Question $5$: Your company has signed a consignment contract to supply a chemical compound to a major customer. This compound costs you $$32\mathrm{.}50/$gallon and generates a contribution margin of $$\frac{12\mathrm{.}50}{?}$ gallon. Delivery costs you $$175$ in shipping and handling. Demand for the year is estimated at $2000$ gallons with a standard deviation of $400$ gallons. The contract says that the compound is inventoried at your client's facility: he will draw the material from that stock as needed and pay you at such time.

The contract also says that if the client finds that inventory empty when needed, you are subject to a fee of $$\frac{0\mathrm{.}325}{?}$ gallon$/$day short. ${?}^2$ Inventory carrying cost is $0\mathrm{.}20\frac{\text{\$}}{\text{\$}}\frac{?}{?}$ year. ${?}^3$ Assume that senior management has set a minimum safety factor of $k=1\mathrm{.}25$ for all of these questions.

a$.$ Determine the optimal $(s,Q)$ policy $($reorder point and order quantity$)$ assuming that the shipment size is the EOQ and lead time is stochastic with an average of $30$ days and a standard deviation of $6$ days.

b$.$ Your customer has realized that controlling the stockout charge according to the contract terms is too burdensome. Two alternative charges have been proposed:

A flat charge of $$1,000$ per stockout occasion ${?}^4$

A $$65$ charge per gallon per short per cycle ${?}^5$

Under each alternative, what would your new $(s,Q)$ policy be$?$ Which alternative would your company prefer?