Solve the inventory problem given in Prob. 19.7-6, but assume that the policy is to be used for only 1 year (a 12-period model). Shortages are backlogged each month, except that any shortages remaining at the end of the year are made up by purchasing similar items at a unit cost of $2. Any remaining inventory at the end of the year can be sold at a unit price of $2.

Prob. 19.7-6

Determine the optimal inventory policy when the goods are to be ordered at the end of every month from now on. The cost of bringing the inventory level up to y when x already is available is given by 2(y  x). Similarly, the cost of having the monthly demand D exceed y is given by 5(D  y). The probability density function for D is given by  The holding cost when y exceeds D is given by y  D. A monthly discount factor of 0.95 is used.