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Notation

  • $\text{src}_{h,s},\text{dst}_{h,s},\text{ch}_{h,s}$ are constants.

  • $a_{h,s},x_{i,j,s}$ are binary variables.

  • $\text{wt}_{h,s}$ are continuous variables.

Problem

\begin{align}\min.&\qquad\sum_{h \in H}\sum_{s\in S}(\text{src}_{h,s}+\text{ch}_{h,s}+\text{dst}_{h,s}+\text{wt}_{h,s})\times a_{h,s}\\\text{s.t.}&\qquad{\forall i,j\in H,\,\forall s\in S}:\text{wt}_{j,s}\geq((\text{src}_{i,s}+\text{ch}_{i,s}+\text{wt}_{i,s})-\text{src}_{j,s})\times x_{i,j,s}\end{align}

Now $x_{i,j,s} = 1$ only when vehicle $i$ charges before vehicle $j$. (Finding minimum time for vehicle to reach to its destination) for reference.

Vehicle $i$ charges before $j$ only when $\text{src}_{i,s} < \text{src}_{j,s}$ so how could I force $x_{i,j,s} = 1$ when this condition meets?

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    $\begingroup$ If $src$ are constants, then you know in advance whether $i$ charges before $j$, and you can just force $x_{ijs} = 1$ in this case (via a constraint or via treating it like a constant) — or am I missing something? $\endgroup$ – LarrySnyder610 Aug 3 '19 at 21:42
  • $\begingroup$ yes, you are right. $\endgroup$ – anoop yadav Aug 4 '19 at 8:50
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    $\begingroup$ In that case I will write it as an answer in case it is useful to future readers. $\endgroup$ – LarrySnyder610 Aug 5 '19 at 0:41
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Since the $\text{src}$ are constants, you know in advance whether $i$ charges before $j$, and you can just force $x_{ijs}=1$ in this case (via a constraint or by treating it like a constant).

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Add two indicator constraints:

  • when $x_{i,j,s} = 1$, the condition must be true ($i$ charges before $j$)
  • when $x_{i,j,s} = 0$, the condition must be false ($i$ charges after $j$)

Most commercial solvers have simple APIs that allow you to add indicator constraints directly, without reformulation. For example, here's the documentation for Gurobi's addGenConstrIndicator function.

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