By Len Testa
[Update: Our statistician Fred Hazelton suggested we post the computed costs of dining plan components from previous years, to show how prices have gone up. Prices from 2012 are now included in the Results section.]
Last week we did a presentation to Furman University math students on the math and science behind a Walt Disney World vacation. (You can see my presentation to the class here.)
One of the things we discussed was how you could use a system of linear equations to determine the value of each component in Disney’s dining plans. And then I realized that I hadn’t run the numbers for 2014. So here they are. (If you want to follow along in Excel, here’s the spreadsheet. You’ll need to load Excel’s Solver add-in.)
Disney has 3 versions of its dining plan:
The Standard Plan, with 1 quick-service meal, 1 table-service meal, and 1 snack per person per night of your stay, plus a refillable mug good for the length of your stay, for $60.64 per night
The Quick Service Plan, with 2 quick-service meals and 1 snack per person per night, plus a refillable mug, for $41.99 per night
The Deluxe Dining Plan, with 3 meals at any combination of quick-service and table-service locations, 2 snacks, and 1 refillable mug, for $109.53 per night
A first attempt at modeling the Standard and Quick Service plans as math equations might look like this:
Version 1
(Standard Plan) 1Q + 1T + 1S + 1M = $60.64
(Quick Service) 2Q + 0T + 1S + 1M = $41.99
In English, the first line says that the cost of one quick-service meal (Q) plus the cost of one table-service meal (T), plus the cost of one snack (S), plus the cost of one refillable mug (M) is $60.64. The second line is similar, and reflects the fact that there are zero table-service meals.
But we need to tweak those equations a little bit. You’ll notice that the equations refer to the per-night cost of the quick-service, table-service, and snack components, but the mug is one per stay. Thus, cost of the mug should be spread out over every night of your stay.
That requires us to make an assumption as to how many nights are in the average Disney World hotel stay. Let’s say it’s 4 nights. In that case, you’re really using 1/4th of the mug’s cost each night, so the equations should look like this:
Version 2
(Standard Plan) 1Q + 1T + 1S + 0.25M = $60.64
(Quick Service) 2Q + 0T + 1S + 0.25M = $41.99
Now the equations more accurately reflect the amortized cost of the refillable mug, and every variable is expressed in per-night costs.
Next we’ve got to add an equation for the Deluxe Dining Plan. The tricky part here is that the Deluxe Plan’s 3 meals can be in any combination of quick-service and table-service restaurants, so we have to make an assumption about how many times we’re likely to eat at each per day.
Let’s make a starting assumption that if you’re buying the Deluxe Plan, you’re really interested in eating at Disney, and more specifically, at sit-down restaurants at least 2 times per day. (This is a safe assumption, since there are other dining plans for people who want to visit sit-down restaurants 0 or 1 times per day. If you wanted fewer than 2 sit-down meals per day, you could just choose one of the other plans.)
One of the things I decided to do in Excel was let Solver try to figure out the right ratio of counter-service and table-service meals in the Deluxe Dining Plan. In my runs, Solver generated reasonable results with around 0.5 counter-service meals and 2.5 table-service meals per day. That’s one counter-service meal every other day. That seems like a lot of food, but keep in mind that if that’s not what people want, they’d choose other dining plans.
Here’s the updated version with the Deluxe Plan and its assumptions:
Version 3
(Standard Plan) 1Q + 1T + 1S + 0.25M = $60.64
(Quick Service) 2Q + 0T + 1S + 0.25M = $41.99
(Deluxe Plan) 0.5Q + 2.5T + 2S + 0.25M = $109.53
Next, it helps to give solver a set of constraints about the various costs. Examples of constraints include:
Snacks cost less than counter-service meals
Counter-service meals cost less than table service meals
Snacks cost more than $1 and less than $7
The value of the refillable mug is at least $3 but not more than $18 (because that’s what a length-of-stay mug costs at retail)
A counter-service meal costs at least $5 and less than $25
These kinds of constraint guides Solver towards a reasonable answer. You can see all of the constraints I used in the spreadsheet.
The next step is to run Solver a few times with different random starting values for Q, T, S, and M. Why? I’m not exactly sure, other than Solver picks a point in the search space to start looking for valid solutions, and random prices decreases the chances of Solver getting stuck.
Results
Solver came back with several solutions whose values of Q, T, S, and M got to within $0.01 of the exact price of the Standard and Quick-Service dining plans, and within $1.40 of the Deluxe Plan. Using these values of each component in the plan, the total error for pricing every plan was between $0.01 and $1.50:
Each counter service meal is worth $14.75 to $18.07 [in 2012: $14.60]
Each table service meal is worth $33.50 to $36.72 [in 2012: $33.15]
Every snack is worth $4.24 to $4.50 [in 2012: $4.18]
The refillable mug is worth $8 to $12 [in 2012: $8]
How to Use These Numbers
You can use these numbers to determine whether a dining plan is right for you. Take a look at our Walt Disney World restaurant menus and ask yourself whether you’d spend between $14.75 and $18.07 for a quick-service meal, or $4.24 to $4.50 for a snack, or around $33 to $37 for a sit-down meal.
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What Each Disney Dining Plan Component Costs For 2014 is a post from the TouringPlans.com Blog. Signup for a premium subscription today! Or get news via Email, Twitter, & Facebook.