Lecture Introduction to Management Science with Spreadsheets: Chapter 3 - Stevenson, Ozgur

Chapter 3 Linear programming: basic concepts and graphical solution, after completing this chapter, you should be able to: Explain what is meant by the terms constrained optimization and linear programming, list the components and the assumptions of linear programming and briefly explain each, name and describe at least three successful applications of linear programming,...

Introduction to Management Science with Spreadsheets Stevenson and Ozgur First Edition Part Introduction to Management Science and Forecasting Chapter 3 Linear Programming: Basic Concepts and  Graphical Solution McGraw­Hill/Irwin Copyright © 2007 by The McGraw­Hill Companies, Inc. All rights reserved Learning Objectives After completing this chapter, you should be able to: Explain what is meant by the terms constrained optimization and linear programming List the components and the assumptions of linear programming and briefly explain each Name and describe at least three successful applications of linear programming Identify the type of problems that can be solved using linear programming Formulate simple linear programming models Identify LP problems that are amenable to graphical solutions Copyright © 2007 The McGraw­Hill  McGraw­ Companies. All rights reserved.   Hill/Irwin  3–2 Learning Objectives (cont’d) After completing this chapter, you should be able to: Explain these terms: optimal solution, feasible solution space, corner point, redundant constraint slack, and surplus Solve two-variable LP problems graphically and interpret your answers Identify problems that have multiple solutions, problems that have no feasible solutions, unbounded problems, and problems with redundant constraints Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–3 Decisions Decisions and and Linear Linear Programming Programming • Constrained optimization – Finding the optimal solution to a problem given that certain constraints must be satisfied by the solution – A form of decision making that involves situations in which the set of acceptable solutions is somehow restricted – Recognizes scarcity—the limitations on the availability of physical and human resources – Seeks solutions that are both efficient and feasible in the allocation of resources Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–4 Linear Linear Programming Programming • Linear Programming (LP) – A family of mathematical techniques (algorithms) that can be used for constrained optimization problems with linear relationships • Graphical method • Simplex method • Karmakar’s method – The problems must involve a single objective, a linear objective function, and linear constraints and have known and constant numerical values Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–5 Example Example33––11 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–6 Table Table3–1 3–1 Successful SuccessfulApplications ApplicationsofofLinear LinearProgramming ProgrammingPublished Published ininInterfaces Interfaces Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–7 Table Table3–2 3–2 Characteristics Characteristicsof ofLP LPModels Models Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–8 Formulating Formulating LP LP Models Models • Formulating linear programming models involves the following steps: Define the decision variables Determine the objective function Identify the constraints Determine appropriate values for parameters and determine whether an upper limit, lower limit, or equality is called for Use this information to build a model Validate the model Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–9 Example Example33––22 x1 = quantity of server model to produce x2 = quantity of server model to produce maximize Z = 60x1+50x2 Subject to: Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–10 Graphing Graphing— —Objective Objective Function Function Approach Approach Graph the constraints Identify the feasible solution space Set the objective function equal to some amount that is divisible by each of the objective function coefficients After identifying the optimal point, determine which two constraints intersect there Substitute the values obtained in the previous step into the objective function to determine the value of the objective function at the optimum Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–24 Figure Figure3–8 3–8 AAComparison ComparisonofofMaximization Maximizationand andMinimization MinimizationProblems Problems Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–25 Example Example3-3 3-3 Minimization Minimization Determine the values of decision variables x1 and x2 that will yield the minimum cost in the following problem Solve using the objective function approach Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–26 Figure Figure3–9 3–9 Graphing Graphingthe theFeasible FeasibleRegion Regionand andUsing Usingthe theObjective Objective Function FunctiontotoFind Findthe theOptimum Optimumfor forExample Example3-3 3-3 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–27 Example Example3-4 3-4 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–28 Table Table3–3 3–3 Summary Summaryof ofExtreme ExtremePoint PointAnalysis Analysisfor forExample Example3-4 3-4 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–29 Table Table3–5 3–5 Computing Computingthe theAmount Amountof ofSlack Slackfor forthe theOptimal OptimalSolution Solutionto to the theServer ServerProblem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–30 Some Some Special Special Issues Issues • No Feasible Solutions – Occurs in problems where to satisfy one of the constraints, another constraint must be violated • Unbounded Problems – Exists when the value of the objective function can be increased without limit • Redundant Constraints – A constraint that does not form a unique boundary of the feasible solution space; its removal would not alter the feasible solution space • Multiple Optimal Solutions – Problems in which different combinations of values of the decision variables yield the same optimal value Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–31 Figure Figure3–10 3–10 Infeasible InfeasibleSolution: Solution:No NoCombination Combinationofofx1 x1and andx2, x2,Can Can Simultaneously SimultaneouslySatisfy SatisfyBoth BothConstraints Constraints Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–32 Figure Figure3–11 3–11 An AnUnbounded UnboundedSolution SolutionSpace Space Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–33 Figure Figure3–12 3–12 Examples ExamplesofofRedundant RedundantConstraints Constraints Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–34 Figure Figure3–13 3–13 Multiple MultipleOptimal OptimalSolutions Solutions Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–35 Figure Figure3–14 3–14 Constraints Constraintsand andFeasible FeasibleSolution SolutionSpace Spacefor for Solved SolvedProblem Problem22 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–36 Figure Figure3–15 3–15 AAGraph Graphfor forSolved SolvedProblem Problem33 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–37 Figure Figure3–16 3–16 Graph Graphfor forSolved SolvedProblem Problem44 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  3–38 ... theOptimum Optimumfor forExample Example3 -3 3 -3 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?3? ??27 Example Example 3-4 3- 4 Copyright © 2007 The McGraw­Hill ... Hill/Irwin ? ?3? ? ?36 Figure Figure3–15 3? ??15 AAGraph Graphfor forSolved SolvedProblem Problem 33 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?3? ? ?37 Figure Figure3–16 3? ??16... Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?3? ??28 Table Table3? ?3 3? ?3 Summary Summaryof ofExtreme ExtremePoint PointAnalysis Analysisfor forExample Example 3-4 3- 4 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.