Linear Programming McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

You should be able to: 1. Describe the type of problem that would lend itself to solution using linear programming 2. Formulate a linear programming model from a description of a problem 3. Solve simple linear programming problems using the graphical method 4. Interpret computer solutions of linear programming problems 5. Do sensitivity analysis on the solution of a linear programming problem 19-2 Student Slides

LP A powerful quantitative tool used by operations and other manages to obtain optimal solutions to problems that involve restrictions or limitations Applications include: Establishing locations for emergency equipment and personnel to minimize response time Developing optimal production schedules Developing financial plans Determining optimal diet plans 19-3 Student Slides

LP Models Mathematical representations of constrained optimization problems LP Model Components: Objective function A mathematical statement of profit (or cost, etc.) for a given solution Decision variables Amounts of either inputs or outputs Constraints Limitations that restrict the available alternatives Parameters Numerical constants 19-4 Student Slides

1. List and define the decision variables (D.V.) These typically represent quantities 2. State the objective function (O.F.) It includes every D.V. in the model and its contribution to profit (or cost) 3. List the constraints Right hand side value Relationship symbol (≤, ≥, or =) Left Hand Side The variables subject to the constraint, and their coefficients that indicate how much of the RHS quantity one unit of the D.V. represents 4. Non-negativity constraints 19-5 Student Slides

MS Excel can be used to solve LP problems using its Solver routine Enter the problem into a worksheet Where there is a zero in Figure 19.15, a formula was entered Solver automatically places a value of zero after you input the formula You must designate the cells where you want the optimal values for the decision variables 19-6 Student Slides

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In Excel 2010, click on Tools on the top of the worksheet, and in that menu, click on Solver Begin by setting the Target Cell This is where you want the optimal objective function value to be recorded Highlight Max (if the objective is to maximize) The changing cells are the cells where the optimal values of the decision variables will appear 19-8 Student Slides

Add a constraint, by clicking add For each constraint, enter the cell that contains the left-hand side for the constraint Select the appropriate relationship sign (≤, ≥, or =) Enter the RHS value or click on the cell containing the value Repeat the process for each system constraint 19-9 Student Slides

For the nonnegativity constraints, enter the range of cells designated for the optimal values of the decision variables Click OK, rather than Add You will be returned to the Solver menu Click on Options In the Options menu, Click on Assume Linear Model Click OK; you will be returned to the solver menu Click Solve Student Slides

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The Solver Results menu will appear You will have one of two results A Solution In the Solver Results menu Reports box Highlight both Answer and Sensitivity Click OK An Error message Make corrections and click solve Student Slides

Solver will incorporate the optimal values of the decision variables and the objective function into your original layout on your worksheets Student Slides 19-13

Student Slides 19-14

Student Slides 19-15