# Modelling stronger binary expression

$$\delta_1, \delta_2, ..., \delta_k, W$$ are binary variables and the constraint $$δ_1 + δ_2 + \ldots + δ_k ≤ W$$ holds.

Is it better to write $$\delta_1 + \delta_2 + \ldots +\delta_k \leq W$$ or $$\begin{gathered} \delta_1 \leq W \\ \delta_2 \leq W \\ ...\\ \delta_k \leq W \\ \end{gathered}$$

and why?

Since $$W$$ is a binary variable, it follows that $$\sum_k \delta_k \le W \le 1$$ And so you are in the presence of a clique constraint.
A simple example : take $$\delta_k = 0.9$$ for every $$k$$. It is easy to see that such a solution is valid with the second group, but not with the first one. So the first one leads to a tighter relaxation.