# Reformulate constraints

I have the following constraints and am wondering whether I can formulate the whole thing more narrowly and with fewer constraints. $$x_{itk}$$ is binary and $$u_{it}, v_{itk}\in [0,1]$$. $$M$$ is a Big-M constant. These are my constraints.

\begin{align} &v_{itk} \geq u_{it} - M\cdot (1-x_{itk})& \forall i \in I, t\in T, k\in K \\ &v_{itk} \leq u_{it} + M\cdot (1-x_{itk})& \forall i \in I, t\in T, k\in K\\ &v_{itk} \leq x_{itk}& \forall i \in I, t\in T, k\in K \end{align}

• Does this answer your question? How to linearize the product of a binary and a non-negative continuous variable? Commented Apr 21 at 14:24
• If you really want fewer constraints, you can instead impose the nonlinear constraints $v_{itk}=u_{it}x_{itk}$ or omit these constraints and just replace $v_{itk}$ throughout the rest of your model. Commented Apr 21 at 14:28

It looks like You want to enforce for all $$i,t,k$$: $$x_{itk}=1 \implies v_{itk} = u_{it} \quad$$ and $$x_{itk}=0 \implies v_{itk} =0$$
\begin{align} x_{itk} + u_{it} - 1 &\le v_{itk} \le u_{it} \quad &\forall i,t,k\\ 0 &\le v_{itk} \le x_{itk} \quad &\forall i,t,k\\ \end{align}
To be more precise, it is equivalent to OP's formulation with $$M=1$$ from the first constraint, and $$M=0$$ from the second constraint.