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A rental car company has three locations in Lexington. One at the airport, one downtown, and one on Nicholasville road. 95% of cars rented at the airport are returned there, 2% are returned downtown and 3% on Nicholasville road. Of cars rented downtown, 80% are returned there, 15% are returned at the airport and 5% are returned on Nicholasville road. Of cars rented at the Nicholasville road location, 90% are returned there and the remaining 10% are retuned at the airport.

(a) What matrix describes the movement of cars between locations?

(b) If the company has a fleet of 90 cars and there are 30 at each location on Monday morning, how many cars are at the airport on Thursday morning?

I'm confused on how to actually implement this word problem in to a matrix.. For part $a)$ I was thinking something like

$\begin{bmatrix}95&2&3\\15&80&56\\10&0&90\end{bmatrix}$

Though, I am not really sure if that is correct?

For part $b)$ I'm a bit confused because all of the case(s) we are given have to do with specifically if the car was rented or not rented at the specific location.

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1 Answer

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Make the assumption that all cars available for rental are indeed rented every day without fail. (Equivalently, assume that cars not rented in a particular day count as a part of those "returned to the same location") Else, there is not enough information. Further assume that cars are rented for a period of one day, say for example rented at $10$am and returned at $10$pm daily. Without knowledge of the duration of each rental, there is again not enough information.

Let $a_0,d_0,r_0$ represent the number of cars currently at the airport, downtown, and nicholasville road locations respectively.

Let $a_1,d_1,r_1$ represent the number of cars at the airport, downtown, and nicholasville road locations after one day.

The problem statement suggests the following system of equations:

$$\begin{cases} a_1 = .95 a_0 + .15d_0 + .10r_0\\ d_1 = .02 a_0 + .80 d_0 + 0 r_0\\ r_1 = .03 a_0 + .05 d_0 + .90 r_0\end{cases}$$

Let us look at one of these lines in more detail, say the first line. The number $a_1$ is the number of cars at the airport after one day. Some of those cars will have have originated from the airport from the current day. Specifically $.95a_0$ cars will have come from the airport to the airport. Similarly, some of the cars will have come from downtown. $0.15d_0$ of the cars will have come from downtown. Finally, $0.10 r_0$ of the cars will have come from nicholasville road.

The above system of equations can be described in matrix form as:

$$\begin{bmatrix}0.95&0.15&0.10\\0.02&0.80&0\\0.03&0.05&0.90\end{bmatrix}\begin{bmatrix}a_0\\d_0\\r_0\end{bmatrix}=\begin{bmatrix}a_1\\d_1\\r_1\end{bmatrix}$$

Refer to the matrices appearing in the above as $A,v_0,v_1$ repsectively.

We recognize also that to find the amount of cars at each location after two days, it suffices to calculate $Av_1=A(Av_0)=A^2v_0$. In general, to find the amount of cars at each location after $n$ days, you can calculate $A(v_{n-1})=A^nv_0$

With $30$ cars at each location on monday morning, say for example at $6$am, how many times will cars change location by thursday morning $6$am? $3$ times, yes? So then, letting $v_0=\begin{bmatrix}30\\30\\30\end{bmatrix}$, we have $A^3v_0$ will describe the number of cars at each location on thursday.

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Note: in the above notation I use, I use column matrices to represent distribution vectors and left-multiplication by the stochastic matrix to move from one state to the other. As such, the matrix describing the markov chain will be column stochastic, i.e. columns adding to one. If you prefer to use row vectors, you may do so by transposing everything used above and do right-multiplication by the matrices instead. In doing so, the matrix describing the markov chain will instead be row stochastic. Both are equally correct, and which you use depends on what you are comfortable with.

I personally teach using column stochastic out of the book finite mathematics by goldstein et al, however I know popular youtube teacher patrickjmt teaches using row stochastic matrices.

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