I am trying to formulate a problem that goes the following
Min $f(.)$
This is a generalized objective function.
Subject to,
$x_{i}^{(t+1)} = x_{i}^{(t)} + r_{i}^{(t)} - x_{i}^{(t)}z_{i}^{(t)}$
$x_{i}^{(t)}=$ amount of goods node $i$ at time $t$.
$r_{i}^{(t)}=$ increase in amount of goods of node $i$ at time $t$
$z_{i}^{(t)}=$ the decision variable that equals to 1 if the node is $i$ is attacked at time $t$
$ $ $ $
The logic of this problem is that for every time period, there is an increase of goods in node $i$. The decision variable determines if node $i$ is consumed or not at time $t$.If the node is consumed, the amount of goods is decreased by $x_{i}^{(t)}$.
$ $ My problem is that $r_{i}^{(t)}$ depends on how long does it take before node is consumed. For example,
$ $$r_i(1)=$ replenish rate 1 time period after consumption
$r_i(2)=$ replenish rate 2 time period after consumption
And so on. We also assume during the period of consumption, $r_{i}(0)$ is 0. (Replenish rate 0 time after consumption)
$ $
My planning horizan is say $t=1,2,3,4,5$. Consider 2 nodes, $A$ and $B$ as an example, say at time $t=0$ both nodes are consumed, then at time $t=1$, both nodes replenish by the amount $r_{i}(1)$ and no node is consumed. Then at time $t=2$, only node $A$ is consumed. Thus node $B$ increases by $r_{i}(2)$.
$ $
This wouldnt be a problem is my time horizon is small, but I am trying to solve this for at least 10 time periods, and $r_{i}(j)$ for $j=1,,,10$ all have different values!
Please help! I need to figure out how I can explicitly model this into the constraints!