Numeric Simulation, Apple Pie Sales, and Thanksgiving

Accurately determining the appropriate stocking levels of perishable items is very important. Stock too few items and you will have disappointed customers. Stock too many and you’ll have unsold product - which has a high probability of turning into a loss.     

A number of industries face this dilemma each and every day.

For example, how many seats should an airline sell for a flight? If a carrier sell exactly the same number of seats that are on the plane, then they are almost certain to have empty seats. If they oversell too much, they are likely to strand passengers and create some serious angst...especially around this time of year when people are trying to get home for Thanksgiving and their favorite apple pie.

There are a number of ways to model and address these questions, perhaps the easiest way is to simply do a numeric simulation.

Let’s say that a bakery has 1,000 customers and they expect 90% of their customers to purchase an apple pie. In this case, the number of customers that actually pick up an apple pie might look like this:

apple pieTen thousand cases were simulated for the above graph and not surprisingly, the average number of apple pies picked up is 900 (i.e. the number of customers multiplied by the likelihood that each customer will pick up an apple pie). But, the real interesting questions tie to how much variation there is in this distribution, because that determines how many pies the bakery needs to make. That is, if we just rearrange the data and plot it as a cumulative probability (see figure below), we can see that 99.99% of the time, customers picked up fewer than 937 apple pies.

applie pie2And, if you do these numeric simulations enough, for different number of customers (n) and differently probabilities (p), you will notice that the average number of apple pies picked up (x-bar) is:

apple pie3 And the standard deviation of the distribution (σ) is:

apple pie4 which is simply a binomial process in action. 

Happy Thanksgiving, everyone. Enjoy your apple pie!


There is 1 Comment

Luc Audibert's picture

Hi Michael,

I think that with such binomial distribution we can approximate with the normal law. Then at average + 2 SD = 920 pies, we might have in average 17 unhappy customers. Or 98.2 customer satisfaction, or again a sigma capability of 3.59. At 99.9 % we have all possibilities and then 100% customer satisfaction. But we do not need statistic for that. We need statistic to model the best stock level to achieve best customer satisfaction and best supplier's interest.

Questions we should have as well are if the result of pie sells is due to limited offer or not? In real life the distribution might not be close to normal as pie availability is just one of the factors defining this response.

I just discovered your blog and this is a very good one!