The demand for a particular spare part appears to be decreasing rapidly. Any demand occurs only in the month of December each year. Because of warranty considerations, the company has guaranteed the availability of spare parts through to (and including) the 2017 year.

The inventory of the item in mid-November 2016 is 10 units. The demands in December 2016 and December 2017 are assumed to be independent random variables x and y with probability mass functions as follows:

x0 0 10 20 30 px(x0) 0.2-.-

0.4 0.3 0.1 y0 0 10 20 py(y0) 0.4 0.5 0.1 There is a setup cost of $50 associated with producing any units of the item. The unit variable cost of a produced piece is $20/unit and it costs $4.50 to carry each unit in inventory from December 2016 to December 2017. In any month of December when the total demand exceeds the stock available, additional spare parts are purchased from a competitor at a cost in dollars of 60 + 35z, where z is the number of units purchased (i.e., the size of the shortage). Any unit left over at the end of December 2017 will have a salvage value of $5/unit. Production each year must take place before the December demand is known.

a. Suppose at the beginning of December 2017 there were no units in stock. Determine the quantity that should be produced at that time and the associated expected costs.

b. What are two complicating factors that make a standard newsvendor analysis inappropriate for determining the best quantity to produce prior to the December 2016 season (i.e., in late November, 2016).

c. Briefly outline (but do not attempt to carry out any of the numerical details of ) a method for determining the best quantity to produce prior to the December 2016 season.