a positivity preserving numerical method for stochastic r d model abstract the stochastic research and development r d model plays an important role in the growth rates of technological progress and capital accumulation however we can not obtain the explicit solution of stochastic r d model to explore a numerical approximate method preserving positivity for the r d model with uncertainty from the population growth we first construct an explicit euler maruyama em method and then verify it converges to the true solution with strong order 1 2 1 1 p but the em method may lead to a negative approximation which is unrealistic so we establish the balanced explicit method bem to overcome the defect of em method and give sufficient conditions to preserve the positivity of bem then the convergence order of bem is obtained finally numerical simulations are carried out to verify our theoretical work

brief paper robustness of exponential stability of a class of stochastic functional differential equations with infinite delay we regard the stochastic functional differential equation with infinite delay dx t f x t dt g x t dw t as the result of the effects of stochastic perturbation to the deterministic functional differential equation email protected t f x t where x t x t q c 0 r n is defined by x t q x email protected email protected 0 we assume that the deterministic system with infinite delay is exponentially stable in this paper we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable

computing non stationary s s policies using mixed integer linear programming abstract this paper addresses the single item single stocking location non stationary stochastic lot sizing problem under the s s control policy we first present a mixed integer non linear programming minlp formulation for determining near optimal s s policy parameters to tackle larger instances we then combine the previously introduced minlp model and a binary search approach these models can be reformulated as mixed integer linear programming milp models which can be easily implemented and solved by using off the shelf optimization software computational experiments demonstrate that optimality gaps of these models are less than 0 3 of the optimal policy cost and computational times are reasonable

almost sure exponential stability of the backward euler maruyama discretization for highly nonlinear stochastic functional differential equation abstract stability of numerical solutions to stochastic differential equations have received an increasing attention but there is so far no work on stability of numerical solutions to nonlinear stochastic functional differential equations sfdes to close the gap the paper develops a criterion on stability of numerical solutions to highly nonlinear sfdes by using of the discrete semi martingale convergence theorem we show that backward euler maruyama method can reproduce almost sure exponential stability of the exact solutions to highly nonlinear sfdes several high order examples are provided to illustrate the main results which implies that a wide class of nonlinear systems obeys the new criterion

numerical solutions of stochastic differential delay equations under the generalized khasminskii type conditions the classical existence and uniqueness theorem of the solution to a stochastic differential delay equation sdde requires the local lipschitz condition and the linear growth condition see e g 11 12 and 20 the numerical solutions under these conditions have also been discussed intensively see e g 4 10 13 16 17 18 21 22 and 24 recently mao and rassias 14 and 15 established the generalized khasminskii type existence and uniqueness theorems for sddes where the linear growth condition is no longer imposed these generalized khasminskii type theorems cover a wide class of highly nonlinear sddes but these nonlinear sddes do not have explicit solutions whence numerical solutions are required in practice however there is so far little numerical theory on sddes under these generalized khasminskii type conditions the key aim of this paper is to close this gap

a sigmoid approximation of the standard normal integral abstract this paper presents a simple sigmoid function for approximation of cumulative standard normal probabilities the approximation has an error of 0 000043 for 8 z 8

stability analysis and optimization of an inventory system with bounded orders this paper addresses the control of a one item inventory system subject to random order lead time and random demand the key parameter of the control policy is the objective inventory in each period the order to be placed brings the inventory position as close as possible to the objective inventory the order of each period is kept between a lower bound and an upper bound we show that the distribution of the inventory level converges to its stationary distribution provided that the lower bound is smaller than the average demand the upper bound is greater than the average demand and some regularity conditions hold the average inventory cost is shown to be a convex function of the objective inventory level a simulation based approach is proposed for the determination of the optimal objective inventory a method of bisection with derivative is then used to determine the optimal objective inventory the derivatives needed in various iterations of this method are estimated using a single sample path with respect to a given objective inventory numerical results are provided

exponential stability of numerical solution to neutral stochastic functional differential equation stability of the solution to neutral stochastic functional differential equation nsfde has received a great deal of attention but there is so far little work on stability of numerical solution to close the gap the paper develops new criteria on stability of numerical solutions to linear and nonlinear nsfdes we show that the backward euler maruyama em method can reproduce the almost surely exponential stability of the exact solution to highly nonlinear nsfde and em method can preserve the almost surely exponential stability of nsfde with linear growing coefficients two examples are provided to illustrate the main results

numerical approximation for nonlinear stochastic pantograph equations with markovian switching the main aim of the paper is to prove that the implicit numerical approximation can converge to the true solution to highly nonlinear hybrid stochastic pantograph differential equation after providing the boundedness of the exact solution the paper proves that the backward euler maruyama numerical method can preserve boundedness of moments and the numerical approximation converges strongly to the true solution finally the exponential stability criterion on the backward euler maruyama scheme is given and a high order example is provided to illustrate the main result

sample path derivatives for s s inventory systems for s s inventory systems we derive sample path derivatives of performance measures with respect to the two parameters s and s these derivatives yield derivative estimators which can be estimated from a single sample path or simulation of the inventory system in some cases not even requiring actual knowledge of the underlying demand distribution such derivative estimates would be useful in sensitivity analysis or in gradient based optimization techniques we consider the nondiscounted periodic review system with general independent and identically distributed i i d continuous demands full backlogging and general holding and shortage costs for the infinite horizon model consistency proofs are given for some special cases although we argue why the estimators should be correct for the more general case

optimality of s s policies in inventory models with markovian demand this paper is concerned with a generalization of classical inventory models with fixed ordering costs that exhibit s s policies in our model the distribution of demands in successive periods is dependent on a markov chain the model includes the case of cyclic or seasonal demand the model is further extended to incorporate some other realistic features such as no ordering periods and storage and service level constraints both finite and infinite horizon nonstationary problems are considered we show that s s policies are also optimal for the generalized model as well as its extensions

stochastic kolmogorov type system with infinite delay abstract the paper considers a stochastic functional kolmogorov type population system with infinite delay under the general probability measures main aim is to show that the environment noise will not only suppress a potential population explosion but also make the solution be stochastically ultimately bounded and asymptotic stable moreover two stochastic functional lotka volterra equations as examples are provided to illustrate the main results

almost sure exponential stability of backward euler maruyama discretizations for hybrid stochastic differential equations this is a continuation of the first author s earlier paper 1 jointly with pang and deng in which the authors established some sufficient conditions under which the euler maruyama em method can reproduce the almost sure exponential stability of the test hybrid sdes the key condition imposed in 1 is the global lipschitz condition however we will show in this paper that without this global lipschitz condition the em method may not preserve the almost sure exponential stability we will then show that the backward em method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid sdes

double precision rational approximation algorithms for the standard normal first and second order loss functions abstract we present double precision algorithms based upon piecewise rational approximations for the standard normal first and second order loss functions these functions are used frequently in inventory management no direct approximation or closed formulation exists for the standard first and second order loss functions current state of the art algorithms require intermediate computations of the cumulative normal distribution or tabulations and they do not compute to full double precision we deal with both these issues and present direct double precision accurate algorithms which are valid in the full range of double precision floating point numbers

robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay this paper establishes the existence and uniqueness theorem of neutral stochastic functional differential equations with infinite delay and examines the almost sure stability of this solution with general decay rate this result may be used to examine almost sure robust stability to illustrate our idea more carefully we carefully discuss a scalar stochastic integro differential equation with neutral type and its asymptotic stability including the exponential stability and the polynomial stability

constraint programming for computing non stationary r s inventory policies abstract this paper proposes a constraint programming model for computing the finite horizon single item inventory problem with stochastic demands in discrete time periods with service level constraints under the non stationary version of the periodic review order up to level policy i e non stationary r s or simply r n s n it is observed that the modeling process is more natural and the required number of variables is smaller compared to the mip formulation of the same problem the computational tests show that the cp approach is more tractable than the conventional mip formulation two different domain reduction methods are proposed to improve the computational performance of solution algorithms the numerical experiments confirmed the effectiveness of these methods

a uniform approximation to the right normal tail integral this paper presents two simple formulas for approximation of the standard normal right tail probabilities and indicates the method of computing approximations of higher order one of the approximations relies on two simple numerical constants has absolute error of 0 00071 and relative error 0 023 the other uses four numerical constants has absolute error of 0 000019 the relative error is 0 005 and gives at least two significant digits for probabilities over the entire range 0

finding optimal s s policies is about as simple as evaluating a single policy in this paper a new algorithm for computing optimal s s policies is derived based upon a number of new properties of the infinite horizon cost function c s s as well as a new upper bound for optimal order up to levels s and a new lower bound for optimal reorder levels s the algorithm is simple and easy to understand its computational complexity is only 2 4 times that required to evaluate a specific single s s policy the algorithm applies to both periodic review and continuous review inventory systems

new bounds and heuristics for q r policies to clarify the impact of demand variability on single item stochastic inventory systems with setup costs we subsume the distributional information of the lead time demand into its mean and variance and solve the resulting problem against the worst possible distribution in this class for q r policies we obtain in closed form a distribution free solution for q and r and upper bounds on the optimal long run average cost and on the optimal batch size as a byproduct we develop a robust distribution free batch size heuristic that causes a relative cost increase of no more than 6 07 in addition when the newsvendor cost is known we obtain sharper lower and upper bounds on the long run average cost these bounds clarify in an exceedingly simple way the cost impact of fixed setup costs demand variability and constraints on the batch size we illustrate our bounds and heuristics on problems with poisson and compound poisson demands

an efficient algorithm for computing optimal s s policies this paper presents an algorithm to compute an optimal s s policy under standard assumptions stationary data well behaved one period costs discrete demand full backlogging and the average cost criterion the method is iterative starting with an arbitrary given s s policy and converging to an optimal policy in a finite number of iterations any of the available approximations can thus be used as an initial solution each iteration requires only modest computations also a lower bound on the true optimal cost can be computed and used in a termination test empirical testing suggests very fast convergence

piecewise linear lower and upper bounds for the standard normal first order loss function the first order loss function and its complementary function are extensively used in practical settings when the random variable of interest is normally distributed the first order loss function can be easily expressed in terms of the standard normal cumulative distribution and probability density function however the standard normal cumulative distribution does not admit a closed form solution and cannot be easily linearised several works in the literature discuss approximations for either the standard normal cumulative distribution or the first order loss function and their inverse however a comprehensive study on piecewise linear upper and lower bounds for the first order loss function is still missing in this work we initially summarise a number of distribution independent results for the first order loss function and its complementary function we then extend this discussion by focusing first on random variables featuring a symmetric distribution and then on normally distributed random variables for the latter we develop effective piecewise linear upper and lower bounds that can be immediately embedded in milp models these linearisations rely on constant parameters that are independent of the mean and standard deviation of the normal distribution of interest we finally discuss how to compute optimal linearisation parameters that minimise the maximum approximation error

positive solution and its asymptotic behaviour of stochastic functional kolmogorov type system abstract in general population systems are often subject to environmental noise this paper considers the stochastic functional kolmogorov type system d x t diag x 1 t x n t f x t d t g x t d w t under the traditionally diagonally dominant condition we study existence and uniqueness of the global positive solution of this stochastic system and its asymptotic bound properties and moment average boundedness in time these properties are natural requirements from the biological point of view as the special cases we also discuss some stochastic lotka volterra systems

numerical solutions of neutral stochastic functional differential equations this paper examines the numerical solutions of neutral stochastic functional differential equations nsfdes d x t u x t f x t dt g x t dw t t geq 0 the key contribution is to establish the strong mean square convergence theory of the euler maruyama approximate solution under the local lipschitz condition the linear growth condition and contractive mapping these conditions are generally imposed to guarantee the existence and uniqueness of the true solution so the numerical results given here are obtained under quite general conditions although the way of analysis borrows from x mao lms j comput math 6 2003 pp 141 161 to cope with u x t several new techniques have been developed

on the objective function behavior in s s inventory models the most common measure of effectiveness used in determining the optimal s s inventory policies is the total cost function per unit time es s s in stationary analysis this function is constructed through the limiting distribution of on hand inventory and it involves some renewal theoretic elements for 0 given es turns out to be convex in s so that the corresponding optimal reorder point s1 can be characterized easily however es1 is not in general unimodal on 0 this requires the use of complicated search routines in computations as there is no guarantee that a local minimum is global both for periodic and continuous review systems with constant lead times full backlogging and linear holding and shortage costs we prove in this paper that e s1 0 0 is both necessary and sufficient for a global minimum es1 is pseudoconvex on 0 if the underlying renewal function is concave the optimal stationary policy can then be computed efficiently by a one dimensional search routine the renewal function in question is that of the renewal process of periodic demands in the periodic review model and of demand sizes in the continuous review model

an epq model with inflation in an imperfect production system in this paper a production inventory model is considered for stochastic demand with the effect of inflation generally every manufacturing system wants to produce perfect quality items however due to real life problems labor problems machine breakdown etc a certain percentage of products are of imperfect quality the imperfect items are reworked at a cost the lifetime of a defective item follows a weibull distribution due to the production of imperfect quality items a product shortage occurs the profit function is derived by using both a general distribution of demand and the uniform rectangular distribution of demand computational experiments along with graphical illustrations are presented to discuss the optimality of the probability functions

strong convergence of implicit numerical methods for nonlinear stochastic functional differential equations abstract the main aim of this work is to prove that the backward euler maruyama approximate solutions converge strongly to the true solutions for stochastic functional differential equations with superlinear growth coefficients the paper also gives the boundedness and mean square exponential stability of the exact solutions and shows that the backward euler maruyama method can preserve the boundedness of mean square moments finally a highly nonlinear example is provided to illustrate the main results

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

inventory models with general backorder costs abstract under the prevalent assumption of linear inventory holding and backorder costs important results have been established in many basic inventory models however the assumption is often too restrictive for many applications in this paper we study stochastic inventory models under more general backorder costs namely the penalty costs for each unit of backorder consisting of fixed and proportional components under this backorder cost structure we derive a necessary and sufficient condition for the quasiconvexity of the one step loss function this condition is satisfied by a wide spectrum of demand distributions but it is not distribution free consequently existing results based on quasiconvex loss functions are applicable in many but not all models with the above cost structure

double precision rational approximation algorithm for the inverse standard normal second order loss function abstract we present a double precision algorithm based upon rational approximations for the inverse standard normal second order loss function this function is used frequently in inventory management no direct approximation or closed formulation exists for the inverse standard normal second order loss function calculations are currently based on root finding methods and intermediate computations of the cumulative normal distribution or tabulations results then depend on the accuracy and valid range of that underlying function we deal with these issues and present a direct double precision accurate algorithm valid in the full range of double precision floating point numbers

a fully polynomial time approximation scheme for single item stochastic inventory control with discrete demand the single item stochastic inventory control problem is to find an inventory replenishment policy in the presence of independent discrete stochastic demands under periodic review and finite time horizon in this paper we prove that this problem is intractable and design for it a fully polynomial time approximation scheme

scaling at the rod to flexible chain crossover in the stiff limit the authors critically examine the present theoretical understanding of the conformational rod to flexible chain crossover of a linear chain polymer in dilute solution with a good solvent there is good evidence for weak universality in this crossover while strong universality seems unlikely from available evidence in addition this system should exhibit a second crossover from a gaussian to an excluded volume they present a scaling hypothesis for the behaviour of this second crossover and compare with the initial results of numerical simulations and with the experiment of murakami et al 1980

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

scaling at the conformational rod to coil transition the authors present evidence for the partial suppression of self avoidance effects in the rod to coil transition scaling observed recently in numerical simulations the scaling ansatz is examined critically and also checked explicitly in a soluble model on a partially directed lattice

a simple heuristic for computing nonstationary s s policies nonstationary inventory problems with set up costs proportional ordering costs and stochastic demands occur in a large number of industrial distribution and service contexts it is well known that nonstationary s s policies are optimal for such problems in this paper we propose a simple myopic heuristic for computing the policies the heuristic involves approximating the future problem at each period by a stationary one and obtaining the solution to the corresponding stationary problem we numerically compare our heuristic with an earlier myopic heuristic and the optimal dynamic programming solution procedure over all problems tested the new heuristic averaged 1 7 error compared with 2 0 error for the old procedure and is on average 399 times as fast as the d p and 2062 as fast as the old heuristic moreover our heuristic owing to its myopic nature requires the demand data only a few periods into the future while the dynamic programming solution needs the demand data for the entire time horizon which are typically not available in most practical situations

the asymptotic behaviour of directed self avoiding walks chakrabarti and manna 1983 have made a numerical study of directed self avoiding walks on a square lattice in which the random walker is forbidden to move upward extrapolating from an enumeration of n step walks up to n 14 they claimed that the mean end to end displacement rn behaves asymptotically for large n as nnu with nu 0 86 or 0 02 it is shown that a simple application of the generating function method yields the behaviour of n step walks of this type exactly for all n and the result nu 1 is rigorously proved

modelling and computing rn sn policies for inventory systems with non stationary stochastic demand abstract this paper addresses the single item non stationary stochastic demand inventory control problem under the non stationary r s policy in non stationary r s policies two sets of control parameters the review intervals which are not necessarily equal and the order up to levels for replenishment periods are fixed at the beginning of the planning horizon to minimize the expected total cost it is assumed that the total cost is comprised of fixed ordering costs and proportional direct item inventory holding and shortage costs with the common assumption that the actual demand per period is a normally distributed random variable about some forecast value a certainty equivalent mixed integer linear programming model is developed for computing policy parameters the model is obtained by means of a piecewise linear approximation to the non linear terms in the cost function numerical examples are provided

nervousness in inventory management comparison of basic control rules abstract using a rolling horizon planning framework in inventory control leads to nervousness in the planning system caused by instability of order release decisions in successive planning cycles for a single stage inventory system with arbitrary stochastic demand it is shown analytically how planning stability is affected by policy parameters if s nq s s and t s control rules are applied it turns out that the reorder point s does not influence stability whereas the lot size determining parameters q s s and t can have a considerable impact however this influence turns out to be quite different for different measures of stability regarding order setup or order quantity deviations respectively using these planning stability results the priority of the different control rules under the aspect of nervousness is discussed

an operational calculus for matrix exponential distributions with applications to a brownian q q inventory model a distribution g on math not displayed is called matrix exponential if the density has the form etz s where is a row vector t a square matrix and s a column vector equivalently the laplace transform is rational for such distributions we develop an operator calculus where the key step is manipulation of analytic functions f z extended to matrix arguments the technique is illustrated via an inventory model moving according to a reflected brownian motion with negative drift such that an order of size q is placed when the stock process down crosses some level q explicit formulas for the stationary density are found under the assumption that the leadtime z has a matrix exponential distribution and involve expressions of the form f t where math not displayed

sunoptimal policies with bounds for parameter adaptive decision processes a parameter adaptive decision process is a sequential decision process where some parameter or parameter set impacting the rewards and or transitions of the process is not known with certainty signals from the performance of the system can be processed by the decision maker as time progresses yielding information regarding which parameter set is operative active learning is an essential feature of these processes and the decision maker must choose actions that simultaneously guide the system in a preferred direction as well as yield information that can be used to better prescribe future actions if the operative parameter set is known with certainty the parameter adaptive problem reduces to a conventional stochastic dynamic program which is presumed solvable previous authors have shown how to use these solutions to generate suboptimal policies with performance bounds for the parameter adaptive problem here it is shown that some desirable characteristics of those bounds are shared by a larger cla