Supplement Linear Programming McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc All Rights Reserved Supplement 6: Learning Objectives • You should be able to: – Describe the type of problem that would be appropriately solved using linear programming – Formulate a linear programming model – Solve simple linear programming problems using the graphical method – Interpret computer solutions of linear programming problems – Do sensitivity analysis on the solution of a linear programming problem 6S-2 Linear Programming (LP) • 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 6S-3 Model Formulation List and define the decision variables (D.V.) – These typically represent quantities State the objective function (O.F.) – It includes every D.V in the model and its contribution to profit (or cost) 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 Non-negativity constraints 6S-4 Computer Solutions • MS Excel can be used to solve LP problems using its Solver routine – Enter the problem into a worksheet – You must designate the cells where you want the optimal values for the decision variables 6S-5 Computer Solutions • Click on Tools on the top of the worksheet, and in the drop-down 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 6S-6 Computer Solutions • Add the 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 6S-7 Computer Solutions • 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 6S-8 Solver Results • 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 6S-9 Solver Results • Solver will incorporate the optimal values of the decision variables and the objective function into your original layout on your worksheets 6S-10 Sensitivity Analysis • Sensitivity Analysis – Assessing the impact of potential changes to the numerical values of an LP model – Three types of changes • Objective function coefficients • Right-hand values of constraints • Constraint coefficients 6S-11 O.F Coefficient Changes • 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 6S-12 Basic and Non-Basic Variables • Basic variables – Decision variables whose optimal values are non-zero • Non-basic variables – Decision variables whose optimal values are zero – Reduced cost • Unless the non-basic variable’s coefficient increases by more than its reduced cost, it will continue to be nonbasic 6S-13 RHS Value Changes • 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 6S-14 Binding vs Non-binding Constraints • Non-binding constraints – have shadow price values that are equal to zero – have slack (≤ constraint) or surplus (≥ constraint) – Changing the RHS value of a non-binding constraint (over its range of feasibility) will have no effect on the optimal solution • Binding constraint – have shadow price values that are non-zero – have no slack (≤ constraint) or surplus (≥ constraint) – Changing the RHS value of a binding constraint will lead to a change in the optimal decision values and to a change in the value of the objective function 6S-15 ... sensitivity analysis on the solution of a linear programming problem 6S-2 Linear Programming (LP) • LP – A powerful quantitative tool used by operations and other manages to obtain optimal solutions to... the cells where the optimal values of the decision variables will appear 6S -6 Computer Solutions • Add the constraint, by clicking add – For each constraint, enter the cell that contains the left-hand... non-basic variable’s coefficient increases by more than its reduced cost, it will continue to be nonbasic 6S-13 RHS Value Changes • Shadow price – Amount by which the value of the objective function