Is working with equations in an MILP more efficient than working with inequaltities?

I have the following model \begin{align}\max&\quad B_{1} + B_{2}\\\text{s.t.}&\quad 0 \leq B_{1} \leq c_{1}\cdot Y_{1} \\&\quad 0 \leq B_{2} \leq c_{2}\cdot Y_{2} \\&\quad \end{align} where $$B_{1}, B_{2}$$ are reals, $$Y_{1}, Y_{2}$$ are binary, and $$c_{1}, c_{2}$$ are positive constants.

Is the following transformation of advantage when thinking about speeding up B&B?

\begin{align}\max&\quad B_{1} + B_{2} - M_{1} - M_{2}\\\text{s.t.}&\quad B_{1} + M_{1} = c_{1}\cdot Y_{1} \\&\quad B_{2} + M_{2} = c_{2}\cdot Y_{2} \\&\quad\end{align} where $$B_{1}, B_{2},M_{1}, M_{2}$$ are reals, $$Y_{1}, Y_{2}$$ are binary, and $$c_{1}, c_{2}$$ are positive constants.

• Let $M_1$ be negative. Now we improve the objective and violate the original inequality by as muih as we want. How is that equivalent? Even if $M_1$ is constrained to be nonnegative, then what happened to $0 \le B_1$, and how is penalizing the slack ($M_1$)in the objective a "proper" thing to do? Of course, same with $M_2, B_2$. May 2 '20 at 12:25
• You cannot get your exact model by eliminating the given inequalities using the equalities proposed by you. May 2 '20 at 17:08