# Is there a preference to use Convex-hull reformulation instead of the BigM constraints?

I am trying to work on a scheduling problem based on its polyhedron reformulations. For that, I would like to reformulate a BigM model into its equivalent C-hull formulation. The transforming map is depicted in the following picture:

As an example, suppose there is a simple scheduling model:

\begin{align*} \text{minimize} \quad & s_1 + s_2 \\ \text{subject to} \quad & LB \leq s_1 \leq UB \\ \ & LB \leq s_2 \leq UB \\ \ & (s_1 \geq s_2+\alpha) \lor (s_2 \geq s_1+\beta)\\ & s_1, s_2 \geq 0. \end{align*}

It's linearized as:

\begin{align*} \text{minimize} \quad & s_1 + s_2 \\ \text{subject to} \quad & LB \leq s_1 \leq UB \\ \ & LB \leq s_2 \leq UB \\ \ & s_1 - s_2 + (M+\beta)y_1 \leq M \\ \ & s_2 - s_1 + (M+\alpha)y_2 \leq M \\ \ & y_1 + y_2 = 1 \\ & s_1, s_2 \geq 0, y_1, y_2 \in \{0,1\}. \end{align*}

and its C-hull equivalent would be:

\begin{align*} \text{minimize} \quad & s_1 + s_2 \\ \text{subject to} \quad & s_1 = w_1 + w_2 \\ \ & s_2 = w_3 + w_4 \\ \ & LB \leq s_1 \leq UB \\ \ & LB \leq s_2 \leq UB \\ \ & y_1 + y_2 = 1\\ \ & w_1 - w_3 + \beta y_1 \leq 0\\ \ & w_4 - w_2 + \alpha y_2 \leq 0\\ \ & w_{i^s} - UBy_{i^d} \leq 0 ,\ \forall i \in I \\ & s_1, s_2 \geq 0, y_{i^d} \in \{0,1\}, w_{i^s} \geq 0. \end{align*}

Now, my questions are:

• Is it proven that a C-hull formulation always has a better relaxation than a BigM or time-indexed formulation?

• Is the time-indexed formulation a kind of C-hull reformulation?

• Can we infer a C-hull formulation may lead to reducing the solving process in an MIP?

• I have tried to solve a large instance of the above formulation. In that case, it seems the BigM model has a better performance against the C-hull formulation. May it back to the internal mechanism of the modern MIP solvers to deal with BigM formulation rather than whose C-hull model?

• What actually is the benefit of the convex hull reformulation if we again need to deal with BigM constraints? Does it really provide a tighter relaxation? If so, why in large instances the BigM formulation seem to solve faster than whose c-hull?