It costs $12 to make a single unit using regular production and $15 to make a single unit using overtime production. Total overtime production is limited to 500 units for the five month period. The manufacturing plant has a regular production capacity of 250 units per month and 50 units in inventory at the start of the planning period. There is a $5 per unit charge for holding inventory at the end of each month and a limit of 250 units ending inventory for any period. What is the total number of units to be produced using overtime throughout the entire planning period if the forecast must be met and costs are to be minimized with a zero ending inventory each month? $$\begin{array} { | l | c | }
\hline { \text { Month } } & \text { Forecast } \
\hline \text { Tanuary } & 250 \
\hline \text { February } & 200 \
\hline \text { March } & 300 \
\hline \text { April } & 400 \
\hline \text { May } & 500 \
\hline
\end{array}$$
A)450
B)500
C)400
D)350

Question

It costs $12 to make a single unit using regular production and $15 to make a single unit using overtime production. Total overtime production is limited to 500 units for the five month period. The manufacturing plant has a regular production capacity of 250 units per month and 50 units in inventory at the start of the planning period. There is a $5 per unit charge for holding inventory at the end of each month and a limit of 250 units ending inventory for any period. What is the total number of units to be produced using overtime throughout the entire planning period if the forecast must be met and costs are to be minimized with a zero ending inventory each month? $$\begin{array} { | l | c | } 
\hline { \text { Month } } & \text { Forecast } \
\hline \text { Tanuary } & 250 \
\hline \text { February } & 200 \
\hline \text { March } & 300 \
\hline \text { April } & 400 \
\hline \text { May } & 500 \
\hline
\end{array}$$
A)450
B)500
C)400
D)350

Quizplus · Accepted Answer

To minimize costs and achieve a zero ending inventory each month, we need to determine the optimal production plan for each month. We can start by calculating the net requirements for each month, which is the forecasted demand minus the beginning inventory:

- January: 250 - 50 = 200 units
- February: 200 - 0 = 200 units
- March: 300 - 0 = 300 units
- April: 400 - 0 = 400 units
- May: 500 - 0 = 500 units

Next, we can calculate the production plan for each month using regular production and overtime production as needed. Let's start with January:

- Regular production: We can produce up to 250 units using regular production, which would cost $12 per unit. However, we only need 200 units, so we'll use up to 200 units of regular production.
- Overtime production: We need an additional 200 - 50 = 150 units, which we can produce using overtime production at a cost of $15 per unit. However, we only have 500 units of overtime production available for the entire planning period, so we can't use all of it in January. Let's assume we use x units of overtime production in January, then we'll have 500 - x units remaining for the rest of the planning period. Therefore, our production plan for January is:

Regular production: 200 units at $12 per unit = $2,400
Overtime production: (150 - x) units at $15 per unit = $(2,250 - 15x)

To minimize costs, we want to produce as much as possible using regular production before using overtime production. However, we also need to make sure that we don't exceed the 250 units ending inventory limit for January. Let's assume we produce y units using regular production, then we'll have:

Ending inventory = 250 - (200 - y) = 50 + y units

We can set this equal to the $5 per unit holding cost and solve for y:

5(50 + y) = 12(y + 200 - 250)
250 + 5y = 12y - 300
7y = 550
y = 78.57

Since we can't produce a fraction of a unit, we'll produce y = 78 units using regular production, which will leave us with:

Ending inventory = 250 - (200 - 78) = 128 units

Now we can calculate the optimal value of x by comparing the costs of regular production and overtime production:

Regular production: 78 units at $12 per unit = $936
Overtime production: (150 - x) units at $15 per unit = $(2,250 - 15x)

We want to minimize costs, so we'll choose the production method with the lowest cost. We can set the costs equal to each other and solve for x:

936 + 12x = 2,250 - 15x
27x = 1,314
x = 48.7

Again, we can't produce a fraction of a unit, so we'll use x = 48 units of overtime production in January. Therefore, our final production plan for January is:

Regular production: 78 units at $12 per unit = $936
Overtime production: 102 units at $15 per unit = $1,530

We can use a similar approach to calculate the production plans for February through May:

- February:

Net requirements: 200 units
Regular production: 200 units at $12 per unit = $2,400
Overtime production: 0 units
Ending inventory: 50 units
Total cost: $2,400

- March:

Net requirements: 300 units
Regular production: 250 units at $12 per unit = $3,000
Overtime production: 50 units at $15 per unit = $750
Ending inventory: 0 units
Total cost: $3,750

- April:

Net requirements: 400 units
Regular production: 250 units at $12 per unit = $3,000
Overtime production: 150 units at $15 per unit = $2,250
Ending inventory: 0 units
Total cost: $5,250

- May:

Net requirements: 500 units
Regular production: 250 units at $12 per unit = $3,000
Overtime production: 250 units at $15 per unit = $3,750
Ending inventory: 0 units
Total cost: $6,750

Adding up the total costs for each month, we get:

$936 + $1,530 + $2,400 + $3,750 + $5,250 + $6,750 = $20,616

Therefore, the total number of units to be produced using overtime throughout the entire planning period is:

48 + 0 + 50 + 150 + 250 = 498 units

Since we're limited to 500 units of overtime production for the entire planning period, we can't produce any more units using overtime without exceeding that limit. Therefore, the best choice is A) 450 units, which is the closest option to the optimal value of 498 units.

It Costs $12 to Make a Single Unit Using Regular