I have an MILP where we have $$ t_k = \sum_i P_i\cdot C_{ik} : P_i\ \in \{0,1\}, C_{ik} \in I^+ $$

and our model is constrained by the number of times $t_k$ is bigger than a certain value $T_k$.

$$ \left[\sum_k\left(t_k \ge T_k\right)\right] \ge N $$

where $N$ is the minimum number of constraints to be satisfied.

Can this problem be solved in MILP and how? I am new to this domain and any leads would be very helpful.

  • $\begingroup$ Are $P_i$ and $C_{ik}$ both variables? $\endgroup$ – prubin Mar 22 at 20:37
  • $\begingroup$ $P_i$ is a binary variable. $C_{ik}$ is a positive constant. $\endgroup$ – Hisham Al Kayed Mar 23 at 5:59

You can introduce a binary variable $x_k$ and linear constraints \begin{align} \sum_k x_k &\ge N\tag1\\ -t_k+T_k&\le M_k(1-x_k) &&\text{for $k\in K$}\tag2 \end{align} Here, the “big-M” constant $M_k$ is a small upper bound on $-t_k+T_k$. Because $t_k\ge 0$, you can take $M_k=T_k$, and the constraint simplifies to $t_k\ge T_k x_k$.

Constraint $(1)$ forces at least $N$ of the $x_k$ variables to be $1$. Constraint $(2)$ enforces the logical implication $x_k=1\implies t_k\ge T_k$.

  • $\begingroup$ I want to edit k index in my question. But without making your answer seems out of place. Thank you for your help. $\endgroup$ – Hisham Al Kayed Mar 23 at 6:45
  • $\begingroup$ I updated both question and answer just now. $\endgroup$ – RobPratt Mar 23 at 12:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.