Let's say that I know that:

$\displaystyle \sum_{i,j} a_{ij} \leq r$, where $\displaystyle 0 \leq a_{ij} \leq 0.5$ and $\displaystyle r \geq 0$ and $\displaystyle 1 \leq i,j \leq n$

and I also know that

$\displaystyle \sum_{i,j} y_{ij} \leq d$, where $\displaystyle y_{ij} \geq 0$ and $\displaystyle d \geq 0$ and $\displaystyle y$ is zero-diagonal and $\displaystyle 1 \leq i,j \leq n$

What do we know about?

$\displaystyle \sum_{i,j} a_{ij} y_{ij} \leq ?$

So far the best I can prove is

$\displaystyle \sum_{i,j} a_{ij} y_{ij} \leq \frac{d}{2}$

Is there a tighter bound?