Lower Bounds for the Multiperiod Capacitated Minimal Spanning Tree with Node Outage Cost Design Problem

The Multiperiod Capacitated Minimal Spanning Tree With Node Outage Costs (MCMSTWOC) Design problem consists of scheduling the installation of links in a communication 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 costs. The cost of the network is the sum of link layout cost and node outage costs. The link capacities limit the number of terminal nodes sharing a link. Node outage cost associated with each terminal node is the economic cost incurred by the network user whenever the terminal node is disabled due to failure of a link. In the network some of the terminal nodes are active at the beginning of the planning horizon while others are activated over time. The problem is formulated as an integer-programming problem. A Lagrangian relaxation method is used to find a lower bound for the optimal objective function value. Subgradient optimization method is used to find good lower bounds. This lower bound can be used to estimate the quality of the solution given by a heuristic

Satellite Network, Design, Optimization, and Management

We introduce several network design and planning problems that arise in the context of commercial satellite networks. At the heart of most of these problems we deal with a traffic routing problem over an extended planning horizon. In satellite networks route changes are associated with significant monetary penalties that are usually in the form of discounts (up to 40%) offered by the satellite provider to the customer that is affected. The notion of these rerouting penalties requires the network planners to consider management problems over multiple time periods and introduces novel challenges that have not been considered previously in the literature. Specifically, we introduce a multiperiod traffic routing problem and a multiperiod network design problem that incorporate rerouting penalties. For both of these problems we present novel path-based reformulations and develop branch-and-price-and-cut approaches to solve them. The pricing problems in both cases present new challenges and we develop special purpose approaches that can deal with them. We also show how these results can be extended to deal with traffic routing and network design decisions in other settings with much more general rerouting penalties. Our computational work demonstrates the benefits of using the branch-and-price-and-cut procedure developed that can deal with the multiperiod nature of the problem as opposed to straightforward, myopic period-by-period optimization approaches. In order to deal with cases in which future demand is not known with certainty we present the stochastic version of the multiperiod traffic routing problem and formulate it as a stochastic multistage recourse problem with integer variables at all stages. We demonstrate how an appropriate path-based reformulation and an associated branch-and-price-and-cut approach can solve this problem and other more general multistage stochastic integer multicommodity flow problems. Finally, we motivate the notion of reload costs that refer to variable (i.e., per unit of flow) costs for the usage of pairs of edges, as opposed to single edges. We highlight the practical and theoretical significance of these cost structures and present two extended graphs that allow us to easily capture these costs and generate strong formulations

Models and algorithms for decomposition problems

This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms. This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced. The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity

Models for planning the evolution of local telecommunication networks

Includes bibliographical references.Research initiated through a grant from GTE Laboratories, Inc. Supported in part by an AT&T research award. Supported in part by the Systems Theory and Operations Research Program of the National Science Foundation. ECS-8316224 Supported in part by ONR. N0000-14-86-0689A. Balakrishnan ... [et al.]

Models for planning the evolution of local telecommunication networks

Includes bibliographical references.Research initiated through a grant from GTE Laboratories, Inc. Supported in part by an AT&T research award. Supported in part by the Systems Theory and Operations Research Program of the National Science Foundation. ECS-8316224 Supported in part by ONR. N0000-14-86-0689A. Balakrishnan ... [et al.]

Robust optimization criteria: state-of-the-art and new issues

Uncertain parameters appear in many optimization problems raised by real-world applications. To handle such problems, several approaches to model uncertainty are available, such as stochastic programming and robust optimization. This study is focused on robust optimization, in particular, the criteria to select and determine a robust solution. We provide an overview on robust optimization criteria and introduce two new classifications criteria for measuring the robustness of both scenarios and solutions. They can be used independently or coupled with classical robust optimization criteria and could work as a complementary tool for intensification in local searches

Multicommodity capacitated network design

Network design models have wide applications in telecommunications and transportation planning; see, for example, the survey articles by Magnanti and Wong (1984), Minoux (1989), Chapter 16 of the book by Ahuja, Magnanti and Orlin (1993), Section 13 of Ahuja et al. (1995). In particular, Gavish (1991) and Balakrishnan et al. (1991) present reviews of important applications in telecommunications. In many of these applications, it is required to send flows (which may be fractional) to satisfy demands given arcs with existing capacities, or to install, in discrete amounts, additional facilities with fixed capacities. In doing so, one pays a price not only for routing flows, but also for using an arc or installing additional facilities. The objective is then to determine the optimal amounts of flows to be routed and the facilities to be installed. Document type: Part of book or chapter of boo

The incremental connected facility location problem

We consider the incremental connected facility location problem (incremental ConFL), in which we are given a set of potential facilities, a set of interconnection nodes, a set of customers with demands, and a planning horizon. For each time period, we have to select a set of facilities to open, a set of customers to be served, the assignment of these customers to the open facilities, and a network that connects the open facilities. Once a customer is served, it must remain served in subsequent periods. Furthermore, in each time period the total demand of all customers served must be at least equal to a given minimum coverage requirement for that period. The objective is to minimize the total cost for building the network given by the investment and maintenance costs for the facilities and the network summed up over all time periods. We propose a mixed integer programming approach in which, in each time period, a single period ConFL with coverage restrictions has to be solved. For this latter problem, which is of particular interest in itself, new families of valid inequalities are proposed: these are set union knapsack cover (SUKC) inequalities, which are further enhanced by lifting and/or combined with cut-set inequalities, which are primarily used to ensure connectivity requirements. Details of an efficient branch-and-cut implementation are presented and computational results on a benchmark set of large instances are given, including examples of telecommunication networks in German

Horizontal Cooperation in Network Expansion: An Empirical Evaluation of Gas Transportation Networks

This research presents a coordination approach for the expansion of gas transportation networks to serve an increasing customer base. An empirical study of natural gas markets in the southeastern United States shows that horizontal cooperation among transportation service providers (i.e., pipeline companies) allows for expanding the gas transportation networks efficiently to serve new customers. The benefits of coordination are identified through key structural elements such as number and location of additional pipeline links, lower infrastructure expansion costs, and demand segmentation for the gas transportation service providers