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Spring 2024
MATH**3
Case Studies in Mathematical Modeling
Group Project

Instruction: Form a group of 3 and complete the following two tasks. 

Task 1: Choose one topic from those attached at the end of this document, follow the instructions and requirements, and submit your written solutions (including specific requirements of the chosen topic, e.g., poster) via the designated link on iSpace by 23:59, 14 May 2024. Your submissions will then be distributed to your peers for Task 2.

Task 2: Prepare an oral presentation (12 minutes presentation + 3 minutes Q&A) to report the following aspects of another group’s project:
Background of the chosen topic
Model(s)
Methodology
Results 
Comments and suggestions for improvements
Your final judgement (give a score) on the project 
The presentation will be tentatively scheduled on Week 14 (the week of 20 May 2024). Please submit your PowerPoint slides by 23:59, 19 May 2024 to the designated link on iSpace.

Plagiarism is strictly prohibited. If found, you will get zero marks for the project.

Problem 1: Jet Lag
Organizing international meetings is not easy in many ways, including the problem that some of the participants may experience the effects of jet lag after recent travel from their home country to the meeting location which may be in a different time-zone, or in a different climate and time of year, and so on. All these things may dramatically affect the productivity of the meeting.
The International Meeting Management Corporation (IMMC) has asked your expert group (your team) to help solve the problem by creating an algorithm that suggests the best place(s) to hold a meeting given the number of participants, their home cities, approximate dates of the meeting and other information that the meeting management company may request from its clients.
The participants are usually from all corners of the Earth, and the business or scientific meeting implies doing hard intellectual team work for three intensive days, with the participants contributing approximately equally to the end result. Assume that there are no visa problems or political limitations, and so any country or city can be a potential meeting location.
The output of the algorithm should be a list of recommended places (regions, zones, or specific cities) that maximize the overall productivity of the meeting. The questions of costs are not of primary importance, but the IMMC, just as any other company, has a limited budget. So the costs may be considered as a secondary criterion. And the IMMC definitely cannot afford bringing the participants in a week before the meeting to acclimatize or give them the time to rest after a long exhausting journey.
Test your algorithm at least on the two following datasets:
Scenario 1) “Small Meeting”:
Time: mid-June
Participants: 6 individuals from:
oMonterey CA, USA
oZutphen, Netherlands
oMelbourne, Australia
oShanghai, China
oHong Kong (SAR), China
oMoscow, Russia
Scenario 2) “Big meeting”:
Time: January
Participants: 11 individuals from:
oBoston MA, USA (2 people)
oSingapore
oBeijing, China
oHong Kong (SAR), China (2 people)
oMoscow, Russia
oUtrecht, Netherlands
oWarsaw, Poland
oCopenhagen, Denmark
oMelbourne, Australia
Your solution cannot exceed 20 pages. (The appendices and references should appear at the end of the paper and do not count toward the 20 page limit)

Problem 2: The Best Hospital
Almost everyone will seek health care at some point in life. In an emergency, an individual will most likely go to the closest hospital, but for non-emergencies we may have a choice of where to seek treatment. Suppose there are 4 or 5 hospitals reasonably accessible to your residence. You wish to pick the “best” hospital. How would you measure and choose the “best” of these local hospitals? Suppose the gravity of your situation is such that you are willing to travel for your health care and you want to pick the “best” of 50 or so hospitals. What variables do you use and how well can you measure each? Certainly mortality is an important variable. Measuring death rates has the advantage that death is a definite unique event. The total number of deaths may not be a good measure of the quality of the hospital at all, but the number of evitable deaths could be a very good measure. How do we decide whether a death is evitable or inevitable? Each death case can be coded with data to include, for example, primary diagnosis, age, gender, urgency of admission, comorbidity, length of stay, social deprivation, and other factors. With large sample sizes, the performance of different hospitals could possibly be measured by comparing cases with similar characteristics. In addition to mortality there are other factors that one might want to use in measuring the overall quality of a hospital. 
A few possible variables include: 
The experience of the doctors 
The amount of attention one expects to receive from the staff and attending physician 
Your team is tasked to use mathematical modeling to address three requirements.
Develop a model that uses mortality to measure the quality of a hospital. 
Develop a model that uses other factors, in addition to mortality, to measure the quality of a hospital. Based on the factors you include from particular hospitals, your model must result in information to make a decision of which hospital is the best. 
In addition to the mathematical analysis you provide in your report, include a two-page “user-friendly” memo that a person without much mathematical expertise or computing ability can use to choose a hospital. 
Your submission should consist of
Two-page memo 
Your solution of no more than 20 pages.
Note: Reference list and any appendices do not count toward the 23-page limit and should appear after your completed solution. 

Glossary 
Evitable: capable of being avoided. 
Comorbidity: the presence of one or more medical conditions co-occurring with a primary condition. 
User-friendly: easy to learn, use, understand, or deal with. 
Social deprivation: hardship caused by a lack of the ordinary material benefits of life in society

Problem 3: Boarding and Disembarking a Plane
Background
In air transportation, efficiency is time and time is money. Even small delays in the schedules of passenger airplanes result in lost time for both air carriers and their passengers. During any passenger flight, there are two time-consuming operations that depend mostly on human behavior: boarding and disembarking the aircraft.
In commercial passenger air travel, airlines use various boarding and disembarking methods from completely unstructured (passengers board or leave the plane without guidance) to structured (passengers board or leave the plane using a prescribed method). Prescribed methods may be based on row numbers, seat positions, or priority groups. In practice, however, even when the prescribed method is announced, not all passengers follow the instructions.
The boarding process includes the movement of passengers from the entrance of the aircraft to their assigned seats. This movement can be hindered by aisle and seat interference. For example, many passengers have carry-on bags which they stow into the overhead bins before taking their seats. Each time a passenger stops to stow a bag, the queue of other passengers stops because narrow aircraft aisles allow only one passenger to pass at a time. Another hindrance is that some seats (e.g., window seats) are unreachable if other seats (e.g., aisle seats) are already occupied. When this occurs, some passengers must stand up and move into the aisle so other passengers can reach their seats.
The disembarking process is the opposite of boarding with its own possible hindrances to passenger movement. Some passengers are simply slow getting out of their seat and row, or slow moving to the exit. Passengers also block the aisle while collecting their belongings from either their seat or from the overhead bin forcing passengers behind them in the aircraft to wait.
Requirements
Your team is to create plane boarding and disembarking methods that will be the most time- effective in real practice.
1.Construct a mathematical model or models to calculate total aircraft boarding and disembarking times. Ensure your model is adaptable to various prescribed boarding/disembarking methods and varying numbers of carry-on bags to be stowed, as well as accounts for passengers who do not follow the prescribed boarding/disembarking methods.
2.Apply your model to the standard “narrow-body” aircraft shown in Figure 1.
a.Compare the average, practical maximum (95th percentile) and practical minimum (5th percentile) boarding times for the following widely used boarding methods:
•Random (unstructured) boarding.
•Boarding by Section: Examine varying the order of aft section (rows 23-33), middle section (rows 12-22), and bow section (rows **11).
•Boarding by Seat: In the order of window seats (A and F), middle seats (B and E), and aisle seats (C and D).
b.Analyze how these times vary based on the percentage of passengers not following the prescribed boarding method and on the average number of carry-on bags per flight

(i.e., perform a basic sensitivity analysis). Based on your analysis, which of the above boarding methods is the best?
c.Consider the situation when passengers carry more luggage than normal and stow all their carry-ons in the overhead bins. How does this change affect the results?
d.Describe two additional possible boarding methods. Explain and justify your recommended optimal boarding method (from your two and the three in part 2.a.).
e.Explain and justify your optimal disembarking method.
3.Modify your model for the following passenger aircraft and recommend your optimal boarding and disembarking methods for each aircraft.
•The Flying Wing aircraft with relatively wide and short passenger cabins as shown in Figure 2.
•A Two-Entrance, Two-Aisle aircraft as shown in Figure 3.
4.Due to the pandemic situation, capacity limitations are sometimes implemented on passenger airliners. Will your recommended prescribed methods for boarding and disembarking of the three aircraft change if the number of passengers is limited to 70%, 50%, or 30% of the number of seats?
5.Write a one-page letter to an airline executive describing and explaining your results, recommendations, and rationale about passenger aircraft boarding and disembarking in a non- mathematical way.
Note that it is not sufficient to simply re-present any of these models or discussions, even if properly cited. Any successful solution MUST include development and analysis of your own team’s model and a clear explanation of the difference between your model and any referenced aircraft boarding and disembarking models.
Your PDF submission should consist of:
•One-page letter to an airline executive.
•Your solution of no more than 20 pages (A4 or letter size). Note that your font size must be no smaller than 12-point type.
Note: Reference List and any appendices do not count toward the page limit and should appear after your completed solution. You should not make use of unauthorized images and materials whose use is restricted by copyright laws. Ensure you cite the sources for your ideas and the materials used in your report.

Glossary
Carry-On Bag – a piece of luggage a passenger carries onto an airplane with dimensions such that it can fit in the overhead bin.
Disembarking – leaving (an airplane).
Overhead Bins – storage compartments attached to the ceilings of aircraft for baggage stowage during a flight.

Airline Figures

Figure 1. “Narrow-Body” Passenger Aircraft

Figure 2. “Flying Wing” Passenger Aircraft

Figure 3. “Two-Entrance, Two Aisle” Passenger Aircraft

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