1. Table of Contents Chapter 6 (Network Optimization Problems)
Minimum-Cost Flow Problems (Section 6.1) 6.2–6.12
A Case Study: The BMZ Maximum Flow Problem (Section 6.2) 6.13–6.16
Maximum Flow Problems (Section 6.3) 6.17–6.21
Shortest Path Problems: Littletown Fire Department (Section 6.4) 6.22–6.25
Shortest Path Problems: General Characteristics (Section 6.4) 6.26–6.27
Shortest Path Problems: Minimizing Sarah’s Total Cost (Section 6.4) 6.28–6.31
Shortest Path Problems: Minimizing Quick’s Total Time (Section 6.4) 6.32–6.36
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

2. Distribution Unlimited Co. Problem
The Distribution Unlimited Co. has two factories producing a product that needs to be shipped to two warehouses
Factory 1 produces 80 units.
Factory 2 produces 70 units.
Warehouse 1 needs 60 units.
Warehouse 2 needs 90 units.
There are rail links directly from Factory 1 to Warehouse 1 and Factory 2 to Warehouse 2.
Independent truckers are available to ship up to 50 units from each factory to the distribution center, and then 50 units from the distribution center to each warehouse.
Question: How many units (truckloads) should be shipped along each shipping lane?
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3. The Distribution Network
Figure 6.1 The distribution network for the Distribution Unlimited Co. problem, where each feasible shipping lane is represented by an arrow.
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4. Data for Distribution Network
Figure 6.2 The data for the distribution network for the Distribution Unlimited Co. problem.
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5. A Network Model
Figure 6.3 A network model for the Distribution Unlimited Co. problem as a minimum-cost flow problem.
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6. The Optimal Solution
Figure 6.4 The optimal solution for the Distribution Unlimited Co. problem, where the shipping amounts are shown in parentheses over the arrows.
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7. Terminology for Minimum-Cost Flow Problems
The model for any minimum-cost flow problem is represented by a network with flow passing through it.
The circles in the network are called nodes.
Each node where the net amount of flow generated (outflow minus inflow) is a fixed positive number is a supply node.
Each node where the net amount of flow generated is a fixed negative number is a demand node.
Any node where the net amount of flow generated is fixed at zero is a transshipment node. Having the amount of flow out of the node equal the amount of flow into the node is referred to as conservation of flow.
The arrows in the network are called arcs.
The maximum amount of flow allowed through an arc is referred to as the capacity of that arc.
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8. Assumptions of a Minimum-Cost Flow Problem
At least one of the nodes is a supply node.
At least one of the other nodes is a demand node.
All the remaining nodes are transshipment nodes.
Flow through an arc is only allowed in the direction indicated by the arrowhead, where the maximum amount of flow is given by the capacity of that arc. (If flow can occur in both directions, this would be represented by a pair of arcs pointing in opposite directions.)
The network has enough arcs with sufficient capacity to enable all the flow generated at the supply nodes to reach all the demand nodes.
The cost of the flow through each arc is proportional to the amount of that flow, where the cost per unit flow is known.
The objective is to minimize the total cost of sending the available supply through the network to satisfy the given demand. (An alternative objective is to maximize the total profit from doing this.)
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9. Properties of Minimum-Cost Flow Problems
The Feasible Solutions Property: Under the previous assumptions, a minimum-cost flow problem will have feasible solutions if and only if the sum of the supplies from its supply nodes equals the sum of the demands at its demand nodes.
The Integer Solutions Property: As long as all the supplies, demands, and arc capacities have integer values, any minimum-cost flow problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its flow quantities.
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10. Spreadsheet Model
Figure 6.5 A spreadsheet model for the Distribution Unlimited Co. minimum-cost flow problem, including the target cell Total Cost (D11). The changing cells Ship (D4:D9) show the optimal shipping quantities through the distribution network obtained by the Solver.
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11. =SUMIF(“From labels”, x, “Flow”) – SUMIF(“To labels”, x, “Flow”)
The SUMIF Function
The SUMIF formula can be used to simplify the node flow constraints.
=SUMIF(Range A, x, Range B)
For each quantity in (Range A) that equals x, SUMIF sums the corresponding entries in (Range B).
The net outflow (flow out – flow in) from node x is then
=SUMIF(“From labels”, x, “Flow”) – SUMIF(“To labels”, x, “Flow”)
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12. Typical Applications of Minimum-Cost Flow Problems
Kind of Application
Supply Nodes
Transshipment Nodes
Demand Nodes
Operation of a distribution network
Sources of goods
Intermediate storage facilities
Customers
Solid waste management
Sources of solid waste
Processing facilities
Landfill locations
Operation of a supply network
Vendors
Intermediate warehouses
Coordinating product mixes at plants
Plants
Production of a specific product
Market for a specific product
Cash flow management
Sources of cash at a specific time
Short-term investment options
Needs for cash at a specific time
Table 6.1 Typical kinds of applications of minimum-cost flow problems.
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13. The BMZ Maximum Flow Problem
The BMZ Company is a European manufacturer of luxury automobiles. Its exports to the United States are particularly important.
BMZ cars are becoming especially popular in California, so it is particularly important to keep the Los Angeles center well supplied with replacement parts for repairing these cars.
BMZ needs to execute a plan quickly for shipping as much as possible from the main factory in Stuttgart, Germany to the distribution center in Los Angeles over the next month.
The limiting factor on how much can be shipped is the limited capacity of the company’s distribution network.
Question: How many units should be sent through each shipping lane to maximize the total units flowing from Stuttgart to Los Angeles?
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14. The BMZ Distribution Network
Figure 6.6 The BMZ Co. distribution network from its main factory in Stuttgart, Germany, to a distribution center in Los Angeles.
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15. A Network Model for BMZ
Figure 6.7 A network model for the BMZ Co. problem as a maximum flow problem, where the number in square brackets below each arc is the capacity of that arc.
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16. Spreadsheet Model for BMZ
Figure 6.8 A spreadsheet model for the BMZ Co. maximum flow problem.
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17. Assumptions of Maximum Flow Problems
All flow through the network originates at one node, called the source, and terminates at one other node, called the sink. (The source and sink in the BMZ problem are the factory and the distribution center, respectively.)
All the remaining nodes are transshipment nodes.
Flow through an arc is only allowed in the direction indicated by the arrowhead, where the maximum amount of flow is given by the capacity of that arc. At the source, all arcs point away from the node. At the sink, all arcs point into the node.
The objective is to maximize the total amount of flow from the source to the sink. This amount is measured in either of two equivalent ways, namely, either the amount leaving the source or the amount entering the sink.
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18. BMZ with Multiple Supply and Demand Points
BMZ has a second, smaller factory in Berlin.
The distribution center in Seattle has the capability of supplying parts to the customers of the distribution center in Los Angeles when shortages occur at the latter center.
Question: How many units should be sent through each shipping lane to maximize the total units flowing from Stuttgart and Berlin to Los Angeles and Seattle?
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19. Network Model for The Expanded BMZ Problem
Figure 6.9 A network model for the expanded BMZ Co. problem as a maximum flow problem, where the number in square brackets below each arc is the capacity of that arc.
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20. Spreadsheet Model
Figure A spreadsheet model for the expanded BMZ Co. problem as a variant of a maximum flow problem with sources in both Stuttgart and Berlin and sinks in both Los Angeles and Seattle. Using the target cell Maximum Flow (D21) to maximize the total flow from the two sources to the two sinks, the Solver yields the optimal shipping plan shown in the changing cells Ship (D4:D19).
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21. Some Applications of Maximum Flow Problems
Maximize the flow through a distribution network, as for BMZ.
Maximize the flow through a company’s supply network from its vendors to its processing facilities.
Maximize the flow of oil through a system of pipelines.
Maximize the flow of water through a system of aqueducts.
Maximize the flow of vehicles through a transportation network.
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22. Littletown Fire Department
Littletown is a small town in a rural area.
Its fire department serves a relatively large geographical area that includes many farming communities.
Since there are numerous roads throughout the area, many possible routes may be available for traveling to any given farming community.
Question: Which route from the fire station to a certain farming community minimizes the total number of miles?
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23. The Littletown Road System
Figure The road system between the Littletown Fire Station and a certain farming community, where A, B, … , H are junctions and the number next to each road shows its distance in miles.
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24. The Network Representation
Figure The network representation of Figure 6.11 as a shortest path problem.
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25. Spreadsheet Model
Figure A spreadsheet model for the Littletown Fire Department shortest path problem, including the target cell Total Distance (D29). The values of 1 in the changing cells On Route (D4:D27) reveal the optimal solution obtained by the Solver for the shortest path (19 miles) from the fire station to the farming community.
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26. Assumptions of a Shortest Path Problem
You need to choose a path through the network that starts at a certain node, called the origin, and ends at another certain node, called the destination.
The lines connecting certain pairs of nodes commonly are links (which allow travel in either direction), although arcs (which only permit travel in one direction) also are allowed.
Associated with each link (or arc) is a nonnegative number called its length. (Be aware that the drawing of each link in the network typically makes no effort to show its true length other than giving the correct number next to the link.)
The objective is to find the shortest path (the path with the minimum total length) from the origin to the destination.
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27. Applications of Shortest Path Problems
Minimize the total distance traveled.
Minimize the total cost of a sequence of activities.
Minimize the total time of a sequence of activities.
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28. Minimizing Total Cost: Sarah’s Car Fund
Sarah has just graduated from high school.
As a graduation present, her parents have given her a car fund of $21,000 to help purchase and maintain a three-year-old used car for college.
Since operating and maintenance costs go up rapidly as the car ages, Sarah may trade in her car on another three-year-old car one or more times during the next three summers if it will minimize her total net cost. (At the end of the four years of college, her parents will trade in the current used car on a new car for Sarah.)
Question: When should Sarah trade in her car (if at all) during the next three summers?
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29. Sarah’s Cost Data Operating and Maintenance Costs for Ownership Year
Trade-in Value at End of Ownership Year
Purchase Price 1 2 3 4
$12,000
$2,000
$3,000
$4,500
$6,500
$8,500
Table 6.2 Sarah’s data each time she purchases a three-year-old car
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30. Shortest Path Formulation
Figure Formulation of the problem of when Sarah should trade in her car as a shortest path problem. The node labels measure the number of years from now. Each arc represents purchasing a car and then trading it in later.
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31. Spreadsheet Model
Figure A spreadsheet model that formulates Sarah’s problem as a shortest path problem where the objective is to minimize the total cost instead of the total distance. After applying the Solver, the values of 1 in the changing cells On Route (D12:D21) identify the shortest (least expensive) path for scheduling trade-ins.
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32. Minimizing Total Time: Quick Company
The Quick Company has learned that a competitor is planning to come out with a new kind of product with great sales potential.
Quick has been working on a similar product that had been scheduled to come to market in 20 months.
Quick’s management wishes to rush the product out to meet the competition.
Each of four remaining phases can be conducted at a normal pace, at a priority pace, or at crash level to expedite completion. However, the normal pace has been ruled out as too slow for the last three phases.
$30 million is available for all four phases.
Question: At what pace should each of the four phases be conducted?
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33. Time and Cost of the Four Phases
Level
Remaining Research
Development
Design of Mfg. System
Initiate Production and Distribution
Normal
5 months —
Priority
4 months
3 months
2 months
Crash
1 month
$3 million
6 million
$6 million
$9 million
9 million
12 million
Table 6.3 Time required for the phases of preparing Quick Company’s new product.
Table 6.4 Cost for the phases of preparing Quick Company’s new product.
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34. Shortest Path Formulation
Figure Formulation of the Quick Company problem as a shortest path problem. Except for the dummy destination, the arc labels indicate, first, the number of phases completed and, second, the amount of money left (in millions of dollars) for the remaining phases. Each arc length gives the time (in months) to perform that phase.
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35. Spreadsheet Model
Figure A spreadsheet model that formulates the Quick Company problem as a shortest path problem where the objective is to minimize the total time instead of the total distance, so the target cell is Total Time (D32). The values of 1 in the changing cells On Route (D4:D30) reveal the shortest (quickest) path obtained by the Solver.
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36. The Optimal Solution Phase Level Time Cost Remaining research Crash
2 months
$9 million
Development
Priority
3 months
6 million
Design of manufacturing system
12 million
Initiate production and distribution
3 million
Total
10 months
$30 million
Table 6.5 The optimal solution obtained by the Excel Solver for Quick Company’s shortest path problem.
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