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Let $z_{ij}\in\{0,1\}$ be a binary variable and $t_{ij}\geqslant0$ be a continuous variable. I have the following equivalence: $$z_{ij}=1\iff t_{ij}>0.$$

Since we have an equivalence between $z_{ij}$ and $t_{ij}$, is it possible to write the following constraints $$\sum_{i=1}^nz_{ij}\leqslant B,\forall j,$$ in terms of $t_{ij}$?

Like for example $$\sum_{i=1}^nt_{ij}\leqslant C,\forall j?$$

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  • $\begingroup$ Is there some restriction, perhaps implied by other constraints, on the values $t$ can take when it is positive? $\endgroup$ – Rob Pratt Oct 29 '19 at 23:26
  • $\begingroup$ Yes, when $t_{ij}$ is positive, we have: $t_{ij}\leqslant T_i$ for all $i, j$ $\endgroup$ – zdm Oct 29 '19 at 23:30
  • $\begingroup$ I don’t think you can get a constraint like $\sum t_{i,j}\le C$ unless they all take some common value when positive. Do you already know how to express the relationship between $z$ and $t$ via linear constraints? $\endgroup$ – Rob Pratt Oct 29 '19 at 23:36
  • $\begingroup$ I don't see the motivation here. What is wrong with $\sum_{i=1}^n z_{ij}\leqslant B\ \forall j$? $\endgroup$ – Math1000 Oct 29 '19 at 23:46
  • $\begingroup$ @Math1000 I am trying to remove binary variables and reduce the number of variables I have in a formulation. But it seems, as Rob Pratt said, it is not possible to do so. $\endgroup$ – zdm Oct 30 '19 at 0:05
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The standard way to limit the number of positive values to $B$ is to introduce a binary variable $z_{i,j}$ with $\sum_i z_{i,j} \le B$, as you have done. The remaining part is to enforce $t_{i,j} >0 \implies z_{i,j}=1$, which is accomplished via linear constraint $t_{i,j} \le T_i z_{i,j}$.

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