Academic Commons Search Results
http://academiccommons.columbia.edu/catalog.rss?f%5Bsubject_facet%5D%5B%5D=Operations+research&q=&rows=500&sort=record_creation_date+desc
Academic Commons Search Resultsen-usFrom Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options
http://academiccommons.columbia.edu/catalog/ac:171186
Cao, Menghuihttp://dx.doi.org/10.7916/D8PG1PS1Fri, 28 Feb 2014 15:26:30 +0000This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options.
The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one formula.
In Chapter 3 we introduce new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal-dual simulation algorithm (Andersen and Broadie 2004), we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.Operations research, FinanceOperations Research, Industrial Engineering and Operations ResearchDissertationsInfrastructure Scaling and Pricing
http://academiccommons.columbia.edu/catalog/ac:171000
Gocmen, Fikret Canerhttp://dx.doi.org/10.7916/D8SQ8XFFTue, 18 Feb 2014 12:15:29 +0000Infrastructure systems play a crucial role in our daily lives. They include, but are not limited to, the highways we take while we commute to work, the stadiums we go to watch games, and the power plants that provide the electricity we consume in our homes. In this thesis we study infrastructure systems from several different perspectives with a focus on pricing and scalability. The pricing aspect of our research focuses on two industries: toll roads and sports events. Afterwards, we analyze the potential impact of small modular infrastructure on a wide variety of industries. We start by analyzing the problem of determining the tolls that maximize revenue for a managed lane operator -- that is, an operator who can charge a toll for the use of some lanes on a highway while a number of parallel lanes remain free to use. Managing toll lanes for profit is becoming increasingly common as private contractors agree to build additional lane capacity in return for the opportunity to retain toll revenue. We start by modeling the lanes as queues and show that the dynamic revenue-maximizing toll is always greater than or equal to the myopic toll that maximizes expected revenue from each arriving vehicle. Numerical examples show that a dynamic revenue-maximizing toll scheme can generate significantly more expected revenue than either a myopic or a static toll scheme. An important implication is that the revenue-maximizing fee does not only depend on the current state, but also on anticipated future arrivals. We discuss the managerial implications and present several numerical examples. Next, we relax the queueing assumption and model traffic propagation on a highway realistically by using simulation. We devise a framework that can be used to obtain revenue maximizing tolls in such a context. We calibrate our framework by using data from the SR-91 Highway in Orange County, CA and explore different tolling schemes. Our numerical experiments suggest that simple dynamic tolling mechanisms can lead to substantial revenue improvements over myopic and time-of-use tolling policies. In the third part, we analyze the revenue management of consumer options for tournaments. Sporting event managers typically only offer advance tickets which guarantee a seat at a future sporting event in return for an upfront payment. Some event managers and ticket resellers have started to offer call options under which a customer can pay a small amount now for the guaranteed option to attend a future sporting event by paying an additional amount later. We consider the case of tournament options where the event manager sells team-specific options for a tournament final, such as the Super Bowl, before the finalists are determined. These options guarantee a final game ticket to the bearer if his team advances to the finals. We develop an approach by which an event manager can determine the revenue maximizing prices and amounts of advance tickets and options to sell for a tournament final. Afterwards, for a specific tournament structure we show that offering options is guaranteed to increase expected revenue for the event. We also establish bounds for the revenue improvement and show that introducing options can increase social welfare. We conclude by presenting a numerical application of our approach. Finally, we argue that advances made in automation, communication and manufacturing portend a dramatic reversal of the ``bigger is better'' approach to cost reductions prevalent in many basic infrastructure industries, e.g. transportation, electric power generation and raw material processing. We show that the traditional reductions in capital costs achieved by scaling up in size are generally matched by learning effects in the mass-production process when scaling up in numbers instead. In addition, using the U.S. electricity generation sector as a case study, we argue that the primary operating cost advantage of large unit scale is reduced labor, which can be eliminated by employing low-cost automation technologies. Finally, we argue that locational, operational and financial flexibilities that accompany smaller unit scale can reduce investment and operating costs even further. All these factors combined argue that with current technology, economies of numbers may well dominate economies of unit scale.Business, Operations research, Applied mathematicsBusinessDissertationsPricing, Trading and Clearing of Defaultable Claims Subject to Counterparty Risk
http://academiccommons.columbia.edu/catalog/ac:169814
Kim, Jinbeomhttp://dx.doi.org/10.7916/D8319SWWMon, 03 Feb 2014 12:12:22 +0000The recent financial crisis and subsequent regulatory changes on over-the-counter (OTC) markets have given rise to the new valuation and trading frameworks for defaultable claims to investors and dealer banks. More OTC market participants have adopted the new market conventions that incorporate counterparty risk into the valuation of OTC derivatives. In addition, the use of collateral has become common for most bilateral trades to reduce counterparty default risk. On the other hand, to increase transparency and market stability, the U.S and European regulators have required mandatory clearing of defaultable derivatives through central counterparties. This dissertation tackles these changes and analyze their impacts on the pricing, trading and clearing of defaultable claims. In the first part of the thesis, we study a valuation framework for financial contracts subject to reference and counterparty default risks with collateralization requirement. We propose a fixed point approach to analyze the mark-to-market contract value with counterparty risk provision, and show that it is a unique bounded and continuous fixed point via contraction mapping. This leads us to develop an accurate iterative numerical scheme for valuation. Specifically, we solve a sequence of linear inhomogeneous partial differential equations, whose solutions converge to the fixed point price function. We apply our methodology to compute the bid and ask prices for both defaultable equity and fixed-income derivatives, and illustrate the non-trivial effects of counterparty risk, collateralization ratio and liquidation convention on the bid-ask prices. In the second part, we study the problem of pricing and trading of defaultable claims among investors with heterogeneous risk preferences and market views. Based on the utility-indifference pricing methodology, we construct the bid-ask spreads for risk-averse buyers and sellers, and show that the spreads widen as risk aversion or trading volume increases. Moreover, we analyze the buyer's optimal static trading position under various market settings, including (i) when the market pricing rule is linear, and (ii) when the counterparty -- single or multiple sellers -- may have different nonlinear pricing rules generated by risk aversion and belief heterogeneity. For defaultable bonds and credit default swaps, we provide explicit formulas for the optimal trading positions, and examine the combined effect of heterogeneous risk aversions and beliefs. In particular, we find that belief heterogeneity, rather than the difference in risk aversion, is crucial to trigger a trade. Finally, we study the impact of central clearing on the credit default swap (CDS) market. Central clearing of CDS through a central counterparty (CCP) has been proposed as a tool for mitigating systemic risk and counterpart risk in the CDS market. The design of CCPs involves the implementation of margin requirements and a default fund, for which various designs have been proposed. We propose a mathematical model to quantify the impact of the design of the CCP on the incentive for clearing and analyze the market equilibrium. We determine the minimum number of clearing participants required so that they have an incentive to clear part of their exposures. Furthermore, we analyze the equilibrium CDS positions and their dependence on the initial margin, risk aversion, and counterparty risk in the inter-dealer market. Our numerical results show that minimizing the initial margin maximizes the total clearing positions as well as the CCP's revenue.Operations research, Financejk3071Industrial Engineering and Operations ResearchDissertationsTwo Papers of Financial Engineering Relating to the Risk of the 2007--2008 Financial Crisis
http://academiccommons.columbia.edu/catalog/ac:167143
Zhong, Haowenhttp://dx.doi.org/10.7916/D8CC0XMGFri, 15 Nov 2013 17:04:33 +0000This dissertation studies two financial engineering and econometrics problems relating to two facets of the 2007-2008 financial crisis. In the first part, we construct the Spatial Capital Asset Pricing Model and the Spatial Arbitrage Pricing Theory to characterize the risk premiums of futures contracts on real estate assets. We also provide rigorous econometric analysis of the new models. Empirical study shows there exists significant spatial interaction among the S&P/Case-Shiller Home Price Index futures returns. In the second part, we perform empirical studies on the jump risk in the equity market. We propose a simple affine jump-diffusion model for equity returns, which seems to outperform existing ones (including models with Levy jumps) during the financial crisis and is at least as good during normal times, if model complexity is taken into account. In comparing the models, we made two empirical findings: (i) jump intensity seems to increase significantly during the financial crisis, while on average there appears to be little change of jump sizes; (ii) finite number of large jumps in returns for any finite time horizon seem to fit the data well both before and after the crisis.Operations research, Statisticshz2193Industrial Engineering and Operations ResearchDissertationsData-driven System Design in Service Operations
http://academiccommons.columbia.edu/catalog/ac:163306
Lu, Yinahttp://hdl.handle.net/10022/AC:P:21080Tue, 16 Jul 2013 14:18:37 +0000The service industry has become an increasingly important component in the world's economy. Simultaneously, the data collected from service systems has grown rapidly in both size and complexity due to the rapid spread of information technology, providing new opportunities and challenges for operations management researchers. This dissertation aims to explore methodologies to extract information from data and provide powerful insights to guide the design of service delivery systems. To do this, we analyze three applications in the retail, healthcare, and IT service industries. In the first application, we conduct an empirical study to analyze how waiting in queue in the context of a retail store affects customers' purchasing behavior. The methodology combines a novel dataset collected via video recognition technology with traditional point-of-sales data. We find that waiting in queue has a nonlinear impact on purchase incidence and that customers appear to focus mostly on the length of the queue, without adjusting enough for the speed at which the line moves. We also find that customers' sensitivity to waiting is heterogeneous and negatively correlated with price sensitivity. These findings have important implications for queueing system design and pricing management under congestion. The second application focuses on disaster planning in healthcare. According to a U.S. government mandate, in a catastrophic event, the New York City metropolitan areas need to be capable of caring for 400 burn-injured patients during a catastrophe, which far exceeds the current burn bed capacity. We develop a new system for prioritizing patients for transfer to burn beds as they become available and demonstrate its superiority over several other triage methods. Based on data from previous burn catastrophes, we study the feasibility of being able to admit the required number of patients to burn beds within the critical three-to-five-day time frame. We find that this is unlikely and that the ability to do so is highly dependent on the type of event and the demographics of the patient population. This work has implications for how disaster plans in other metropolitan areas should be developed. In the third application, we study workers' productivity in a global IT service delivery system, where service requests from possibly globally distributed customers are managed centrally and served by agents. Based on a novel dataset which tracks the detailed time intervals an agent spends on all business related activities, we develop a methodology to study the variation of productivity over time motivated by econometric tools from survival analysis. This approach can be used to identify different mechanisms by which workload affects productivity. The findings provide important insights for the design of the workload allocation policies which account for agents' workload management behavior.Operations researchyl2494BusinessDissertationsApproximate dynamic programming for large scale systems
http://academiccommons.columbia.edu/catalog/ac:169790
Desai, Vijay V.Fri, 28 Jun 2013 10:51:16 +0000Sequential decision making under uncertainty is at the heart of a wide variety of practical problems. These problems can be cast as dynamic programs and the optimal value function can be computed by solving Bellman's equation. However, this approach is limited in its applicability. As the number of state variables increases, the state space size grows exponentially, a phenomenon known as the curse of dimensionality, rendering the standard dynamic programming approach impractical. An effective way of addressing curse of dimensionality is through parameterized value function approximation. Such an approximation is determined by relatively small number of parameters and serves as an estimate of the optimal value function. But in order for this approach to be effective, we need Approximate Dynamic Programming (ADP) algorithms that can deliver `good' approximation to the optimal value function and such an approximation can then be used to derive policies for effective decision-making. From a practical standpoint, in order to assess the effectiveness of such an approximation, there is also a need for methods that give a sense for the suboptimality of a policy. This thesis is an attempt to address both these issues. First, we introduce a new ADP algorithm based on linear programming, to compute value function approximations. LP approaches to approximate DP have typically relied on a natural `projection' of a well studied linear program for exact dynamic programming. Such programs restrict attention to approximations that are lower bounds to the optimal cost-to-go function. Our program -- the `smoothed approximate linear program' -- is distinct from such approaches and relaxes the restriction to lower bounding approximations in an appropriate fashion while remaining computationally tractable. The resulting program enjoys strong approximation guarantees and is shown to perform well in numerical experiments with the game of Tetris and queueing network control problem. Next, we consider optimal stopping problems with applications to pricing of high-dimensional American options. We introduce the pathwise optimization (PO) method: a new convex optimization procedure to produce upper and lower bounds on the optimal value (the `price') of high-dimensional optimal stopping problems. The PO method builds on a dual characterization of optimal stopping problems as optimization problems over the space of martingales, which we dub the martingale duality approach. We demonstrate via numerical experiments that the PO method produces upper bounds and lower bounds (via suboptimal exercise policies) of a quality comparable with state-of-the-art approaches. Further, we develop an approximation theory relevant to martingale duality approaches in general and the PO method in particular. Finally, we consider a broad class of MDPs and introduce a new tractable method for computing bounds by consider information relaxation and introducing penalty. The method delivers tight bounds by identifying the best penalty function among a parameterized class of penalty functions. We implement our method on a high-dimensional financial application, namely, optimal execution and demonstrate the practical value of the method vis-a-vis competing methods available in the literature. In addition, we provide theory to show that bounds generated by our method are provably tighter than some of the other available approaches.Operations research, Mathematicsvvd2101Operations Research, BusinessDissertationsStochastic Models of Limit Order Markets
http://academiccommons.columbia.edu/catalog/ac:161685
Kukanov, Arseniyhttp://hdl.handle.net/10022/AC:P:20511Thu, 30 May 2013 16:40:28 +0000During the last two decades most stock and derivatives exchanges in the world transitioned to electronic trading in limit order books, creating a need for a new set of quantitative models to describe these order-driven markets. This dissertation offers a collection of models that provide insight into the structure of modern financial markets, and can help to optimize trading decisions in practical applications. In the first part of the thesis we study the dynamics of prices, order flows and liquidity in limit order markets over short timescales. We propose a stylized order book model that predicts a particularly simple linear relation between price changes and order flow imbalance, defined as a difference between net changes in supply and demand. The slope in this linear relation, called a price impact coefficient, is inversely proportional in our model to market depth - a measure of liquidity. Our empirical results confirm both of these predictions. The linear relation between order flow imbalance and price changes holds for time intervals between 50 milliseconds and 5 minutes. The inverse relation between the price impact coefficient and market depth holds on longer timescales. These findings shed a new light on intraday variations in market volatility. According to our model volatility fluctuates due to changes in market depth or in order flow variance. Previous studies also found a positive correlation between volatility and trading volume, but in order-driven markets prices are determined by the limit order book activity, so the association between trading volume and volatility is unclear. We show how a spurious correlation between these variables can indeed emerge in our linear model due to time aggregation of high-frequency data. Finally, we observe short-term positive autocorrelation in order flow imbalance and discuss an application of this variable as a measure of adverse selection in limit order executions. Our results suggest that monitoring recent order flow can improve the quality of order executions in practice. In the second part of the thesis we study the problem of optimal order placement in a fragmented limit order market. To execute a trade, market participants can submit limit orders or market orders across various exchanges where a stock is traded. In practice these decisions are influenced by sizes of order queues and by statistical properties of order flows in each limit order book, and also by rebates that exchanges pay for limit order submissions. We present a realistic model of limit order executions and formalize the search for an optimal order placement policy as a convex optimization problem. Based on this formulation we study how various factors determine investor's order placement decisions. In a case when a single exchange is used for order execution, we derive an explicit formula for the optimal limit and market order quantities. Our solution shows that the optimal split between market and limit orders largely depends on one's tolerance to execution risk. Market orders help to alleviate this risk because they execute with certainty. Correspondingly, we find that an optimal order allocation shifts to these more expensive orders when the execution risk is of primary concern, for example when the intended trade quantity is large or when it is costly to catch up on the quantity after limit order execution fails. We also characterize the optimal solution in the general case of simultaneous order placement on multiple exchanges, and show that it sets execution shortfall probabilities to specific threshold values computed with model parameters. Finally, we propose a non-parametric stochastic algorithm that computes an optimal solution by resampling historical data and does not require specifying order flow distributions. A numerical implementation of this algorithm is used to study the sensitivity of an optimal solution to changes in model parameters. Our numerical results show that order placement optimization can bring a substantial reduction in trading costs, especially for small orders and in cases when order flows are relatively uncorrelated across trading venues. The order placement optimization framework developed in this thesis can also be used to quantify the costs and benefits of financial market fragmentation from the point of view of an individual investor. For instance, we find that a positive correlation between order flows, which is empirically observed in a fragmented U.S. equity market, increases the costs of trading. As the correlation increases it may become more expensive to trade in a fragmented market than it is in a consolidated market. In the third part of the thesis we analyze the dynamics of limit order queues at the best bid or ask of an exchange. These queues consist of orders submitted by a variety of market participants, yet existing order book models commonly assume that all orders have similar dynamics. In practice, some orders are submitted by trade execution algorithms in an attempt to buy or sell a certain quantity of assets under time constraints, and these orders are canceled if their realized waiting time exceeds a patience threshold. In contrast, high-frequency traders submit and cancel orders depending on the order book state and their orders are not driven by patience. The interaction between these two order types within a single FIFO queue leads bursts of order cancelations for small queues and anomalously long waiting times in large queues. We analyze a fluid model that describes the evolution of large order queues in liquid markets, taking into account the heterogeneity between order submission and cancelation strategies of different traders. Our results show that after a finite initial time interval, the queue reaches a specific structure where all orders from high-frequency traders stay in the queue until execution but most orders from execution algorithms exceed their patience thresholds and are canceled. This "order crowding" effect has been previously noted by participants in highly liquid stock and futures markets and was attributed to a large participation of high-frequency traders. In our model, their presence creates an additional workload, which increases queue waiting times for new orders. Our analysis of the fluid model leads to waiting time estimates that take into account the distribution of order types in a queue. These estimates are tested against a large dataset of realized limit order waiting times collected by a U.S. equity brokerage firm. The queue composition at a moment of order submission noticeably affects its waiting time and we find that assuming a single order type for all orders in the queue leads to unrealistic results. Estimates that assume instead a mix of heterogeneous orders in the queue are closer to empirical data. Our model for a limit order queue with heterogeneous order types also appears to be interesting from a methodological point of view. It introduces a new type of behavior in a queueing system where one class of jobs has state-dependent dynamics, while others are driven by patience. Although this model is motivated by the analysis of limit order books, it may find applications in studying other service systems with state-dependent abandonments.Operations research, Finance, Statisticsak2870Industrial Engineering and Operations ResearchDissertationsFinancial Portfolio Risk Management: Model Risk, Robustness and Rebalancing Error
http://academiccommons.columbia.edu/catalog/ac:161415
Xu, Xingbohttp://hdl.handle.net/10022/AC:P:20382Mon, 20 May 2013 15:59:07 +0000Risk management has always been in key component of portfolio management. While more and more complicated models are proposed and implemented as research advances, they all inevitably rely on imperfect assumptions and estimates. This dissertation aims to investigate the gap between complicated theoretical modelling and practice. We mainly focus on two directions: model risk and reblancing error. In the first part of the thesis, we develop a framework for quantifying the impact of model error and for measuring and minimizing risk in a way that is robust to model error. This robust approach starts from a baseline model and finds the worst-case error in risk measurement that would be incurred through a deviation from the baseline model, given a precise constraint on the plausibility of the deviation. Using relative entropy to constrain model distance leads to an explicit characterization of worst-case model errors; this characterization lends itself to Monte Carlo simulation, allowing straightforward calculation of bounds on model error with very little computational effort beyond that required to evaluate performance under the baseline nominal model. This approach goes well beyond the effect of errors in parameter estimates to consider errors in the underlying stochastic assumptions of the model and to characterize the greatest vulnerabilities to error in a model. We apply this approach to problems of portfolio risk measurement, credit risk, delta hedging, and counterparty risk measured through credit valuation adjustment. In the second part, we apply this robust approach to a dynamic portfolio control problem. The sources of model error include the evolution of market factors and the influence of these factors on asset returns. We analyze both finite- and infinite-horizon problems in a model in which returns are driven by factors that evolve stochastically. The model incorporates transaction costs and leads to simple and tractable optimal robust controls for multiple assets. We illustrate the performance of the controls on historical data. Robustness does improve performance in out-of-sample tests in which the model is estimated on a rolling window of data and then applied over a subsequent time period. By acknowledging uncertainty in the estimated model, the robust rules lead to less aggressive trading and are less sensitive to sharp moves in underlying prices. In the last part, we analyze the error between a discretely rebalanced portfolio and its continuously rebalanced counterpart in the presence of jumps or mean-reversion in the underlying asset dynamics. With discrete rebalancing, the portfolio's composition is restored to a set of fixed target weights at discrete intervals; with continuous rebalancing, the target weights are maintained at all times. We examine the difference between the two portfolios as the number of discrete rebalancing dates increases. We derive the limiting variance of the relative error between the two portfolios for both the mean-reverting and jump-diffusion cases. For both cases, we derive ``volatility adjustments'' to improve the approximation of the discretely rebalanced portfolio by the continuously rebalanced portfolio, based on on the limiting covariance between the relative rebalancing error and the level of the continuously rebalanced portfolio. These results are based on strong approximation results for jump-diffusion processes.Operations research, Finance, Mathematicsxx2126Industrial Engineering and Operations Research, BusinessDissertationsOptimization Algorithms for Structured Machine Learning and Image Processing Problems
http://academiccommons.columbia.edu/catalog/ac:158764
Qin, Zhiweihttp://hdl.handle.net/10022/AC:P:19648Fri, 05 Apr 2013 10:47:07 +0000Optimization algorithms are often the solution engine for machine learning and image processing techniques, but they can also become the bottleneck in applying these techniques if they are unable to cope with the size of the data. With the rapid advancement of modern technology, data of unprecedented size has become more and more available, and there is an increasing demand to process and interpret the data. Traditional optimization methods, such as the interior-point method, can solve a wide array of problems arising from the machine learning domain, but it is also this generality that often prevents them from dealing with large data efficiently. Hence, specialized algorithms that can readily take advantage of the problem structure are highly desirable and of immediate practical interest. This thesis focuses on developing efficient optimization algorithms for machine learning and image processing problems of diverse types, including supervised learning (e.g., the group lasso), unsupervised learning (e.g., robust tensor decompositions), and total-variation image denoising. These algorithms are of wide interest to the optimization, machine learning, and image processing communities. Specifically, (i) we present two algorithms to solve the Group Lasso problem. First, we propose a general version of the Block Coordinate Descent (BCD) algorithm for the Group Lasso that employs an efficient approach for optimizing each subproblem exactly. We show that it exhibits excellent performance when the groups are of moderate size. For groups of large size, we propose an extension of the proximal gradient algorithm based on variable step-lengths that can be viewed as a simplified version of BCD. By combining the two approaches we obtain an implementation that is very competitive and often outperforms other state-of-the-art approaches for this problem. We show how these methods fit into the globally convergent general block coordinate gradient descent framework in (Tseng and Yun, 2009). We also show that the proposed approach is more efficient in practice than the one implemented in (Tseng and Yun, 2009). In addition, we apply our algorithms to the Multiple Measurement Vector (MMV) recovery problem, which can be viewed as a special case of the Group Lasso problem, and compare their performance to other methods in this particular instance; (ii) we further investigate sparse linear models with two commonly adopted general sparsity-inducing regularization terms, the overlapping Group Lasso penalty l1/l2-norm and the l1/l_infty-norm. We propose a unified framework based on the augmented Lagrangian method, under which problems with both types of regularization and their variants can be efficiently solved. As one of the core building-blocks of this framework, we develop new algorithms using a partial-linearization/splitting technique and prove that the accelerated versions of these algorithms require $O(1 sqrt(epsilon) ) iterations to obtain an epsilon-optimal solution. We compare the performance of these algorithms against that of the alternating direction augmented Lagrangian and FISTA methods on a collection of data sets and apply them to two real-world problems to compare the relative merits of the two norms; (iii) we study the problem of robust low-rank tensor recovery in a convex optimization framework, drawing upon recent advances in robust Principal Component Analysis and tensor completion. We propose tailored optimization algorithms with global convergence guarantees for solving both the constrained and the Lagrangian formulations of the problem. These algorithms are based on the highly efficient alternating direction augmented Lagrangian and accelerated proximal gradient methods. We also propose a nonconvex model that can often improve the recovery results from the convex models. We investigate the empirical recoverability properties of the convex and nonconvex formulations and compare the computational performance of the algorithms on simulated data. We demonstrate through a number of real applications the practical effectiveness of this convex optimization framework for robust low-rank tensor recovery; (iv) we consider the image denoising problem using total variation regularization. This problem is computationally challenging to solve due to the non-differentiability and non-linearity of the regularization term. We propose a new alternating direction augmented Lagrangian method, involving subproblems that can be solved efficiently and exactly. The global convergence of the new algorithm is established for the anisotropic total variation model. We compare our method with the split Bregman method and demonstrate the superiority of our method in computational performance on a set of standard test images.Operations research, Computer science, Statisticszq2107Industrial Engineering and Operations ResearchDissertationsModels for managing surge capacity in the face of an influenza epidemic
http://academiccommons.columbia.edu/catalog/ac:157364
Zenteno, Anahttp://hdl.handle.net/10022/AC:P:19200Fri, 01 Mar 2013 10:01:02 +0000Influenza pandemics pose an imminent risk to society. Yearly outbreaks already represent heavy social and economic burdens. A pandemic could severely affect infrastructure and commerce through high absenteeism, supply chain disruptions, and other effects over an extended and uncertain period of time. Governmental institutions such as the Center for Disease Prevention and Control (CDC) and the U.S. Department of Health and Human Services (HHS) have issued guidelines on how to prepare for a potential pandemic, however much work still needs to be done in order to meet them. From a planner's perspective, the complexity of outlining plans to manage future resources during an epidemic stems from the uncertainty of how severe the epidemic will be. Uncertainty in parameters such as the contagion rate (how fast the disease spreads) makes the course and severity of the epidemic unforeseeable, exposing any planning strategy to a potentially wasteful allocation of resources. Our approach involves the use of additional resources in response to a robust model of the evolution of the epidemic as to hedge against the uncertainty in its evolution and intensity. Under existing plans, large cities would make use of networks of volunteers, students, and recent retirees, or borrow staff from neighboring communities. Taking into account that such additional resources are likely to be significantly constrained (e.g. in quantity and duration), we seek to produce robust emergency staff commitment levels that work well under different trajectories and degrees of severity of the pandemic. Our methodology combines Robust Optimization techniques with Epidemiology (SEIR models) and system performance modeling. We describe cutting-plane algorithms analogous to generalized Benders' decomposition that prove fast and numerically accurate. Our results yield insights on the structure of optimal robust strategies and on practical rules-of-thumb that can be deployed during the epidemic. To assess the efficacy of our solutions, we study their performance under different scenarios and compare them against other seemingly good strategies through numerical experiments. This work would be particularly valuable for institutions that provide public services, whose operations continuity is critical for a community, especially in view of an event of this caliber. As far as we know, this is the first time this problem is addressed in a rigorous way; particularly we are not aware of any other robust optimization applications in epidemiology.Operations research, Public healthacz2103Industrial Engineering and Operations ResearchDissertationsChance Constrained Optimal Power Flow: Risk-Aware Network Control under Uncertainty
http://academiccommons.columbia.edu/catalog/ac:156182
Bienstock, Daniel; Chertkov, Michael; Harnett, Seanhttp://hdl.handle.net/10022/AC:P:18933Tue, 05 Feb 2013 10:34:34 +0000When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely used by the electric power industry to re-dispatch hourly controllable generation (coal, gas and hydro plants) over control areas of transmission networks, can result in grid instability, and, potentially, cascading outages. This risk arises because OPF dispatch is computed without awareness of major uncertainty, in particular fluctuations in renewable output. As a result, grid operation under OPF with renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by simulations of real grids, would likely resulting in automatic line tripping to protect lines from thermal stress, a risky and undesirable outcome which compromises stability. Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our Chance Constrained (CC) OPF corrects the problem and mitigates dangerous renewable fluctuations with minimal changes in the current operational procedure. Assuming availability of a reliable wind forecast parameterizing the distribution function of the uncertain generation, our CC-OPF satisfies all the constraints with high probability while simultaneously minimizing the cost of economic re-dispatch. CC-OPF allows efficient implementation, e.g. solving a typical instance over the 2746-bus Polish network in 20 seconds on a standard laptop.Industrial engineering, Operations researchdb17, srh2144Industrial Engineering and Operations Research, Applied Physics and Applied MathematicsArticlesChance Constrained Optimal Power Flow: Risk-Aware Network Control under Uncertainty
http://academiccommons.columbia.edu/catalog/ac:153902
Bienstock, Daniel; Chertkov, Michael; Harnett, Seanhttp://hdl.handle.net/10022/AC:P:15118Mon, 29 Oct 2012 09:19:08 +0000When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely used by the electric power industry to redispatch hourly controllable generation (coal, gas and hydro plants) over control areas of transmission networks, can result in grid instability, and, potentially, cascading outages. This risk arises because OPF dispatch is computed without awareness of major uncertainty, in particular fluctuations in renewable output. As a result, grid operation under OPF with renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by simulations of real grids, would likely resulting in automatic line tripping to protect lines from thermal stress, a risky and undesirable outcome which compromises stability. Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our Chance Constrained (CC) OPF corrects the problem and mitigates dangerous renewable fluctuations with minimal changes in the current operational procedure. Assuming availability of a reliable wind forecast parameterizing the distribution function of the uncertain generation, our CCOPF satisfies all the constraints with high probability while simultaneously minimizing the cost of economic redispatch. CCOPF allows efficient implementation, e.g. solving a typical instance over the 2746bus Polish network in 20s on a standard laptop.Industrial engineering, Operations researchdb17Industrial Engineering and Operations Research, Applied Physics and Applied MathematicsArticlesPrice competition and the impact of service attributes: Structural estimation and analytical characterizations of equilibrium behavior
http://academiccommons.columbia.edu/catalog/ac:153522
Pierson, Margaret Parkerhttp://hdl.handle.net/10022/AC:P:14979Wed, 17 Oct 2012 10:56:33 +0000This dissertation addresses a number of outstanding, fundamental questions in operations management and industrial organization literature. Operations management literature has a long history of studying the competitive impact of operational, firm-level strategic decisions within oligopoly markets. The first essay reports on an empirical study of an important industry, the drive-thru fast-food industry. We estimate a competition model, derived from an underlying Mixed MultiNomial Logit (MNML) consumer choice model, using detailed empirical data. The main goal is to measure to what extent waiting time performance, along with price levels, brand attributes, geographical and demographic factors, impacts competing firms' market shares. The primary goal of our second essay is to characterize the equilibrium behavior of price competition models with Mixed Multinomial Logit (MMNL) demand functions under affine cost structures. In spite of the huge popularity of MMNL models in both the theoretical and empirical literature, it is not known, in general, whether a Nash equilibrium (in pure strategies) of prices exists, and whether the equilibria can be uniquely characterized as the solutions to the system of First Order Condition (FOC) equations. The third essay, which is the most general in its context, we establish that in the absence of cost efficiencies resulting from a merger, aggregate profits of the merging firms increase as do equilibrium prices for general price competition models with general nonlinear demand and cost functions as long as the models are supermodular, with two additional structural conditions: (i) each firm's profit function is strictly quasi-concave in its own price(s), and (ii) markets are competitive, i.e., in the pre-merger industry, each firm's profits increase when any of his competitors increases his price, unilaterally. Even the equilibrium profits of the remaining firms in the industry increase, while the consumer ends up holding the bag, i.e., consumer welfare declines. As demonstrated by this essay, the answers to these sorts of strategy questions have implications not only for the firms and customers but also the policy makers policing these markets.Operations research, BusinessBusinessDissertationsContingent Capital: Valuation and Risk Implications Under Alternative Conversion Mechanisms
http://academiccommons.columbia.edu/catalog/ac:152933
Nouri, Behzadhttp://hdl.handle.net/10022/AC:P:14800Fri, 28 Sep 2012 11:11:35 +0000Several proposals for enhancing the stability of the financial system include requirements that banks hold some form of contingent capital, meaning equity that becomes available to a bank in the event of a crisis or financial distress. Specific proposals vary in their choice of conversion trigger and conversion mechanism, and have inspired extensive scrutiny regarding their effectivity in avoiding costly public rescues and bail-outs and potential adverse effects on market dynamics. While allowing banks to leverage and gain a higher return on their equity capital during the upturns in financial markets, contingent capital provides an automatic mechanism to reduce debt and raise the loss bearing capital cushion during the downturns and market crashes; therefore, making it possible to achieve stability and robustness in the financial sector, without reducing efficiency and competitiveness of the banking system with higher regulatory capital requirements. However, many researchers have raised concerns regarding unintended consequences and implications of such instruments for market dynamics. Death spirals in the stock price near the conversion, possibility of profitable stock or book manipulations by either the investors or the issuer, the marketability and demand for such hybrid instruments, contagion and systemic risks arising from the hedging strategies of the investors and higher risk taking incentives for issuers are among such concerns. Though substantial, many of such issues are addressed through a prudent design of the trigger and conversion mechanism. In the following chapters, we develop multiple models for pricing and analysis of contingent capital under different conversion mechanisms. In Chapter 2 we analyze the case of contingent capital with a capital-ratio trigger and partial and on-going conversion. The capital ratio we use is based on accounting or book value to approximate the regulatory ratios that determine capital requirements for banks. The conversion process is partial and on-going in the sense that each time a bank's capital ratio reaches the minimum threshold, just enough debt is converted to equity to meet the capital requirement, so long as the contingent capital has not been depleted. In Chapter 3 we simplify the design to all-at-once conversion however we perform the analysis through a much richer model which incorporates tail risk in terms of jumps, endogenous optimal default policy and debt rollover. We also investigate the case of bail-in debt, where at default the original shareholders are wiped out and the converted investors take control of the firm. In the case of contingent convertibles the conversion trigger is assumed as a contractual term specified by market value of assets. For bail-in debt the trigger is where the original shareholders optimally default. We study incentives of shareholders to change the capital structure and how CoCo's affect risk incentives. Several researchers have advocated use of a market based trigger which is forward looking, continuously updated and readily available, while some others have raised concerns regarding unintended consequences of a market based trigger. In Chapter 4 we investigate one of these issues, namely the existence and uniqueness of equilibrium when the conversion trigger is based on the stock price.Finance, Operations researchbn2164Industrial Engineering and Operations Research, BusinessDissertationsThree Essays on Dynamic Pricing and Resource Allocation
http://academiccommons.columbia.edu/catalog/ac:151966
Nur, Cavdarogluhttp://hdl.handle.net/10022/AC:P:14492Thu, 23 Aug 2012 11:27:07 +0000This thesis consists of three essays that focus on different aspects of pricing and resource allocation. We use techniques from supply chain and revenue management, scenario-based robust optimization and game theory to study the behavior of firms in different competitive and non-competitive settings. We develop dynamic programming models that account for pricing and resource allocation decisions of firms in such settings. In Chapter 2, we focus on the resource allocation problem of a service firm, particularly a health-care facility. We formulate a general model that is applicable to various resource allocation problems of a hospital. To this end, we consider a system with multiple customer classes that display different reactions to delays in service. By adopting a dynamic-programming approach, we show that the optimal policy is not simple but exhibits desirable monotonicity properties. Furthermore, we propose a simple threshold heuristic policy that performs well in our experiments. In Chapter 3, we study a dynamic pricing problem for a monopolist seller that operates in a setting where buyers have market power, and where each potential sale takes the form of a bilateral negotiation. We review the dynamic programming formulation of the negotiation problem, and propose a simple and tractable deterministic "fluid" analogue for this problem. The main emphasis of the chapter is in expanding the formulation to the dynamic setting where both the buyer and seller have limited prior information on their counterparty valuation and their negotiation skill. In Chapter 4, we consider the revenue maximization problem of a seller who operates in a market where there are two types of customers; namely the "investors" and "regular-buyers". In a two-period setting, we model and solve the pricing game between the seller and the investors in the latter period, and based on the solution of this game, we analyze the revenue maximization problem of the seller in the former period. Moreover, we study the effects on the total system profits when the seller and the investors cooperate through a contracting mechanism rather than competing with each other; and explore the contracting opportunities that lead to higher profits for both agents.Operations researchIndustrial Engineering and Operations ResearchDissertationsModeling Customer Behavior for Revenue Management
http://academiccommons.columbia.edu/catalog/ac:151773
Bansal, Matulyahttp://hdl.handle.net/10022/AC:P:14424Fri, 17 Aug 2012 12:02:54 +0000In this thesis, we model and analyze the impact of two behavioral aspects of customer decisionmaking upon the revenue maximization problem of a monopolist firm. First, we study the revenue maximization problem of a monopolist firm selling a homogeneous good to a market of risk-averse, strategic customers. Using a discrete (but arbitrary) valuation distribution, we show how the dynamic pricing problem with strategic customers can be formulated as a mechanism design problem, thereby making it more amenable to analysis. We characterize the optimal solution, and solve the problem for several special cases. We perform asymptotic analysis for the low risk-aversion case and show that it is asymptotically optimal to offer at most two products. Second, we consider a revenue-maximizing monopolist firm that serves a market of customers that are heterogeneous with respect to their valuations and desire for a quality attribute. Instead of optimizing the net utility that results from an appropriate combination of product price and quality, as in the traditional model of customer behavior, we consider a setting where customers purchase the cheapest product subject to its quality exceeding a customer specific quality threshold. We call such preferences threshold preferences. We solve the firm’s product design problem in this setting, and contrast with the traditional model of customer choice behavior. We consider several scenarios where such preferences might arise, and identify the optimal solution in each case. In addition to these product design problems, we study the problem of identifying the optimal putting strategy for a golfer. We develop a model of golfer putting skill, and combine it with a putt trajectory and holeout model to identify a golfer’s optimal putting strategy. The problem of identifying the optimal putting strategy is shown to be equivalent to a two-dimensional stochastic shortest path problem, with continuous state and control space, and solved using approximate dynamic programming. We calibrate the golfer model to professional and amateur player data, and use the calibrated model to answer several interesting questions, e.g., how does green reading ability affect golfer performance, how do professional and amateur golfers differ in their strategy, how do uphill and downhill putts compare in difficulty, etc.Business, Operations researchmb2431BusinessDissertationsStrategic Models in Supply Network Design
http://academiccommons.columbia.edu/catalog/ac:147203
Lederman, Rogerhttp://hdl.handle.net/10022/AC:P:13314Thu, 24 May 2012 12:15:32 +0000This dissertation contains a series of essays intended to introduce strategic modeling techniques into the network design problem. While investment in production capacity has long been approached as a critical strategic decision, the increasing need for robust, responsive supply capabilities has made it essential to take a network view, where multiple products and sites are considered simultaneously. In traditional network planning, models have rarely accounted for the behavior of additional players - customers, competitors, suppliers - on whom a firm can exert only a limited influence. We analyze a set of models that account for the dynamics of the firm's interaction with these outside actors. In Chapters 2 and 3, we develop game-theoretic models to characterize the allocation of resources in a network context. In Chapter 2, we use series-parallel networks to model the arrangement of producers whose output is bundled. This structure may arise, for example, when various components of the production process are outsourced individually. We study supply-function mechanisms through which producers strategically manage scarce capacity. Our results show how network structure can be analyzed to measure producers' market power and its effect on equilibrium markups. Chapter 3 looks at the network design problem of a vertically integrated firm with the ability to flexibly allocate resources across markets. We consider optimal design of the firm's production network as an upper-level decision to be optimized with respect to competitive outcomes in the lower stage. We find that optimal strategies regarding the location and centralization of production will differ across firms, depending on their competitive position in the market. The final two chapters discuss practical issues regarding the availability of model inputs in a multi-product context. In Chapter 4, we propose a method to construct competitor sets through estimation of a latent-segment choice model. We present a case study in a hotel market, where demand is distributed both spatially and temporally. We show how widely available data on market events can be used to drive identification of customer segments, providing a basis to assess competitive interactions. Chapter 5 provides a further example, in the setting of urban transportation networks, of how user behavior on a network can be estimated from partially observed data. We present a novel two-phase approach for performing this estimation in real time.Operations research, Businessrdl2102BusinessDissertationsEssays on Inventory Management and Object Allocation
http://academiccommons.columbia.edu/catalog/ac:144769
Lee, Thiam Huihttp://hdl.handle.net/10022/AC:P:12623Fri, 17 Feb 2012 15:52:21 +0000This dissertation consists of three essays. In the first, we establish a framework for proving equivalences between mechanisms that allocate indivisible objects to agents. In the second, we study a newsvendor model where the inventory manager has access to two experts that provide advice, and examine how and when an optimal algorithm can be efficiently computed. In the third, we study classical single-resource capacity allocation problem and investigate the relationship between data availability and performance guarantees. We first study mechanisms that solve the problem of allocating indivisible objects to agents. We consider the class of mechanisms that utilize the Top Trading Cycles (TTC) algorithm (these may differ based on how they prioritize agents), and show a general approach to proving equivalences between mechanisms from this class. This approach is used to show alternative and simpler proofs for two recent equivalence results for mechanisms with linear priority structures. We also use the same approach to show that these equivalence results can be generalized to mechanisms where the agent priority structure is described by a tree. Second, we study the newsvendor model where the manager has recourse to advice, or decision recommendations, from two experts, and where the objective is to minimize worst-case regret from not following the advice of the better of the two agents. We show the model can be reduced to the class machine-learning problem of predicting binary sequences but with an asymmetric cost function, allowing us to obtain an optimal algorithm by modifying a well-known existing one. However, the algorithm we modify, and consequently the optimal algorithm we describe, is not known to be efficiently computable, because it requires evaluations of a function v which is the objective value of recursively defined optimization problems. We analyze v and show that when the two cost parameters of the newsvendor model are small multiples of a common factor, its evaluation is computationally efficient. We also provide a novel and direct asymptotic analysis of v that differs from previous approaches. Our asymptotic analysis gives us insight into the transient structure of v as its parameters scale, enabling us to formulate a heuristic for evaluating v generally. This, in turn, defines a heuristic for the optimal algorithm whose decisions we find in a numerical study to be close to optimal. In our third essay, we study the classical single-resource capacity allocation problem. In particular, we analyze the relationship between data availability (in the form of demand samples) and performance guarantees for solutions derived from that data. This is done by describing a class of solutions called epsilon-backwards accurate policies and determining a suboptimality gap for this class of solutions. The suboptimality gap we find is in terms of epsilon and is also distribution-free. We then relate solutions generated by a Monte Carlo algorithm and epsilon-backwards accurate policies, showing a lower bound on the quantity of data necessary to ensure that the solution generated by the algorithm is epsilon-backwards accurate with a high probability. Combining the two results then allows us to give a lower bound on the data needed to generate an Î±-approximation with a given confidence probability 1-delta. We find that this lower bound is polynomial in the number of fares, M, and 1/Î±.Operations researchthl2102Industrial Engineering and Operations ResearchDissertationsMultiproduct Pricing Management and Design of New Service Products
http://academiccommons.columbia.edu/catalog/ac:144706
Wang, Ruxianhttp://hdl.handle.net/10022/AC:P:12603Fri, 17 Feb 2012 12:45:47 +0000In this thesis, we study price optimization and competition of multiple differentiated substitutable products under the general Nested Logit model and also consider the designing and pricing of new service products, e.g., flexible warranty and refundable warranty, under customers' strategic claim behavior. Chapter 2 considers firms that sell multiple differentiated substitutable products and customers whose purchase behavior follows the Nested Logit model, of which the Multinomial Logit model is a special case. In the Nested Logit model, customers make product selection decision sequentially: they first select a class or a nest of products and subsequently choose a product within the selected class. We consider the general Nested Logit model with product-differentiated price coefficients and general nest-heterogenous degrees. We show that the adjusted markup, which is defined as price minus cost minus the reciprocal of the price coefficient, is constant across all the products in each nest. When optimizing multiple nests of products, the adjusted nested markup is also constant within a nest. By using this result, the multi-product optimization problem can be reduced to a single-dimensional problem in a bounded interval, which is easy to solve. We also use this result to simplify the oligopolistic price competition and characterize the Nash equilibrium. Furthermore, we investigate its application to dynamic pricing and revenue management. In Chapter 3, we investigate the flexible monthly warranty, which offers flexibility to customers and allow them to cancel it at anytime without any penalty. Frequent technological innovations and price declines severely affect sales of extended warranties as product replacement upon failure becomes an increasingly attractive alternative. To increase sales and profitability, we propose offering flexible-duration extended warranties. These warranties can appeal to customers who are uncertain about how long they will keep the product as well as to customers who are uncertain about the product's reliability. Flexibility may be added to existing services in the form of monthly-billing with month-by-month commitments, or by making existing warranties easier to cancel, with pro-rated refunds. This thesis studies flexible warranties from the perspectives of both the customer and the provider. We present a model of the customer's optimal coverage decisions under the objective of minimizing expected support costs over a random planning horizon. We show that under some mild conditions the customer's optimal coverage policy has a threshold structure. We also show through an analytical study and through numerical examples how flexible warranties can result in higher profits and higher attach rates. Chapter 4 examines the designing and pricing of residual value warranty that refunds customers at the end of warranty period based on customers' claim history. Traditional extended warranties for IT products do not differentiate customers according to their usage rates or operating environment. These warranties are priced to cover the costs of high-usage customers who tend to experience more failures and are therefore more costly to support. This makes traditional warranties economically unattractive to low-usage customers. In this chapter, we introduce, design and price residual value warranties. These warranties refund a part of the upfront price to customers who have zero or few claims according to a pre-determined refund schedule. By design, the net cost of these warranties is lower for light users than for heavy users. As a result, a residual value warranty can enable the provider to price-discriminate based on usage rates or operating conditions without the need to monitor individual customers' usage. Theoretic results and numerical experiments demonstrate how residual value warranties can appeal to a broader range of customers and significantly increase the provider's profits.Operations research, Industrial engineeringrw2267Industrial Engineering and Operations ResearchDissertationsA Simulation Model to Analyze the Impact of Golf Skills and a Scenario-based Approach to Options Portfolio Optimization
http://academiccommons.columbia.edu/catalog/ac:143076
Ko, Soonminhttp://hdl.handle.net/10022/AC:P:12166Tue, 10 Jan 2012 14:41:51 +0000A simulation model of the game of golf is developed to analyze the impact of various skills (e.g., driving distance, directional accuracy, putting skill, and others) on golf scores. The golf course model includes realistic features of a golf course including rough, sand, water, and trees. Golfer shot patterns are modeled with t distributions and mixtures of t and normal distributions since normal distributions do not provide good fits to the data. The model is calibrated to extensive data for amateur and professional golfers. The golf simulation is used to assess the impact on scores of distance and direction, determine what factors separate pros from amateurs, and to determine the impact of course length on scores. In the second part of the thesis, we use a scenario-based approach to solve a portfolio optimization problem with options. The solution provides the optimal payoff profile given an investor's view of the future, his utility function or risk appetite, and the market prices of options. The scenario-based approach has several advantages over the traditional covariance matrix method, including additional flexibility in the choice of constraints and objective function.Engineering, Operations researchsk2822Industrial Engineering and Operations Research, BusinessDissertationsSupply Chain Management: Supplier Financing Schemes and Inventory Strategies
http://academiccommons.columbia.edu/catalog/ac:142635
Wang, Minhttp://hdl.handle.net/10022/AC:P:11857Wed, 30 Nov 2011 11:40:42 +0000This dissertation addresses a few fundamental questions on the interface between supplier financing schemes and inventory management. Traditionally, retailers finance their inventories through an independent financing institution or by drawing from their own cash reserves, without any supplier involvement (Independent Financing). However, suppliers may reduce their buyers' costs and stimulate sales and associated revenues and profits, by either (i) adopting the financing function themselves (Trade Credit), or (ii) subsidizing the inventory costs (Inventory Subsidies). In the first part (Chapter 2) we analyze and compare the equilibrium performance of supply chains under these three basic financing schemes. The objective is to compare the equilibrium profits of the individual chain members, the aggregate supply chain profits, the equilibrium wholesale price, the expected sales volumes and the average inventory levels under the three financing options, and thus provide important insights for the selection and implementation of supply chain financing mechanisms. Several of the financing schemes introduce a new type of inventory control problem for the retailers in response to terms specified by their suppliers. In Chapter 3 we therefore consider the inventory management problem of a firm which incurs inventory carrying costs with a general shelf age dependent structure and, even more generally, that of a firm with shelf age and delay dependent inventory and backlogging costs. Beyond identifying the structure of optimal replenishment strategies and corresponding algorithms to compute them, it is often important to understand how changes in various primitives of the inventory model impact on the optimal policy parameters and performance measures. In spite of a voluminous literature over more than fifty years, very little is known about this area. In Chapter 4, we therefore study monotonicity properties of stochastic inventory systems governed by an (r; q) or (r; nq) policy and apply the results in our general theorems both to standard inventory models and to those with general shelf age and delay dependent inventory costs.Business, Operations researchmw2426BusinessDissertationsAlgorithms for Sparse and Low-Rank Optimization: Convergence, Complexity and Applications
http://academiccommons.columbia.edu/catalog/ac:137539
Ma, ShiqianMon, 22 Aug 2011 11:53:09 +0000Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the "length" of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these "sparse" optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10-5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/ε) iterations to obtain an ε-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/√ε) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/ε) and O(1/√ε) iteration complexity bounds for obtaining an ε-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.Operations researchsm2756Industrial Engineering and Operations ResearchDissertationsMany-Server Queues with Time-Varying Arrivals, Customer Abandonment, and non-Exponential Distributions
http://academiccommons.columbia.edu/catalog/ac:136569
Liu, Yunanhttp://hdl.handle.net/10022/AC:P:10801Tue, 02 Aug 2011 15:07:23 +0000This thesis develops deterministic heavy-traffic fluid approximations for many-server stochastic queueing models. The queueing models, with many homogeneous servers working independently in parallel, are intended to model large-scale service systems such as call centers and health care systems. Such models also have been employed to study communication, computing and manufacturing systems. The heavy-traffic approximations yield relatively simple formulas for quantities describing system performance, such as the expected number of customers waiting in the queue. The new performance approximations are valuable because, in the generality considered, these complex systems are not amenable to exact mathematical analysis. Since the approximate performance measures can be computed quite rapidly, they usefully complement more cumbersome computer simulation. Thus these heavy-traffic approximations can be used to improve capacity planning and operational control. More specifically, the heavy-traffic approximations here are for large-scale service systems, having many servers and a high arrival rate. The main focus is on systems that have time-varying arrival rates and staffing functions. The system is considered under the assumption that there are alternating periods of overloading and underloading, which commonly occurs when service providers are unable to adjust the staffing frequently enough to economically meet demand at all times. The models also allow the realistic features of customer abandonment and non-exponential probability distributions for the service times and the times customers are willing to wait before abandoning. These features make the overall stochastic model non-Markovian and thus thus very difficult to analyze directly. This thesis provides effective algorithms to compute approximate performance descriptions for these complex systems. These algorithms are based on ordinary differential equations and fixed point equations associated with contraction operators. Simulation experiments are conducted to verify that the approximations are effective. This thesis consists of four pieces of work, each presented in one chapter. The first chapter (Chapter 2) develops the basic fluid approximation for a non-Markovian many-server queue with time-varying arrival rate and staffing. The second chapter (Chapter 3) extends the fluid approximation to systems with complex network structure and Markovian routing to other queues of customers after completing service from each queue. The extension to open networks of queues has important applications. For one example, in hospitals, patients usually move among different units such as emergency rooms, operating rooms, and intensive care units. For another example, in manufacturing systems, individual products visit different work stations one or more times. The open network fluid model has multiple queues each of which has a time-varying arrival rate and staffing function. The third chapter (Chapter 4) studies the large-time asymptotic dynamics of a single fluid queue. When the model parameters are constant, convergence to the steady state as time evolves is established. When the arrival rates are periodic functions, such as in service systems with daily or seasonal cycles, the existence of a periodic steady state and the convergence to that periodic steady state as time evolves are established. Conditions are provided under which this convergence is exponentially fast. The fourth chapter (Chapter 5) uses a fluid approximation to gain insight into nearly periodic behavior seen in overloaded stationary many-server queues with customer abandonment and nearly deterministic service times. Deterministic service times are of applied interest because computer-generated service times, such as automated messages, may well be deterministic, and computer-generated service is becoming more prevalent. With deterministic service times, if all the servers remain busy for a long interval of time, then the times customers enter service assumes a periodic behavior throughout that interval. In overloaded large-scale systems, these intervals tend to persist for a long time, producing nearly periodic behavior. To gain insight, a heavy-traffic limit theorem is established showing that the fluid model arises as the many-server heavy-traffic limit of a sequence of appropriately scaled queueing models, all having these deterministic service times. Simulation experiments confirm that the transient behavior of the limiting fluid model provides a useful description of the transient performance of the queueing system. However, unlike the asymptotic loss of memory results in the previous chapter for service times with densities, the stationary fluid model with deterministic service times does not approach steady state as time evolves independent of the initial conditions. Since the queueing model with deterministic service times approaches a proper steady state as time evolves, this model with deterministic service times provides an example where the limit interchange (limiting steady state as time evolves and heavy traffic as scale increases) is not valid.Operations researchyl2342Industrial Engineering and Operations ResearchDissertationsFirst Order Methods for Large-Scale Sparse Optimization
http://academiccommons.columbia.edu/catalog/ac:135750
Aybat, Necdet Serhathttp://hdl.handle.net/10022/AC:P:10735Fri, 15 Jul 2011 12:00:39 +0000In today's digital world, improvements in acquisition and storage technology are allowing us to acquire more accurate and finer application-specific data, whether it be tick-by-tick price data from the stock market or frame-by-frame high resolution images and videos from surveillance systems, remote sensing satellites and biomedical imaging systems. Many important large-scale applications can be modeled as optimization problems with millions of decision variables. Very often, the desired solution is sparse in some form, either because the optimal solution is indeed sparse, or because a sparse solution has some desirable properties. Sparse and low-rank solutions to large scale optimization problems are typically obtained by regularizing the objective function with L1 and nuclear norms, respectively. Practical instances of these problems are very high dimensional (~ million variables) and typically have dense and ill-conditioned data matrices. Therefore, interior point based methods are ill-suited for solving these problems. The large scale of these problems forces one to use the so-called first-order methods that only use gradient information at each iterate. These methods are efficient for problems with a "simple" feasible set such that Euclidean projections onto the set can be computed very efficiently, e.g. the positive orthant, the n-dimensional hypercube, the simplex, and the Euclidean ball. When the feasible set is "simple", the subproblems used to compute the iterates can be solved efficiently. Unfortunately, most applications do not have "simple" feasible sets. A commonly used technique to handle general constraints is to relax them so that the resulting problem has only "simple" constraints, and then to solve a single penalty or Lagrangian problem. However, these methods generally do not guarantee convergence to feasibility. The focus of this thesis is on developing new fast first-order iterative algorithms for computing sparse and low-rank solutions to large-scale optimization problems with very mild restrictions on the feasible set - we allow linear equalities, norm-ball and conic inequalities, and also certain non-smooth convex inequalities to define the constraint set. The proposed algorithms guarantee that the sequence of iterates converges to an optimal feasible solution of the original problem, and each subproblem is an optimization problem with a "simple" feasible set. In addition, for any eps > 0, by relaxing the feasibility requirement of each iteration, the proposed algorithms can compute an eps-optimal and eps-feasible solution within O(log(1/eps)) iterations which requires O(1/eps) basic operations in the worst case. Algorithm parameters do not depend on eps > 0. Thus, these new methods compute iterates arbitrarily close to feasibility and optimality as they continue to run. Moreover, the computational complexity of each basic operation for these new algorithms is the same as that of existing first-order algorithms running on "simple" feasible sets. Our numerical studies showed that only O(log(1/eps)) basic operations, as opposed to O(1/eps) worst case theoretical bound, are needed for obtaining eps-feasible and eps-optimal solutions. We have implemented these new first-order methods for the following problem classes: Basis Pursuit (BP) in compressed sensing, Matrix Rank Minimization, Principal Component Pursuit (PCP) and Stable Principal Component Pursuit (SPCP) in principal component analysis. These problems have applications in signal and image processing, video surveillance, face recognition, latent semantic indexing, and ranking and collaborative filtering. To best of our knowledge, an algorithm for the SPCP problem that has O(1/eps) iteration complexity and has a per iteration complexity equal to that of a singular value decomposition is given for the first time.Operations research, Applied mathematicsnsa2106Industrial Engineering and Operations ResearchDissertationsEssays in Consumer Choice Driven Assortment Planning
http://academiccommons.columbia.edu/catalog/ac:131420
Saure, Denis R.http://hdl.handle.net/10022/AC:P:10232Thu, 28 Apr 2011 16:02:19 +0000Product assortment selection is among the most critical decisions facing retailers: product variety and relevance is a fundamental driver of consumers' purchase decisions and ultimately of a retailer's profitability. In the last couple of decades an increasing number of firms have gained the ability to frequently revisit their assortment decisions during a selling season. In addition, the development and consolidation of online retailing have introduced new levels of operational flexibility, and cheap access to detailed transactional information. These new operational features present the retailer with both benefits and challenges. The ability to revisit the assortment decision frequently over time allows the retailer to introduce and test new products during the selling season, and adjust on the fly to unexpected changes in consumer preferences, and use customer profile information to customize (in real time) online shopping experience. Our main objective in this thesis is to formulate and solve assortment optimization models addressing the challenges present in modern retail environments. We begin by analyzing the role of the assortment decision in balancing information collection and revenue maximization, when consumer preferences are initially unknown. By considering utility maximizing consumers, we establish fundamental limits on the performance of any assortment policy whose aim is to maximize long run revenues. In addition, we propose adaptive assortment policies that attain such performance limits. Our results highlight salient features of this dynamic assortment problem that distinguish it from similar problems of sequential decision making under model uncertainty. Next, we extend the analysis to the case when additional consumer profile information is available; our primary motivation here is the emerging area of online advertisement. As in the previous setup, we identify fundamental performance limits and propose adaptive policies attaining these limits. Finally, we focus on the effects of competition and consumers' access to information on assortment strategies. In particular, we study competition among retailers when they have access to common products, i.e., products that are available to the competition, and where consumers have full information about the retailers' offerings. Our results shed light on equilibrium properties in such settings and the effect common products have on this behavior.Operations researchdrs2114BusinessDissertations