The Accidental Tourist

As you may have read in previous blog posts, our class recently participated in an activity called the Traveling Tourist Problem (TTP). This was a Math and the Mouse version of the Amazing Race, of sorts, and we, the professors, would like to take a turn at blog writing and and share some details about this activity.  

One purpose of the activity is to give students the chance to understand the complexities of decision making. The main topic of the course itself is the mathematics of decision making, and this activity gave them first hand experience.

Some decisions lend themselves nicely to a "brute force" approach --- let me consider all possible options and then choose the best one (whatever "best" means to me). Most decisions, however, have too many options to be able to consider all possible choices, and the Traveling Tourist Problem is one of these examples.

Teams of three students each (plus one professor team) were given the task to visit and ride twelve different attractions in the Magic Kingdom --- and to do so as quickly as possible. The twelve attractions were ...

• Big Thunder Mountain
• Buzz Lightyear’s Space Ranger Spin
• Dumbo
• Haunted Mansion
• It’s a Small World
• Jungle Cruise
• Peter Pan’s Flight
• Pirates of the Caribbean
• Many Adventures of Winnie the Pooh
• Seven Dwarfs Mine Train
• Space Mountain
• Under the Sea—Voyage of the Little Mermaid
  

On the day before the big race, teams were given data regarding expected wait times for each ride (for every 15-minute interval of the day) and the walking times between attractions. Using this information, teams could plan their strategies for the next day. Which ordering of these twelve rides would mean the fastest completion?

If a team thought that they could just check all possible orderings and choose the fastest one based on the data, then they quickly realized such a plan would not work. There are 12! (=12 * 11 * 9 * ... * 3 * 2 * 1) different ways to order the rides. This is almost 480 million variations!

This activity is related to a famous problem in mathematics called the Traveling Salesman Problem. In this problem, a salesman needs to find the shortest route connecting all of his target cities. As the number of cities increases, finding an optimal solution quickly becomes intractable (i.e. very difficult for even the fastest computers). The problem our students worked on is actually harder than this one, since the wait times at each attraction vary throughout the day. So, while it was not realistic for any of us to find (and know we found) the fastest route, it was still fun to do our best and see what we learned.

There are a number of strategies that teams could have implemented. Among them are ...

• try to minimize walking distance
• try to minimize total waiting time
• go to most popular rides first
• get as many low-wait rides as possible early

Teams could also wait to see what opportunities arise on the day itself. Since there are lots of unexpected things that can happen (especially rides going down for a period), it is sometimes beneficial to be flexible.

The conventional best first move in the Magic Kingdom is to go immediately to Seven Dwarfs Mine Train --- the ride that typically has the longest average waiting time throughout the day. Interestingly, only one of our teams made this move. Other teams made different decisions. One team went first to Peter Pan's Flight (another ride with very long waits), one team went to Space Mountain. One challenge with these rides is that they are in an area of the park that was open 30 minutes earlier than we could enter (open early for guests staying in Disney hotels). So, even when they were the first ones in our group to enter, there were already people in line when they got there. Several other groups headed first to Adventureland and Frontierland, areas of the park that had popular rides AND that were empty immediately prior to the teams' arrival.   

In the image below, you will see all of the first rides that were chosen by the groups --- and the numbers in the circles represent the finishing place of the corresponding team.

This year's first place team (Emily, Savannah, and Virginia) started at Space Mountain, picked up the nearby Buzz Lightyear afterwards, and then went to the popular Seven Dwarfs Mine Train third (waiting only 16 minutes for this popular ride). Interestingly, they waited the exact same amount of time for this ride as the team who went there first. So it seems that the trip to Space Mountain first was a wise one. The winning team did save a few longer rides until the end, waiting 25 and 35 minutes (respectively) for Pirates of the Caribbean and Jungle Cruise. Overall, the average wait time for the first place team over all twelve rides was 12 minutes and 35 seconds. Impressive! You can see their winning path in the figure below with colors following rainbow order showing the passage of time (red shows the first hour of the day, followed by orange, etc.).

The professor team, keeping the non-winning streak alive, finished in fifth place. We waited an average of 13 minutes and 20 seconds in line over the twelve rides. That isn't terribly different from the winning team's average wait, but since that team spent less time walking between rides, they finished 30 minutes before the professors did.

Interestingly, the team who finished in a close second place (Hannah, Tyler, and Will) waited an average of 14 minutes and 15 seconds in line --- nearly two minutes longer per line than the winners. They finished the activity just four minutes after the first place team, and this suggests that they made up their time by walking shorter distances than the first place team. Indeed, the first place team walked around 2.3 miles, and the second place team walked just over 2 miles. By comparison, the fifth place professor team walked over 2.7 miles.

Two days after the activity was complete, our students had the chance to meet Len Testa, president of touringplans.com --- a company that provides efficient plans for visiting attractions in Walt Disney World and in other theme parks. Len enjoyed hearing about the students' adventures, and they, in turn, enjoyed hearing about all of the things that go into the ability to provide personalized plans for tens of thousands of subscribers.

All teams have enjoyed recounting their strategies and stories. It was really rewarding to us as professors to see our students engage in skills that are important in all aspects of life: planning, adapting, executing, and having fun. Through this activity, students saw firsthand the importance of careful decision making, the reality of uncertainty, the benefits of thinking on your feet, and the significance of collaboration and teamwork.

First place team: Emily, Virginia, and Savannah