A Dual-Based Algorithm for Multi-Level Network Design

Abstract

Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels or grades, the Multi-level Network Design (MLND) problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher grade. This problem generalizes the well-known Steiner network problem and the hierarchical network design problem, and has applications in telecommunication, transportation, and electric power distribution network design. In a companion paper we studied alternative model formulations for a two-level version of this problem, and analyzed the worst-case performance of several heuristics based on Steiner network and spanning tree solutions. This paper develops a dual-based algorithm for the MLND problem. The method first performs problem preprocessing to fix certain design variables, and then applies a dual ascent procedure to generate upper and lower bounds on the optimal value. We report extensive computational results on large, random two-level test problems (containing up to 500 nodes, and 5,000 edges) with varying cost structures. The integer programming formulation of the largest of these problems has 20,000 integer variables and over 5 million constraints. Our tests indicate that the dual-based algorithm is very effective, producing solutions that are within 0.9% of optimality.network design, integer programming, dual ascent algorithm

Modeling and Heuristic Worst-Case Performance Analysis of the Two-Level Network Design Problem

“…This paper considers modeling issues for the TLND problem, and develops worst-case bounds for a combined heuristic based on Steiner and spanning tree solutions. A companion paper (Balakrishnan, Magnanti and Mirchandani [1992]) develops and tests an algorithm that combines problem preprocessing, dual ascent, and local improvement to approximately solve the MLND problem. Using this method, we have solved large-scale TLND problems containing up to 500 nodes and 5000 edges to within 0.9% of optimality; the mixed integer formulation for our largest test problem contains 20,000 integer variables and over 5 million constraints.…”

“…In Section 3, we transform the undirected problem into a directed problem, and prove that the linear programming relaxation of the directed formulation has the same optimal objective function value as the linear programming relaxation for the enhanced undirected formulation. This result enables us to apply a dual ascent method for the directed problem, which is easier to describe and implement (Balakrishnan et al [1992]). …”

“…As we illustrate in Section 2.4, the basic undirected flow-based formulation [BUF] has a relatively weak linear programming relaxation, making it unsuitable for LP-based solution methods such as the dual ascent procedure we develop in Balakrishnan et al [1992]. To strengthen this relaxation, we describe a class of additional valid inequalities which we call the bidirectional commodity-pair forcing constraints.…”

“…However, since these constraints apply to every pair of commodities, the enhanced formulation for a network with n nodes and m edges contains O(mn 2 ) forcing constraints rather than the O(mn) forcing constraints in the basic formulation. For the largest network size that we tested in our computational study (Balakrishnan et al [1992]), formulation [EUF] contains more than 450 million constraints.…”

“…Since the directed formulation [DF] and the enhanced undirected formulation [EUF] are LP-equivalent, in Balakrishnan et al [1992] we use the directed formulation to develop the dual ascent algorithm. This algorithm is much easier to describe and implement than its equivalent undirected version.…”