# Logical Constraints Modelling using Big-M formulation

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.

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 $$s$$.
• RHS is $d_1$ for constraints 3-4 and $d_2$ for constraints 5-6. RHS $\le1$ means the only variable is equal to $1$. E.g. if $d_1$ is $3$ these constraints won't be suitable. The aim of these constraints is that the gap of assignment places between two related variables should be $\min d_1$ and $\max d_2$. E.g. if the gap minimum between variable $x_1$ and $x_7$ is $2$ the assignment $x_{15}=1$ and $x_{71}=1$ is proper but if $x_{15}=1$ and $x_{74}$ this is not proper as the assignment place of $x_1$ is $5$ and $x_7$ is $4$ and the gap is $5-4=1<2$. A similar situation is valid for maximum gap. – memop Oct 18 '19 at 16:31
• Thanks Rob. I will try your suggestion. It is important to define $D_{is}$ by using ILOG expression. Regards. – memop Oct 18 '19 at 17:10