The Genetics of Touring

Each student has written three blog posts at this point, so it's now the professors' turn to take over! Yesterday, we started the day with an optional trip to Hollywood Studios to allow students to collect project data. Unfortunately, the park had Extra Magic Hours that morning, so WDW resort guests could enter the park an hour before we could, which made for long lines early in the day. After some time to work on projects back in the hotel in the afternoon, we headed to Animal Kingdom to enjoy another evening in the park. 

The Genetics of Touring

Last week, the students participated in a race to see which team could accomplish a list of 19 tasks in the Magic Kingdom most efficiently.  The tasks ranged from riding an attraction to getting a picture taken with a character.  Differing from the Traveling Tourist Problem at Epcot two years ago, we allowed teams some flexibility in the attractions that they visited.  Here is a list of possible events:

The winning group's tour of the Magic Kingdom.

Required                                

Space Mountain       

Buzz Lightyear’s Spin            

Seven Dwarfs Mine Train 

Dumbo   

Haunted Mansion 

Peter Pan’s Flight           

Under the Sea - Voyage of the Little Mermaid

Splash Mountain              

Big Thunder Mountain Railroad

Pirates of the Caribbean           

Jungle Cruise         

Group Photo with a costumed character  

Group Photo with Walt Disney 

Fantasyland (Pick 3)
It’s a Small World
Winnie the Pooh

The winning group, Zack, Mary Lib, and Johanna.

Mad Tea Party
Hall of Presidents
Enchanted Tales with Belle
Barnstormer
Prince Charming’s Royal Carousel
Mickey’s Philharmagic

Tomorrowland (Pick 2)
Astro Orbitors
Stitch’s Great Escape
Monster’s Inc. Laugh Floor
Carousel of Progress
Tomorrowland Speedway

Adventureland (Pick 1)
Magic Carpets of Aladdin
Country Bear Jamboree
Enchanted Tiki Room

The students were assigned into groups of three and were given data on the expected wait, walk, and ride times in the Magic Kingdom that day (supplied by tourningplans.com).  They had an afternoon to design a tour of the chosen 19 attractions and were allowed time in the park to perform reconnaissance.  The following day we raced, and (drumroll, please) the winning team was Johanna, Mary Lib, and Zack who finished the tour in just under 6 hours.  Their tour is shown above.

Johanna, Mary Lib, and Zack flying with Dumbo.

After the race, the students learned to model the problem using networks and also learned that the problem they were trying to solve is a variation of the Traveling Salesman Problem.  This problem of trying to find an efficient tour of a given list of locations is one that is researched by logistics companies like FedEx and UPS.  A variety of algorithms exist to try to produce optimal, or near-optimal, solutions to these types of problems.  One such method is to use what are known as genetic algorithms.  These algorithms mimic natural selection in that a variety of solutions are produced (called a population) and their tour length (called their fitness) are calculated.  A solution for this problem is just a sequence of attractions in the order that they are visited (called a tour).  Solutions are then subjected to one of two types of operations to produce new, child solutions.  The first type of operation is called a mutator.  These operators take a member of the population and augment it in some way, such as switching the order in which you visit two (or more) attractions.  The second type of operation is called a crossover operator.  These operators take two members of the population and produce a child that resembles both parents.  For instance, a crossover operator might take two members of the population and find the attractions that are sequenced in a common position in both parents and include those attractions in those positions in the child.  The remaining attractions are then randomly placed in the remaining sequence positions in the child.  Every time a child is produced its fitness is calculated and if it is better fit than the least-fit solution in the population, that child replaces the least-fit solution in the population.  These operations reoccur for a fixed number of iterations (usually quite large) and the most-fit solution from the population is chosen as the “optimal” solution. 

The winning group at the finish line and
home of Dole Whip, Aloha Isle.

The second project for the students in the course involved inventing mutators and crossovers for a genetic algorithm to find the optimal solution of an abbreviated Traveling Tourist Problem involving only ten rides.  In fact, the example mutation and crossover operators described above are ones that students came up with.  Here are examples of other inventive operators that the students generated.

Crossover Example

1. Define Parent 1 as the parent with the single highest wait time.
2. Find the sequenced attractions that match in P1 and P2.  Include these attractions in these sequenced slots in the child.  Then take P1’s highest non-matching wait time and swap it with other attractions until it is in the location in P2’s sequence of the highest non-matching wait time.
3. Continue to find a nonmatching ride whose wait time is the highest and switch it with the lowest remaining wait time of the other parent. 

Mutator Example (Frame Shift)

Move an attraction from sequence spot j to sequence spot k and then shift all of the attractions in between sequence spots j and k one slot to the right or left depending on j’s relative position to k.

After running their genetic algorithms on the subset of rides that they were given, the group consisting of Johana, Mary Lib, Alyssa, and Molly found a tour that could be traversed in 284 minutes.  The sequence of attractions in this tour was

1. Seven Dwarfs Mine Train
2. Peter Pan's Flight
3. Haunted Mansion
4. Jungle Cruise
5. Buzz Lightyear
6. Dumbo
7. Space Mountain
8. Splash Mountain
9. Pirates of the Caribbean
10. Big Thunder Mountain Railroad

After studying these genetic algorithms, our group met with Len Testa, President of Touring Plans and co-author of The Unofficial Guide to Walt Disney World, which has sold more than 4 million copies worldwide.  Testa’s company, Touring Plans, employs an analytic approach to travel, helping its subscribers not only find good park tours, but also finding affordable options for park tickets, finding the quietest hotel rooms, etc.  His company employs mathematicians, statisticians, and computer scientists to model and produce solutions to many problems related to travel.  Len spoke with our students about the evolutionary algorithms (such as genetic algorithms) that his company employ to produce tours for users in a quick amount of time.  The time that the students put into to developing an intuition about the problem and creating mutation and crossover operators paid off when they saw how Touring Plans employs these types of algorithms to produce solutions to real-world problems.  They came away with an appreciation for the mathematical sophistication that Touring Plans brings to their solution approaches.  For some students this was an experience that caused them to remark that they wished they had brought a resume to the talk to give to Testa.  For the professors, we give many thanks to Len Testa for inspiring a group of students to continue to develop their mathematical creativity and problem-solving skills to make a difference for others. 

Day 17: Just Look at the Time!

Posing together outside one of our
group's favorite rides, Tower of Terror. 

What do Midway Mania, Markov Chains, Mission Space, and matrices all have in common? They're all integral parts of our magical studies here at Math and the Mouse! Mickey and Minnie Mouse have helped us explore mutations, modeling, mapping, and more - you wouldn't believe how much math can be found from observing the Mary Poppins line, watching MuppetVision 3D, and even riding Mad Tea Party or exploring the Maharajah Jungle Trek! We've enjoyed all of our time and studies in each of the four parks (although, if they have to choose, Molly and McKenna say that their personal favorite park is Magic Kingdom) and it has certainly been a marvelous, mathemagical, and alliterative May X! And today's update on our group's latest adventures is brought to you by Maria and Mary Lib!

Hollywood Studios

This morning we arrived in Hollywood Studios for rope drop! We started our day off at Midway Mania and Tower of Terror, and then continued collecting data at Rock ‘n' Roller Coaster. As mentioned in previous blogs, a couple of the groups are interested in Rock ‘n' Roller Coaster for various research reasons: one group is looking at the efficiency and worth of the FastPass, while the other is concerned with the value of the single rider line. While standing outside the ride itself, we collected entrance and exit data in 15-minute intervals, which will help us determine arrival and departure rates of the ride. While in line, groups recorded data such as the times group members entered the ride, when the FastPass and Standby lines merged, non-single rider line party sizes, when group members boarded the ride, how far apart each consecutive boarding time was, and when each group member exited the ride.

We were so lucky to have had the opportunity to meet
 with a Disney engineer this morning!

After collecting data, we reconvened as a group in front of Rock 'n' Roller Coaster at 11:00 AM to meet with an electrical engineer for Disney that has a family connection to Furman. He talked to us about the mechanical operation and troubleshooting of two park headline attractions, Rock ‘n’ Roller Coaster and Tower of Terror, as well as a few others. Everything he discussed was very interesting, and one of the highlights of his talk was his discussion of the random sequencing of Tower of Terror and Star Tours. Although it’s not obvious, each ride on Tower of Terror and Star Tours is not the same every time. Tower of Terror has a set of possible drop and rise sequences, and one of these is randomly selected for an elevator each time it runs. In a similar manner, the ride sequence for Star Tours is randomized, where each ride includes four of the thirteen possible segments. The random sequencing of both attractions allows for a unique and surprising experience for guests, even if they have ridden the rides repeatedly. Just another example of Disney going above and beyond!

After meeting with our guest speaker, we left the parks and the professors treated us to lunch - the break from PB&Js was greatly appreciated by students! The Chipotle and Panera Bread near our hotel are within a few feet of each other, so we split up and reconvened back at the hotel to eat together.

The Physics Behind Disney World

This afternoon in class, we were able to look more closely at the application of science in Disney World; specifically, we learned more about the physics behind the park.  Dr. Bouzarth gave a very interesting presentation that overviewed some fundamentals of physics and ways that they apply to the operation of Disney - this connected very well to our session with the guest speaker this morning, as we were able to explore in even more depth several ideas that he had discussed.  In the context of discussing such ideas as force, energy, magnetism, and Newton's laws of motion, Dr. Bouzarth helped us to make connections between these concepts and the park operation.  For example, we discussed the Test Track ride in Epcot, which our guest speaker had told us was the fastest ride among the four Disney World parks, in light of the idea of energy transfer.  We also learned that Rock ‘n’ Roller Coaster (the second fastest ride in the park) is able to accelerate from 0 to 57 miles per hour in less than three seconds because it utilizes more than 100 electromagnets which are underneath the tracks.  The quick acceleration is one of the signature features of the ride and one that we have all experienced, and it was interesting to learn more about how it actually works!  We finished the lesson by reviewing limits, derivatives, and integrals (concepts that we had all covered in calculus) and their applications to calculating position, velocity, and acceleration of objects.   

Animal Kingdom After Dark!

After riding Everest seven times tonight, we became pros
at having fun with the coaster's camera!

After class this afternoon, we left the hotel at 4:30 PM and headed to Animal Kingdom - this was very exciting, as it had been almost a week since we were in this specific park.  Also, today was an unusually special day in the park - for the first time ever, Animal Kingdom was open until 11:00 PM (a big difference from its traditional 6:30 closing time)!  Tonight was also the unofficial premiere (or “soft opening”) of the Jungle Book show in Animal Kingdom, so there was an extra amount of excitement within the park this evening which only added to the fun of the atmosphere. Once we had arrived at the park, our entire class rode the Dinosaur ride, a thrill ride that takes groups of 12 on a dark and bumpy journey through a path filled with a variety of moving dinosaur figures.  All of the students in our class were able to fit on the same car, which turned out to be quite the adventure!  The ride, which can be fairly unnerving at the best of times, became extra scary today when our class got stuck on it, and we were in complete darkness for what seemed like quite a while (though probably was only about two minutes).  After this unexpected adventure, the group had free time to visit other attractions in the park and to continue collecting data at Expedition Everest for the single rider line project group. We also had evening FastPasses for the Kilamanjaro Safari; while we had done this safari once before in the daytime, it was an even more fun activity around dusk, and we were able to see a wide variety of animals.  

Tonight definitely had the shortest wait times for rides we have ever experienced in any of the parks, and as a result we were able to ride Kali River Rapids five times and Expedition Everest seven times over the course of the day!  Undoubtedly, one of the highlights of the evening was getting to ride Expedition Everest at night.  Although approximately half of the ride takes place indoors and the other half takes place outdoors, the darkness outside made the ride transitions feel even more seamless than usual - plus, it was extra fun to know that we were some of the first park visitors to get to ride the coaster at night! After closing out the evening by riding Everest as a class for five consecutive times, we headed out right at closing at 11:00 PM.

Looking Ahead
Tomorrow, we will be heading to Epcot at 9:00 AM for rope drop and will spend the afternoon continuing some FastPass data collection in Magic Kingdom.  On Sunday, we will be back in Animal Kingdom for part of the day - and we also have FastPasses to see the Jungle Book!  While we’re certainly sad that the trip is coming to a close, we are doing our best to enjoy every minute left and we know that there is definitely still much more excitement ahead!         

Written by: Maria Bartlett & Mary Lib Saine

Day 16: Happy Happy Birthday

Hey guys, its Alyssa and Zack blogging our way through day 16 of our 21 day adventure. Today we had some choices for our morning activities (Animal Kingdom or sleeping), a birthday party, and spent the night in Hollywood Studios.

Morning Choices

This morning our group had the choice to be early birds and go to Animal Kingdom for rope drop or to act like Sleepy at the hotel.

We both chose to arrive at rope drop, we saw the pre-show for opening the park where a flock of brightly colorful parrots flew over us. We were able to ride Expedition Everest three times, Primeval Whirl, and Dinosaur in about an hour and a half. Time well spent if you ask us! Three of the four students that chose to go to the park this morning are working on the single rider project, collecting data to compare the single rider lines of Expedition Everest in Animal Kingdom and Rock 'n' Roller Coaster in Hollywood Studios. We found that Everest's single rider line is easily observable while you are in it and we gathered a large number of data points to use in our project.

Birthday Cake Time!

After our morning adventure it was time to celebrate Dr. Bouzarth’s son’s birthday! William turned a whopping four years old today, and we had a Cars themed birthday bash full of cake, gifts, and festive party hats.

After the party we met to sort through all the data we had been collecting for the past few days. Each group met to evaluate the data they had, look for potential anomalies or patterns, and establish marching orders for tonight in Hollywood Studios. 

Our group is still playing with different hypotheses we might test once we finish data collection. One thing we've found interesting is that Rock 'n' Roller Coaster and Expedition Everest both have two people per row, but the single rider lines move at different speeds. There are many obvious reasons, such as larger train capacity at Everest, but it is evident that Rock 'n' Roller Coaster's single rider line takes more time to go through.

Hollywood Studios

After we made it to Hollywood Studios, project groups split up to collect data in the park or ride a few rides before dinner at Backlot Express. Dr. Harris and Dr. Hutson tried out the newly opened third track on Toy Story Mania and gave positive reviews of it. The FastPass project group worked on Rock 'n' Roller Coaster, collecting data on the relative speeds of the FastPass and standby lines and the number of riders entering each line. Working on the character meet project, the third group of students gathered data from the Disney Junior characters.

We met for dinner at Backlot Express where I (Alyssa) got the opportunity to fight the dark side by eating a Darth Vader waffle. After dinner we split up again to collect data and ride some more attractions. Our group worked on counting party sizes at Rock 'n' Roller Coaster. I (Alyssa) also ran into one of my Furman friends Elise, who went to all four parks in one day! 

Since Hollywood Studios closed at 8:30, we had time to stop at Twistee Treat and eat some delicious ice cream. We returned to the hotel with enough time to enjoy the pool before resting up for the fun that awaits us tomorrow. We are headed right back to Hollywood Studios at rope drop for data collection and a guest speaker!

Sitting in the (medium sized) giant chair.

Written By: Alyssa Ciurlik and Zack Miller

What Are the Odds?

We have crossed the halfway point here at Disney World on day 11 of our MayX. Blogging today are Mary Lib and Zack! We started our day later than normal, giving us a chance to make up for the sleep that we missed out on for our Keys to the Kingdom tour yesterday. After class and lunch we spent our afternoon in Animal Kingdom and are currently writing this blog from Panera at dinner.

Morning Class Time

In our class time this morning, we discussed binomial experiments, probability distributions, and applied the idea of conditional probability to the game Liars Dice which we played yesterday. 

A binomial experiment uses a binomial random variable and allows you to calculate the likelihood of an outcome after performing n-trials in which each results in a success or failure and the probability is fixed and independent for each trial.  This technique can give the probability of a coin landing on heads 5 times if you flip it 7 times. The expected value is the weighted average of outcomes which could tell you how many times you would expect heads if you flip a coin a certain number of times.

Related to the binomial random variable is the geometric random variable. It gives the probability of the first success happening on trial k. This can be used in Disney to see when the first person will give up on a line that isn't moving.

In our games of Liars Dice last night, we could see our dice and had to make a guess on how many of any number there were based on what we had in our hands. Conditional probabilities use this kind of information to calculate the probability of A, given B. Knowing our dice, conditional probability would allow us to make a more accurate bet on the total number of dice displaying a certain number.

Animal Kingdom

In the park today we had FastPasses for Finding Nemo the Musical, Expedition Everest, and DINOSAUR! 

Many of our group took the time to ride Expedition Everest, Primeval Whirl, and watch the It's Tough to be a Bug show before our FastPasses. The It's Tough to be a Bug show was a lot more interactive than any of us remembered. We were sprayed with bug spray, poked in the back by bees, and had bugs crawling on the seats under us. This show is actually inside the Tree of Life, so the line to get to it brings you close to the animals carved into the tree.

While a little cheesy at parts, Finding Nemo the Musical was a cool adaptation of the movie. The actors are much more talented than either of us - singing, dancing, and puppeteering - all at the same time. Our favorite scene was with Squirt the turtle who was flying and flipping over the stage. Thanks to Crush and Squirt, Mary Lib will have "Go With the Flow" stuck in her head for the next few days. 

We finished up our park time on Kali River Rapids. The warning that "you will get wet and may get soaked" was quite the understatement. Based on our experience, the warning should have just said, "Plan to get soaked." However, it was a nice refresher in the heat.

Tonight

Tonight our groups are working on our second project of the trip. On Monday groups tested strategies for our traveling tourist problem. This project asks us to design new solutions to the problem by using genetic algorithms. A genetic algorithm creates new tours based on your current best solutions to the problem. Each group has to create their own operators for creating new tours based on altering one parent tour or a crossover tour from two parents. 

Written by: Mary Lib Saine and Zack Miller

Day 4: Lions, Tigers and Yetis, Oh My!

Jambo, readers, Molly and Alyssa here giving you the highlights of our day exploring Disney's Animal Kingdom. Jambo means "hello" in Swahili, which we learned from our tour guide during the Kilimanjaro Safari.

Our mandatory group photo in front of the Tree of Life

Our Morning:
We arrived at the park as soon as it opened in order to have as much time at the park before our math portion of the day started. Since our first fast pass was not until 11:00, we were given free time to roam the park. For our first ride we took advantage of the short wait time at Expedition Everest, the popular attraction where riders encounter the infamous Yeti as they venture through Mount Everest. From there we went our separate ways to explore the parts of the park that we were most excited about.

Molly:
After riding Expedition Everest, my group decided to journey through the Maharajah Jungle Trek, a wildlife trail in the Asian Section of Animal Kingdom. On the trail we came across many animals native to Asia such as a komodo dragon, a tiger, some water buffalo and various Asian birds. We were very surprised to see that the tiger, water buffalo and deer are all kept in the same enclosure, thinking that there was no way that was safe for the other animals. However we were assured that the tiger is kept very well-fed and usually just keeps to herself in the shade. On our way over to Dinoland, USA we got distracted by some monkeys right outside the Maharajah Trek. We stayed and watched them for a while, fascinated by the way they moved around, swinging from one rope to another. Once we reached Dinoland we decided to ride Primeval Whirl and Dinosaur, which are two very popular, but both very different rides. On Primeval Whirl we were screaming with excitement, while on Dinosaur we were screaming with fear.

Alyssa: 
Some of us had so much fun at Mount Everest we decided to partake in the adventure again. Since we were now experienced expeditioners, we knew this time to look behind us during the two instances where the ride stops to see the track spin and redirect our course.Viewing this Disney “secret” was perhaps even more exciting than the Yeti sightings Disney wants you look at. After round two of Everest, we decided we needed our spines realigned so we ventured to Primeval Whirl and received a healthy dose of whiplash. It turns out that great math minds think alike, because we all reunited at Dinosaur. Together we enjoyed the exciting, albeit a bit terrifying end to our Animal Kingdom free time. After the ride Dr. Hutson was sure to point out the ridiculousness of a dinosaur being kind enough to help us with our time traveling mission by holding up the log that let us escape a deadly meteorite shower. This, however, did not phase us because we concluded that any dinosaur wanted by Disney must be of the friendly, life saving variety.

Reunited (and it felt so good): 
To ease our fears from riding Dinosaur, we boarded the Kilimanjaro Safari for a relaxing ride through the Harambe Wildlife Reserve, which is dedicated to the protection of many different African animals. On the ride our tour guide pointed out to us a wide variety of animals, ranging from hyenas and flamingos to rhinos and giraffes. The crowd favorite was the giraffe who almost stuck his head in the jeep. We had so much fun trying to spot animals as we rode along and by the end of the ride we had filled the storage on our phones with countless images of every animal.

This giraffe came right up to the side of our vehicle

A crash (group) of rhinos

Following the end of the safari, we made our way to the Festival of the Lion King, a show filled with singing, dancing and tumbling monkeys celebrating King Simba. The energy was so high that we couldn't keep ourselves from singing along to our favorite songs, such as "Circle of Life", "I Just Can't Wait to be King", "Hakuna Matata" and "Can You Feel the Love Tonight".

The finale of Festival of the Lion King 

We ended our day at the park in the exact same way we started, a final ride on Expedition Everest. This marked fourth time for some of us who rode it twice after lunch in the single rider line. With some guidance from Dr. Hutson we were all able to find the Yeti and our reaction was priceless.

Post Animal Kingdom Fun 

To celebrate the completion of visiting all four parks we all went to 7-Eleven to cheer our accomplishment with Slurpees. After that it was back to business, all of the attractions and adventure left us itching to do some math. In the previous days we learned how to model operation problems, but today we finally got to learn the most important part--how to solve them. First, we were introduced to a software called Mosel which basically does all of the hard calculating work for us. The user inputs the objective function, variable declarations, and constraints and the Mosel returns the optimal solution almost instantaneously! 

To introduce how to think about solving this type of  problem, Caroline graciously volunteered to step outside the “classroom” and be blindfolded. We then placed a Clif Bar on the floor, and when she re-entered the room she was allowed ask yes/no questions to direct herself to the prize. From this activity we learned that there are three types of questions you can ask when solving a snack finding problem, “Can I move?, Should I move?, How far can I move?”. It turns out that trying to find a snack while blindfolded is pretty similar to finding an optimal solution, because those are the exact same questions you ask when searching a graph for the right intersection of our constraints. 

For the rest of the night, we were given time to work on our group project. In this problem we are trying to create a work schedule for a Disney restaurant, where each group has to wrestle a different set of constraints. It may to be hard to believe now, but this time next week each group will have a solution to what now seems like a Mount Everest of a problem.

Written by:

Molly

Alyssa