1. Reid & Sanders, Operations Management © Wiley 2002 Linear Programming B SUPPLEMENT

2. Reid & Sanders, Operations Management © Wiley 2002 Page 2 Learning Objectives Understand the role of mathematical modeling Describe constrained optimization models Understand the advantages & disadvantages of LPs Know the assumptions underlying LPs Formulate linear programs Describe the geometry of linear programs Prepare graphical LP solutions

3. Reid & Sanders, Operations Management © Wiley 2002 Page 3 Mathematical Models A model represents the essential features of an object, system or problem without unimportant details Mathematical models are cheaper, faster and safer than constructing and manipulating real systems.

4. Reid & Sanders, Operations Management © Wiley 2002 Page 4 Constrained Optimization Constrained optimization models find the best solution according to a pre- established evaluation criteria

5. Reid & Sanders, Operations Management © Wiley 2002 Page 5 Model Components Constrained optimization models consist of three types of components: –Decision variables: Physical quantities controlled by the decision maker –Objective function: The evaluation criteria (often maximizing profit or minimizing cost) –Constraints: Physical, economic, technical, or other limits on the numeric value assigned to the decision variable

6. Reid & Sanders, Operations Management © Wiley 2002 Page 6 Advantages of Math Models Provides structure: –Modeling acts as a way of organizing and clarifying the thought process Increases objectivity: –All assumptions and criteria must be clearly specified Makes complex problems tractable Leverages computer power Facilitates ‘what if’ analysis: –Clarifies how sensitive the solution is to the assumptions underlying the model

7. Reid & Sanders, Operations Management © Wiley 2002 Page 7 Disadvantages of Math Models Potential for poor modeling: –If an important decision variable or relationship is omitted, the model may not reflect the actual situation under study Math models only consider quantitative issues: –Don’t ignore or forget potential qualitative issues that don’t lend themselves to math modeling

8. Reid & Sanders, Operations Management © Wiley 2002 Page 8 Linear Programs A linear program must satisfy three requirements: –The decision variables must be continuous. –The objective function must be linear. –The left-hand sides of the constraints must be linear functions.

9. Reid & Sanders, Operations Management © Wiley 2002 Page 9 LP Form

10. Reid & Sanders, Operations Management © Wiley 2002 Page 10 Step-by-Step Identify and define the decision variables for the problem Define the objective function Identify and express mathematically all of the relevant constraints

11. Reid & Sanders, Operations Management © Wiley 2002 Page 11 Terminology A solution: –Any set of numeric values for all of the variables in an LP A feasible solution: –A solution that satisfies all of the constraints The feasible set: –The set containing all of the feasible solutions The optimal solution: –The feasible solution that provides the best numeric value for the objective function

12. Reid & Sanders, Operations Management © Wiley 2002 Page 12 Geometry of LP The characteristic that makes LPs easy to solve is their geometric structure. Extreme points are formed by the intersection of two or more constraints. These extreme points form the corners of the feasible set. If a finite optimal solution exists, it will be found at least one of the extreme points.

13. Reid & Sanders, Operations Management © Wiley 2002 Page 13 Special Issues Multiple optima: –Optimal solutions may not be unique –Occasionally two or more adjacent extreme points tie (then the entire edge between them is optimal) Infeasible problems: –Sometimes all constraints cannot be satisfied simultaneously (the modeler needs to identify a constraint that may be relaxed) Unbounded problems: –If the objective function approaches positive infinity, the modeler has probably forgot to include a relevant constraint

14. Reid & Sanders, Operations Management © Wiley 2002 Page 14 Solution Algorithms The Simplex Method developed by George Danzig in 1949 is the most widely used method for solving linear programs. Narendra Karmarkar developed a competing, but more mathematically complex algorithm in 1984.

15. Reid & Sanders, Operations Management © Wiley 2002 Page 15 The End Copyright © 2002 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United State Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.