Chapter 9, Problem 30P

To determine

The alternative to be selected.

Answer to Problem 30P

The alternative to be selected is Alternative $B$.

Explanation of Solution

Given:

MARR is $10\%$.

Useful life is $5$ 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 $P$, 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 $\$158.3$ for $A$, $10\%$ for $i$ and $5$ for $n$ in Equation (I).

$\begin{array}{c}P=\$158.3\left(\frac{{\left(1+0.10\right)}^5-1}{0.10{\left(1+0.10\right)}^5}\right)\\ =\$158.3\left(3.79\right)\\ =\$600\end{array}$

Calculate the present worth benefit of the alternative $B$

Substitute $\$138.7$ for $A$, $10\%$ for $i$ and $5$ for $n$ in Equation (I).

$\begin{array}{c}P=\$138.7\left(\frac{{\left(1+0.10\right)}^5-1}{0.10{\left(1+0.10\right)}^5}\right)\\ =\$138.7\left(3.79\right)\\ =\$525.673\end{array}$

Calculate the present worth benefit of the alternative $C$

Substitute $\$58.3$ for $A$, $10\%$ for $i$ and $5$ for $n$ in Equation (I).

$\begin{array}{c}P=\$58.3\left(\frac{{\left(1+0.10\right)}^5-1}{0.10{\left(1+0.10\right)}^5}\right)\\ =\$58.3\left(3.79\right)\\ =\$220.957\end{array}$

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 $\$599.957$ for $B$, $\$600$ for $C$, Equation (I).

$\begin{array}{c}B/C=\frac{\$600}{\$600}\\ =1\end{array}$

For Alternative $B$.

Substitute $\$525.673$ for $B$, $\$500$ for $C$, Equation (I).

$\begin{array}{c}B/C=\frac{\$525.63}{\$500}\\ =1.05126\end{array}$

For Alternative $C$.

Substitute $\$220.957$ for $B$, $\$200$ for $C$, Equation (I).

$\begin{array}{c}B/C=\frac{\$220.957}{\$200}\\ =1.104\end{array}$

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

Calculate the ${\Delta }_{COST}$ 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 $\$200$ for $P{W}_C$ and $\$500$ for $P{W}_B$ in Equation (III).

$\begin{array}{c}{\Delta }_{\mathrm{Cos}t}=\left(\$200-\$500\right)\\ =-\$300\end{array}$

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 $\$220.957$ for $P_C$ and $\$525.673$ for $P_B$ in Equation (IV).

$\begin{array}{c}{\Delta }_{\text{Benefit}}=\$220.957-\$525.673\\ =-304.716\end{array}$

Calculate the incremental benefit-cost ratio

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

Substitute $-\$304.716$ for and $-\$300$ for ${\Delta }_{Cost}$ in Equation (V).

$\begin{array}{c}\Delta \left(B/C\right)=\frac{-\$304.716}{-\$300}\\ =1.0572\end{array}$

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

Calculate the ${\Delta }_{COST}$ 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 $\$200$ for $P{W}_A$ and $\$500$ for $P{W}_B$ in Equation (III).

$\begin{array}{c}{\Delta }_{\text{Cost}}=\left(\$600-\$500\right)\\ =\$100\end{array}$

Calculate the ${\Delta }_{Benefit}$ 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 $\$600$ for $P_A$ and $\$525.673$ for $P_B$ in Equation (IV).

$\begin{array}{c}{\Delta }_{\text{Benefit}}=\$600-\$525.673\\ =\$74.327\end{array}$

Calculate the incremental benefit-cost ratio

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

Substitute $-\$74.327$ for ${\Delta }_{\text{Benefit}}$ and $-\$100$ for ${\Delta }_{\text{Cost}}$ in Equation (V).

$\begin{array}{c}\Delta \left(B/C\right)=\frac{\$74.327}{\$100}\\ =0.74327\end{array}$

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 $B$.