February 2021

Question:

When a forest fire occurs, how many fire fighter should be sent to fight the fire so that the total cost is kept minimum?

Answer:

Suppose the fire starts at $t=0$ ; at $t=t_1$ , the fire fighter arrives; the fire is extinguished later at $t=t_2$ .

Let $c_1, c_2, c_3$ and $x$ represent the monetary damage per square footage burnt, the hourly wage per fire fighter, the one time cost per fire fighter, and the number of fire fighters sent respectively. The total cost consists damage caused by the fire, the wage paid to the fire fighters, and the one time cost of the fire fighters:

$c(x) = c_1\cdot$ (total square footage burnt) $+\;c_2\cdot (t_2-t_1)\cdot x + c_3\cdot x.$

Notice $t_2-t_1$ , the duration of fire fighting.

Assume the fire ignites at a single point and quickly spreads in all directions with flame velocity $v_*$ , the growing square footage of engulfed circular area is a function of time. Namely,

$b(t) = \pi r^2 = \pi (v_* t) ^2 = \pi v_*^2 t^2 \overset{k=\pi v_*^2}{=} k t^2.$

Its burning rate

$v_b(t) = b'(t) = 2 k t \overset{\alpha = 2 k}{=} \alpha t.\quad\quad\quad(1)$

However, after the arrival of $x$ fire fighters, $\alpha$ is reduced by $\beta x$ , an amount that is directly proportional to the number of fire fighters on the scene. The reduction of $\alpha$ reflects the fire fighting efforts exerted by the crew. As a result, for $t > t_1, v_b(t)$ declines along the line described by

$v_b(t) = (\alpha-\beta x)t + d$

where $\alpha-\beta x <0$ . Or equivalently,

$\beta x - \alpha >0.\quad\quad\quad(2)$

Moreover, the fire is extinguished at $t_2$ suggests that

$v_b(t_2) = 0 \implies (\alpha-\beta x) t_2 + d =0 \implies d = -(\alpha-\beta x)t_2.$

i.e.,

$\forall t_1< t \le t_2, v_b(t) = (\alpha-\beta x) t - (\alpha-\beta x) t_2=(\alpha-\beta x) (t-t_2).\quad\quad\quad(3)$

Combine (1) and (3),

$v_b(t) = \begin{cases} \alpha t, \quad\quad\quad\quad\quad\quad\quad\quad0\le t \le t_1\\ (\alpha-\beta x)(t-t_2),\quad\quad t_1< t \le t_2 \end {cases}.$

It is further assumed that

$v_b(t)$ is continuous at $t=t_1.\quad\quad\quad(4)$

We illustrate $v_b(t)$ in Fig. 1.

Fig. 1

The fact that $v_b(t)$ is continuous at $t_1$ means

$\lim\limits_{t \to t_1^+}v_b(t)=(\alpha-\beta x)(t_1-t_2) = \lim\limits_{t \to t_1^-}v_b(t) = h \implies (\alpha-\beta x) (t_1-t_2)=h.$

That is,

$t_2-t_1=\frac{h}{\beta x - \alpha}.\quad\quad\quad(5)$

The area of triangle in Fig. 1 represents $b(t_2)$ , the total square footage damaged by the fire. i.e.,

$b(t_2) =\frac{1}{2}t_1 h + \frac{1}{2}(t_2-t_1)h\overset{(5)}{=}\frac{1}{2}t_1 h+ \frac{1}{2}\frac{h^2}{\beta x-\alpha}.\quad\quad\quad(6)$

Consequently, the total cost

$c(x)= c_1 b(t_2) + c_2 x (t_2-t_1) + c_3 x \overset{(6), (5)}{=}\frac{c_1 t_1 h}{2} + \frac{c_1 h^2}{2(\beta x-\alpha)} + \frac{c_2 h x}{\beta x-\alpha} + c_3 x.$

To minimize the total cost, we seek a value $x$ where function $c$ attains its minimum value $c(x)$ .

From expressing $x$ as

$x = \frac{1}{\beta}\beta x = \frac{1}{\beta}({\beta x-\alpha +\alpha} )= \frac{1}{\beta}(\beta x -\alpha) + \frac{\alpha}{\beta},$

we obtain

$\frac{c_2 h x}{\beta x-\alpha} = \frac{c_2 h}{\beta x-\alpha}(\frac{1}{\beta}(\beta x - \alpha) + \frac{\alpha}{\beta})=\frac{c_2 h}{\beta} + \frac{c_2 \alpha h}{\beta(\beta x-\alpha)},\quad\quad\quad(7)$

$c_3 x = \frac{c_3}{\beta}(\beta x-\alpha) + \frac{c_3 \alpha}{\beta}.\quad\quad\quad(8)$

It follows that

$c(x) = \underbrace{\frac{c_1 t_1 h}{2} + \frac{c_1 h^2}{2(\beta x-\alpha)}}_{c_1\cdot(6)} + \underbrace{\frac{c_2 h}{\beta}+\frac{c_2 \alpha h}{\beta(\beta x - \alpha)}}_{(7)} + \underbrace{\frac{c_3}{\beta}(\beta x-\alpha) + \frac{c_3 \alpha}{\beta}}_{(8)}$

$=\frac{c_1 t_1 h}{2} + \frac{c_2 h}{\beta} + \frac{c_3 \alpha}{\beta} + \underline{\frac{c_1 \beta h^2+2 c_2 \alpha h}{2\beta(\beta x - \alpha)} + \frac{c_3(\beta x-\alpha)}{\beta}}.$

Hence, to minimize $c(x)$ is to find the $x$ that minimizes $\frac{c_1\beta h^2+2 c_2\alpha h }{2\beta(\beta x - \alpha)} + \frac{c_3(\beta x - \alpha)}{\beta}.$

Since $\frac{c_1\beta h^2+2 c_2\alpha h }{2\beta(\beta x - \alpha)} \cdot \frac{c_3(\beta x - \alpha)}{\beta}=\frac{(c_1 \beta h^2+2 c_2 \alpha h)c_3}{2\beta^2}$ , a constant, by the following theorem:

For positive quantities $a_1, a_2, ..., a_n, c_1, c_2, ..., c_n$ and positive rational quantities $p_1, p_2, ..., p_n$ , if $a_1^{p_1}a_2^{p_2}...a_k^{p_k}$ is a constant, then $c_1a_1+c_2a_2+...+c_na_n$ attains its minimum if $\frac{c_1a_1}{p_1} = \frac{c_2a_2}{p_2} = ... = \frac{c_na_n}{p_n}.$

(see “Solving Kepler’s Wine Barrel Problem without Calculus“), we solve equation

$\frac{c_1\beta h^2+2 c_2\alpha h }{2\beta(\beta x - \alpha)} = \frac{c_3(\beta x - \alpha)}{\beta}$

for $x$ :

Fig. 2

From Fig. 2, we see that $x =\frac{\alpha}{\beta} - \frac{\sqrt{\beta c_1 c_3 h^2+2\alpha c_2 c_3 h}}{\sqrt{2}\beta c_3} \implies \beta x - \alpha < 0$ , contradicts (2).

However, when $x =\frac{\sqrt{\beta c_1 c_3 h^2+2\alpha c_2 c_3 h}}{\sqrt{2}\beta c_3} + \frac{\alpha}{\beta}=\sqrt{\frac{\beta c_1 h^2 + 2 \alpha c_2 h}{2\beta^2 c_3}} + \frac{\alpha}{\beta}, \beta x-\alpha$ is a positive quantity. Therefore, it is

$x \overset{h=\alpha t_1}{=} \alpha \sqrt{\frac{\beta c_1 t_1^2 + 2 c_2 t_1}{2\beta^2 c_3}} + \frac{\alpha}{\beta}$

that minimizes the total cost.

Consider function $f$ that has first and second derivatives at each point.

There are two finite difference approximations to its derivatives at a particular point $x$ :

[1] $\frac{d}{dx}f(x) \approx \frac{ f(x+\frac{h}{2})-f(x-\frac{h}{2})}{h}$

Fig. 1

Intuitively (see Fig. 1),

$\frac{d}{dx} f(x) \approx \frac{ f(x+\frac{h}{2})-f(x-\frac{h}{2})}{h}\quad\quad\quad(1)$

[2] $\frac{d^2}{dx}f(x) \approx \frac{f(x+h)-2f(x)+f(x-h)}{h^2}$

Let

$g(x) = \frac{d}{dx}f(x) \overset{(1)}{\approx} \frac{f(x+\frac{h}{2})-f(x-\frac{h}{2})}{h}\quad\quad\quad(2)$

so that

$\frac{d^2}{dx}f(x) = \frac{d}{dx}(\frac{d}{dx}f(x))=\frac{d}{dx}(g(x)) \overset{(1)}{\approx} \frac{g(x+\frac{h}{2})-g(x-\frac{h}{2})}{h}.\quad\quad\quad(3)$

Substituting

$g(x+\frac{h}{2}) \overset{(2)}{=} \frac{f((x+\frac{h}{2})+\frac{h}{2})-f((x+\frac{h}{2})-\frac{h}{2})}{h}=\frac{f(x+h)-f(x)}{h}$

and

$g(x-\frac{h}{2}) \overset{(2)}{=} \frac{f((x-\frac{h}{2})+\frac{h}{2})-f((x-\frac{h}{2})-\frac{h}{2})}{h}=\frac{f(x)-f(x-h)}{h}$

into (3) gives

$\frac{d^2}{dx}f(x) \approx \frac{\frac{f(x+h)-f(x)}{h}-\frac{f(x)-f(x-h)}{h}}{h}=\frac{f(x+h)-f(x)-f(x)+f(x-h)}{h^2}=\frac{f(x+h)-2f(x)+f(x-h)}{h^2}.$

i.e.,

$\frac{d^2}{dx}f(x) \approx \frac{f(x+h)-2f(x)+f(x-h)}{h^2}.$

Month: February 2021

Solving Difference Equations using Omega CAS Explorer

Fire & Water

Seeing Finite Difference Approximations of the Derivatives