A company produces three types of items. A single machine is used to produce the three items on a cyclical basis. The company has the policy that every item is produced once during each cycle, and it wants to determine the number of production cycles per year that will minimize the sum of holding and setup costs (no shortages allowed). The following data are given:

P_i = number of units of product i that could be produced per year if the machine were entirely devoted to producing product i

D_i = annual demand for product i

K_i = cost of setting up production for product i

h_i = cost of holding one unit of product i in inventory for one year

a. Suppose there are N cycles per year. Assuming that during each cycle, a fraction 1/N of all demand for each product is met, determine the annual holding cost and the annual setup cost.

b. Let q_i* be the number of units of product i produced during each cycle. Determine the optimal value of N (call it N*) and qi*.

c. Let EROQi be the optimal production run size for product i if the cyclical nature of the problem is ignored. Suppose qi* is much smaller than EROQi. What conclusion could be drawn?

d. Under certain circumstances, it might not be desirable to produce every item during each cycle. Explain which of the following factors would tend to make it undesirable to produce product i during each cycle:

i) Demand is relatively low.

ii) The setup cost is relatively high.

iii) The holding cost is relatively high.