Markov Chains, Matrices, Mutations, Magic, and More!

Hi everyone, Maria and Courtney here! Today was another exciting day of both math and park exploration, and it included further study of probability concepts, project work, and time in Epcot and Hollywood Studios.  Here’s a quick look at all that we learned and did today:    

Markov Chains & Matrices

An example tree diagram that we studied
this morning in class.  The branch values
display the probabilities of each
event in the chain and the values to the
right show conditional probabilities.

Our class this morning provided a more in-depth look at probability, and today’s material focused most on a topic called Markov Chains. Essentially, a Markov Chain is a series of phases in which the outcome of each phase is one of a set number of possibilities, and the probability of a certain outcome occurring given the event of a prior state can be determined exclusively from the data from these two phases.  A Markov Chain qualifies as a stochastic process, meaning that at each phase of the chain there is more than one possible outcome, and the chosen outcome is variable.  


For example, this can be more easily visualized by studying a tree diagram; with this type of display, the tree begins with a single state (or event) and has a set number of offshoots of different possible states (see picture).  The probability that one event occurs given the previous state is calculated by multiplying the probability values along the two branches (this is known as conditional probability).  Additionally, we were able to utilize matrices in order to learn how to work with Markov Chains that have more states than can be easily displayed on a tree diagram, and we learned that matrix multiplication can be used to find the conditional probabilities of a Markov Chain.  Questions that Markov Chains may ask are ones like: “Given the fact that a person rides Rock ‘n' Roller Coaster once, what is the probability that that person gets back in line for the ride? What is the probability that they choose to ride another particular attraction in the park?”  These are the types of ideas Markov chains can describe and matrices can help us to solve!    

Genetic Algorithms

As the genetic project presentations are continuing to approach, each group spent several hours making progress on their respective projects this morning after class. To complete this project, we are using the same data that we were given for the traveling tourist problem and are asked to meet the same basic objective; however, this time we only need a path that completes 10 rides in the shortest time instead of 19 attractions. The time it takes to complete the path is called the "tour fitness", and the goal is to find a tour with as minimal a time as possible to complete all the attractions. The project requires that we first choose a set number of parent tours that represent different paths that could be potentially used to complete the 10 rides. We then use different operators to mutate a parent or cross over two parents to come up with a new child tour. Ideally, the child will have a better fitness than at least one of the parents in the population, and it will be able to replace the parent in the new population. After performing 30 mutations or crossovers (called iterations), we will have achieved a population of paths of the best fitness we have calculated, and the goal is to have a population of more fit tours than the ones with which we began.

Data Collection in the Parks   

There is always time for data collection in the parks, even on
Rock 'n' Roller Coaster!

As many of us began to express interest in using our own data for our future projects, the professors thought that today would be a good time to have us begin collecting data in the parks. For our final project, students will have the option to either work on a project involving queuing theory or a project of their own choosing. Based on the ideas that we brainstormed this morning, the professors decided that we would spend part of the day in Hollywood Studios instead of only in Epcot as we had previously planned.

Some of us have expressed interest in a project involving the fastpass line. How many people with fastpasses do Disney employees let onto the rides in comparison to the number of people in the standby line? Is it variable to the ride, the employee, or the time of day? To begin collecting data for this project, as we neared the front of a line, we attempted to count the number of people from each line that the employees let into the next segment of the line.  

Another set of students wants to look into how accurate Disney’s posted wait times are. Therefore, for each attraction we rode we recorded the ride, the time of day, the type of line, the posted wait time, the time we entered the line, and the time we got on the ride.

A third group of people is interested in the single rider line and measuring how beneficial the single rider line is for each of the rides that offer it. Test Track in Epcot, Rock ‘n’ Roller Coaster in Hollywood Studios, and Expedition Everest in Animal Kingdom all utilize a single rider line.

Epcot

The simulator at Sum of All Thrills that visitors ride in order
to experience the coasters they have designed.

After project time at the hotel, our group loaded the van and headed to Epcot for the afternoon. As stated, the time in the park provided a valuable opportunity to begin the data collection for our upcoming projects.  Because official groups haven’t yet been decided on for each project, we all worked collectively on gathering data throughout the day to help with all of the potential future projects.  For example, I (Maria) worked on gathering data about single rider lines; specifically, at Test Track (and later at Rock ‘n’ Roller Coaster), I observed the group size of people that were let onto the rides consecutively (which is usually called out by the worker heading the line) and how often single riders are asked to board the ride.  I also worked on collecting data for the project that seeks to compare posted wait times with actual wait times; with a group of several other people, I visited multiple attractions in order to time my wait in each queue and compare them to the posted times.


During our time at Epcot, we were also able to visit an attraction called Sum of All Thrills for the first time, which involves pairs of people designing their own rollercoasters on a computer and entering a simulator to ride the coaster they have designed.  This provided a perfect example of the prevalence of math and science in the park, and as the ride voiceover reminded us before we worked on our designs: “Control the math, control the ride!”  When my group (Courtney) was designing our rollercoaster, we were able to choose what types of thrills we wanted the track to do, the height of the track, as well as the speed. A small crash robot on our design screen would test the ride to make sure it was safe. After our first try, the robot reminded us that our speed had to be compatible with the height of the track. We manipulated our design for another minute and discovered that the higher we made the track, the faster the coaster needed to travel. It is exciting to see how Disney makes an effort to not only give children a fun experience on a ride, but also make it an educational experience.


Before leaving Epcot, we got to eat dinner in the World Showcase - the variety of cuisine among all the countries in the park always makes for a delicious event!

Hollywood Studios

After our Epcot dinner, we reconvened as a group and headed toward Hollywood Studios. This is the first occasion that we have visited two parks in the same day, and the change provided a good way to continue to diversify our data collection.  Our group spread throughout the park and were able to visit and collect information from a number of attractions.  I (Maria) visited Rock ‘n’ Roller Coaster twice over the course of the evening; the first time, I went with a large group and we spread throughout the line in order to broaden the amount of consecutive data we could collect.  As explained above, we focused on recording the group sizes called onto the ride; by spreading ourselves throughout the ride, we could collect information about group sizes for a long line of people.  Many people from the group were also able to visit Tower of Terror, Star Tours, a Frozen Sing-Along - and Dr. Hutson even reached a new high score at Toy Story Midway Mania!  Of course, no evening at Disney World is complete without a fireworks show, and our group enjoyed getting to see the impressive Star Wars-themed fireworks display before leaving the park.



Tomorrow, each group will present its genetic algorithm project results in the morning, and after class we’ll head back to Hollywood Studios to continue our data collection!

Written By:

Maria & Courtney

Day 7: Takeovers, Rainstorms, and Bratwurst

Hello, everyone! Caroline and Johanna here to take you through our 7th day at Disney. Well recovered from our borderline traumatic day at Magic Kingdom, we began the day with a great lesson on graph theory and then visited Epcot for the second time.

Pick the Right Path

Courtney mapping out a path for the class

Today, we built upon our earlier work with network models to create shortest path models. Basically, we wanted to find the best path to get around a certain number of vertices based on different constraints. One constraint could have us not trying to hit every vertex, but every edge. Another could have us trying to hit every vertex only once and end up at our starting vertex. Finally, the professors introduced a problem where certain vertices and pathways had certain values, where the goal was to find the non-repeating path with the lowest total value. The vertices had different values depending on their order in the sequence, which made the problem simulate walk and wait times in a park. In one example, the vertices had different values depending on whether they were among your first four or last four, which resembled increasing wait times throughout a day at Disney. At the end of the session, Dr. Hutson introduced how the Traveling Salesman Problem, the basis of our Traveling Tourist Problem from yesterday, relies on a lot of the same data we used in our practice problems, and how companies like Touring Plans can make good solutions quickly for their clients.

OR We There Yet?
After class, we continued working on our OR problems. My group (Caroline) has made a lot of progress, and was actually able to start putting the pieces of our constraints together. Because each section of the constraint can be very long, we incorporated some Mouse into our math by naming each different section after a Disney character. Hopefully tomorrow we can begin consolidating our "characters" and have a model ready to present! My group (Johanna) has also made good progress. We are currently working on building generalizations from our constraints that make the strings of different variables we are summing more concise for the sake of both the computer and our classmates. Though we won't have a solution until we use a program like Mosel to calculate the problem, we have built a good bit of our model for Thursday, which is quite exciting!

Takeover Time
So, today, I (Caroline) had the opportunity to take over the Furman Snapchat as part of their showcase of different MayX courses. It was a lot of fun to be able to explain and update our activities in a way that would reach a lot of people. If you would like to see my story, follow furmanu on Snapchat, or look for it on YouTube!

Rain, Rain, Go Away!

The crew sporting their fashionable rain gear in Epcot's Germany

Unfortunately, we had our first spot of bad weather today, but that did not keep us away from the parks. After class and lunch, we set out for Epcot, where we took advantage of the single rider line in Test Track, in what was probably one of the most efficient uses of the single rider line thus far in the trip. We made it on the ride in less than 5 minutes, just before inclement weather halted it indefinitely. From there, some of the group decided to test ourselves and our stomachs at Mission Space, where we learned that Zack is the last person you want on your space shuttle due to his inability to push buttons at the correct time. After Mission Space, a few of us ended up taking pictures with Baymax from Big Hero 6, who is as adorable as he is squishy, and who gave great hugs. After wandering around Epcot for a while, we all met up at the Land Pavilion to ride Living With the Land. Despite its slow pace, it was really interesting to see how Disney is a prominent facility for ecologically smart farming techniques. The food grown in the Epcot facility is actually used in the park. After the ride, we met with Furman student Haley Cottingham, who is just finishing an internship with Disney. Haley is a computer science major, and her internship actually deals with workforce scheduling, the same issue as our projects. She was able to give us a lot of information about the Disney internship program in general, and told us about all the possible career opportunities available at Disney. When we finished our visit with Haley, we had some time to kill before our dinner reservation, so we tried the ride in the Mexico Pavilion, and almost lost Dr. Harris when he nearly fell out of the boat!

We had a great time talking with Haley (third from right, bottom row) about her experiences at Disney

Did Someone Say Bratwurst?

Jamie enjoying some
delicious German food

We ended the day with a delicious dinner at the German restaurant Biergarten. While we waited for the table, we explored the nearby shops, including a toy store and a glass and jewelry shop. When we finally entered the restaurant, we were floored. The restaurant is perfectly decorated and even featured a performance by a German band. Because it's buffet-style, we were able to try a whole spectrum of German foods, and we could get as much as we pleased. Some favorites were the schnitzel, the bratwurst and sauerkraut, red cabbage, and apple strudel. Due to the rain, we decided to opt out on a 2nd viewing of Illuminations, and made our way to the exit. We didn't let the rain dampen our fun, and we splashed and laughed our way to the van, ready to get some rest for another day.



Written by:
            Johanna Swab                                                         Caroline Trammell

Day 2: Modeling and Epcot

Morning:
Hi, everyone, Caroline and Jamie here! Today we had our first full day in Florida, and we began bright and early exploring some of the math aspects of Math and the Mouse. We spent the morning learning about the basics of Operations Research (OR) Modeling. This form of modeling seeks to accomplish a main objective under a certain set of constraints. For example, a bakery would seek to optimize profits with the constraints of limited ingredients or time. At Disney, OR Modeling is used to design incredibly detailed schedules for the cast members in the parks, who often are moving every 15-30 minutes. After looking at several examples of different OR Modeling problems, we split into groups to solve some problems on our own. One of the problems we examined in detail was the Knapsack Problem, which seeks to fit as many items as possible, with the greatest utility, or usefulness, in a set space. Later in the day, we learned that this problem would be a critical part of tomorrow's activity. After working through and presenting the problems, we set out for Epcot.

The required photo in front of the Daisy Duck topiary

Epcot:
Our first stop at Epcot was in the big sphere itself, as we experienced Spaceship Earth. Soon, we moved to the much more exciting Test Track. We were a little parched after Test Track, so we went to the Coca-Cola Club Cool, where Dr. Harris and Dr. Hutson convinced several unsuspecting students to try the infamous drink Beverly. Luckily, the Flower and Garden Festival is going on currently, and there were many great exhibits and unique stops, including a Butterfly Garden.

Some of the beautiful flowers in the Butterfly Garden

Our next stop took us to the relaxing and almost sleep-inducing Journey into Imagination and The Seas with Nemo and Friends rides, though we did enjoy the air conditioning and seeing a dolphin. Following our time at the rides, we received our assignment for tomorrow, and split into our groups to formulate plans. From here, our smaller groups split up to see more of what Epcot had to offer.

Caroline:

After quickly finishing our project, my group decided to take advantage of the dinnertime short ride lines. Johanna and I had a great time on Mission Space and then enjoyed the quick wait for a second run on Test Track. After we had our fill of rides, we set out for the World Showcase, but not before I had my very first Dole Whip. We slowly made our way through all the pavilions, and ate our way through quite a few of them. We ended up enjoying teriyaki beef at Japan, gelato at Italy, and funnel cakes just in time for Illuminations. Even though Dr. Bouzarth assured me that the pavilions had not been renovated recently, I couldn't help but feel that they were so much bigger than when I last visited Disney! I could easily spend an hour in each pavilion, exploring the various shops and restaurants. Illuminations was just as exciting as I remembered, and was a great way to close out another day in Disney.

The Italy Pavilion, home to gelato and delicious food 

Jamie:
After a more tedious attempt at strategy, my group also went to Mission Space. While I was a bit queasy afterwards, it was quite fun.  Following that, we began our tour of the countries. We stopped for fish and chips in England, Bratwurst in Germany, and churros in Mexico. The national pavilions are fantastic with a variety of sights and tastes unique to each one. Following our walk, we regrouped with the others to watch Illuminations, a spectacular display of laser optics and pyrotechnics that left our ears ringing and our eyes dazzled. After so much excitement, we took the long slow walk back to the van to get some much needed beauty sleep.

The Illuminations Finale

Written by:

Caroline

Jamie