If c_1 and O are decision variables, then c_1O_t = I_{1,t} would no longer be a linear constraint.
It can be useful to try and write down the structural matrix. You should be able to do so in all cases. Variables can be additionally marked integer‑valued or binary, given your solver supports MILP. As an example using this case — which was as far as I could get. HTH, R.
