Operations management 12th stevenson ch19 linear programming

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