1. McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 6S Linear Programming

2. 6S-2 Learning Objectives  Describe the type of problem tha would lend itself to solution using linear programming  Formulate a linear programming model from a description of a problem  Solve linear programming problems using the graphical method  Interpret computer solutions of linear programming problems  Do sensitivity analysis on the solution of a linear progrmming problem

3. 6S-3  Used to obtain optimal solutions to problems that involve restrictions or limitations, such as:  Materials  Budgets  Labor  Machine time Linear Programming

4. 6S-4  Linear programming (LP) techniques consist of a sequence of steps that will lead to an optimal solution to problems, in cases where an optimum exists Linear Programming

5. 6S-5  Objective Function: mathematical statement of profit or cost for a given solution  Decision variables: amounts of either inputs or outputs  Feasible solution space: the set of all feasible combinations of decision variables as defined by the constraints  Constraints: limitations that restrict the available alternatives  Parameters: numerical values Linear Programming Model

6. 6S-6  Linearity: the impact of decision variables is linear in constraints and objective function  Divisibility: noninteger values of decision variables are acceptable  Certainty: values of parameters are known and constant  Nonnegativity: negative values of decision variables are unacceptable Linear Programming Assumptions

7. 6S-7 1.Set up objective function and constraints in mathematical format 2.Plot the constraints 3.Identify the feasible solution space 4.Plot the objective function 5.Determine the optimum solution Graphical Linear Programming Graphical method for finding optimal solutions to two-variable problems

8. 6S-8  Objective - profit Maximize Z=60X 1 + 50X 2  Subject to Assembly 4X 1 + 10X 2 <= 100 hours Inspection 2X 1 + 1X 2 <= 22 hours Storage3X 1 + 3X 2 <= 39 cubic feet X 1, X 2 >= 0 Linear Programming Example

9. 6S-9 Linear Programming Example

10. 6S-10 Linear Programming Example

11. 6S-11 Assembly Storage Inspection Feasible solution space Linear Programming Example

12. 6S-12 Z=300 Z=900 Z=600 Linear Programming Example

13. 6S-13  The intersection of inspection and storage  Solve two equations in two unknowns 2X1 + 1X2 = 22 3X1 + 3X2 = 39 X1 = 9 X2 = 4 Z = $740 Solution

14. 6S-14  Redundant constraint: a constraint that does not form a unique boundary of the feasible solution space  Binding constraint: a constraint that forms the optimal corner point of the feasible solution space Constraints

15. 6S-15 Solutions and Corner Points  Feasible solution space is usually a polygon  Solution will be at one of the corner points  Enumeration approach: Substituting the coordinates of each corner point into the objective function to determine which corner point is optimal.

16. 6S-16  Surplus: when the optimal values of decision variables are substituted into a greater than or equal to constraint and the resulting value exceeds the right side value  Slack: when the optimal values of decision variables are substituted into a less than or equal to constraint and the resulting value is less than the right side value Slack and Surplus

17. 6S-17  Simplex: a linear-programming algorithm that can solve problems having more than two decision variables Simplex Method

18. 6S-18 Figure 6S.15 MS Excel Worksheet for Microcomputer Problem

19. 6S-19 Figure 6S.17 MS Excel Worksheet Solution

20. 6S-20  Range of optimality : the range of values for which the solution quantities of the decision variables remains the same  Range of feasibility : the range of values for the fight-hand side of a constraint over which the shadow price remains the same  Shadow prices : negative values indicating how much a one-unit decrease in the original amount of a constraint would decrease the final value of the objective function Sensitivity Analysis