How to solve a knapsack problem with increased weight limit?
That guess is obviously wrong. If all items have weight 666 and value 1, then you can fit 3 items into a bin of size 2000, and only one into a bin of size 1000. 

If all weights are W &lt; $w_i$ ≤ 2W, then O = 0 and O' = highest value of a single item. 

There is very little that you can say except the obvious O ≤ O'. You might have "valuable" items with a total weight ≤ W, and all other items are worthless. Then O' is not or only very little greater than O. Or all items with weight ≤ W might be worthless with some items with weight W &lt; $w_i$ ≤ 2W very valuable, then the ratio O' / O can be arbitrarily large.