Mary Custard's is a pie shop that specializes in custard and fruit pies.  It makes delicious pies and sells them at reasonable prices so that it can sell all the pies it makes in a day.  Every dozen custard pies nets Mary Custard's $15 and requires 12 pounds of flour, 50 eggs, 5 pounds of sugar and no frut mixture.  Every dozen fruit pies nets a $25 profit and uses 10 pounds of flour, 40 eggs, 10 pounds of sugar, and 15 pounds of fruit mixture.

On a given day, the bakers at Mary Custard's found that they had 150 pounds of flour, 500 eggs, 90 pounds of sugar, and 120 pounds of fruit mixture with which to make pies.

a. Formulate and solve a linear program that will give the optimal production schedule of pies for the day.

b. If Mary Custard's  could double its profit on custard pies, should more custard pies be produced?  Explain.

c. If Mary Custard's raised the price (and hence the profit) on all pies by $0.25 ($3 per dozen), would the optimal production schedule for the day change? Would the profit change?

d. Suppose Mary Custard's found that 10% of its fruit mixture had been stored in containers that were not air-tight. For quality and health reasons, it decided that it would be unwise to use any of this portion of the fruit mixture.  How would this affect the optimal production schedule?  Explain.

e. Mary Custard's currently pays $2.50 for a five pound bag of sugar from its bakery supply vendor. (The $0.50 per pound price of sugar is included in the unit profits given earlier.)  Its vendor has already made its deliveries for the day.  If Mary Custard's wishes to purchase additional sugar, it must buy it from Donatelli's Market, a small local independent grocery store that sells sugar in  one-pound boxes for $2.25 a box.  Should Mary Custard's purchase any boxes of sugar from Donatelli's Market? Explain.