Multiple knapsack problem with equal profit and different weight I am doing a research about the load balancing problem in 5G system, but I am not sure if my problem is a NP-complete problem.

The problem is:


given a set of n items and a set of m knapsack
capacity of knapsacks are equal
the weight of item j in knapsack i is w[i][j],that means weight of a item in each knapsack are different
each profit of items are equal


I am not trying to put all item in least knapsack like bin packing problem.
I saw some similar question answered, but no one is identical to this case.
In this case, the goal is to put as more as possible item with m knapsacks.
Is the problem a NP-complete problem?
This problem can be shown to be NP-complete via reduction from PARTITION. Simply take $m=2$, the weights of each item to be the same across both knapsacks, and the capacities of each knapsack to be half the total weight across all items.