Chapter 10, Problem 30P

To determine

If the embankment should be built and if yes, to which of the five heights above the roadway should it be built.

Answer to Problem 30P

Yes, the embankment can be built.

The lowest total expected annual cost is at $3\text{m}$ embankment height.

Explanation of Solution

Given:

$\$300,000$ of damage occurs.

The rate of return is $12\%$.

Concept used:

Write the expression for the annual probability of flood damage.

$P_A=\frac{1}{F}$ ...... (I).

Here, the annual probability of flood damage is $P_A$ and the average frequency that flood level will exceed height in years is $F$.

Write the expression for the expected annual damage cost.

$C=P_A\times \$300,000$ ...... (II).

Here, the expected annual damage cost is $C$ and the annual probability of flood damage is $P_A$.

Write the expression for the factor $\left(\frac{A}{P},i,N\right)$.

$\left(\frac{A}{P},i,N\right)=\frac{i\times {\left(i+1\right)}^N}{{\left(1+i\right)}^N-1}$ ...... (III).

Here, the rate of interest is $i$ and the time period is $N$.

Write the expression for the equivalent uniform annual cost for the embankment.

$EUAC=C_i\times \left(\frac{A}{P},i,N\right)$ ...... (IV).

Here, the initial cost is $C_i$.

Write the expression for the total expected Annual cost.

$C_T=EUAC+C$ ...... (V).

Here, the total expected Annual cost is $C_T$, the expression for the equivalent uniform annual cost for the embankment is $EUAC$ and the expected annual damage cost is $C$.

Calculation:

Calculate the annual probability of flood damage and the expected annual damage cost.

Embankment height above roadway $\left(\text{m}\right)$	Initial cost $\left(C_i\right)$	Average frequency that will exceed height $\left(F\right)$ in years	Annual probability of flood damages $\left(P_A\right)$	Expected annual damage cost $\left(C\right)$
$2.0$	$\$100,000$	$3$	$0.333$	$\$99,900$
$2.5$	$\$165,000$	$8$	$0.125$	$\$37,500$
$3.0$	$\$300,000$	$25$	$0.04$	$\$12,000$
$3.5$	$\$400,000$	$50$	$0.02$	$\$6000$
$4.0$	$\$550,000$	$100$	$0.01$	$\$3000$

Here, calculate the annual probability of flood damage and the expected annual damage cost from Equation-(I) and Equation-(II).

Calculate the factor $\left(\frac{A}{P},i,N\right)$.

Substitute $0.12$ for $i$ and $50$ year for $N$ in Equation-(III).

$\begin{array}{c}\left(\frac{A}{P},i,N\right)=\frac{i\times {\left(i+1\right)}^N}{{\left(1+i\right)}^N-1}\\ =\frac{0.12\times {\left(0.12+1\right)}^{50}}{{\left(1+0.12\right)}^{50}-1}\\ =0.1204\end{array}$

Calculate the equivalent uniform annual cost for the embankment and total expected annual cost.

Embankment height above roadway $\left(\text{m}\right)$	Initial cost $\left(C_i\right)$	$\left(\frac{A}{P},0.12,50\right)$	$EUAC$	Expected annual damage cost $\left(C\right)$	Total Expected Annual cost $\left(C_T\right)$
$2.0$	$\$100,000$	$0.1204$	$\$12,040$	$\$99,900$	$\$111,940$
$2.5$	$\$165,000$	$0.1204$	$\$19,866$	$\$37,500$	$\$57,366$
$3.0$	$\$300,000$	$0.1204$	$\$36,120$	$\$12,000$	$\$48,120$
$3.5$	$\$400,000$	$0.1204$	$\$48,160$	$\$6,000$	$\$54,160$
$4.0$	$\$550,000$	$0.1204$	$\$66,220$	$\$3,000$	$\$69,220$

Here, calculate the equivalent uniform annual cost for the embankment and total expected annual cost from Equation-(IV) and Equation-(V) respectively.

The lowest total expected annual cost is at $3\text{m}$ embankment height.

Conclusion:

Yes, the embankment can be built.

Thus, the lowest total expected annual cost is at $3\text{m}$ embankment height.