I have three binary variables $x_{i,j}^{m,r}$ , $y_i^{m,r}$, and $z_i^{m,r}$. There is another integer variable $w_i^r$. And I want to linearize the following logic:

$$ \sum_{m} x_{i,j}^{m,r} \ge 1 \implies w_j^r = w_i^r + \sum_{m} y_i^{m,r} - \sum_{m} z_i^{m,r} \qquad \forall r, i, j $$

I think to linearize the above I need to introduce another binary. But could we do it without any new variables?

I have three binary variables $x_{i,j}^{m,r}$ , $y_i^{m,r}$, and $z_i^{m,r}$. There is another integer variable $w_i^r$. And I want to linearize the following logic:

$$ \sum_{m} x_{i,j}^{m,r} \ge 1 \implies w_j^r = w_i^r + \sum_{m} y_j^{m,r} - \sum_{m} z_j^{m,r} \qquad \forall r, i, j $$

I think to linearize the above I need to introduce another binary. But could we do it without any new variables?

I have three binary varibles $x_{i,j}^{m,r}$ , $y_i^{m,r}$, and $z_i^{m,r}$. There is another integer varible $w_i^r$. And I want to linearize the following logic:

$$ \sum_{m} x_{i,j}^{m,r} \ge 1 \implies w_j^r = w_i^r + \sum_{m} y_i^{m,r} - \sum_{m} z_i^{m,r} \qquad \forall m,r , i, j $$

I think to linearize the above I need to introduce another binary. But could we do it without any new variables?

I have three binary variables $x_{i,j}^{m,r}$ , $y_i^{m,r}$, and $z_i^{m,r}$. There is another integer variable $w_i^r$. And I want to linearize the following logic:

$$ \sum_{m} x_{i,j}^{m,r} \ge 1 \implies w_j^r = w_i^r + \sum_{m} y_i^{m,r} - \sum_{m} z_i^{m,r} \qquad \forall r, i, j $$

I think to linearize the above I need to introduce another binary. But could we do it without any new variables?