In an LP model, I have a decision variable $x$ with a composite index, i.e. $x_{i \delta}$, where $i \in \mathbb{N}$ and $\delta \in \{ -, + \}$.
In the following constraint of the problem, I want to add the condition that $i \delta$ is not equal to $j \theta$, (i.e. the two $x$ variables do not have the same composite index -- please see examples below):
$A_{i \delta}x_{i \delta} + A_{j \theta}x_{j \theta} \leq C$, where $A_{i \delta}, A_{j \theta}, C$ are constants.
I considered expressing the constraint as:
$A_{i \delta}x_{i \delta} + A_{j \theta}x_{j \theta} \leq C, i \delta \neq j \theta$,
but I am not sure that is correct.
Another approach could be to express it as:
$A_{i \delta}x_{i \delta} + A_{j \theta}x_{j \theta} \leq C, i \neq j \lor \delta \neq \theta$,
but I am not sure if logical OR conditions can be included in the conditions of a linear constraint in an LP problem.
Any help would be greatly appreciated.
Example cases where $i \delta$ is not considered equal to $j \theta$:
- $x_{1 +}$ and $x_{1 -}$
- $x_{1 +}$ and $x_{2 +}$
- $x_{1 +}$ and $x_{2 -}$
Example case where $i \delta$ is considered equal to $j \theta$:
- $x_{1 +}$ and $x_{1 +}$
Yes, your logical OR is acceptable in an index set for constraints. I think the following is more readable: $(i,\delta)\not=(j,\theta)$. But your proposed $i\delta\not=j\theta$ looks instead like multiplication.