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Understanding Linear Programming: Models and Applications

School
University of Toronto**We aren't endorsed by this school
Course
CHM 136
Subject
Industrial Engineering
Date
Dec 11, 2024
Pages
40
Uploaded by LieutenantCoyotePerson1151
Chapter 19 Linear Programming© McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
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19-2© McGraw-Hill Education.Learning Objective 19.1Linear 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:oEstablishing locations for emergency equipment and personnel to minimize response timeoDeveloping optimal production schedulesoDeveloping financial plansoDetermining optimal diet plans
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19-3© McGraw-Hill Education.Learning Objective 19.1LP Models•LP Models–Mathematical representations of constrained optimization problems–LP model components:▪Objective functionoA mathematical statement of profit (or cost, etc.) for a given solution▪Decision variablesoAmounts of either inputs or outputs▪ConstraintsoLimitations that restrict the available alternatives▪ParametersoNumerical constants
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19-4© McGraw-Hill Education.Learning Objective 19.1LP 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
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19-5© McGraw-Hill Education.Learning Objective 19.2Model Formulation1.List and define the decision variables (D.V.)–These typically represent quantities2.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. represents4.Non-negativity constraints
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19-6© McGraw-Hill Education.Learning Objective 19.2Example –LP Formulation)(03,2,1units1011Productpounds100352617Material)(hours250382412LabortoSubject)((profit)342815Maximizeproduceto3productofQuantity3produceto2productofQuantity2produceto1productofQuantity1VariablesDecisionsconstraintityNonnegativsConstraintfunctionObjective++++++===xxxxxxxxxxxxxxxx
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19-7© McGraw-Hill Education.Learning Objective 19.3Graphical LP•Graphical LP–A method for finding optimal solutions to two-variable problems–Procedure1.Set up the objective function and the constraints in mathematical format2.Plot the constraints3.Identify the feasible solution spaceoThe set of all feasible combinations of decision variables as defined by the constraints4.Plot the objective function5.Determine the optimal solution
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19-8© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 102,1feetcubic392313Storagehours222112Inspectionhours10021014AssemblytoSubject250160Maximizeproduceto2typeofquantity2produceto1typeofquantity1VariablesDecision++++==xxxxxxxxxxxx
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19-9© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 2 (1 of 6)•Plotting constraints:–Begin by placing the nonnegativity constraints on a graph
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19-10© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 2 (2 of 6)•Plotting constraints:1. Replace the inequality sign with an equal sign2.Determine where the line intersects each axis3.Mark these intersection on the axes, and connect them with a straight line4. Indicate by shading, whether the inequality is greater than or less than5.Repeat steps 1 –4 for each constraint
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19-11© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 2 (3 of 6)
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19-12© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 2 (4 of 6)
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19-13© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 2 (5 of 6)
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19-14© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 2 (6 of 6)
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19-15© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 3•Feasible solution space–The set of points that satisfy all constraints simultaneously
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19-16© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 4 (1 of 3)•Plotting the objective function line–This follows the same logic as plotting a constraint line–There is no equal sign, so we simply set the objective function to some quantity (profit or cost)–The profit line can now be interpreted as an isoprofit line▪Every point on this line represents a combination of the decision variables that result in the same profit (in this case, to the profit you selected)
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19-17© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 4 (2 of 3)
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19-18© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 4 (3 of 3)•As we increase the value for the objective function:–The isoprofit line moves further away from the origin–The isoprofit lines are parallel
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19-19© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 5 (1 of 2)•Where is the optimal solution?–The optimal solution occurs at the furthest point (for a maximization problem) from the origin the isoprofit can be moved and still be touching the feasible solution space–This optimum point will occur at the intersection of two constraints:▪Solve for the values of x1andx2where this occurs
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19-20© McGraw-Hill Education.Learning Objective 19.3Example –Graphical LP: Step 5 (2 of 2)
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19-21© McGraw-Hill Education.Learning Objective 19.3Redundant Constraints (1 of 2)•Redundant constraints–A constraint that does not form a unique boundary of the feasible solution space–Test:▪A constraint is redundant if its removal does not alter the feasible solution space
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19-22© McGraw-Hill Education.Learning Objective 19.3Redundant Constraints (2 of 2)
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19-23© McGraw-Hill Education.Learning Objective 19.3Solutions and Corner Points•The solution to any problem will occur at one of the feasible solution space corner points•Enumeration approach–Determine the coordinates for each of the corner points of the feasible solution space▪Corner points occur at the intersections of constraints–Substitute the coordinates of each corner point into the objective function–The corner point with the maximum (or minimum, depending on the objective) value is optimal
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19-24© McGraw-Hill Education.Learning Objective 19.3Slack and Surplus•Binding constraint–If a constraint forms the optimal corner point of the feasible solution space, it is binding–It effectively limits the value of the objective function–If the constraint could be relaxed, the objective function could be improved•Surplus–When the value of decision variables are substituted into a ≥ constraint the amount by which the resulting value exceeds the right-hand side value•Slack–When the values of decision variables are substituted into a ≤ constraint, the amount by which the resulting value is less than the right-hand side
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19-25© McGraw-Hill Education.Learning Objective 19.4Computer Solutions (1 of 6)•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
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19-26© McGraw-Hill Education.Learning Objective 19.4Computer Solutions (2 of 6)
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19-27© McGraw-Hill Education.Learning Objective 19.4Computer Solutions (3 of 6)•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
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19-28© McGraw-Hill Education.Learning Objective 19.4Computer Solutions (4 of 6)•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
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19-29© McGraw-Hill Education.Learning Objective 19.4Computer Solutions (5 of 6)•For the non-negativity constraints, check the checkbox to Make Unconstrained Variables Non-Negative•Select Simplex LP as the Solving Method•Click Solve
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19-30© McGraw-Hill Education.Learning Objective 19.4Computer Solutions (6 of 6)
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19-31© McGraw-Hill Education.Learning Objective 19.4Solver Results (1 of 2)•The Solver Results menu will appear–You will haveone of two results▪A SolutionoIn the Solver Results menu Reports box➢Highlight both Answer and Sensitivity➢Click OK▪An ErrormessageoMake corrections and click solve
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19-32© McGraw-Hill Education.Learning Objective 19.4Solver Results (2 of 2)•Solver will incorporate the optimal values of the decision variables and the objective function into your original layout on your worksheets
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19-33© McGraw-Hill Education.Learning Objective 19.4Answer Report
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19-34© McGraw-Hill Education.Learning Objective 19.5Sensitivity Report
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19-35© McGraw-Hill Education.Learning Objective 19.5Sensitivity Analysis•Sensitivity Analysis–Assessing the impact of potential changes to the numerical values of an LP model–Three types of changes▪We will consider theseoObjective function coefficientsoRight-hand values of constraintsoConstraint coefficients
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19-36© McGraw-Hill Education.Learning Objective 19.5O.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 optimalityoThe range of O.F. coefficient values for which the optimal values of the decision variables will not change
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19-37© McGraw-Hill Education.Learning Objective 19.5Basic 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
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19-38© McGraw-Hill Education.Learning Objective 19.5RHS 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
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19-39© McGraw-Hill Education.Learning Objective 19.5Binding versus Non-Binding Constraints (1 of 2)•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
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19-40© McGraw-Hill Education.Learning Objective 19.5Binding versus Non-Binding Constraints (2 of 2)•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
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