A production process consists of a three step operation. The scrap rate is 10% for the first step and 6 percent for each of the other two steps. a. If the desired daily output is 450 units, how many units must be started to allow for loss due to scrap?
b. If the scrap rate for each step could be cut in half, how many units would this save in terms of the scrap allowance?c. If the scrap represents a cost of $10 per unit, how much is it costing the company per day for the original scrap rate?

a. The number of units that must be started to allow for loss due to scrap is 566 units.

b. The number of units that this would save in terms of the scrap allowance is 63 units.

c. The scrap is costing the company $1,160 per day.

Explanation:

a. If the desired daily output is 450 units, how many units must be started to allow for loss due to scrap?

Let x represent the original quantity.

Therefore, we have:

Scrap from the first step operation = 0.10x

Remaining quantity after the first step operation = Original quantity - Scrap of the first step operation = x – 0.10x = (1 – 0.10)x = 0.90x

Scrap from the second step operation = Remaining quantity after the first step operation * 0.06 = 0.90x * 0.06 = 0.054x

Remaining quantity after the second step operation = Remaining quantity after the first step operation - Scrap from the second step operation = 0.90x - 0.054x = (0.90 - 0.054)x = 0.846x

Scrap from the third step operation = Remaining quantity after the second step operation * 0.06 = 0.846x * 0.06 = 0.05076x

Remaining quantity after the third step operation = Remaining quantity after the second step operation - Scrap from the third step operation = 0.846x - 0.05076x = (0.846 - 0.05076)x = 0.79524x

Since the desired daily output is 450 units, we can solve for x as follows:

0.79524x = 450

x = 450 / 0.79524 = 565.866908103214, or 566 units

Therefore, the number of units that must be started to allow for loss due to scrap is 566 units.

b. If the scrap rate for each step could be cut in half, how many units would this save in terms of the scrap allowance?

Scrap rate of first operation is = 10%/2 = 5%, or 0.05

Scrap rate of second and third operations = 6%/2 = 3%, or 0.03 each

Let x also represent the original quantity.

Therefore, we have:

Scrap from the first step operation = 0.05x

Remaining quantity after the first step operation = Original quantity - Scrap of the first step operation = x - 0.05x = (1 – 0.05)x = 0.95x

Scrap from the second step operation = Remaining quantity after the first step operation * 0.03 = 0.95x * 0.03 = 0.0285x

Remaining quantity after the second step operation = Remaining quantity after the first step operation - Scrap from the second step operation = 0.95x - 0.0285x = (0.95 - 0.0285)x = 0.9215x

Scrap from the third step operation = Remaining quantity after the second step operation * 0.03 = 0.9215x * 0.03 = 0.027645x

Remaining quantity after the third step operation = Remaining quantity after the second step operation - Scrap from the third step operation = 0.9215x - 0.027645x = (0.9215 - 0.027645)x = 0.893855x

Since the desired daily output is 450 units, we can solve for x as follows:

0.893855x = 450

x = 450 / 0.893855 = 503.437358408243, or 503 units

Number units to save = Calculated x from part a above - Calculated x in part 2 here = 566 – 503 = 63

Therefore, the number of units that this would save in terms of the scrap allowance is 63 units.

c. If the scrap represents a cost of $10 per unit, how much is it costing the company per day for the original scrap rate?

Quantity to be introduced at the original scrap rate = Calculated x from part a above = 566

Remaining quantity after the third step operation = 450

Scrap = Quantity to be introduced at the original scrap rate - Remaining quantity after the third step operation = 566 – 450 = 166 units

Total cost of scrap = Scrap * Cost of each unit of scrap = 166 * $10 = $1,160

Therefore, the scrap is costing the company $1,160 per day.