# Linear constraint

I am having trouble to formulate the following constraint:

I am looking to formulate a constraint that if a facility was opened at some point in time, then $$Y[j,t]$$ does always have to be $$1$$ from that point in time onwards (see values below). How do I do that?

O[j,t] Y[j,t]
0 0
0 0
1 1
0 1
0

You can activate variable $$Y[j,t]$$ when $$O[j,t]$$ is active ($$O[j,t] \; \Longrightarrow \; Y[j,t]$$) with: $$O[j,t] \le Y[j,t] \tag{1}$$ And then make sure $$Y[j,t]$$ remains active ($$Y[j,t] \; \Longrightarrow \; Y[j,t+1]$$): $$Y[j,t] \le Y[j,t+1] \tag{2a}$$
I suppose there is some cost on variables $$Y[j,t]$$ ? If so, this cost should preclude the solver from activating $$Y[j,t]$$ when $$O[j,t]=Y[j,t-1]=0$$.
Now, if you can close an exising facility, then $$Y[j,t]$$ should be active until the facility is closed, so you can adjust constraint $$(2a)$$ as follows: $$Y[j,t] \le Y[j,t+1] + S[j,t+1] \tag{2b}$$ So if $$Y[j,t]$$ is active, in the following period you either maintain activity, or close the facility. And if you cannot reopen the facility, you need to impose: $$S[j,t] \le 1 - O[j,k] \quad \forall k=t,t+1,t+2,... \tag{3}$$