$(13$ points total$)$

$3.$ Suppose an industrial firm orders a liquid ingredient from their supplier using a continuous review policy. The average demand is $150$ gallons per day with a standard deviation of $40$ per day. Suppose the lead time from their supplier varies with an average of $8$ days with a standard deviation of $2$ days. The firm operates $365$ days a year.

$(7$ pts$.)$ a$)$ If a $95\%$ service level is desired, then what would be the reorder point?

Show all your work.

Reorder point $=q,$

$1508+-\sqrt[\phantom{2}]{(8)({40}^2{)}^2+({150}^2)(3)}$

$(3$ pts$.)$ b$)$ Suppose each order from the firm to its supplier is for $1500$ gallons $($not your answer to part a$).$ With that order size, on average how many orders do they make per year? Show all your work.

Number of orders per year $=q,$

$50365$

$\frac{1500}{?}=36\mathrm{.}50$

$(3$ pts$.)$ c$)$ Suppose each order from the firm to its supplier is for $1500$ gallons $($not your answer to part a$).$ With that order size, and with a $90\%$ service level, on average how many stockouts do they expect in a year? Show all your work.

Stockouts per year $=$

$(1-\mathrm{.}9)36\mathrm{.}50=3\mathrm{.}65$