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Showing papers on "Stochastic programming published in 2011"


BookDOI
27 Jun 2011
TL;DR: This textbook provides a first course in stochastic programming suitable for students with a basic knowledge of linear programming, elementary analysis, and probability to help students develop an intuition on how to model uncertainty into mathematical problems.
Abstract: The aim of stochastic programming is to find optimal decisions in problems which involve uncertain data. This field is currently developing rapidly with contributions from many disciplines including operations research, mathematics, and probability. At the same time, it is now being applied in a wide variety of subjects ranging from agriculture to financial planning and from industrial engineering to computer networks. This textbook provides a first course in stochastic programming suitable for students with a basic knowledge of linear programming, elementary analysis, and probability. The authors aim to present a broad overview of the main themes and methods of the subject. Its prime goal is to help students develop an intuition on how to model uncertainty into mathematical problems, what uncertainty changes bring to the decision process, and what techniques help to manage uncertainty in solving the problems.In this extensively updated new edition there is more material on methods and examples including several new approaches for discrete variables, new results on risk measures in modeling and Monte Carlo sampling methods, a new chapter on relationships to other methods including approximate dynamic programming, robust optimization and online methods.The book is highly illustrated with chapter summaries and many examples and exercises. Students, researchers and practitioners in operations research and the optimization area will find it particularly of interest. Review of First Edition:"The discussion on modeling issues, the large number of examples used to illustrate the material, and the breadth of the coverage make'Introduction to Stochastic Programming' an ideal textbook for the area." (Interfaces, 1998)

5,398 citations


Journal ArticleDOI
TL;DR: A detailed review of the basic concepts of DE and a survey of its major variants, its application to multiobjective, constrained, large scale, and uncertain optimization problems, and the theoretical studies conducted on DE so far are presented.
Abstract: Differential evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms in current use. DE operates through similar computational steps as employed by a standard evolutionary algorithm (EA). However, unlike traditional EAs, the DE-variants perturb the current-generation population members with the scaled differences of randomly selected and distinct population members. Therefore, no separate probability distribution has to be used for generating the offspring. Since its inception in 1995, DE has drawn the attention of many researchers all over the world resulting in a lot of variants of the basic algorithm with improved performance. This paper presents a detailed review of the basic concepts of DE and a survey of its major variants, its application to multiobjective, constrained, large scale, and uncertain optimization problems, and the theoretical studies conducted on DE so far. Also, it provides an overview of the significant engineering applications that have benefited from the powerful nature of DE.

4,321 citations


Book
01 Jan 2011
TL;DR: This chapter discusses Optimization Techniques, which are used in Linear Programming I and II, and Nonlinear Programming II, which is concerned with One-Dimensional Minimization.
Abstract: Preface. 1 Introduction to Optimization. 1.1 Introduction. 1.2 Historical Development. 1.3 Engineering Applications of Optimization. 1.4 Statement of an Optimization Problem. 1.5 Classification of Optimization Problems. 1.6 Optimization Techniques. 1.7 Engineering Optimization Literature. 1.8 Solution of Optimization Problems Using MATLAB. References and Bibliography. Review Questions. Problems. 2 Classical Optimization Techniques. 2.1 Introduction. 2.2 Single-Variable Optimization. 2.3 Multivariable Optimization with No Constraints. 2.4 Multivariable Optimization with Equality Constraints. 2.5 Multivariable Optimization with Inequality Constraints. 2.6 Convex Programming Problem. References and Bibliography. Review Questions. Problems. 3 Linear Programming I: Simplex Method. 3.1 Introduction. 3.2 Applications of Linear Programming. 3.3 Standard Form of a Linear Programming Problem. 3.4 Geometry of Linear Programming Problems. 3.5 Definitions and Theorems. 3.6 Solution of a System of Linear Simultaneous Equations. 3.7 Pivotal Reduction of a General System of Equations. 3.8 Motivation of the Simplex Method. 3.9 Simplex Algorithm. 3.10 Two Phases of the Simplex Method. 3.11 MATLAB Solution of LP Problems. References and Bibliography. Review Questions. Problems. 4 Linear Programming II: Additional Topics and Extensions. 4.1 Introduction. 4.2 Revised Simplex Method. 4.3 Duality in Linear Programming. 4.4 Decomposition Principle. 4.5 Sensitivity or Postoptimality Analysis. 4.6 Transportation Problem. 4.7 Karmarkar's Interior Method. 4.8 Quadratic Programming. 4.9 MATLAB Solutions. References and Bibliography. Review Questions. Problems. 5 Nonlinear Programming I: One-Dimensional Minimization Methods. 5.1 Introduction. 5.2 Unimodal Function. ELIMINATION METHODS. 5.3 Unrestricted Search. 5.4 Exhaustive Search. 5.5 Dichotomous Search. 5.6 Interval Halving Method. 5.7 Fibonacci Method. 5.8 Golden Section Method. 5.9 Comparison of Elimination Methods. INTERPOLATION METHODS. 5.10 Quadratic Interpolation Method. 5.11 Cubic Interpolation Method. 5.12 Direct Root Methods. 5.13 Practical Considerations. 5.14 MATLAB Solution of One-Dimensional Minimization Problems. References and Bibliography. Review Questions. Problems. 6 Nonlinear Programming II: Unconstrained Optimization Techniques. 6.1 Introduction. DIRECT SEARCH METHODS. 6.2 Random Search Methods. 6.3 Grid Search Method. 6.4 Univariate Method. 6.5 Pattern Directions. 6.6 Powell's Method. 6.7 Simplex Method. INDIRECT SEARCH (DESCENT) METHODS. 6.8 Gradient of a Function. 6.9 Steepest Descent (Cauchy) Method. 6.10 Conjugate Gradient (Fletcher-Reeves) Method. 6.11 Newton's Method. 6.12 Marquardt Method. 6.13 Quasi-Newton Methods. 6.14 Davidon-Fletcher-Powell Method. 6.15 Broyden-Fletcher-Goldfarb-Shanno Method. 6.16 Test Functions. 6.17 MATLAB Solution of Unconstrained Optimization Problems. References and Bibliography. Review Questions. Problems. 7 Nonlinear Programming III: Constrained Optimization Techniques. 7.1 Introduction. 7.2 Characteristics of a Constrained Problem. DIRECT METHODS. 7.3 Random Search Methods. 7.4 Complex Method. 7.5 Sequential Linear Programming. 7.6 Basic Approach in the Methods of Feasible Directions. 7.7 Zoutendijk's Method of Feasible Directions. 7.8 Rosen's Gradient Projection Method. 7.9 Generalized Reduced Gradient Method. 7.10 Sequential Quadratic Programming. INDIRECT METHODS. 7.11 Transformation Techniques. 7.12 Basic Approach of the Penalty Function Method. 7.13 Interior Penalty Function Method. 7.14 Convex Programming Problem. 7.15 Exterior Penalty Function Method. 7.16 Extrapolation Techniques in the Interior Penalty Function Method. 7.17 Extended Interior Penalty Function Methods. 7.18 Penalty Function Method for Problems with Mixed Equality and Inequality Constraints. 7.19 Penalty Function Method for Parametric Constraints. 7.20 Augmented Lagrange Multiplier Method. 7.21 Checking the Convergence of Constrained Optimization Problems. 7.22 Test Problems. 7.23 MATLAB Solution of Constrained Optimization Problems. References and Bibliography. Review Questions. Problems. 8 Geometric Programming. 8.1 Introduction. 8.2 Posynomial. 8.3 Unconstrained Minimization Problem. 8.4 Solution of an Unconstrained Geometric Programming Program Using Differential Calculus. 8.5 Solution of an Unconstrained Geometric Programming Problem Using Arithmetic-Geometric Inequality. 8.6 Primal-Dual Relationship and Sufficiency Conditions in the Unconstrained Case. 8.7 Constrained Minimization. 8.8 Solution of a Constrained Geometric Programming Problem. 8.9 Primal and Dual Programs in the Case of Less-Than Inequalities. 8.10 Geometric Programming with Mixed Inequality Constraints. 8.11 Complementary Geometric Programming. 8.12 Applications of Geometric Programming. References and Bibliography. Review Questions. Problems. 9 Dynamic Programming. 9.1 Introduction. 9.2 Multistage Decision Processes. 9.3 Concept of Suboptimization and Principle of Optimality. 9.4 Computational Procedure in Dynamic Programming. 9.5 Example Illustrating the Calculus Method of Solution. 9.6 Example Illustrating the Tabular Method of Solution. 9.7 Conversion of a Final Value Problem into an Initial Value Problem. 9.8 Linear Programming as a Case of Dynamic Programming. 9.9 Continuous Dynamic Programming. 9.10 Additional Applications. References and Bibliography. Review Questions. Problems. 10 Integer Programming. 10.1 Introduction 588. INTEGER LINEAR PROGRAMMING. 10.2 Graphical Representation. 10.3 Gomory's Cutting Plane Method. 10.4 Balas' Algorithm for Zero-One Programming Problems. INTEGER NONLINEAR PROGRAMMING. 10.5 Integer Polynomial Programming. 10.6 Branch-and-Bound Method. 10.7 Sequential Linear Discrete Programming. 10.8 Generalized Penalty Function Method. 10.9 Solution of Binary Programming Problems Using MATLAB. References and Bibliography. Review Questions. Problems. 11 Stochastic Programming. 11.1 Introduction. 11.2 Basic Concepts of Probability Theory. 11.3 Stochastic Linear Programming. 11.4 Stochastic Nonlinear Programming. 11.5 Stochastic Geometric Programming. References and Bibliography. Review Questions. Problems. 12 Optimal Control and Optimality Criteria Methods. 12.1 Introduction. 12.2 Calculus of Variations. 12.3 Optimal Control Theory. 12.4 Optimality Criteria Methods. References and Bibliography. Review Questions. Problems. 13 Modern Methods of Optimization. 13.1 Introduction. 13.2 Genetic Algorithms. 13.3 Simulated Annealing. 13.4 Particle Swarm Optimization. 13.5 Ant Colony Optimization. 13.6 Optimization of Fuzzy Systems. 13.7 Neural-Network-Based Optimization. References and Bibliography. Review Questions. Problems. 14 Practical Aspects of Optimization. 14.1 Introduction. 14.2 Reduction of Size of an Optimization Problem. 14.3 Fast Reanalysis Techniques. 14.4 Derivatives of Static Displacements and Stresses. 14.5 Derivatives of Eigenvalues and Eigenvectors. 14.6 Derivatives of Transient Response. 14.7 Sensitivity of Optimum Solution to Problem Parameters. 14.8 Multilevel Optimization. 14.9 Parallel Processing. 14.10 Multiobjective Optimization. 14.11 Solution of Multiobjective Problems Using MATLAB. References and Bibliography. Review Questions. Problems. A Convex and Concave Functions. B Some Computational Aspects of Optimization. B.1 Choice of Method. B.2 Comparison of Unconstrained Methods. B.3 Comparison of Constrained Methods. B.4 Availability of Computer Programs. B.5 Scaling of Design Variables and Constraints. B.6 Computer Programs for Modern Methods of Optimization. References and Bibliography. C Introduction to MATLAB(R) . C.1 Features and Special Characters. C.2 Defining Matrices in MATLAB. C.3 CREATING m-FILES. C.4 Optimization Toolbox. Answers to Selected Problems. Index .

3,283 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider a point-to-point data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating in a wireless fading channel, and they consider two objectives: maximizing the throughput by a deadline, and minimizing the transmission completion time of the communication session.
Abstract: Wireless systems comprised of rechargeable nodes have a significantly prolonged lifetime and are sustainable. A distinct characteristic of these systems is the fact that the nodes can harvest energy throughout the duration in which communication takes place. As such, transmission policies of the nodes need to adapt to these harvested energy arrivals. In this paper, we consider optimization of point-to-point data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating in a wireless fading channel. We consider two objectives: maximizing the throughput by a deadline, and minimizing the transmission completion time of the communication session. We optimize these objectives by controlling the time sequence of transmit powers subject to energy storage capacity and causality constraints. We, first, study optimal offline policies. We introduce a directional water-filling algorithm which provides a simple and concise interpretation of the necessary optimality conditions. We show the optimality of an adaptive directional water-filling algorithm for the throughput maximization problem. We solve the transmission completion time minimization problem by utilizing its equivalence to its throughput maximization counterpart. Next, we consider online policies. We use stochastic dynamic programming to solve for the optimal online policy that maximizes the average number of bits delivered by a deadline under stochastic fading and energy arrival processes with causal channel state feedback. We also propose near-optimal policies with reduced complexity, and numerically study their performances along with the performances of the offline and online optimal policies under various different configurations.

1,130 citations


Posted Content
TL;DR: This paper considers optimization of point-to-point data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating in a wireless fading channel, and introduces a directional water-filling algorithm which provides a simple and concise interpretation of the necessary optimality conditions.
Abstract: Wireless systems comprised of rechargeable nodes have a significantly prolonged lifetime and are sustainable. A distinct characteristic of these systems is the fact that the nodes can harvest energy throughout the duration in which communication takes place. As such, transmission policies of the nodes need to adapt to these harvested energy arrivals. In this paper, we consider optimization of point-to-point data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating in a wireless fading channel. We consider two objectives: maximizing the throughput by a deadline, and minimizing the transmission completion time of the communication session. We optimize these objectives by controlling the time sequence of transmit powers subject to energy storage capacity and causality constraints. We, first, study optimal offline policies. We introduce a directional water-filling algorithm which provides a simple and concise interpretation of the necessary optimality conditions. We show the optimality of an adaptive directional water-filling algorithm for the throughput maximization problem. We solve the transmission completion time minimization problem by utilizing its equivalence to its throughput maximization counterpart. Next, we consider online policies. We use stochastic dynamic programming to solve for the optimal online policy that maximizes the average number of bits delivered by a deadline under stochastic fading and energy arrival processes with causal channel state feedback. We also propose near-optimal policies with reduced complexity, and numerically study their performances along with the performances of the offline and online optimal policies under various different configurations.

950 citations


Journal ArticleDOI
TL;DR: A two-stage stochastic programming model for committing reserves in systems with large amounts of wind power outperforms common reserve rules and is tested on a model of California consisting of 122 generators.
Abstract: We present a two-stage stochastic programming model for committing reserves in systems with large amounts of wind power. We describe wind power generation in terms of a representative set of appropriately weighted scenarios, and we present a dual decomposition algorithm for solving the resulting stochastic program. We test our scenario generation methodology on a model of California consisting of 122 generators, and we show that the stochastic programming unit commitment policy outperforms common reserve rules.

587 citations


Journal ArticleDOI
TL;DR: In this article, a robust optimization model for handling the inherent uncertainty of input data in a closed-loop supply chain network design problem is proposed, and the robust counterpart of the proposed mixed-integer linear programming model is presented by using the recent extensions in robust optimization theory.

571 citations



Journal ArticleDOI
TL;DR: This paper discusses statistical properties and convergence of the Stochastic Dual Dynamic Programming method applied to multistage linear stochastic programming problems, and argues that the computational complexity of the corresponding SDDP algorithm is almost the same as in the risk neutral case.

399 citations


Journal ArticleDOI
TL;DR: It is proved that, if constraints in the SP problem are optimally removed—i.e., one deletes those constraints leading to the largest possible cost improvement—, then a precise optimality link to the original chance-constrained problem CCP in addition holds.
Abstract: In this paper, we study the link between a Chance-Constrained optimization Problem (CCP) and its sample counterpart (SP). SP has a finite number, say N, of sampled constraints. Further, some of these sampled constraints, say k, are discarded, and the final solution is indicated by $x^{\ast}_{N,k}$ . Extending previous results on the feasibility of sample convex optimization programs, we establish the feasibility of $x^{\ast}_{N,k}$ for the initial CCP problem. Constraints removal allows one to improve the cost function at the price of a decreased feasibility. The cost improvement can be inspected directly from the optimization result, while the theory here developed permits to keep control on the other side of the coin, the feasibility of the obtained solution. In this way, trading feasibility for performance is put on solid mathematical grounds in this paper. The feasibility result here obtained applies to a vast class of chance-constrained optimization problems, and has the distinctive feature that it holds true irrespective of the algorithm used to discard k constraints in the SP problem. For constraints discarding, one can thus, e.g., resort to one of the many methods introduced in the literature to solve chance-constrained problems with discrete distribution, or even use a greedy algorithm, which is computationally very low-demanding, and the feasibility result remains intact. We further prove that, if constraints in the SP problem are optimally removed—i.e., one deletes those constraints leading to the largest possible cost improvement—, then a precise optimality link to the original chance-constrained problem CCP in addition holds.

397 citations


Posted Content
01 Jan 2011
TL;DR: In this paper, robust linear optimization problems with uncertainty regions defined by o-divergences (for example, chi-squared, Hellinger, Kullback-Leibler) are studied.
Abstract: Samenvatting In this paper we focus on robust linear optimization problems with uncertainty regions defined by o-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on o-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with o-divergence uncertainty is tractable for most of the choices of o typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.

Journal ArticleDOI
TL;DR: This paper studies two problems which often occur in various applications arising in wireless sensor networks, and provides a diminishing step size algorithm which guarantees asymptotic convergence of the consensus problem and the problem of cooperative solution to a convex optimization problem.
Abstract: In this paper, we study two problems which often occur in various applications arising in wireless sensor networks. These are the problem of reaching an agreement on the value of local variables in a network of computational agents and the problem of cooperative solution to a convex optimization problem, where the objective function is the aggregate sum of local convex objective functions. We incorporate the presence of a random communication graph between the agents in our model as a more realistic abstraction of the gossip and broadcast communication protocols of a wireless network. An added ingredient is the presence of local constraint sets to which the local variables of each agent is constrained. Our model allows for the objective functions to be nondifferentiable and accommodates the presence of noisy communication links and subgradient errors. For the consensus problem we provide a diminishing step size algorithm which guarantees asymptotic convergence. The distributed optimization algorithm uses two diminishing step size sequences to account for communication noise and subgradient errors. We establish conditions on these step sizes under which we can achieve the dual task of reaching consensus and convergence to the optimal set with probability one. In both cases we consider the constant step size behavior of the algorithm and establish asymptotic error bounds.

Journal ArticleDOI
TL;DR: A novel divide-and-conquer algorithm for generating p-efficient points, used to transform the problem into a set of disjunctive, convex MIPs and handle dual-bounded chance constraints, is proposed.
Abstract: Astochastic, mixed-integer program (MIP) involving joint chance constraints is developed that generates least-cost vehicle redistribution plans for shared-vehicle systems such that a proportion of all near-term demand scenarios are met. The model aims to correct short-term demand asymmetry in shared-vehicle systems, where flow from one station to another is seldom equal to the flow in the opposing direction. The model accounts for demand stochasticity and generates partial redistribution plans in circumstances when demand outstrips supply. This stochastic MIP has a nonconvex feasible region. A novel divide-and-conquer algorithm for generating p-efficient points, used to transform the problem into a set of disjunctive, convex MIPs and handle dual-bounded chance constraints, is proposed. Assuming independence of random demand across stations, a faster cone-generation method is also presented. In a real-world application for a system in Singapore, the potential of redistribution as a fleet management strategy and the value of accounting for inherent stochasticities are demonstrated.

Journal ArticleDOI
Thomas Alerstam1
TL;DR: Using optimality perspectives is now regarded as an essential way of analysing and understanding adaptations and behavioural strategies in bird migration using stochastisch-dynamischer Programmierung, Jahresroutinemodellen and multiobjektiver Optimierung.
Abstract: Using optimality perspectives is now regarded as an essential way of analysing and understanding adaptations and behavioural strategies in bird migration Optimization analyses in bird migration research have diversified greatly during the two recent decades with respect to methods used as well as to topics addressed Methods range from simple analytical and geometric models to more complex modeling by stochastic dynamic programming, annual routine models and multiobjective optimization Also, game theory and simulation by selection algorithms have been used A wide range of aspects of bird migration have been analyzed including flight, fuel deposition, predation risk, stopover site use, transition to breeding, routes and detours, daily timing, fly-and-forage migration, wind selectivity and wind drift, phenotypic flexibility, arrival time and annual molt and migration schedules Optimization analyses have proven to be particularly important for defining problems and specifying questions and predictions about the consequences of minimization of energy, time and predation risk in bird migration Optimization analyses will probably also be important in the future, when predictions about bird migration strategies can be tested by much new data obtained by modern tracking techniques and when the importance of new trade-offs, associated with, eg, digestive physiology, metabolism, immunocompetence and disease, need to be assessed in bird migration research

Journal ArticleDOI
TL;DR: In this article, the authors proposed a methodology to generate a robust logistics plan that can mitigate demand uncertainty in humanitarian relief supply chains by applying robust optimization for dynamically assigning emergency response and evacuation traffic flow problems with time dependent demand uncertainty.
Abstract: This paper proposes a methodology to generate a robust logistics plan that can mitigate demand uncertainty in humanitarian relief supply chains. More specifically, we apply robust optimization (RO) for dynamically assigning emergency response and evacuation traffic flow problems with time dependent demand uncertainty. This paper studies a Cell Transmission Model (CTM) based system optimum dynamic traffic assignment model. We adopt a min–max criterion and apply an extension of the RO method adjusted to dynamic optimization problems, an affinely adjustable robust counterpart (AARC) approach. Simulation experiments show that the AARC solution provides excellent results when compared to deterministic solution and sampling based stochastic programming solution. General insights of RO and transportation that may have wider applicability in humanitarian relief supply chains are provided.

Journal ArticleDOI
TL;DR: This paper proposes an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction: it applies the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program.
Abstract: Linear stochastic programming provides a flexible toolbox for analyzing real-life decision situations, but it can become computationally cumbersome when recourse decisions are involved. The latter are usually modeled as decision rules, i.e., functions of the uncertain problem data. It has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of decision rules to those that exhibit a linear data dependence. In this paper, we propose an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction: we apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program. By employing techniques that are commonly used in modern robust optimization, we show that both arising approximate problems are equivalent to tractable linear or semidefinite programs of moderate sizes. The gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation. Our method remains applicable if the stochastic program has random recourse and multiple decision stages. It also extends to cases involving ambiguous probability distributions.

Journal ArticleDOI
TL;DR: A Monte-Carlo simulation-based algorithm is described that integrates a sample average approximation scheme with a Benders decomposition algorithm to solve problems having stochastic independent transportation costs.

Journal ArticleDOI
TL;DR: A novel two-stage stochastic mixed-integer programming model is presented to minimize total expected operating cost given that scheduling decisions are made before the resolution of uncertainty in surgery durations.
Abstract: Operating room (OR) scheduling is an important operational problem for most hospitals. In this study, we present a novel two-stage stochastic mixed-integer programming model to minimize total expected operating cost given that scheduling decisions are made before the resolution of uncertainty in surgery durations. We use this model to quantify the benefit of pooling ORs as a shared resource and to illustrate the impact of parallel surgery processing on surgery schedules. Decisions in our model include the number of ORs to open each day, the allocation of surgeries to ORs, the sequence of surgeries within each OR, and the start time for each surgeon. Realistic-sized instances of our model are difficult or impossible to solve with standard stochastic programming techniques. Therefore, we exploit several structural properties of the model to achieve computational advantages. Furthermore, we describe a novel set of widely applicable valid inequalities that make it possible to solve practical instances. Based on our results for different resource usage schemes, we conclude that the impact of parallel surgery processing and the benefit of OR pooling are significant. The latter may lead to total cost reductions between 21% and 59% on average.

Journal ArticleDOI
TL;DR: Based on a stochastic programming with recourse framework, the authors incorporates different probabilistic scenarios in the rolling horizon decision process to recognize the input data uncertainty associated with predicted segment running times and segment recovery times and the possibilities of rescheduling decisions after receiving status updates.
Abstract: After a major service disruption on a single-track rail line, dispatchers need to generate a series of train meet-pass plans at different decision times of the rescheduling stage. The task is to recover the impacted train schedule from the current and future disturbances and minimize the expected additional delay under different forecasted operational conditions. Based on a stochastic programming with recourse framework, this paper incorporates different probabilistic scenarios in the rolling horizon decision process to recognize (1) the input data uncertainty associated with predicted segment running times and segment recovery times and (2) the possibilities of rescheduling decisions after receiving status updates. The proposed model periodically optimizes schedules for a relatively long rolling horizon, while selecting and disseminating a robust meet-pass plan for every roll period. A multi-layer branching solution procedure is developed to systematically generate and select meet-pass plans under different stochastic scenarios. Illustrative examples and numerical experiments are used to demonstrate the importance of robust disruption handling under a dynamic and stochastic environment. In terms of expected total train delay time, our experimental results show that the robust solutions are better than the expected value-based solutions by a range of 10–30%.

Journal ArticleDOI
TL;DR: In this article, a decision-making framework based on stochastic programming is proposed for a retailer to determine the sale price of electricity to the customers based on time-of-use (TOU) rates, and manage a portfolio of different contracts in order to procure its demand and to hedge against risks, within a medium-term period.
Abstract: This paper proposes a decision-making framework, based on stochastic programming, for a retailer: 1) to determine the sale price of electricity to the customers based on time-of-use (TOU) rates, and 2) to manage a portfolio of different contracts in order to procure its demand and to hedge against risks, within a medium-term period. Supply sources include the pool, self-production facilities and several instruments such as forward contracts, call options, and interruptible contracts. The objective is to maximize the profit and simultaneously to minimize the risks in terms of a multi-period risk measure. Moreover, the risks are measured using conditional value at risk (CVaR) methodology. The reaction of the customers to the retailers' selling prices as well as the competition between the retailers is modeled through a market share function. The problem is formulated as a mixed-integer stochastic programming. It is solved by a decomposition technique, and the decomposed parts are solved by a branch-and-bound algorithm.

Journal ArticleDOI
TL;DR: By using a finite set to define the visitation angle of a vehicle over a target, this work poses the integrated problem of task assignment and path optimization in the form of a graph, and proposes genetic algorithms for the stochastic search of the space of solutions.

01 Jan 2011
TL;DR: Based on a stochastic programming with recourse framework, the authors incorporates different probabilistic scenarios in the rolling horizon decision process to recognize the input data uncertainty associated with predicted segment running time and segment recovery times, and the possibilities of rescheduling decisions after receiving status updates.
Abstract: After a major service disruption, dispatchers need to continuously generate train meet-pass plans that can recover the impacted train schedule from the current and future disturbances and minimize the expected additional delay under different forecasted operational conditions. Based on a stochastic programming with recourse framework, this paper incorporates different probabilistic scenarios in the rolling horizon decision process to recognize (1) the input data uncertainty associated with predicted segment running time and segment recovery times, and (2) the possibilities of rescheduling decisions after receiving status updates. The proposed model periodically optimizes schedules for a relatively long rolling horizon, while selecting and disseminating a robust meet-pass plan for every roll period. A multi-layer branching solution procedure is developed to systematically generate and select meet-pass plans under different stochastic scenarios. Illustrative examples and numerical experiments are used to demonstrate the importance of robust disruption handling under a dynamic and stochastic environment.

Journal ArticleDOI
TL;DR: This model optimally allocates defensive resources among facilities to minimize the worst-case impact of an intentional disruption and proposes pre-processing techniques based on the computation of valid lower and upper bounds to expedite the solution of instances of realistic size.

Journal ArticleDOI
TL;DR: Resource allocation issues are investigated in this paper for multiuser wireless transmissions based on orthogonal frequency division multiplexing (OFDM) using convex and stochastic optimization tools.
Abstract: Resource allocation issues are investigated in this paper for multiuser wireless transmissions based on orthogonal frequency division multiplexing (OFDM). Relying on convex and stochastic optimization tools, the novel approach to resource allocation includes: i) development of jointly optimal subcarrier, power, and rate allocation for weighted sum-average-rate maximization; ii) judicious formulation and derivation of the optimal resource allocation for maximizing the utility of average user rates; and iii) development of the stochastic resource allocation schemes, and rigorous proof of their convergence and optimality. Simulations are also provided to demonstrate the merits of the novel schemes.

Book
18 Aug 2011
TL;DR: In this article, it was shown that if performance measures in stochastic and dynamic scheduling problems satisfy generalized conservation laws, then the feasible region of achievable performance is a polyhedron called an extended polymatroid that generalizes the classical polymatroids introduced by Edmonds.
Abstract: We show that if performance measures in stochastic and dynamic scheduling problems satisfy generalized conservation laws, then the feasible region of achievable performance is a polyhedron called an extended polymatroid, that generalizes the classical polymatroids introduced by Edmonds. Optimization of a linear objective over an extended polymatroid is solved by an adaptive greedy algorithm, which leads to an optimal solution having an indexability property indexable systems. Under a certain condition the indices possess a stronger decomposition property decomposable systems. The following problems can be analyzed using our theory: multiarmed bandit problems, branching bandits, scheduling of multiclass queues with or without feedback, scheduling of a batch of jobs. Consequences of our results include: 1 a characterization of indexable systems as systems that satisfy generalized conservation laws, 2 a sufficient condition for indexable systems to be decomposable, 3 a new linear programming proof of the decomposability property of Gittins indices in multiarmed bandit problems, 4 an approach to sensitivity analysis of indexable systems, 5 a characterization of the indices of indexable systems as sums of dual variables, and an economic interpretation of the branching bandit indices in terms of retirement options, 6 an analysis of the indexability of undiscounted branching bandits, 7 a new algorithm to compute the indices of indexable systems in particular Gittins indices, as fast as the fastest known algorithm, 8 a unification of Klimov's algorithm for multiclass queues and Gittms' algorithm for multiarmed bandits as special cases of the same algorithm, 9 a closed formula for the maximum reward of the multiarmed bandit problem, with a new proof of its submodularity and 10 an understanding of the invariance of the indices with respect to some parameters of the problem. Our approach provides a polyhedral treatment of several classical problems in stochastic and dynamic scheduling and is able to address variations such as: discounted versus undiscounted cost criterion, rewards versus taxes, discrete versus continuous time, and linear versus nonlinear objective functions.

Proceedings ArticleDOI
09 Oct 2011
TL;DR: The experimental results show that stochastic implementations tolerate more noise and consume less hardware than their conventional counterparts, and the validity of the present stoChastic computational elements is demonstrated through four basic digital image processing algorithms.
Abstract: As device scaling continues to nanoscale dimensions, circuit reliability will continue to become an ever greater problem. Stochastic computing, which performs computing with random bits (stochastic bits streams), can be used to enable reliable computation using those unreliable devices. However, one of the major issues of stochastic computing is that applications implemented with this technique are limited by the available computational elements. In this paper, first we will introduce and prove a stochastic absolute value function. Second, we will demonstrate a mathematical analysis of a stochastic tanh function, which is a key component used in a stochastic comparator. Third, we will present a quantitative analysis of a one-parameter linear gain function, and propose a new two-parameter version. The validity of the present stochastic computational elements is demonstrated through four basic digital image processing algorithms: edge detection, frame difference based image segmentation, median filter based noise reduction, and image contrast stretching. Our experimental results show that stochastic implementations tolerate more noise and consume less hardware than their conventional counterparts.

Journal ArticleDOI
TL;DR: This paper analyzes the corresponding problem arising in the daily operation of the Austrian Red Cross and proposes four different modifications of metaheuristic solution approaches for this problem as a dynamic stochastic dial-a-ride problem with expected return transports.

Journal ArticleDOI
TL;DR: This analysis constitutes so far the most realistic attempt to better understand and approach the real PSO dynamics from a stochastic point of view.
Abstract: Particle swarm optimization (PSO) can be interpreted physically as a particular discretization of a stochastic damped mass-spring system. Knowledge of this analogy has been crucial to derive the PSO continuous model and to introduce different PSO family members including the generalized PSO (GPSO) algorithm, which is the generalization of PSO for any time discretization step. In this paper, we present the stochastic analysis of the linear continuous and generalized PSO models for the case of a stochastic center of attraction. Analysis of the GPSO second order trajectories is performed and clarifies the roles of the PSO parameters and that of the cost function through the algorithm execution: while the PSO parameters mainly control the eigenvalues of the dynamical systems involved, the mean trajectory of the center of attraction and its covariance functions with the trajectories and their derivatives (or the trajectories in the near past) act as forcing terms to update first and second order trajectories. The similarity between the oscillation center dynamics observed for different kinds of benchmark functions might explain the PSO success for a broad range of optimization problems. Finally, a comparison between real simulations and the linear continuous PSO and GPSO models is shown. As expected, the GPSO tends to the continuous PSO when time step approaches zero. Both models account fairly well for the dynamics (first and second order moments) observed in real runs. This analysis constitutes so far the most realistic attempt to better understand and approach the real PSO dynamics from a stochastic point of view.

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TL;DR: A hierarchy of near-optimal polynomial disturbance-feedback control policies is introduced, and it is shown how these can be computed by solving a single semidefinite programming problem.
Abstract: In this paper, we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty, applicable to multistage robust optimization and stochastic programming. We introduce a hierarchy of near-optimal polynomial disturbance-feedback control policies, and show how these can be computed by solving a single semidefinite programming problem. The approach yields a hierarchy parameterized by a single variable (the degree of the polynomial policies), which controls the trade-off between the optimality gap and the computational requirements. We evaluate our framework in the context of three classical applications-two in inventory management, and one in robust regulation of an active suspension system-in which very strong numerical performance is exhibited, at relatively modest computational expense.

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TL;DR: In this article, a two-stage stochastic programming (SP) model and a chance-constrained programming (CCP) model are developed to determine a minimal set of suppliers and optimal order quantities with consideration of business volume discounts.