A Lagrangean Approach to Network Design Problems KAJ HOLMBERG * and DI YUAN Department of Mathematics, LinkoÈping Institute of Technology, S-581 83 LinkoÈping, Sweden Network design is a very important issue in the area of telecommunications and computer networks, where there is a large need for construction of new networks. This is due to technological development (®ber optics for telecommunication) and new ways of usage (Internet for computer networks). Optimal design of such networks requires formulation and solution of new optimization models. In this paper, we formulate several ®xed charge network design models, capacitated or uncapacitated, directed or undirected, possibly with staircase costs, and survivability requirements. We propose a common sol- ution approach for all these problems, based on Lagrangean relaxation, subgradient optimization and primal heuristics, which together form a Lagrangean heuristic. The Lagrangean heuristic can be incor- porated into a branch-and-bound framework, if the exact optimal solution must be found. The approach has been tested on problems of various structures and sizes, and computational results are presented. # 1998 IFORS. Published by Elsevier Science Ltd. All rights reserved Key words: Network design, mixed-integer programming, lagrangean heuristic, branch-and-bound. 1. INTRODUCTION The network design optimization problem is of great importance. It occurs especially in the areas of telecommunications and of computer networks. Due to technological development (®ber optics for telecommunication) and new ways of usage (Internet for computer networks), there is a large need for construction of new networks. Such networks involve large investments, and it is important to make the designs as good as possible. The new networks must satisfy other requirements than the existing ones, and this poses new restrictions on their design. Old planning models for design of copper cable networks often assume that the network should have the form of a tree, in order to minimize the costs. This is not desirable for the ®ber optic networks. Other models could be based on pure linear costs, and thereby favor links with small capacity. Considering the huge investments involved, such models are no longer satisfactory in achieving cost-ecient design. Furthermore, modern telecommunication networks are under constant dimension expansion, and dierences in long-term and short-term planning should be captured in network planning models. For example, while designing a long-term communication backbone network using ®ber optics, the main cost component concerns canalization for ®ber optic cables; for short-term planning in meeting increased demand on bandwidth, costs are associated with the increment in ®ber capacity. Conforming to other important issues in ®ber optic network planning, such as survivability, gives rise to more complex design models. Clearly, these planning tasks can no longer be carried out manually using rules of thumb, nor by simple models failing to capture the essence of the situation. However, they should be accomplished by network planning tools, which adopt optimization techniques for eciently providing good solutions. The fast develop- ment of computers also enables solving of models of earlier prohibiting size and complexity. Int. Trans. Opl Res. Vol. 5, No. 6, pp. 529±539, 1998 # 1998 IFORS. Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0969-6016/98 $19.00 + 0.00 PII: S0969-6016(98)00040-9 Paper presented at the Seventh International Special Conference of IFORS: `Information Systems in Logistics and Transportation, Gothenburg, Sweden, 16±18 June 1997. *Corresponding author. Tel.: 0046-13-282-867; fax: 0046-13-100-746; e-mail: kahol@math.liu.se. 529