The achievable region method in the optimal control of queueing systems : formulations, bounds and policies

Cover title.Includes bibliographical references (p. 44-48).Supported in part by a Presidential Young Investigator Award, with matching funds from Draper Laboratory. DDM-9158118Dimitris Bertsimas

The achievable region method in the optimal control of queueing systems : formulations, bounds and policies

Cover title.Includes bibliographical references (p. 44-48).Supported in part by a Presidential Young Investigator Award, with matching funds from Draper Laboratory. DDM-9158118Dimitris Bertsimas

A mathematical programming approach to stochastic and dynamic optimization problems

Includes bibliographical references (p. 46-50).Supported by a Presidential Young Investigator Award. DDM-9158118 Supported by matching funds from Draper Laboratory.Dimitris Bertsimas

A mathematical programming approach to stochastic and dynamic optimization problems

Includes bibliographical references (p. 46-50).Supported by a Presidential Young Investigator Award. DDM-9158118 Supported by matching funds from Draper Laboratory.Dimitris Bertsimas

Reformulation Techniques and Solution Approaches for Fractional 0-1 Programs and Applications

Fractional binary programs (FPs) form a broad class of nonlinear integer optimization problems, where the objective is to optimize the sum of ratios of (linear) binary functions. FPs arise naturally in a number of important real-life applications such as scheduling, retail assortment, facility location, stochastic service systems, and machine learning, among others. This dissertation studies methods that improve the performance of solution approaches for fractional binary programs in their general structure. In particular, we first explore the links between equivalent mixed-integer linear programming (MILP) and conic quadratic programming reformulations of FPs. Thereby, we show that integrating the ideas behind these two types of reformulations of FPs allows us to push further the limits of the current state-of-the-art results and tackle larger-size problems. In practice, the parameters of an optimization problem are often subject to uncertainty. To deal with uncertainties in FPs, we extend the robust methodology to fractional binary programming. In particular, we study robust fractional binary programs (RFPs) under a wide-range of disjoint and joint uncertainty sets, where the former implies separate uncertainty sets for each numerator and denominator, and the latter accounts for different forms of inter-relatedness between them. We demonstrate that, unlike the deterministic case, single-ratio RFP is NP-hard under general polyhedral uncertainty sets. However, if the uncertainty sets are imbued with a certain structure - variants of the well-known budgeted uncertainty - the disjoint and joint single-ratio RFPs are polynomially-solvable when the deterministic counterpart is. We also propose MILP formulations for multiple-ratio RFPs and evaluate their performances by using real and synthetic data sets. One interesting application of FPs arises in feature selection which is an essential preprocessing step for many machine learning and pattern recognition systems and involves identification of the most characterizing features from the data. Notably, correlation-based and mutual-information-based feature selection problems can be reformulated as single-ratio FPs. We study approaches that ensure globally optimal solutions for medium- and reasonably large-sized instances of the aforementioned problems, where the existing MILPs in the literature fail. We perform computational experiments with diverse classes of real data sets and report encouraging results

Location Problems in Supply Chain Design: Concave Costs, Probabilistic Service Levels, and Omnichannel Distribution

Location of facilities such as plants, distribution centers in a supply chain plays critical role in efficient management of logistics activities. Real-life supply chains are generally large in size with multiple echelons, prone to disruptions and uncertainties, and constantly evolving to meet customer demands in a fast and reliable way. Therefore, it is quite challenging to identify these locations while balancing the trade-off between costs and service levels. In this thesis, we investigate three supply chain design problems addressing various issues that complicate the location of facilities in a supply chain. The first paper investigates a multilevel capacitated facility location problem. Such problems commonly arise in large scale production-distribution supply chains with plants at one echelon, and distribution centers / warehouse at another, and there is hierarchy of flow between facilities and to the end customers such as retail stores. The operating costs at facilities and transportation costs on arcs are assumed to be concave. The concave functions model economies of scale in operations (such as production, handling, transportation) performed at large scale and emission of green house gases from transportation activities. The mathematical model for our problem is nonlinear (concave) for which we present two formulations. The first formulation is a prevalent mixed-integer nonlinear program, and second is a purely nonlinear programming problem. Extensive computations are performed to measure the efficiency of two formulations, and managerial insights are provided to understand the behavior of the model under different scenarios of concavities. The second work focuses on e-commerce supply chains that have a common objective of providing fast and reliable deliveries of customers’ orders. The order delivery time primarily depends on the time taken to process the order at the facilities and travel time from facilities to customers. These two times are uncertain in practice, therefore, to capture the combined effect of both uncertainties, we introduce a mathematical model with a requirement that all customer orders should be delivered within a committed time with some probabilistic guarantee. The problem is formulated as a dynamic (multiperiod) capacitated facility location problem with modular capacities. The probabilistic service level constraints make the problem nonconvex. We present two linear binary programming reformulations, and develop an exact branch-and-cut algorithm utilizing the reformulations to solve large size instances. We also include sensitivity analysis to study the change in network configuration under various modeling parameters. An increase in online sales every year is driving many brick-and-mortar retailers to follow an omni-channel retailing approach that would integrate their online sales channel with store sales. Omnichannel retailing requires a considerable change in current practices. For instance, a retailer generally decides if there is a need of new distribution facilities, which stores should be used as fulfillment centers as well, where to keep safety stocks, from where to serve online demand, among others. To study these aspect, in the third paper, we propose a novel mathematical model for the design of omnichannel distribution network along with allocation of safety stock to the facilities. The original problem is nonlinear which can be reformulated as conic quadratic mixed integer programming problem. The problem is solved using a branch-and-cut solution algorithm. Further, we present several managerial insights related to fulfillment and safety stock decisions using a small example

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations