Rakesh Kawatra

A multiperiod planning model for the capacitated minimal spanning tree problem

The Multiperiod Capacitated Minimal Spanning Tree (MCMST) Problem consists of scheduling the installation of links in a network so as to connect a set of terminal nodes S=[2,3,…,N] to a central node (node 1) with minimal present value of expenditures, where link capacities limit the number of terminal nodes sharing a link. Some of the terminal nodes are active at the beginning of the planning horizon while others are activated over time. We formulate this problem as an integer programming problem. A branch exchange heuristic procedure for solving the problem is presented. We also present a Lagrangian relaxation method to find a lower bound for the optimal objective function value. This lower bound may be used to estimate the quality of the solution given by the branch exchange heuristic. Experimental results over a wide range of problem structures show that the branch exchange heuristic method yields verifiably good solutions to this problem.

A multiperiod degree constrained minimal spanning tree problem

Abstract The multiperiod degree constrained minimal spanning tree (MDCMST) problem consists of scheduling the installation of links in a network so as to connect a set of terminal nodes to a central node with minimal present value of expenditures. The network design is subject to degree constraint, which limits the number of links incident on each terminal node to a prespecified number due to the number of ports available on it. Some of the terminal nodes in the network are active at the beginning of the planning horizon while others are activated over time. We formulate this problem as an integer programming problem. We suggest a Lagrangian-based heuristic to solve the integer programming formulation of the network topology problem. Lower bounds found as a byproduct of the solution procedure are used to estimate the quality of the solution given by the heuristic. Experimental results over a wide range of problem structures show that the Lagrangian-based heuristic method yields verifiably good solutions to this hard problem.

Topological design of a centralized communication network with unreliable links and node outage costs

Abstract In this paper, we present a mathematical formulation of a terminal layout problem in the design of a centralized communication network with unreliable links and node outage costs. The node outage cost associated with a terminal node is a cost incurred by the network user whenever that terminal node is unable to communicate with the central node due to failure of a link. We suggest a two-phase heuristic with a time complexity of O( N 3 ) to solve the problem. We also present a Lagrangean relaxation method to find the lower bound of the objective function value. The lower bound given by the Lagrangean relaxation method is used to estimate the quality of the solution given by the two-phase heuristic. Experimental results over a wide range of problem structures show that the average solution given by the two-phase heuristic is within 10% of the optimal objective function value.

Design of a degree-constrained minimal spanning tree with unreliable links and node outage costs

The degree-constrained minimal spanning tree (DCMST) problem with unreliable links and node outage costs consists of finding links in a network to connect a set of terminal nodes to a central node while minimizing the expected annual expenditure. The number of ports available on each terminal node limits the number of incident links (the degree constraint). Each terminal node in the network has an associated node outage cost, which is the economic cost incurred by the network user whenever that node is disabled due to failure of a link. We formulate this problem as an integerprogramming problem and present a Lagrangian relaxation method which, for each choice of Lagrangian multipliers, provides a lower bound for the optimal objective function value. A subgradient optimization method is used to search for multipliers which yield good lower bounds. A branch exchange heuristic procedure makes modifications to each infeasible solution of the Lagrangian relaxation in order to find good feasible solutions. The quality of these heuristic solutions is estimated using the best obtained lower bounds. Experimental results over a wide range of problem structures show that the branch exchange heuristic method yields verifiably good solutions to this problem. 2003 Elsevier B.V. All rights reserved.

A Lagrangian Based Heuristic for the Design of Multipoint Linkages in a Communication Network with Unreliable Links and Node Outage Costs

This paper studies the capacitated minimal spanning tree with unreliable links and node outage costs problem. Tree topologies appear in the design of centralized communication networks. In these topologies the number of nodes in a subtree rooted at the central node is limited to a predefined number due to polling, loading, and response time restrictions. The links in a communication network are prone to failure. Whenever a link in these networks fails all the terminal nodes connected to the central node through that link are unable to communicate till the faulty link is repaired. In some networks such failures can have adverse economic effect on the network user. The economic effect on the network user due to inability of a terminal to communicate with the central node due to link failure is called node outage cost. The sum of expected yearly node outage costs for a network depends on the topology of the network. In this paper we suggest a Lagrangean based heuristic to solve the integer programming formulation of the network topology problem. The objective of the problem is to minimize the sum of link costs and node outage costs. Our computational results on a set of test data with up to 80 nodes show that compared to the previously developed greedy heuristic, our method gives solution that are better by up to 6 percent. The gaps between our heuristic solutions and the lower bounds found as a byproduct of the solution procedure are in the 2–17 percent range.