I would be thankful if anyone can help me to find the best approach to solve the following MINLP model. In this model, $p_{js}$, $p_{js}$ and $Q$ are parameters. $x_{js}$ is a binary variable and $y_i$ is a continuous variable between 0 and 1 (Note the variable $y_i$ shows how much the right-hand-side of the $i^{th}$ constraint can be tight).
$maximize ~~ \sum_{j \in \mathcal{J}}\sum_{s \in \mathcal{S}} p_{js} x_{js} $
Subject to:
$\sum_{j \in \mathcal{J}}\sum_{s \in \mathcal{S}} q_{ijs} x_{js} \leqslant Q^{y_i} ~~~~~~~~~~~~~~~~~~~~~ \forall i \in \mathcal{I}$
$\sum_{s \in \mathcal{S}} x_{js}=1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \forall j \in \mathcal{J}$
$\sum_{i \in \mathcal{I}} y_{i}=1$
$x_{js} \in \lbrace 0,1 \rbrace ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \forall j \in \mathcal{J}; s \in \mathcal{S}$
$y_{i} \in (0,1) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \forall i \in \mathcal{I} $
For example, the simplest case is when we only have $y_1$ and $y_2$ as follows:
$maximize ~~ \sum_{j \in \mathcal{J}}\sum_{s \in \mathcal{S}} p_{js} x_{js} $
Subject to:
$\sum_{j \in \mathcal{J}}\sum_{s \in \mathcal{S}} q_{1js} x_{js} \leqslant Q^{y_1}$
$\sum_{j \in \mathcal{J}}\sum_{s \in \mathcal{S}} q_{2js} x_{js} \leqslant Q^{y_2}$
$\sum_{s \in \mathcal{S}} x_{js}=1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \forall j \in \mathcal{J}$
$y_{1}+y_{2}=1$
$x_{js} \in \lbrace 0,1 \rbrace ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \forall j \in \mathcal{J}; s \in \mathcal{S}$
$y_{1} \in (0,1) ~~ and ~~ y_{2} \in (0,1) $
What is coming to my mind is that there should be a method to remove $y_i$ from the model. Any help?