I have encountered a variant of the knapsack problem with shrinking items.
Effectively, it is a 0-1 knapsack problem where the initial weight of each item is $W(n)+V(n)$ and their value is $V(N)$, but immediately after being put into the container the item shrinks by $W(n)$. This means that for every item IN the knapsack, weight and value are the same. (It is therefore also possible to look at this as a value-irrelevant "best fill only" 0-1 knapsack.)
So, in this case, one also needs to determine the proper order of items to insert and whether a specific item is legal in the combination before being put in the container.
I have already determined that order of insertion, given a valid collection, should be based on $W(n)+V(n)$ descending—but thus far I am failing to create a set of valid collections in anything under $O(2^n)$.
Dynamic programming solutions are failing me both because of the apparent interdependence of elements which makes memoization seemingly impossible.
I realize I have not given much here, but I could really use some help. How does one go about approaching this variant?
If curious, the actual application is that we have a large number of different electrical appliances that all cause a known power surge when turned on and afterwards draw a set amount of power. We never want to trip a breaker, so the surge needs to be taken into account when figuring out how many at most we can turn on.