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 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. n For instance, 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. Par, Inc. manufactures golf bags Par, Inc. manufactures golf bags Two Models: Standard (S) and Deluxe (D) Two Models: Standard (S) and Deluxe (D) n Four Operations Required for each Bag Cutting and Dyeing Cutting and Dyeing Sewing Sewing Finishing Finishing Inspection and Packaging Inspection and Packaging Example: Production Application n Par, Inc.

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. Example: Production Application n Assume that the demand for Standard (S) and Deluxe (D) golf bags are (projected sales – cost differential): S = 2250 – 15Ps S = 2250 – 15Ps D = 1500 – 5Pd D = 1500 – 5Pd where Ps = the price of a Standard bag Pd = the price of a Deluxe bag. Pd = the price of a Deluxe bag. We will need to isolate Ps and Pd: - 15Ps = 2250 – S5Pd = 1500 – D - Ps = 2250/15 –S/15Pd = 1500/5 – 1/5D - Ps = 150 – 1/15SPd = 300 – 1/5D

7. Example: Production Application n Profit Contribution as a Function of Demand n The profit contributions (revenue – cost) are: PsS – 70S (Standard bags) PsS – 70S (Standard bags) PdD – 150D (Deluxe bags) PdD – 150D (Deluxe bags) Solving for Ps we get:Solving for Ps we get: PsS – 70SPsS – 70S (150 – 1/15S)S – 70S(150 – 1/15S)S – 70S 80S – 1/15S^280S – 1/15S^2 Solving for Pd we getSolving for Pd we get PdD – 150DPdD – 150D (300 – 1/5D)D – 150D(300 – 1/5D)D – 150D 150D – 1/5D^2150D – 1/5D^2

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. Example: Production Application n Total Profit Contribution Total Profit Contrib. = 80S – 1/15S^2 + 150D – 1/5D^2 Total Profit Contrib. = 80S – 1/15S^2 + 150D – 1/5D^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.

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. Par Inc, Unconstrained Solution n If we were to just solve the optimization equation, then we would find that it is: n S = 600, D = 375, Ps = 110, Pd = 225 n BUT WE HAVENT INCLUDED THE CONSTRAINTS! 7/10s + 1d <= 630 (Cutting and Dyeing)7/10s + 1d <= 630 (Cutting and Dyeing) 1/2s + 5/6d <= 600 (Sewing)1/2s + 5/6d <= 600 (Sewing) 1s + 2/3d <= 708 (Finishing)1s + 2/3d <= 708 (Finishing) 1/10s + 1/4d <= 135 (Inspecting and Packing)1/10s + 1/4d <= 135 (Inspecting and Packing) s, d >= 0s, d >= 0

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. Par Inc

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. Par Inc

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. Par Inc

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. 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.

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. 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.

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. 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?

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: 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.

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. Grand Strand

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. 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

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. 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

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. 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)

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. Example: Blending - The Pooling Problem n Define the 11 Constraints Components 1 and 2 consumed Similar to a “flow-in-flow-out” constraint: 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

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. Example: Blending - The Pooling Problem n Define the 11 Constraints (continued) Regular gasoline specifications: We have to measure the proportion of a component in the pool when calculating the specification constraints 6)7)8)

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. 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

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. Example: Blending - The Pooling Problem

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. Example: Blending - The Pooling Problem n Solution (with pooling optimal sol. = $5831.43 n Without Pooling optimal sol. = $7100

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. 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.

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. 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.

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. 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)

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. 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

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. 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

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. 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

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: 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).

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. 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

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. Forecasting (General Form)

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. 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

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: 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)

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. 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

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. 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.

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 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

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. n Subscribers versus Forecasts Example: Forecasting New-Product Adoption Subscribers Month Subscribers (1000s) 54321 1 2 3 4 5 6 7 8 9 Forecast

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. End of Chapter 8