The maximal total social welfare Q = [[summation].sub.m[member of]M] [e.sub.m][d.sub.m] - [[summation].sub.j[member of]J] [c.sub.j][q.sub.j] is achieved by solving VWDP. The buyers and sellers are selfish and tend to maximize their individual profits.
Let Q be the social welfare resulting from the optimal solution of VWDP. Let M be a mechanism that satisfies absolute and relative fairness conditions.
We will call the solution [d.sup.u] = ([d.sup.u.sub.m]), [q.sup.u] = ([q.sup.u.sub.j]) of the VWDP with network constraints neglected the unconstrained solution.