Problem: Unloading Commuter Trains
Trains arrive often at a central Station, the nexus for many commuter trains from suburbs of larger cities on a “commuter” line. Most trains are long (perhaps 10 or more cars long). The distance a passenger has to walk to exit the train area is quite long. Each train car has only two exits, one near each end so that the cars can carry as many people as possible. Each train car has a center aisle and there are two seats on one side and three seats on the other for each row of seats.
To exit a typical station of interest, passengers must exit the car, and then make their way to a stairway to get to the next level to exit the station. Usually these trains are crowded so there is a “fan” of passengers from the train trying to get up the stairway. The stairway could accommodate two columns of people exiting to the top of the stairs.
Most commuter train platforms have two tracks adjacent to the platform. In the worst case, if two fully occupied trains arrived at the same time, it might take a long time for all the passengers to get up to the main level of the station.
Build a mathematical model to estimate the amount of time for a passenger to reach the street level of the station to exit the complex. Assume there are n cars to a train, each car has length d. The length of the platform is p, and the number of stairs in each staircase is q.
Use your model to specifically optimize (minimize) the time traveled to reach street level to exit a station for the following:
Requirement 1. One fully occupied train’s passengers to exit the train, and ascend the stairs to reach the street access level of the station
Requirement 2. Two fully occupied trains’ passengers (all passengers exit onto a common platform) to exit the trains, and ascend the stairs to reach the street access level of the station.
Requirement 3. If you could redesign the location of the stairways along the platform, where should these stairways be placed to minimize the time for one or two trains’ passengers to exit the station?
Requirement 4. How does the time to street level vary with the number s of stairways that one builds?
Requirement 5. How does the time vary if the stairways can accommodate k people, k an integer greater than one?
In addition to the HiMCM format, prepare a short non-technical article to the director of transportation explaining why they should adopt your model to improve exiting a station.
Problem: The Next Plague?
In 2014, the world saw the infectious Ebola virus spreading in western Africa. Throughout human history, epidemics have come and gone with some infecting and/or killing thousands and lasting for years and others taking less of a human toll. Some believe these events are just nature’s way of controlling the growth of a species while others think they could be a conspiracy or deliberate act to cause harm. This problem will most likely come down to how to expend (or not expend) scarce resources (doctors, containment facilities, money, research, serums, etc…) to deal with a crisis.
Situation: A routine humanitarian mission on an island in Indonesia reported a small village where almost half of its 300 inhabitants are showing similar symptoms. In the past week, 15 of the “infected” have died. This village is known to trade with nearby villages and other islands. Your modeling team works for a major center of disease control in the capital of your country (or if you prefer, for the International World Health Organization).
Requirement 1: Develop a mathematical model(s) that performs the following functions as well as how/when to best allocate these scarce resources and…
• Determines and classifies the type and severity of the spread of the disease
• Determines if an epidemic is contained or not
• Triggers appropriate measures (when to treat, when to transport victims, when to restrict movement, when to let a disease run its course, etc…) to contain a disease
Note: While you may want to start with the well-known “SIR” family of models for parts of this problem, consider others, modifications to the SIR, multiple models, or creating your own.
Requirement 2: Based on the information given, your model, and the assumptions your team has made, what initial recommendations does your team have for your country’s center for disease control? (Give 3-5 recommendations with justifications)
Additional Situational Information: A multi-national research team just returned to your country’s capital after spending 7 days gathering information in the infected village.
Requirement 3: You can ask them up to 3 questions to improve your model. What would you ask and why?
Additional Situational Information: The multi-national research team concluded that the disease:
• Appears to spread through contact with bodily fluids of an infected person
• The elderly and children are more likely to die if infected
• A nearby island is starting to show similar signs of infection
• One of the researchers that returned to your capital appears infected
Requirement 4: How does the additional information above change/modify your model?
Requirement 5: Write a one-page synopsis of your findings for your local non-technical news outlet.