Problem:
Given an unlimited amount of empty boxes and a limited amount of items N, each with a specific value but not necessarily an unique value. A box has to be filled with at least 1 item and the total value of the items in the box has to be at least a given value Vmin to count as a filled box F. The value of a filled box minus the minimum value Vmin for a box is called the overvalue of a box denoted by O. The sum of overvalues of all filled boxes is denoted by TO
Example:
Items (N = 5):
Item 1: 1.97
Item 2: 2.32
Item 3: 4.18
Item 4: 6.76
Item 5: 8.05
Each box has to be filled with at least 1 item. The total value of items in the box has to be at least 5.00 (Vmin = 5.00).
Filled boxes: (F = 3)
box 1: item 1 and 3 (box value: 6.15, overvalue O: 1.15)
box 2: item 4 (box value: 6.76, overvalue O: 1.76)
box 3: item 5 (box value: 8.05, overvalue O: 3.05)
The maximum possible amount of filled boxes F is obviously 3. The total overvalue TO is equal to 1.15 + 1.76 + 3.05 = 5.96.
Question:
Given a considerably large amount of items. How can one determine the maximum amount of filled boxes F while having TO as small as possible?
