You have just accepted a job with Brentwood Pharma (BP). Your first challenge is to decide how much capacity BP should reserve to produce a new drug for upcoming clinical trials. This new drug is unlike any other that BP has ever developed, and is produced using a revolutionary biological process. BP needs to produce a batch of 10,000 doses of the drug, but given the biological nature of the process, nobody knows exactly how long it will take to produce that batch. BP assumes the total production time will follow a normal distribution, with a mean of 25 days, but an unknown standard deviation. BioCon is a contract manufacturer that operates the biological process that BP needs; they rent out capacity to various pharmaceutical firms. BP needs to reserve capacity from BioCon; the charge is $5,000/day for the capacity itself, and another $5,000 per day in variable costs. In total, BP has to pay $10,000 for each day of capacity they reserve. If the production takes longer than expected, they can buy additional capacity for a total of $20,000 per day. If production takes shorter than expected, BP is reimbursed for the unused variable costs, but not for the costs of the unused capacity.

a) If BP assumes the standard deviation of the processing time is 10 days, how much capacity should they reserve? (It is possible to reserve fractions of days.)

b) In (a), what is the probability that BP will have to buy additional capacity? Now suppose that BP decides to produce twice the quantity of the drug. For quality reasons, they plan to achieve in two separate batches. However, they can reserve the total capacity for the two batches back to a-back (i.e., as one stretch of time). The times taken to do the batches can be taken to be independent random variables.

c) How much capacity should BP reserve at BioCon to produce two batches? What does doing the batches back-to-back gain BP, over doing them at different times?