ΠΜΣ: Μαθηματικά των Υπολογιστών και των Αποφάσεων

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4. TURNER AIRLINES Fred Turner is a successful entrepreneur in Atlanta, Georgia who has just hit upon his next moneymaking idea. Recently, Fred was planning to fly to Washington, DC early one morning for a business meeting. Fred's own personal Leer jet happened to have a mechanical problem that morning, which forced Fred to try to book a flight on a commercial carrier. Much to his dismay, Fred learned that all the flights to DC were overbooked but he was told might be able to get a seat if he waited at the gate as a standby passenger. When Fred arrived at the gate, he was surprised to find several people waiting, hoping to get on a flight as a standby passenger. When Fred saw this situation, he immediately saw a business opportunity. He knew that the route from Atlanta to Washington, DC was a popular one as these airports are two of the busiest in the world. And the frustration he saw on the faces of those waiting in line with him convinced him that there was money to be made here. Fred knew that it would be difficult to compete head to head against the established, full-service, international airlines. But he figured he could begin a flight service to fill the niche to business travelers wanting to travel between Atlanta and DC. Fred called some of his business associates and put some people to work on this idea. A month later, their marketing research revealed that there were several times during every business day when travelers want to hop a plane going between Atlanta and DC. They also discovered that many business travelers would prefer to fly to Dulles Airport in Washington, whereas most airlines fly to Washington's Reagan Airport. The following table summarizes the flight schedules where there is considerable unmet demand and the estimate revenues and variable costs that would be incurred in operating these flights. In addition to the costs given, the airport in Atlanta charges a $600 servicing charge for planes left there overnight, whereas Dulles Airport charges $700. Of course, there are also the costs of buying or leasing aircraft, hiring pilots, etc. Before pursuing this any further, Fred wants to know if the proposed flight schedule is profitable and how many planes it would take to implement it. And he has turned to you for help. Draw the graph of the network flow problem Fred should solve to determine the minimum number of planes required to implement the proposed flight schedule in the most efficient manner. Implement the model for this problem in software and solve it. Describe the optimal solution to the problem. Suppose there was adequate demand to run two flights from Atlanta to Washington leaving Atlanta at 4 p.m. and arriving at Dulles at 6 p.m. Would this be more profitable than the current schedule? Should Fred be willing to sacrifice any of the current flights to add this second flight out of Atlanta at 4 P.M.? 6
5. MANEUVERS IN THE SENATE Sherill Wiley, the only reporter from Channel Four covering United States congressional proceedings, knows that the public's perception of partisan politics in the Senate is not entirely correct. Although Republicans and Democrats are often at loggerheads over many national issues—such as the crime bill, health care reform, and others—members of the two parties frequently vote together on bills that are considered beneficial to both. Sherill was surprised at the level of cooperation between the parties when passing legislation that benefits a particular region. For example, if a Democratic senator from the west was sponsoring a bill that would provide funding to divert water from a river so that part of his sundrenched state would eventually receive it, quite a few Republican senators would vote for such a bill. The Republican senators know that when they need votes for a project that would be advantageous for their states, the Democratic senators would return the favor and vote for that bill. Richard Kennely, the leading Republican in the Senate and a strong conservative from a Midwest state, has little if any direct interaction with Robin Doll. Doll, a senator from a New England state, is the Senate's most liberal Democrat. The two senators have never served on a committee together or participated in a joint task force. Yet the two often rely on each other and their respective parties when voting on many fiscal issues. Senator Doll recently added an amendment to a finance bill that would provide funding for research on the effects of using a mouse when working on a computer. Several universities from Doll's state submitted grant proposals to do this research, citing the widespread use of the mouse and the growing number of people who complain of wrist and arm fatigue as a result. If the bill and the attached amendment pass, the universities in Doll's state would be the main beneficiaries of the funding. Sherill knows that Senator Kennely, who was recently on national television railing against what he referred to as "excessive spending for trivial research," would be the main opponent to the bill. Senator Doll cannot directly influence Senator Kennely's position or vote. But she can contact other senators with whom she has influence, pleading her case and hoping that they, in turn, will influence others who might eventually influence Senator Kennely to vote in favor of the bill. After many years as a reporter in Washington, Sherill has developed a sense of the amount of influence one senator has on another, based on party affiliation, school ties, past favors, mutual interests, and so on. She even developed a numerical system, on a scale from 0 to 10 that quantifies the amount of influence one senator has on another. A10 means that a particular senator is almost guaranteed to secure the vote of another senator and a 0 means that there is no interaction at all between two senators. Sherill created a table that shows the senators' names and the amount of influence each has on the other. Sherill saw Senators Abbott, Bartle, Crumb, Dodge, and Fernandez leave Senator Doll's office one morning last week. Looking at her influence table, she notes that Doll's influence on Abbott rates a 7, on Bartle a 6, a 9 for Crumb, only a 3 for Dodge, and a 4 for Fernandez. Later that week, Senator Abbott called in Senators Bartle and Dodge for a strategy meeting. Abbott's influence on Bartle is a 7 and on 7
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ΠΜΣ: Μαθηματικά των Υπολογιστών και των Αποφάσεων

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