I need to formulate an Integer Linear Programming model for a Minimum cost-flow problem over a graph without constraints on edges. The cost over edges isn't linear but piecewise, given by:
$c(x_{ij})=\begin{cases} c_{ij}^{1}x_{ij} & \mbox{}\mbox{$0 \leq x_{ij} \leq h_1$} \\ c_{ij}^{k+1}x_{ij} + \sum_{r=1}^{k}(c_{ij}^{r}-c_{ij}^{r+1})h_r & \mbox{}\mbox{$h_k \leq x_{ij} \leq h_{k+1},\;\; k=1,..,T-1$} \\ c_{ij}^{T+1}x_{ij} + \sum_{r=1}^{T}(c_{ij}^{r}-c_{ij}^{r+1})h_r & \mbox{}\mbox{$x_{ij} \geq h_T$} \\ \end{cases} $
T is a given parameter,
$c^k_{ij} > c^{k+1}_{ij} \;\; per \;\; k=1,..,T$
$h^k < h^{k+1}$, $\forall k=1,..,T-1$ are constants
I tried the linearization of the minimax:
min $\sum_{(i,j)\in A}y_{ij}$
$y_{ij}\geq c^1_{ij}x_{ij}$ $\forall (i,j)\in A$
$y_{ij}\geq c^{k+1}_{ij}x_{ij}+\sum_{r=1}^{k}(c_{ij}^{r}-c_{ij}^{r+1})h_r$ $\forall\: k\leq T-1$, $\forall (i,j)\in A$
$y_{ij}\geq c_{ij}^{T+1}x_{ij} + \sum_{r=1}^{T}(c_{ij}^{r}-c_{ij}^{r+1})h_r$ $\forall (i,j)\in A$
$\sum_{j:(i,j)\in A}x_{ij}-\sum_{j:(j,i)\in A}x_{ji}=b_i$ $\forall i\in V$
$0\leq x_{ij},\:integers$ $\forall (i,j)\in A$
with each variable $y_{ij}$ representing the cost over one edge, but it seem to be unsuitable for my problem. I also tried with the maximin but it is still wrong. How can I define this Problem?