10 During month t, a firm faces a demand for dt units of a product. The firm’s production cost during month t consists of two components. First, for each unit produced during month t, the firm incurs a variable production cost of ct.

Second, if the firm’s production level during month t  1 is xt1 and the firm’s production level during month t is xt, then during month t, a smoothing cost of 5xt  xt1 will be incurred (see Section 16.12 for an explanation of smoothing costs). At the end of each month, a holding cost of ht per unit is incurred. Formulate a recursion that will enable the firm to meet (on time) its demands over the next 12 months. Assume that at the beginning of the first month, 20 units are in inventory and that last month’s production was 20 units. (Hint: The state during each month must consist of two quantities.)