onsider a product produced from raw material using a single machine as shown in the figure below. The machine has production rate p 10,000 units per year and the product has a constant demand rate D 5,000 units per year. At the beginning of a production cycle a batch of raw material is delivered to the stocking point 1 from which the machine is directly fed. The output from this machine is immediately (unit for unit) transferred to stocking point 2 which faces the constant customer demand. The ordering cost of the raw material is 1 A = $300. The machines have setup costs 2 A =$50. The unit variable cost of the raw material is $3 1 v . After being processed in the machine, the unit variable cost increases to $30 2 v . The carrying charge has been established as 0.25 $/$/year. Assume one unit of raw material is used to produce one unit of finished good. i) Using graph, show the inventory history of both the raw material and that of the finished product. Write a total relevant cost expression and determine the optimal order quantity. ii) Taking the fact that the raw material has a larger ordering and a smaller unit carrying costs than that of the finished product, the production manager suggested that the order quantity of the raw material should be large enough to cover two consecutive production runs. Write a total relevant cost expression and determine the corresponding optimal order quantity following the suggestion of the production manager. Also determine the cost saving that may result by following his/her suggestion. (Using graph, show the inventory history of both the raw material and that of the finished product.) iii) Assume you have been hired as an inventory analyst in this company and you think that the suggestion given by the production manager is just an intuitive and not based on analytical analysis. Rather, you decided to determine an integer number k such that the replenishment quantity of the raw material is equal to the replenishment quantity of the finished product times this integer number. Write the total relevant cost expression and drive a relation for the optimal value of replenishment quantity of the finishes product as the function of the unknown integer value k.