A pharmaceutical manufacturer must supply 30 batches of its new medication in the next quarter, then 25, 10, and 35 in successive quarters.

Each quarter in which the company makes product requires a $100,000 setup, plus $3000 per batch produced. There is no limit on production capacity.

Batches can be held in inventory, but the cost is a high $5000 per batch per quarter. The company seeks a minimum total cost production plan.

(a) Explain why this problem can be approached by discrete dynamic programming, with states k = 1,

c, 5 representing the reaching of quarter k with all earlier demand fulfilled and no inventory on hand.

(b) Sketch the digraph corresponding to the dynamic program structure of part (a).

Include costs on all arcs.

(c) Explain why the feasible production plans correspond exactly to the paths from node k = 1 to node k = 5 in your digraph.

(d) Solve a shortest path problem on your digraph to compute an optimal production plan.

(e) Use your computations in part

(d) to compute an optimal production plan for the first two quarters.