Reformulate constraints

Question

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}

Answer 1

It looks like You want to enforce for all $i,t,k$: $$ x_{itk}=1 \implies v_{itk} = u_{it} \quad $$ and $$ x_{itk}=0 \implies v_{itk} =0 $$

You can get rid of the big-M as follows:

\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.