Decomposition Algorithms for Multicommodity Network Design Problems with Penalized Constraints Tolga Bektas ¸, Universit´ e du Qu´ ebec ` a Montr´ eal, Montr´ eal, QC, Canada Teodor Gabriel Crainic, Universit´ e du Qu´ ebec ` a Montr´ eal, Montr´ eal, QC, Canada Keywords: nonlinear multicommodity network design, Lagrangean relaxation, decomposition. 1. Introduction Network design models are extensively used to represent a wide range of planning and operation management issues in transportation, telecommunications, logistics and production-distribution. In a very general sense, the problem consists of designing a network by selecting links to connect a set of nodes and to determine the amount of flow on each link such that the demand of each node for a number of commodities is satisfied. The objective is to minimize the total cost of establishing the links and flows. This basic variant is usually referred to as the uncapacitated network design problem. The problem has extensions that arise when addi- tional restrictions are incorporated, such as imposing capacity limits on the amount of demand that may be transported on the links (referred to as the capacitated network design problem). Interested readers on the problem may consult the surveys by Magnanti and Wong [7] and Minoux [8]. Network design formulation provide a good modelling framework for service network design problems, usually at the strategic or the tactical level, for which Crainic [3] gives an overview and a classification of formulations. Consider the graph G =(N , A) where N is the set of nodes and A is the set of links. For the links activated in the network, there is a fixed-charge cost vector denoted by f =[f ij ]. There exists a set of commodities denoted by P . Let y = {y ij |(i, j ) ∈ A} denote the vector of design variables with y ∈Y , where Y = {0, 1} |A| and x = {x p ij |(i, j ) ∈A,p ∈ P} denote the vector of flow variables with x ∈X = N |A||P| + .A generic formulation for the multicommodity network design problem can be given as follows: Minimize cx + fy (1) subject to Nx = d (2) Ax ≤ by (3) Dx ≤ ey (4) where N is an arc-node incidence matrix, A and D are matrices and b and e are column vectors of ap- propriate dimensions. Constraints (2) are network flow constraints and (3) and (4) are additional relations such as linking or capacity restrictions. This paper focuses on multicommodity network design formulations incorporating penalized constraints. In specific, let us assume that constraints (3) are allowed to be violated (penalized) at the expense of additional cost. Then, we are interested in the problems of the following form, Minimize cx + fy + p[max(0, Ax - by)] n (5) subject to (2), (4) where the last component of (5) is the penalty term imposing an additional cost whenever the constraint set is violated, with p being the vector of penalty cost coefficients. The idea of penalizing various types of constraints, such as capacity, was discussed by Crainic [2] in the context of freight transportation. Considering capacity constraints as an example, the author argues that for a tactical model, “one is generally less concerned with the specific vehicle capacity, the emphasis rather being on determining the frequency of the service, which determines its capacity, and the distribution of the freight traffic, which determines how this capacity is to be used”. Such constraints can be allowed to be violated, although at the expense of additional cost. It is therefore more appropriate to treat such relations as utilization targets as opposed to strict constraints, as this would provide a better modelling framework in terms of planning. However, these penalized structures