XYZ sells cups of organic peanut butters at the local market in Mexico City on Mondays. He is equally likely to sell 200 or 400 cups at each week. Each time XYZ places an order, he pays $600 plus $5 for each cup he orders. Each cup sells for $12. A holding cost of $3 per cup is assessed against each cup left at the end of a market day. XYZ can store at most 400 cups after market. Assume that the number of cups ordered by XYZ must be a multiple of 100, and any leftover cups have a value of $8. Also assume that he begins his first market day with no inventories on hand.

To determine an ordering policy that maximizes expected profits earned during his first three market days via dynamic programming let:

ft(x) = maximum profit earned during the market days t, t + 1, ..., 3 given that x cups are on hand at the beginning of t th market day (before an order is placed).

Here, x may equal 0, 100, 200, 300, or 400. We assume that before placing an order for 1 st market day no cups are on hand.

a) Let c(0) = 0 and for s>0, c(s) = 600 + 5s where s is the amount cups that are ordered before market day t. and c(s) is the related cost. Then:

f3(x) = max {-c(s) - 1/2*[3*max(0, x + s - 200) + 3*max(0, x + s - 400)]+ 1/2*[8*max(0, x + s - 200) + 8* max(0, x + s - 400)] + 1/2*[12*min(x + s, 200) + 12*min(x + s, 400)]}

and s must satisfy x + s 200 400 which is equivalent to s 600 - x.

Explain the formula for calculating f3(x) in words explicitly. Also explain the reason why s must satisfy the inequalities s 600 - x.

b) What does f3(0) mean?

c) For the first stage what will be the appropriate state (or states)? Why?