Chapter 20, Problem 27P

Calen Company manufactures and sells three products in a factory of three departments. Both labor and machine time are applied to the products as they pass through each department. The nature of the machine processing and of the labor skills required in each department is such that neither machines nor labor can be switched from one department to another.

Calen’s management is attempting to plan its production schedule for the next several months. The planning is complicated by the fact that labor shortages exist in the community and some machines will be down several months for repairs.

Following is information regarding available machine and labor time by department and the machine hours and direct labor hours required per unit of product. These data should be valid for at least the next six months.

Calen believes that the monthly demand for the next six months will be as follows:

Inventory levels will not be increased or decreased during the next six months. The unit cost and price data for each product are as follows:

Required:

1. 1. Calculate the monthly requirement for machine hours and direct labor hours for producing Products 401, 402, and 403 to determine whether or not the factory can meet the monthly sales demand.
2. 2. Determine the quantities of 401, 402, and 403 that should be produced monthly to maximize profits. Prepare a schedule that shows the contribution to profits of your product mix.
3. 3. Assume that the machine hours available in Department 3 are 1,500 instead of 2,700. Calculate the optimal monthly product mix using the graphing approach to linear programming. Prepare a schedule that shows the contribution to profits from this optimal mix. (CMA adapted)

To determine

Calculate the monthly machine hours and direct labor hours for product 401, 402 and 403 and indicate whether the factory can meet monthly sales demand or not.

Explanation of Solution

Contribution margin: Contribution margin is a measurement of performance where only revenue and variable costs are taken into consideration. Hence, this measurement is useful in the evaluation of the probable outcomes of decisions including pricing decisions and other marketing strategies that affect primarily revenue and variable costs.

Calculate the monthly machine hours and direct labor hours for product 401, 402 and 403 and indicate whether the factory can meet monthly sales demand or not as follows:

Direct labor hours:

Particulars	Department 1	Department 2	Department 3	Total
Product 401:
Labor hours per unit (A)	2	3	3
Unit sold (B)	500 units	500 units	500 units
Labor hours for Product 401$\left(A\times B\right)$	1,000	1,500	1,500	4,000 hours
Product 402:
Labor hours per unit (A)	1	2	0
Unit sold (B)	400 units	400 units	400 units
Labor hours for Product 401$\left(A\times B\right)$	400	800	0	1,200 hours
Product 403:
Labor hours per unit (A)	2	2	2
Unit sold (B)	1,000 units	1,000 units	1,000 units
Labor hours for Product 401$\left(A\times B\right)$	2,000	2,000	2,000	6,000 hours
Totals	3,400 hours	4,300 hours	3,500 hours	11,200 hours

Table (1)

Machine hours:

Particulars	Department 1	Department 2	Department 3	Total
Product 401:
Machine hours per unit (A)	1	1	2
Unit sold (B)	500 units	500 units	500 units
Machine hours for Product 401$\left(A\times B\right)$	500	500	1,000	2,000 hours
Product 402:
Machine hours per unit (A)	1	1	0
Unit sold (B)	400 units	400 units	400 units
Machine hours for Product 401$\left(A\times B\right)$	400	400	0	800 hours
Product 403:
Machine hours per unit (A)	2	2	1
Unit sold (B)	1,000 units	1,000 units	1,000 units
Machine hours for Product 401$\left(A\times B\right)$	2,000	2,000	1,000	5,000 hours
Totals	2,900 hours	2,900 hours	2,000 hours	7,800 hours

Table (2)

In this case, company can meet the demand in all departments expect Department 3, because available labor hours for department 3 (2,750 hours) is less than the monthly requirement of 3,500 hours.

To determine

Prepare a schedule of contribution to profits for the given product mix.

Explanation of Solution

Prepare a schedule of contribution to profit for the given product mix as follows:

In this case, department 3 in product 401 has more labor hour than the actual available hours. Hence, after meeting the demand the additional labor hours of Department 3 is used to produce the product 1 as a subsidy of department 1. Thus, department 1 would produce only 250 units$\left(\frac{2,750-2,000}{3}\right)$.

Particulars	Optimal output (A)	Contribution margin per unit (B)	Total contribution $\left(A\times B\right)$
Product 401	250 units	$93 (1)	$23,250
Product 402	400 units	$50 (2)	$20,000
Product 403	1,000 units	$70 (3)	$70,000
Total contribution margin			$113,250

Table (3)

Working note (1):

Calculate the contribution margin per unit for product 401.

$\begin{array}{c}\text{Contribution margin = }\left(\text{Selling price }-\left(\begin{array}{l}\text{Total varaible cost}-\\ \text{Fixed cost}\end{array}\right)\right)\\ =\$196-\left(\$118-\$15\right)\\ =\$93\end{array}$

Working note (2):

Calculate the contribution margin per unit for product 402.

$\begin{array}{c}\text{Contribution margin = }\left(\text{Selling price }-\left(\begin{array}{l}\text{Total varaible cost}-\\ \text{Fixed cost}\end{array}\right)\right)\\ =\$123-\left(\$83-\$10\right)\\ =\$50\end{array}$

Working note (3):

Calculate the contribution margin per unit for product 403.

$\begin{array}{c}\text{Contribution margin = }\left(\text{Selling price }-\left(\begin{array}{l}\text{Total varaible cost}-\\ \text{Fixed cost}\end{array}\right)\right)\\ =\$167-\left(\$129-\$32\right)\\ =\$70\end{array}$

To determine

Compute the optimal monthly product mix using the graphical approach to linear program and calculate the contribution profit for the optimal mix.

Explanation of Solution

Compute the optimal monthly product mix using the graphical approach to linear program and calculate the contribution profit for the optimal mix as follows:

Machine constraint:

$\text{2X+Y}\le \text{1,500}$

Direct labor constraint:

$\text{3X+2Y}\le 2,750$

Demand constraint for each product:

$\text{X}\le \text{500}$

$\text{Y}\le \text{1,000}$

$\text{Z=400}$

Note: X denotes number of product produced in Product 401, Y denotes number of product produced in Product 402, and Z denotes number of product produced in Product 403.

Corner point	X-value	Y-value	W-value	$Z=\$96X+\$70Y+\$50W$
A	0	0	400	$20,000
B	500	0	400	$66,500
C	500	500	400	$101,500
D	250	1,000	400	$113,250
E	0	1,000	400	$90,000

Table (1)

Figure (1)

Hence, contribution margin under optimal output is $113,250.