As Math and the Mouse 2022 comes to an end, we, the professors, would like to wrap up the experience by describing and emphasizing the academic components in the course.
As the world becomes more data-centric and quantitative, understanding how industries use mathematics in the decision-making process is important in terms of preparing the students for the future. This course was an introduction to that very topic.
The content of the course was divided into three parts: (i) data analytics, (ii) optimization modeling and algorithm design, and (iii) dealing with uncertainty. Not only did the students learn about these areas from us, but the students also met with professionals from Disney (and other companies) who work in these areas. Over the course of just three weeks, the students encountered material that appears in a variety mathematics courses at Furman (MTH 151—Calculus II, MTH 160—Vectors and Matrices, MTH 250—Multivariable Calculus, MTH 330—Combinatorics and Graph Theory, MTH 335—Mathematical Modeling, MTH 337—Operations Research, MTH 340—Probability, MTH 341—Mathematical Statistics, MTH 345—Statistical Modeling). Prior experience with these courses varied from student to student, and we hope that exposure to the ideas served as both reinforcement and motivation.
We started the course by introducing the students to some foundational topics in data analytics. Two problems we worked on heavily were the classification problem and the prediction problem. Given a data set, the classification problem asks one to partition or cluster the data into different segments. For example, the current Genie system that Disney designed recommends attractions based on survey characteristics of users. Recommendation systems, in general, will try to make recommendations to users based on how similar a user’s characteristics are to other users. The prediction problem involves the use of past data to forecast a future result. For instance, Disney might try to predict how much a family with certain characteristics will spend in the parks each day. Our students’ first project involved using a process called “k-means clustering” to pinpoint where to put a mobile Mickey Bar stand throughout the day to interact with the highest traffic areas of the park. The process was a huge success. The students worked very hard, and their presentation to Disney professionals on this project was fantastic!
The second main topic in the course was optimization modeling and algorithm design. The students learned to model real-world optimization problems such as those involving logistics, scheduling, and resource allocation. These types of problems are not only applicable to Disney’s operations but serve as foundational material for professionals working in the areas of industrial and systems engineering and operations management. The students completed a sophisticated and difficult modeling project that involved assigning workers to shifts at Yak & Yeti (a restaurant in Animal Kingdom) in such a way that the demand throughout the day was satisfied. Although the problem was much smaller than ones solved by Disney, the concepts learned by students through the project were the essential concepts any company uses to schedule workers. The students did a great job on this challenging project, and their presentation to Disney on this project was better than a lot of professional conference presentations the professors have heard.
Once the students learned how to form a mathematical model, we showed them how to exploit the mathematical structure present in their created models to design algorithms to produce optimal (or near optimal) solutions. For fun, we designed in-park competitions to help the students explore algorithms to search for optimal solutions to their optimization models. The first activity was the Traveling Tourist activity where the students tried to solve an instance of the Time-Dependent Traveling Salesman Problem by finding a tour of 14 different Epcot attractions that took the shortest amount of time to complete. This is the same problem that companies like Amazon, UPS, FedEx, and touringplans.com solve on a daily basis. We enjoyed watching the students discover well-known heuristic algorithms that industry professionals use to produce solutions to this problem. After the contest, the students were able to interact with Len Testa (President of Touring Plans), and they gained a better understanding of how Testa’s company designs algorithms to solve the Disney touring problem for its customers. The second algorithm design activity was a seating problem that Disney faces: how to efficiently fill a ride vehicle based on the guests that appear in the queue. This is known as an online bin packing problem, and the students used an app, designed by Craig Bouzarth (Liz’s dad), to seat a queue of numbers into seats on Tower of Terror. Students were delighted to hear one of the Disney professionals describe a similar app that was developed to help Disney employees seat people on the Star Tours ride.
In the final stage of the course, we introduced students to some basic concepts in probability and statistics. These topics helped to reinforce the idea that most models are imperfect, since inputs to these models often have high variability and are not always known with certainty. We discussed how Disney and other companies use simulation to test different possible cases that could arise from models as the likelihood of the inputs of the models change. Almost every activity we worked on in the class, from the Traveling Tourist Problem to the Tower of Terror Bin Packing Problem, asked the students to deal with uncertainty. Using expected wait times or average party size is good in forecasting a decision, but the day-to-day variability that is encountered in most business scenarios force real-time decisions to made that may be different from expected. The biggest random process that the students encountered in the course is standing in queues. We showed the students how to simulate a queue in order to discover expected wait times and queue lengths. Each of the final projects that students worked on interacted in some way with the variability of throughputs of rides or queue wait times.
Speaking of final projects, let us describe the four wonderful (student-designed) projects that were the culmination of the course.
- An Investigation of Single Rider Lines: For some rides, Disney has created a single-rider line to try to fill each ride vehicle to capacity and thus maximizing their throughput throughout the day. For instance, if a party of size five is being seated in a six-person ride vehicle, the ride workers will pull a guest from the single rider line to complete the capacity on the vehicle. The group of Jalen, Joey, and Taylor looked at whether the Remy’s Ratatouille Adventure ride could benefit from a single rider line by comparing its characteristics with other six-person ride vehicle attractions, like Test Track and Space Mountain. They were able to find that the Remy ride sends vehicles out at only 72% of capacity compared to Test Track’s 91% of capacity and Space Mountain’s 92% of capacity. They also performed a Chi-Squared Test to determine that the party size distribution of both rides is similar. Given this they concluded that Remy would see a huge increase in the number of people who could ride the attraction each day by incorporating a single rider line.
- A Knapsack Problem at Hollywood Studios: Whether you know it or not, you have encountered the knapsack problem. When packing for vacation, you make decisions regarding what to put in your suitcase. Each item you could take has a value to you but takes up a certain amount of space in the limited suitcase. The Knapsack Problem essentially involves determining how to get the most value in the suitcase without the suitcase bursting. The group of Elizabeth, Joshua, Katie, and Rhett tried to determine how much value they could get at Hollywood Studios within a four-hour time window. For this they employed some regression analysis and class surveying to determine the values they would place on the rides in the park. The four-hour time limit served as the suitcase in that each ride would take up walk, wait, and ride time subtracting from this limited resource. They designed a competition where teams went into the park to test their approaches to solving this problem. The group modeled the problem and solved it using the optimization techniques we had talked about in the class. This solution was generated with expected wait times for the attractions, and since the actual simulation (contest) used actual waiting times on that day, there was a great deal of variability. Because of this variability, a couple of teams did find better solutions than the computer by exploiting bargain wait times.
- An Analysis of the Queue of Big Thunder Mountain Railroad: One of the favorite attractions of our class (especially during fireworks) was Big Thunder Mountain Railroad. The queue for this attraction asks guest to choose between going in the left queue or the right queue to get seated on the roller coaster. Grace, Maddie, Nathan, and Sydney wondered whether there was a difference in the wait of the two lines. The students designed a matched pairs statistical experiment to gather data by having pairs of students simultaneously join the left and right queues and report times as they passed certain checkpoints in the line. They used some nice data visualization in Tableau to show the average wait from each line. Their analysis showed that there was not a statistical difference in the waits between the two lines, even though the left side is physically shorter.
- A Network Analysis of Congestion in the Magic Kingdom: Navigating the Magic Kingdom efficiently involves a lot of split-second decisions. There are several areas in the park where people
are almost always shoulder-to-shoulder and traveling in these areas can result in delays. Lily, Louisa, Regan, and Sam modeled the Magic Kingdom as a network and used a network analysis technique called betweenness to find areas in the network where they expected congestion to occur. The technique finds shortest paths between every pair of nodes (attractions and other points of interest) and sends a unit of flow along each of these paths. The edges (walkways) of the network accumulate flow on them, and the highest accumulation indicates walkway areas where congestion is likely to occur. They also weighted this flow based on the popularity of attractions. It was really cool to see that the walkways identified through the network analysis were actual areas of highest congestion in the park!
We are very proud of the projects that these groups produced. They did a fantastic job!
As you can see, we threw a lot at these students in terms of academic content, and we were impressed with how much they absorbed. While we can’t expect all of it to stick, we do hope that what they heard will pique their interests and inspire them to learn more by taking more classes in the mathematics curriculum. We also hope that exposing them to professionals (Disney and others) will increase their appreciation for how the concepts they learned are used in industry. Hopefully, as a result of the course, the students will discover that at Furman they are building the tools they will need to succeed. Beyond the academics, the opportunity for students and faculty to spend three weeks learning together, communicating together, and helping each other has made us all grow as individuals.
We are extremely proud of this impressive group of students and are grateful we get to share this experience with them!
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| Signing off the blog for the last time, Drs. Hutson, Bouzarth, and Harris |
