$18\mathrm{.}7-7$ Reconsider Prob. $18\mathrm{.}7-5.$ The bakery owner, Ken Swanson, now has developed a new plan to decrease the size of shortages.

The bread will be baked twice a day, once before the bakery opens $($as before$)$ and the other during the day after it becomes clearer what the demand for that day will be$.$ The first baking will produce $300$ loaves to cover the minimum demand for the day. The size of the second baking will be based on an estimate of the remaining demand for the day. This remaining demand is assumed to have a uniform distribution from a to b$,$ where the values of a and b are chosen each day based on the sales so far. It is anticipated that $($b $-$ a$)$ typically will be approximately $75,$ as opposed to the range of $300$ for the distribution of demand in Prob. $18\mathrm{.}7-5.$

$($a$)$ Ignoring any cost of the loss of customer goodwill $[$as in parts

$$to $($d$)$ of Prob. $18\mathrm{.}7-5],$ write a formula for how many loaves should be produced in the second baking in terms of a and b$.$

$$What is the probability of still incurring a shortage of fresh bread on any given day? How should this answer compare to the corresponding probability in Prob. $18\mathrm{.}7-5?$

$$When b $-$ a $=75,$ what is the maximum size of a shortage that

can occur? What is the maximum number of loaves of fresh bread that will not be sold? How do these answers compare to the corresponding numbers for the situation in Prob. $18\mathrm{.}7-5$ where only one $($early morning$)$ baking occurs per day?

$($d$)$ Now consider just the cost of underordering and the cost of overordering. Given your answers in part $($c$),$ how should the