6s-1Linear Programming William J. Stevenson Operations Management 8 th edition

6s-2Linear Programming CHAPTER 6s Linear Programming McGraw-Hill/Irwin Operations Management, Eighth Edition, by William J. Stevenson Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

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

6s-4Linear Programming  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

6s-5Linear Programming  Objective: the goal of an LP model is maximization or minimization  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

6s-6Linear Programming  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

6s-7Linear Programming 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

6s-8Linear Programming  Objective - profit Maximize Z=60X X 2  Subject to Assembly 4X X 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

6s-9Linear Programming Linear Programming Example

6s-10Linear Programming Linear Programming Example

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

6s-12Linear Programming Z=300 Z=900 Z=600 Linear Programming Example

6s-13Linear Programming  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

6s-14Linear Programming  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

6s-15Linear Programming  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

6s-16Linear Programming  Simplex: a linear-programming algorithm that can solve problems having more than two decision variables Simplex Method

6s-17Linear Programming Figure 6S.15 MS Excel Worksheet for Microcomputer Problem

6s-18Linear Programming Figure 6S.17 MS Excel Worksheet Solution

6s-19Linear Programming  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