Question

A biotechnology manufacturing firm can produce diagnostic test kits at a cost of $20. Each kit for which there is a demand in the week of production can be sold for $100. However, the half-life of components in the kit requires the kit to be scrapped if it is not sold in the week of production. The cost of scrapping the kit is $5. The weekly demand is summarized as follows:

Number of units
0 50 100 200
Probability of demand
0.05 0.4 0.3 0.25
How many kits should be produced each week to maximize the mean earnings of the firm?

Answer 1

To maximize the mean earnings of the firm, the optimal quantity of test kits to produce each week is 200 units. This quantity yields the highest expected earnings of $47.5.

Step-by-step explanation:

To maximize the mean earnings of the firm, we need to determine the optimal quantity of test kits to produce each week. This can be done by calculating the expected earnings for different production quantities and selecting the quantity that yields the highest mean earnings.

To calculate the expected earnings, we multiply the demand for each quantity by the selling price and subtract the cost of production and the cost of scrapping for any unsold kits:

1. For 0 units, the expected earnings = (0.05)(0) - (0.05)(5) = -0.25
2. For 50 units, the expected earnings = (0.4)(50) - (0.4)(5) = 18
3. For 100 units, the expected earnings = (0.3)(100) - (0.3)(5) = 27.5
4. For 200 units, the expected earnings = (0.25)(200) - (0.25)(5) = 47.5

The quantity of kits that maximizes mean earnings is 200 units, as it yields the highest expected earnings of $47.5.