Is There Another Way To Code The Idea of a MAX Constraint Without The Use of Binary Variables?

Question

I have a constraint of the following form that describes the growth of trees, where the population of trees in period $t$ is the previous period's population minus some trees infected with a virus:

$$ S_{t}^{ij} = S_{t-1}^{ij} - I_{t-1}^{ij} $$ $$ S_{t}\ge0,I_{t} \ge 0$$

From this constraint, you can see that it is possible for $ I_{t-1}^{ij} $ to be larger than $S_{t-1}^{ij}$ , but I need to be able to incorporate the fact that in any period $S_{t}^{ij} $ can never be less than 0 because you cannot have negative trees.

I am wondering if there is any way to incorporate this besides including a max constraint of the form :$$ S_{t}^{ij} = \max\{S_{t-1}^{ij} - I_{t-1}^{ij}, 0 \}$$

When I include the max constraint and then linearize it, my problem becomes very challenging to solve, so I am wondering if there's maybe some other way I can try to reformulate the constraint so to avoid the binary variables.

Answer 1

Why not just impose $S_t^{ij} \ge 0$?

Answer 2

As @RobPratt states why not impose $S^{ij}_t\geq0$? Especially it seems rather reasonable to inforce this through $S^{ij}_t\geq I^{ij}_t $ given that there probably cannot be more infected trees than there are trees.