Stochastic model for the nonlinear point reactor kinetics equations in the presence Newtonian temperature feedback effects

A stochastic model for the nonlinear point reactor kinetics equations with Newtonian temperature feedback and multi-group of precursor delayed neutrons is presented. This model is a couple of the stiff stochastic nonlinear differential equations. The matrix formula of this stochastic nonlinear model is solved by the analytical exponential technique (AET). This proposed technique is based on the integration factor, Euler’s method and the exponential function of the coefficient matrix. This exponential function is determined via the eigenvalues and corresponding eigenvectors of the coefficient matrix. The mean neutron population of the stochastic nonlinear model in the presence Newtonian temperature feedback and six-groups of delayed neutrons is computed for various cases of the external reactivity. The numerical results of the analytical exponential technique are compared with the results of the Euler–Maruyama method and the deterministic results. This comparison confirms that the AET for stochastic nonlinear model is efficient to study the natural behavior of neutron population in the presence temperature feedback effects and multi-group of precursor delayed neutrons.     

Chaos prediction and control of Goodwin’s nonlinear accelerator model

Chaos prediction and its control of the Goodwin model under the deterministic or stochastic excitation are studied theoretically and numerically. Applying the Melnikov technique, the threshold conditions for the occurrence of chaos are obtained theoretically. The stable and unstable manifolds of saddle are computed to verify the effectiveness of the analytical prediction in the deterministic case. Also, the safe basins are introduced to show how the externally stochastic perturbation affects the safety of the economic system as the noise amplitude increases. Finally, the analytical criterion of controlling chaos is derived via the delayed feedback control method. Numerical investigations including the top Lyapunov exponent, Poincare section, and phase portraits are carried out to demonstrate the validity and effectiveness of the theoretical results.     

Exponential stabilization by delay feedback control for highly nonlinear hybrid stochastic functional differential equations with infinite delay

Given an unstable hybrid stochastic functional differential equation, how to design a delay feedback controller to make it stable? Some results have been obtained for hybrid systems with finite delay. However, the state of many stochastic differential equations are related to the whole history of the system, so it is necessary to discuss the feedback control of stochastic functional differential equations with infinite delay. On the other hand, in many practical stochastic models, the coefficients of these systems do not satisfy the linear growth condition, but are highly nonlinear. In this paper, the delay feedback controls are designed for a class of infinite delay stochastic systems with highly nonlinear and the influence of switching state.     

Postoptimality for mean-risk stochastic mixed-integer programs and its application

The mean-risk stochastic mixed-integer programs can better model complex decision problems under uncertainty than usual stochastic (integer) programming models. In order to derive theoretical results in a numerically tractable way, the contamination technique is adopted in this paper for the postoptimality analysis of the mean-risk models with respect to changes in the scenario set, here the risk is measured by the lower partial moment. We first study the continuity of the objective function and the differentiability, with respect to the parameter contained in the contaminated distribution, of the optimal value function of the mean-risk model when the recourse cost vector, the technology matrix and the right-hand side vector in the second stage problem are all random. The postoptimality conclusions of the model are then established. The obtained results are applied to two-stage stochastic mixed-integer programs with risk objectives where the objective function is nonlinear with respect to the probability distribution. The current postoptimality results for stochastic programs are improved.     

Dynamical behavior of a stochastic model of gene expression with distributed delay and degenerate diffusion

This article is concerned with a stochastic model of gene expression with distributed delay and degenerate diffusion. We transform the model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is of degenerate type, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the solutions can converge in L¹ to an invariant density. The existence of the stationary distribution implies stochastic weak stability. Numerical simulation is introduced to illustrate the analytical result.     

Persistence and extinction of a stochastic Gilpin–Ayala model with jumps

This paper considers a stochastic Gilpin–Ayala model with jumps. First, we show the model that has a unique global positive solution. Then we establish the sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence of the solution. The threshold between weak persistence and extinction is obtained. Finally, we make simulations to conform our analytical results. The results show that the jump process can change the properties of the population model significantly. Copyright © 2014 John Wiley & Sons, Ltd.     

Probabilistic verification of a biodiesel production system using statistical model checking

Biochemical system designers are increasingly using formal modelling, simulation, and verification methods to improve the understanding of complex systems. Probabilistic models can incorporate realistic stochastic dynamics, but creating and analysing probabilistic models in a formal way is challenging. In this work, we present a stochastic model of biodiesel production that incorporates an inexpensive test of fuel quality, and we validate the model using statistical model checking, which can be used to evaluate simple or complex temporal properties efficiently. We also describe probabilistic simulation and analysis techniques for stochastic hybrid system (SHS) models to demonstrate the properties of our model. We introduce a variety of properties for various configurations of the reactor as well as results of testing our model against the properties.     

Modeling and analysis of a supply–assembly–store chain

We consider a supply–assembly–store chain with produce-to-stock strategy, which comprises a set of component suppliers, a mixed-model assembly line with a constantly moving conveyor linking a set of workstations in series, and a set of product storehouses. Each supplier provides components of a specified family, which are assembled at a corresponding workstation. Units belonging to different models of products are sequentially fed onto the conveyor, and pass through the workstations to generate finished products. Each storehouse stores finished products belonging to a specific model for satisfying customer demands. The suppliers deliver components according to a just-in-time supply policy with stochastic leadtimes. Customer demands for a particular model of products arrive at the corresponding storehouse according to a Poisson stream. The paper conducts a modeling and performance analysis in the design stage of the system in the sense of “long-term-behavior”. A rolling technique is constructed for analyzing stationary probability distributions of the numbers of components. A two-dimensional Markov chain with infinite states is introduced for analyzing stationary probability distributions of inventories of finished products. Based on these distributions, performance measures of the system, such as work-in-process of components, inventory amounts of finished products, as well as service levels for customers, can be easily obtained. Managerial insights are obtained from both analytical and numerical results.     

Analytical formulation for explaining the variations in traffic states: A fundamental diagram modeling perspective with stochastic parameters

Despite the simplicity and practicality of (deterministic) fundamental diagram models in highway traffic flow theory, the wide scattering effect observed in empirical data remains highly controversial, particularly for explaining traffic state variations. Owing to the analytical properties of the fundamental diagram modeling approach, in this study, we proposed an analytical and quantitative method for analyzing traffic state variations. We investigated the scattering effect in the fundamental diagram and proposed two stochastic fundamental diagram (SFD) models with lognormal and skew-normal distributions to explain the variations in traffic states. The first SFD model assumes that the scattering effect results from stochasticity in both the free-flow speed and the speed at critical density. Both random variables were assumed to follow the lognormal distribution. In the second SFD model, an integrated error term that was assumed to follow the skew-normal distribution over different density ranges was appended to the deterministic fundamental diagram. The properties of these two SFD models were analyzed and compared, and the parameters in these SFD models were calibrated using real-world loop detector data. The observed scatters from the empirical data were reproduced well by the simulated fundamental diagram model, indicating the validity of the proposed SFD models for explaining traffic state variations. Using these two analytical SFD models, we can analyze the stochastic capacity of freeways with closed forms. More importantly, the sources of stochasticity in freeway capacity can be traced in terms of randomly distributed parameters in fundamental diagram models.     

Sales and operations planning in systems with order configuration uncertainty

This paper addresses the problem of aligning demand and supply in configure-to-order systems. Using stochastic programming methods, this study demonstrates the value of accounting for the uncertainty associated with how orders are configured. We also demonstrate the value of component supply flexibility in the presence of order configuration uncertainty. We present two stochastic models: an explosion problem model and an implosion problem model. These models are positioned sequentially within a popular business process called sales and operations planning. Both models are formulated as two-stage stochastic programs with recourse and are solved using the sample average approximation method. Computational analyses were performed using data obtained from IBM System and Technology Group. The problem sets used in our analysis are created from actual industry data and our results show that significant improvements in revenue and serviceability can be achieved by appropriately accounting for the uncertainty associated with order configurations.     

Stationary distribution and persistence of a stochastic predator-prey model with a functional response

A stochastic predator-prey model with a functional response is investigated in this paper. The asymptotic properties of the stochastic model are considered here. Under some conditions, we show that the stochastic model is persistent in mean. Moreover, the existence of stationary distribution to the model is obtained. Simulations are also carried out to confirm our analytical results.     

A Lagrangian relaxation approach for network inventory control of stochastic revenue management with perishable commodities

Airline seat inventory control is the allocation of seats in the same cabin to different fare classes such that the total revenue is maximized. Seat allocation can be modelled as dynamic stochastic programs, which are computationally intractable in network settings. Deterministic and probabilistic mathematical programming models are therefore used to approximate dynamic stochastic programs. The probabilistic model, which is the focus of this paper, has a nonlinear objective function, which makes the solution of large-scale practical instances with off-the-shelf solvers prohibitively time consuming. In this paper, we propose a Lagrangian relaxation (LR) method for solving the probabilistic model by exploring the fact that LR problems are decomposable. We show that the solutions of the LR problems admit a simple analytical expression which can be resolved directly. Both the booking limit policy and the bid-price policy can be implemented using this method. Numerical simulations demonstrate the effectiveness of the proposed method.     

Criticality of viscous Hamilton–Jacobi equations and stochastic ergodic control

This paper deals with nonlinear additive eigenvalue problems for viscous Hamilton–Jacobi equations which appear in stochastic ergodic control. Certain qualitative properties of principal eigenvalues and associated eigenfunctions are studied. Such analysis plays a key role in studying the recurrence and transience of feedback diffusions for the corresponding stochastic control problems. Our results can be regarded as a nonlinear extension of the criticality theory for Schrödinger operators with decaying potentials.     

Project dynamics and emergent complexity

This paper presents a theoretical analysis of project dynamics and emergent complexity in new product development (NPD) projects subjected to the management concept of concurrent engineering. To provide a comprehensive study, the complexity frameworks, theories and measures that have been developed in organizational theory, systematic engineering design and basic scientific research are reviewed. For the evaluation of emergent complexity in NPD projects, an information-theory quantity—termed “effective measure complexity” (EMC)—is selected from a variety of measures, because it can be derived from first principles and therefore has high construct validity. Furthermore, it can be calculated efficiently from dynamic generative models or purely from historical data, without intervening models. The EMC measures the mutual information between the infinite past and future histories of a stochastic process. According to this principle, it is particularly interesting to evaluate the time-dependent complexity in NPD and to uncover the relevant interactions. To obtain analytical results, a model-driven approach is taken and a vector autoregression (VAR) model of cooperative work is formulated. The formulated VAR model provided the foundation for the calculation of a closed-form solution of the EMC in the original state space. This solution can be used to analyze and optimize complexity based on the model’s independent parameters. Moreover, a transformation into the spectral basis is carried out to obtain more expressive solutions in matrix form. The matrix form allows identification of the surprisingly few essential parameters and calculation of two lower complexity bounds. The essential parameters include the eigenvalues of the work transformation matrix of the VAR model and the correlations between components of performance fluctuations.     

Parametric modeling of protein–DNA binding kinetics: A discrete event based simulation approach

To understand the stochastic behavior of biological systems, we adopt an “in silico” stochastic event based simulation methodology that can determine the temporal dynamics of different molecules. The main requirement for this technique is the event execution time models for the different biological functions of the system. This paper presents a parametric model to determine the execution time of one such biological function: protein–DNA binding. This biological function is modeled as a combination of microlevel biological events using a coarse-grained probability measure to estimate the stochastic parameters of the function. Our model considers the actual binding mechanism along with some approximated protein and DNA structural information. We use a collision theory based approach to transform the thermal and concentration gradients of this biological function into their corresponding probability measure. This information theoretic approach significantly removes the complexity of the classical protein sliding along the DNA model, improves the speed of computation and can bypass the speed-stability paradox. This model can produce acceptable estimates of DNA–protein binding time necessary for our event based stochastic system simulator where the higher order (more than second order statistics) uncertainties can be ignored. The results show good correspondence with available experimental estimates. The model depends very little on experimentally generated rate constants and brings the important biological parameters and functions into consideration. We also present some “in silico” results showing the effects of protein–DNA binding on gene expression in prokaryotic cells.     

Optimal PID controller of robots under stochastic uncertainty

In practice often it is not possible to specify exact model parameters. Hence, precomputed controller based on some parameter estimates can produce bad results. In this presentation the aim is to combine classical PID control theory and stochastic optimisation methods in order to obtain robust optimal feedback control. The method works with cost functions being minimized and takes into account stochastic parameter varations. After Taylor expansion to calculate expected cost functions and a few transformations an approximate deterministic substitute PID control problem follows. Here, retaining only linear terms, approximation of expectations and variances of the expected cost functions can be calculated explicitly. By means of splines, numerical approximations of the objective function and the differential equations are obtained then. Using stochastic optimization methods, random parameter variations are incorporated into the optimal control process. Hence, robust optimal feedback controls are obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Component commonality via hierarchical orthogonal arrays and detailing economic decision matrices

Component commonality (CC) implies products using many common parts, desensitized to the range of product applications (noise), and meeting the functionality objectives of the product line. This paper lists a nine-step methodology for developing CC and applies it to a problem. These steps utilize the major concepts of analytical modeling, economic decision matrices (EDM), quality loss functions (QLS) for variates and weighted utilities, stochastic models, finite element (FE) simulations for concurrent engineering, and statistical design of experiments (DOE) for uncertainty in either application, statistics or managerial decisions. The details of the first six steps were illustrated in a previous paper by application to a problem involving a slider link subjected to an extreme range of “noise” (various inertia/pressure loadings). Six candidate designs of steel, aluminum and titanium were generated using an analytical model and a sensitivity study. The DOE utilized Taguchi's orthogonal arrays. These designs were ranked using cost, weight, and factors of safety with respect to yielding. A refining EDM with a three-part robustness criteria selected two candidates (best was steel, followed by aluminum) considering inner noise in the managerial decisions. In the current paper, the last three steps of the nine-step methodology are applied to these two candidates in order to obtain the “optimal” part for CC. The FE stress results are used with a modified Goodman fatigue criteria, and a stochastic model is developed based upon beta (strength) and three-parameter Weibull (stress) distributions. The model is then used in a detailing EDM to determine the stochastic reliability associated with a QLS defined with respect to fatigue reliability. A “fine-tuned” aluminum candidate is shown to meet a priori reliability requirements and have low-quality losses. However, both original candidates exhibited some high-quality losses, even though such losses were acceptable in the preceding refining EDM. The authors demonstrate that this loss of quality can be prevented if a fatigue criteria is used in both the refining and detailing EDM stages of the design process and, “warranty failures” are based on stochastic rather than deterministic definitions of maximum environmental conditions.     

An optimal regulation of a storage level with application to the water level regulation of a lake

In [10] a system of stochastic programming models was introduced for the optimal control of a storage level. Each model in this system serves to determine the optimal policy for only one period ahead though the time horizon consists of many future periods. The optimal control thus obtained can be considered an open loop control methodology. The main purpose of this paper is to present an application by giving an optimal control method for the regulation of the water level of Lake Balaton in Hungary. By solving almost 600 stochastic programming problems we analyze what would have happened if we had controlled the water level using our method between 1922 and 1970, where one decision period is one month. The numerical results show that the proposed control methodology works quite well in this case.