Chapter 9, Problem 88P

To determine

The plan which is preferable by benefit cost ratio analysis.

Answer to Problem 88P

The Plan 3 is preferable as benefit cost ratio is maximum for the Plan 3.

Explanation of Solution

Given:

The cost saving percentage per vehicle minute for autos is $\$0.0375$.

The cost saving percentage per vehicle minute for trucks is $\$0.1875$.

The annual maintenance for existing highway is $\$1875$.

The average accident cost is $\$1500$.

For plan $1$ the cost per lane is $\$562500$ per mile.

The annual maintenance cost is $\$1560$.

Estimated accident rate is $3.1$ per MVM.

For plan $2$ the cost per lane is $\$812500$ per mile.

The annual maintenance cost is $\$1250$.

Estimated accident rate is $3.1$ per MVM.

For plan $3$ the cost per lane is $\$1000000$ per mile.

The annual maintenance cost is $\$1250$.

Estimated accident rate is $3$ per MVM.

There are average $25000$ vehicles per day over next $20$ years.

Capital improvement investment can be secured at $6.25\%$.

Concept used:

Write the expression for annualized capital cost.

$A_C=C\left[\frac{i{\left(1+i\right)}^n}{{\left(1+i\right)}^n-1}\right]$ ...... (I)

Here, the total cost is $C$, the interest rate is $i$ and time period is $n$.

Write the expression for cost saving.

$\left(\text{Cost saving}\right)=P\times V\times T\times 365$ ...... (II)

Here, the cost saving percentage per vehicle minute is $P$, number of vehicle is $V$ and time saved is $T$.

Write the expression for saving in maintenance.

${\left(\text{saving}\right)}_M=\left(M_E-M_P\right)\times L\times \text{miles}$ ...... (III)

Here, the annual maintenance cost for existing highway is $M_E$, the annual maintenance cost for the plan is $M_P$ and number of lanes is $L$.

Write the expression for saving in accidents.

${\left(\text{saving}\right)}_A=\left(A_E-A_P\right)\times \left(V\times 365\times \text{miles}\right)\times A_C$ ...... (IV)

Here, the annual accident for existing highway is $A_E$, the annual accident for the plan is $A_P$ and average accident cost is $A_C$.

Calculations:

Consider Plan 1.

Calculate operating cost of auto.

$\begin{array}{c}{A}_O=\$0.15\times \left(6\text{ mile}\right)\times 365\\ =\$328.5\end{array}$

Calculate operating cost of truck.

$\begin{array}{c}{A}_T=\$0.225\times \left(6\text{ mile}\right)\times 365\\ =\$492.75\end{array}$

Calculate total cost saving for auto due to reduction in travel time.

Substitute $\$0.0375$ for $P$, $23437.5$ for $V$ and $2.5\text{ min}$ for $T$ in Equation (II).

$\begin{array}{c}{\left(\text{Cost saving}\right)}_A=\$0.0375\times \left(23437.5\text{ vehicles}\right)\times \left(2.5\text{ min}\right)\times \text{365}\\ =\$802001.95\end{array}$

Calculate total cost saving for truck due to reduction in travel time.

Substitute $\$0.1875$ for $P$, $1562.5$ for $V$ and $1.25\text{ min}$ for $T$ in Equation (II).

$\begin{array}{c}{\left(\text{Cost saving}\right)}_T=\$0.1875\times \left(1562.5\text{ vehicles}\right)\times \left(1.25\text{ min}\right)\times \text{365}\\ =\$1\text{33667}\end{array}$

Calculate total saving in maintenance cost.

Substitute $\$1875$ for $M_E$, $\$1560$ for $M_P$, $4$ for $L$ and $6$ for miles in Equation (III).

$\begin{array}{c}{\text{Saving}}_M=\left(\$1875-\$1560\right)\times 4\times 6\\ =\$7560\end{array}$

Calculate saving in accident reduction.

Substitute $\$5.725$ for $A_E$, $\$3.125$ for $A_P$, $25000$ for $V$, $\$1500$ for $A_C$ and $6$ for miles in Equation (IV).

$\begin{array}{c}{\text{Saving}}_A=\left(\$5.725-\$3.125\right)\times \left(\frac{25000\times 365\times 6}{{10}^6}\right)\times \$1500\\ =\$213525\end{array}$

Calculate the value of $C$.

$\begin{array}{c}C=\$562500\times 6\\ =\$3375000\end{array}$

Calculate the annualized capital cost.

Substitute $\$562500\times 6$ for $C$, $6.25\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}{A}_C=\$3375000\left[\frac{0.062{\left(1+0.0625\right)}^{20}}{{\left(1+0.062\right)}^{20}-1}\right]\\ =\$3375000\left[0.0889\right]\\ =\$300247.66\end{array}$

Calculate total annual cost.

$\begin{array}{c}\text{Total annual cost}=\$300247.66+\$328.5+\$492.75+\left(\$1560\times 4\times 6\right)\\ =\$338508.91\end{array}$

Calculate total annual benefit.

$\begin{array}{c}\text{Total annual benefit}=\$213525+\$7560+\$802001.95+\$133667\\ =\$1156753.95\end{array}$

Calculate benefit cost ratio.

$\begin{array}{c}B-C\text{ ratio}=\frac{\$1156753.95}{\$338508.91}\\ =3.4172\end{array}$

Consider Plan 2.

Calculate operating cost of auto.

$\begin{array}{c}{A}_O=\$0.15\times \left(6\text{ mile}\right)\times 365\\ =\$328.5\end{array}$

Calculate operating cost of truck.

$\begin{array}{c}{A}_T=\$0.225\times \left(6\text{ mile}\right)\times 365\\ =\$492.75\end{array}$

Calculate total cost saving for auto due to reduction in travel time.

Substitute $\$0.0375$ for $P$, $23437.5$ for $V$ and $3.75\text{ min}$ for $T$ in Equation (II).

$\begin{array}{c}{\left(\text{Cost saving}\right)}_A=\$0.0375\times \left(23437.5\text{ vehicles}\right)\times \left(3.75\text{ min}\right)\times \text{365}\\ =\$\text{1203002}\text{.93}\end{array}$

Calculate total cost saving for truck due to reduction in travel time.

Substitute $\$0.1875$ for $P$, $1562.5$ for $V$ and $3.75\text{ min}$ for $T$ in Equation (II).

$\begin{array}{c}{\left(\text{Cost saving}\right)}_T=\$0.1875\times \left(1562.5\text{ vehicles}\right)\times \left(3.75\text{ min}\right)\times \text{365}\\ =\$40\text{1000}\text{.98}\end{array}$

Calculate total saving in maintenance cost.

Substitute $\$1875$ for $M_E$, $\$1250$ for $M_P$, $4$ for $L$ and $6$ for miles in Equation (III).

$\begin{array}{c}{\text{Saving}}_M=\left(\$1875-\$1250\right)\times 4\times 6\\ =\$15000\end{array}$

Calculate saving in accident reduction.

Substitute $\$5.725$ for $A_E$, $\$3.1$ for $A_P$, $25000$ for $V$, $\$1500$ for $A_C$ and $6$ for miles in Equation (IV).

$\begin{array}{c}{\text{Saving}}_A=\left(\$5.725-\$3.1\right)\times \left(\frac{25000\times 365\times 6}{{10}^6}\right)\times \$1500\\ =\$215578\end{array}$

Calculate the value of $C$.

$\begin{array}{c}C=\$812500\times 6\\ =\$4875000\end{array}$

Calculate the annualized capital cost.

Substitute $\$812500\times 6$ for $C$, $6.25\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}{A}_C=\$4875000\left[\frac{0.062{\left(1+0.0625\right)}^{20}}{{\left(1+0.062\right)}^{20}-1}\right]\\ =\$4875000\left[\frac{0.0625{\left(1.0625\right)}^{20}}{{\left(1.0625\right)}^{20}-1}\right]\\ =\$433691.06\end{array}$

Calculate total annual cost.

$\begin{array}{c}\text{Total annual cost}=\$433691.06+\$328.5+\$492.75+\left(\$1250\times 4\times 6\right)\\ =\$464512.31\end{array}$

Calculate total annual benefit.

$\begin{array}{c}\text{Total annual benefit}=\$215578+\$15000+\text{\$1203002}\text{.93}+\$401000.98\\ =\$1834581.91\end{array}$

Calculate benefit cost ratio.

$\begin{array}{c}B-C\text{ ratio}=\frac{\$1834581.91}{\$464512.31}\\ =3.9495\end{array}$

Consider Plan 3.

Calculate operating cost of auto.

$\begin{array}{c}{A}_O=\$0.15\times \left(6.5\text{ mile}\right)\times 365\\ =\$355.875\end{array}$

Calculate operating cost of truck.

$\begin{array}{c}{A}_T=\$0.225\times \left(6.5\text{ mile}\right)\times 365\\ =\$533.8125\end{array}$

Calculate total cost saving for auto due to reduce in travel time.

Substitute $\$0.0375$ for $P$, $23437.5$ for $V$ and $6.25\text{ min}$ for $T$ in Equation (II).

$\begin{array}{c}{\left(\text{Cost saving}\right)}_A=\$0.0375\times \left(23437.5\text{ vehicles}\right)\times \left(6.25\text{ min}\right)\times \text{365}\\ =\$\text{2005004}\text{.883}\end{array}$

Calculate total cost saving for truck due to reduce in travel time.

Substitute $\$0.1875$ for $P$, $1562.5$ for $V$ and $5\text{ min}$ for $T$ in Equation (II).

$\begin{array}{c}{\left(\text{Cost saving}\right)}_T=\$0.1875\times \left(1562.5\text{ vehicles}\right)\times \left(5\text{ min}\right)\times \text{365}\\ =\$5\text{34667}\text{.97}\end{array}$

Calculate total saving in maintenance cost.

Substitute $\$1875$ for $M_E$, $\$1250$ for $M_P$, $4$ for $L$ and $6.5$ for miles in Equation (III).

$\begin{array}{c}{\text{Saving}}_M=\left(\$1875-\$1250\right)\times 4\times 6.5\\ =\$16250\end{array}$

Calculate saving in accident reduction.

Substitute $\$5.725$ for $A_E$, $\$3.1$ for $A_P$, $25000$ for $V$, $\$1500$ for $A_C$ and $6.5$ for miles in Equation (IV).

$\begin{array}{c}{\text{Saving}}_A=\left(\$5.725-\$3.1\right)\times \left(\frac{25000\times 365\times 6.5}{{10}^6}\right)\times \$1500\\ =\$242440\end{array}$

Calculate the value of $C$.

$\begin{array}{c}C=\$1000000\times 6.\\ =\$6500000\end{array}$

Calculate the annualized capital cost.

Substitute $\$6500000$ for $C$, $6.25\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}{A}_C=\$6500000\left[\frac{0.062{\left(1+0.0625\right)}^{20}}{{\left(1+0.062\right)}^{20}-1}\right]\\ =\$6500000\left[\frac{0.0625{\left(1.0625\right)}^{20}}{{\left(1.0625\right)}^{20}-1}\right]\\ =\$578254.75\end{array}$

Calculate total annual cost.

$\begin{array}{c}\text{Total annual cost}=\$578254.75+\$355.875+\$533.8125+\left(\$1250\times 4\times 6\right)\\ =\$611644.44\end{array}$

Calculate total annual benefit.

$\begin{array}{c}\text{Total annual benefit}=\$242440+\$16250+\text{\$2005004}\text{.883}+\text{\$534667}\text{.97}\\ =\$2798362.853\end{array}$

Calculate benefit cost ratio.

$\begin{array}{c}B-C\text{ ratio}=\frac{\$2798362.853}{\$611644.44}\\ =4.58\end{array}$.

Conclusion:

Thus, Plan 3 is preferable as benefit cost ratio is maximum for the Plan 3.