(s,S) Inventory System with Postponed Demands

Abstract

This paper analyzes an (s, S) Inventory system where arrivals of customers form a Poisson process. When inventory level reaches zero due to demands, further demands are sent to a pool which has capacity M(<∞). Service to the pooled customers will be provided after replenishment against the order placed on reaching that level s. Further they are served only if the inventory level is at least s + 1. The lead-time is exponentially distributed. The joint probability distribution of the number of customers in the pool and the Inventory level is obtained in both the transient and steady state cases. Some measures of the system performance in the steady state are derived and some numerical illustrations are provided.     

Two-Commodity Inventory System with Individual and Joint Ordering Policies and Renewal Demands

Abstract

This article analyzes a two-commodity continuous review inventory system with renewal demands. The ordering policy is a combination of policies namely ordering individual commodities and ordering jointly both commodities. The steady state probability distribution for the joint inventory levels is computed. Various system performance measures in the steady state are derived. The results are illustrated numerically.     

一类跳扩散需求存贮系统(s,S)库存控制策略研究

考虑的是连续检查库存,需求为一个常时间函数和-个复合Poison跳扩散随机过程的和的存贮系统最优库存控制问题.基于期望折扣成本最小建立了无穷时间区间具有固定订购成本的最优库存模型,确定可采用(s,S)策略进行库存控制,给出了最优(s,S)策略的充要条件--HJB方程Ⅰ、Ⅱ.我们采用猜测的方法确定了最优(s,S)策略对应的值函数形式,建立了确定库存参数的最优化模型.     

Analysis of a Discrete-Time Queueing System with a Single Server and Heterogeneous Markovian Arrivals

We consider a discrete-time queueing system with a single deterministic server, heterogeneous Markovian arrivals and finite capacity. Most existing techniques model the queueing system using a direct bivariate Markov chain which requires a state space that grows rapidly as the number of customer types increases. In this paper, we define renewal cycles in terms of the input process and model the system occupancy level on each renewal cycle using a one-dimensional Markov chain. We derive the exact joint steady-state probability distribution of both states of input and system occupancy with a considerably reduced state space, which leads to the efficient calculation of overall/individual performance measures such as loss probability and average delay.     

Inventory control with fixed cost and price optimization in continuous time

We continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper [8], we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see [11]), i.e., the (s, S, p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s, S, p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Se- lecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy.     

快速补货模式下的动态定价与库存联合策略

在供应有限的情况下,研究常规补货和快速补货下商品动态定价问题.首先,建立了动态规划模型,理论证明了最优库存策略是基于(s,S)策略下改进的基本库存策略.其次,提出了一种启发式策略求复杂系统的最优策略,启发式算法能够求出最优价格和最优库存水平.最后,数值算例研究表明,库存管理中采用快速补货提高了零售商的利润;初始库存水平越高零售商的利润越高.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

相依修理的可修排队系统 MAP/PH(M/PH)/2

系统地研究了两个不同并行服务台的可修排队系统MAP/PH(M/PH)/2,其中两个不同的服务台拥有一个修理工.若其中一台处于修理状态,则另一台失效后就处于待修状态.利用拟生灭过程理论,我们首先讨论了两个服务台的广义服务时间的相依性,然后给出了系统的稳态可用度和稳态故障度,最后得到了系统首次失效前的时间分布及其均值.     

Optimal Hysteretic Control for the BMAP/G/ 1 System with Single and Group Service Modes

In this paper, we consider a single server queuing model with an infinite buffer in which customers arrive according to a batch Markovian arrival process (BMAP). The services are offered in two modes. In mode 1, the customers are served one at a time and in mode 2 customers are served in groups of varying sizes. Various costs for holding, service and switching are imposed. For a given hysteretic strategy, we derive an expression for the cost function from which an optimal hysteretic control can be obtained. Illustrative numerical examples are presented.     

Findings about the BMMPP for modeling dependent and simultaneous data in reliability and queueing systems

The batch Markov‐modulated Poisson process (BMMPP) is a subclass of the versatile batch Markovian arrival process (BMAP), which has been widely used for the modeling of dependent and correlated simultaneous events (as arrivals, failures, or risk events). Both theoretical and applied aspects are examined in this paper. On one hand, the identifiability of the stationary BMMPP_m(K ) is proven, where K is the maximum batch size and m is the number of states of the underlying Markov chain. This is a powerful result for inferential issues. On the other hand, some novelties related to the correlation and autocorrelation structures are provided.     

Average Cost Optimality in Inventory Models with Markovian Demands

This paper is concerned with long-run average cost minimization of a stochastic inventory problem with Markovian demand, fixed ordering cost, and convex surplus cost. The states of the Markov chain represent different possible states of the environment. Using a vanishing discount approach, a dynamic programming equation and the corresponding verification theorem are established. Finally, the existence of an optimal state-dependent (s, S) policy is proved.     

基于排队的库存服务系统最优控制策略

研究一个连续盘点的(s,Q)补货的库存服务系统。基于排队理论建立库存水平状态平衡方程,并推导出库存水平稳态概率分布以及作为库存控制的系统稳态性能指标。以库存成本最小化为目标,构建服务水平约束的库存控制模型。针对模型的非线性约束与整数型变量的特征,采用一种改进的遗传算法(IGA)用于决策变量的寻优。数值实验表明,当目标服务水平大于库存系统内生的服务水平时,实施服务水平约束能够降低库存控制成本。     

求解硬时间窗约束下随机需求库存-路径问题的优化算法

随机需求库存-路径问题(Stochastic Demand Inventory Routing Problem, SDIRP)即考虑随机需求环境下供应链中库存与配送的协调优化问题,是实施供应商管理库存策略过程中的关键所在,也是典型的NP难题之一。文章以具有硬时间窗约束的随机需求库存-路径问题(Stochastic Demand Inventory Routing Problem with Hard Time Windows, SDIRPHTW)为研究对象,将SDIRPHTW分解为直接配送的随机库存-路径问题和具有硬时间窗约束的路径优化问题两个子问题,并以最小化系统运行成本和用车数量为目标,设计了一个基于(s,S)库存策略和修正C-W节约法的启发式算法。最后,通过相应的数值算例验证了算法的有效性。     

具有不耐烦顾客和多重工作休假的排队库存系统

研究具有不耐烦顾客和多重工作休假的M/M/1/N排队库存系统模型,分别考虑了系统中库存为零时服务员休假和系统中顾客数为零时服务员休假两种休假方式,基于(s,S)库存策略,运用矩阵迭代方法得到了系统稳态概率分布,并给出系统相关性能指标,进而建立系统平均库存费用函数.通过数值算例对比分析了两种休假方式下的系统主要参数变化对...     

BMAP/G/1 queue with correlated arrivals of customers and disasters

We consider a single server queue with disasters where the arrivals of customers and disasters are correlated. When a disaster occurs, it removes all the customers in the system and there requires repair time for the system to be operated normally. The stationary queue length distribution at the embedded points and at an arbitrary time are presented.     

具有批到达的滞后排队系统分析

本文考虑一个基本模型的M^[x]/G^[r]/1/∞的排队系统，引入了(r,N）－策略（也称为滞后系统），首先分析了嵌入马氏链{Qn,n∈N}的平稳分布，得到其概率母函数Ｐ(z)，再利用半再生过程理论得到原系统的稳态队长分布的概率母函数π（ｚ）。     

On the Counting Process for a Class of Markovian Arrival Processes with an Application to a Queueing System

In this paper, we consider the counting process for a class of Markovian arrival processes (MAPs). We assume that the representing matrices in such MAPs are expanded in terms of matrix representations of the standard generators in the Lie algebra of the special linear group. The primary purpose of this paper is to construct an explicit solution of the time-dependent distribution and factorial moments of the number of arrival events in (0,t] of the counting process for this class of MAPs. Our construction relies on the Baker–Hausdorff lemma and the specific structure of the representing matrices. To investigate the efficiency of CPU usage with the explicit solution, we have conducted numerical experiments on computing the time-dependent distribution of the counting process through the explicit solution and uniformization-based method. We show that the CPU times required to compute the time-dependent distribution of the number of arrival events in (0,t] through the explicit solution have little sensitivity to t, while the consumption of CPU times with the uniformization-based method becomes greater as t increases. For illustrative purposes, we present a system performance analysis of a queueing system for possible use in automatic call distribution (ACD) systems. As an application of the explicit solution, we use it to express the waiting time distribution of the queueing system. Some numerical examples are also given with comparisons to computer simulations.AMS subject classification: 60K20, 60K25, 68M20This revised version was published online in June 2005 with corrected coverdate     

一种改进企业在框架协议下库存管理的方法

企业为了货源的稳定而与供应商签订一定期限内的订购框架协议,但如何实施订购策略以保障协议量的完成,保证满足需求的同时最小化企业的库存成本成为企业必须考虑的一个问题.传统的企业库存管理方法如(r,Q)、(s,S)等订购策略在一定程度上只考虑了如何满足需求,避免缺货,而没有考虑供应商的供货能力,在协议期内的总订购量只考虑了需求的影响,因而不能保证框架协议的完成.本文在(s,S)策略的基础上,考虑了框架协议的约束,提出了一种既能够保障框架协议的完成,又能够最大限度满足需求,同时降低企业库存成本的方法一“双表作业法”,并验证了方法的有效性.     

离散时间服务台可修的排队系统MAP／PH（PH／PH）／1

本文研究离散时间可修排队系统,其中顾客的输入过程为离散马尔可夫到达过程（MAP）,服务台的寿命,服务台的顾客的服务时间和修理时间均为离散位相型（PH）变量,首先我们考虑广义服务过程,证明它是离散MAP,然后运用阵阵几何解理论,我们给出了系统的稳态队长分布和稳态等待时间分布,同时给出了系统的稳态可用度这一可靠性指标。