Resource usage in continous time modeling
AnsweredHello Gurobi community,
I wonder how I can express the following constraint in Gurobi.
I have the following variables:
- S_j : starting time of event j (no preemption). with P_j is a parameter for processing time of event j.
- Y_kt : number of resources of type k used at time-slot t.
- Z_jk : number of resources of type k used by event j. (A choice between group of resources).
I don't want to have an extra variable with 3 indices (X_jkt) in order to linearize this constraint because it will explose the memory since I have a big number of timeslots.
-
Official comment
-
Gurobi Staff
This post is more than three years old. Some information may not be up to date. For current information, please check the Gurobi Documentation or Knowledge Base. If you need more help, please create a new post in the community forum. Or why not try our AI Gurobot?. -
-
-
Hi Amine,
I assume the \(\texttt{if}\)-statement means that \(Z_{jk}\) shall equal 1 when the inequalities are fulfilled and 0 otherwise, correct?
One possible way would be to introduce the relationships
if \(S_j \leq t\) then \(x_j=1\) else \(x_j=0\)
if \(t \leq S_j + P_j -1\) then \(y_j=1\) else \(y_j=0\)
This is well described in the stackexchange post How to write if else statement in Linear programming?
You can then use the binary variables \(x_j, y_j\) and the addGenConAnd() function to model
\(Z_{jk} = x_j \land y_j\) which means that \(Z_{jk}\) is 1 only if both inequalities hold and 0 otherwise.This is just one possible way to avoid the introduction of a triple indexed variable. There might be different and/or better ways to achieve this.
Best regards,
Jaromił0 -
-
Hello Jaromił, Thank you for your quick answer,
My problem is that \(S_j \) is a decision variable which represents the starting time and can take values in \( [ 1 ... H ] \). On the other hand, \(Z_{jk}\) is not binary but an integer decision variable that represents the allocation of resource k to event j. And I want to determine \(Y_{kt}\) (resource usage) which is equal to the sum over events j of \(Z_{jk}\) but only when event is scheduled.
Best regards.
0 -
-
-
Hi Amine,
Could you explain what exactly the \(\texttt{if}\)-statement means in this case?
Best regards,
Jaromił0 -
-
Hi again Jaromił,
The if statement means that we take into account the value of \(Z_{jk}\) only in the timeslots where the event j is scheduled ( \(S_{j} <= t <= S_{j} + P_{j} - 1 \) ). Otherwise we don't take it into account.
Best regards.
0 -
-
-
I understand, so to achieve that, you could use an additional \(\texttt{if-then}\)-statement as
if \(z_j\geq1\) then \(w_{jk} = Z_{jk}\) else \(w_{jk}=0\)where \(z_j = x_j \land y_j\) with \(x_j, y_j\) as defined in my previous message. \(w_{jk}\) would then represent the value of the term in the summation which either equals \(Z_{jk}\) or is ignored and equals \(0\).
0 -
-
Hello,
That's a really interesting technique, I will try it out and let you know so the community can benefit from it. Thanks a lot for your time and quick answers Jaromił.
Have a good day.
0 -
Post is closed for comments.
Comments
7 comments