Suppose that we have a stochastic vector $\psi$ and $S$ realisations of $\psi$ given by $\psi_1,\dots,\psi_S$ with equal probability of occurrence. In addition, we have constraints of the form \begin{equation} h_i(x,\psi)\leq b_i,\quad \forall i=1,...,m \end{equation} for the decision vector $x$.
A joint chance constraint is then given by \begin{equation} P(\ h_i(x,\psi)\leq b_i , \quad \forall i=1,..,m\ )\geq \alpha \end{equation} stating that we can accept that some (or all) of these constraints are violation with a probability of $1-\alpha$. We could also write single chance constraints as follows \begin{equation} P(\ h_i(x,\psi)\leq b_i \ )\geq \alpha, \quad \forall i=1,..,m \end{equation} stating that we will accept violations of the individual constraints with a probability of $1-\alpha$. Using binary variables $z^s$ equalling 0 iff all constraints are satisfied in realisation $s$, we can formulate the joint chance constraint as the MIP \begin{align} h_i(x,\psi_s)\leq b_i+Mz^s,&&\forall i=1,...,m,s=1,...,S\\ \sum_{s=1}^Sz^s\leq \lfloor (1-\alpha)S\rfloor \end{align}
Using binary variables $z^s_i$ equalling 0 iff constraint $i$ is statisfied in realisation $s$ we can formulate the single chance constraint version as follows: \begin{align} &h_i(x,\psi_s)\leq b_i+Mz^s_i,&&\forall i=1,...,m,s=1,...,S\\ &\sum_{s=1}^Sz^s_i\leq \lfloor (1-\alpha)S\rfloor,&&\forall i=1,...,m \end{align}
My question is, what is the interpretation of the following MIP \begin{align} &h_i(x,\psi_s)\leq b_i+Mz^s_i,&&\forall i=1,...,m,s=1,...,S\\ &\sum_{i=1}^m\sum_{s=1}^Sz^s_i\leq \lfloor (1-\alpha)S\rfloor \end{align} Does it have some sensible interpretation?
