IE 423 HW#3 page 1 of 1

IE 423 Engineering OR II
Homework #3
Due Thursday, March 3, 2016

Part I: (s,S) Inventory without Backorder: Consider a reorder-point/order-up-to type of
inventory control system, sometimes referred to as ( s , S ) Inventory control system.
Suppose that the inventory is counted at the end of the week (Saturday evening), and if s is 2 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 day:
P{D=0}= 0.15 P{D=1}= 0.25 P{D=2}=0.1 P{D=3}= 0.5
(a) Construct a one-step transition probability matrix using the following 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 (3 points).
Xn = 1 2 3 4 5 6 7 8 9
SOH = 0 1 2 3 4 5 6 7 8
(b) Over a long period of time, what is the percent of the weeks in which you would expect there
to be a stockout (zero inventory)? (2 points)
Part II: (s,S) Inventory with Backorder:
At the beginning of each day, a company observes its inventory level. Then an order maybe
placed (and is instantaneously received). The probability distribution of demand is the same
each day:
P{D=1}= 0.3 P{D=2}= 0.2 P{D=3}=0.15 P{D=4}= 0.35
If the on-hand inventory is 1 unit or less, enough is ordered to bring the on-hand inventory level
up to 5.
(a) Construct a one-step transition probability matrix using the following state of the system
which is defined according to the stock-on-hand (SOH) at the beginning of each day before
replenishment occurs (3 points).
State = 1 2 3 4 5 6 7
SOH = -2 -1 0 1 2 3 4
(b) Over a long period of time, what is the percent of the days in which backorders occur?
(2 points)