We consider the production of a single product over T periods. If we decide to produce at period t, a setup cost c_t is incurred. For t = 1, ...,T, let d_t be the demand for this product in period t, and let p_t, h_t be the unit production cost and unit storage cost (per period), respectively.

(a) Formulate an integer programming problem in order to minimize the total cost of production, storage, and setup.

(b) Suppose we allow demand to be lost in every period except for period T, at a cost of b_t per unit of lost demand. Show how to modify the model to handle this option.

(c) Suppose that production can occur in at most five periods, but no two such periods can be consecutive. Show how to modify the model to handle this option.