times the number of miles involved, where the distance (in miles) between every pair of offices is as follows:
NETWORK OPTIMIZATION PROBLEMS
1.
The diagram below depicts a system of aqueducts that originate at three rivers (nodes R1, R2, and R3) and terminate at a major city (node T), where the other nodes are junction points in the system. Using units of thousands of acre feet, the following tables show the maximum amount of water that can be pumped through each aqueduct per day.
Management wishes to determine which pairs of offices should be directly connected by special phone lines in order to connect every branch office (directly or indirectly) to the main office at a minimum total cost. a. b.
3.
The city water manager wants to determine a flow plan that will maximize the flow of water to the city. a.
b.
2.
Describe how this problem fits the network description of the minimum spanning tree problem. How should the connection be done?
You need to take a trip by car to another town that you have never visited before. Therefore, you are studying a map to determine the shortest route to your destination. Depending on which route you choose, there are five other towns (call them A, B, C, D, E) that you might pass through on the way. The map shows the mileage along each road that directly connects two towns without any intervening towns. These numbers are summarized in the following table, where a dash indicates that there is no road directly connecting these two towns without going through any other towns.
Formulate this problem as a maximum flow problem by identifying a source, a sink, and the transshipment nodes, and then drawing the complete network that shows the capacity of each arc. Determine the maximum flow of water to the city.
The Premiere Bank soon will be hooking up computer terminals at each of its branch offices to the computer at its main office using special phone lines with telecommunications devices. The phone line from a branch office need not be connected directly to the main office. It can be connected indirectly by being connected to another branch office that is connected (directly or indirectly) to the main office. The only requirement is that every branch office be connected by some route to the main office. The charge for the special phone lines is $100
a.
Formulate this problem as a shortest-path problem by drawing a network where nodes represent towns, links represent roads, and numbers indicate the length of each link in miles.
b.
If each number in the table represented your cost (in tens of pesos) for driving your car from one town to the next, determine your minimum cost route.
6.
During the early 1970s, the political scandal Watergate shook the United States and toppled a presidency. While there were many aspects t o the episode (robbery, enemies lists, abuse of power, cover-ups, etc.), a key component was the ‘laundering’’ of funds from big money contributors to campaign coffers. This practice consists of channeling a large ‘‘gift’’ of money through various banks and
4.
5.
Water is to be transported through a network of pipelines from the big dam to the low valley for irrigation. The network is shown below where arcs represent pipelines and the number on each arc represents the maximum permitted rate of water flow in kilo-tons per hour. It is desired to determine the maximum rate of flow from the big dam to the low valley.
individuals so that its source cannot be traced. Unfortunately, such activities continue today as evidenced by congressional investigations beginning in 1997. Suppose millionaire I. S. Halverson has $5000 (in reality, he would probably have 10 or 100 times this amount) that he would like to donate ‘‘anonymously’’
to the Independent National Party (INP). He might first split the money up in smaller units and deposit the money in several bank accounts spread throughout the world. Money from these accounts could be mixed or further divided and sent to other accounts or individuals, who, in turn, would do the same, until several checks for $1000 or less eventually arrive at party headquarters. To avert suspicion, a limit has been placed on the amount of each transaction between intermediaries. These limits are given in the following network depicting I. S. Halverson, the intermediaries, and the INP. Given these limitations, how much of the $5000 can I. S. Halverson launder to the INP? (Note: The federal government employs management scientists who also use such models to help determine transaction limits that should be monitored.)
One of Speedy Airlines’ flights is about to take off from Seattle for a nonstop flight to London. There is some flexibility in choosing the precise route to be taken, depending upon weather conditions. The following network depicts the possible routes under consideration, where SE and LN are Seattle and London, respectively, and the other nodes represent various intermediate locations. The winds along each arc greatly affect the flying time (and so the fuel consumption). Based on current meteorological reports, the flying times (in hours) for this particular flight are shown next to the arcs. Because the fuel consumed is so expensive, the management of Speedy Airlines has established a policy of choosing the route that minimizes the total flight time.
7.
Thirteen Savage Beasts is a popular rock group that has toured all over North America. It is now beginning its Japanese tour and will be playing to a sold-out house in a major sports stadium in Osaka. After strategically positioning 12 banks of loudspeakers, the manager for the group has found that the local government requires all cables and wires be housed in specially insulated rigid casings. (In the United States, the group simply
lays the cables along the ground or across rafters, but this is unacceptable to the Japanese authorities.) A diagram of the stage area and the 12 banks of loudspeakers is shown in the following figure.
The yearly operating costs include insurance, license, and normal repairs and reflect the fact that the first year carries a full warranty, the second year a limited warranty, and in the fifth year (when the car is four years old at the beginning of the year) there is a major 60,000-mile service. Since John has been a valued lease customer, the dealership will allow him to purchase his current two-year-old vehicle (which cost $36,000 new) for the two-year trade-in price of .50($36,000) = $18,000. a. Complete the following shortest path representation of this problem. What is the minimum amount of the insulated casings the group must purchase before the rock concert can proceed? (Note: Loudspeakers may be connected to one another or directly to the stage.)
8.
John Stanford is at the end of a two-year lease on his Lincoln Town Car, and, although he is determined to drive a Lincoln Town Car for the next four years (until his twins go to college), he simply refuses to lease another car, claiming, ‘‘Ownership is the only way.’’ John can either purchase his two -year-old Lincoln or purchase a new one. At the start of any subsequent year, he can trade in his Lincoln for a new one. At the end of the fourth year, however, he will definitely trade in his Lincoln for a Porsche, which he and his wife will share. John would like to determine the optimal purchase/trade-in policy for the next four years. To aid him in his decision process, the salesperson at the LincolnMercury/Porsche dealership (in whom John places complete trust) has given him the following information. b.
Solve this shortest path problem to determine the optimal purchase/trade-in policy.
