Chapter 6 - Supplement Transportation and assignment solution procedures, after completing this chapter, you should be able to: Use the transportation method to solve problems manually, deal with special cases in solving transportation problems, use the assignment (Hungarian) method to solve problems manually. deal with special cases in solving assignment problems.
Introduction to Management Science with Spreadsheets Stevenson and Ozgur First Edition Chapter 6 Supplement Transportation and Assignment Solution Procedures McGrawHill/Irwin Copyright © 2007 by The McGrawHill Companies, Inc. All rights reserved Learning Objectives After completing this chapter, you should be able to: Use the transportation method to solve problems manually Deal with special cases in solving transportation problems Use the assignment (Hungarian) method to solve problems manually Deal with special cases in solving assignment problems Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S–2 Table Table6S–1 6S–1 Transportation TransportationTable Tablefor forHarley’s Harley’sSand Sandand andGravel Gravel Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S–3 Figure Figure6S–1 6S–1 Overview Overviewofofthe theTransportation TransportationMethod Method Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S–4 Finding Finding an an Initial Initial Feasible Feasible Solution: Solution: The The Northwest-Corner Northwest-Corner Method Method • The Northwest-Corner Method – is a systematic approach for developing an initial feasible solution – is simple to use and easy to understand – does not take transportation costs into account – gets its name because the starting point for the allocation process is the upper-left-hand (northwest) corner of the transportation table Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S–5 Table Table6S–2 6S–2 Initial InitialFeasible FeasibleSolution Solutionfor forHarley HarleyUsing UsingNorthwest-Corner Northwest-Corner Method Method Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S–6 Finding Finding an an Initial Initial Feasible Feasible Solution: Solution: The The Intuitive Intuitive Approach Approach Identify the cell that has the lowest unit cost Cross out the cells in the row or column that has been exhausted (or both, if both have been exhausted), and adjust the remaining row or column total accordingly Identify the cell with the lowest cost from the remaining cells Repeat steps and until all supply and demand have been allocated Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S–7 Table Table6S–3a 6S–3a Find Findthe theCell CellThat ThatHas Hasthe theLowest LowestUnit UnitCost Cost Table Table6S 6S–3b –3b Allocate Allocate150 150Units Unitsto toCell CellB–2 B–2 Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S–8 Table Table6S–4 6S–4 200 200Units UnitsAre AreAssigned Assignedto toCell CellC–3 C–3and and50 50Units UnitsAre Are Assigned Assignedtotocell cellA–1 A–1 Table Table6S 6S–5 –5 Completion Completionofofthe theInitial InitialFeasible FeasibleSolution Solutionfor forthe theHarley Harley Problem ProblemUsing Usingthe theIntuitive IntuitiveApproach Approach Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S–9 Table Table6S 6S–6 –6 Vogel’s Vogel’sApproximation ApproximationInitial InitialAllocation AllocationTableau Tableauwith withPenalty Penalty Costs Costs Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 10 Table Table6S–13 6S–13 Cell CellEvaluations Evaluationsfor forNorthwest-Corner Northwest-CornerSolution Solutionfor forthe theHarley Harley Problem Problem Table Table6S 6S–14 –14 Stepping-Stone Stepping-StonePath Pathfor forCell CellA–3 A–3 Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 15 Table Table6S–15 6S–15 Distribution DistributionPlan Planafter afterReallocation Reallocationof of50 50Units Units Table Table6S–16 6S–16 Index IndexNumbers Numbersand andCell CellEvaluations Evaluations Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 16 Summary Summary of of the the Transportation Transportation Method Method Obtain an initial feasible solution Use either the northwest-corner method, the intuitive method, or the Vogel’s approximation method Generally, the intuitive method and Vogel’s approximation are the preferred approaches Evaluate the solution to determine if it is optimal Use either the stepping-stone method or MODI The solution is not optimal if any unoccupied cell has a negative cell evaluation If the solution is not optimal, select the cell that has the most negative cell evaluation Obtain an improved solution using the stepping-stone method Repeat steps and until no cell evaluations (reduced costs) are negative Once you have identified the optimal solution, compute its total cost McGraw Copyright © 2007 The McGrawHill Companies. All rights reserved. Hill/Irwin 6S– 17 Special Special Issues Issues Determining if there are alternate optimal solutions Recognizing and handling degeneracy (too few occupied cells to permit evaluation of a solution) Avoiding unacceptable or prohibited route assignments Dealing with problems in which supply and demand are not equal Solving maximization problems Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 18 Table Table6S–17a 6S–17a Index IndexNumbers Numbersand andCell CellEvaluations Evaluations Table Table6S–17b 6S–17b Alternate AlternateOptimal OptimalSolution Solution Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 19 Table Table6S–18 6S–18 Harley HarleyAlternate AlternateSolution SolutionModified Modifiedfor forDegeneracy Degeneracy Table Table6S–19 6S–19 Solution SolutiontotoHarley HarleyProblem Problemwith withaaProhibited ProhibitedRoute Route Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 20 Table Table6S–20 6S–20 AADummy DummyOrigin OriginIsIsAdded Addedto toMake MakeUp Up80 80Units Units Table Table6S–21 6S–21 Solution SolutionUsing Usingthe theDummy DummyOrigin Origin Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 21 Table Table6S–21 6S–21 Solution SolutionUsing Usingthe theDummy DummyOrigin Origin Table Table6S–22 6S–22 Solution SolutionUsing Usingthe theDummy DummyOrigin Origin Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 22 Table Table6S–23 6S–23 Row RowReduction Reduction •The Hungarian Method • provides a simple heuristic that can be used to find the optimal set of assignments It is easy to use, even for fairly large problems It is based on minimization of opportunity costs that would result from potential pairings These are additional costs that would be incurred if the lowest-cost assignment is not made, in terms of McGraw either jobs (i.e., rows) or employees (i.e., columns) Copyright © 2007 The McGrawHill Companies. All rights reserved. Hill/Irwin 6S– 23 The The Hungarian Hungarian Method Method • Provides a simple heuristic that can be used to find the optimal set of assignments • Is easy to use, even for fairly large problems • Is based on minimization of opportunity costs that would result from potential pairings – These additional costs would be incurred if the lowestcost assignment is not made, in terms of either jobs (i.e., rows) or employees (i.e., columns) Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 24 Requirements Requirements for for Use Use of of the the Hungarian Hungarian Method Method • Situations in which the Hungarian method can be used are characterized by the following: There needs to be a one-for-one matching of two sets of items The goal is to minimize costs (or to maximize profits) or a similar objective (e.g., time, distance, etc.) The costs or profits (etc.) are known or can be closely estimated McGraw Copyright © 2007 The McGrawHill Hill/Irwin 6S– Companies. All rights reserved. 25 Special Special Situations Situations • Special Situations – Certain situations can arise in which the model deviates slightly from that previously described • Among those situations are the following: – The number of rows does not equal the number of columns – The problem involves maximization rather than minimization – Certain matches are undesirable or not allowed – Multiple optimal solutions exist McGraw Copyright © 2007 The McGrawHill Hill/Irwin 6S– Companies. All rights reserved. 26 Table Table6S–24 6S–24 Column ColumnReduction ReductionofofOpportunity Opportunity(Row (RowReduction) Reduction)Costs Costs Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 27 Table Table6S–25 6S–25 Determine Determinethe theMinimum MinimumNumber NumberofofLines LinesNeeded NeededtotoCover Cover the theZeros Zeros Table Table6S 6S–26 –26 Further FurtherRevision Revisionofofthe theCost CostTable Table Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 28 Table Table6S–27 6S–27 Optimal OptimalAssignments Assignments Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 6S– 29 ... Hill/Irwin ? ?6S? ??8 Table Table6S–4 6S? ??4 200 200Units UnitsAre AreAssigned Assignedto toCell CellC–3 C–3and and50 50Units UnitsAre Are Assigned Assignedtotocell cellA–1 A–1 Table Table6S 6S? ??5 –5 Completion... Table Table6S–12 6S? ??12 Index IndexNumbers Numbersfor forInitial InitialNorthwest-Corner Northwest-CornerSolution Solutiontotothe the Harley HarleyProblem Problem Rules for Tracing Stepping-Stone Paths... CellEvaluations Evaluationsfor forNorthwest-Corner Northwest-CornerSolution Solutionfor forthe theHarley Harley Problem Problem Table Table6S 6S? ??14 –14 Stepping-Stone Stepping-StonePath Pathfor forCell CellA–3