The weighted linear optimal distribution problem and applications (Q805491)

Let G be a connected network where each arc has a capacity interval \(C(j)=[c^-(j),c^+(j)]\) and each node i has a supply b(i), where \(b(N)=0\). It assumes that the cost of the flux \(x(j)\in [c^- (j),c^+(j)]\) is given by a linear expression \(d(j)x(j)+r(j)\), where d(j) and r(j) are constants associated with the arc j. The following optimality problem is studied: \[ \text{ minimize } \sum_{j\in A}d(j)x(j)+r(j)\quad =\quad dx\quad +\quad (text{constant}) \] over all flows x such that \(c^-(j)\leq x(j)\leq c^+(j)\) for all \(j\in A\) and \[ \sum_{j\in A}e(i,j)p(j)x(j)\quad =\quad b(i)\text{ for all } i\in N, \] where A is the set of arcs and p(j) is a constant associated with the arc j.