if statement of objective in linear programming

Question

I am trying to code a Linear Programming model with Pyomo. I have a question about the if statement. Let objective is minimum the time of all jobs done.

          Job1  Job2  Setup  
machine1  25    30    30  
machine2  40    50    50  
machine3  35    59    60  

Each machine needs to be set up at least once, if machine 1 and machine 2 both are being used, we need extra 20 mins.

I have read some posts about the big-M method that can solve this problem. But I don't know how to do it when it related to the objective. Thanks!

Answer 1

Let m(i)∈{0,1} indicating if machine i is used. Then the additional setup time can look like:

  both12Time ≥ 20*(m(1) + m(2) - 1)
  both12Time ≥ 0

We assume here total time is minimized.