Lecture Introduction to Management Science with Spreadsheets: Chapter 5 - Stevenson, Ozgur

Chapter 5 Linear programming: Sensitivity analysis and duality, after completing this chapter, you should be able to: Explain how sensitivity analysis can be useful to a decision maker, explain why it can be useful for a decision maker to extend the analysis of a linear programming problem beyond determination of the optimal solution, explain how to analyze graphically and interpret the impact of a change in the value of the objective function coefficient,...

Introduction to Management Science with Spreadsheets Stevenson and Ozgur First Edition Part Deterministic Decision Models Chapter 5 Linear Programming: Sensitivity Analysis and  Duality McGraw­Hill/Irwin Copyright © 2007 by The McGraw­Hill Companies, Inc. All rights reserved Learning Objectives After completing this chapter, you should be able to: Explain how sensitivity analysis can be useful to a decision maker Explain why it can be useful for a decision maker to extend the analysis of a linear programming problem beyond determination of the optimal solution Explain how to analyze graphically and interpret the impact of a change in the value of the objective function coefficient Explain how to graphically analyze and interpret the impact of a change in the right-hand-side value of a constraint Copyright © 2007 The McGraw­Hill  McGraw­ Companies. All rights reserved.   Hill/Irwin  5–2 Learning Objectives (cont’d) After completing this chapter, you should be able to: Explain what a dual is Formulate the dual of a problem Read and interpret the solution to a dual problem and relate the dual solution to the primal solution Explain in economic terms the interpretation of dual variables and the dual solution 10.Determine if adding another variable to a problem will change the optimal solution mix of the original problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–3 Sensitivity Sensitivity Analysis Analysis • Benefits of sensitivity analysis – Enables the decision maker to determine how a change in one of the values of a model will impact the optimal solution and the optimal value of the objective function while holding all other parameters constant – Provides the decision maker with greater insight about the sensitivity of the optimal solution to changes in various parameters of a problem – Permits quick examination of changes due to improved information relating to a problem or because of the desire to know the potential impact of changes that are contemplated Copyright © 2007 The McGraw­Hill  McGraw­ Companies. All rights reserved.   Hill/Irwin  5–4 Changes Changes in in Parameter Parameter Values Values • Categories of model parameters subject to potential changes – The value of an objective function coefficient – The right-hand side (RHS) value of a constraint – A coefficient of a constraint • Concerns about ranges of changes – Which range pertains to a given situation? – How can the range be determined? – What impact on the optimal solution does a change that is within the range have? Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–5 Optimality Optimality and and the the Objective Objective Function Function Coefficient Coefficient • Range of optimality – Finding the range of objective function values for which the optimal values of the decision variables would not change – A value of the objective function that falls within the range of optimality will not change the optimal solution, although the optimal value of the objective function will change Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–6 Feasibility Feasibility • Range of feasibility – The range of values over which the right-hand-side (RHS) value can change without causing the shadow price to change – Within this range of feasibility, the same decision variables will remain optimal, although their values and the optimal value of the objective function will change – Analysis of RHS changes begins with determination of a constraint’s shadow price in the optimal solution Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–7 Figure Figure5–1 5–1 AAGraph Graphofofthe theServer ServerProblem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–8 Figure Figure5–2 5–2 Graphical GraphicalRepresentation Representationof ofaaChange Changeininthe theObjective Objective Function FunctionCoefficients Coefficients Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–9 Example Example5-1 5-1 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–10 Example Example5-3 5-3(cont’d) (cont’d) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–24 Table Table5–1 5–1 Summary Summaryof ofResults ResultsofofChanges ChangesThat ThatAre Arewithin withinRanges Rangesofof Optimality Optimalityand andFeasibility Feasibility Copyrightâ2007TheMcGrawưHill Companies.Allrightsreserved. McGrawư Hill/Irwin525 Duality Duality ã The Dual – An alternate formulation of a linear programming problem as either the original problem or its mirror image, the dual, which can be solved to obtain the optimal solution – Its variables have a different economic interpretation than the original formulation of the linear programming problem (the primal) – It can be easily used to determine if the addition of another variable to a problem will change the optimal Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–26 Formulation Formulation of of aa Dual Dual • Dual – The number of decision variables in the primal is equal to the number of constraints in the dual – The number of decision variables in the dual is equal to the number of constraints in the primal – Since it is computationally easier to solve problems with less constraints in comparison to solving problems with less variables, the dual gives us the flexibility to choose which problem to solve Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–27 Example Example5-4 5-4 A comparison of these two versions of the problem will reveal why the dual might be termed the “mirror image” of the primal Table 5-2 shows how the primal problem is transformed into its dual We can see in Table 5-2 that the original objective was to minimize, whereas the objective of the dual is to maximize In addition, the coefficients of the primal’s objective function become the right-handside values for the dual’s constraints, whereas the primal’s right-hand side values become the coefficients of the dual’s objective function Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–28 Table Table5–2 5–2 Transforming Transformingthe thePrimal Primalinto intoIts ItsDual Dual Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–29 Example Example5-5 5-5 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–30 Economic Economic Interpretation Interpretation of of The The Dual Dual • Economic interpretation of dual solution results – Analysis enables a manager to evaluate the potential impact of a new product – Analysis can determine the marginal values of resources (i.e., constraints) to determine how much profit one unit of each resource is equivalent to – Analysis helps the manager to decide which of several alternative uses of resources is the most profitable Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–31 Example Example5-7 5-7 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–32 Exhibit Exhibit5–6 5–6 Excel ExcelWorksheet Worksheetfor forArc ArcManufacturing ManufacturingInc Inc Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–33 Exhibit Exhibit5–7 5–7 Excel ExcelBasic BasicOutput OutputReport Reportfor forArc ArcManufacturing ManufacturingInc Inc Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–34 Exhibit Exhibit5–8 5–8 Excel ExcelSensitivity SensitivityAnalysis Analysisfor forArc ArcManufacturing ManufacturingInc Inc Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–35 Exhibit Exhibit5–9 5–9 Excel ExcelSensitivity SensitivityReport Reportfor forSolved SolvedProblem Problem33 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–36 Table Table5–3 5–3 Excel ExcelReports Reportsfor forProblem Problem14 14 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–37 Table Table5–3 5–3 Excel ExcelReports Reportsfor forProblem Problem14 14(cont’d) (cont’d) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  5–38 ... Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?5? ??22 Example Example 5-3 5- 3 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?5? ??23 Example Example 5-3 5- 3(cont’d) (cont’d) Copyright © 2007 The McGraw­Hill ... McGraw­ Hill/Irwin ? ?5? ?? 15 Example Example 5-2 5- 2(cont’d) (cont’d) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?5? ??16 Figure Figure5–7 5? ??7 Range RangeofofFeasibility... Hill/Irwin ? ?5? ??11 Example Example 5-2 5- 2 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?5? ??12 Figure Figure5–4 5? ??4 The TheUpper UpperLimit Limiton onUsing