1. 10-1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Nonlinear Programming Chapter 10

2. 10-2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■Nonlinear Profit Analysis ■Constrained Optimization ■Solution of Nonlinear Programming Problems with Excel ■Nonlinear Programming Model with Multiple Constraints ■Nonlinear Model Examples Chapter Topics

3. 10-3 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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. Overview

4. 10-4 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Profit function, Z, with volume independent of price: Z = vp - c f - vc v where v = sales volume p = price c f = unit fixed cost c v = unit variable cost Add volume/price relationship: v = 1,500 - 24.6p Nonlinear Profit Analysis Optimal Value of a Single Nonlinear Function Figure 10.1 Linear relationship of volume to price

5. 10-5 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall With fixed cost (c f = $10,000) and variable cost (c v = $8): Profit, Z = 1,696.8p - 24.6p 2 - 22,000 Optimal Value of a Single Nonlinear Function Figure 10.2 The nonlinear profit function

6. 10-6 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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 10.3 Maximum profit for the profit function Optimal Value of a Single Nonlinear Function Maximum Point on a Curve

7. 10-7 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 10.4 Maximum profit, optimal price and optimal volume Optimal Value of a Single Nonlinear Function Solution Using Calculus Z = 1,696.8p - 24.6p 2 - 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 Z = $7,259.45

8. 10-8 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■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. Constrained Optimization in Nonlinear Problems Definition

9. 10-9 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Effect of adding constraints to nonlinear problem: Figure 10.5 Nonlinear profit curve for the profit analysis model Constrained Optimization in Nonlinear Problems Graphical Interpretation (1 of 3)

10. 10-10 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 10.6 A constrained optimization model Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3)

11. 10-11 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 10.7 A constrained optimization model with a solution point not on the constraint boundary Constrained Optimization in Nonlinear Problems Graphical Interpretation (3 of 3)

12. 10-12 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■Unlike linear programming, the 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. Constrained Optimization in Nonlinear Problems Characteristics

13. 10-13 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.1 Western Clothing Problem Solution Using Excel (1 of 3) Formula for profit =1500-24.6*C5

14. 10-14 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.2 Western Clothing Problem Solution Using Excel (2 of 3) Click on “GRG Nonlinear”

15. 10-15 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.3 Western Clothing Problem Solution Using Excel (3 of 3)

16. 10-16 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Maximize Z = $(4 - 0.1x 1 )x 1 + (5 - 0.2x 2 )x 2 subject to: x 1 + 2x 2 = 40 Where: x 1 = number of bowls produced x 2 = number of mugs produced 4 – 0.1X 1 = profit ($) per bowl 5 – 0.2X 2 = profit ($) per mug Beaver Creek Pottery Company Problem Solution Using Excel (1 of 6)

17. 10-17 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.4 Beaver Creek Pottery Company Problem Solution Using Excel (2 of 6) =C5+2*C6 =SUMPRODUCT (C5:C6,D5:D6)

18. 10-18 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.5 Beaver Creek Pottery Company Problem Solution Using Excel (3 of 6)

19. 10-19 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.6 Beaver Creek Pottery Company Problem Solution Using Excel (4 of 6)

20. 10-20 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.7 Beaver Creek Pottery Company Problem Solution Using Excel (5 of 6) Select “Sensitivity”

21. 10-21 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.8 Beaver Creek Pottery Company Problem Solution Using Excel (6 of 6) Lagrange multiplier for labor

22. 10-22 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Maximize Z = (p 1 - 12)x 1 + (p 2 - 9)x 2 subject to: 2x 1 + 2.7x 2  6,000 3.6x 1 + 2.9x 2  8,500 7.2x 1 + 8.5x 2  15,000 where: x 1 = 1,500 - 24.6p 1 x 2 = 2,700 - 63.8p 2 p 1 = price of designer jeans p 2 = price of straight jeans Western Clothing Company Problem Solution Using Excel (1 of 4)

23. 10-23 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.9 Western Clothing Company Problem Solution Using Excel (2 of 4) =D5-12 =SUMPRODUCT (C5:C6,E5:E6) =2*C5+2.7*C6

24. 10-24 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.10 Western Clothing Company Problem Solution Using Excel (3 of 4)

25. 10-25 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.11 Western Clothing Company Problem Solution Using Excel (4 of 4)

26. 10-26 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Centrally locate a facility that serves several customers or other facilities in order to minimize distance or miles traveled (d) between facility and customers. Where: (x,y) = coordinates of proposed facility (x i,y i ) = coordinates of customer or location facility i Minimize total miles d =  d i t i Where: d i = distance to town i t i =annual trips to town i Facility Location Example Problem Problem Definition and Data (1 of 2)

27. 10-27 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Facility Location Example Problem Problem Definition and Data (2 of 2)

28. 10-28 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.12 Facility Location Example Problem Solution Using Excel =SQRT((B6-C14)^2+(C6-C15)^2)

29. 10-29 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Figure 10.8 Rescue squad facility location Facility Location Example Problem Solution Map

30. 10-30 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 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.

31. 10-31 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Minimize S = x 1 2 s 1 2 + x 2 2 s 2 2 + … +x n 2 s n 2 +  x i x j r ij s i s j where: S = variance of annual return of the portfolio x i,x j = the proportion of money invested in investments i or j s i 2 = the variance for investment i r ij = the correlation between returns on investments i and j s i,s j = the std. dev. of returns for investments i and j subject to: r 1 x 1 + r 2 x 2 + … + r n x n  r m x 1 + x 2 + …x n = 1.0 where: r i = expected annual return on investment i r m = the minimum desired annual return from the portfolio Investment Portfolio Selection Example Problem Definition and Model Formulation (2 of 2) i≠j

32. 10-32 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Investment Portfolio Selection Example Problem Solution Using Excel (1 of 5)

33. 10-33 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Four stocks, desired annual return of at least 0.11. Minimize Z = S = x 1 2 (.009) + x 2 2 (.015) + x 3 2 (.040) + X 4 2 (.023) + x 1 x 2 (.4)(.009) 1/2 (0.015) 1/2 + x 1 x 3 (.3)(.009) 1/2 (.040) 1/2 + x 1 x 4 (.6)(.009) 1/2 (.023) 1/2 + x 2 x 3 (.2)(.015) 1/2 (.040) 1/2 + x 2 x 4 (.7)(.015) 1/2 (.023) 1/2 + x 3 x 4 (.4)(.040) 1/2 (.023) 1/2 + x 2 x 1 (.4)(.015) 1/2 (.009) 1/2 + x 3 x 1 (.3)(.040) 1/2 + (.009) 1/2 + x 4 x 1 (.6)(.023) 1/2 (.009) 1/2 + x 3 x 2 (.2)(.040) 1/2 (.015) 1/2 + x 4 x 2 (.7)(.023) 1/2 (.015) 1/2 + x 4 x 3 (.4)(.023) 1/2 (.040) 1/2 subject to:.08x 1 +.09x 2 +.16x 3 +.12x 4  0.11 x 1 + x 2 + x 3 + x 4 = 1.00 x i  0 Investment Portfolio Selection Example Problem Solution Using Excel (2 of 5)

34. 10-34 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.13 Investment Portfolio Selection Example Problem Solution Using Excel (3 of 5) Doubling covariances will include all investment pairs =SUMPRODUCT9B6:B9,E6:E9) =SUM(E6:E9)

35. 10-35 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 10.14 Investment Portfolio Selection Example Problem Solution Using Excel (4 of 5) All money is invested constraint Investment return constraint

36. 10-36 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Investment Portfolio Selection Example Problem Solution Using Excel (5 of 5) Exhibit 10.15

37. 10-37 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall The Hickory Cabinet and Furniture Company makes chairs and tables. The company has developed the following nonlinear programming model to determine the optimal number of chairs and tables to produce each day to maximize profit. Determine the solution using Excel. Model: Maximize Z = $280x 1 - 6x 1 2 + 160x 2 - 3x 2 2 subject to: 20x 1 + 10x 2 = 800 board ft. Where: x 1 = number of chairs x 2 = number of tables Hickory Cabinet and Furniture Company Example Problem and Solution (1 of 2)

38. 10-38 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Hickory Cabinet and Furniture Company Example Problem and Solution (2 of 2)

39. 10-39 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.