Linear Programming Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

19-2 You should be able to: LO 19.1Describe the type of problem that would lend itself to solution using linear programming LO 19.2Formulate a linear programming model from a description of a problem LO 19.3Solve simple linear programming problems using the graphical method LO 19.4Interpret computer solutions of linear programming problems LO 19.5Do sensitivity analysis on the solution of a linear programming problem

19-3 In order for LP models to be used effectively, certain assumptions must be satisfied: Linearity The impact of decision variables is linear in constraints and in the 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 LO 19.1

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 LO 19.2

19-5 Graphical LP A method for finding optimal solutions to two-variable problems Procedure 1. Set up the objective function and the constraints in mathematical format 2. Plot the constraints 3. Identify the feasible solution space The set of all feasible combinations of decision variables as defined by the constraints 4. Plot the objective function 5. Determine the optimal solution LO 19.3

19-6 LO 19.4

19-7 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 LO 19.4

19-8 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 LO 19.4

19-9 For the non-negativity constraints, check the checkbox to Make Unconstrained Variables Non-Negative Select Simplex LP as the Solving Method Click Solve LO 19.4

19-10 LO 19.4

19-11 Solver will incorporate the optimal values of the decision variables and the objective function into your original layout on your worksheets LO 19.4

19-12 LO 19.4

19-13 LO 19.5

19-14 A change in the value of an O.F. coefficient can cause a change in the optimal solution of a problem Not every change will result in a changed solution Range of Optimality The range of O.F. coefficient values for which the optimal values of the decision variables will not change LO 19.5

19-15 Shadow price Amount by which the value of the objective function would change with a one-unit change in the RHS value of a constraint Range of feasibility Range of values for the RHS of a constraint over which the shadow price remains the same LO 19.5