Nonlinear Programming Chapter 10 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10-1 Chapter Topics ■Nonlinear Profit Analysis ■Constrained Optimization ■Solution of Nonlinear Programming Problems with Excel ■Nonlinear Programming Model with Multiple Constraints ■Nonlinear Model Examples Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10-2 Overview ■ 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 ■ Determining an optimal solution is often difficult, if not impossible ■ Solution techniques generally involve searching a solution surface for high or low points requiring the use of advanced mathematics Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10-3 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 = unit fixed cost cv = unit variable cost Add volume/price relationship: Figure 10.1 Linear Relationship of v = 1,500 - 24.6p Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10-4 Optimal Value of a Single Nonlinear Function With fixed cost (cf = $10,000) and variable cost (cv = $8): Profit, Z = 1,696.8p - 24.6p2 - 22,000 Figure 10.2 The Nonlinear Profit Function Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10-5 Optimal Value of a Single Nonlinear Function Maximum Curve ■ The slope of aPoint curve aton anyapoint is equal to the derivative of the curve’s function ■ The slope of a curve at its highest point equals zero Figure 10.3 Maximum profit for the profit function Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10-6 Optimal Value of a Single Nonlinear Function Solution Using Calculus Z = 1,696.8p - 24.6p2 2,000 dZ/dp = 1,696.8 49.2p =0 p = 1696.8/49.2 = $34.49 v = 1,500 - 24.6p v = 651.6 pairs of jeans Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Figure 10.4 Z = $7,259.45 10-7 Constrained Optimization in Nonlinear Problems Definition ■ A nonlinear problem containing one or more constraints becomes a constrained optimization model or a nonlinear programming (NLP) 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 and no guaranteed procedure exists for all NLP models Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10-8 Constrained Optimization in Nonlinear Problems Graphical Interpretation (1 of 3) Effect of adding constraints to nonlinear problem: Figure 10.5 Nonlinear Profit Curve for the Profit Copyright © 2010 Pearson Education, Inc Publishing as Analysis Model Prentice Hall 10-9 Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3) Figure 10.6 A Constrained Copyright © 2010 Pearson Education, Inc Publishing asModel Optimization Prentice Hall 10- Western Clothing Company Problem Solution Using Excel (4 of 4) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 10.11 10- 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] Where: (x,y) = coordinates of proposed facility (xi,yi) = coordinates of customer or location facility i Minimize total miles d = Σ diti Where: di = distance to town i Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall t =annual trips to town i 10- Facility Location Example Problem Problem Definition and Data (2 of 2) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- Facility Location Example Problem Solution Using Excel Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 10.12 10- Facility Location Example Problem Solution Map Figure 10.8 Rescue Squad Facility Location Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- Investment Portfolio Selection Example Problem Definition and Model Formulation (1 of the portfolio selection model is to: ofObjective 2) ■ minimize some measure of portfolio risk (variance in the return on investment) ■ while achieving some specified minimum return on the total portfolio investment Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- Investment Portfolio Selection Example Problem Definition and Model Formulation (2 2 2 2 Minimize S = x1 s1 + x2 s2 + … +xn sn + Σxixjrijsisj of 2) i≠j 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 x©1 2010 +x …xn Inc = Publishing 1.0 as Copyright Pearson +Education, Prentice Hall 10- Investment Portfolio Selection Example Problem Solution Using Excel (1 of 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- Investment Portfolio Selection Example Problem Solution Using Excel (2 of 5) Four stocks, desired annual return of at least 0.11 Minimize Z = S = x12(.009) + x22(.015) + x32(.040) + X42(.023) + x1x2 (.4)(.009)1/2(0.015)1/2 + x1x3(.3)(.009)1/2(.040)1/2 + x1x4(.6) (.009)1/2(.023)1/2 + x2x3(.2)(.015)1/2(.040)1/2 + x2x4(.7)(.015)1/2(.023)1/2 + x3x4(.4)(.040)1/2(.023)1/2 + x2x1(.4)(.015)1/2(.009)1/2 + x3x1(.3)(.040)1/2 + (.009)1/2 + x4x1(.6)(.023)1/2(.009)1/2 + x3x2(.2) (.040)1/2(.015)1/2 + x4x2(.7)(.023)1/2(.015)1/2 + x4x3(.4)(.023)1/2(.040)1/2 subject to: Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- Investment Portfolio Selection Example Problem Solution Using Excel (3 of 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 10.13 10- Investment Portfolio Selection Example Problem Solution Using Excel (4 of 5) Exhibit 10.14 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- Investment Portfolio Selection Example Problem Solution Using Excel (5 of 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 10.15 10- 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- Hickory Cabinet and Furniture Company Example Problem and Solution (2 of 2) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10- ... total miles d = Σ diti Where: di = distance to town i Copyright © 2 010 Pearson Education, Inc Publishing as Prentice Hall t =annual trips to town i 10- Facility Location Example Problem Problem... Prentice Hall Exhibit 10. 4 10- Beaver Creek Pottery Company Problem Solution Using Excel (3 of 6) Copyright © 2 010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 10. 5 10- Beaver Creek... 2 010 Pearson Education, Inc Publishing as Prentice Hall 10- Western Clothing Problem Solution Using Excel (1 of 3) Copyright © 2 010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 10. 1