Consider a discrete$-$time periodic review inventory model.

Let $\backslash$xi n be the total demand in period n with the following probability

distribution

Let Xn be the inventory quantity on hand at the end of period n$.$ Suppose

$($q$,$Q$)=(2,2)$ inventory rule is applied, and backordering is not allowed.

a$)$ Determine the transition probability matrix

b$)$ In the long run, during what fraction of periods are orders placed?

c$)$ If setup cost of placing an order is $$10,$ what is the expected setup

cost per period?

d$)$ Suppose for every item carried in the stocks from one period to the

next period costs $$1.$ What is the expected inventory holding cost per

period?

e$)$ Suppose for every demand lost, there is a cost of $$2.$ What is the

expected shortage cost per period?

$($Hint: first calculate the expected demand lost per period$)$