This is (an example of) an initial flow network for your example.

If you repeat finding the max-flow and then subtracting off the items until you get zero items, the number of times you repeat Ford–Fulkerson is your answer. However, for Ford–Fulkerson, this can be simplified to,

That is, it doesn't matter what the strategy for individual items is; what matters is the sum. Thus you should be able to solve this problem in O(number of items).