# In a textile plant, cloth is manufactured in rolls of length L. Defects sometimes occur along the length of the cloth. Consider a specifi

In a textile plant, cloth is manufactured in rolls of length L. Defects sometimes occur along the length of the cloth. Consider a specific roll with (N − 1) defects appearing at distances y1, y2, . . . , yN−1 from the start of the roll (yi+1 > yfor all i). Denote the start of the roll by y0, the end by yN .

The roll is cut into pieces for sale. The value of a piece depends on its length and the number of defects. Let v(x, m) = Value of a piece of length x having m defects. Assume that all cuts are made through defects and that such cutting removes the defect. Specify how to determine where to cut the cloth to maximize total value. 18. A manufacturing company, receiving an order for a special product, has worked out a production plan for the next 5 months. All components will be manufactured internally except for one electronic part that must be purchased. The purchasing manager in charge of buying the electronic part must meet the requirements schedule established by the production department. After negotiating with several suppliers, the purchasing manager has determined the best possible price for the electronic part for each of the five months in the planning horizon. Table summarizes the requirement schedule and purchase price information.

The storage capacity for this item is limited to 12,000 units; there is no initial stock, and after the five-month period the item will no longer be needed. Assume that the orders for the electronic part are placed once every month (at the beginning of each month) and that the delivery lead time is very short (delivery is made practically instantaneously). No back-ordering is permitted. a) Derive the monthly purchasing schedule if total purchasing cost is to be minimized. b) Assume that a storage charge of \$250 is incurred for each 1000 units found in inventory at the end of a month. What purchasing schedule would minimize the purchasing and storage costs?