The notation is as follows. Let the network (N, A) be an oriented graph consisting of a number of nodes and edges with a given direction, indexed by $n∈ N = {1, ..., N}$ and $(n,m)∈ A ⊆ {(n,m) :n,m∈ N,n≠m}$}. A node corresponds to a location in the network and an edge to a transmission connection (ie a cable) between two nodes. Each node has a given net supply such that $d_n> 0$ (measured in MWh) if the node offers more than it demands, $d_n <0$ if inverse and $d_n = 0$ if the node is merely for transit. Each edge (n, m) is connected to a transmission cost $c_{nm}$ (DKK(danish money) / MWh). The net supply and costs are parameters, whereas the transmission on the edges is represented by decision variables. The transmission on edge (n, m) is denoted as $l_{nm}$ (MWh). I have to solve the following instance of the network issue:
The graph indicates data for the network problem. The number on the edge (n, m) indicates costs, $c_{nm}$, and the number at the node indicates supply or demand, $d_n$ ($d_n$> 0 indicates supply and $d_n$ <0 indicates demand). A stretchy tree is marked with bold edges.
I think I then get the LP-problem: $$Min\\\ 3x_{ab}+2x_{ad}+4x_{bc}+4x_{be}+x_{ca}+2x_{cd}+7x_{ce}+5x_{de}+3x_{ea}$$ $$ubb \\\ x_{ca}+x_{ea}-x_{ab}-x_{ad}= 2$$ $$ x_{ab}-x_{bc}-x_{be}= -8$$ $$ x_{bc}-x_{ca}-x_{cd}-x_{ce}= 5$$ $$ x_{ad}+x_{cd}-x_{de}=-6$$ $$ x_{be}+x_{ce}+x_{de}-x_{ea}=7$$
We find $Ax=-b$
A=\begin{pmatrix} -1 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ 1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & -1 & -1 & -1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & -1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & -1\\ \end{pmatrix} -b=\begin{pmatrix} -2\\ 8\\ -5\\ 6\\ -7\\ \end{pmatrix}
with rows a,b,c,d,e and columns (a,b),(a,d),(b,c),(b,e),(c,a),(c,d),(c,e),(d,e),(e,a). What to do now? Can someone help me with the rest? can be solved by future substitution?