A multiperiod min-sum arborescence problem

In this paper we present a mathematical formulation of the multiperiod min-sum arborescence problem which requires scheduling the installation of directed links to connect a set of terminal nodes N = {2,3,…,n} to a central node. The selection of links and the timing of their installation should be such that the present value of expenditures including cost of installing all the links and maintaining them till the end of the planning horizon should be minimal. The set of links selected for installation should ensure that (i) each terminal node j has exactly one entering link; and (ii) for each terminal node j, a unique path exists from the central node to node j from the period when j installed till the end of the planning horizon. Some of the terminal nodes are included in the network at the beginning of the planning horizon while others are added to the network over time. We present a branch exchange heuristic embedded in the Lagrangian relaxation method to find a solution to this problem. The lower bound given by our solution method is used to estimate the quality of the solution given by the heuristic. Test results over a wide range of problem structures show that for networks with up to 140 nodes our Lagrangian-based heuristic method gives solutions that are within 5–15 % of optimality. © Operational Research Society of India 2013.