MCM 2006 A: Positioning and Moving Sprinkler Systems for Irrigation

There are a wide variety of techniques available for irrigating a field. The technologies range from advanced drip systems to periodic flooding. One of the systems that is used on smaller ranches is the use of “hand move” irrigation systems. Lightweight aluminum pipes with sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals to insure that the whole field receives an adequate amount of water. This type of irrigation system is cheaper and easier to maintain than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time and effort to move and set up the equipment at regular intervals.

Given that this type of irrigation system is to be used, how can it be configured to minimize the amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are asked to find an algorithm to determine how to irrigate the rectangular field that minimizes the amount of time required by a rancher to maintain the irrigation system. One pipe set is used in the field. You should determine the number of sprinklers and the spacing between sprinklers, and you should find a schedule to move the pipes, including where to move them.

A pipe set consists of a number of pipes that can be connected together in a straight line. Each pipe has a 10 cm inner diameter with rotating spray nozzles that have a 0.6 cm inner diameter. When put together the resulting pipe is 20 meters long. At the water source, the pressure is 420 Kilo- Pascal’s and has a flow rate of 150 liters per minute. No part of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at least 2 centimeters of water every 4 days. The total amount of water should be applied as uniformly as possible.

MCM 2006 B: Wheelchair Access at Airports

One of the frustrations with air travel is the need to fly through multiple airports, and each stop generally requires each traveler to change to a different airplane. This can be especially difficult for people who are not able to easily walk to adifferentflight’s waiting area. Oneof the ways that an airline canmake the transition easier is to provide a wheelchair and an escort to those people who ask for help. It is generally known well in advance which passengers require help, but it is not uncommon to receive notice when a passenger first registers at the airport. In rare instances an airline may not receive notice from a passenger until just prior to landing.

Airlines are under constant pressure to keep their costs down. Wheelchairs wear out and are expensive and require maintenance. There is also a cost for making the escorts available. Moreover, wheelchairs and their escorts must be constantly moved around the airport so that they are available to people when their flight lands. In some large airports, the time required to move across the airport is nontrivial. The wheelchairs must be stored somewhere, but space is expensive and severely limited in an airport terminal. Also, wheelchairs left in high traffic areas represent a liability risk as people try to move around them.

Finally, one of the biggest costs is the cost of holding a plane if someone must wait for an escort and becomes late for their flight. The latter cost is especially troubling because it can affect the airline’s average flight delay which can lead to fewer ticket sales as potential customers may choose to avoid an airline.

Epsilon Airlines has decided to ask a third party to help them obtain a detailed analysis of the issues and costs of keeping and maintaining wheelchairs and escorts available for passengers. The airline needs to find a way to schedule themovement of wheelchairs throughout each day in a cost effective way. They also need to find and define the costs for budget planning in both the short term and in the long term.

Epsilon Airlines has asked your consultant group to put together a bid to help them solve their problem. Your bid should include an overview and analysis of the situation to help them decide if you fully understand their problem. They require a detailed description of an algorithm that you would like to implement which can determine where the escorts and wheelchairs should be and how they should move throughout each day. The goal is to keep the total costs as low as possible. Your bid is one of many that the airline will consider. You must make a strong case as to why your solution is the best and show that it will be able to handle a wide range of airports under a variety of circumstances.

Your bid should also include examples of how the algorithm would work for a large (at least four concourses), a medium (at least two concourses), and a small airport (one concourse) under high and low traffic loads. You should determine all potential costs and balance their respective weights. Finally, as populations begin to include a higher percentage of older people who have more time to travel but may require more aid, your report should include projections of potential costs and needs in the future with recommendations to meet future needs.

ICM 2006 C: The HIV/AIDS Problem

AsAs the HIV/AIDS pandemic enters its 25th year, both the number of infections and number of deaths due to the disease continue to rise. Despite an enormous amount of effort, our global society remains uncertain on how to most effectively allocate resources to fight this epidemic.

You are a team of analysts advising the United Nations (UN) on how to manage the available resources for addressing HIV/AIDS. Your job is to model several scenarios of interest and to use your models to recommend the allocation of financial resources. The narrative below provides some background information, and outlines specific tasks.

Task 1: For each of the continents (Africa, Asia, Europe, North America, Australia, and South America), choose the country you believe to be most critical in terms of HIV/AIDS. Build a model to approximate the expected rate of change in the number of HIV/AIDS infections for these countries from 2006 to 2050, in the absence of any additional interventions. Fully explain your model and the assumptions that underlie your model. In addition, explain how you selected the countries to model.

Use as a list of countries for inclusion in your analysis the countries included in the attached spreadsheet, which include all member states of the World Health Organization (WHO) as of 2003.

Data: “list_WHO_member_states.xls”.

Reliable data on HIV prevalence rates by county are generally difficult to obtain. The attached spreadsheet includes several worksheets of data which you may use in your analysis. Data: “hiv_aids_data.xls”.

a. “Global HIV- AIDS cases, 1999”: These data come from UNAIDS (the Joint United Nations Program on HIV/AIDS) and report the estimated number of HIV positive 0 to 49 year olds by country at the end of 1999.

b. “HIV- AIDS in Africa over time”: These data come from the US government and give some piecemeal time series data on measured HIV prevalence rates among women of childbearing age, in urban areas, over time for some African countries.

c. “HIV- AIDS subtypes”: These data come from UNAIDS and give the geographic distribution of HIV-1 subtypes by country. Also attached, for your use, are some basic population and demographic data.


(1) “fertility_data.xls”: These data come from the UN and give age-specific fertility rates by major area, region, and country, 1995-2050 (births per thousand women)

a. Estimates for 1995-2005.

b. Projections (under the assumption of medium fertility levels) for 2005-2050.

(2) “population_data.xls”: These data come from the UN and give total population (both sexes combined) by major area, region, and country, annually for 1950-2050 (thousands).

a. Estimates for 1950-2005.

b. Projections (under the assumption of medium fertility levels) for 2006-2050.

(3) “age_data.xls”: These data come from the UN and give population (for both sexes, and by gender) by five-year age groups, major area, region, and country, 1950-2050 (thousands).

a. Estimates, 1950-2005.

b. Projections (under the assumption of medium fertility levels) for 2010-2050.

(4) “birth_rate_data.xls”: These data come from the UN and give crude birth rates by major area, region, and country, 1950-2050 (births per thousand population)

a. Estimates, 1950-2005.

b. Projections (under the assumption of medium fertility levels) for 2005-2050.

(5) “life_expectancy_0_data.xls”: These data come from the UN and give life expectancy at birth (by sex and both sexes combined) by major area, region, and country, 1950-2050 (years).

a. Estimates, 1950-2005.

b. Projections (under the assumption of medium fertility levels) for 2005-2050.

There are a number of interventions that HIV/AIDS funding could be directed towards -- including prevention interventions (voluntary counseling and testing, condom social marketing, school-based AIDS education, medicines to prevent mother-to-child transmission, etc.) and care interventions (treating other untreated sexually transmitted diseases, treating opportunistic infections, etc.). You should focus on only two potential interventions: provision of antiretroviral (ARV) drug therapies, and provision of a hypothetical HIV/AIDS preventative vaccine.

Task 2: First, estimate the level of financial resources from foreign aid donors that you realistically expect to be available to address HIV/AIDS, by year, from 2006 to 2050, for the countries you selected in Task 1. Then use the model you developed in Task #1 and these estimates of financial resources to estimate the expected rate of change in the number of HIV/AIDS infections for your selected countries from 2006 to 2050 under realistic assumptions for the following three scenarios:

(1) Antiretroviral (ARV) drug therapy.

(2) A preventative HIV/AIDS vaccine.

(3) Both ARV provision and a preventative HIV/AIDS vaccine.

Assume in these scenarios that there is no risk of emergence of drug-resistant strains of HIV (you will examine this issue in Task 3).

Be sure to carefully describe the assumptions that underlie your model.

You can choose whether these scenarios should be implemented for all of the countries you selected in Task 1, or for certain subsets of countries based on income cut-offs, disease burden, etc. Available for use if you wish is a spreadsheet of country-level income data.

Data: “income_data.xls” .

These data are from the World Bank (2002) and give per-capita gross national product (GNP) data as well as broad income classifications that you are free to use in your analysis if you wish.

ARV drug therapies can have tremendous benefits in terms of prolonging the lives of individuals infected with HIV/AIDS. ARVs are keeping a high proportion of HIV/AIDS-infected individuals in rich countries alive, and policy makers and international institutions are facing tremendous political pressure to increase access to ARVs for individuals in poor countries. Health budgets in low-income countries are very limited, and it seems unlikely that poor countries will be able to successfully expand these programs to the majority of their populations using their own resources. Appendix 1 presents country-specific data from UNAIDS on current access to ARVs for a number of countries.

The efficacy of ARVs depends in large part on adherence to the treatment regimen and to proper monitoring. The most favorable conditions for ARVs are structured programs with extensive counseling and physician care, as well as regular testing to monitor for disease progression and the onset of opportunistic infections. Non-adherence or inadequate treatment carries with it two very serious consequences. First, the treatment may not be effective for the individual undergoing treatment. Second, partial or inadequate treatments are thought to directly lead to the emergence of drug-resistant strains of HIV.

The price of the drugs initially used to treat patients has come down to several hundred dollars a year per patient, but delivering them and providing the necessary accompanying medical care and further treatment is the key administrative and financial challenge. It is estimated that purchasing and delivering antiretroviral using the clinically-recommended approach (DOTS, or directly observed short course treatments) which is intended to minimize the emergence of drug-resistant strains would cost less than $1,100 per person per year. (Adams, Gregor et al. [2001]. “Consensus Statement on Antiretroviral Treatment for AIDS in Poor Countries,” )

For a preventative HIV vaccine, make assumptions you feel are reasonable about the following (in addition to other factors you may choose to include in your model):

(1) The year in which an HIV/AIDS preventative vaccine might be available.

(2) How quickly vaccination rates might reach the following steady-state levels of vaccination:

a. If you wish to immunize new cohorts (infants), assume the steady-state level for new cohorts of the country-by-country immunization rates for the third dose of the diphtheria-pertussis-tetanus vaccine (DTP3), as reported by the WHO (2002).

Data: “vaccination_rate_data.xls”.

b. If you wish to immunize adults (any group over age 5), assume the steady-state level for older cohorts is the second dose of the tetanus toxoid (TT2) rate, as reported by the WHO (2002).

Data: “vaccination_rate_data.xls”.

(3) The efficacy and duration of protection of the vaccine.

(4) Whether there would be epidemiological externalities from vaccination.

(5) Assume the vaccine is a three-dose vaccine, and can be added to the standard package of vaccines delivered under the WHO’s Expanded Programme on Immunization (EPI) at an incremental cost of addition of $0.75.

Task 3: Re-formulate the three models developed in Task 2, taking into consideration the following assumptions about the development of ARV-resistant disease strains.

Current estimates suggest that patients falling below 90-95 percent adherence to ARV treatment are at a “substantial risk” of producing drug resistant strains. Use as an assumption for your analysis that a person receiving ARV treatment with adherence below 90 percent has a 5 percent chance of producing a strain of HIV/AIDS which is resistant to standard first-line drug treatments.

Second- and third-line ARV drug therapies are available, but assume for your analysis that these drugs are prohibitively expensive to implement in countries outside of Europe, Japan, and the United States.

Task 4: Write a white paper to the United Nations providing your team’s recommendations on the following:

(1) Your recommendations for allocation of the resources available for HIV/AIDS among ARV provision and a preventative HIV vaccine.

(2) Your argument for how to weigh the importance of HIV/AIDS as an international concern relative to other foreign policy priorities.

(3) Your recommendations for how to coordinate donor involvement for HIV/AIDS.

For (1): assume that between now and 2010 the available financial resources could be allocated so as the speed the development of a preventative HIV vaccine – through directly-financing vaccine research and development (R&D), or through other mechanisms. Any gains from such spending would move the date of development you assumed in Task 2 to some earlier date.

The solutions and analysis refer to《Analysis and Research of the American Mathematical Contest in Modeling: 2nd part》