )) ( m – 11.92) 6) F 6 = ( p + q (15.36/ m )) ( m – 15.36) 7) F 7 = ( p + q (18.12/ m )) ( m – 18.12) 8) F 8 = ( p + q (19.94/ m )) ( m – 19.94) 9) F 9 = ( p + q (20.87/ m )) ( m – 20.87)

63. 63 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Forecasting New-Product Adoption n Define the Constraints (continued) Define the forecast error for each time period: 10) E 1 = F 1 – 0.53 11) E 2 = F 2 – 2.94 12) E 3 = F 3 – 3.60 13) E 4 = F 4 – 4.85 13) E 4 = F 4 – 4.85 14) E 5 = F 5 – 3.44 15) E 6 = F 6 – 2.76 16) E 7 = F 7 – 1.82 17) E 8 = F 8 – 0.93 18) E 9 = F 9 – 0.61

64. 64 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Forecasting New-Product Adoption n Optimal Forecast Parameter Values Parameter Value Parameter Value p 0.08 p 0.08 q 0.62 q 0.62 m 21.26 m 21.26 The value of the imitation parameter q =.62 is The value of the imitation parameter q =.62 is substantially larger than the value of the innovation parameter p =.08. Subscriptions gain momentum over time due mainly to very favorable word-of- mouth.

65. 65 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Optimal Solution Example: Forecasting New-Product Adoption Month Forecast Subscribers Error 11.770.53 1.24 11.770.53 1.24 22.052.94-0.89 22.052.94-0.89 33.293.60-0.31 33.293.60-0.31 44.124.85-0.73 44.124.85-0.73 54.033.44 0.59 54.033.44 0.59 63.142.76 0.38 63.142.76 0.38 71.931.82 0.11 71.931.82 0.11 80.880.93-0.05 80.880.93-0.05 90.270.61-0.34 90.270.61-0.34

66. 66 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Subscribers versus Forecasts Example: Forecasting New-Product Adoption Subscribers Month Subscribers (1000s) 54321 1 2 3 4 5 6 7 8 9 Forecast

67. 67 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 8