Network Loading Problem: Valid inequalities from 5- and higher partitions

• Published in: Computers & operations research Vol. 99; pp. 123 - 134
• Main Author: Agarwal, Y.K.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01-11-2018; Pergamon Press Inc
• Subjects: Branch & bound algorithms; Branch and bound; Chemical partition; Graph theory; Inequalities; Integer programming; Network flow problem; Network loading; Operations research; p-Partition; Partitions; Studies; Valid inequalities; Integer programming; Valid inequalities; Branch and bound; Network loading; p-Partition
• ISSN: 0305-0548; 1873-765X; 0305-0548
• DOI: 10.1016/j.cor.2018.06.013

•Three new classes of p-partition based valid inequalities for the Network Loading Problem are described.•Total Capacity Inequality is obtained by recursive application of C-G procedure on the well-known cut inequalities.•One-Two inequalities are computed by solving a sequence of LPs.•Several characteristics of problem instances are identified that influence the effectiveness of these inequalities.•Computational results demonstrate that effectiveness of these inequalities in solving benchmark instances much faster than published results.
This paper addresses the well-known Network Loading Problem, and develops several large new classes of valid inequalities based on p-partitions of the graph. The total capacity inequalities are obtained by recursively applying a simple C-G procedure to the cut inequalities, and can be efficiently computed for p-partitions with p ≤ 10. Another class of inequalities called one-two inequalities are separated by solving a sequence of LPs, heuristically modifying the LHS coefficients of the inequality until a violated inequality is found. The spanning tree inequalities are based on a simple observation that if the demand graph of a problem is connected, the solution must be at least a tree, i.e. have (n−1) or more edges. The conditions under which these inequalities are facet-defining are analyzed. The effectiveness of these inequalities in obtaining substantially tighter bounds and reduced solutions times is demonstrated via several computational experiments. More than 10-fold reduction in the solution time is obtained on several benchmark problems.