Lecture Introduction to Management Science with Spreadsheets: Chapter 4S - Stevenson, Ozgur

Chapter 4 - Supplement Linear programming: The simplex method, after completing this chapter, you should be able to: Explain the ways in which the simplex method is superior to the graphical method for solving linear programming problems, solve small maximization problems manually using the simplex method, interpret simplex solutions,...

Introduction to Management Science with Spreadsheets Stevenson and Ozgur First Edition Part Deterministic Decision Models Chapter 4 Supplement Linear Programming: The Simplex Method McGraw­Hill/Irwin Copyright © 2007 by The McGraw­Hill Companies, Inc. All rights reserved Learning Objectives After completing this chapter, you should be able to: Explain the ways in which the simplex method is superior to the graphical method for solving linear programming problems Solve small maximization problems manually using the simplex method Interpret simplex solutions Convert = and > constraints into standard form Solve maximization problems that have mixed constraints and interpret those solutions Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S–2 Learning Objectives (cont’d) After completing this chapter, you should be able to: Solve minimization problems and interpret those solutions Discuss unbound solutions, degeneracy, and multiple optimal solutions in terms of the simplex method and recognize infeasibility in a simplex solution Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S–3 Overview Overview of of the the Simplex Simplex Method Method • Advantages and Characteristics – More realistic approach as it is not limited to problems with two decision variables – Systematically examines basic feasible solutions for an optimal solution – Based on the solutions of linear equations (equalities) using slack variables to achieve equality • Rule – Linear programming models have fewer equations than variables; unless the number of equations equals the number of variables, a unique solution cannot be found Copyright © 2007 The McGraw­Hill  McGraw­ Companies. All rights reserved.   Hill/Irwin  4S–4 Developing Developing the the Initial Initial Simplex Simplex Tableau Tableau • Notation used in the simplex tableau: Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S–5 Figure Figure4S–1 4S–1 Comparison ComparisonofofServer ServerModel Modeland andGeneral GeneralSimplex SimplexNotation Notation Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S–6 Table Table4S–1 4S–1 Comparison ComparisonofofServer ServerModel Modeland andGeneral GeneralSimplex SimplexNotation Notation(cont’d) (cont’d) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S–7 Table Table4S–2 4S–2 Completed CompletedInitial InitialTableau Tableaufor forthe theServer ServerProblem Problem Unit Vector Each tableau represents a basic feasible solution to the problem A simplex solution in a maximization problem is optimal if the C–Z row consists entirely of zeros and negative numbers (i.e., there are no positive values in the bottom row) When this has been achieved, there is no opportunity for improving the solution Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S–8 Table Table4S–3 4S–3 Determining Determiningthe theEntering Enteringand andExiting ExitingVariables Variables Select the leaving variable as the one that has the smallest nonnegative ratio of quantity divided by substitution rate Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S–9 Figure Figure4S–2 4S–2 The TheNext NextCorner CornerPoint PointIsIsDetermined Determinedby bythe theMost MostLimiting Limiting Constraint Constraint Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 10 Figure Figure4S–6 4S–6 Graph Graphofofthe theProblem ProblemininExample Example4S-2 4S-2 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 32 Table Table4S–19 4S–19 Initial InitialTableau Tableaufor forExample Example4S-2 4S-2 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 33 Table Table4S–20 4S–20 Second SecondTableau Tableaufor forExample Example4S-2 4S-2 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 34 Table Table4S–21 4S–21 Third ThirdTableau Tableaufor forExample Example4S-2 4S-2 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 35 Figure Figure4S–7 4S–7 Sequence SequenceofofTableaus Tableausfor forSolution Solutionof ofExample Example4S-2 4S-2 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 36 Some Some Special Special Issues Issues • Unbounded Solutions – A solution is unbounded if the objective function can be improved without limit – An unbounded solution will exist if there are no positive values in the pivot column • Degeneracy – A conditions that occurs when there is a tie for the lowest nonnegative ratio which, theoretically, makes it possible for subsequent solutions to cycle (i.e., to return to previous solutions) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 37 Example Example4S 4S––33 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 38 Table Table4S–22 4S–22 Second SecondTableau Tableau Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 39 Table Table4S–23 4S–23 Final FinalSimplex SimplexTableau Tableau Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 40 Some Some Special Special Issues Issues (cont’d) (cont’d) • Multiple Optimal Solutions – Occur when the same maximum value of the objective function might be possible with a number of different combinations of values of the decision variables because the objective function is parallel to a binding constraint Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 41 Table Table4S–24 4S–24 Final FinalTableau Tableaufor forModified ModifiedServer ServerProblem Problemwith withan an Alternative AlternativeOptimal OptimalSolution Solution Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 42 Table Table4S–25 4S–25 The TheAlternate AlternateOptimal OptimalSolution Solutionfor forthe theModified Modified Server ServerProblem Problem Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 43 Some Some Special Special Issues Issues (cont’d) (cont’d) • Infeasibility – A problem in which no combination of decision and slack/surplus variables will simultaneously satisfy all constraints – Can be the result of an error in formulating a problem or it can be because the existing set of constraints is too restrictive to permit a solution – Recognized by the presence of an artificial variable in a solution that appears optimal (i.e., a tableau in which the signs of the values in row C – Z indicate optimality), and it has a nonzero quantity McGraw­ Copyright © 2007 The McGraw­Hill  Hill/Irwin  4S– Companies. All rights reserved.   44 Table Table4S–26 4S–26 Simplex SimplexTableaus Tableausfor forInfeasibility InfeasibilityProblem Problemfor forExample Example4S-4 4S-4 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 45 Example Example4S 4S––44 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  4S– 46 ... Example4S-2 4S- 2 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?4S? ?? 34 Table Table4S–21 4S? ??21 Third ThirdTableau Tableaufor forExample Example4S-2 4S- 2 Copyright © 2007 The McGraw­Hill ... Example4S-1 4S- 1 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?4S? ?? 26 Table Table4S–16 4S? ??16 The TheSecond SecondTableau Tableaufor forExample Example4S-1 4S- 1... Hill/Irwin ? ?4S? ?? 24 Figure Figure4S–4 4S? ??4 Graph Graphfor forExample Example4S-1 4S- 1 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?4S? ?? 25 Table Table4S–15 4S? ??15