Chapter 9, Problem 31P

To determine

The alternative to be selected.

Answer to Problem 31P

The alternative to be selected is Alternative $B$.

Explanation of Solution

Given:

MARR is $12\%$.

Useful life is $15$ years.

Calculation:

Write the expression to calculate the present worth of the benefit.

$P=A\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)$ ....... (I)

Here, the annual benefit is A, the present worth is $P$, the rate of interest is $i$ and the number of period is $n$.

Calculate the present worth benefit of the alternative $A$.

Substitute $\$130$ for $A$, $12\%$ for $i$ and $15$ for $n$ in Equation (I).

$\begin{array}{c}P=\$130\left(\frac{{\left(1+0.12\right)}^{15}-1}{0.12{\left(1+0.12\right)}^{15}}\right)\\ =\$130\left(6.81\right)\\ =\$885.3\end{array}$

Calculate the present worth benefit of the alternative $B$.

Substitute $\$60$ for $A$, $12\%$ for $i$ and $15$ for $n$ in Equation (I).

$\begin{array}{c}P=\$60\left(\frac{{\left(1+0.12\right)}^{15}-1}{0.12{\left(1+0.12\right)}^{15}}\right)\\ =\$60\left(6.81\right)\\ =\$408.6\end{array}$

Calculate the present worth benefit of the alternative $C$.

Substitute $\$35$ for $A$, $12\%$ for $i$ and $15$ for $n$ in Equation (I).

$\begin{array}{c}P=\$35\left(\frac{{\left(1+0.12\right)}^{15}-1}{0.12{\left(1+0.12\right)}^{15}}\right)\\ =\$35\left(6.81\right)\\ =\$238.35\end{array}$

Write the expression to calculate the benefit-cost ratio.

$B/C=\frac{B}{C}$ ....... (II)

Here, the present worth benefit is $B$, the present worth cost is $C$.

For Alternative $A$.

Substitute $\$885.3$ for $B$, $\$800$ for $C$, Equation (I).

$\begin{array}{c}B/C=\frac{\$885.3}{\$800}\\ =1.11\end{array}$

For Alternative $B$.

Substitute $\$408.6$ for $B$, $\$300$ for $C$, Equation (I).

$\begin{array}{c}B/C=\frac{\$408.6}{\$300}\\ =1.362\end{array}$

For Alternative $C$.

Substitute $\$238.35$ for $B$, $\$150$ for $C$, Equation (I).

$\begin{array}{c}B/C=\frac{\$238.35}{\$150}\\ =1.589\end{array}$

Since all the ratios are positive hence we need to make comparison be increment method.

Calculate the ${\Delta }_{\text{Cost}}$ for alternative $B-C$.

${\Delta }_{\text{Cost}}=\left(P{W}_C-P{W}_B\right)$ ...... (III)

Here the incremental cost between the alternative $B$ and $C$ is ${\Delta }_{\text{Cost}}$, the present worth of cost for alternative $C$ is $P{W}_C$ and the present worth of cost for alternative $B$ is $P{W}_B$.

Substitute $\$150$ for $P{W}_C$ and $\$300$ for $P{W}_B$ in Equation (III).

$\begin{array}{c}{\Delta }_{\text{Cost}}=\left(\$300-\$150\right)\\ =\$150\end{array}$

Calculate the ${\Delta }_{\text{Benefit}}$ for alternative $B-C$.

${\Delta }_{\text{Benefit}}=\left(P_C-P_B\right)$ ...... (IV)

Here the incremental benefit between the alternative $B$ and $C$ is ${\Delta }_{\text{Benefit}}$, the present worth of benefit for alternative $C$ is $P_C$ and the present worth of benefit for alternative $B$ is $P_B$.

Substitute $\$238.35$ for $P_C$ and $\$408.6$ for $P_B$ in Equation (IV).

$\begin{array}{c}{\Delta }_{\text{Benefit}}=\$408.6-\$238.35\\ =\$170.25\end{array}$

Calculate the incremental benefit-cost ratio

$\Delta \left(B/C\right)=\frac{{\Delta }_{\text{Benefit}}}{{\Delta }_{\text{Cost}}}$ ...... (V)

Substitute $\$170.25$ for ${\Delta }_{\text{Benefit}}$ and $\$150$ for ${\Delta }_{\text{Cost}}$ in Equation (V).

$\begin{array}{c}\Delta \left(B/C\right)=\frac{\$170.25}{\$150}\\ =1.135\end{array}$

Since, $\Delta \left(B/C\right)$ ratio greater than $1$ therefore alternative $B$ is desirable.

Calculate the ${\Delta }_{\text{Cost}}$ for alternative $A-B$.

${\Delta }_{\text{Cost}}=\left(P{W}_A-P{W}_B\right)$ ...... (VI)

Here the incremental cost between the alternative $B$ and $A$ is ${\Delta }_{\text{Cost}}$, the present worth of cost for alternative $B$ is $P{W}_B$ and the present worth of cost for alternative $A$ is $P{W}_A$.

Substitute $\$800$ for $P{W}_A$ and $\$300$ for $P{W}_B$ in Equation (III).

$\begin{array}{c}{\Delta }_{\text{Cost}}=\left(\$800-\$300\right)\\ =\$500\end{array}$

Calculate the ${\Delta }_{\text{Benefit}}$ for alternative $A-B$.

${\Delta }_{\text{Benefit}}=\left(P_A-P_B\right)$ ...... (VII)

Here the incremental benefit between the alternative $B$ and $A$ is ${\Delta }_{\text{Benefit}}$, the present worth of benefit for alternative $A$ is $P_A$ and the present worth of benefit for alternative $B$ is $P_B$.

Substitute $\$885.3$ for $P_A$ and $\$408.6$ for $P_B$ in Equation (IV).

$\begin{array}{c}{\Delta }_{\text{Benefit}}=\$885.3-\$408.6\\ =\$476.7\end{array}$

Calculate the incremental benefit-cost ratio

$\Delta \left(B/C\right)=\frac{{\Delta }_{\text{Benefit}}}{{\Delta }_{\text{Cost}}}$ ...... (V)

Substitute $\$476.7$ for ${\Delta }_{\text{Benefit}}$ and $\$500$ for ${\Delta }_{\text{Cost}}$ in Equation (V).

$\begin{array}{c}\Delta \left(B/C\right)=\frac{\$476.7}{\$500}\\ =0.9534\end{array}$

Since, $\Delta \left(B/C\right)$ ratio less than $1$ therefore alternative $A$ is not desirable.

Hence, alternative $B$ is desirable.

In both the cases alternative $B$ is desirable.

Hence select alternative $B$.

Conclusion:

The alternative to be selected is Alternative $B$.