Our firm operates 5 high-end music and video equipment stores spread across a metropolitan area.  For a particular kind of speaker system, we estimate that each store sells an average of 3 units a month, with a Poisson distribution, and that sales each month at the 5 stores are independent.  The financial data are as follows:

• Each speaker system sells for $750
• Ordering the systems costs $300 per order, plus $500 per system ordered
• Delivering a system from a store to a customer costs you $20.

Each store has room for up to 20 units of inventory.  If a customer decides to buy a system, and his or her local store is out of stock, assume that you lose the sale to a competitor.  15 months from now, the manufacturer will replace the system with a new model, at which time you will sell your inventory to a discount dealer for $400 per unit. 

1. Assume a holding cost of $20 per unit per month (assessed on month-end inventories), and that all stores currently have an inventory of 3 units.  Find an optimal 15-month inventory stocking policy for each store (they are all identical).  What is the optimal profit for each store and for the full 5-store chain?
    
2. Instead of holding inventory at the stores, assume that you hold inventory at a central depot with room for 50 units.  Your unit holding costs are the same as in part (a), but it now costs $40 to deliver each speaker system.  Assuming a starting inventory of 15 units, develop an optimal 15-month stocking policy for the depot.  Is the total profit bigger or smaller than the total 5-store chain profit in part (a)?  Why?
    
3. Repeat parts (a) and (b), but instead of a holding cost of $20 per unit per month, assume a discount rate of 2% per month.