Howling Cow needs to figure out the production of its ice cream for the Creamery throughout the next week. We assume that production decisions can be made for each flavor independently, and therefore consider only one flavor in this problem. For the flavor considered, they anticipate a demand of d??t gallons for each day t (Sunday through Friday; the Creamery is closed on Saturdays.) If they decide to produce any of the flavor on day t, a setup cost of c?t is incurred. The production and storage costs (for unsold ice cream that is saved for the next day) are pt and s?t dollars per gallon, respectively, on day t. Unsold ice cream can survive throughout the week, and is thrown out at closing on Friday.
(a) Formulate an mixed-integer linear program to minimize the total cost of production and storage, while meeting the expected demand every day.
(b) Suppose we allow unsatisfied demand, at a cost of b?t dollars per gallon on day t. Show how to modify the model to handle this option.