Markov Chain Problem Each month, customers are equally likely to demand 1 or 2 c
ID: 3042670 • Letter: M
Question
Markov Chain Problem
Each month, customers are equally likely to demand 1 or 2
computers from a dealer. All orders must be met from current stock. Two ordering policies
are under consideration:
(a) Policy 1 If ending inventory is 2 units or less, order enough to bring next month's
beginning inventory to 4 units.
(b) Policy 2 If ending inventory is 1 unit or less, order enough to bring next month's begin-
ning inventory up to 3 units.
2
The following costs are incurred by dealer:
It costs $4,000 to order a computer.
It costs $100 to hold a computer in inventory for a month.
It costs $500 to place an order for computers. This is in addition to the per-customer
cost of $4,000.
Which ordering policy has a lower expected monthly cost?
Hint: You need to determine the long-run cost of each policy separately and then choose the
one with lower average cost per year using the limiting probabilities. Each policy will induce
its own Markov chain.
Explanation / Answer
Cost per unit, Cp = 4000
Order placing cost, Co = 500
Carrying cost per unit per year, Cc = 1200
policy 1
Total number of orders per year = 12
beginning inventory = 4
ending inventory = 2
Average inventory maintained at any point beginning + ending inventory / 2 = 3
Total Inventory management cost = Ordering costs+ Carrying costs
carrying cost for a year = Cost to hold one unit inventory for a year * average inventory maintained = Cc* 3 = 3600
Ordering cost = 6000
Inventory cost= Yearly Carrying Cost + Yearly Ordering Cost = 9600
Policy 2
Total number of orders per year = 12
Beginning inventory = 3
Ending inventory = 1
Average inventory maintained at any point beginning + ending invenotry / 2 = 2
Total Inventory management cost = Ordering costs+ Carrying costs
Carrying cost for a year = Cost to hold one unit inventory for a year * average inventory maintained =Cc* 2 = 2400
Ordering cost = 6000
Inventory cost= Yearly Carrying Cost + Yearly Ordering Cost = 8400.
Hence, policy 2 with low inventory costs is better.