Periodic control of intermittent demand items: theory and empirical analysis

In this paper we propose a modification to the standard forecasting, periodic order-up-to-level inventory control approach to dealing with intermittent demand items, when the lead-time length is shorter than the average inter-demand interval. In particular, we develop an approach that relies upon the employment of separate estimates of the inter-demand intervals and demand sizes, when demand occurs, directly for stock control purposes rather than first estimating mean demand and then feeding the results in the stock control procedure. The empirical performance of our approach is assessed by means of analysis on a large demand data set from the Royal Air Force (RAF, UK). Our work allows insights to be gained on the interactions between forecasting and stock control as well as on demand categorization-related issues for forecasting and inventory management purposes.     

可变抽样区间的二项EWMA控制图

在制造过程中,对产品的不合格品数进行监控时,通常选用计数性控制图-np图,它是基于过程服从二项分布建立的,一般对于过程中出现的较大波动效果明显。为了提高控制图对不合格品数较小波动的监控效果,本文设计了产品不合格品数服从二项分布的EWMA控制图。提出可变抽样区间的二项EWMA控制图,并采用马可夫链法计算其平均报警时间。对固定抽样区间以及可变抽样区间二项EWMA控制图对比研究,表明当过程失控时,可变抽样区间二项EWMA控制图具有较小的失控平均报警时间,能够迅速监测出过程中的异常波动,明显优于固定抽样区间的二项EWMA控制图。     

On the categorization of demand patterns

The categorization of alternative demand patterns facilitates the selection of a forecasting method and it is an essential element of many inventory control software packages. The common practice in the inventory control software industry is to arbitrarily categorize those demand patterns and then proceed to select an estimation procedure and optimize the forecast parameters. Alternatively, forecasting methods can be directly compared, based on some theoretically quantified error measure, for the purpose of establishing regions of superior performance and then define the demand patterns based on the results. It is this approach that is discussed in this paper and its application is demonstrated by considering EWMA, Croston's method and an alternative to Croston's estimator developed by the first two authors of this paper. Comparison results are based on a theoretical analysis of the mean square error due to its mathematically tractable nature. The categorization rules proposed are expressed in terms of the average inter-demand interval and the squared coefficient of variation of demand sizes. The validity of the results is tested on 3000 real-intermittent demand data series coming from the automotive industry.     

Forecasting of compound Erlang demand

Intermittent demand items dominate service and repair inventories in many industries and they are known to be the source of dramatic inefficiencies in the defence sector. However, research in forecasting such items has been limited. Previous work in this area has been developed upon the assumption of a Bernoulli or a Poisson demand arrival process. Nevertheless, intermittent demand patterns may often deviate from the memory-less assumption. In this work we extend analytically previous important results to model intermittent demand based on a compound Erlang process, and we provide a comprehensive categorisation scheme to be used for forecasting purposes. In a numerical investigation we assess the benefit of departing from the memory-less assumption and we provide insights into how the degree of determinism inherent in the process affects forecast accuracy. Operationalised suggestions are offered to managers and software manufacturers dealing with intermittent demand items.     

EWMA control charts for autoregressive processes

Many processes must be monitored by using observations that are correlated. An approach called algorithmic statistical process control can be employed in such situations. This involves fitting an autoregressive/moving average time series model to the data. Forecasts obtained from the model are used for active control, while the forecast errors are monitored by using a control chart. In this paper we consider using an exponentially weighted moving average (EWMA) chart for monitoring the residuals from an autoregressive model. We present a computational method for finding the out-of-control average run length (ARL) for such a control chart when the process mean shifts. As an application, we suggest a procedure and provide an example for finding the control limits of an EWMA chart for monitoring residuals from an autoregressive model that will provide an acceptable out-of-control ARL. A computer program for the needed calculations is provided via the World Wide Web.     

具有可变样本容量的非正态EWMA控制图

为了提高控制图的监控效率,本文研究非正态分布下,EWMA控制图的可变样本容量设计问题。首先利用Burr分布近似各种非正态分布,构造可变样本容量的非正态EWMA控制图;其次运用马尓科夫链法计算可变样本容量非正态EWMA控制图的平均运行长度;然后与传统的非正态EWMA控制图进行比较得出:当过程中出现小波动时,可变样本容量的非正态EWMA控制图能够更快地发现过程中的异常波动,具有较小的平均运行长度,其监控效率明显优于传统的非正态EWMA控制图。     

基于APL的EWMA控制图经济统计优化设计

为了提高指数加权移动平均(EWMA)控制图监控效率评价的精确性和全面性,提出用一个质量周期内的平均产品个数替代平均抽样个数来进行控制图的经济性和统计性综合评价,建立了一种基于平均产品长度(APL)的更为精确的EWMA控制图经济统计多目标优化设计模型,并采用具体的算例说明了采用NSGA-Ш算法对该模型进行计算的步骤。最后,用该方法优化设计的EWMA控制图与已有的几种EWMA控制图优化设计进行比较,结果表明:本文提出的经济统计优化设计方法显著优于只考虑经济性能的经济设计和只考虑统计性能的统计设计方案。     

Style goods pricing with demand learning

For many industries (e.g., apparel retailing) managing demand through price adjustments is often the only tool left to companies once the replenishment decisions are made. A significant amount of uncertainty about the magnitude and price sensitivity of demand can be resolved using the early sales information. In this study, a Bayesian model is developed to summarize sales information and pricing history in an efficient way. This model is incorporated into a periodic pricing model to optimize revenues for a given stock of items over a finite horizon. A computational study is carried out in order to find out the circumstances under which learning is most beneficial. The model is extended to allow for replenishments within the season, in order to understand global sourcing decisions made by apparel retailers. Some of the findings are empirically validated using data from U.S. apparel industry.     

预测组合权重更新的过程控制及分析

源于与决策分析的相关性,预测组合已经逐渐形成了一个重要的研究领域。为此,本文引进EWMA技术对预测组合权重更新的过程进行控制,从而提出一种能够应用于实际且简单有效的EWMA赋权方法。这种赋权方法能够确定预测组合权重应该何时更新,而不是机械地更新预测组合权重。本文额外针对各种赋权方法在旅游预测组合模型中的预测性能(全面预测性能和总均方根误差)和预测效率(权重更新频率)进行了经验评估。结果显示:EWMA赋权方法的预测性能优于传统的赋权方法,并与CUSUM赋权方法相似,同时该赋权方法获得了最小的权重更新频率。综合考虑预测性能和预测效率,EWMA赋权方法相比于其他赋权方法在旅游实际应用过程中更具优势。     

Joint economic design of EWMA control charts for mean and variance

Control charts with exponentially weighted moving average (EWMA) statistics (mean and variance) are used to jointly monitor the mean and variance of a process. An EWMA cost minimization model is presented to design the joint control scheme based on pure economic or both economic and statistical performance criteria. The pure economic model is extended to the economic-statistical design by adding constraints associated with in-control and out-of-control average run lengths. The quality related production costs are calculated using Taguchi’s quadratic loss function. The optimal values of smoothing constants, sampling interval, sample size, and control chart limits are determined by using a numerical search method. The average run length of the control scheme is computed by using the Markov chain approach. Computational study indicates that optimal sample sizes decrease as the magnitudes of shifts in mean and/or variance increase, and higher values of quality loss coefficient lead to shorter sampling intervals. The sensitivity analysis results regarding the effects of various inputs on the chart parameters provide useful guidelines for designing an EWMA-based process control scheme when there exists an assignable cause generating concurrent changes in process mean and variance.     

An examination of the size of orders from customers,their characterisation and the implications for inventory control of slow moving items

This paper examines half a million observations of the size of orders from customers at an electrical wholesaler. It notes: the distribution of the size of customer orders for a single item (stock keeping unit or SKU) is very skewed and resembles a geometric distribution; while the average size of an order is different for different items, for one SKU the mean order size is effectively the same at different branches even when the branches have very different demand rates; across a range of SKUs there is a strong relationship linking the mean and the variance of order size. The general results above are shown to apply to even the slowest movers. This extension is important because for items with intermittent demand the size of customer orders is required to produce an unbiased estimate of demand. Also a knowledge of the distribution of demand is important for setting maximum and minimum stock levels and the scheme employed is described.     

监测均值变动的多重控制图方法

本文证明了当受控平均运行长度充分大时, 多重控制图有两个优点: 一是相比较GLR (广义似然比) 和GEWMA (广义指数权重移动平均)控制图它可以大大降低运算的复杂性; 二是能够较快地监测均值变化的大小. 数值模拟也表明: 多重控制图不仅优于其构成的单个控制图, 而且在监测未知的均值变动方面也优于单个的CUSUM, EWMA, 多重EWMA和GLR控制图.     

Selecting the best periodic inventory control and demand forecasting methods for low demand items

The (s,S) form of the periodic review inventory control system has been claimed theoretically to be the best for the management of items of low and intermittent demand. Various heuristic procedures have been put forward, usually justified on the basis of generated data with known properties. Some stock controllers also have other simple rules which they employ and which are rarely seen in the literature. Determining how to forecast future demands is also a major problem in the area. The research described in this paper compares various periodic inventory policies as well as some forecasting methods and attempts to determine which are best for low and intermittent demand items. It evaluates the alternative methods on some long series of daily demands for low demand items for a typical spare parts depot.     

可变抽样区间几何EWMA控制图的经济设计

为了降低生产过程周期成本,本文对单位缺陷数服从几何分布时,可变抽样区间的指数加权移动平均(EWMA)控制图进行经济设计。首先建立可变抽样区间几何EWMA控制图的经济模型,使单位时间期望费用最小来确定参数的最优值;其次用遗传算法来寻找经济模型的最优解;最后对可变抽样区间几何EWMA控制图的经济模型进行灵敏度分析和最优性分析。研究结果表明单位时间期望费用分别随着异常原因发生的频率、过程失控时单位时间的质量费用、发现异常原因的时间期望值和纠正过程的时间期望值的增大而增大。     

A modified (S − 1, S) inventory system for deteriorating items with Poisson demand and non-zero lead time

An inventory system is considered for continuous decaying items with non-zero lead time and stochastic demand when shortages are allowed and all unsatisfied demands are backlogged. In this research we consider orders as separate packages where replenishment is one-for-one and a modified base stock policy is applied. In this paper, a penalty cost is introduced for stochastic inventory models with decaying items when less than one unit of the product is delivered to the customers. The objective of the warehouse is to maximize his average profit. Since the concavity analysis of the model is extremely complicated, an upper bound is introduced and an algorithm is presented for finding the optimal solution. Finally, a numerical example is presented and sensitivity analysis is carried out for a number of important parameters.     

Some properties of a simple moving average when applied to forecasting a time series

Simple (equally weighted) moving averages are frequently used to estimate the current level of a time series, with this value being projected as a forecast for future observations. A key measure of the effectiveness of the method is the sampling error of the estimator, which this paper defines in terms of characteristics of the data. This enables the optimal length of the average for any steady state model to be established and the lead time forecast error derived. A comparison of the performance of a simple moving average (SMA) with an exponentially weighted moving average (EWMA) is made. It is shown that, for a steady state model, the variance of the forecast error is typically less than 3% higher than the appropriate EWMA. This relatively small difference may explain the inconclusive results from the empirical studies about the relative predictive performance of the two methods.     

An Application of Double Exponential Smoothing Technique to Quality Control

休哈特控制图和ＳＵＳＵＭ控制图技术，在制造业得到广泛应用．Ｊ．Ｓ．Ｈｕｎｔｅｒ提出用指数加权滑动平均（ＥＷＭＡ）作为质量管理工具．然而，众所周知存在趋势项的情况下，用ＥＷＭＡ预测往往不是过低就是过高．本文的目的是表明利用双指数平滑（ＤＥＳ）技术可以减少此缺点．文中给出一个例子，它表明双指数平滑法比单指数平滑法，能更好地监控对于目标的偏离．     

Determining order-up-to levels under periodic review for compound binomial (intermittent) demand

We propose a new method for determining order-up-to levels for intermittent demand items in a periodic review system. Contrary to existing methods, we exploit the intermittent character of demand by modelling lead time demand as a compound binomial process. In an extensive numerical study using Royal Air Force (RAF) data, we show that the proposed method is much better than existing methods at approximating target service levels and also improves inventory-service efficiency. Furthermore, the proposed method can be applied for both cost and service oriented systems, and is easy to implement.     

An adaptively managed dynamic portfolio selection model using a time-varying investment target according to the market forecast

In this paper, we propose an adaptive investment strategy (AIS) based on a dynamic portfolio selection model (DPSM) that uses a time-varying investment target according to the market forecast. The DPSM allows for flexible investments, setting relatively aggressive investment targets when market growth is expected and relatively conservative targets when the market is expected to be less attractive. The model further allows investments to be liquidated into risk-free assets when the market forecast is pessimistic. By dynamically determining the investment target, the DPSM allows construction of portfolios that are more responsive to market changes, while eliminating the possibility of the model becoming infeasible under certain market conditions. When the proposed DPSM is implemented in real-life investment scenarios using the AIS, the portfolio is rebalanced according to a predefined rebalancing cycle and the model’s input parameters are estimated on each rebalancing date using an exponentially weighted moving average (EWMA) estimator. To evaluate the performance of the proposed approach, a 7-year investment experiment was conducted using historical stock returns data from 10 different stock markets around the world. Performance was assessed and compared using diverse measures. Superior performance was achieved using the AIS proposed herein compared with various benchmark approaches for all performance measures. In addition, we identified a converse relationship between the average trading volume of a market and the value of the weighting parameter prescribed to the EWMA estimator, which maximizes cumulative returns in each market.     

Forecasting for intermittent demand: the estimation of an unbiased average

The majority of the range of items held by many stockists exhibit intermittent demand. Accurate forecasting of the issue rate for such items is important and several methods have been developed, but all produce biased forecasts to a greater or lesser degree. This paper derives the bias expected when the order arrivals follows a Poisson process, which leads to a correction factor for application in practice. Extensions to some other arrival processes are briefly considered.