Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 9 Nonlinear Programming Part 2 Deterministic Decision Models

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–2 Learning Objectives 1.Explain the difference between optimization problems that can be solved using linear programming methods and those that require nonlinear programming or calculus-based methods. 2.Find the optimal values of the decision variable and the objective function in problems that involve one decision variable (unconstrained and constrained). 3.Solve one-decision variable problems with a nonlinear objective function using Excel (unconstrained or constrained). 4.Find the optimal values of the decision variables and the objective function in unconstrained problems that involve two decision variables. After completing this chapter, you should be able to:

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–3 Learning Objectives (cont’d) 5.Solve one-decision-variable unconstrained problems with a nonlinear objective function using Excel. 6.Use the Lagrange multiplier to find the optimal values of two decision variables and the objective function in problems that involve equality constraints. 7.Solve two-decision-variable problems with a nonlinear objective function and an equality constraint using Excel. 8.Find the optimal values of the decision variables and the objective function in problems that involve two decision variables and one inequality constraint. 9.Solve two-decision-variable problems with a nonlinear objective function and multiple constraints using Excel. After completing this chapter, you should be able to:

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–4 Nonlinear Programming Characteristics of Nonlinear Models –Have one or more nonlinear components which cannot be handled by linear programming techniques. –Require nonlinear programming procedures which involves obtaining the first derivative of the objective function, finding all solutions for which the first derivative is equal to zero, and then checking second derivative conditions to ascertain the nature of the zero points (e.g., a local maximum or a local minimum).

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–5 Figure 9–1A U-Shaped Cost Function

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–6 Figure 9–2Illustrations of Local Maximum and Local Minimum Points

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–7 Figure 9–2Illustrations of Local Maximum and Local Minimum Points (cont’d)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–8 Figure 9–3Volume–Price Relationship for the X-Tech Inc. Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–9 Figure 9–4The Graph of the Profit Function for the X-Tech Inc. Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–10 Exhibit 9-1Worksheet for the One-Decision-Variable, Unconstrained Problem (X-Tech Inc., Example 9-3)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–11 Exhibit 9-2Parameter Specification Screen for the One-Decision- Variable, Unconstrained Problem (X-Tech Inc., Example 9-3)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–12 Figure 9–5aThe Optimal Point Lies within the Feasible Solution Space

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–13 Figure 9–5bThe Optimal Point is on the Boundary of the Feasible Solution Space

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–14 Figure 9–6The Global Maximum Can Be at the Point Where x = C

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–15 Figure 9–7The Graph of the Profit Function and the Feasible Space for the X-Tech Inc. Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–16 Exhibit 9–3Worksheet for the One-Decision-Variable, Constrained Problem (X-Tech Inc., Example 9-5)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–17 Exhibit 9–4Parameter Specification Screen for the One-Decision- Variable, Constrained Problem (X-Tech Inc. Example 9-5)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–18 Figure 9–8The Graph of the Profit Function and the Feasible Space for the X-Tech Inc. Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–19 Exhibit 9–5Worksheet for the Two-Decision-Variable, Unconstrained Problem (Example, 9-6, Imaging Equipment)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–20 Exhibit 9–6Parameter Specification Screen for the Two-Decision- Variable, Unconstrained Problem (Example 9-6, Imaging Equipment)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–21 Exhibit 9–7Worksheet for the Problem with Two Decision Variables and a Single Equality Constraint (Example 9-8, Daisy-Fresh Company Deodorant Problem)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–22 Exhibit 9–8Parameter Specification Screen for Two Decision Variables and a Single Equality Constraint Problem (Example 9-8, Daisy-Fresh Co. Deodorant Problem)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–23 Exhibit 9–9Worksheet for a Two-Decision-Variable, Multiple-Constraint Problem (Example 9-10, East-West Company Computer Storage Problem)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–24 Exhibit 9–10Parameter Specification Screen for the Two-Decision- Variable, Multiple-Constraints Problem (Example 9-10, East- West Company Computer Storage Problem)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–25 Exhibit 9–11Worksheet for Solved Problem 1: One-Decision-Variable, Unconstrained Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–26 Exhibit 9–12Parameter Specification Screen for Solved Problem 1: One- Decision-Variable, Unconstrained Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–27 Exhibit 9–13Worksheet for Solved Problem 2: One-Decision-Variable, Constrained Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–28 Exhibit 9–14Parameter Specification Screen for Solved Problem 2: One-Decision-Variable, Unconstrained Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–29 Exhibit 9–15Worksheet for Solved Problem 3: Two Decision Variables and a Single Equality Constraint

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–30 Exhibit 9–16Parameter Specification Screen for Solved Problem 3: Two Decision Variables and a Single Equality Constraint Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–31 Exhibit 9–17Worksheet for Solved Problem 4: Two Decision Variables and a Single Inequality Constraint

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 9–32 Exhibit 9–18Parameter Specification Screen for Solved Problem 4: Two Decision Variables and a Single Inequality Constraint Problem