I have $i$ items and I should pre-packed $m$ knapsacks with identical items where only $K<n$ items can be packed. Also, we should have only one of each item in each sack. The time capacity for knapsacks preparation is limited to $C_{m}$ for each sack. The $t_{i}$ is the packaging time of item $i$. The objective is to minimize the packaging costs. I believe it is a multiply constrained knapsack problem and it is also bounded. However, It has some similarities and differences with bin packing problem as well. I am wondering if this problem is a variant of knapsack problem. Is it still NP-hard? If so, how can I prove it? Thanks.
The formulation is:
$$\min \sum_{i \in I}c_{i}z_{i}x_{m} $$ $$\sum_{i \in I}z_{i} \le K $$ $$\sum_{i \in I} t_{i} x_{m} \le C_{m} \hspace{0.35in}\forall m \in M $$ $$x_{m} \le u \hspace{0.35in}\forall m \in M $$ $$x_{m} \in \mathbb{Z}_{+} \hspace{0.35in}\forall m \in M $$ $$z_{i} \in \{0,1\} \hspace{0.35in}\forall i \in I $$