Introduction to
Management Science
with Spreadsheets
Stevenson and Ozgur
First Edition
Chapter 6 Supplement
Transportation and
Assignment Solution
Procedures
McGraw-Hill/Irwin
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Learning Objectives
After completing this chapter, you should be able to:
1. Use the transportation method to solve problems
manually.
2. Deal with special cases in solving transportation
problems.
3. Use the assignment (Hungarian) method to solve
problems manually.
4. Deal with special cases in solving assignment
problems.
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McGraw-Hill/Irwin 6S–2
Table 6S–1
Transportation Table for Harley’s Sand and Gravel
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McGraw-Hill/Irwin 6S–3
Figure 6S–1
Overview of the Transportation Method
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Finding an Initial Feasible Solution:
The Northwest-Corner 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.
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Table 6S–2
Initial Feasible Solution for Harley Using Northwest-Corner
Method
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Finding an Initial Feasible Solution:
The Intuitive Approach
1. Identify the cell that has the lowest unit cost.
2. 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.
3. Identify the cell with the lowest cost from the
remaining cells.
4. Repeat steps 2 and 3 until all supply and demand
have been allocated.
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McGraw-Hill/Irwin 6S–7
Table 6S–3a
Find the Cell That Has the Lowest Unit Cost
Table 6S–3b
Allocate 150 Units to Cell B–2
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Table 6S–4
200 Units Are Assigned to Cell C–3 and 50 Units Are
Assigned to cell A–1
Table 6S–5
Completion of the Initial Feasible Solution for the Harley
Problem Using the Intuitive Approach
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McGraw-Hill/Irwin 6S–9
Table 6S–6
Vogel’s Approximation Initial Allocation Tableau with Penalty
Costs
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Table 6S–7
Initial Feasible Solution Obtained Using the Northwest-Corner
Method
Table 6S–8
Evaluation Path for Cell B–1
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McGraw-Hill/Irwin 6S–11
Table 6S–9
Evaluation Path for Cell C–1
Table 6S–10
Evaluation Paths for Cells A–3 and C–2
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McGraw-Hill/Irwin 6S–12
Table 6S–11
Initial Feasible Solution Obtained Using the Northwest-Corner
Method
Evaluation Using the MODI Method
The 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.
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McGraw-Hill/Irwin 6S–13
Table 6S–12
Index Numbers for Initial Northwest-Corner Solution to the
Harley Problem
Rules for Tracing Stepping-Stone Paths
1. All unoccupied cells must be evaluated. Evaluate cells one at a time.
2. 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.)
3. A path will consist of only horizontal and vertical moves, starting and
ending with the empty cell that is being evaluated.
4. Alternate + and - signs, beginning with a + sign in the cell being evaluated.
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Table 6S–13
Cell Evaluations for Northwest-Corner Solution for the Harley
Problem
Table 6S–14
Stepping-Stone Path for Cell A–3
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Table 6S–15
Distribution Plan after Reallocation of 50 Units
Table 6S–16
Index Numbers and Cell Evaluations
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Summary of the Transportation Method
1. 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.
2. 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.
3. 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.
4. Repeat steps 2 and 3 until no cell evaluations (reduced costs) are
negative. Once you have identified the optimal solution, compute
its total cost.
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Special Issues
1. Determining if there are alternate optimal solutions.
2. Recognizing and handling degeneracy (too few
occupied cells to permit evaluation of a solution).
3. Avoiding unacceptable or prohibited route
assignments.
4. Dealing with problems in which supply and demand
are not equal.
5. Solving maximization problems.
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Table 6S–17a Index Numbers and Cell Evaluations
Table 6S–17b
Alternate Optimal Solution
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Table 6S–18
Harley Alternate Solution Modified for Degeneracy
Table 6S–19
Solution to Harley Problem with a Prohibited Route
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Table 6S–20
A Dummy Origin Is Added to Make Up 80 Units
Table 6S–21
Solution Using the Dummy Origin
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Table 6S–21
Solution Using the Dummy Origin
Table 6S–22
Solution Using the Dummy Origin
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McGraw-Hill/Irwin 6S–22
Table 6S–23
Row 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
either jobs (i.e., rows) or employees (i.e., columns).
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McGraw-Hill/Irwin 6S–23
The Hungarian 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).
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Requirements for Use of
the Hungarian Method
• Situations in which the Hungarian method can
be used are characterized by the following:
1. There needs to be a one-for-one matching of two
sets of items.
2. The goal is to minimize costs (or to maximize
profits) or a similar objective (e.g., time, distance,
etc.).
3. The costs or profits (etc.) are known or can be
closely estimated.
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Special 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.
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McGraw-Hill/Irwin 6S–26
Table 6S–24
Column Reduction of Opportunity (Row Reduction) Costs
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McGraw-Hill/Irwin 6S–27
Table 6S–25
Determine the Minimum Number of Lines Needed to Cover
the Zeros
Table 6S–26
Further Revision of the Cost Table
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McGraw-Hill/Irwin 6S–28
Table 6S–27
Optimal Assignments
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McGraw-Hill/Irwin 6S–29