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

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

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

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

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

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

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

6S-8  Objective - profit Maximize Z=60X X 2  Subject to Assembly 4X X 2 <= 100 hours Materials3X 1 + 3X 2 <= 39 cubic feet X 1, X 2 >= 0 Linear Programming Example

6S-9 Solution

6S-10  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-11 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.

6S-12 GM operates a plant that assembles and finishes cars and trucks. It takes 5 man-days to assemble a truck and 2 man-days to assemble a car. It takes 3 man-days to finish each type of vehicle. Because of man-power limitations, assembly can take no more than 180 man-days per week and finishing no more than 135 man-days per week. If the profit on each truck is $300 and $200 on each car, how many of each should they produce to maximize profit? Example No.1

6S-13 Flair Furniture Co. Data Example Tables (per table) Chairs (per chair) Hours Available Profit Contribution $7$5 Carpentry3 hrs4 hrs2400 Painting2 hrs1 hr1000 Other Limitations: Make no more than 450 chairs Make at least 100 tables