Rent A Car Forecasting: Calculate the # of Bookings for July

[evinda]

Hello! (Wave)

Given a rent a car company, suppose that it has 411 cars for each day.

So far there are 246 bookings totally for July.

The total days of rental are 2625 and the total days/reservations 170.24.

Which formula can we use to make a forecast how many bookings there will be in July?

 

[evinda]

If there are for example 700 cars and the average per rental is 9 days, then we muliply $2.3$ by $700$.

So is the expected number of bookings equal to $700 \cdot \frac{700}{9 \cdot 30} $ ?

If so, why does this happen? Could you explain to me the formula? (Thinking)

 

[I like Serena]

Hello! (Wave)

Given a rent a car company, suppose that it has 411 cars for each day.

Hey evinda!

I'm trying to understand what your numbers mean.

Does this one mean that the company simply has 411 cars that can be rented each day? (Wondering)

So far there are 246 bookings totally for July.
The total days of rental are 2625 and the total days/reservations 170.24.

July has 31 days, so there are 31 x 411 = 12741 days that a car can be rented, isn't it?

Is 2625 than the number of days of those 246 bookings in July? (Wondering)
Or does it means something different?
If so, than we have 2625/246 = 10.67 days/booking, don't we?

But what is that 170.24 then? (Wondering)

Is a booking the same as a reservation?

Which formula can we use to make a forecast how many bookings there will be in July?

A typical approach would be to take the historical average of the number of bookings per day, and multiply by the number of days in the month.
If we have the historical number of days per booking instead, we have do divide by it instead of multiply.
Is that perhaps what your 170.24 is? (Wondering)

If there are for example 700 cars and the average per rental is 9 days, then we muliply $2.3$ by $700$.

Where does 2.3 come from? (Wondering)

So is the expected number of bookings equal to $700 \cdot \frac{700}{9 \cdot 30} $ ?

If so, why does this happen? Could you explain to me the formula?

Where did that 30 come from? (Wondering)

 

[evinda]

Does this one mean that the company simply has 411 cars that can be rented each day? (Wondering)

Yes.

July has 31 days, so there are 31 x 411 = 12741 days that a car can be rented, isn't it?

Is 2625 than the number of days of those 246 bookings in July? (Wondering)
Or does it means something different?
If so, than we have 2625/246 = 10.67 days/booking, don't we?

But what is that 170.24 then? (Wondering)

Yes, 2625 is the number of days of those 246 bookings in July.

We have the number of days which each cargroup is booked and 170.24 is probably the sum of these numbers.

Is a booking the same as a reservation?

Yes.

A typical approach would be to take the historical average of the number of bookings per day, and multiply by the number of days in the month.
If we have the historical number of days per booking instead, we have do divide by it instead of multiply.
Is that perhaps what your 170.24 is? (Wondering)

You mean the number of bookings of the previous year? But this year, the number of cars has been increased... (Thinking)

Where does 2.3 come from? (Wondering)

We have that $\frac{700}{9 \cdot 30}=2.59$ and in order not to have that high expectations, we make it $2.3$...

Where did that 30 come from? (Wondering)

$30$ is the average number of days of a month.
Is it right to find expected number of bookings by the formula $$\text{ number of cars} \cdot \frac{\text{number of cars}}{\text{average number of rental length} \cdot \text{average number of days in the month}}$$ ? (Thinking)
If so, why?

 

[I like Serena]

We have the number of days which each cargroup is booked and 170.24 is probably the sum of these numbers.

Car groups?
It's not clear to me yet what the number represents. Are we supposed to do something with it?
Or can we just ignore it? (Wondering)

You mean the number of bookings of the previous year? But this year, the number of cars has been increased...

Perhaps we could compensate for the increase in cars... (Thinking)

Then again, looking at your final formula, I think there is an assumption in there.
I think we assume that every car is rented out every day. Can that be the case? (Wondering)
If so, then historical numbers are irrelevant.

We have that $\frac{700}{9 \cdot 30}=2.59$ and in order not to have that high expectations, we make it $2.3$...

That doesn't sound like mathematics does it?
Or are we talking about a real life situation where we apply some rules of thumb? (Wondering)

Whatever the case, such assumptions and rules of thumb should be clearly stated. (Nerd)

Is it right to find expected number of bookings by the formula $$\text{ number of cars} \cdot \frac{\text{number of cars}}{\text{average number of rental length} \cdot \text{average number of days in the month}}$$ ?
If so, why?

It looks a bit strange...
Let's see what we can expect... (Thinking)

Suppose we assume there is sufficient demand and that we are merely limited by the number of available cars.
Then we can expect that if we double the number of cars, that the bookings will also double.
Note that if we have many more cars than there is demand, that then the number of cars would not affect the bookings at all.

If we look at a period that is twice as long, we can expect that the number of bookings will be twice as many as well.

If the rental length is twice as long, we can only honor half the number of bookings.

With these assumptions, I get:
$$\text{expected number of bookings in July} \sim \frac{\text{number of cars}\cdot \text{number of days in July}}{\text{average rental length} }$$
where $\sim$ indicates a linear correlation.

If we assume that every car is rented out on every day, then the $\sim$ becomes an equality.

How does that look? (Wondering)

 

[evinda]

Car groups?
It's not clear to me yet what the number represents. Are we supposed to do something with it?
Or can we just ignore it? (Wondering)

I think that we can ignore it.

Perhaps we could compensate for the increase in cars... (Thinking)

Then again, looking at your final formula, I think there is an assumption in there.
I think we assume that every car is rented out every day. Can that be the case? (Wondering)
If so, then historical numbers are irrelevant.

(Thinking)

That doesn't sound like mathematics does it?
Or are we talking about a real life situation where we apply some rules of thumb? (Wondering)

Whatever the case, such assumptions and rules of thumb should be clearly stated. (Nerd)

It's a real life situation.

It looks a bit strange...
Let's see what we can expect... (Thinking)

Suppose we assume there is sufficient demand and that we are merely limited by the number of available cars.
Then we can expect that if we double the number of cars, that the bookings will also double.
Note that if we have many more cars than there is demand, that then the number of cars would not affect the bookings at all.

If we look at a period that is twice as long, we can expect that the number of bookings will be twice as many as well.

If the rental length is twice as long, we can only honor half the number of bookings.

With these assumptions, I get:
$$\text{expected number of bookings in July} \sim \frac{\text{number of cars}\cdot \text{number of days in July}}{\text{average rental length} }$$
where $\sim$ indicates a linear correlation.

If we assume that every car is rented out on every day, then the $\sim$ becomes an equality.

How does that look? (Wondering)

In April there were 319 cars and 654 bookings.

With your formula we would get about $1063$ as the excpected number of bookings, and with the formula $319 \cdot \frac{319}{9\cdot 30}$ the number $376,8$.

By multiplying $319$ by $2.3$ we would get $733.7$, which is the closer approximation.

So this should be the formula... But why? (Thinking)

 

[I like Serena]

It's a real life situation.In April there were 319 cars and 654 bookings.

With your formula we would get about $1063$ as the excpected number of bookings, and with the formula $319 \cdot \frac{319}{9\cdot 30}$ the number $376,8$.

By multiplying $319$ by $2.3$ we would get $733.7$, which is the closer approximation.

So this should be the formula... But why?

My formula is a linear approximation.
Typically we need to multiply it with some correlation constant between 0 and 1, since not every car will be rented out every day of the month... (Thinking)
Do you have the numbers of a number of months?
Then we can see which formula fits best, or if we need to compensate somewhere.
And then we can do an actual linear correlation.

Either way, I find it hard to believe that bookings would go up with the square of the number of cars.
Then adding more cars would become a semi-infinite source of income, which it cannot be. (Worried)

If it is a real life situation, I presume that the demand would vary over the year.
Holiday periods in summer (like July) and Christmas periods are typically different from other periods.
So typically we would look at the demand in the previous year in the same month... (Thinking)

 

[evinda]

Do you have the numbers of a number of months?

What do you mean? (Thinking)

If it is a real life situation, I presume that the demand would vary over the year.
Holiday periods in summer (like July) and Christmas periods are typically different from other periods.
So typically we would look at the demand in the previous year in the same month... (Thinking)

The previous July there where 466 bookings.

In April the last year there were 129 bookings, and this year 654...

But I don't the number of cars of the last year... (Wasntme)

 

[I like Serena]

What do you mean?

The previous July there where 466 bookings.

In April the last year there were 129 bookings, and this year 654...

But I don't the number of cars of the last year... (Wasntme)

Something like:
\begin{array}{|c|c|c|c|}\hline
\textbf{Year} & \textbf{Month} & \textbf{Cars} & \textbf{Average rental length} & \textbf{Bookings} \\
\hline
2019 & \text{July} & 441 & 10.6 \text{ (so far)} & 246 \text{ (so far)} \\
2019 & \text{April} & 319 & ? & 654 \\
2018 & \text{July} & \text{n/a} & ? & 466 \\
2018 & \text{April} & \text{n/a} & ? & 129 \\
\hline
\end{array}
Preferably with a couple more numbers... (Thinking)

 

[evinda]

Something like:
\begin{array}{|c|c|c|c|}\hline
\textbf{Year} & \textbf{Month} & \textbf{Cars} & \textbf{Average rental length} & \textbf{Bookings} \\
\hline
2019 & \text{July} & 441 & 10.6 \text{ (so far)} & 246 \text{ (so far)} \\
2019 & \text{April} & 319 & ? & 654 \\
2018 & \text{July} & \text{n/a} & ? & 466 \\
2018 & \text{April} & \text{n/a} & ? & 129 \\
\hline
\end{array}
Preferably with a couple more numbers... (Thinking)

The average rental lenth for April 2018 was 8,4 days and this for April 2019 8,8 days.

The average rental length for July 2018 was 10,3 days and for July 2019 10,9 days.

Which correlation constant do you suggest me to suggest for each month? (Thinking)

 

[I like Serena]

The average rental lenth for April 2018 was 8,4 days and this for April 2019 8,8 days.

The average rental length for July 2018 was 10,3 days and for July 2019 10,9 days.

Which correlation constant do you suggest me to suggest for each month? (Thinking)

It seems we have:

Your formula (Ev):
$$\text{expected number of bookings} =\text{ number of cars} \cdot \frac{\text{number of cars}}{\text{average rental length} \cdot \text{average days in the month}}\tag{Ev}$$
My proposed formula (KvA):
$$\text{expected number of bookings} =\gamma \cdot \frac{\text{number of cars} \cdot \text{number of days in the month}}{\text{average rental length}}\tag{KvA}$$
where $\gamma$ is an as yet unknown constant between 0 and 1.

\begin{array}{|c|c|c|c|}\hline
\textbf{Year} & \textbf{Month} & \textbf{Cars} & \textbf{Average rental length} & \textbf{Bookings} & \text{Ev} &\text{KvA}\\
\hline
2019 & \text{July} & 441 & 10.9 \text{ (so far)} & 246 \text{ (so far)} \\
2019 & \text{April} & 319 & 8.8 & 654 \\
2018 & \text{July} & \text{n/a} & 10.3 & 466 \\
2018 & \text{April} & \text{n/a} & 8.4 & 129 \\
\hline
\end{array}
We could calculate the predicted numbers and see how well they fit...
We really need more numbers though to get any confidence. (Thinking)

Also, since the number of cars is increasing, I would expect to hit some saturation point in real life sooner or later.
That is difficult to predict, and may mean that whatever we come up with, may not be reliable anyway.

 

[evinda]

It seems we have:

Your formula (Ev):
$$\text{expected number of bookings} =\text{ number of cars} \cdot \frac{\text{number of cars}}{\text{average rental length} \cdot \text{average days in the month}}\tag{Ev}$$
My proposed formula (KvA):
$$\text{expected number of bookings} =\gamma \cdot \frac{\text{number of cars} \cdot \text{number of days in the month}}{\text{average rental length}}\tag{KvA}$$
where $\gamma$ is an as yet unknown constant between 0 and 1.

\begin{array}{|c|c|c|c|}\hline
\textbf{Year} & \textbf{Month} & \textbf{Cars} & \textbf{Average rental length} & \textbf{Bookings} & \text{Ev} &\text{KvA}\\
\hline
2019 & \text{July} & 441 & 10.9 \text{ (so far)} & 246 \text{ (so far)} \\
2019 & \text{April} & 319 & 8.8 & 654 \\
2018 & \text{July} & \text{n/a} & 10.3 & 466 \\
2018 & \text{April} & \text{n/a} & 8.4 & 129 \\
\hline
\end{array}
We could calculate the predicted numbers and see how well they fit...
We really need more numbers though to get any confidence. (Thinking)

Also, since the number of cars is increasing, I would expect to hit some saturation point in real life sooner or later.
That is difficult to predict, and may mean that whatever we come up with, may not be reliable anyway.

So $\gamma$ for April 2019 is $0,6$.

For July we cannot calculate it... Maybe we should guess how it could be ?

I noticed that from 01.01.2018-04.05.2019 there were 180 bookings for June 2018 and from 01.01.2019-04.05.2019 there are 313 bookings for June 2019...

So we have an increase of $73,8$ %... (Thinking)

 

[I like Serena]

So $\gamma$ for April 2019 is $0,6$.

For July we cannot calculate it... Maybe we should guess how it could be ?

We can't use the data of July until July is over. We can only make predictions for July based on a $\gamma$ from historical data. (Thinking)

I noticed that from 01.01.2018-04.05.2019 there were 180 bookings for June 2018 and from 01.01.2019-04.05.2019 there are 313 bookings for June 2019...

So we have an increase of $73,8$ %...

The increase for April is much much higher.
It suggests that we will also have a high increase for July.
It also shows that the numbers are "off the chart". That is, we cannot make reliable predictions. (Tmi)

 

[evinda]

We can't use the data of July until July is over. We can only make predictions for July based on a $\gamma$ from historical data. (Thinking)

So we cannot make a prediction if we do not know the number of cars of the previous year, right? (Thinking)

The increase for April is much much higher.
It suggests that we will also have a high increase for July.
It also shows that the numbers are "off the chart". That is, we cannot make reliable predictions. (Tmi)

Neither if we know the number of cars of the previous year? (Thinking)

 

[I like Serena]

So we cannot make a prediction if we do not know the number of cars of the previous year, right?

Not with a formula that includes the number cars no. (Shake)

Neither if we know the number of cars of the previous year?

That would help. It's just that it seems unlikely to me that it will explain the huge fluctuations that we see. (Thinking)

 

[evinda]

I got to know that we get the number $2.5$, which we multiply by the number of cars, by considering that we get a reservation for 9 days, then the car gets returned and then we give the car again for rental after 9 days...

Have you understood this? If so could you explain it to me in order to ponder over it? (Thinking)

 

[I like Serena]

I got to know that we get the number $2.5$, which we multiply by the number of cars, by considering that we get a reservation for 9 days, then the car gets returned and then we give the car again for rental after 9 days...

Have you understood this? If so could you explain it to me in order to ponder over it?

I'm not clear about that $2.5$ yet... anyway, here is what I can come up with... (Thinking)

In a month with 30 days with an average reservation of 9 days, we can rent out the same car $\frac{30}{9}\approx 3.3$ times.
That is, we can accept 3.3 bookings for it in that month.
In practice this will be a bit lower, since there will be days in between reservations for which no booking fits.
So the number of bookings we can accept in total is 'at most' $3.3$ times the number of cars. (Thinking)

 

[evinda]

I'm not clear about that $2.5$ yet... anyway, here is what I can come up with... (Thinking)

In a month with 30 days with an average reservation of 9 days, we can rent out the same car $\frac{30}{9}\approx 3.3$ times.
That is, we can accept 3.3 bookings for it in that month.
In practice this will be a bit lower, since there will be days in between reservations for which no booking fits.
So the number of bookings we can accept in total is 'at most' $3.3$ times the number of cars. (Thinking)

I see... but if we suppose that a car is rented for 9 days, and then after it will be returned, it will be rented again in $9$ days, how do we calculate how many times we can rent out the same car in one month? (Thinking)

 

[I like Serena]

I see... but if we suppose that a car is rented for 9 days, and then after it will be returned, it will be rented again in $9$ days, how do we calculate how many times we can rent out the same car in one month?

Do you mean that we rent it out for 9 days, then we have no one to rent for 8 days, followed by 9 days of rent again?
That would be pretty much the worst case scenario where the bookings are made in the worst possible way. (Thinking)

 

[evinda]

Do you mean that we rent it out for 9 days, then we have no one to rent for 8 days, followed by 9 days of rent again?
That would be pretty much the worst case scenario where the bookings are made in the worst possible way. (Thinking)

Yes, I mean this case. Supposing this, how do we find $2,5$ ? (Thinking)

 

[I like Serena]

Yes, I mean this case. Supposing this, how do we find $2,5$ ?

On average we can rent out the car every $9+8$ days.
In a month of $30$ days that is $\frac{30}{9+8}\approx 1.76$ times in this worst case scenario.

In the best case scenario we can rent out the car $\frac{30}{9}\approx 3.33$ times.

If we assume a uniform probability distribution, then the expectation is that we can rent out the car $\frac 12\left(\frac{30}{9+8}+\frac{30}{9}\right) \approx 2.55$ times. (Thinking)

 

[evinda]

On average we can rent out the car every $9+8$ days.

According to the assumption above, right? (Thinking)

In a month of $30$ days that is $\frac{30}{9+8}\approx 1.76$ times in this worst case scenario.

In the best case scenario we can rent out the car $\frac{30}{9}\approx 3.33$ times.

If we assume a uniform probability distribution, then the expectation is that we can rent out the car $\frac 12\left(\frac{30}{9+8}+\frac{30}{9}\right) \approx 2.55$ times. (Thinking)

So in order to find an approximation of how many times we can rent out a car in a month we calculate the median of the best and the worst case? (Thinking)And something else... I have also thought of a formula for a forecast based on the sites that there were reservations and I wanted to ask you if it is right... (Thinking)Suppose for example that for August 2018 there were the last year 118 bookings with reservation date till the 11.05.2018 and this year there are for August 2019, 263 bookings with reservation date till the 11.05.2019.

From the common sites from where we had bookings this and last year, we have an increase of $48.31$ %.

This year, there are also new sites from which we had bookings so far.

So in order to find the percentage of bookings that we get from the new sites, I thought to compute the sum of number of bookings that we got so far from them, say $y$.Then

$\text{number of bookings of common sites}=\text{number of bookings of last year}+\frac{48.31}{100} \cdot \text{number of bookings of last year}$.Then the expected number of bookings of this year for August, say $E$, is given by the following formula.

$$E=\text{number of bookings of common sites}+\frac{y}{263}\cdot 411. $$Is the formula above correct or doesn't it make sense? (Blush)(Thinking)

 

[I like Serena]

According to the assumption above, right?

Yes. (Nod)

So in order to find an approximation of how many times we can rent out a car in a month we calculate the median of the best and the worst case?

Yes. However, I realized that the uniform distribution would actually be for the day that we get the next booking.
That is, we should take the mean of the 0 to 8 days after the last booking.
So on average there will be 4 days between bookings.
We can then rent out a single car $\frac{30}{9+4}\approx 2.3$ times per 30-day month on average. (Nerd)

And something else... I have also thought of a formula for a forecast based on the sites that there were reservations and I wanted to ask you if it is right...

Suppose for example that for August 2018 there were the last year 118 bookings with reservation date till the 11.05.2018 and this year there are for August 2019, 263 bookings with reservation date till the 11.05.2019.

From the common sites from where we had bookings this and last year, we have an increase of $48.31$ %.

This year, there are also new sites from which we had bookings so far.

What are 'common sites' and 'new sites'? (Wondering)

So in order to find the percentage of bookings that we get from the new sites, I thought to compute the sum of number of bookings that we got so far from them, say $y$.

Then

$\text{number of bookings of common sites}=\text{number of bookings of last year}+\frac{48.31}{100} \cdot \text{number of bookings of last year}$.

Do you mean to calculate the expected number of bookings of common sites?
Because we don't know the actual number yet, do we? (Wondering)

Then the expected number of bookings of this year for August, say $E$, is given by the following formula.

$$E=\text{number of bookings of common sites}+\frac{y}{263}\cdot 411. $$

Is the formula above correct or doesn't it make sense?

What is $411$?
Is it the actual number of bookings in August last year? (Wondering)

 

[evinda]

Yes. However, I realized that the uniform distribution would actually be for the day that we get the next booking.
That is, we should take the mean of the 0 to 8 days after the last booking.
So on average there will be 4 days between bookings.
We can then rent out a single car $\frac{30}{9+4}\approx 2.3$ times per 30-day month on average. (Nerd)

Could you explain this further to me?

What are 'common sites' and 'new sites'? (Wondering)

We have a list of the sites from which we had bookings done till one date of this year and an other one from the bookings done last year till the respective date. So this year, some sites that appear did also appear the last year and some others are new.

Do you mean to calculate the expected number of bookings of common sites?
Because we don't know the actual number yet, do we? (Wondering)

Yes, that's what I mean... No, we are looking for a forecast... (Thinking)

What is $411$?
Is it the actual number of bookings in August last year? (Wondering)

$411$ is the total number of cars that the company has.

 

[I like Serena]

Could you explain this further to me?

If we want to find the expected number of bookings in a month we either have to know (or assume) what the probability distribution is that new bookings are made.
Or we need to use the historical distribution.

We are already assuming that a car cannot be booked every day of the month when it is available.
In this case I've made the assumption that there is a practically unlimited demand, just at an unpredictable day.
It's a bit shaky to be honest.
Better would be to calculate the average number of bookings based on historical data. (Nerd)

We have a list of the sites from which we had bookings done till one date of this year and an other one from the bookings done last year till the respective date. So this year, some sites that appear did also appear the last year and some others are new.

Yes, that's what I mean... No, we are looking for a forecast...

$411$ is the total number of cars that the company has.

I don't really know what we can do... (Sadface)

We are making assumptions to be able to predict the future.
Assumptions about demand, comparibility of old and new sites, how assignment of bookings depends on the number of cars, and so on.
Forecasts are always already tricky, since we cannot know what the future will bring.
To make them anyway, at the very least we need to verify the assumptions that we make.
Either they should be supported by known relations found by researchers, or they need to be validated by 'testing' them against historical data. (Thinking)