Chapter 9, Problem 41P

To determine

(a)

The best alternative using Benefit cost ratio.

Answer to Problem 41P

Using cost benefit ratio alternative A is beneficial.

Explanation of Solution

Given:

	A	B
User benefits $\left(\$M/\text{yr}\right)$	$2.1$	$2.6$
User dis-benefits $\left(\$M/\text{yr}\right)$	$1.2$	$2.1$
First cost $\left(\$M\right)$	$6.9$	$9.9$
Operations and maintenance $\left(\$M/\text{yr}\right)$	$0.75$	$0.825$

Calculation:

Calculate the cost benefit ratio.

$\text{Cost-benefit}\text{ratio}=\frac{\Delta \text{EUAB}}{\Delta \text{EUAC}}$ ...... (I)

Here, EUAB is annualised uniform benefits and EUAC is annualized uniform costs.

Calculate the EUAC for alternative A.

${\text{EUAC}}_{\text{A}}=\left(\text{First}\text{cost}+\text{User}\text{disbenefits}\right)\left(\frac{A}{P},8\%,15\text{years}\right)+\text{Annual}\text{Cost}$ ...... (II)

Substitute $6.9$ for first cost. $1.2$ for user disbenefits and $0.75$ for annual cost in Equation (II).

$\begin{array}{c}{\text{EUAC}}_{\text{A}}=\left(\$6.9+\$1.2\right)\left(\frac{A}{P},8\%,15\text{years}\right)+\$0.75\\ =\left(\$6.9+\$1.2\right)\left(\frac{0.08{\left(1+0.08\right)}^{15}}{{\left(1+0.08\right)}^{15}-1}\right)+\$0.75\\ =\$0.946+\$0.75\\ =\$1.696\end{array}$

EUAB is annual benefits which are $\$2.1\text{million}$.

Calculate the EUAC of alternative B.

Substitute $9.9$ for first cost. $2.1$ for user dis-benefits and

$0.825$ for annual cost in Equation (II).

$\begin{array}{c}{\text{EUAC}}_{\text{B}}=\left(\$9.9+\$2.1\right)\left(\frac{A}{P},8\%,15\text{years}\right)+\$0.825\\ =\left(\$9.9+\$2.1\right)\left(\frac{0.08{\left(1+0.08\right)}^{15}}{{\left(1+0.08\right)}^{15}-1}\right)+\$0.825\\ =\$1.4016+\$0.825\\ =\$2.226\text{million}\end{array}$

EUAB is $\$2.6\text{million}$.

Calculate the difference between the EUAC’s of two alternatives.

$\text{ΔEUAC}=EUA{C}_B-EUA{C}_A$ ...... (III)

Substitute $\$2.226\text{million}$ for $EUA{C}_B$ and $\$1.696$ for $EUA{C}_A$ in Equation (III).

$\begin{array}{c}\text{ΔEUAC}=\$2.226-\$1.696\\ =0.53\end{array}$

Calculate the difference between the EUAB’s of two alternatives.

$\begin{array}{c}\text{ΔEUAB}=EUA{B}_B-EUA{B}_A\\ =\$2.6-\$2.1\\ =\$0.5\text{million}\end{array}$

Calculate the benefit-cost ratio.

Substitute $\$0.5\text{million}$ for $\text{ΔEUAB}$ and $0.53$ for $\text{ΔEUAC}$ in Equation (I).

$\begin{array}{c}\frac{B}{C}\text{ratio}=\frac{\$0.5}{\$0.53}\\ =\$0.94\end{array}$

Thus the incremental cost ratio is less than $1$ alternative A must be chosen.

Conclusion:

Using cost benefit ratio alternative A is beneficial.

To determine

(b)

The best alternative using Modified benefit cost ratio.

Answer to Problem 41P

Using Modified cost benefit ratio alternative A is beneficial.

Explanation of Solution

Calculation:

In modified benefit cost ratio the salvage value is not added to the benefits. It is deducted from the costs. Since there is no salvage value in this case, thus, the answer will remain same as of part $\left(a\right)$.

Conclusion:

Using Modified cost benefit ratio alternative A is beneficial.

To determine

(c)

The best alternative using the public/government version of the $B/C$ ratio.

Answer to Problem 41P

The best alternative using the public/government version of the $B/C$ ratio is alternative B..

Explanation of Solution

Calculation:

Alternative A:

Benefits of public and government are calculated separately.

Calculate the present worth of the consumer’s benefits and dis-benefits.

$PW=-\text{User}\text{benefits}+\text{Annual}\text{benefits}\left(\frac{P}{A},8\%,15\right)$ ...... (IV)

Substitute $1.2$ for User dis-benefits and $\$2.1$ for Annual benefits in Equation (IV).

$\begin{array}{c}PW=-\$1.2+\$2.1\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)\\ =-\$1.2+\$2.1\left(8.559\right)\\ =\$16.7739\text{million}\end{array}$

Calculate the present worth of governments cost.

$PW=\text{First}\text{cost}+\text{Annual}\text{cost}\left(\frac{P}{A},8\%,15\right)$ ...... (V)

Substitute $\$6.9$ for First cost and $\$0.75$ for Annual cost in Equation (IV).

$\begin{array}{c}PW=\$6.9+\$0.75\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)\\ =\$6.9+\$0.75\left(8.559\right)\\ =\$13.319\text{million}\end{array}$

Calculate the benefit cost ratio.

$\frac{B}{C}=\frac{\text{Present}\text{worth}\text{of}\text{customer's}\text{benefits}}{\text{Present}\text{worth}\text{of}\text{Government's}\text{cost}}$ ...... (VI)

Substitute $\$16.7739\text{million}$ for $\text{Present}\text{worth}\text{of}\text{customer's}\text{benefits}$ and $\$13.319\text{million}$ for $\text{Present}\text{worth}\text{of}\text{Government's}\text{cost}$ in Equation (VI).

$\begin{array}{c}\frac{B}{C}=\frac{\$16.7739\text{million}}{\$13.319\text{million}}\\ =1.259\end{array}$

Thus, alternative can be accepted.

Alternative B:

Benefits of public and government are calculated separately.

Calculate the present worth of the consumer’s benefits and dis-benefits.

Substitute $2.1$ for User dis-benefits and $\$2.6$ for Annual benefits in Equation (IV).

$\begin{array}{c}PW=-\$2.1+\$2.6\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)\\ =-\$2.1+\$2.6\left(8.559\right)\\ =\$20.1534\text{million}\end{array}$

Calculate the present worth of governments cost.

Substitute $\$9.9$ for First cost and $\$0.825$ for Annual cost in Equation (IV).

$\begin{array}{c}PW=\$9.9+\$0.825\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)\\ =\$9.9+\$0.825\left(8.559\right)\\ =\$16.9612\text{million}\end{array}$

Calculate the benefit cost ratio.

Substitute $\$20.1534\text{million}$ for $\text{Present}\text{worth}\text{of}\text{customer's}\text{benefits}$ and $\$16.9612\text{million}$ for $\text{Present}\text{worth}\text{of}\text{Government's}\text{cost}$ in Equation (VI).

$\begin{array}{c}\frac{B}{C}=\frac{\$20.1534\text{million}}{\$16.9612\text{million}}\\ =1.188\end{array}$

Project B is also acceptable.

To chose between alternatives incremental analysis is performed.

Calculate the difference between the present worth of consumer’s benefits and disbenefits.

$\begin{array}{c}P{W}_{A-B}=\left[-\left(\text{Difference}\text{in}\text{User}\text{benefits}\right)+\text{Difference}\text{in}\text{annual}\text{benefits}\left(\frac{P}{A},8\%,15\right)\right]\\ =-\left(1.2-2.1\right)+\left(2.1-2.6\right)\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)\\ =0.9-4.2795\\ =-\$3.3795\end{array}$

Calculate the difference between the present worth of government’s cost.

$\begin{array}{c}P{W}_{A-B}=\left[-\left(\text{Difference}\text{in}\text{First}\text{cost}\right)+\text{Difference}\text{in}\text{annual}\text{cost}\left(\frac{P}{A},8\%,15\right)\right]\\ =-\left(6.9-9.9\right)+\left(0.75-0.825\right)\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)\\ =-3-0.642\\ =-\$3.642\text{million}\end{array}$

Calculate the benefit cost ratio.

$\begin{array}{c}\frac{B}{C}=\frac{\text{Present}\text{worth}\text{of}\text{customer's}\text{benefits}}{\text{Present}\text{worth}\text{of}\text{government's}\text{cost}}\\ =\frac{-\$3.3795}{-\$3.642}\\ =0.927\end{array}$

Since the ratio is less than $1$ so alternative B is selected.

Conclusion:

The best alternative using the public/government version of the $B/C$ ratio is alternative B.

To determine

(d)

The best alternative using present worth index.

Answer to Problem 41P

The best alternative using present worth index is alternative A..

Explanation of Solution

Calculation:

Alternative A:

Calculate the present worth of A.

Write the expression to calculate the present worth for the alternative $expand$.

$PW=-P+A\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)-A_{OM}\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)-D$ ...... (VII)

Here, the present worth is, $PW$, the initial cost is $P$, the annual benefit is $A$, the annual operation and maintenance cost is $A_{OM}$ and the user dis-benefit is $D$.

Substitute $\$6.9$ for, $P$, $\$2.1$ for $A$, $\$0.75$ for $A_{OM}$, $1.2$ for $D$. $8\%$ for, $i$ and $15$ for $n$ in Equation (VII).

$\begin{array}{c}PW=-\$6.9+\$2.1\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)-\$0.75\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)-\$1.2\\ =-\$6.9+\$2.1\left(8.559\right)-\$0.75\left(8.559\right)-\$1.2\\ =3.45\end{array}$

Calculate the present worth of B.

Substitute $\$9.9$ for, $P$, $\$2.6$ for $A$, $\$0.825$ for $A_{OM}$, $2.1$ for $D$. $8\%$ for, $i$ and $15$ for $n$ in Equation (VII).

$\begin{array}{c}PW=-\$9.9+\$2.6\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)-\$0.825\left(\frac{{\left(1+0.08\right)}^{15}-1}{0.08{\left(1+0.08\right)}^{15}}\right)-\$2.1\\ =-\$9.9+\$2.6\left(8.559\right)-\$0.825\left(8.559\right)-\$2.1\\ =3.16\end{array}$

Since the present worth of A is more thus, project A should be selected.

Conclusion:

The best alternative using present worth index is alternative A.

To determine

(e)

The consistency of data from $a$ to $d$ and the largest value.

Explanation of Solution

The result remains consistent in all options except in the public/government version of $B/C$ ratio.

The alternative A must be chosen as the highest value of modified $B/C$ ratio is obtained. Because in this method all the operation and the maintenance cost and user dis-benefit are included as a deduction in Benefits.