Brown (1967) has suggested a procedure for allocating a total order (of size W dollars)

among a group of n items (i = 1, 2, ... , n). A particular time interval (perhaps until the next order arrives) of duration T is considered. The allocation is made so that the probability that item i runs out during T is proportional to the fraction of the group’s sales that item i contributes. (Brown shows that this procedure, at least approximately, minimizes the total stock remaining at the end of the interval.) Suppose that it is reasonable to assume that demand in period T for item i is normally distributed with mean xˆi and standard deviation σTi .

a. Introducing whatever symbols are necessary, develop a routine for allocating the order among the items according to the above criterion.

b. Illustrate for the following 3 item example, in which W = $900:

Initial

(Before Allocation)

xˆi σT vi Inventory Ii Item i (Units) (Units) ($/Unit) (Units)

1 100 40 1.00 50 2 300 70 2.00 100 3 250 100 1.20 250

c. Repeat part b but with I1 = 250 I2 = 150 I3 = 0?  P-96