How to use the appointment overbooking calculator
- Enter the number of slots you can serve and the chance that a booked person shows up.
- Enter the income you earn from each person you serve and the cost of turning a person away.
- Optionally open the advanced options and add a cost for each slot that stays empty.
- Read the number of bookings with the highest expected profit and compare it with not overbooking at all.
Formula
people who show up out of n bookings follow a binomial distribution with chance q
expected served = sum over k of P(k) x min(k, slots)
expected profit = income x expected served - overflow cost x expected turned away - empty slot cost x expected empty slots
best bookings = the n from slots upward with the highest expected profit
The probabilities are computed in a way that stays accurate for large numbers of bookings and for show-up chances very near 0 or 100%.
Ten slots and an 85% show-up chance
With 10 slots, an 85% chance that each booked person shows up, $120 of income per person served and $90 for each person turned away, booking exactly 10 gives expected profit of $1,020.00 and never turns anyone away.
Booking 12 raises the expected profit to $1,101.00, because 12 x 0.85 = 10.2 people are expected and the extra bookings usually fill slots that would otherwise be empty. The cost is a 44.3% chance that at least one person has to be turned away. If a person turned away costs $240, the best level drops to 11 bookings with an expected profit of $1,061.76.
The trade-off
Every extra booking fills a slot that would otherwise stay empty with some probability, and risks turning someone away with another probability. The best number balances the two. If turning someone away costs a lot, the best level is close to the number of slots. If an empty slot costs a lot, it is higher.
The same logic of too much against too little drives the Newsvendor Order Quantity Calculator. To look at the odds of an exact number of people showing, use the Binomial Probability Calculator.
Cautions
People are assumed to decide independently, but families and groups come together. Some fields restrict or forbid overbooking, and a turned-away person may not come back. Use the result as one input and set a limit you are comfortable with.
Assumptions and limits
- Each booked person shows up independently with the same chance.
- People turned away are not replaced, and late cancellations are not refilled.
- The table tries bookings from the number of slots up to double that, at most 1,200 in total.
- On a tie the lower number of bookings is chosen.
- The result is a planning aid and you should check any rules that apply to your field.
Frequently asked questions
How do I find my show-up chance?
Use your own records: the share of booked people who attended over a few months, ideally by day of the week.
What if I cannot measure the cost of turning someone away?
Try a few values, such as a refund, a discount and the value of a lost customer, and see how the best level moves.
Is overbooking always profitable?
No. If turning someone away is very costly, the best level is simply the number of slots.
Why is the chance of overflow so high at the best level?
The best level accepts some risk of overflow in return for fewer empty slots. If that risk is too high for you, choose a lower number from the table.