Q$12\mathrm{.}9($ CPG Bagels $)$ CPG Bagels starts the day with a large production run of bagels. Through$-$

out the morning, additional bagels are produced as needed. The last bake is completed at

$3$ P$.$M$.$ and the store closes at $8$ P$.$M$.$ It costs approximately $$0\mathrm{.}20$ in materials and labor to

make a bagel. The price of a fresh bagel is $$0\mathrm{.}60.$ Bagels not sold by the end of the day

are sold the next day as $$day old$$ bagels in bags of six, for $$0\mathrm{.}99$ a bag. About two$-$thirds

of the day$-$old bagels are sold; the remainder are just thrown away. There are many bagel

flavors, but for simplicity, concentrate just on the plain bagels. The store manager predicts

that demand for plain bagels from $3$ P$.$M$.$ until closing is normally distributed with mean of

$54$ and standard deviation of $21.$

a$.$ How many bagels should the store have at $3$ P$.$M$.$ to maximize the store$$s expected profit

$($from sales between $3$ P$.$M$.$ until closing$)?($ Hint: Assume day$-$old bagels are sold for

$$0\mathrm{.}99/6$ $$0\mathrm{.}165$ each; i$.$e$.,$ don$$t worry about the fact that day$-$old bagels are sold in

bags of six.$)$

b$.$ Suppose that the store manager is concerned that stockouts might cause a loss of future

business. To explore this idea, the store manager feels that it is appropriate to assign

a stockout cost of $$5$ per bagel that is demanded but not filled. $($Customers frequently

purchase more than one bagel at a time. This cost is per bagel demanded that is not

satisfied rather than per customer that does not receive a complete order.$)$ Given the

additional stockout cost, how many bagels should the store have at $3$ P$.$M$.$ to maximize

the store$$s expected profit?

c$.$ Suppose the store manager has $101$ bagels at $3$ P$.$M$.$ How many bagels should the store

manager expect to have at the end of the day?