I have a simple problem involving a variable $x_{it}$ representing the amount of a resource allotted to a task $i$ in time $t$.
The quantity of the (renewable) resource is constrained at a value $R$ per period: $$ \sum_{i=1}^{I} x_{it}\leq R\,\,\,\forall t \in T $$
and I have to allott a given quantity $A_i$ of $R$ to each task: $$ \sum_{t=1}^{T} x_{it}\geq A_{i}\,\,\,\forall i \in I $$
My question is: is it possible with this formulation to get the highest and lowest $t$s such as $x_{it}$ is not zero, keeping the problem linear? $$ t_{i}^{MIN}=\{ \tau \;|\; x_{i\tau}>0 \wedge x_{it}=0\;\forall t<\tau\} $$ $$ t_{i}^{MAX}=\{ \tau \;|\; x_{i\tau}>0 \wedge x_{it}=0\;\forall t>\tau\} $$
What I am trying to get is the starting time ($t_{i}^{MIN}$) and the completion time ($t_{i}^{MAX}$) of each task. The objective function could be to minimize the maximum of the $t_{i}^{MAX}$, or to minimize the difference between the maximum $t_{i}^{MAX}$and the minimum $t_{i}^{MIN}$ (that is the makespan).
I think this formulation is not the ideal one, but I can't figure out a suitable, simple alternative. The fact is that the duration of the task is dependent upon the amount of resources allotted in each period, so a formulation using variables representing the starting time and the completion time seems not ok either.