Periodic review inventory systems with a service level criterion

Full-cost inventory models are mostly studied in the literature, whereas service level constraints are more common to be observed in practical settings. In this paper, we consider periodic review inventory systems with service level restrictions. The control of such inventory systems is limited to (s, S)-type policies in the literature. To the best of our knowledge, we are the first authors to compare such policies with optimal replenishment policies, and illustrate an average cost difference of 0.64%. This justifies the use of these popular (s, S) policies in practice. Furthermore, we propose a new one-dimensional search procedure that is bounded to set the reorder level s and order-up-to level S, whereas the solution space is unbounded and two dimensional. Our heuristic procedure is guaranteed to satisfy the service level constraint and numerical experiments illustrate that it results in an average cost deviation of 1–2% compared with the best (s, S) policy. Consequently, it significantly outperforms all existing procedures from literature, both in service and costs.     

Stochastic decomposition in production inventory with service time

We study an (s, S) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on, are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s.     

Approximate steady-state distribution for a large repairable item inventory system

This paper examines a model for a repairable inventory system where some items are condemned upon failure. These condemnations require an ordering policy which we take to be an (s, S) policy. We model this system as a version of the classical machine repair problem using continuous time Markov processes. We develop solution procedures based on approximations to the steady state probability distribution, which enable us to obtain the operating characteristics of (s, S) policies very easily, with or without an ordering leadtime. The approximate models are found to be very accurate and computationally very efficient.     

Analytical and computational solution of adaptive inventory processes

H. Scarf (Ann. Math. Statist.30 (2) (1959)) has discussed a technique for the computation of optimal inventory level in the case where the demand distribution contains an unknown statistical parameter. It was assumed that initially the parameter could be described by a priori distribution, which would be subsequently revised on the basis of additional demand information. By assuming that the demand distribution of a cumulative observed demand is a sufficient statistic for the unknown parameter, it was shown that the optimal inventory levels could be obtained by the recursive computation of a sequence of function of two variables. He also showed (N.R.L.A.7 (8) (1960)) that if the demand distributions are gamme distributions and if the holding and penalty costs are linear, then the sequence of function of two variables may be solved by a related set of equations requiring tabulation of functions of only one variable. The main part of the argument presented in this paper is that an analytical consideration and numerical examples of equation requiring tabulation of function of only one variable given by H. Scarf were given with special choices of the factors, and were compared with solution when demand distribution is assumed exactly known and unknown.     

Inventory control for point-of-use locations in hospitals

Most inventory management systems at hospital departments are characterised by lost sales, periodic reviews with short lead times, and limited storage capacity. We develop two types of exact models that deal with all these characteristics. In a capacity model, the service level is maximised subject to a capacity restriction, and in a service model the required capacity is minimised subject to a service level restriction. We also formulate approximation models applicable for any lost-sales inventory system (cost objective, no lead time restrictions etc). For the capacity model, we develop a simple inventory rule to set the reorder levels and order quantities. Numerical results for this inventory rule show an average deviation of 1% from the optimal service levels. We also embed the single-item models in a multi-item system. Furthermore, we compare the performance of fixed order size replenishment policies and (R,?s,?S) policies.     

Capacity and Production Managment in a Single Product Manufacturing System

In planning and managing production systems, manufacturers have two main strategies for responding to uncertainty: they build inventory to hedge against periods in which the production capacity is not sufficient to satisfy demand, or they temporarily increase the production capacity by “purchasing” extra capacity. We consider the problem of minimizing the long-run average cost of holding inventory and/or purchasing extra capacity for a single facility producing a single part-type and assume that the driving uncertainty is demand fluctuation. We show that the optimal production policy is of a hedging point policy type where two hedging levels are associated with each discrete state of the system: a positive hedging level (inventory target) and a negative one (backlog level below which extra capacity should be purchased). We establish some ordering of the hedging levels, derive equations satisfied by the steady-state probability distribution of the inventory/backlog, and give a more detailed analysis of the optimal control policy in a two state (high and low demand rate) model.     

Linear equations over commutative rings

A generalized rank (McCoy rank) of a matrix with entries in a commutative ring R with identity is discussed. Some necessary and sufficient conditions for the solvability of the linear equation Ax = b are derived, where x, b are vectors and A is a matrix with entries in either a Noetherian full quotient ring or a zero dimensional ring.     

Un modèle mathématique de cyclone

The source waves are some particular solutions to genuinely non linear hyperbolic systems with a source term, whose propagation velocity is a constant determined by the roots of this source term. We propose a system of 2 equations on a 2 dimension space, which model the velocity field of the atmosphere near the ground. The source term is made of three parts: a given pressure gradient, a friction or aspiration effect and the Coriolis force. In the case where these parameters are constant, we build a solution which is a constant outside a circular crown. The internal circle represents the eye's wall of the hurricane and corresponds to a share shock wave. The external circle is a set generating the bifurcation which actually models the hurricane. To cite this article: A.-Y. LeRoux et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On a stochastic inventory model with a generalized holding costs

This paper is concerned with finding the optimal replenishment policy for an inventory model that minimizes the total expected discounted costs over an infinite planning horizon. The demand is assumed to be driven by a Brownian motion with drift and the holding costs (inventory and shortages) are assumed to take some general form. This generalizes the earlier work where holding costs were assumed linear. It turns out that problem of finding the optimal replenishment schedule reduces to the problem of solving a Quasi-Variational Inequality Problem (QVI). This QVI is then shown to lead to an (s, S) policy, where s and S are determined uniquely as a solution of some algebraic equations.     

On properties of discrete (r, q) and (s, T) inventory systems

We consider single-item (r, q) and (s, T) inventory systems with integer-valued demand processes. While most of the inventory literature studies continuous approximations of these models and establishes joint convexity properties of the policy parameters in the continuous space, we show that these properties no longer hold in the discrete space, in the sense of linear interpolation extension and L^?-convexity. This nonconvexity can lead to failure of optimization techniques based on local optimality to obtain the optimal inventory policies. It can also make certain comparative properties established previously using continuous variables invalid. We revise these properties in the discrete space.     

Reuse,Recycle, Reduce (3R) – strategies for the calculation of transient magnetic fields

The discretization of transient magneto-dynamic field problems with geometric discretization schemes such as the Finite Integration Technique or the Finite-Element Method based on Whitney form functions results in nonlinear differential-algebraic systems of equations of index 1. Their time integration with embedded s-stage singly diagonal implicit Runge–Kutta methods requires the solution of s nonlinear systems within one time step. Accelerated solution of these schemes is achieved with techniques following so-called 3R-strategies (“reuse, recycle, reduce”). This involves e.g. the solution of the linear(-ized) equations in each time step where the solution process of the iterative preconditioned conjugate gradient method reuses and recycles spectral information of linear systems from previous stages. Additionally, a combination of an error controlled spatial adaptivity and an error controlled implicit Runge–Kutta scheme is used to reduce the number of unknowns for the algebraic problems effectively and to avoid unnecessary fine grid resolutions both in space and time. First numerical results for 2D nonlinear magneto-dynamic problems validate the presented approach and its implementation. The space discretization in the numerical examples is done by Lagrangian nodal finite elements but the presented algorithms also work in combination with other discretization schemes for the Maxwell equations such as the Whitney vector finite elements.     

On the stabilizability problem in Banach space

The model is a linear system defined on Banach (state and control) spaces, with the operator acting on the state only the infinitesimal generator of a strongly continuous semigroup. The stabilizability problem of expressing the control through a bounded operator acting on the state as to make the resulting feedback system globally asymptotically stable is considered. On the negative side, and in contrast with the finite dimensional theory, a few counter examples are given of systems which are densely controllable in the space and yet are not stabilizable, even if some further “nice properties” hold. Use is made of the notion of essential spectrum and its stability under relatively compact perturbations. On the positive side, it is shown, however, that for large classes of systems of physical interest (classical selfadjoint boundary value problems, delay equations, etc.) controllability on a suitable finite dimensional subspace still yields stabilizability on the whole space.     

Using aggregate fill rate for dynamic scheduling of multi-class systems

For dynamic scheduling of multi-class systems where backorder cost is incurred per unit backordered regardless of the time needed to satisfy backordered demand, the following models are considered: the cost model to minimize the sum of expected average inventory holding and backorder costs and the service model to minimize expected average inventory holding cost under an aggregate fill rate constraint. Use of aggregate fill rate constraint in the service model instead of an individual fill rate constraint for each class is justified by deriving equivalence relations between the considered cost and service models. Based on the numerical investigation that the optimal policy for the cost model is a base-stock policy with switching curves and fixed base-stock levels, an alternative service model is considered over the class of base-stock controlled dynamic scheduling policies to minimize the total inventory (base-stock) investment under an aggregate fill rate constraint. The policy that solves this alternative model is proposed as an approximation of the optimal policy of the original cost and the equivalent service models. Very accurate heuristics are devised to approximate the proposed policy for given base-stock levels. Comparison with base-stock controlled First Come First Served (FCFS) and Longest Queue (LQ) policies and an extension of LQ policy (Δ policy) shows that the proposed policy performs much better to solve the service models under consideration, especially when the traffic intensity is high.     

Periodic review (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">S</Emphasis>) inventory model with permissible delay in payments

This paper investigates the effect of permissible delay in payments on ordering policies in a periodic review (s, S) inventory model with stochastic demand. A new mathematical model is developed, which is an extension to that of Veinott and Wagner (Mngt Sci 1965; 11: 525) who applied renewal theory and stationary probabilistic analysis to determine the equivalent average cost per review period. The performance of the model is validated using a custom-built simulation programme. In addition, two distribution-free heuristic methods of reasonable accuracy develop approximate optimal policies for practical purposes based only on the mean and the standard deviation of the demand. Numerical examples are presented with results discussed.     

Monotonicity properties of a class of stochastic inventory systems

We consider inventory systems which are governed by an (r,q) or (r,nq) policy. We derive general conditions for monotonicity of the three optimal policy parameters, i.e., the optimal reorder level, order quantity and order-up-to level, as well as the optimal cost value, as a function of the various model primitives, be it cost parameters or complete cost rate functions or characteristics of the demand and leadtime processes. These results are obtained as corollaries from a few general theorems, with separate treatment given to the case where the policy parameters are continuous variables and that where they need to be restricted to integer values. The results are applied both to standard inventory models and to those with general shelf age and delay dependent inventory costs.     

Non-gaussian Test Models for Prediction and State Estimation with Model Errors

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems: the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for the real time prediction and the state estimation.     

Multiple solutions of two-point nonlinear boundary value problems

A quasi-linear boundary value problem has a solution with properties induced by oscillatory properties of the linear part of an equation. This result is proved for two dimensional systems. Consequences for Φ-Laplacian equations and problems with resonant linear parts are discussed.     

Admissibilities of linear estimator in a class of linear models with a multivariate t error variable

This paper discusses admissibilities of estimators in a class of linear models,which include the following common models:the univariate and multivariate linear models,the growth curve model,the extended growth curve model,the seemingly unrelated regression equations,the variance components model,and so on.It is proved that admissible estimators of functions of the regression coefficient β in the class of linear models with multivariate t error terms,called as Model II,are also ones in the case that error terms have multivariate normal distribution under a strictly convex loss function or a matrix loss function.It is also proved under Model II that the usual estimators of β are admissible for p 2 with a quadratic loss function,and are admissible for any p with a matrix loss function,where p is the dimension of β.     

Degenerate KAM theory for partial differential equations

This paper deals with degenerate KAM theory for lower dimensional elliptic tori of infinite dimensional Hamiltonian systems, depending on one parameter only. We assume that the linear frequencies are analytic functions of the parameter, satisfy a weak non-degeneracy condition of Rüssmann type and an asymptotic behavior. An application to nonlinear wave equations is given.     

Roundoff-error analysis of a new class of conjugate-gradient algorithms

We perform the rounding-error analysis of the conjugate-gradient algorithms for the solution of a large system of linear equations Ax=b where Ais an hermitian and positive definite matrix. We propose a new class of conjugate-gradient algorithms and prove that in the spectral norm the relative error of the computed sequence {x_k} (in floating-point arithmetic) depends at worst on ζк^{$\text{3}\text{2}$}, where ζ is the relative computer precision and к is the condition number of A. We show that the residual vectors r_k=Ax_k-b are at worst of order ζк?vA?v ?vx_k?v. We p oint out that with iterative refinement these algorithms are numerically stable. If ζк ² is at most of order unity, then they are also well behaved.