Limiting the number of changes in the operating mode of a machine in a scheduling problem can be seen as a direct extension of limiting the number of start-ups.
Suppose we have a machine scheduling problem, where we use the decision variable \[x_t = \begin{cases} 1 & \text{if machine is operating at time slot $t$}\\ 0 & \text{if machine is idle at time slot $t$}\end{cases}\] To limit the number number of start-ups we introduce a binary variable \(\mathit{start}_t \in \{0,1\}\) and write: \[\begin{align} & \mathit{start}_t \ge x_t - x_{t-1}\\ & \sum_t \mathit{start}_t \le U\end{align}\] The idea is as follows:
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Actually we are only sure that \[x_{t-1}=0, x_t = 1 \Rightarrow \mathit{start}_t =1\] so a better picture is:
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Notes:
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Limit number of start-ups
Actually we are only sure that \[x_{t-1}=0, x_t = 1 \Rightarrow \mathit{start}_t =1\] so a better picture is:
as the \(\mathit{start}_t\) is undetermined when we do not switch from 0 to 1.
- 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: \[\begin{align} & \mathit{switch}_t \ge \mathit{mode}_{m,t-1} - \mathit{mode}_{m,t} && \forall m\\ & \sum_t \mathit{switch}_t \le U\end{align}\] 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