You are a purchasing manager in charge of stocking a certain type of transformer for a large electric utility. Weekly demand among your field crews for these transformers is normally distributed, with a mean of $100$ and a standard deviation of $50.$ Holding costs are $25$ percent, and you must hold a level of inventory corresponding to a cycle service level of $95$ percent. You are faced with two suppliers, Reliable Components and Value Electric, that offer the following terms. Reliable sells the transformer for $$5,000$ with a minimum order of $100,$ and a lead time of $1$ week with a standard deviation of $0\mathrm{.}1$ week. Value sells the transformer for $$4,800,$ has a minimum batch of $1,000,$ a lead time of $5$ weeks, and a lead$-$time standard deviation of $4$ weeks.

Imagine that you have chosen Reliable as your supplier. Value Electric wants your business very much and offers you the choice of three mutually exclusive alternatives: reduce lead time by $1$ week, reduce the minimum batch to $800,$ or reduce the standard deviation of lead time to $3$ weeks.

a$.$ What are the expected annual costs of undertaking each of these options?

b$.$ What is the expected annual cost if all three could be put into effect?

c$.$ Would you change your decision to go with Reliable for any of these options?