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Chapter TopicsComputer SolutionSensitivity Analysis1Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisEarly linear programming used lengthy manual mathematical solution procedure called the Simplex Method(See CD-ROM Module A).Steps of the Simplex Method have been programmed in software packages designed for linear programming problems.Many such packages available currently.Used extensively in business and government.Text focuses on Excel Spreadsheets and QM for Windows.Computer Solution2Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisBeaver Creek Pottery ExampleExcel Spreadsheet Data Screen(1 of 5)Exhibit 3.13Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisBeaver Creek Pottery Example“Solver”Parameter Screen(2 of 5)Exhibit 3.24Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisExhibit 3.3Beaver Creek Pottery ExampleAdding Model Constraints(3 of 5)5Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisExhibit 3.4Beaver Creek Pottery ExampleSolution Screen(4 of 5)6Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisBeaver Creek Pottery ExampleAnswer Report(5 of 5)Exhibit 3.57Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisLinear Programming Problem Standard FormStandard form requires all variables in the constraint equations to appear on the left of the inequality(or equality)and all numeric values to be on the right-hand side.Examples:x3 x1+x2 must be converted to x3-x1-x2 0 x1/(x2+x3)2 becomes x1 2(x2+x3)and then x1-2x2-2x3 0 8Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisBeaver Creek Pottery ExampleQM for Windows Data Screen(1 of 3)Exhibit 3.69Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisBeaver Creek Pottery ExampleModel Solution Screen(2 of 3)Exhibit 3.710Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisBeaver Creek Pottery ExampleGraphical Solution Screen(3 of 3)Exhibit 3.811Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisBeaver Creek Pottery ExampleSensitivity Analysis(1 of 4)Sensitivity analysis determines the effect on optimal solutions of changes in parameter values of the objective function and constraint equations.Changes may be reactions to anticipated uncertainties in the parameters or to new or changed information concerning the model.12Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisMaximize Z=$40 x1+$50 x2subject to:1x1+2x2 40 4x2+3x2 120 x1,x2 0Figure 3.1Optimal Solution PointBeaver Creek Pottery ExampleSensitivity Analysis(2 of 4)13Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisMaximize Z=$100 x1+$50 x2subject to:1x1+2x2 40 4x2+3x2 120 x1,x2 0Figure 3.2Changing the x1 Objective Function CoefficientBeaver Creek Pottery ExampleChange x1 Objective Function Coefficient(3 of 4)14Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisMaximize Z=$40 x1+$100 x2subject to:1x1+2x2 40 4x2+3x2 120 x1,x2 0Figure 3.3Changing the x2 Objective Function CoefficientBeaver Creek Pottery ExampleChange x2 Objective Function Coefficient(4 of 4)15Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisThe sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point will remain optimal.The sensitivity range for the xi coefficient is designated as ci.Objective Function CoefficientSensitivity Range(1 of 3)16Chapter 3-Linear Programming:Computer Solution and Sensitivity Analysisobjective function Z=$40 x1+$50 x2 sensitivity range for:x1:25 c1 66.67 x2:30 c2 80Figure 3.4Determining the Sensitivity Range for c1Objective Function CoefficientSensitivity Range for c1 and c2(2 of 3)17Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisMinimize Z=$6x1+$3x2subject to:2x1+4x2 164x1+3x2 24 x1,x2 0sensitivity ranges:4 c1 0 c2 4.5Objective Function CoefficientFertilizer Cost Minimization Example(3 of 3)Figure 3.5Fertilizer Cost Minimization Example18Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisExhibit 3.9Objective Function Coefficient RangesExcel“Solver”Results Screen(1 of 3)19Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisExhibit 3.10Objective Function Coefficient RangesBeaver Creek Example Sensitivity Report(2 of 3)20Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisExhibit 3.11Objective Function Coefficient RangesQM for Windows Sensitivity Range Screen(3 of 3)21Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisChanges in Constraint Quantity ValuesSensitivity Range(1 of 4)The sensitivity range for a right-hand-side value is the range of values over which the quantity values can change without changing the solution variable mix,including slack variables.22Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisChanges in Constraint Quantity ValuesIncreasing the Labor Constraint(2 of 4)Maximize Z=$40 x1+$50 x2subject to:1x1+2x2 40 4x2+3x2 120 x1,x2 0Figure 3.6Increasing the Labor Constraint Quantity23Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisChanges in Constraint Quantity ValuesSensitivity Range for Labor Constraint(3 of 4)Sensitivity range for:30 q1 80 hrFigure 3.7Determining the Sensitivity Range for Labor Quantity24Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisChanges in Constraint Quantity ValuesSensitivity Range for Clay Constraint(4 of 4)Sensitivity range for:60 q2 160 lbFigure 3.8Determining the Sensitivity Range for Clay Quantity25Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisExhibit 3.12Constraint Quantity Value Ranges by ComputerExcel Sensitivity Range for Constraints(1 of 2)26Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisExhibit 3.13Constraint Quantity Value Ranges by ComputerQM for Windows Sensitivity Range(2 of 2)27Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisChanging individual constraint parametersAdding new constraintsAdding new variablesOther Forms of Sensitivity AnalysisTopics(1 of 4)28Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisOther Forms of Sensitivity AnalysisChanging a Constraint Parameter(2 of 4)Maximize Z=$40 x1+$50 x2subject to:1x1+2x2 40 4x2+3x2 120 x1,x2 0Figure 3.9Changing the x1 Coefficient in the Labor Constraint29Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisAdding a new constraint to Beaver Creek Model:0.20 x1+0.10 x2 5 hours for packaging Original solution:24 bowls,8 mugs,$1,360 profitExhibit 3.13Other Forms of Sensitivity AnalysisAdding a New Constraint(3 of 4)30Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisAdding a new variable to the Beaver Creek model,x3,a third product,cupsMaximize Z=$40 x1+50 x2+30 x3subject to:x1+2x2+1.2x3 40 hr of labor 4x1+3x2+2x3 120 lb of clay x1,x2,x3 0Solving model shows that change has no effect on the original solution(i.e.,the model is not sensitive to this change).Other Forms of Sensitivity AnalysisAdding a New Variable(4 of 4)31Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisDefined as the marginal value of one additional unit of resource.The sensitivity range for a constraint quantity value is also the range over which the shadow price is valid.Shadow Prices(Dual Values)32Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisMaximize Z=$40 x1+$50 x2 subject to:x1+2x2 40 hr of labor4x1+3x2 120 lb of clay x1,x2 0Exhibit 3.14Excel Sensitivity Report for Beaver Creek PotteryShadow Prices Example(1 of 2)33Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisExcel Sensitivity Report for Beaver Creek PotterySolution Screen(2 of 2)Exhibit 3.1534Chapter 3-Linear Programming:Computer Solution and Sensitivity AnalysisTwo airplane parts:no.1 and no.2.Three manufacturing stages:stamping,drilling,milling.Decision variables:x1(number of part no.1 to produce)x2(number of part no.2 to produce)Model:Maximize Z=$650 x1+910 x2 subject to:4x1+7.5x2 105(stamping,hr)6.2x1+4.9x2 90(drilling,hr)9.1x1+4.1x2 110(finishing,hr)x1,x2 0 Example ProblemProblem Statement(1 of 3)35Chapter 3-Linear Programming:Computer Solution and Sensitivity Analysisp经常不断地学习,你就什么都知道。你知道得越多,你就越有力量pStudyConstantly,AndYouWillKnowEverything.TheMoreYouKnow,TheMorePowerfulYouWillBe写在最后感谢聆听不足之处请大家批评指导Please Criticize And Guide The Shortcomings结束语讲师：XXXXXX XX年XX月XX日
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