#### Limit number of start-ups

- This formulation works for \(\sum_t \mathit{start}_t \le U\) or minimize \(\sum_t \mathit{start}_t \)
- It is not suited for \(\sum_t \mathit{start}_t = U\), \(\sum_t \mathit{start}_t \ge U\) or maximize \(\sum_t \mathit{start}_t \)
- We can relax \(\mathit{start}_t\) to \(\mathit{start}_t \in [0,1]\), i.e. continuous between 0 and 1

#### Limit the number of mode changes

Let's now assume that the machine can operate in different modes. One could be inclined to use an integer variable: \(\mathit{mode}_t \in \{0,1,2,\dots,M\}\). Easier is to use a binary variable \[\mathit{mode}_{m,t} = \begin{cases} 1 & \text{if the machine is operating in mode \(m\) in time slot \(t\)} \\ 0 & \text{otherwise}\end{cases}\] A mode switch from \(m1\) to \(m2\) means:

In other words, by just focusing on the mode that is being activated, limiting the number of mode switches can be interpreted as limiting the number start-ups for each mode: \[\begin{align} & \mathit{switch}_t \ge \mathit{mode}_{m,t} - \mathit{mode}_{m,t-1} && \forall m,t\\ & \sum_t \mathit{switch}_t \le U\end{align}\] Again, this works whether we have an upper limit on the number of mode changes or just minimize this number.

#### References

- Maximizing number of consecutive days with the same operation mode for a scheduling problem, https://stackoverflow.com/questions/64012796/maximizing-number-of-consecutive-days-with-the-same-operation-mode-for-a-schedul

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