Bi Knapsack Optimisation Problem

Definition

Bi Knapsack Optimisation Problem

Instance: Indexed items $1,\dots ,n$ with positive integer sizes $(a_i,b_i)$ and nonnegative integer capacities $S_1,S_2$, encoded in binary.

Feasible solutions: Index sets $J\subseteq \{1,\dots ,n\}$ with
${\sum }_{i\in J}{a}_i\le S_1,{\sum }_{i\in J}{b}_i\le S_2.$

Objective: Maximise $|J|$, the number of selected items.
Each selected item consumes capacity in both dimensions simultaneously. Unlike Two-Bag Knapsack, there is no choice of which bag receives an item. Equal size pairs at different indices remain distinct items.