Stochastic regression model with heteroscedastic disturbance

This paper discusses some properties of stochastic regression model with continuous form of heteroscedastic disturbance. The strong consistency and asymptotic normality of a generalized weighted least squares estimate will be investigated under certain conditions on the stochastic regressors and errors. More, the linear hypothesis testing problem also be discussed and an example to be demonstrated to reestablish the results of Cheng and Chang (1990, Tech. Report, National Tsing Hua University).Supported by the National Science Council Grant No. 810208M763 at National Tsing Hua University.     

Channel coordination using product returns for a supply chain with stochastic salvage capacity

We examine returns policy in a Newsboy framework. Unlike the prior literature, however, we assume that both supplier and retailer have limited and stochastic salvage capacities. We first analyze the case of integrated supply chain in which the agents’ decisions are fully coordinated for the joint profits. The result prescribes a partial returns policy, in which the retailer returns a part of the leftovers to the supplier and liquidates the remainder through its clearance sale. In a decentralized system, the supplier should motivate the retailer to duplicate the outcome of the integrated system in choosing order and returns quantities. We propose three coordination contracts, of which each uses two benefit transfer schemes as an incentive to the retailer, instead of using a single benefit scheme as in the prior literature. All three effectively coordinate the supply chain. The supplier, as a Stakelberg leader, chooses the most profitable one since each contract yields the different shares of the agents’ profits.     

An approach to constructing mathematical models of moving mechanical systems under uncertainty

The paper outlines an approach to constructing a mathematical model of moving mechanical systems under uncertainty. Generalized fuzzy differential equations are used to prove that the model is mathematically correct. The motion of a control oscillator under uncertainty is considered as an example __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 108–119, October 2007.     

A linear Bayesian stochastic approximation to update project duration estimates

By relaxing the unrealistic assumption of probabilistic independence on activity durations in a project, this paper develops a hierarchical linear Bayesian estimation model. Statistical dependence is established between activity duration and the amount of resource, as well as between the amount of resource and the risk factor. Upon observation or assessment of the amount of resource required for an activity in near completion, the posterior expectation and variance of the risk factor can be directly obtained in the Bayesian scheme. Then, the expected amount of resources required for and the expected duration of upcoming activities can be predicted. We simulate an application project in which the proposed model tracks the varying critical path activities on a real time basis, and updates the expected project duration throughout the entire project. In the analysis, the proposed model improves the prediction accuracy by 38.36% compared to the basic PERT approach.     

On the Finite-Time Dynamics of Ant Colony Optimization

An analytical framework for investigating the finite-time dynamics of ant colony optimization (ACO) under a fitness-proportional pheromone update rule on arbitrary construction graphs is developed. A limit theorem on the approximation of the stochastic ACO process by a deterministic process is demonstrated, and a system of ordinary differential equations governing the process dynamics is identified. As an example for the application of the presented theory, the behavior of ACO on three different construction graphs for subset selection problems is analyzed and compared for some basic test functions. The theory enables first rough theoretical predictions of the convergence speed of ACO. AMS 2000 Subject classification 68W20, 68W40     

A generalized Duffing equation with the Coulomb’s friction law and Signorini–type contact conditions

This work provides mathematical and numerical analyses for a spring–mass system, in which Signorini–type contact conditions and Coulomb’s friction law with thermal effects are taken into consideration. The motion of a mass attached to a viscoelastic (Kelvin–Voigt type) nonlinear spring is described by a generalized Duffing equation. Signorini contact conditions are understood as extended complementarity conditions (CCs), where convolution is incorporated, allowing to consider thermal aspects of an obstacle. We prove the existence of global weak solutions for the highly nonlinear differential equation system with all the conditions, based on the regularized differential equation and the normal compliance condition with the standard mollifier. In addition, we investigate what side effects produce higher singularities of contact forces in dynamic contact problems, which is also supported by numerical evidences. Numerical schemes are proposed and then several groups of data are selected for the display of our numerical simulations.     

Non-Linear Self-Excited Random Vibrations and Stability of an SDOF System with Parametric Noises

Abstract. The paper presents the solution to the properties of stochastic response of a system with random parametric noises, which is prone to the loss of aerodynamical stability. The system is described by an equation of van der Pol type with the negative linear, and with the positive cubic dampings. The coefficients of the linear damping and of the stiffness include the multiplicative random perturbations, the external excitation being given as a sum of a deterministic function and of an additive perturbation. All three input random processes are supposed to be Gaussian and centered, with the non-zero mutual stochastic parameters, as it corresponds to the properties of real systems. The solution has been based on the method of stochastic linearisation and of the subsequent solution of the Fokker–Planck–Kolmogorov equation in the sense of the first and second stochastic moments for the transient and stationary states. There have been demonstrated several effects, which are typical for systems with parametric noises, differentiating them from the systems with constant coefficients. The principal attention has been devoted to the properties of the spectral density of the response, the character of which changes abruptly with the degree of non-linearity of the damping and of the level of random perturbations.Sommario. La presente memoria studia le proprietà della risposta stocastica di un sistema con eccitazione casuale parametrica, che tende alla perdita della stabilità aerodinamica. Il sistema è descritto mediante un'equazione del tipo di van der Pole con il termine lineare dello smorzamento negativo e il termine cubico positivo. Poichá l'eccitazione esterna è la somma di una funzione deterministica e di una perturbazione additiva, i coefficienti dello smorzamento lineare e della rigidezza comprendono le perturbazioni casuali moltiplicative. I tre processi stocastici di eccitazione sono assunti gaussiani e a media nulla con parametri stocastici incrociati diversi da zero, come si verifica per le proprietà dei sistemi reali. La soluzione è basata sul metodo della linearizzazione stocastica e della successiva soluzione dell'equazione di Fokker-Planck-Kolmogorov studiando i primi e i secondi momenti statistici per gli stati transitori e stazionari. Vengono mostrati diversi effetti, tipici dei sistemi con eccitazione parametrica, differenziandoli dai sistemi a coefficienti costanti. Particolare attenzione è rivolta alle proprietà della densità spettrale della risposta le cui caratteristiche cambiano bruscamente con il grado di non linearità dello smorzamento e del livello di casualità delle perturbazioni.     

A decoupled modal analysis for nonlinear dynamics of an orthotropic thin laminate in a simply supported boundary condition subject to thermal mechanical loading

In this paper, we analyze the nonlinear dynamic response of an orthotropic laminate in a simply supported boundary condition subject to thermal and mechanical loading. The equation of motion for the laminate’s deflection is obtained in a decoupled Duffing equation by means of a Galerkin-type method without Berger’s approximations. The Duffing equation incorporates an arbitrary thermal field, with both the in-plane and transverse temperature variations in a steady-state and a transient state. The formulation indicates that the transverse temperature variation produces an additional pressure load, while the in-plane temperature variation affects the system frequency. The equation allows for characterization of the laminate behaviors in nonlinear thermal buckling, thermal vibration and thermal mechanical response.     

Stability of noisy nonlinear auto-parametric systems

The purpose of this work is to examine the stationary motion and stability properties of stationary motion of two degree-of-freedom noisy auto-parametric systems We shall use analytical techniques to extend the existing results to examine such multi-dimensional nonlinear systems with noise, and in particular additive white noise. We obtain an approximation for the top Lyapunov exponent, the exponential growth rate, of the response of the so-called single-mode stationary motion. We show analytically that the top Lyapunov exponent is positive, and for small values of noise intensity ɛ and dissipation ɛ² the exponent grows in proportion with ɛ^2/3.