Let $ A_j=\{(a^i_j,b^i_j)~:~ 0 \leq i \leq n,\text{and } a^i_j,b^i_j \in \mathbb{Z}^+\}$

Given sets $ A_1,\ldots, A_{p}$ and a positive integer $ k$ , the problem is to check whether there exists one element $ (a^{i_j}_j,b^{i_j}_j)$ from each $ A_j$ such that $ \sum_{j}^{} a^{i_j}_j \geq k$ and $ \sum_{j}^{} b^{i_j}_j \geq k$ .

It looks like the problem is related to set partitioning problem, however, I am not sure how to get a reduction from set partitioning problem. Can someone help me to find the algorithm to solve this problem?