## A Lower Bound for Uncapacitated, Multicommodity Fixed Charge Network Design ProblemsInstitute of Transportation Studies, University of California, Irvine, 1986 - Branch and bound algorithms - 45 pages |

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aggregate and disaggregate aggregate formulation Balakrishnan 1984 Benders Decomposition bound to z*[FCND capacitated Capacity Improvement capacity parameter vector charge network design CI procedure commodity K(p compute current improved capacity current lower bound decision variables denote the optimal denote the set determined DFCND disaggregate formulations disaggregate LP relaxations Facilities Location Problem feasible region fixed charge network Hagnanti and Wong i z*[FCND improved capacity parameter intermediate capacity parameter iterative procedure Jt(t Knapsack Problem least as tight Lemma linearized knapsack programs marginal cost maximum flow coefficient network design problem network flow program NF(u NP-hard objective function value Operations Research optimal objective function optimal solution optimality gap parameter for arc program FCND program LKb(t program NF R.T. Wong relaxation of FCND relaxation of program routing cost sequence of target shortest path simplex algorithm solution to program subprograms target values decrease tighter lower bound tion ub(t ugax umax uncapacitated value of program wb(t