Knapsack, but divided by summation

Question

For a given set $S = \{1, 2, ... , N \}$, each component $i\in S$ can be represented by $(a_i, b_i, c_i, w_i)$. Is there any technique for solving the following problem?

$$\max_{S' \subseteq S} \frac{ \left(\sum_{k\in S'} a_k \right) \cdot \left(\sum_{k\in S'} b_k \right) }{\left(\sum_{k\in S'} c_k \right)}$$ subject to $$\sum_{k\in S'} w_k \leq C. $$

If the objective function is not divided by $\left(\sum_{k\in S'} c_k \right)$, it's QUADRATIC-KNAPSACK, which can be solved. How can this be solved?

Answer 1

First linearize the numerator as shown in my answer to NP hard (like KNAPSACK)- any approximation scheme?. Then apply Charnes-Cooper as shown in my answer to Doing a Charnes-Cooper transformation with matrices and an zero-one constraint.