Chapter 19 Linear Programming McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc All rights reserved Chapter 19: Learning Objectives You should be able to: Describe the type of problem that would lend itself to solution using linear programming Formulate a linear programming model from a description of a problem 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 Instructor Slides 19-2 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 Instructor Slides 19-3 LP Models 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 Instructor Slides 19-4 Linear Programming Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials Budgets Labor Machine time 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 LP Assumptions 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 Instructor Slides 19-6 Model Formulation List and define the decision variables (D.V.) State the objective function (O.F.) These typically represent quantities 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 Instructor Slides 19-7 Example– LP Formulation x1 = Quantity of product to produce Decision Variables x2 = Quantity of product to produce x = Quantity of product to produce Maximize x1 + x2 + x3 (profit) Subject to Labor x1 + x2 + x3 ≤ 250 hours Material x1 + x2 + x3 ≤ 100 pounds Product x1 (Constraints) ≤ 10 units x1 , x2 , x3 ≥ Instructor Slides (Objective function) (Nonnegativity constraints) 19-8 Graphical LP Graphical LP A method for finding optimal solutions to two-variable problems Procedure Set up the objective function and the constraints in mathematical format Plot the constraints Indentify the feasible solution space The set of all feasible combinations of decision variables as defined by the constraints Instructor Slides Plot the objective function Determine the optimal solution 19-9 Linear Programming Example Assembly Time/Unit Inspection Time/Unit Storage Space/Unit Model A $ 60 Model B 10 $ 50 Available 100 hours 22 hours 39 cubic feet Profit/Unit Computer Solutions 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 Instructor Slides 19-40 Computer Solutions Instructor Slides 19-41 Computer Solutions 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 Instructor Slides 19-42 Computer Solutions 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 Instructor Slides 19-43 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 Instructor Slides 19-44 Computer Solutions Instructor Slides 19-45 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 Instructor Slides 19-46 Solver Results Solver will incorporate the optimal values of the decision variables and the objective function into your original layout on your worksheets Instructor Slides 19-47 Answer Report Instructor Slides 19-48 Sensitivity Report Instructor Slides 19-49 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 Instructor Slides We will consider these 19-50 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 Instructor Slides 19-51 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 non-basic Instructor Slides 19-52 RHS Value Changes Shadow price Amount by which the value of the objective function would change with a oneunit 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 Instructor Slides 19-53 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 Instructor Slides 19-54 ... using linear programming Formulate a linear programming model from a description of a problem Solve simple linear programming problems using the graphical method Interpret computer solutions of linear. .. linear programming problems Do sensitivity analysis on the solution of a linear programming problem Instructor Slides 19-2 Linear Programming (LP) LP A powerful quantitative tool used by operations. .. Slides 19-4 Linear Programming Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials Budgets Labor Machine time Linear programming