manufacturer uses a particular product at a constant rate of 10,000 pieces per year, and wants to produce this product with a setup cost of $5,000 which is incurred with every lot he produces. Manufacturer's oroduction rate is 20,000 pieces per year, and the unit manufacturing cost (independent from the setup cost) is $40 per piece. The annual inventory carrying cost is $8 per unit, and the manufacturer wants no shortages of the part (i.e. stockouts are not allowed). (a) What is the optimal production quantity? What is the optimal cycle time? What is the optimal number of production runs in a year? What is the optimal length of a production run? What is the average inventory level if the optimal production quantity is used? (b) What is the optimal total annual cost (including setup, inventory carrying and manufacturing costs)? (c) On a graph, illustrates the change in inventory level over time for 2 cycles. Indicate the values you found in part (a) on the graph. (d) Suppose that the manufacturer has a warehouse where the maximum number of pieces that can be stored is at most 2,000 pieces at any point in time. Check whether the capacity of the warehouse is enough to apply the optimal policy determined in part (a) for the unconstrained problem. Using the Lagrange Multiplier method, determine the optimal production quantity for the production alternative in which the warehouse constraint is active