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Showing papers on "Stochastic programming published in 2004"


Journal ArticleDOI
TL;DR: In this paper, the authors propose an approach that attempts to make this trade-off more attractive by flexibly adjusting the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations.
Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.

3,364 citations


Journal ArticleDOI
TL;DR: A variation on the traditional PSO algorithm, called the cooperative particle swarm optimizer, or CPSO, employing cooperative behavior to significantly improve the performance of the original algorithm.
Abstract: The particle swarm optimizer (PSO) is a stochastic, population-based optimization technique that can be applied to a wide range of problems, including neural network training. This paper presents a variation on the traditional PSO algorithm, called the cooperative particle swarm optimizer, or CPSO, employing cooperative behavior to significantly improve the performance of the original algorithm. This is achieved by using multiple swarms to optimize different components of the solution vector cooperatively. Application of the new PSO algorithm on several benchmark optimization problems shows a marked improvement in performance over the traditional PSO.

2,038 citations


Journal ArticleDOI
TL;DR: This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty and discusses and contrast the classical recourse-based stochastic programming, robust stochastics programming, probabilistic (chance-constraint) programming, fuzzy programming, and stochastically dynamic programming.

1,145 citations


Journal ArticleDOI
TL;DR: A new theory of pedestrian behavior under uncertainty based on the concept of utility maximization is put forward, which proposes a trade-off between the utility gained from performing activities at a specific location and the predicted cost of walking subject to the physical limitations of the pedestrians and the kinematics of the pedestrian.
Abstract: Among the most interesting and challenging theoretical and practical problems in describing pedestrians behavior are route choice and activity scheduling. Compared to other modes of transport, a characteristic feature of pedestrian route choice is that routes are continuous trajectories in time and space: since a pedestrian chooses a route from an infinite set of alternatives, dedicated theories and models describing pedestrian route choice are required. This article puts forward a new theory of pedestrian behavior under uncertainty based on the concept of utility maximization. The main behavioral assumption is that pedestrians optimize some predicted pedestrian-specific utility function, representing a trade-off between the utility gained from performing activities at a specific location, and the predicted cost of walking subject to the physical limitations of the pedestrians and the kinematics of the pedestrian. The uncertainty reflects the randomness of the experienced traffic conditions. Based on this normative theory, route choice, activity area choice, and activity scheduling are simultaneously optimized using dynamic programming for different traffic conditions and uncertainty levels. Throughout the article, the concepts are illustrated by examples.

757 citations


Journal ArticleDOI
TL;DR: This study proposes a two-stage stochastic programming model to plan the transportation of vital first-aid commodities to disaster-affected areas during emergency response, and a multi-commodity, multi-modal network flow formulation is developed to describe the flow of material over an urban transportation network.
Abstract: This study proposes a two-stage stochastic programming model to plan the transportation of vital first-aid commodities to disaster-affected areas during emergency response. A multi-commodity, multi-modal network flow formulation is developed to describe the flow of material over an urban transportation network. Since it is difficult to predict the timing and magnitude of any disaster and its impact on the urban system, resource mobilization is treated in a random manner, and the resource requirements are represented as random variables. Furthermore, uncertainty arising from the vulnerability of the transportation system leads to random arc capacities and supply amounts. Randomness is represented by a finite sample of scenarios for capacity, supply and demand triplet. The two stages are defined with respect to information asymmetry, which discloses uncertainty during the progress of the response. The approach is validated by quantifying the expected value of perfect and stochastic information in problem instances generated out of actual data.

696 citations


Journal ArticleDOI
TL;DR: A general classification of mathematical optimization problems is provided, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering.

566 citations


Journal ArticleDOI
TL;DR: This work explicitly characterize the robust counterpart of a linear programming problem with uncertainty set described by an arbitrary norm as well as providing guarantees for constraint violation under probabilistic models that allow arbitrary dependencies in the distribution of the uncertain coefficients.

489 citations


Proceedings ArticleDOI
01 Jan 2004
TL;DR: In this paper, a stochastic dynamic programming (SDP) approach was used to obtain the optimal supervisory control strategy for hybrid vehicles with random Markov processes. But the resulting control strategy was often inherently cycle-beating and lacked a guaranteed level of optimality.
Abstract: The supervisory control strategy of a hybrid vehicle coordinates the operation of vehicle sub-systems to achieve performance targets such as maximizing fuel economy and reducing exhaust emissions. This high-level control problem is commonly referred as the power management problem. In the past, many supervisory control strategies were developed on the basis of a few pre-defined driving cycles, using intuition and heuristics. The resulting control strategy was often inherently cycle-beating and lacked a guaranteed level of optimality. In this study, the power management problem is tackled from a stochastic viewpoint. An infinite-horizon stochastic dynamic optimization problem is formulated. The power demand from the driver is modeled as a random Markov process. The optimal control strategy is then obtained by using stochastic dynamic programming (SDP). The obtained control law is in the form of a stationary full-state feedback and can be directly implemented. Simulation results over standard driving cycles and random driving cycles are presented to demonstrate the effectiveness of the proposed stochastic approach. It was found that the obtained SDP control algorithm outperforms a sub-optimal rule-based control strategy trained from deterministic DP results.

488 citations



Journal ArticleDOI
TL;DR: In this article, the robust design of structures with stochastic parameters is studied using optimization techniques and the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints.

296 citations


Journal ArticleDOI
TL;DR: The structure of the value function of the second-stage integer problem is exploited to develop a novel global optimization algorithm that avoids explicit enumeration of the search space while guaranteeing finite termination.
Abstract: This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature.

Journal ArticleDOI
TL;DR: This work considers the optimal investment and operational planning of gas field developments under uncertainty in gas reserves and presents a novel stochastic programming model that incorporates the decision-dependence of the scenario tree.

Journal ArticleDOI
TL;DR: This work forms a Markov decision process model of the stochastic inventory routing problem and proposes approximation methods to find good solutions with reasonable computational effort and indicates how the proposed approach can be used for other Markov decisions involving the control of multiple resources.
Abstract: This work is motivated by the need to solve the inventory routing problem when implementing a business practice called vendor managed inventory replenishment (VMI). With VMI, vendors monitor their customers' inventories and decide when and how much inventory should be replenished at each customer. The inventory routing problem attempts to coordinate inventory replenishment and transportation in such a way that the cost is minimized over the long run. We formulate a Markov decision process model of the stochastic inventory routing problem and propose approximation methods to find good solutions with reasonable computational effort. We indicate how the proposed approach can be used for other Markov decision processes involving the control of multiple resources.

Journal ArticleDOI
TL;DR: A branch-and-price method to solve special structured multistage stochastic integer programming problems and computational results suggest that both classes of problems can be solved using relatively few nodes of a branch- and-price tree.
Abstract: In this paper, we present a branch-and-price method to solve special structured multistage stochastic integer programming problems. We validate our method on two different versions of a multistage stochastic batch-sizing problem (SBSP). One version adopts a recourse formulation, and the other is based on probabilistic constraints. Our algorithmic approach is applicable to both formulations. Our computational results suggest that both classes of problems can be solved using relatively few nodes of a branch-and-price tree. The success of our approach calls for extensions in methodology as well as applications.

Journal ArticleDOI
TL;DR: In this paper, a methodology for financial risk management in the framework of two-stage stochastic programming for planning under uncertainty is presented, where a known probabilistic definition of financial risk is adapted to be used in this framework and its relation to downside risk is analyzed.
Abstract: A methodology is presented to include financial risk management in the framework of two-stage stochastic programming for planning under uncertainty. A known probabilistic definition of financial risk is adapted to be used in this framework and its relation to downside risk is analyzed. Using these definitions, new two-stage stochastic programming models that manage financial risk are presented. Computational issues related to these models are also discussed. © 2004 American Institute of Chemical Engineers AIChE J, 50: 963–989, 2004

Journal ArticleDOI
TL;DR: In this paper, three approaches are presented for generating scenario trees for 3nancial portfolio problems based on simulation, optimization and hybrid simulation/optimization.

Journal ArticleDOI
TL;DR: A stochastic programming model with probabilistic constraints aimed to solve both the location and the dimensioning problems in order to achieve a reliable level of service and minimize the overall costs is developed.

Journal ArticleDOI
TL;DR: In this article, a two-stage integer stochastic program with recourse where the first stage variables determine which products to produce and how much to produce, and the second stage variables decide how the products are allocated to satisfy the realized demand is considered.
Abstract: In this paper we consider a single period multi-product inventory problem with stochastic demand, setup cost for production, and one-way product substitution in the downward direction. We model the problem as a two-stage integer stochastic program with recourse where the first stage variables determine which products to produce and how much to produce, and the second stage variables determine how the products are allocated to satisfy the realized demand. We exploit structural properties of the model and utilize a combination of optimization techniques including network flow, dynamic programming, and simulation-based optimization to develop effective heuristics. Through a computational study, we evaluate the performance of our heuristics by comparison with the corresponding optimal solution obtained from a large scale mixed integer linear program. The computational study indicates that our solution methodology can be very effective (98.8% on average) and can handle industrial-sized problems efficiently. We...

Journal ArticleDOI
TL;DR: This work proposes the use of sequences of separable, piecewise linear approximations for solving nondifferentiable stochastic optimization problems, and proves the convergence of several versions of such methods when the objective function is separable and has integer break points.
Abstract: We propose the use of sequences of separable, piecewise linear approximations for solving nondifferentiable stochastic optimization problems. The approximations are constructed adaptively using a combination of stochastic subgradient information and possibly sample information on the objective function itself. We prove the convergence of several versions of such methods when the objective function is separable and has integer break points, and we illustrate their behavior on numerical examples. We then demonstrate the performance on nonseparable problems that arise in the context of two-stage stochastic programming problems, and demonstrate that these techniques provide near-optimal solutions with a very fast rate of convergence compared with other solution techniques.

Journal ArticleDOI
TL;DR: For a particular class of minimax stochastic programming models, it is shown that the problem can be equivalently reformulated into a standard stochastics programming problem, which permits the direct use of standard decomposition and sampling methods developed for stochastically programming.
Abstract: For a particular class of minimax stochastic programming models, we show that the problem can be equivalently reformulated into a standard stochastic programming problem. This permits the direct use of standard decomposition and sampling methods developed for stochastic programming. We also show that this class of minimax stochastic programs is closely related to a large family of mean-risk stochastic programs where risk is measured in terms of deviations from a quantile.

Book ChapterDOI
Yaochu Jin1, Bernhard Sendhoff1
TL;DR: This paper suggests a method for constructing dynamic optimization test problems using multi-objective optimization (MOO) concepts that is computationally efficient, easily tunable and functionally powerful.
Abstract: Dynamic optimization using evolutionary algorithms is receiving increasing interests. However, typical test functions for comparing the performance of various dynamic optimization algorithms still lack. This paper suggests a method for constructing dynamic optimization test problems using multi-objective optimization (MOO) concepts. By aggregating different objectives of an MOO problem and changing the weights dynamically, we are able to construct dynamic single objective and multi-objective test problems systematically. The proposed method is computationally efficient, easily tunable and functionally powerful. This is mainly due to the fact that the proposed method associates dynamic optimization with multi-objective optimization and thus the rich MOO test problems can easily be adapted to dynamic optimization test functions.

Book
01 Jan 2004
TL;DR: This book emphasises the dos and don'ts of stochastic calculus, cautioning the reader that certain results and intuitions cherished by many economists do not extend to stochastics models.
Abstract: List of figures Preface 1. Probability theory 2. Wiener processes 3. Stochastic calculus 4. Stochastic dynamic programming 5. How to solve it 6. Boundaries and absorbing barriers Appendix. Miscellaneous applications and exercises Bibliography Index.

Book ChapterDOI
07 Jun 2004
TL;DR: In this paper, the authors studied the design of approximation algorithms for stochastic combinatorial optimization problems, and formulated the problems in the framework of two-staged optimization and provided nearly tight approximations.
Abstract: We study the design of approximation algorithms for stochastic combinatorial optimization problems. We formulate the problems in the framework of two-stage stochastic optimization, and provide nearly tight approximations. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios.

Journal ArticleDOI
TL;DR: This paper shows that, under arbitrary measures for variability, the robust optimization approach might lead to suboptimal solutions to the second-stage planning problem, and proposes sufficient conditions on the variability measure to remedy this problem.
Abstract: Robust-optimization models belong to a special class of stochastic programs, where the traditional expected cost minimization objective is replaced by one that explicitly addresses cost variability. This paper explores robust optimization in the context of two-stage planning systems. We show that, under arbitrary measures for variability, the robust optimization approach might lead to suboptimal solutions to the second-stage planning problem. As a result, the variability of the second-stage costs may be underestimated, thereby defeating the intended purpose of the model. We propose sufficient conditions on the variability measure to remedy this problem. Under the proposed conditions, a robust optimization model can be efficiently solved using a variant of the L-shaped decomposition algorithm for traditional stochastic linear programs. We apply the proposed framework to standard stochastic-programming test problems and to an application that arises in auctioning excess electric power.

Proceedings ArticleDOI
17 Oct 2004
TL;DR: This work gives the first approximation algorithms for 2-stage discrete stochastic optimization problems with recourse for which the underlying random data is given by a "black box" and no restrictions are placed on the costs in the two stages, based on an FPRAS for the LP relaxation of the Stochastic problem (which has exponentially many variables and constraints).
Abstract: Stochastic optimization problems attempt to model uncertainty in the data by assuming that (part of) the input is specified in terms of a probability distribution. We consider the well-studied paradigm of 2-stage models with recourse: first, given only distributional information about (some of) the data one commits on initial actions, and then once the actual data is realized (according to the distribution), further (recourse) actions can be taken. We give the first approximation algorithms for 2-stage discrete stochastic optimization problems with recourse for which the underlying random data is given by a "black box" and no restrictions are placed on the costs in the two stages, based on an FPRAS for the LP relaxation of the stochastic problem (which has exponentially many variables and constraints). Among the range of applications we consider are stochastic versions of the set cover, vertex cover, facility location, multicut (on trees), and multicommodity flow problems.

Journal ArticleDOI
TL;DR: A new algorithm for the stochastic unit commitment problem which is based on column generation approach is proposed which continues adding schedules from the dual solution of the restricted linear master program until the algorithm cannot generate new schedules.

Journal ArticleDOI
TL;DR: This paper describes a framework for modeling significant financial planning problems based on multi-stage optimization under uncertainty based on interior-point methods and possesses a special structure that lends itself to parallel and distributed optimization algorithms.

Journal ArticleDOI
TL;DR: A new splitting approach is developed to these models, optimality conditions and duality theory, which is used to construct special decomposition methods for stochastic dominance constraints of second order.
Abstract: We consider a new class of optimization problems involving stochastic dominance constraints of second order. We develop a new splitting approach to these models, optimality conditions and duality theory. These results are used to construct special decomposition methods.

Journal ArticleDOI
TL;DR: In this paper, a robust optimization model is developed to solve the aggregate production planning problems in an environment of uncertainty in which the production cost, labour cost, inventory cost, and hiring and layoff cost are minimized.
Abstract: The aggregate production planning (APP) problem considers the medium-term production loading plans subject to certain restrictions such as production capacity and workforce level. It is not uncommon for management to often encounter uncertainty and noisy data, in which the variables or parameters are stochastic. In this paper, a robust optimization model is developed to solve the aggregate production planning problems in an environment of uncertainty in which the production cost, labour cost, inventory cost, and hiring and layoff cost are minimized. By adjusting penalty parameters, decision-makers can determine an optimal medium-term production strategy including production loading plan and workforce level while considering different economic growth scenarios. Numerical results demonstrate the robustness and effectiveness of the proposed model. The proposed model is realistic for dealing with uncertain economic conditions. The analysis of the tradeoff between solution robustness and model robustness is al...

Journal ArticleDOI
TL;DR: This paper presents a solution approach to the dynamic vehicle scheduling problem, where a "cluster-reschedule" heuristic is used, where trips are assigned to depots by solving the static problem and then solved dynamic single-depot problems.
Abstract: This paper presents a solution approach to the dynamic vehicle scheduling problem. This approach consists of solving a sequence of optimization problems, where we take into account different scenarios for future travel times. We discuss the potential benefit of our approach compared to the traditional one, where the vehicle scheduling problem is solved only once for a whole period and the travel times are assumed to be fixed. Because in the multiple-depot case we cannot solve the problem exactly within reasonable computation time, we use a "cluster-reschedule" heuristic where we first assign trips to depots by solving the static problem and then solve dynamic single-depot problems. We use new mathematical formulations of these problems that allow fast solution by standard optimization software. Results of a computational study with real-life data are presented, in which we compare different variants of our approach and perform a sensitivity analysis with respect to deviations of the actual travel times from estimated ones.