Introduction to
Management Science
with Spreadsheets
Stevenson and Ozgur
First Edition
Part 2 Deterministic Decision Models
Chapter 6
Transportation,
Transshipment, and
Assignment Problems
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. Describe the nature of transportation,
transshipment, and assignment problems.
2. Formulate a transportation problem as a linear
programming model.
3. Use the transportation method to solve problems
with Excel.
4. Solve maximization transportation problems,
unbalanced problems, and problems with prohibited
routes.
5. Solve aggregate planning problems using the
transportation model.
Copyright © 2007 The McGraw-Hill Companies. All rights reserved.
McGraw-Hill/Irwin 6–2
Learning Objectives (cont’d)
After completing this chapter, you should be able to:
6. Formulate a transshipment problem as a linear
programming model.
7. Solve transshipment problems with Excel.
8. Formulate an assignment problem as a linear
programming model.
9. Use the assignment method to solve problems with
Excel.
Copyright © 2007 The McGraw-Hill Companies. All rights reserved.
McGraw-Hill/Irwin 6–3
Transportation Problems
• Transportation Problem
–A distribution-type problem in which supplies of goods
that are held at various locations are to be distributed
to other receiving locations.
–The solution of a transportation problem will indicate
to a manager the quantities and costs of various
routes and the resulting minimum cost.
–Used to compare location alternatives in deciding
where to locate factories and warehouses to achieve
the minimum cost distribution configuration.
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Formulating the Model
• A transportation problem
– Typically involves a set of sending locations, which
are referred to as origins, and a set of receiving
locations, which are referred to as destinations.
– To develop a model of a transportation problem, it is
necessary to have the following information:
1. Supply quantity (capacity) of each origin.
2. Demand quantity of each destination.
3. Unit transportation cost for each origin-destination route.
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Transshipment Problems
• Transshipment Problems
–A transportation problem in which some locations are
used as intermediate shipping points, thereby serving
both as origins and as destinations.
–Involve the distribution of goods from intermediate
nodes in addition to multiple sources and multiple
destinations.
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Assignment Problems
• The Assignment-type Problems
–Involve the matching or pairing of two sets of items
such as jobs and machines, secretaries and reports,
lawyers and cases, and so forth.
–Have different cost or time requirements for different
pairings.
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Figure 6–1
Schematic of a Transportation Problem
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Table 6–1
Transportation Table for Harley’s Sand and Gravel
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Special Cases of Transportation Problems
• Maximization
–Transportation-type problems that concern profits or
revenues rather than costs with the objective to
maximize profits rather than to minimize costs.
• Unacceptable Routes
–Certain origin-destination combinations may be
unacceptable due to weather factors, equipment
breakdowns, labor problems, or skill requirements that
either prohibit, or make undesirable, certain
combinations (routes).
Copyright © 2007 The McGraw-Hill Companies. All rights reserved.
McGraw-Hill/Irwin 6–10
Special Cases of Transportation Problems
(cont’d)
• Unequal Supply and Demand
–Situations in which supply and demand are not equal
such that it is necessary to modify the original problem
so that supply and demand are equalized.
–Quantities in dummy routes in the optimal solution are
not shipped and serve to indicate which supplier will
hold the excess supply, and how much, or which
destination will not receive its total demand, and how
much it will be short.
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McGraw-Hill/Irwin 6–11
Exhibit 6-1
Input and Output Worksheet for the Transportation (topsoil) Problem
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McGraw-Hill/Irwin 6–12
Exhibit 6-2
Parameter Specification Screen for the Topsoil Transportation
Problem
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McGraw-Hill/Irwin 6–13
Exhibit 6–3
Solver Options Screen
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McGraw-Hill/Irwin 6–14
Exhibit 6–4
Solver Results
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McGraw-Hill/Irwin 6–15
Exhibit 6–5
Answer Report for the Topsoil Transportation Problem
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McGraw-Hill/Irwin 6–16
Exhibit 6–6
Sensitivity Report for the Topsoil Transportation Problem
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McGraw-Hill/Irwin 6–17
Exhibit 6–7
Input and Output Sheet for the Revised Transportation (topsoil)
Problem When the Shipping Route between Farm B and Project 1 Is
Prohibited
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Figure 6–2
A Network Diagram of a Transshipment Problem
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Example 6-2
Transshipment Problem
The manager of Harley’s Sand and Gravel Pit has decided to utilize two
intermediate nodes as transshipment points for temporary storage of topsoil.
The revised diagram of the transshipment problem is given in Figure 6-3.
Table 6–2
Cost of Shipping One Unit from the Farms to Warehouses
Table 6–2
Cost of Shipping One Unit from the Warehouses to Projects
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Figure 6–3
A Network Diagram of Harley’s Sand and Gravel Pit
Transshipment Example
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Exhibit 6–8
Excel Input and Output Screen for the Transshipment Problem
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McGraw-Hill/Irwin 6–22
Exhibit 6–9
Parameter Specifications Screen for the Transshipment
Problem
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McGraw-Hill/Irwin 6–23
Using the Transportation Problem to Solve
Aggregate Planning Problems
• Aggregate Planning
–Involves creating a long-term production plan for
achieving a demand-supply balance.
–Aggregate planners usually avoid in terms of thinking
of individual products.
–Planners are concerned about the quantity and timing
of production to meet the expected demand.
–Aggregate planners attempt to minimize the
production cost over the planning horizon.
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Table 6–4
Transportation Table for Aggregate Planning Purposes
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McGraw-Hill/Irwin 6–25
Example 6-3
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Table 6–5
Transportation Table for the Aggregate Planning Problem of
Example 6-3
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Using the Transportation Problem to Solve
Location Planning Problems
• Location Analysis
–Comparing transportation costs for alternative
locations for new facilities to minimize total cost.
–Provides planners an opportunity to assess the impact
of each warehouse location on the total distribution
costs for the system.
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Table 6–6
System with Chicago Warehouse
Table 6–7
System with Detroit Warehouse
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Example 6-4
A manager has prepared a table that shows the cost of performing
each of five jobs by each of five employees (see Table 6-8). According
to this table, job I will cost $15 if done by Al. $20 if it is done by Bill,
and so on. The manager has stated that his goal is to develop a set of
job assignments that will minimize the total cost of getting all four
jobs done. It is further required that the jobs be performed
simultaneously, thus requiring one job being assigned to each
employee.
In the past, to find the minimum-cost set of assignments, the
manager has resorted to listing all of the different possible
assignments (i.e., complete enumeration) for small problems such as
this one. But for larger problems, the manager simply guesses
because there are too many possibilities to try to list them. For
example, with a 5X5 table, there are 5! = 120 different possibilities; but
with, say, a 7X7 table, there are 7! = 5,040 possibilities.
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Table 6–8
Numerical Example for the Assignment Problem
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Exhibit 6–10
Excel Input and Output Worksheet for the Assignment Problem
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Exhibit 6–11
Parameter Specifications Screen for the Assignment Problem
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Exhibit 6–12
Excel Worksheet for the Transportation Problem in Solved Problem 1
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Exhibit 6–13
Parameter Specification Screen for Solved Problem 1
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Exhibit 6–14
Excel Worksheet for the Assignment Problem in Solved Problem 2
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Exhibit 6–15
Parameter Specification Screen for Solved Problem 2
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Exhibit 6–16
Excel Worksheet for the Transportation Problem in Solved Problem 3
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Exhibit 6–17
Parameter Specification Screen for Solved Problem 3
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