Consider the following inventory model, which is a single-period model with known density of demand D() e for

  0 and D() 0 elsewhere. There are two costs connected with the model. The first is the purchase cost, given by c( y  x).

The second is a cost p that is incurred once if there is any unsatisfied demand (independent of the amount of unsatisfied demand).

(a) If x units are available and goods are ordered to bring the inventory level up to y (if x y), write the expression for the expected loss and describe completely the optimal policy.

(b) If a fixed cost K is also incurred whenever an order is placed, describe the optimal policy.