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Chapter 11 Nonlinear Programming

Introduction to Management Science 8th Edition by Bernard W. Taylor III. Chapter 11 Nonlinear Programming. Chapter Topics. Nonlinear Profit Analysis Constrained Optimization Solution of Nonlinear Programming Problems with Excel A Nonlinear Programming Model with Multiple Constraints

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Chapter 11 Nonlinear Programming

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  1. Introduction to Management Science 8th Edition by Bernard W. Taylor III Chapter 11 Nonlinear Programming Chapter 11 - Nonlinear Programming

  2. Chapter Topics • Nonlinear Profit Analysis • Constrained Optimization • Solution of Nonlinear Programming Problems with Excel • A Nonlinear Programming Model with Multiple Constraints • Nonlinear Model Examples Chapter 11 - Nonlinear Programming

  3. Overview • Many business problems can be modeled only with nonlinear functions. • Problems that fit the general linear programming format but contain nonlinear functions are termed nonlinear programming (NLP) problems. • Solution methods are more complex than linear programming methods. • Often difficult, if not impossible, to determine optimal solution. • Solution techniques generally involve searching a solution surface for high or low points requiring the use of advanced mathematics. • Computer techniques (Excel) are used in this chapter. Chapter 11 - Nonlinear Programming

  4. Optimal Value of a Single Nonlinear Function Basic Model Profit function, Z, with volume independent of price: Z = vp - cf - vcv where v = sales volume p = price cf = fixed cost cv = variable cost Add volume/price relationship: v = 1,500 - 24.6p Figure 11.1 Linear Relationship of Volume to Price Chapter 11 - Nonlinear Programming

  5. Optimal Value of a Single Nonlinear Function Expanding the Basic Model to a Nonlinear Model With fixed cost (cf = $10,000) and variable cost (cv = $8): Z = 1,696.8p - 24.6p2 - 22,000 Figure 11.2 The Nonlinear Profit Function Chapter 11 - Nonlinear Programming

  6. Optimal Value of a Single Nonlinear Function Maximum Point on a Curve • The slope of a curve at any point is equal to the derivative of the curve’s function. • The slope of a curve at its highest point equals zero. Figure 11.3 Maximum profit for the profit function Chapter 11 - Nonlinear Programming

  7. Optimal Value of a Single Nonlinear Function Solution Using Calculus nonlinearity! Z = 1,696.8p - 24.6p2 - 22,000 dZ/dp = 1,696.8 - 49.2p p = $34.49 v = 1,500 - 24.6p v = 651.6 pairs of jeans Z = $7,259.45 Figure 11.4 Maximum Profit, Optimal Price, and Optimal Volume Chapter 11 - Nonlinear Programming

  8. Constrained Optimization in Nonlinear Problems Definition • If a nonlinear problem contains one or more constraints it becomes a constrained optimization model or a nonlinear programming model. • A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s) are nonlinear. • Solution procedures are much more complex andno guaranteed procedure exists. Chapter 11 - Nonlinear Programming

  9. Constrained Optimization in Nonlinear Problems Graphical Interpretation (1 of 3) • Effect of adding constraints to nonlinear problem: Figure 11.5 Nonlinear Profit Curve for the Profit Analysis Model Chapter 11 - Nonlinear Programming

  10. Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3) Figure 11.6 A Constrained Optimization Model Chapter 11 - Nonlinear Programming

  11. Constrained Optimization in Nonlinear Problems Graphical Interpretation (3 of 3) Figure 11.7 A Constrained Optimization Model with a Solution Point Not on the Constraint Boundary Chapter 11 - Nonlinear Programming

  12. Constrained Optimization in Nonlinear Problems Characteristics • Unlike linear programming, solution is often not on the boundary of the feasible solution space. • Cannot simply look at points on the solution space boundary but must consider other points on the surface of the objective function. • This greatly complicates solution approaches. • Solution techniques can be very complex. Chapter 11 - Nonlinear Programming

  13. Western Clothing Problem Solution Using Excel (1 of 3) Exhibit 11.1 Chapter 11 - Nonlinear Programming

  14. Western Clothing Problem Solution Using Excel (2 of 3) Exhibit 11.2 Chapter 11 - Nonlinear Programming

  15. Western Clothing Problem Solution Using Excel (3 of 3) Exhibit 11.3 Chapter 11 - Nonlinear Programming

  16. Beaver Creek Pottery Company Problem Solution Using Excel (1 of 6) Maximize Z = $(4 - 0.1x1)x1 + (5 - 0.2x2)x2 subject to: x1 + x2 = 40 Where: x1 = number of bowls produced x2 = number of mugs produced Chapter 11 - Nonlinear Programming

  17. Beaver Creek Pottery Company Problem Solution Using Excel (2 of 6) Exhibit 11.4 Chapter 11 - Nonlinear Programming

  18. Beaver Creek Pottery Company Problem Solution Using Excel (3 of 6) Exhibit 11.5 Chapter 11 - Nonlinear Programming

  19. Beaver Creek Pottery Company Problem Solution Using Excel (4 of 6) Exhibit 11.6 Chapter 11 - Nonlinear Programming

  20. Beaver Creek Pottery Company Problem Solution Using Excel (5 of 6) Exhibit 11.7 Chapter 11 - Nonlinear Programming

  21. Beaver Creek Pottery Company Problem Solution Using Excel (6 of 6) Exhibit 11.8 Chapter 11 - Nonlinear Programming

  22. Western Clothing Company Problem Solution Using Excel (1 of 4) Maximize Z = (p1 - 12)x1 + (p2 - 9)x2 subject to: 2x1 + 2.7x2 6,00 3.6x1 + 2.9x2 8,500 7.2x1 + 8.5x2 15,000 where: x1 = 1,500 - 24.6p1 x2 = 2,700 - 63.8p p1 = price of designer jeans p2 = price of straight jeans Chapter 11 - Nonlinear Programming

  23. Western Clothing Company Problem Solution Using Excel (2 of 4) Exhibit 11.9 Chapter 11 - Nonlinear Programming

  24. Western Clothing Company Problem Solution Using Excel (3 of 4) Exhibit 11.10 Chapter 11 - Nonlinear Programming

  25. Western Clothing Company Problem Solution Using Excel (4 of 4) Exhibit 11.11 Chapter 11 - Nonlinear Programming

  26. Facility Location Example Problem Problem Definition and Data (1 of 2) Centrally locate a facility that serves several customers or other facilities in order to minimize distance or miles traveled (d) between facility and customers. di = sqrt((xi - x)2 + (yi - y)2) (= straight-line distance) Where: (x,y) = coordinates of proposed facility (xi,yi) = coordinates of customer or location facility i Minimize total miles d =  diti Where: di = distance to town i ti =annual trips to town i Chapter 11 - Nonlinear Programming

  27. Facility Location Example Problem Problem Definition and Data (2 of 2) Chapter 11 - Nonlinear Programming

  28. Facility Location Example Problem Solution Using Excel Exhibit 11.12 Chapter 11 - Nonlinear Programming

  29. Facility Location Example Problem Solution Map Figure 11.8 Rescue Squad Facility Location Chapter 11 - Nonlinear Programming

  30. Investment Portfolio Selection Example Problem Definition and Model Formulation (1 of 2) • Objective of the portfolio selection model is to minimize some measure of portfolio risk (variance in the return on investment) while achieving some specified minimum return on the total portfolio investment. • Since variance is the sum of squares of differences, it is mathematically identical to the “straight-line distance”! Thus, it is possible to visualize variances as such distances, and minimizing the overall variance is then mathematically identical to minimizing such distances. Chapter 11 - Nonlinear Programming

  31. Investment Portfolio Selection Example Problem Definition and Model Formulation (2 of 2) Minimize S = x12s12 + x22s22 + … +xn2sn2 + xixjrijsisj where: S = variance of annual return of the portfolio xi,xj = the proportion of money invested in investments i or j si2 = the variance for investment i rij = the correlation between returns on investments i and j si,sj = the std. dev. of returns for investments i and j subject to: r1x1 + r2x2 + … + rnxn rm x1 + x2 + …xn = 1.0 where: ri = expected annual return on investment i rm = the minimum desired annual return from the portfolio “straight-line distance” Chapter 11 - Nonlinear Programming

  32. Investment Portfolio Selection Example Problem Solution Using Excel (1 of 5) Four stocks, desired annual return of at least 0.11. Minimize Z = S = xA2(.009) + xB2(.015) + xC2(.040) + XD2(.023) + xAxB (.4)(.009)1/2(0.015)1/2 + xAxC(.3)(.009)1/2(.040)1/2 + xAxD(.6)(.009)1/2(.023)1/2 + xBxC(.2)(.015)1/2(.040)1/2 + xBxD(.7)(.015)1/2(.023)1/2 + xCxD(.4)(.040)1/2(.023)1/2 + xBxA(.4)(.015)1/2(.009)1/2 + xCxA(.3)(.040)1/2 + (.009)1/2 + xDxA(.6)(.023)1/2(.009)1/2 + xCxB(.2)(.040)1/2(.015)1/2 + xDxB(.7)(.023)1/2(.015)1/2 + xDxC (.4)(.023)1/2(.040)1/2 subject to: .08x1 + .09x2 + .16x3 + .12x4 0.11 x1 + x2 + x3 + x4 = 1.00 xi 0 Chapter 11 - Nonlinear Programming

  33. Investment Portfolio Selection Example Problem Solution Using Excel (2 of 5) Chapter 11 - Nonlinear Programming

  34. Investment Portfolio Selection Example Problem Solution Using Excel (3 of 5) Exhibit 11.13 Chapter 11 - Nonlinear Programming

  35. Investment Portfolio Selection Example Problem Solution Using Excel (4 of 5) Exhibit 11.14 Chapter 11 - Nonlinear Programming

  36. Investment Portfolio Selection Example Problem Solution Using Excel (5 of 5) Exhibit 11.15 Chapter 11 - Nonlinear Programming

  37. Hickory Cabinet and Furniture Company Example Problem and Solution (1 of 2) Model: Maximize Z = $280x1 - 6x12 + 160x2 - 3x22 subject to: 20x1 + 10x2 = 800 board ft. Where: x1 = number of chairs x2 = number of tables Chapter 11 - Nonlinear Programming

  38. Hickory Cabinet and Furniture Company Example Problem and Solution (2 of 2) Chapter 11 - Nonlinear Programming

  39. Chapter 11 - Nonlinear Programming

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