Given a product set $P$, where each product $p_i \in P$ has a cost $p_i.c$ and a value $p_i.v$. Therefore, $\forall p_i \in P$, $p_i.c > 0$ and $p_i.v > 0$. The cost-efficient of a product is defined as $\frac{p_i.v}{p_i.c}$. Given a subset of product $P' \subseteq P$, the cost-efficient of $P'$ is defined as $\frac{\sum_{p_i \in P'} p_i.v}{\sum_{p_i \in P'} p_i.c}$.
My question is that, given a fixed integer $1 \leq k \leq |P|$, for any $P' \subseteq P$ such that the cardinality $|P'|=k$, what is the maximum cost-efficient?
Eventually, this problem is to simultaneously maximize the total value while minimizing the total cost. It has been shown to be an NP-hard problem to get the exact maximum cost-efficient without the k-size constraint. My question is more simple, as I only want to get the rough upper-bound with the k-size constraint.
Let $v_i = p_i.v$ and $c_i = p_i.c$. You can solve the problem via integer linear programming as follows. Let binary decision variable $x_i$ indicate whether product $p_i$ is selected to be in $P'$. The problem is to maximize $$\frac{\sum_{i\in P} v_i \cdot x_i}{\sum_{i\in P} c_i \cdot x_i} \tag1$$ subject to linear constraint $$\sum_{i\in P} x_i = k.$$ You can linearize the fractional objective $(1)$ via a Charnes-Cooper transformation. The LP relaxation (even without linearizing the product of binary and continuous variables) then yields an upper bound.
For a cheaper but cruder upper bound, you can divide the sum of the largest $k$ values of $v_i$ by the sum of the smallest $k$ values of $c_i$.