Suppose I have $ n$ items, each with value $ v(j)$ and weight $ w(j)$ , and $ m$ knapsacks each with capacity $ c(i)$ . If I make the assumption that $ w(j-1)$ evenly divides $ w(j)$ , then there’s a nice optimal packing algorithm outlined in Detti, A polynomial algorithm for the multiple knapsack problem with divisible item sizes.
I have a slight variant to this problem, where the value depends on the knapsack I put the item in: $ v(j,i)$ . Is there any paper or book detailing an exact optimal solution to this problem? In my case $ v(j,i)$ takes at most 2 distinct values, but I’m not sure if that matters.