Decatron Mills has contracted to deliver 20 tons of a special coarsely ground wheat flour at the end of the current month, and 140 tons at the end of the next month. The production cost, based on which the Sales Department has bargained with prospective customers, is c1(x1) = 7500 + (x1 - 50)2 per ton for the first month, and c2(x2) = 7500 + (x2 - 40) 2 per ton for the second month; x1 and x2are the number of tons of the flour produced in the first and second months, respectively. If the company chooses to produce more than 20 tons in the first month, any excess production can be carried to the second month at a storage cost of $3 per ton.
Assuming that there is no initial inventory and that the contracted demands must be satisfied in each month (that is, no back-ordering is allowed), derive the production plan that minimizes total cost. Solve by both backward and forward induction. Consider x1 and x2 as continuous variables, since any fraction of a ton may be produced in either month.