Consider the following three-period inventory problem. At the beginning of each period, a firm must determine how many units should be produced during the current period. During a period in which x units are produced, a production cost c(x) is incurred, where c(0) 

0, and for x  0, c(x)  3  2x. Production during each period is limited to at most 4 units. After production occurs, the period’s random demand is observed. Each period’s demand is equally likely to be 1 or 2 units. After meeting the current period’s demand out of current production and inventory, the firm’s end-of-period inventory is evaluated, and a holding cost of $1 per unit is assessed. Because of limited capacity, the inventory at the end of each period cannot exceed 3 units. It is required that all demand be met on time.

Any inventory on hand at the end of period 3 can be sold at $2 per unit. At the beginning of period 1, the firm has 1 unit of inventory. Use dynamic programming to determine a production policy that minimizes the expected net cost incurred during the three periods.