The average on-hand inventories under (R,Q) and (T,S) policies

Incredible Machines is a company that sells machinery to 6 distinct geographic mar- kets around the world. You are the inventory manager for the Super X milling machine. Weekly demand in each geographic market is normally distributed with mean 80 and standard deviation 20, and demands in each market are independent. The company fulfills all demand out of a single distribution center. Once an order is shipped from the supplier it takes four weeks to reach the distribution center. The company estimates the holding cost per unit inventory as $105 per week and also incurs a fixed cost of $60,000 per order. Assume you control the inventory in the distribution center with a continuous review (R,Q) policy. Lastly, assume that the company pays the supplier when its orders reach the distribution center and unsatisfied orders are backordered, i.e., the ownership of the in-transit (pipeline) inventory does not belong to Incredible Machines.

The company operates with a target of fulfilling 97% of orders from on-hand inventory. What reorder point R and order quantity Q should be used? What is the average on-hand inventory for Super X?
The company has the option of switching to a shipper that that reduces the lead-time from four weeks to three weeks. However, this shipping company charges $k extra per product. For what values of $k should the company switch to this shipping company?
Instead of operating a single distribution center, the company decides to hold inventory in each of the geographic markets. Assume that the company still uses an (R,Q) policy and it takes four weeks for orders to reach the warehouse in each geographic market. You are ordered to operate this distributed system with the same level of total average on-hand inventory as in part (a). What service level should the company anticipate being able to provide? Is the service level higher or lower? Why?
Assume the company keeps the centralized warehouse. If the company switches to a periodic review, order-up-to (T,S) policy with a review period of four weeks, what base-stock level S should be used?

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What is the average on- hand inventory in this case?

Compare the average on-hand inventories under (R,Q) and (T,S) policies: Which one is higher? Does this finding align with your intuition?

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