Wednesday, February 17, 2016

Strange restriction

In this post the question is posed how to model the restriction
\[x \in \{1,2,3,4,5,6,8,10,12,..,100\}.\]
Note that this is \(\{1,2,3,4,5\} \cup \{6,8,10,..,98,100\}\).  Never seen a thing like this, but here is a way to model this in a MIP model:
\[\begin{align}
    & 6 - (1-\delta)M \le x \le  5 + \delta M\\
    & 2y -(1-\delta)M \le x \le 2y+(1-\delta)M \\
    & 1 \le x \le 100 \\
    & x \> \text{integer}\\
    & y \> \text{integer}\\
    & \delta \> \text{binary}
\end{align}\]
Note that this essentially means:
\[\begin{align}
  & \delta=0 \implies x\le 5 \\
  & \delta=1 \implies x\ge 6, x = 2y
\end{align}\]
We can use \(M=100\) (or even refine that a little bit).