0
$\begingroup$

I would like to create a rota that repeats every 28 days and adheres to the usual rules. These include the minimum/maximum number of consecutive working days and the break days. I have created a model that adheres perfectly to these rules within the period, but has problems with the transitions from $t=28$ to $t=1$. How do I need to adjust my constraints to make it work? These are my constraints.

$$\sum_{d=t-6}^{t}\sum_{k\in K}^{}x_{idk}\le 6\quad\forall i\in I,~t\in \left\{ 7,\ldots,T \right\}$$

$$\sum_{d=t+1}^{t+4}y_{id}\ge 4\cdot(y_{i(t+1)}-y_{it})\quad\forall i\in I,~t\in \left\{ 1,\ldots,\mid T\mid -4\right\}$$

$$1+y_{it}\ge y_{i(t-1)}+y_{is} \quad\forall i\in I, s\in \left\{ t+1,\ldots,t+2 \right\},t\in\left\{ 2,\ldots,\mid T\mid-2 \right\}$$

$\endgroup$
2
  • $\begingroup$ Should the first summation's upper index limit be $t$ rather than $d?$ Also, what should repeat every 28 days: just $x$; just $y$; or both? If both, why not just solve for the first 28 days? $\endgroup$
    – prubin
    Jan 30 at 16:45
  • $\begingroup$ You're right. I fixed the OP. $x$ determines $y$ in another constraint (which i havent mentioned). I don't get your last point. Whenever i solve the model for 28 days, it could very well be that a person works f.e. on $t=1$, but has $t=2-4$ off and starts working again afterwards. Then the person also works on day $t=22-27$ (which doesnt violate any constraint, but then has off on day $t=28$. But then, seeing the whole as a rota, it violates the constraint regarding the minimum number off working days. Of course i could introduce a hard coded constraint, but i want it to behave "dynamically". $\endgroup$
    – lukdooxb1
    Jan 30 at 18:00

1 Answer 1

0
$\begingroup$

You can take time indexes mod $T$ to make it work. Rewrite the first constraint as $$ \sum_{d=t-6}^t \,\sum_{k\in K} x_{i,d\bmod T,k} \le 6 \quad \forall i\in I, t\in \lbrace 1,\dots, T\rbrace.$$

So, for example, for $T=28$ $t=1$ and each $k\in K$ the left side contains $$x_{i,23,k} + x_{i,24,k}+\cdots + x_{i,28,k} + x_{i,1,k}.$$

$\endgroup$

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.