Approximating Connected Facility Location with Lower and Upper Bounds via LP Rounding

We consider a lower- and upper-bounded generalization of the classical facility location problem, where each facility has a capacity (upper bound) that limits the number of clients it can serve and a lower bound on the number of clients it must serve if it is opened. We develop an LP rounding framework that exploits a Voronoi diagram-based clustering approach to derive the first bicriteria constant approximation algorithm for this problem with non-uniform lower bounds and uniform upper bounds. This naturally leads to the the first LP-based approximation algorithm for the lower bounded facility location problem (with non-uniform lower bounds). We also demonstrate the versatility of our framework by extending this and presenting the first constant approximation algorithm for some connected variant of the problems in which the facilities are required to be connected as well

Dual-Based Local Search for Deterministic, Stochastic and Robust Variants of the Connected Facility Location Problem

In this dissertation, we propose the study of a family of network design problems that arise in a wide range of practical settings ranging from telecommunications to data management. We investigate the use of heuristic search procedures coupled with lower bounding mechanisms to obtain high quality solutions for deterministic, stochastic and robust variants of these problems. We extend the use of well-known methods such as the sample average approximation for stochastic optimization and the Bertsimas and Sim approach for robust optimization with heuristics and lower bounding mechanisms. This is particular important for NP-complete problems where even deterministic and small instances are difficult to solve to optimality. Our extensions provide a novel way of applying these techniques while using heuristics; which from a practical perspective increases their usefulness

The optimal location of facilities on a network

Imperial Users onl

Optimizing Strategic Planning With Long-term Sequential Decision Making Under Uncertainty: A Decomposition Approach

The operations research literature has seen decision-making methods at both strategic and operational levels, where high-level strategic plans are first devised, followed by long-term policies that guide future day-to-day operations under uncertainties. Current literature studies such problems on a case-by-case basis, without a unified approach. In this study, we investigate the joint optimization of strategic and operational decisions from a methodological perspective, by proposing a generic two-stage long-term strategic stochastic decision-making (LSSD) framework, in which the first stage models strategic decisions with linear programming (LP), and the second stage models operational decisions with Markov decision processes (MDP). The joint optimization model is formulated as a nonlinear programming (NLP) model, which is then reduced to an integer model through discretization. As expected, the LSSD framework is computationally expensive. Thus, we develop a novel solution algorithm for MDP, which exploit the Benders decomposition with the ``divide-and-conquer\u27\u27 strategy. We further prove mathematical properties to show that the proposed multi-cut L-shaped (MCLD) algorithm is an exact algorithm for MDP. We extend the MCLD algorithm to solve the LSSD framework by developing a two-step backward decomposition (TSBD) method. To evaluate algorithm performances, we adopt four benchmarking problems from the literature. Numerical experiments show that the MCLD algorithm and the TSBD method outperform conventional benchmarks by up to over 90\% and 80\% in algorithm runtime, respectively. The practicality of the LSSD framework is further validated on a real-world critical infrastructure systems (CISs) defense problem. In the past decades, ``attacks\u27\u27 on CIS facilities from deliberate attempts or natural disasters have caused disastrous consequences all over the globe. In this study, we strategically design CIS interconnections and allocate defense resources, to protect the CIS network from sequential, stochastic attacks. The LSSD framework is utilized to model the problem as an NLP model with an alternate integer formulation. We estimate model parameters using real-world CIS data collected from a middle-sized city in the U.S. Previously established algorithms are used to solve the problem with over 45% improvements in algorithm runtime. Sensitivity analyses are conducted to investigate model behaviors and provide insights to practitioners

Supply function equilibrium analysis for electricity markets

The research presented in this Thesis investigates the strategic behaviour of generating firms in bid-based electricity pool markets and the effects of control methods and network features on the electricity market outcome by utilising the AC network model to represent the electric grid. A market equilibrium algorithm has been implemented to represent the bi-level market problem for social welfare maximization from the system operator and utility assets optimisation from the strategic market participants, based on the primal-dual interior point method. The strategic interactions in the market are modelled using supply function equilibrium theory and the optimum strategies are determined by parameterization of the marginal cost functions of the generating units. The AC power network model explicitly represents the active and reactive power flows and various network components and control functions. The market analysis examines the relation between market power and AC networks, while the different parameterization methods for the supply function bids are also investigated. The first part of the market analysis focuses on the effects of particular characteristics of the AC network on the interactions between the strategic generating firms, which directly affect the electricity market outcome. In particular, the examined topics include the impact of transformer tap-ratio control, reactive power control, different locations for a new entry’s generating unit in the system, and introduction of photovoltaic solar power production in the pool market by considering its dependencyon the applied solar irradiance. The observations on the numerical results have shown that their impact on the market is significant and the employment of AC network representation is required for reliable market outcome predictions and for a better understanding of the strategic behaviour as it depends on the topology of the system. The analysis that examines the supply function parameterizations has shown that the resulting market solutions from the different parameterization methods can be very similar or differ substantially, depending on the presence and level of network congestion and on the size and complexity of the examined system. Furthermore, the convergence performance of the implemented market algorithm has been examined and proven to exhibit superior computational efficiency, being able to provide market solutions for large complex AC systems with multiple asymmetric firms, providing the opportunity for applications on practical electricity markets

Solving two-stage stochastic network design problems to optimality

The Steiner tree problem (STP) is a central and well-studied graph-theoretical combinatorial optimization problem which plays an important role in various applications. It can be stated as follows: Given a weighted graph and a set of terminal vertices, find a subset of edges which connects the terminals at minimum cost. However, in real-world applications the input data might not be given with certainty or it might depend on future decisions. For the STP, for example, edge costs representing the costs of establishing links may be subject to inflations and price deviations. In this thesis we tackle data uncertainty by using the concept of stochastic programming and we study the two-stage stochastic version of the Steiner tree problem (SSTP). Thereby, a set of scenarios defines the possible outcomes of a random variable; each scenario is given by its realization probability and defines a set of terminals and edge costs. A feasible solution consists of a subset of edges in the first stage and edge subsets for all scenarios (second stage) such that each terminal set is connected. The objective is to find a solution that minimizes the expected cost. We consider two approaches for solving the SSTP to optimality: combinatorial algorithms, in particular fixed-parameter tractable (FPT) algorithms, and methods from mathematical programming. Regarding the combinatorial algorithms we develop a linear-time algorithm for trees, an FPT algorithm parameterized by the number of terminals, and we consider treewidth-bounded graphs where we give the first FPT algorithm parameterized by the combination of treewidth and number of scenarios. The second approach is based on deriving strong integer programming (IP) formulations for the SSTP. By using orientation properties we introduce new semi-directed cut- and flow-based IP formulations which are shown to be stronger than the undirected models from the literature. To solve these models to optimality we use a decomposition-based two-stage branch&cut algorithm, which is improved by a fast and efficient method for strengthening the optimality cuts. Moreover, we develop new and stronger integer optimality cuts. The computational performance is evaluated in a comprehensive computational study, which shows the superiority of the new formulations, the benefit of the decomposition, and the advantage of using the strengthened optimality cuts. The Steiner forest problem (SFP) is a related problem where sets of terminals need to be connected. On the one hand, the SFP is a generalization of the STP and on the other hand, we show that the SFP is a special case of the SSTP. Therefore, our results are transferable to the SFP and we present the first FPT algorithm for treewidth-bounded graphs and we model new and stronger (semi-)directed cut- and flow-based IP formulations for the SFP. In the second part of this thesis we consider the two-stage stochastic survivable network design problem, an extension of the SSTP where pairs of vertices may demand a higher connectivity. Similarly to the first part we introduce new and stronger semi-directed cut-based models, apply the same decomposition along with the cut strengthening technique, and argue the validity of the newly introduced integer optimality cuts. A computational study shows the benefit, robustness, and good performance of the decomposition and the cut strengthening method

Hub Network Design Problems with Profits

In this thesis we study a new class of hub location problems denoted as \textit{hub network design problems with profits} which share the same feature: a profit oriented objective. We start from a basic model in which only routing and location decisions are involved. We then investigate more realistic models by incorporating new elements such as different types of network design decisions, service commitments constraints, multiple demand levels, multiple capacity levels and pricing decisions. We present mixed-integer programming formulations for each variant and extension and provide insightful computational analyses regarding to their complexity, network topologies and their added value compared to related hub location problems in the literature. Furthermore, we present an exact algorithmic framework to solve two variants of this class of problems. We continue this study by introducing joint hub location and pricing problems in which pricing decisions are incorporated into the decision-making process. We formulate this problem as a mixed-integer bilevel problem and provide feasible solutions using two math-heuristics. The dissertation ends with some conclusions and comments on avenues of future research

Effective Design and Operation of Supply Chains for Remnant Inventory Systems

This research considers a stochastic supply chain problem that (a) has applications in anumber of continuous production industries, and (b) integrates elements of several classicaloperations research problems, including the cutting stock problem, inventory management,facility location, and distribution. The research also uses techniques such as stochasticprogramming and Benders' decomposition. We consider an environment in which a companyhas geographically dispersed distribution points where it can stock standard sizes of a productfrom its plants. In the most general problem, we are given a set of candidate distributioncenters with different fixed costs at the di®erent locations, and we may choose not to operate facilities at one or more of these locations. We assume that the customer demand for smaller sizes comes from other geographically distributed points on a continuing basis and this demand is stochastic in nature and is modeled by a Poisson process. Furthermore, we address a sustainable manufacturing environment where the trim is not considered waste, but rather, gets recycled and thus has an inherent value associated with it. Most importantly, the problem is not a static one where a one-time decision has to be made. Rather, decisions are made on a continuing basis, and decisions made at one point in time have a significant impact on those made at later points. An example of where this problem would arise is a steel or aluminum company that produces product in rolls of standard widths. The decision maker must decide which facilities to open, to find long-run replenishment rates for standard sizes, and to develop long-run policies for cutting these into smaller pieces so as to satisfy customer demand. The cutting stock, facility-location, and transportation problems reside at the heart of the research, and all these are integrated into the framework of a supply chain. We can see that, (1) a decision made at some point in time a®ects the ability to satisfy demand at a later point, and (2) that there might be multiple ways to satisfy demand. The situation is further complicated by the fact that customer demand is stochastic and that this demand could be potentially satisfied by more than one distribution center. Given this background, this research examines broad alternatives for how the company's supply chain should be designed and operated in order to remain competitive with smaller and more nimble companies. The research develops a LP formulation, a mixed-integer programming formulation, and a stochastic programming formulation to model di®erent aspects of the problem. We present new solution methodologies based on Benders' decomposition and the L-shaped method to solve the NP-hard mixed-integer problem and the stochastic problem respectively. Results from duality will be used to develop shadow prices for the units in stock, and these in turn will be used to develop a policy to help make decisions on an ongoing basis. We investigate the theoretical underpinnings of the models, develop new, sophisticated computational methods and interesting properties of its solution, build a simulation model to compare the policies developed with other ones commonly in use, and conduct computational studies to compare the performance of new methods with their corresponding existing methods