# Multiple Controls in Mixed Integer Programming

Cross-posted at Operations Research SE

I am developing a model that operates in the realm of mixed integer programming, although I am fairly unfamiliar with this area of mathematics at the moment. I am hoping to get some clarification on whether my idea can function within the contexts of integer programming. I will provide a picture to an article to help clarify my question.

In the formulation below, the chi variable can be either 0 or 1 to indicate if the cell (i,j) received herbicide treatment or not. I will try to frame the question in relation to this equation for the sake of being on the same page.

So... suppose that in addition to herbicide treatment there was an additional type of treatment that can be applied (for the sake of example suppose it's clearing the invasive species by hand). Whether treatment by herbicide or treatment by hand is chosen would be dependent on a number of various factors, but the new problem is such that you need to somehow incorporate the additional treatment measure into the formulation of equation (9). So do you think that the introduction of a second treatment variable (or perhaps some type of logical conditional treatment variable) would be feasible, or can mixed integer programming models only handle one type of control at a time?

• or.stackexchange.com – Rodrigo de Azevedo Jun 13 at 5:37
• Thank you for this! – D.Gray Jun 13 at 13:47
• You're welcome. Please do note, however, that cross-posting is considered quite impolite in the SE community. – Rodrigo de Azevedo Jun 13 at 20:14
• Thanks for the heads up. It wasn't really meant to try and get an answer faster, but to just put the question where it belongs (which is what I thought you were trying to convey by sending me to the OR SE). Regardless, thanks for informing me on that. – D.Gray Jun 14 at 14:38

There's no reason an integer programming model could not accommodate a second binary variable for whether or not the alternative control was applied to a given cell at a given time. You would need to figure out the revised formula for the $$P$$ variable (whatever that is), and you would also need to decide whether the two controls were mutually exclusive (we either spray or hand-weed but not both) or not (we can do both, and the results will be ). You would then express whatever $$P$$ is with a conditional formula (this if we weed, this other thing if we spray, this third thing if we do neither, and possibly this fourth thing if we do both).