Consider a "reorder-point/order-up-to" type of inventory control system, sometimes referred to as ($, S ) Inventory control system. Suppose that the inventory is counted at the end of the week (Saturday evening), and if s is 3 or fewer items remain, enough is ordered to bring the level up to S=8 before the business reopens on Monday morning. That is, it is assumed that an order may be placed and is instantaneously received. The probability distribution of demand is the same each week: P{D=0} = 0.10 P{D=1} = 0.15 P{D=2} = 0.25 P{D=3} = 0.25 P{D-4} = 0.15 P{D=5) = 0.05 P{D=6} = 0.05.

(a) Construct a one-step transition probability matrix using the state of the system which is defined according to the stock-on-hand (SOH) at the end of the week (Saturday evening) before replenishment occurs. Assume that backorders are not allowed. 

(b) Construct a one-step transition probability matrix under the assumption that demand which cannot be satisfied immediately is backordered. 

(c) Compare the two transition probability matrices from parts (a)

(d). Discuss what has been changed and why?