历年MCM考题

HUASHI3483

MCM1987
Problem B       Parking Lot Design



The owner of a paved, 100' by 200' , corner parking lot in a New England town hires you to design the layout, that is, to design how the ``lines are to be painted.'' You realize that squeezing as many cars into the lot as possible leads to right-angle parking with the cars aligned side by side. However, inexperienced drivers have difficulty parking their cars this way, which can give rise to expensive insurance claims. To reduce the likelihood of damage to parked vehicles, the owner might then have to hire expert drivers for “valet parking”. On the other hand, most drivers seem to have little difficulty in parking in one attempt if there is a large enough ``turning radius'' from the access lane. Of course, the wider the access lane, the fewer cars can be accommodated in the lot, leading to less revenue for the parking lot owner.

HUASHI3483

MCM1988
Problem       Packing Railroad Flatcars



Two railroad flatcars are to be loaded with seven types of packing crates. The crates have the same width and height but varying thickness (t, in cm) and weight (w, in kg). Table 1 gives, for each crate, the thickness, weight, and number available [table omitted]. Each car has 10.2 meters of length available for packing the crates (like slices of toast) and can carry up to 40 metric tons. There is a special constraint on the total number of C_5, C_6, and C_7 crates because of a subsequent local trucking restriction: The total space (thickness) occupied by these crates must not exceed 302.7 cm. Load the two flatcars (see Figure 1) so as to minimize the wasted floor space [figure omitted].

HUASHI3483

MCM1989
Problem       Aircraft Queuing

A common procedure at airports is to assign aircraft (A/C) to runways on a first-come-first-served basis. That is, as soon as an A/C is ready to leave the gate ("push-back"), the pilot calls ground control and is added to the queue. Suppose that a control tower has access to a fast online database with the following information for each A/C:

The time it is scheduled for pushback;

The time it actually pushes back; the number of passengers who are scheduled to make a connection at the next stop, as well as the time to make that connection; and

The schedule time of arrival at its next stop Assume that there are seven types of A/C with passenger capacities varying from 100 to 400 in steps of 50. Develop and analyze a mathematical model that takes into account both the travelers' and airlines' satisfaction.

HUASHI3483

MCM1990
Problem       Snowplow Routing



The solid lines of the map (see Figure 1) represent paved two-lane county roads in a snow removal district in Wicomico County, Maryland [figure omitted]. The broken lines are state highways. After a snowfall, two plow-trucks are dispatched from a garage that is about 4 miles west of each of the two points (*) marked on the map. Find an efficient way to use the two trucks to sweep snow from the county roads. The trucks may use the state highways to access the county roads. Assume that the trucks neither break down nor get stuck and that the road intersections require no special plowing techniques

HUASHI3483

MCM1991
Problem       Water Tank Flow

Some state water-right agencies require from communities data on the rate of water use, in gallons per hour, and the total amount of water used each day. Many communities do not have equipment to measure the flow of water in or out of the municipal tank. Instead, they can measure only the level of water in the tank, within 0.5% accuracy, every hour. More importantly, whenever the level in the tank drops below some minimum level L, a pump fills the tank up to the maximum level, H; however, there is no measurement of the pump flow either. Thus, one cannot readily relate the level in the tank to the amount of water used while the pump is working, which occurs once or twice per day, for a couple of hours each time. Estimate the flow out of the tank f(t) at all times, even when the pump is working, and estimate the total amount of water used during the day. Table 1 gives real data, from an actual small town, for one day[ table omitted]. The table gives the time, in, since the first measurement, and the level of water in the tank, in hundredths of a foot. For example, after 3316 seconds, the depth of water in the tank reached 31.10 feet. The tank is a vertical circular cylinder, with a height of 40 feet and a diameter of 57 feet. Usually, the pump starts filling the tank when the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet.

HUASHI3483

MCM1992
Problem A    Air-Traffic-Control Radar Power

You are to determine the power to be radiated by an air-traffic-control radar at a major metropolitan airport. The airport authority wants to minimize the power of the radar consistent with safety and cost. The authority is constrained to operate with its existing antennae and receiver circuitry. The only option that they are considering is upgrading the transmitter circuits to make the radar more powerful. The question that you are to answer is what power (in watts) must be released by the radar to ensure detection of standard passenger aircraft at a distance of 100 kilometers.

Problem B    Emergency Power Restoration

Power companies serving coastal regions must have emergency response systems for power outages due to storms. Such systems require the input of data that allow the time and cost required for restoration to be estimated and the "value" of the outage judged by objective criteria. In the past, Hypothetical Electric Company (HECO) has been criticized in the media for its lack of a prioritization scheme.

You are a consultant to HECO power company. HECO possesses a computerized database with real time access to service calls that currently require the following information: [several further paragraphs of details omitted]. HECO has hired you to develop the objective criteria and schedule the work for the storm restoration requirements in Table 1 [table omitted].

HECO has asked for a technical report for their purposes and an "executive summary" in laymen's terms that can be presented to the media. Further, they would like recommendations for the future. To determine your prioritized scheduling system, you will have to make additional assumptions. Deal those assumptions. In the future, you may desire additional data. If so, detail the information desired.

HUASHI3483

MCM1993
Problem A    Optimal Composting

An environmentally conscious institutional cafeteria is recycling customers' uneaten food into compost by means of microorganisms. Each day, the cafeteria blends the leftover food into a slurry, mixes the slurry with crisp salad wastes from the kitchen and a small amount of shredded newspaper, and feeds the resulting mixture to a culture of fungi and soil bacteria, which digest slurry, greens, and papers into usable compost. The crisp green provide pockets of oxygen for the fungi culture, and the paper absorbs excess humidity. At times, however, the fungi culture is unable or unwilling to digest as much of the leftovers as customers leave; the cafeteria does not blame the chef for the fungi culture's lack of appetite. Also, the cafeteria has received offers for the purchase of large quantities of it compost. Therefore, the cafeteria is investigating ways to increase its production of compost. Since it cannot yet afford to build a new composting facility, the cafeteria seeks methods to accelerate the fungi culture's activity, for instance, by optimizing the fungi culture's environment (currently held at about 120 F and 100% humidity), or by optimizing the composition of the moisture fed to the fungi culture, or both.

Determine whether any relation exists between the proportions of slurry, greens, and paper in the mixture fed to the fungi culture, and the rate at which the fungi culture composts the mixture. if no relation exists, state so. otherwise, determine what proportions would accelerate the fungi culture's activity. In addition to the technical report following the format prescribed in the contest instructions, provide a one-page no technical recommendation for implementation for the cafeteria manager. Table 1 shows the composition of various mixtures in pounds of each ingredient kept in separate bins, and the time that it took the fungi to culture to compost the mixtures, from the date fed to the date completely composted [table omitted].

Problem B    Coal-Tipple Operations

The Aspen-Boulder Coal Company runs a loading facility consisting of a large coal tipple. When the coal trains arrive, they are loaded from the tipple. The standard coal train takes 3 hours to load, and the tipple's capacity is 1.5 standard trainloads of coal. Each day, the railroad sends three standard trains to the loading facility, and they arrive at any time between 5 A.M. and 8 P.M. local time. Each of the trains has three engines. If a train arrives and sits idle while waiting to be loaded, the railroad charges a special fee, called a demurrage. The fee is $5,000 per engine per hour. In addition, a high-capacity train arrives once a week every Thursday between 11 A.M. and 1 P.M. This special train has five engines and holds twice as much coal as a standard train. An empty tipple can be loaded directly from the mine to its capacity in six hours by a single loading crew. This crew (and its associated equipment) cost $9,000 per hour. A second crew can be called out to increase the loading rate by conducting an additional tipple-loading operation at the cost of $12,000 per hour. Because of safety requirements, during tipple loading no trains can be loaded. Whenever train loading is interrupted to load the tipple, demurrage charges are in effect.

The management of the Coal Company has asked you to determine the expected annual costs of this tipple's loading operations. Your analysis should include the following considerations:

How often should the second crew be called out?

What are the expected monthly demurrage costs?

If the standard trains could be scheduled to arrive at precise times, what daily schedule would minimize loading costs?

Would a third tipple-loading crew at $12,000 per hour reduce annual operations costs?

Can this tipple support a fourth standard train every day?

HUASHI3483

MCM1994
Problem A    Concrete Slab Floors

The U.S. Dept. of Housing and Urban Development (HUD) is considering constructing dwellings of various sizes, ranging from individual houses to large apartment complexes. A principal concern is to minimize recurring costs to occupants, especially the costs of heating and cooling. The region in which the construction is to take place is temperate, with a moderate variation in temperature throughout the year.

Through special construction techniques, HUD engineers can build dwellings that do not need to rely on convection- that is, there is no need to rely on opening doors or windows to assist in temperature variation. The dwellings will be single-story, with concrete slab floors as the only foundation. You have been hired as a consultant to analyze the temperature variation in the concrete slab floor to determine if the temperature averaged over the floor surface can be maintained within a prescribed comfort zone throughout the year. If so, what size/shape of slabs will permit this?

Part 1, Floor Temperature: Consider the temperature variation in a concrete slab given that the ambient temperature varies daily within the ranges given Table 1. Assume that the high occurs at noon and the low at midnight. Determine if slabs can be designed to maintain a temperature averaged over the floor surface within the prescribed comfort zone considering radiation only. Initially, assume that the heat transfer into the dwelling is through the exposed perimeter of the slab and that the top and bottom of the slabs are insulated. Comment on the appropriateness and sensitivity of these assumptions. If you cannot find a solution that satisfies Table 1, can you find designs that satisfy a Table 1 that you propose? [table omitted]

Part 2, Building Temperature: Analyze the practicality of the initial assumptions and extend the analysis to temperature variation within the single-story dwelling. Can the house be kept within the comfort zone?

Part 3, Cost of Construction: Suggest a design that considers HUD's objective of reducing or eliminating heating and cooling costs, considering construction restrictions and costs.



Problem B    Network Design

In your company, information is shared among departments on a daily basis. This information includes the previous day's sales statistics and current production guidance. It is important to get this information out as quickly as possible. [Network diagram (with 5 nodes and 7 capacitated edges) omitted.]

We are interested in scheduling transfers in an optimal way to minimize the total time it takes to complete them all. This minimum total time is called the make span. Consider the three following situations for your company: [Three more network diagrams (on roughly 20 nodes each) omitted.]

HUASHI3483

MCM1995
Problem A    Helix Construction



A small biotechnological company must design, prove, program and test a mathematical algorithm to locate "in real time" all the intersections of a helix and a plane in general positions in space. Design, justify, program and test a method to compute all the intersections of a plane and a helix, both in general positions (at any locations and with any orientations) in space. A segment of the helix may represent, for example, a helicoidal suspension spring or a piece of tubing in a chemical or medical apparatus. Theoretical justification of the proposed algorithm is necessary to verify the solution from several points of view, for instance, through mathematical proofs of parts of the algorithm, and through tests of the final program with known examples. Such documentation and tests will be required by government agencies for medical use.



Problem B    Faculty Compensation

Aluacha Balaclava College, and undergraduate facility, has just hired a new Provost whose first priority is the institution of a fair and reasonable faculty-compensation plan. She has hired your consulting team to design a compensation system that reflects the following circumstances and principles: [Three paragraphs of details omitted] Design a new pay system, first without cost-of-living increases. Incorporate cost-of-living increases, and then finally, design a transition process for current faculty that will move all salaries towards your system without reducing anyone's salary. The Provost requires a detailed compensation system plan for implementation, as well as a brief, clear, executive summary outlining the model, its assumptions, strengths, weaknesses and expected results, which she can present to the Board and faculty. [A detailed table of current salaries is omitted.]

HUASHI3483

2004年美国赛比赛规则
MCM: The Mathematical Contest in Modeling
ICM: The Interdisciplinary Contest in Modeling

Contest Registration and Instructions

(All instructions and rules apply to ICM as well as to MCM, except where otherwise noted.)

To participate in MCM a team must be sponsored by a faculty advisor from their institution. The registration process must be completed by the advisor.

There are several procedures that a team's advisor must go through at various times before, during, and after the contest. Please read these instructions carefully and be sure to complete all the steps involved. It is the advisor's responsibility to make sure that teams are correctly registered and that all steps required for participation in MCM/ICM are completed.

Note that COMAP is in the USA Eastern time zone; all times given in these instructions are in terms of Eastern time.

Before the contest registration deadline at 2pm EST on Thursday February 5, 2004 :

Register your team online:

The registration process will take you through a series of screens that ask you for your email address and contact information. Enter the required information as you step through the screens.

IMPORTANT: Be sure to use a valid current email address so that we can use it to contact you at any point before, during, or after the contest, if necessary.

1. All teams must be registered before 2pm EST on Thursday February 5, 2004 . At that time the registration system will stop accepting new team registrations; any team not registered by that time will not be able to participate in MCM 2004 . No exceptions will be made.

2. To guard against the possibility of interruptions in internet service we recommend that all teams complete the registration process well in advance of the deadline of 2pm EST on Thursday February 5, 2004 . COMAP cannot accept late registrations for MCM/ICM under any circumstances, even if you are unable to reach our web site on the day of the contest. No exceptions will be made.

3. Registration is via the contest web site. To register a team, go to http://www.comap.com/undergraduate/contests/mcm. If you are registering your first team for this year's contest, click on Register for 2004 Contest on the left-hand side of the screen.

If you have already registered a team for this year's contest and want to register an additional team, click on Advisor Login and then login with the email address and password that you used when you registered your first team. Once you're logged in, click on the Register Another Team link near the upper right corner of the page and follow the instructions there.

An advisor may register at most two teams. If you already have two teams registered then the Register Another Team link will not appear and you cannot register another team.

4. There can be no more than two teams from the same department participating in MCM, regardless of whether they have the same or different advisors.

5. There can be at most four teams from the same institution participating in MCM, regardless of whether they are from the same or different departments.

For ICM there can be at most three teams from the same institution.

6. Registration Fee

One of the final steps in the registration process is payment of the $ 45 registration fee per team. We accept payment via Mastercard or Visa, and payment must be made via our secure web site. We regret that we are not able to accept other forms of payment.

The pages that process your credit card payment on our site are secure pages, which means that your credit card number is protected with encryption while being transmitted from your computer to our server. Our system does not store your credit card number; we only use it long enough to process your payment.

7. Once we have received approval from your financial institution (this takes only a few seconds), the system will issue a control number for your team. Your team is not fully registered until you have received a team control number. You should print out the page that gives your team control number; it also contains a reminder of the email address and password that you used when registering, and you will need these to complete the contest procedures.

8. You will not receive any email confirmation of your registration; the only confirmation you will receive will be the screen giving your team's control number.

9. The screen giving your team's control number is your confirmation that your team has been registered. In order to participate in the contest, however, you will need to return to the contest web site several times to enter and confirm information about your team, and to print out your team's control and summary sheets that you will use when preparing your team's solution packet. Please read the instructions below for details on these steps.

If at any point before or during the contest you need to change any of the information (name, address, contact information, etc) that you specified when you registered, you can do so by logging in to the contest web site with the email address and password that you used when registering (click on the Advisor Login link on the left side of the screen). Once logged in, click on the Edit Advisor or Institution Data link near the upper right corner of the page.

10.   Return to the contest web site regularly to check for any updated instructions or announcements about the contest. Except in extreme circumstances, COMAP will not send any confirmation, reminders, or announcements by email. All communication regarding the contest will be via the contest web site.

Before the contest begins at 8pm EST on Thursday February 5, 2004 :

Choose your team members:

1. You must choose your team members before the contest begins at 8pm EST on Thursday February 5, 2004 . Once the contest begins you may not add or change any team members (you may, however, remove a team member, if he or she decides not to participate).

2. Each team may consist of a maximum of three students.

3. No student may be on more than one team.

4. Team members must be enrolled in school at the time of the contest, but do not have to be full-time. They must be enrolled at the same school as the advisor and other team members.

When the contest begins at 8pm EST on Thursday February 5, 2004 :

Teams view the contest problems via the contest web site:

1. The contest problems will become available precisely at 8pm EST on Thursday February 5, 2004 ; team members can view them by visiting http://www.comap.com/undergraduate/contests/mcm. No password will be needed to view the problems; simply go to the contest web site at or after 8pm EST on Thursday February 5, 2004 and you will see a link to view the problems.

2. If for some reason you cannot access our main web site at that time, go to our mirror site at http://www.mirror.comap.com/mcm or click here . The contest site and the mirror site are on two completely different networks in different parts of the USA. If you cannot access either one of them then it probably means that there is a problem with your local internet connection and you should contact your ISP to resolve the issue.

3. The contest consists of a choice of three problems: A, B, and C.
Important:

o      MCM teams should choose either problem A or problem B; an MCM team may submit a solution to only one of the problems. MCM teams should not choose problem C.

o      ICM teams should choose problem C. There is no choice for ICM teams. ICM teams should not choose problem A or B.

Teams prepare solutions:

1. Teams may use any inanimate source of data or materials --- computers, software, references, web sites, books, etc., however all sources used must be credited. Failure to credit a source will result in a team being disqualified from the competition.

2. Team members may not seek help from or discuss the problem with their advisor or anyone else, except other members of the same team. Input of any form from anyone other than student team members is strictly forbidden. This includes email, telephone contact, personal conversation, communication via web chat or other question-answer systems, or any other form of communication.

3. Partial solutions are acceptable. There is no passing or failing cut-off score, nor will numerical scores be assigned. The MCM/ICM judges are primarily interested in the team's approach and methods.

4. Teams should keep in mind the following guidelines and rules while preparing their solution papers:

o      Conciseness and organization are extremely important. Key statements should present major ideas and results.

o      Present a clarification or restatement of the problem, as appropriate.

o      Present a clear exposition of all variables, assumptions, and hypotheses.

o      Present an analysis of the problem, motivating or justifying the modeling to be used.

o      Include a design of the model.

o      Discuss how the model could be tested, including error analysis and stability (conditioning, sensitivity, etc.).

o      Discuss any apparent strengths or weaknesses to your model or approach.

5. Papers must be typed and in English.

6. The solution must consist entirely of written text, and possibly figures, charts, or other written material, on paper only. No non-paper support materials such as computer files or disks will be accepted.

7. Each page of the solution should contain the team control number and the page number at the top of the page; we suggest using a page header on each page, for example:

    Team # 321                        Page 6 of 13
8. The names of the students, advisor, or institution should not appear on any page of the solution. The solution should not contain any identifying information other than the team control number.

9. Any preparation rule not followed is grounds for team disqualification.

After the contest begins at 8pm EST on Thursday February 5, 2004 :

Print Summary Sheet and Control Sheets

While the teams are preparing their solutions, the advisor should

1. Login to the contest web site (go to http://www.comap.com/undergraduate/contests/mcm. and click on Advisor Login and enter your email address and password).

2. Enter the team member names and confirm that they are correctly spelled. This is exactly as the names and institutions will appear on the certificates. COMAP will not make any changes or reprint certificates for any reason.

3. Specify the problem that your team has chosen to solve.

4. Print one copy of the control sheet.

5. Print one copy of the team summary sheet.

When the contest ends at 8pm EST on Monday February 9, 2004 :

Prepare Solution Packet:

1. Have each student sign the control sheet, pledging that they have abided by the contest rules and instructions.

2. Type a summary of your team's solution on the summary sheet that you printed out, and make three copies of it.

3. Make three copies of your team's solution paper. Staple one copy of the summary sheet on top of each copy of the solution paper.

4. Staple the control sheet on top of just one copy of the solution paper.

Mail Solution Packet:

1. After you have prepared your team's solution packet as above, mail it to

MCM/ICM Coordinator
COMAP, Inc.
57 Bedford St., Suite 210
Lexington, MA 02420
USA

2. COMAP must receive your solution on or before Monday February 16, 2004 . It is your responsibility to make sure that your team's solution packet arrives at COMAP by this deadline.

3. Use registered or express mail if necessary to insure that your solution arrives at COMAP by Monday February 16, 2004 .

4. COMAP will not accept late solutions under any circumstances.

5. If you require confirmation that your paper was received by COMAP, send the packet via a carrier that provides package tracking. Due to the number of papers received, COMAP can not answer receipt inquiries or emails.

After the contest is over:

Confirm that your team's solution was received at COMAP:

A few days after mailing your solution packet, you may login to the contest web site using the Advisor Login link to verify that your team's solution was received at COMAP. Please allow several days for us to process your packet before expecting to see this confirmation.

Judging

Judging will be completed by March 31, 2004. The solutions will be recognized as Successful Participant,Honorable Mention, Meritorious, or Outstanding Winner.

Check Results

Return to the contest web site periodically to check for the results of the contest. It will take several weeks for the judges to evaluate the solutions and for COMAP to process the results. We will post the results on the web site as soon as they are available. Please do not call or email COMAP asking when the results will be available; simply visit the contest web site regularly to check for them.

Receive certificate

At some point after the results have been issued, each team that participated successfully will receive a certificate of participation in the mail. The certificate will be mailed to the advisor at the address used during the registration process. Please allow several weeks after the results are posted to the contest web site before expecting to receive your certificate.

Prizes

·   The Institute for Operations Research and the Management Sciences (INFORMS) will designate an Outstanding team from each of the three problems as an INFORMS winner

·   The Society for Industrial and Applied Mathematics (SIAM) will designate one Outstanding team from each problem as a SIAM winner.

·   The Mathematical Association of America (MAA) will designate one Outstanding team from each problem for the MCM as a MAA winner.

Note: COMAP is the final arbiter of all rules and policies, and may disqualify or refuse to register any team that, in its sole discretion, does not follow these contest regulations and procedures.