Nonlinear Programming

Chapter 9Nonlinear ProgrammingPart 2 Deterministic Decision ModelsLearning ObjectivesExplain the difference between optimization problems that can be solved using linear programming methods and those that require nonlinear programming or calculus-based methods.Find the optimal values of the decision variable and the objective function in problems that involve one decision variable (unconstrained and constrained).Solve one-decision variable problems with a nonlinear objective function using Excel (unconstrained or constrained).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:2Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Learning Objectives (cont’d)Solve one-decision-variable unconstrained problems with a nonlinear objective function using Excel.Use the Lagrange multiplier to find the optimal values of two decision variables and the objective function in problems that involve equality constraints.Solve two-decision-variable problems with a nonlinear objective function and an equality constraint using Excel.Find the optimal values of the decision variables and the objective function in problems that involve two decision variables and one inequality constraint.Solve two-decision-variable problems with a nonlinear objective function and multiple constraints using Excel.After completing this chapter, you should be able to:3Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Nonlinear ProgrammingCharacteristics of Nonlinear ModelsHave 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).4Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–1 A U-Shaped Cost Function5Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–2 Illustrations of Local Maximum and Local Minimum Points6Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–2 Illustrations of Local Maximum and Local Minimum Points (cont’d)7Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–3 Volume–Price Relationship for the X-Tech Inc. Problem8Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–4 The Graph of the Profit Function for the X-Tech Inc. Problem9Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9-1 Worksheet for the One-Decision-Variable, UnconstrainedProblem (X-Tech Inc., Example 9-3)10Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9-2 Parameter Specification Screen for the One-Decision-Variable, Unconstrained Problem (X-Tech Inc., Example 9-3)11Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–5a The Optimal Point Lies within the Feasible Solution Space12Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–5b The Optimal Point is on the Boundary of the Feasible Solution Space13Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–6 The Global Maximum Can Be at the Point Where x = C14Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–7 The Graph of the Profit Function and the Feasible Space for the X-Tech Inc. Problem15Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–3 Worksheet for the One-Decision-Variable, ConstrainedProblem (X-Tech Inc., Example 9-5)16Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–4 Parameter Specification Screen for the One-Decision- Variable, Constrained Problem (X-Tech Inc. Example 9-5)17Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 9–8 The Graph of the Profit Function and the Feasible Space forthe X-Tech Inc. Problem18Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–5 Worksheet for the Two-Decision-Variable, UnconstrainedProblem (Example, 9-6, Imaging Equipment)19Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–6 Parameter Specification Screen for the Two-Decision-Variable, Unconstrained Problem (Example 9-6, Imaging Equipment)20Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–7 Worksheet for the Problem with Two Decision Variables and a Single Equality Constraint (Example 9-8, Daisy-Fresh Company Deodorant Problem)21Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–8 Parameter Specification Screen for Two Decision Variables and a Single Equality Constraint Problem (Example 9-8, Daisy-Fresh Co. Deodorant Problem)22Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–9 Worksheet for a Two-Decision-Variable, Multiple-Constraint Problem (Example 9-10, East-West Company Computer Storage Problem)23Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–10 Parameter Specification Screen for the Two-Decision-Variable, Multiple-Constraints Problem (Example 9-10, East-West Company Computer Storage Problem)24Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–11 Worksheet for Solved Problem 1: One-Decision-Variable, Unconstrained Problem25Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–12 Parameter Specification Screen for Solved Problem 1: One-Decision-Variable, Unconstrained Problem26Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–13 Worksheet for Solved Problem 2: One-Decision-Variable, Constrained Problem27Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–14 Parameter Specification Screen for Solved Problem 2:One-Decision-Variable, Unconstrained Problem28Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–15 Worksheet for Solved Problem 3: Two Decision Variables and a Single Equality Constraint29Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–16 Parameter Specification Screen for Solved Problem 3: Two Decision Variables and a Single Equality Constraint Problem30Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–17 Worksheet for Solved Problem 4: Two Decision Variables and a Single Inequality Constraint31Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 9–18 Parameter Specification Screen for Solved Problem 4: Two Decision Variables and a Single Inequality Constraint Problem32Copyright © 2007 The McGraw-Hill Companies. All rights reserved.