Business Decision Analytics under Uncertainty (33:136:400) 
Professor Eckstein
An Inventory Problem with Poisson Demands, Stock-Outs, and Present Value

Your company is about to introduce this year's model of a product, which you expect to offer for sale for the next twelve months.  You have no inventory of the product at present.  You have capacity to hold up to 50 units of the product in inventory, and you can produce up to 45 units per month.  Each month you produce the product, you incur a $10,000 setup cost, which does not apply to months in which you have zero production.  At the beginning of each month, you decide your production level for that month, and you cannot alter your decision.  Your direct cost of producing the product is $2,000 per unit, and you sell it for $3,000 per unit.  

You do not know the exact demand for the product over the coming year.  However, from an analysis of past orders, you have the following estimates of the expected (average) level of demand:

Month 1 2 3 4 5 6 7 8 9 10 11 12
Average Demand 10.0 10.5 11.1 12.2 12.3 12.4 11.8 10.9 9.8 9.0 8.5 8.1

You have a large customer base, and each of your customers tends to order products of this type infrequently.  To make way for next year's model, all units in stock at the end of month 12 will immediately be sold to "" for $1800 per unit.  You do not incur direct holding costs for keeping inventory, but your firm's internal cost of funds is 0.5% per month, compounded monthly.

Customer demand for a given month may be satisfied from that month's starting inventory and that month's production.  Any sales above this amount are lost, with customers buying your competitors' products instead.  Your policy is not to produce any amount of stock that risks overflowing your inventory capacity.  Your management is unwilling to incur any risk, no matter how small, of overflowing the capacity of the inventory storage facility.

  1. Why is a Poisson model appropriate for each month's demand?
  2. Write a computer program that uses dynamic programming to determine the production policy that will optimize your profit for the year.
  3. How does the optimal policy change if we assume that the internal cost of funds is 10% per month?  (Note: this is a very high rate, but should illustrate how the discount rate affects the results of these calculations.)
  4. Suppose that you are willing to run some risk of overflowing your inventory capacity.  If an inventory overflow occurs, you immediately sell any surplus to for the same $1800 per unit price as at the end of the year.  Modify your program to account for this possibility (returning to the 0.5% discount rate).  How do the optimal EMV and policy change?