0
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Please add to your question the definition of every variable. $\endgroup$ Commented Jun 7, 2024 at 15:36
  • $\begingroup$ As mentioned in the comments to the linked question, there is some ambiguity about a switch for the first servicing. Is there anything in the rest of your model that encourages switches? $\endgroup$ Commented Jun 8, 2024 at 12:56

1 Answer 1

Reset to default
1
$\begingroup$

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: $$\rho_{p,t,m}\le 1-s_{p,t,m}-\gamma_{p,t}$$ $$\rho_{p,t,m} \ge d_{p,t,m}-s_{p,t,m} - \gamma_{p,t}.$$

Improve this answer
$\endgroup$
4
  • $\begingroup$ Dear @prubin. Thank you for your anwser. Does this also hold, if $d_{p1m} = 1 \forall p,m$ is true. So does it work if a machine is active on day 1 or not? Could you maybe explain how this works? Doesnt $\gamma_{p,1}=1$ force the machine to be not in production? As you stated in your anwser? $\endgroup$ Commented Jul 8, 2024 at 11:52
  • $\begingroup$ The first three inequalities I listed apply for $t>1$ (so that $t-1$ is meaningful). Yes, it works if the machine is active on day 1. $\gamma_{p,1}=1$ implies that the machine was not in production prior to day 1 (which it cannot be, since day 1 is the first day of the time horizon). $\endgroup$ Commented Jul 8, 2024 at 15:11
  • $\begingroup$ Thanks. I misread that. One final question. With your two lower constraints (the modified ones), they originate from the product $\rho_{ptm}=(1-s_{ptm})d_{ptm}$. What would the non-linear formulation of the modification look like? And I guess the constraints 2-4 from you could also be compromised to $\gamma_{pt}=\gamma_{pt}\cdot x_{pt}$. Correct? $\endgroup$ Commented Jul 8, 2024 at 15:59
  • $\begingroup$ Final answer(s); I assume you mean $\gamma_{p,t}=\gamma_{p,t-1}\cdot x_{p,t},$ which would be equivalent to my constraints 2-4. The nonlinear definition of $\rho$ would be $\rho_{ptm}=(1-s_{ptm}-\gamma_{pt})d_{ptm}.$ $\endgroup$ Commented Jul 8, 2024 at 18:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.