Chapter D, Problem 139P

To determine

The rank of product lines from lowest to highest in terms of manufacturing cost per unit.

Answer to Problem 139P

The correct option is $\left(b\right)\text{3-1-2-4}$.

Explanation of Solution

Given:

Data	Line 1	Line 2	Line 3	Line 3
Annual production	$4000$	$3500$	$9800$	$675$
Cost of direct materials	$800$	$650$	$1200$	$2500$
Cost of direct labor	$3500$	$3750$	$600$	$320$

Calculation:

Write the expression to calculate the direct cost of product lines.

$DC=\left(\text{Cost}\text{of}\text{materials}\right)+\left(\text{Cost}\text{of}\text{direct}\text{labor}\right)$ ...... (I)

Here, the direct cost is DC.

Substitute $\$800,000$ for Cost of materials and $\$3,500,000$ for Cost of direct labor in Equation (I) for line 1.

$\begin{array}{c}D{C}_1=\$800,000+\$3,500,000\\ =\$4,300,000\end{array}$

Substitute $\$650,000$ for cost of materials and $\$3,750,000$ for Cost of direct labor in Equation (I) for line 2.

$\begin{array}{c}D{C}_2=\$650,000+\$3,750,000\\ =\$4,400,000\end{array}$

Substitute $\$1,200,000$ for cost of materials and $\$600,000$ for Cost of direct labor in Equation (I) for line 3.

$\begin{array}{c}D{C}_3=\$1,200,000+\$600,000\\ =\$1,800,000\end{array}$

Substitute $\$2,500,000$ for cost of materials and $\$320,000$ for Cost of direct labor in Equation (I) for line 4.

$\begin{array}{c}D{C}_4=\$2,500,000+\$320,000\\ =\$2,820,000\end{array}$

Write the expression to calculate total cost for line 1.

$T{C}_1=D{C}_1+\$10,800,000\left(\frac{D{C}_1}{D{C}_1+D{C}_2+D{C}_3+D{C}_3}\right)$ ...... (II)

Here, the total cost is TC.

Substitute $\$4,300,000$, for $D{C}_1$, $\$4,400,000$ for $D{C}_2$, $\$1,800,000$ for $D{C}_3$, $\$2,820,000$ for $D{C}_4$ in Equation (II).

$\begin{array}{c}T{C}_1=\$4,300,000+\$10,800,000\left(\frac{\$4,300,000}{\begin{array}{l}\$4,300,000+\$4,400,000\\ +\$1,800,000+\$2,820,000\end{array}}\right)\\ =\$4,300,000+\$10,800,000\left(0.32282\right)\\ =\$4,300,000+\$3,486,486.486\\ =\$7,786,486.486\end{array}$

Write the expression to calculate the manufacturing cost per unit for line 1.

$A{C}_1=\frac{T{C}_1}{\text{Total}\text{units}\text{produced}\text{at}\text{line}\text{1}}$ ...... (III)

Here, the average cost is AC.

Substitute, $\$7,786,486.486$ for $T{C}_1$ and $4000$ for total units produced at line 1 in Equation (III).

$\begin{array}{c}A{C}_1=\frac{\$7,786,486.486}{4,000}\\ =\$1,946.62\end{array}$

Write the expression to calculate total cost for line 2.

$T{C}_2=D{C}_2+\$10,800,000\left(\frac{D{C}_2}{D{C}_1+D{C}_2+D{C}_3+D{C}_3}\right)$ ...... (IV)

Substitute $\$4,300,000$, for $D{C}_1$, $\$4,400,000$ for $D{C}_2$, $\$1,800,000$ for $D{C}_3$, $\$2,820,000$ for $D{C}_4$ in Equation (IV).

$\begin{array}{c}T{C}_2=\$4,400,000+\$10,800,000\left(\frac{\$4,400,000}{\begin{array}{l}\$4,300,000+\$4,400,000\\ +\$1,800,000+\$2,820,000\end{array}}\right)\\ =\$4,400,000+\$10,800,000\left(0.33\right)\\ =\$4,400,000+\$3,567,567.568\\ =\$7,967,567.568\end{array}$

Write the expression to calculate the manufacturing cost per unit for line 2.

$A{C}_2=\frac{T{C}_2}{\text{Total}\text{units}\text{produced}\text{at}\text{line}\text{1}}$ ...... (V)

Substitute $\$7,786,486.486$ for $T{C}_2$ and $4000$ for total units produced at line 2 in Equation (V).

$\begin{array}{c}A{C}_2=\frac{\$7,967,567.568}{3,000}\\ =\$2,655.855\end{array}$

Write the expression to calculate total cost for line 3.

$T{C}_3=D{C}_3+\$10,800,000\left(\frac{D{C}_3}{D{C}_1+D{C}_2+D{C}_3+D{C}_3}\right)$ ...... (VI)

Substitute $\$4,300,000$, for $D{C}_1$, $\$4,400,000$ for $D{C}_2$, $\$1,800,000$ for $D{C}_3$, $\$2,820,000$ for $D{C}_4$ in Equation (VI).

$\begin{array}{c}T{C}_3=\$1,800,000+\$10,800,000\left(\frac{\$1,800,000}{\begin{array}{l}\$4,300,000+\$4,400,000\\ +\$1,800,000+\$2,820,000\end{array}}\right)\\ =\$1,800,000+\$10,800,000\left(0.1351\right)\\ =\$1,800,000+\$1,459,459.45\\ =\$3,259,459.459\end{array}$

Write the expression to calculate the manufacturing cost per unit for line 3.

$A{C}_3=\frac{T{C}_3}{\text{Total}\text{units}\text{produced}\text{at}\text{line}\text{1}}$ ...... (VII)

Substitute $\$7,786,486.486$ for $T{C}_3$ and $4000$ for total units produced at line 3 in Equation (VII).

$\begin{array}{c}A{C}_3=\frac{\$3,259,459.459}{9,800}\\ =\$332.597\end{array}$

Write the expression to calculate total cost for line 4.

$T{C}_4=D{C}_4+\$10,800,000\left(\frac{D{C}_4}{D{C}_1+D{C}_2+D{C}_3+D{C}_3}\right)$ ...... (VIII)

Substitute $\$4,300,000$, for $D{C}_1$, $\$4,400,000$ for $D{C}_2$, $\$1,800,000$ for $D{C}_3$, $\$2,820,000$ for $D{C}_4$ in Equation (VIII).

$\begin{array}{c}T{C}_4=\$2,820,000+\$10,800,000\left(\frac{\$2,820,000}{\begin{array}{l}\$4,300,000+\$4,400,000\\ +\$1,800,000+\$2,820,000\end{array}}\right)\\ =\$2,820,000+\$10,800,000\left(0.2117\right)\\ =\$2,820,000+\$2,286,486.486\\ =\$5,106,486.486\end{array}$

Write the expression to calculate the manufacturing cost per unit for line 4.

$A{C}_4=\frac{T{C}_4}{\text{Total}\text{units}\text{produced}\text{at}\text{line}\text{1}}$ ...... (IX)

Substitute $\$7,786,486.486$ for $T{C}_4$ and $4000$ for total units produced at line 4 in Equation (IX).

$\begin{array}{c}A{C}_4=\frac{\$5,106,486.486}{675}\\ =\$7,565.16\end{array}$

The product lines are ranked on the basis of average cost from lowest to highest.

$3<1<2<4$.

Conclusion:

Therefore, the correct option is $\left(b\right)\text{3-1-2-4}$.