Transportation and Assignment Solution Procedures

Chapter 6 SupplementTransportation and Assignment Solution ProceduresLearning ObjectivesUse 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.After completing this chapter, you should be able to:2Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–1 Transportation Table for Harley’s Sand and Gravel3Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 6S–1 Overview of the Transportation Method4Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Finding an Initial Feasible Solution: The Northwest-Corner MethodThe Northwest-Corner Methodis 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. 5Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–2 Initial Feasible Solution for Harley Using Northwest-Corner Method6Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Finding an Initial Feasible Solution: The Intuitive ApproachIdentify 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 2 and 3 until all supply and demand have been allocated.7Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–3a Find the Cell That Has the Lowest Unit CostTable 6S–3b Allocate 150 Units to Cell B–28Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–4 200 Units Are Assigned to Cell C–3 and 50 Units AreAssigned to cell A–1Table 6S–5 Completion of the Initial Feasible Solution for the Harley Problem Using the Intuitive Approach9Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–6 Vogel’s Approximation Initial Allocation Tableau with Penalty Costs10Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–7 Initial Feasible Solution Obtained Using the Northwest-Corner MethodTable 6S–8 Evaluation Path for Cell B–111Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–9 Evaluation Path for Cell C–1Table 6S–10 Evaluation Paths for Cells A–3 and C–212Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–11 Initial Feasible Solution Obtained Using the Northwest-Corner MethodEvaluation Using the MODI MethodThe MODI (MOdified DIstribution) method of evaluating a transportation solution for optimality involves the use of index numbers that are established for the rows and columns. These are based on the unit costs of the occupied cells. The index numbers can be used to obtain the cell evaluations for empty cells without the use of stepping-stone paths.13Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–12 Index Numbers for Initial Northwest-Corner Solution to the Harley ProblemRules for Tracing Stepping-Stone PathsAll unoccupied cells must be evaluated. Evaluate cells one at a time.Except for the cell being evaluated, only add or subtract in occupied cells. (It is permissible to skip over occupied cells to find an occupied cell from which the path can continue.)A path will consist of only horizontal and vertical moves, starting and ending with the empty cell that is being evaluated.Alternate + and - signs, beginning with a + sign in the cell being evaluated.14Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–13 Cell Evaluations for Northwest-Corner Solution for the Harley ProblemTable 6S–14 Stepping-Stone Path for Cell A–315Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–15 Distribution Plan after Reallocation of 50 UnitsTable 6S–16 Index Numbers and Cell Evaluations16Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Summary of the Transportation MethodObtain 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 2 and 3 until no cell evaluations (reduced costs) are negative. Once you have identified the optimal solution, compute its total cost.17Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Special IssuesDetermining 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.18Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–17a Index Numbers and Cell EvaluationsTable 6S–17b Alternate Optimal Solution19Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–18 Harley Alternate Solution Modified for DegeneracyTable 6S–19 Solution to Harley Problem with a Prohibited Route20Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–20 A Dummy Origin Is Added to Make Up 80 UnitsTable 6S–21 Solution Using the Dummy Origin21Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–21 Solution Using the Dummy OriginTable 6S–22 Solution Using the Dummy Origin22Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–23 Row ReductionThe Hungarian Methodprovides 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 either jobs (i.e., rows) or employees (i.e., columns).23Copyright © 2007 The McGraw-Hill Companies. All rights reserved. The Hungarian MethodProvides 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 lowest-cost assignment is not made, in terms of either jobs (i.e., rows) or employees (i.e., columns).24Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Requirements for Use of the Hungarian MethodSituations 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.25Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Special SituationsSpecial SituationsCertain 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.26Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–24 Column Reduction of Opportunity (Row Reduction) Costs27Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–25 Determine the Minimum Number of Lines Needed to Coverthe ZerosTable 6S–26 Further Revision of the Cost Table28Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 6S–27 Optimal Assignments29Copyright © 2007 The McGraw-Hill Companies. All rights reserved.