Valid Inequalities and Facets for Multi-Module (Survivable) Capacitated Network Design Problem

In this dissertation, we develop new methodologies and algorithms to solve the multi-module (survivable) network design problem. Many real-world decision-making problems can be modeled as network design problems, especially on networks with capacity requirements on arcs or edges. In most cases, network design problems of this type that have been studied involve different types of capacity sizes (modules), and we call them the multi-module capacitated network design (MMND) problem. MMND problems arise in various industrial applications, such as transportation, telecommunication, power grid, data centers, and oil production, among many others. In the first part of the dissertation, we study the polyhedral structure of the MMND problem. We summarize current literature on polyhedral study of MMND, which generates the family of the so-called cutset inequalities based on the traditional mixed integer rounding (MIR). We then introduce a new family of inequalities for MMND based on the so-called n-step MIR, and show that various classes of cutset inequalities in the literature are special cases of these inequalities. We do so by studying a mixed integer set, the cutset polyhedron, which is closely related to MMND. We We also study the strength of this family of inequalities by providing some facet-defining conditions. These inequalities are then tested on MMND instances, and our computational results show that these classes of inequalities are very effective for solving MMND problems. Generalizations of these inequalities for some variants of MMND are also discussed. Network design problems have many generalizations depending on the application. In the second part of the dissertation, we study a highly applicable form of SND, referred to as multi-module SND (MM-SND), in which transmission capacities on edges can be sum of integer multiples of differently sized capacity modules. For the first time, we formulate MM-SND as a mixed integer program (MIP) using preconfigured-cycles (p-cycles) to reroute flow on failed edges. We derive several classes of valid inequalities for this MIP, and show that the valid inequalities previously developed in the literature for single-module SND are special cases of our inequalities. Furthermore, we show that our valid inequalities are facet-defining for MM-SND in many cases. Our computational results, using a heuristic separation algorithm, show that these inequalities are very effective in solving MM-SND. In particular they are more effective than compared to using single-module inequalities alone. Lastly, we generalize the inequalities for MMND for other mixed integer sets relaxed from MMND and the cutset polyhedron. These inequalities also generalize several valid inequalities in the literature. We conclude the dissertation by summarizing the work and pointing out potential directions for future research

The Multilayer Capacitated Survivable IP Network Design Problem : valid inequalities and Branch-and-Cut

Telecommunication networks can be seen as the stacking of several layers like, for instance, IP-over-Optical networks. This infrastructure has to be sufficiently survivable to restore the traffic in the event of a failure. Moreover, it should have adequate capacities so that the demands can be routed between the origin-destinations. In this paper we consider the Multilayer Capacitated Survivable IP Network Design problem. We study two variants of this problem with simple and multiple capacities. We give two multicommodity flow formulations for each variant of this problem and describe some valid inequalities. In particular, we characterize valid inequalities obtained using Chvatal-Gomory procedure from the well known Cutset inequalities. We show that some of these inequalities are facet defining. We discuss separation routines for all the valid inequalities. Using these results, we develop a Branch-and-Cut algorithm and a Branch-and-Cut-and-Price algorithm for each variant and present extensive computational results

Topological Design of Survivable Networks

In the field of telecommunications there are several ways of establishing links between different physical places that must be connected according to the characteristics and the type of service they should provide. Two main considerations to be taken into account and which require the attention of the network planners are, in one hand the economic effort necessary to build the network, and in the other hand the resilience of the network to remain operative in the event of failure of any of its components. A third consideration, which is very important when quality of services required, such as video streaming or communications between real-time systems, is the diameter constrained reliability. In this thesis we study a set of problems that involve such considerations. Firstly, we model a new combinatorial optimization problem called Capacitated m-Two Node Survivable Star Problem (CmTNSSP). In such problem we optimize the costs of constructing a network composed of 2-node-connected components that converge in a central node and whose terminals can belong to these connected 2-node structures or be connected to them by simple edges. The CmTNSSP is a relaxation of the Capacitated Ring Star Problem (CmRSP), where the cycles of the latter can be replaced by arbitrary 2-node-connected graphs. According to previous studies, some of the structural properties of 2-node-connected graphs can be used to show a potential improvement in construction costs, over solutions that exclusively use cycles. Considering that the CmTNSSP belongs to the class of NP-Hard computational problems, a GRASP-VND metaheuristic was proposed and implemented for its approximate resolution, and a comparison of results was made between both problems (CmRSP and CmTNSSP) for a series of instances. Some local searches are based on exact Integer Linear Programming formulations. The results obtained show that the proposed metaheuristic reaches satisfactory levels of accuracy, attaining the global optimum in several instances. Next, we introduce the Capacitated m Ring Star Problem under Diameter Constrained Reliability (CmRSP-DCR) wherein DCR is considered as an additional restriction, limiting the number of hops between nodes of the CmRSP problem and establishing a minimum level of network reliability. This is especially useful in networks that should guarantee minimum delays and quality of service. The solutions found in this problem can be improved by applying some of the results obtained in the study of the CmTNSSP. Finally, we introduce a variant of the CmTNSSP named Capacitated Two-Node Survivable Tree Problem, motivated by another combinatorial optimization problem most recently treated in the literature, called Capacitated Ring Tree Problem (CRTP). In the CRTP, an additional restriction is added with respect to CmRSP, where the terminal nodes are of two different types and tree structures are also allowed. Each node in the CRTP may be connected exclusively in one cycle, or may be part of a cycle or a tree indistinctly, depending on the type of node. In the variant we introduced, the cycles are replaced by 2-node-connected structures. This study proposes and implements a GRASP-VND metaheuristic with specific local searches for this type of structures and adapts some of the exact local searches used in the resolution CmTNSSP. A comparison of the results between the optimal solutions obtained for the CRTP and the CTNSTP is made. The results achieved show the robustness and efficiency of the metaheuristi

A multi-exchange heuristic for formation of balanced disjoint rings

Telecommunication networks form an integral part of life. Avoiding failures on these networks is always not possible. Designing network structures that survive these failures have become important in ensuring the reliability of these network structures. With the introduction of SONET (Synchronous Optical Network) technology, rings have become the preferred survivable network structure. This network configuration has a set of disjoint rings (each node being a part of single ring), and these disjoint rings are connected via another main ring. In this research, we present a mathematical model for the design of such disjoint rings with node number balance criterion among the rings. When, given a set of nodes and distances between them, the Balanced Disjoint Rings (BDR) problem is the minimum total link length clustering of nodes into a given number of disjoint rings in such a way that there is almost the same number of nodes in each ring. The BDR problem is a class of the standard Traveling Salesman Problem (TSP). It is clear from this observation that the BDR problem becomes a TSP when the number of rings required is set to one. Hence BDR is NP-Hard, and we do not expect to obtain a polynomial time algorithm for its solution. To overcome this problem, we developed a set of construction heuristics (Break-MST, Distance Method, Hybrid Method, GRASP-Based Distance Method) and improvement heuristics (Multi-Exchange, Single Move). Different combinations of construction and improvement heuristics were implemented and the quality of solution thus obtained was compared to the standard Branch and Cut Technique. It was found that the algorithm with GRASP-Based Distance Method as the construction heuristic and multi-exchange - single-move combination as the improvement heuristic performed better than other combinations. All combinations performed better in general than the standard Branch and Cut technique in terms of solution time

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Hub & regenerator location and survivable network design

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2010.Thesis (Ph. D.) -- Bilkent University, 2010.Includes bibliographical references leaves 180-184.With the vast development of the Internet, telecommunication networks are employed in numerous different outlets. In addition to voice transmission, which is a traditional utilization, telecommunication networks are now used for transmission of different types of data. As the amount of data transmitted through the network increases, issues such as the survivability and the capacity of the network become more imperative. In this dissertation, we deal with both design and routing problems in telecommunications networks. Our first problem is a two level survivable network design problem. The topmost layer of this network consists of a backbone component where the access equipments that enable the communication of the local access networks are interconnected. The second layer connects the users on the local access network to the access equipments, and consequently to the backbone network. To achieve a survivable network, one that stays operational even under minor breakdowns, the backbone network is assumed to be 2-edge connected while local access networks are to have the star connectivity. Within the literature, such a network is referred to as a 2-edge connected/star network. Since the survivability requirements of networks may change based on the purposes they are utilized for, a variation of this problem in which local access networks are also required to be survivable is also analyzed. The survivability of the local access networks is ensured by providing two connections for every component of the local access networks to the backbone network. This architecture is known as dual homing in the literature. In this dissertation, the polyhedral analysis of the two versions of the two level survivable network design problem is presented; separation problems are analyzed; and branch-and-cut algorithms are developed to find exact solutions. The increased traffic on the telecommunications networks requires the use of high capacity components. Optical networks, composed of fiber optical cables, offer solutions with their higher bandwidths and higher transmission speeds. This makes the optical networks a good alternative to handle the rapid increase in the data traffic. However, due to signal degradation which makes signal regeneration necessary introduces the regenerator placement problem as signal regeneration is a costly process in optical networks. In the regenerator placement problem, we study a location and routing problem together on the backbone component of a given telecommunications network. Survivability is also considered in this problem simultaneously. Exact solution methodologies are developed for this problem: mathematical models and some valid inequalities are proposed; separation problems for the valid inequalities are analyzed and a branch-and-cut algorithm is devised.Özkök, OnurPh.D

Facets for Continuous Multi-Mixing Set and Its Generalizations: Strong Cuts for Multi-Module Capacitated Lot-Sizing Problem

The research objective of this dissertation is to develop new facet-defining valid inequalities for several new multi-parameter multi-constraint mixed integer sets. These valid inequalities result in cutting planes that significantly improve the efficiency of algorithms for solving mixed integer programming (MIP) problems involving multimodule capacity constraints. These MIPs arise in many classical and modern applications ranging from production planning to cloud computing. The research in this dissertation generalizes cut-generating methods such as mixed integer rounding (MIR), mixed MIR, continuous mixing, n-step MIR, mixed n-step MIR, migling, and n-step mingling, along with various well-known families of cuts for problems such as multi-module capacitated lot-sizing (MMLS), multi-module capacitated facility location (MMFL), and multi-module capacitated network design (MMND) problems. More specifically, in the first step, we introduce a new generalization of the continuous mixing set, referred to as the continuous multi-mixing set, where the coefficients satisfy certain conditions. For each n’ ϵ {1; : : : ; n}, we develop a class of valid inequalities for this set, referred to as the n0-step cycle inequalities, and present their facet-defining properties. We also present a compact extended formulation for this set and an exact separation algorithm to separate over the set of all n’-step cycle inequalities for a given n’ ϵ {1; : : : ; n}. In the next step, we extend the results of the first step to the case where conditions on the coefficients of the continuous multi-mixing set are relaxed. This leads to an extended formulation and a generalization of the n-step cycle inequalities, n ϵ N, for the continuous multi-mixing set with general coefficients. We also show that these inequalities are facet-defining in many cases. In the third step, we further generalize the continuous multi-mixing set (where no conditions are imposed on the coefficients) by incorporating upper bounds on the integer variables. We introduce a compact extended formulation and new families of multi-row cuts for this set, referred to as the mingled n-step cycle inequalities (n ϵ N), through a generalization of the n-step mingling. We also provide an exact separation algorithm to separate over a set of all these inequalities. Furthermore, we present the conditions under which a subset of the mingled n-step cycle inequalities are facet-defining for this set. Finally, in the fourth step, we utilize the results of first step to introduce new families of valid inequalities for MMLS, MMFL, and MMND problems. Our computational results show that the developed cuts are very effective in solving the MMLS instances with two capacity modules, resulting in considerable reduction in the integrality gap, the number of nodes, and total solution time