In a MIP, how to force a decision variable to be zero unless the sum of specific other decision variables is equal to a certain number?

Question

In an MIP, how can I formulate a constraint such that a decision variable is only greater (or equal to) zero if (and only if) the sum of different decision variables is equal to something.

I'm working with a path flow formulation model and I want to have a constraint that forces flow q on path p to be zero if not all routes in path p are flown.

Example: flow q on path p, which contains flight A to B ($f_{AB}$) and flight B to C ($f_{BC}$), can be greater than zero if and only if one aircraft flies from A to B and (the same or another aircraft) flies from B to C.

I) q = 0 if $\sum$($f_{AB}$ + $f_{BC}$) $\le$ 1

II) q $\ge$ 0 if $\sum$($f_{AB}$ + $f_{BC}$) = 2

In case there are three flights in one path, constraint I becomes $\le$ 2, and constraint II becomes = 3, etc.

(I know exactly which flow can go over which paths, and I know how many flights are contained in all of the available paths. Moreover, all $f_{ij}$ are binary)

Help with this would be highly appreciated! (I'm writing my problem in python, in case that matters for anything)

Answer 1

Introduce a binary variable $\delta \in \{0,1\}$ to indicate whether $q$ is positive or not. You want: $$ \sum_{(i,j) \in A}f_{ij} \le |A| - 1 \quad \Longrightarrow \quad \delta=0 $$ or the contrapositive: $$ \delta=1 \quad \Longrightarrow \quad \sum_{(i,j) \in A}f_{ij} \ge |A| $$ which you can achieve with: $$ \sum_{(i,j) \in A}f_{ij} \ge |A| \delta $$

Then, assuming you want $q\ge L$ when $\sum_{(i,j) \in A}f_{ij} = |A|$, add the constraints: $$ L\delta \le q \le M \delta $$ $M$ is an upper bound on $q$. The right hand side of this last constraint enforces $q$ to take value $0$ when $\delta=0$.

Answer 2

With CPLEX you can use logical constraints.

For instance with the OPL API you can write:

dvar boolean q;
dvar int a;
dvar int b;

subject to
{
  q==(a+b>=2);
}

q is 0 iff a+b<=1