So, in an ordinary MCK problem, there is just one weight constraint, namely, the weight constraint of the knapsack. I have encountered a special variation of the MCK problem, in which there are more than one weight constraints. Conceptually, this can be thought of as if there are multiple chambers in the knapsack, and each chamber has its own weight constraint. An assumption can be made that all the chambers have the same weight constraint. When taking an item from a class, this item will occupy just one chamber, or it can occupy more than one chambers. Everything else is exactly a MCK problem.
I was wondering if anyone came across similar problems before? If you did, could you please point me to some literature on its analysis or approximation solutions?