A company sells a product line of J items, each facing independent weekly normally distributed demands with mean and standard deviation . All shortages are backlogged. The company wants to maintain a service level of 95%, i.e. the likelihood of being out of stock should be 5% (or less). The manufacturing process consists of two steps: the first (second) step takes L1 (L2) weeks and the company follows an order-up-to policy.

(a) If each item is manufactured and managed by itself, give an expression for the expected inventory at the end of an ordinary period.

(b) Give an expression for the safety stock reduction factor that is achievable due to design for postponement, where in the first stage a common intermediate product is manufactured, but not stored, which is differentiated into one of the J final products, only in the second stage.

(c) Assume now that the product demands are correlated with a common correlation coefficient =0.2 or =0.8. Redo part (b) under these assumptions.

(d) In the absence of any fixed order costs, and an infinite horizon model , what specific order policy do you recommend when each item is manufactured independently, as in part (a) ? Is your recommended policy optimal? What is the expected aggregate inventory level at the end of any given period?

(e) In the absence of any fixed order costs, and an infinite horizon model , what specific order policy do you recommend under postponed differentiation , as in part (b) ? Is your recommended policy optimal? What is the expected aggregate inventory level at the end of any given period?

Please show all steps