Solving the two‐facility network design problem with 3‐partition facets

Abstract

The article studies the problem of designing telecommunication networks using transmission facilities of two different capacities. The point‐to‐point communication demands are met by installing a mix of facilities of both capacities on the edges to minimize total cost. We consider 3‐partitions of the original graph which results in smaller 3‐node subproblems. The extreme points of this subproblem polyhedron are characterized using a set of propositions. A new approach for computing the facets of the 3‐node subproblem is introduced based on polarity theory. The facets of the subproblem are then translated back to those of the original problem using a generalized version of a previously known theorem. The approach has been tested on several randomly generated and real life networks. The computational results show that the new family of facets significantly strengthen the linear programming formulation and reduce the integrality gap. Also, there is a substantial reduction in the size of the branch‐and‐bound (B&B) tree if these facets are used. Problems as large as 37 nodes and 57 edges have been solved to optimality within a few minutes of computer time. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(1), 11–32 2015

Multi-Commodity Multi-Facility Network Design

“…is valid for the original network design problem on G. Hamid and Agarwal (2015) show that if inequality ( 5) is facet-defining for the network design problem on G with c and w, then inequality (6) is facet-defining for P ND provided that α = 0, γ > 0, and node sets N 1 , . .…”

“…Furthermore, as total capacity inequalities have the same form as inequality (24), one can define the corresponding integer knapsack cover set from the stronger one and derive further valid inequalities using the MIR procedure iteratively. Hamid and Agarwal (2015) study the undirected variant of the two-facility network design problem, where the total flow on an arc plus the flow on reverse arc is limited by the capacity of the undirected edge associated with the two arcs. In this case, the authors computationally enumerate the complete list of facets that can be obtained from a given three-partition.…”

A note on capacity models for network design

“…Conversely, if ax + cy ≥ b is valid for B(2T * ), then 2ax + cy ≥ b is valid for U (T ). Therefore, valid inequalities that are defined by coefficients satisfying (10) can easily be translated between U (T ) and B(2T * ).…”

“…As an application of Theorem . Therefore, valid inequalities that are defined by coefficients satisfying (10) can easily be translated between U (T ) and B(2T * ).…”

“…In the literature polyhedral investigations for strengthening mixed-integer formulations are done separately for each model. Magnanti and Mirchandani [11], Magnanti et al [13,14], Mirchandani [15], Agarwal [2], Hamid and Agarwal [10] consider the undirected capacity model. Bienstock and Günlük [7], Günlük [9] study the bidirected capacity model.…”