Stochastic Dynamic Problem Research Articles

Polynomial Correlated Function Expansion for Nonlinear Stochastic Dynamic Analysis

AbstractA new computational tool, referred to as the polynomial correlated function expansion (PCFE), has been developed for predicting response statistics of nonlinear stochastic dynamic problems. This method facilitates a systematic mapping between the input and output by expressing the output as a ranked order of component functions, with higher-order component functions representing higher-order cooperative effects. The component functions are expressed in terms of extended bases, and the unknown coefficients associated with the basis are determined by using a homotopy algorithm. This algorithm considers the hierarchical orthogonality of the component functions as a constraint and determines the coefficients associated with the basis. Implementation of the proposed approach for nonlinear stochastic dynamic problems has been demonstrated with three numerical problems. Results obtained certify the accuracy of the proposed method. Furthermore, the proposed approach significantly reduces the number of cal...

Multimodularity and Its Applications in Three Stochastic Dynamic Inventory Problems

We apply the concept of multimodularity in three stochastic dynamic inventory problems in which state and decision variables are economic substitutes. The first is clearance sales of perishable goods. The second is sourcing from multiple suppliers with different lead times. The third is transshipment under capacity constraints. In all three problems, we establish monotone optimal polices with bounded sensitivity. Multimodularity proves to be an effective tool for these problems because it implies substitutability, it is preserved under minimization, and it leads directly to monotone optimal policies with bounded sensitivity.

Performance Analysis of Simulation of Pedestrian Traffic inside the Hub

Transport hub belongs to the people highly gathering places,the pedestrian traffic within the hub is a nonlinear stochastic dynamic problem with uncertain cluster density. The computer dynamic simulation is one of the effective methods to investigate the dynamic characteristics of pedestrian traffic flow at present within the hub.The article studies the error between pedestrian traffic simulation of the commuter simulation software and the actual pedestrian traffic,and explains the simulation error due to the discrete space, fidelity are introduced to evaluate the accuracy of the simulation model.By comparing the Hohhot underground tunnel experimental data and simulation results and the analysis of pedestrian traffic characteristics, it is concluded that the pedestrian traffic simulation performance is closely related to the environment in the hub and the importance of the fidelity of simulation model.

Research on Traffic Behaviors and Psychological Characteristics of Pedestrians within the Comprehensive Hub

Traffic terminal is a spot of high density for people’s traffic activities,the pedestrian transportation in the traffic terminal can be considered as a bunching density un-sure and nonlinear stochastic dynamics problem of interaction between each individual.pedestrian,as the primary part that composed all activities in the traffic terminal,has a significant influence on traffic behaviors and psychic properties.In this paper,we proposal a series of rationalized improvement measures to relief congestion and promote pedestrian security in the traffic terminal,by analysis on travel environment and traffic behaviors properties of pedestrian and deeply studying their traffic psychic properties

Dimension Reduction of the FPK Equation via an Equivalence of Probability Flux for Additively Excited Systems

AbstractSolving the Fokker-Planck-Kolmogorov (FPK) equation is one of the most important and challenging problems in high-dimensional nonlinear stochastic dynamics, which is widely encountered in various science and engineering disciplines. To date, no method available is capable of dealing with systems of dimensions higher than eight. The present paper aims at tackling this problem in a different way, i.e., reducing the dimension of the FPK equation by constructing an equivalent probability flux. In the paper, two different treatments for multidimensional stochastic dynamical systems, the FPK equation and the probability density evolution method (PDEM), are outlined. Particularly, the FPK equation is revisited in a completely new way by constructing the probability fluxes based on the embedded dynamics mechanism and then invoking the principle of preservation of probability. The FPK equation is then marginalized to reduce the dimension, resulting in a flux-form equation involving unknown probability flux...

PARTITIONS-REQUIREMENTS-MATRICES AS OPTIMAL MARKOV KERNELS OF SPECIAL STOCHASTIC DYNAMIC DISTANCE OPTIMAL PARTITIONING PROBLEMS

The Stochastic Dynamic Distance Optimal Partitioning problem (SDDP problem) is a complex Operations Research problem. The SDDP prob- lem is based on an industrial problem, which contains an optimal conversion of machines. Partitions of integers as states of these stochastic dynamic programming problems involves combinatorial aspects of SDDP problems. Under the as- sumption of identical (in other words of unit distances) and independent and identically distributed requirements we will show (in many cases) by means of combinatorial ideas that decisions for feasible states with least square sums of their parts are optimal solutions. Corresponding Markov kernels are called partitions-Requirements-Matrices (PRMs). Optimal decisions of such problems can be used as approximate solutions of corresponding SDDP problems, in which the basic costs differ only slightly from each other or as starting decisions if corresponding SDDP problems are solved by iterative methods, such as the Howard algorithm.

Control and dynamics of a SDOF system with piecewise linear stiffness and combined external excitations

The paper considers a problem of stochastic control and dynamics of a single-degree-of-freedom system with piecewise linear stiffness subjected to combined periodic and white noise external excitations. To minimize the system response energy a bounded in magnitude control force is applied to the systems. The stochastic optimal control problem is handled through the dynamic programming approach. Based on the solution to the Hamilton–Jacobi–Bellman equation it is proposed to use the dry friction control law in the non-resonant case. In the resonant case the stochastic averaging procedure has been used to derive stochastic differential equations for system response amplitude and phase. The joint PDF of response amplitude and phase is derived analytically and numerically using the Path Integration approach.

Predictive Information in a Nonequilibrium Critical Model

We propose predictive information, that is information between a long past of duration T and the entire infinitely long future of a time series, as a universal order parameter to study phase transitions in physical systems. It can be used, in particular, to study nonequlibrium transitions and other exotic transitions, where a simpler order parameter cannot be identifies using traditional symmetry arguments. As an example, we calculate predictive information for a stochastic nonequilibrium dynamics problem that forms an absorbing state under a continuous change of a parameter. The information at the transition point diverges as log(T), and a smooth crossover to constant away from the transition is observed.

Stochastic capacity planning and dynamic network design

Abstract The present paper proposes a mathematical model and a solution approach to the Stochastic Capacity Planning and Dynamic Network Design Problem. Here, strategic decisions usually comprise developing the necessary capacity — through either incrementing capacity on existing assets (facilities or logistics channels) or establishing new capacity in the form of new assets — in order to satisfy increasing demand. Hence, throughout the planning horizon, decisions on which new assets to establish and where to increment capacity must be taken at minimal cost and in a timely manner. However, when demand varies nonmonotonically, decisions on which assets to temporarily shut down in times of decreasing demand and which of those to reopen when market conditions improve must also be taken into account. We propose a multi-stage stochastic mixed-integer programming approach to the problem as well as a Lagrangian Heuristic procedure to attain reasonably well bounded feasible solutions. The proposed method is evaluated in a Global Mining Supply Chain context which, due to the inherently large capital expenses, could have the outcome of its strategic decision making process significantly improved.

Application of Finite Volume Method to Structural Stochastic Dynamics

The stochastic dynamic problems were becoming more difficult after considering the influences of stochastic factors and the complexity of the dynamic problems. To this background, the finite volume method combined with Perturbation Method was proposed for the stochastic dynamic analysis. The equations of perturbation-finite volume method were derived; the explicit expressions between random response and basic random variables were given; the method of stochastic dynamic analysis was discussed; and the examples were presented to verify the perturbation-finite volume method. The results of perturbation-finite volume method were compared with the results of Monte Carlo Method, which proved that the proposed method was correct and accurate. Because the proposed method was simple and clear, the equations were easy to establish and the efficiency was improved. Meanwhile, the proposed method was successfully applied to the stochastic dynamic analysis of linear multibody system, which was verified through the example in this paper.

An Advanced Heuristic for Multiple‐Option Spare Parts Procurement after End‐of‐Production

After‐sales service is a major source of profit for many original equipment manufacturers in industries with durable products. Successful engagement in after‐sales service improves customer loyalty and allows for competitive differentiation through superior service like an extended service period during which customers are guaranteed to be provided with service parts. Inventory management during this period is challenging due to the substantial uncertainty concerning demand over a long time horizon. The traditional mechanism of spare parts acquisition is to place a large final order at the end of regular production of the parent product, causing major holding costs and a high level of obsolescence risk. With an increasing length of the service period, more flexibility is needed and can be provided by adding options like extra production and remanufacturing. However, coordinating all three options yields a complicated stochastic dynamic decision problem. For that problem type, we show that a quite simple decision rule with order‐up‐to levels for extra production and remanufacturing is very effective. We propose a heuristic procedure for parameter determination which accounts for the main stochastic and dynamic interactions in decision making, but still consists of relatively simple calculations that can be applied to practical problem sizes. A numerical study reveals that the heuristic performs extremely well under a wide range of conditions, and therefore can be strongly recommended as a decision support tool for the multi‐option spare parts procurement problem. A comparison with decision rules adapted from practice demonstrates that our approach offers an opportunity for major cost reductions.

Interactive procedure for a multiobjective stochastic discrete dynamic problem

Multiple objectives and dynamics characterize many sequential decision problems. In the paper we consider returns in partially ordered criteria space as a way of generalization of single criterion dynamic programming models to multiobjective case. In our problem evaluations of alternatives with respect to criteria are represented by distribution functions. Thus, the overall comparison of two alternatives is equivalent to the comparison of two vectors of probability distributions. We assume that the decision maker tries to find a solution preferred to all other solutions (the most preferred solution). In the paper a new interactive procedure for stochastic, dynamic multiple criteria decision making problem is proposed. The procedure consists of two steps. First, the Bellman principle is used to identify the set of efficient solutions. Next interactive approach is employed to find the most preferred solution. A numerical example and a real-world application are presented to illustrate the applicability of the proposed technique.

In This Issue

In This Issue

Analytic modelling and dynamic analysis of pole-changing line-start permanent-magnet motors

Pole-changing of line-start permanent magnet motors at start-up has many advantages and improves starting torque, synchronisation capability and steady-state performance. However, some oscillations may occur after switching which may cause dynamic problems. Depending on the speed and torque angle at switching time, the magnitude of oscillations may be large and even may lead to instability and reoccurrence of the sub-synchronous mode. In this study an analytic modelling is implemented to analyse the dynamic performance of motor with pole-changing method and the results have been verified by the 2-D finite-element method. In the analysis the various impacts of parameters such as switching speed, torque angle, inertia and load torque on the synchronisation capability are investigated. Next, two solutions are presented that overcome the stochastic dynamic problems and cause a suitable synchronisation. Finally, the experimental results are presented to verify the validity of the proposed methods and the overcoming of the dynamic problems.

Small noise methods for risk-sensitive/robust economies

We provide small noise expansions for the value function and decision rule for the recursive risk-sensitive preferences specified by Hansen and Sargent (1995), Hansen et al. (1999), and Tallarini (2000). We use the expansions (1) to provide a fast method for approximating solutions of dynamic stochastic problems and (2) to quantify the effects on decisions of uncertainty and concerns about robustness to misspecification.

Numerical simulation methods for the Rouse model in flow

Simulation of the Rouse model in flow underlies a great variety of numerical investigations of polymer dynamics, in both entangled melts and solutions and in dilute solution. Typically a simple explicit stochastic Euler method is used to evolve the Rouse model. Here we compare this approach to an operator splitting method, which splits the evolution operator into stochastic linear and deterministic nonlinear parts and takes advantage of an analytical solution for the linear Rouse model in terms of the noise history. We show that this splitting method has second-order weak convergence, whereas the Euler method has only first-order weak convergence. Furthermore, the splitting method is unconditionally stable, in contrast to the limited stability range of the Euler method. Similar splitting methods are applicable to a broad class of problems in stochastic dynamics in which noise competes with ordering and flow to determine steady-state order parameter structures.

Solving the stochastic dynamic lot-sizing problem through nature-inspired heuristics

We investigate the dynamic lot-size problem under stochastic and non-stationary demand over the planning horizon. The problem is tackled by using three popular heuristic methods from the fields of evolutionary computation and swarm intelligence, namely particle swarm optimization, differential evolution and harmony search. To the best of the authors' knowledge, this is the first investigation of the specific problem with approaches of this type. The algorithms are properly manipulated to fit the requirements of the problem. Their performance, in terms of run-time and solution accuracy, is investigated on test cases previously used in relevant works. Specifically, the lot-size problem with normally distributed demand is considered for different planning horizons, varying from 12 up to 48 periods. The obtained results are analyzed, providing evidence on the efficiency of the employed approaches as promising alternatives to the established Wagner–Whitin algorithm, as well as hints on their proper configuration.

An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of a class of randomly excited nonlinear oscillators is developed. Specifically, combining the concepts of statistical linearization and of stochastic averaging the evolution of the response amplitude of oscillators with nonlinear damping is captured by a first-order stochastic differential equation (SDE). This equation has nonlinear drift but constant diffusion coefficients. This convenient feature of the SDE along with the concept of the Wiener path integral is utilized in conjunction with a variational formulation to derive an approximate closed form solution for the response amplitude PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes. In this manner, an analytical Wiener path integral based technique is developed for treating certain stochastic dynamics problems for the first time. Further, the technique can be used as a convenient tool for assessing the accuracy of alternative, more general, approximate solution methods. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.

Stochastic dynamic lot-sizing problem using bi-level programming base on artificial intelligence techniques

Simulation is generally used to study non-deterministic problems in industry. When a simulation process finds the solution to an NP-hard problem, its efficiency is lowered, and computational costs increase. This paper proposes a stochastic dynamic lot-sizing problem with asymmetric deteriorating commodity, in which the optimal unit cost of material and unit holding cost would be determined. This problem covers a sub-problem of replenishment planning, which is NP-hard in the computational complexity theory. Therefore, this paper applies a decision system, based on an artificial neural network (ANN) and modified ant colony optimization (ACO) to solve this stochastic dynamic lot-sizing problem. In the methodology, ANN is used to learn the simulation results, followed by the application of a real-valued modified ACO algorithm to find the optimal decision variables. The test results show that the intelligent system is applicable to the proposed problem, and its performance is better than response surface methodology.

Network Cargo Capacity Management

We consider the problem faced by an airline that is flying both passengers and cargo over a network of locations on a fixed periodic schedule. Bookings for many classes of cargo shipments between origin-destination pairs in this network are made in advance, but the weight and volume of aircraft capacity available for cargo as well as the exact weight and volume of each shipment are not known at the time of booking. The problem is to control cargo accept/reject decisions to maximize expected profits while ensuring effective dispatch of accepted shipments through the network. This network stochastic dynamic control problem has very high computational complexity. We propose a linear programming and stochastic simulation-based computational method for learning approximate control policies and discuss their structural properties. The proposed method is flexible and can utilize historical booking data as well as decisions generated by default control policies.