Logical Constraints Modelling using Big-M formulation

Question

I am trying to model some logical constraints in ILOG. Logical constraints could be given such as:

• Constraint 1 or Constraint 2,

• Constraint 3 or Constraint 4,

• Constraint 5 or Constraint 6.

The six constraints in question are listed below. \begin{align}&\,\,\sum_{s=1}^Sx_{is}=1\quad\forall i\in T\\\text{Constraint}\,1:&\,\,\sum_{s=1}^Ss\cdot x_{is}-\sum_{s=1}^Ss\cdot x_{js}&=0&\quad(i,j)\in\text{linked}\\\text{Constraint}\,2:&\,\,\sum_{s=1}^Ss\cdot x_{is}-\sum_{s=1}^Ss\cdot x_{j(2\cdot m-s)}&=0&\quad(i,j)\in\text{linked}\\\text{Constraint}\,3:&\,\,\left\vert\sum_{s=1}^Ss\cdot x_{is}-\sum_{s=1}^Ss\cdot x_{js}\right\vert&\ge d_1&\quad(i,j)\in\min\\\text{Constraint}\,4:&\,\,\left\vert\sum_{s=1}^Ss\cdot x_{is}-\sum_{s=1}^Ss\cdot x_{j(2\cdot m-s)}\right\vert&\ge d_1&\quad(i,j)\in\min\\\text{Constraint}\,5:&\,\,\left\vert\sum_{s=1}^Ss\cdot x_{is}-\sum_{s=1}^Ss\cdot x_{js}\right\vert&\le d_2&\quad(i,j)\in\max\\\text{Constraint}\,6:&\,\,\left\vert\sum_{s=1}^Ss\cdot x_{is}-\sum_{s=1}^Ss\cdot x_{j(2\cdot m-s)}\right\vert&\le d_2&\quad(i,j)\in\max\end{align} Only one of the constraints in each group should be satisfied and active, i.e if Constraint 1 is active, Constraint 2 should not be active, or vice versa.

I have tried some logical constraints definition method, e.g. Big M method, but I could not define the constraints and run the model. Since there are too many sum functions in my model, it is very challenging to build a big-M model. So I need your help.

I would appreciate it if you have any suggestions. Thank you in advance. Regards.

Answer 1

You can enforce constraints 1 and 2 by instead imposing $x_{i,s}= x_{j,s}+ x_{j, 2m-s}$. For the other four, you can impose no-good constraints of the form $$x_{i,s}+ \sum\limits_{(j,t)\in D_{i,s}} x_{j,t}\le 1,$$ where $D_{i,s}$ is the set of disallowed assignments for $j$ if $i$ is assigned to $s$.