Inconsistencies in modeling a binary variable that indicates a switch

Question

this is a follow-up question to this post here and to @RobPratt's answer. I have implemented the whole thing, and now it happened that on day 1 the machine did not run and was only used for the first time on day 2, but a shift change was modeled, even though there was actually none. What could be the reason for this and how do I have to adjust the constraints to prevent this from happening? That is the answer:

\begin{align} \rho_{ptm} &\le 1-s_{ptm} &&\text{for all $p,t,m$} \\ \rho_{ptm} &\le d_{ptm} &&\text{for all $p,t,m$} \\ \rho_{ptm} &\ge d_{ptm} - s_{ptm} &&\text{for all $p,t,m$}\\ \sum_m d_{ptm} + x_{pt} &= 1 &&\text{for all $p,t$} \\ s_{p,t+1,m} &= d_{ptm} + y_{ptm} &&\text{for all $p,t\not=T,m$} \\ y_{ptm} &\le s_{ptm} &&\text{for all $p,t,m$} \\ y_{ptm} &\le x_{pt} &&\text{for all $p,t,m$} \\ y_{ptm} &\ge s_{ptm} + x_{pt} - 1 &&\text{for all $p,t,m$} \end{align}

The variables are defined as:

1. $d_{ptm}$: Binary variable indicating if product $p$ was serviced by machine $m$ in period $t$
2. $\rho_{ptm}$: Binary variable indicating if there was a switch in machines for product $p$ in period $t$ to machine $m$
3. $s_{ptm}$: State variable indicating if the last servicing of product $p$ prior to period $t$ was by machine $m$
4. $y_{ptm}$: Auxiliary variable to assist in defining constraints related to $s_{ptm}$ and $x_{pt}$
5. $x_{pt}$: Binary variable indicating if product $p$ is not being serviced in period $t$

And that is the original question:

I have the following question / concern. I have a modeling problem with the decision variable $d_{ptm}$ which indicates whether product $p$ was serviced by machine $m$ in period $t$. It takes the value 1 if so, and 0 otherwise. A product can be serviced by different machine in different periods. In addition, a product may not be serviced for two days and then again, possibly by a different machine. I now want to introduce a binary variable $\rho$ that indicates such a switch in machines a product was serviced by. It should take the value = 1 if there was a machine switch from the last time a product was serviced till the current period. How can I model this?

Answer 1

I'm going to assume that time indexing starts at 1. You can introduce a new binary variable $\gamma_{p,t}$ together with the constraints $$\gamma_{p,1}=1$$ $$\gamma_{p,t} \le \gamma_{p, t-1}$$ $$\gamma_{p,t} \le x_{p,t-1}$$ and $$\gamma_{p,t} \ge \gamma_{p,t-1} + x_{p,t-1} - 1.$$ Collectively this ensures that $\gamma_{p,t}=1$ if and only if $p$ has not been in production (on any machine) prior to time $t.$

Now adjust your first and third constraints as follows: $$\rho_{p,t,m}\le 1-s_{p,t,m}-\gamma_{p,t}$$ $$\rho_{p,t,m} \ge d_{p,t,m}-s_{p,t,m} - \gamma_{p,t}.$$