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.     

Classification for forecasting and stock control: a case study

Different stock keeping units (SKUs) are associated with different underlying demand structures, which in turn require different methods for forecasting and stock control. Consequently, there is a need to categorize SKUs and apply the most appropriate methods in each category. The way this task is performed has significant implications in terms of stock and customer satisfaction. Therefore, categorization rules constitute a vital element of intelligent inventory management systems. Very little work has been conducted in this area and, from the limited research to date, it is not clear how managers should classify demand patterns for forecasting and inventory management. A previous research project was concerned with the development of a theoretically coherent demand categorization scheme for forecasting only. In this paper, the stock control implications of such an approach are assessed by experimentation on an inventory system developed by a UK-based software manufacturer. The experimental database consists of the individual demand histories of almost 16?000 SKUs. The empirical results from this study demonstrate considerable scope for improving real-world systems.     

On exhibiting inventory systems with Erlangian lifetimes under renewal demands

A continuous revies (s, S) inventory system with renewal demand in which one item is put into operation as an exhibiting piece is analyzed. The lifetime of any operating unit has Erlangian distribution, and on failure is replaced by another one from the stock and the failed item is disposed of. Replenishment of stock is instantaneous. The transient and stationary values of inventory level distribution and the mean reorder rate are obtained using the techniques of semi-regenerative processes. Decision rules for optimums andS that minimize the long-run expected cost rate are derived. The solution for a dual model with the distribution of lifetimes and inter-demand times interchanged is also given.     

Forecasting daily supermarket sales using exponentially weighted quantile regression

Inventory control systems typically require the frequent updating of forecasts for many different products. In addition to point predictions, interval forecasts are needed to set appropriate levels of safety stock. The series considered in this paper are characterised by high volatility and skewness, which are both time-varying. These features motivate the consideration of forecasting methods that are robust with regard to distributional assumptions. The widespread use of exponential smoothing for point forecasting in inventory control motivates the development of the approach for interval forecasting. In this paper, we construct interval forecasts from quantile predictions generated using exponentially weighted quantile regression. The approach amounts to exponential smoothing of the cumulative distribution function, and can be viewed as an extension of generalised exponential smoothing to quantile forecasting. Empirical results are encouraging, with improvements over traditional methods being particularly apparent when the approach is used as the basis for robust point forecasting.     

Control Theory in Production/Inventory Systems: A Case Study in a Food Processing Organization

Production/inventory systems must attempt to cope with varying demand. This is traditionally done by applying the R,t inventory system after forecasting demand. If the demand is not stationary in mean and variance, this system relies on advanced forecasting models and may still be inadequate. A technique is presented which applies feedback control theory to the maintenance of a chosen level of safety stock. A term involving the rate of change of stock on hand is introduced to overcome the deficiencies of the standard R,t reorder policy when dealing with erratic demand. Once the parameters for an individual application are chosen the technique may be automated on any spreadsheet or modelling package. Results are produced which indicate that this is a superior heuristic to the standard R,t system.     

A periodic-review inventory control policy for a two-level supply chain with multiple retailers and stochastic demand

We consider a time-based inventory control policy for a two-level supply chain with one warehouse and multiple retailers in this paper. Let the warehouse order in a fixed base replenishment interval. The retailers are required to order in intervals that are integer-ratio multiples of the base replenishment interval at the warehouse. The warehouse and the retailers each adopt an order-up-to policy, i.e. order the needed stock at a review point to raise the inventory position to a fixed order-up-to level. It is assumed that the retailers face independent Poisson demand processes and no transshipments between them are allowed. The contribution of the study is threefold. First, we assume that when facing a shortage the warehouse allocates the remaining stock to the retailers optimally to minimize system cost in the last minute before delivery and provide an approach to evaluate the exact system cost. Second, we characterize the structural properties and develop an exact optimal solution for the inventory control system. Finally, we demonstrate that the last minute optimal warehouse stock allocation rule we adopt dominates the virtual allocation rule in which warehouse stock is allocated to meet retailer demand on a first-come first-served basis with significant cost benefits. Moreover, the proposed time-based inventory control policy can perform equally well or better than the commonly used stock-based batch-ordering policy for distribution systems with multiple retailers.     

A semi-parametric approach for estimating critical fractiles under autocorrelated demand

Forecasting critical fractiles of the lead time demand distribution is an important problem for operations managers making newsvendor-type inventory decisions. In this paper, we propose a semi-parametric approach to forecasting the critical fractile when demand is serially correlated. Starting from a user-defined but potentially misspecified forecasting model, we use historical demand data to generate empirical forecast errors of this model. These errors are then used to (1) parametrically correct for any bias in the point forecast conditional on the recent demand history and (2) non-parametrically estimate the critical fractile of the demand distribution without imposing distributional assumptions. We present conditions under which this semi-parametric approach provides a consistent estimate of the critical fractile and evaluate its finite sample properties using simulation and real data for retail inventory planning.     

Decision making on stock levels in cases of uncertain demand rate

Inventory control is a typical problem of decision making. In this paper a periodic replenishment of stock, the spare parts being of one kind, is discussed for some cases when the demand rate is uncertain. The first decision, before all others in the sequence, is done by assuming an a priori distribution of demand rate. In time, as the demand process goes on, corrections of parameters of the a priori distribution are made according to the accumulated knowledge about past demand. This Bayesian approach to decision making based on learning about the uncertain demand rate is known for the case when the demand rate is unknown but constant. It is shown that this same approach can be used in some cases when the demand rate is unknown and not constant. Results are given and used for inventory control.     

Inventory control under temporal demand heteroscedasticity

This paper studies an inventory control problem when the variance of demand is time-varying and exhibits temporal heteroscedasticity. We use a first-order autoregressive process to characterize the dynamic changes in the level of demand over time and a GARCH(1, 1) structure to describe the changes in the variance of demand. Under these demand settings, we quantify the effect of a temporal heterogeneous variance on inventory performance for a system controlled via an order-up-to-level policy. We show that the effect of temporal heteroscedasticity on the forecasting accuracy can be additively decomposed from the total forecasting error variance. The decomposition is used to derive the absolute and relative cost deviations when the temporal heteroscedasticity is ignored. The relationship of these cost deviations to demand autocorrelation and replenishment leadtime is investigated. Computational results show that ignoring temporal heteroscedasticity can increase firm’s inventory costs by as much as 30% when demand autocorrelation is highly positive.     

A new approach of forecasting intermittent demand for spare parts inventories in the process industries

Accurate demand forecasting is of vital importance in inventory management of spare parts in process industries, while the intermittent nature makes demand forecasting for spare parts especially difficult. With the wide application of information technology in enterprise management, more information and data are now available to improve forecasting accuracy. In this paper, we develop a new approach for forecasting the intermittent demand of spare parts. The described approach provides a mechanism to integrate the demand autocorrelated process and the relationship between explanatory variables and the nonzero demand of spare parts during forecasting occurrences of nonzero demands over lead times. Two types of performance measures for assessing forecast methods are also described. Using data sets of 40 kinds of spare parts from a petrochemical enterprise in China, we show that our method produces more accurate forecasts of lead time demands than do exponential smoothing, Croston's method and Markov bootstrapping method.     

Optimal pricing and inventory policies: Centralized and decentralized decision making

The paper studies an optimal control problem of pricing and inventory replenishment in a system with serial inventories. Consumer demand for a specific product at a retail outlet depends on price as well as the in-store stock of the product. The hypothesis is that for certain consumer products, a large volume of displayed goods leads consumers to buy more than if the stock is small. In addition to the stock that is on display in the store, there is an inventory of the product in a central warehouse. First we consider a setup in which management of the two stocks is decentralized such that pricing decisions are made by the store manager who also decides on the level of in-store inventory. The warehouse manager makes the replenishment decisions concerning the stock in the warehouse. Next we study the problem where stock management and pricing decisions are centralized. Optimal trajectories for inventories, replenishment rates, and retail price are derived by using phase diagrams and a formal synthesizing procedure.     

Concepts for safety stock determination under stochastic demand and different types of random production yield

We consider a manufacturer’s stochastic production/inventory problem under periodic review and present methods for safety stock determination to cope with uncertainties that are caused by stochastic demand and different types of yield randomness. Following well-proven inventory control concepts for this problem type, we focus on a critical stock policy with a linear order release rule. A central parameter of this type of policy is given by the safety stock value. When non-zero manufacturing lead times are taken into account in the random yield context, it turns out that safety stocks have to be determined that vary from period to period. We present a simple approach for calculating these dynamic safety stocks for different yield models. Additionally, we suggest approaches for determining appropriate static safety stocks that are easier to apply in practice. In a simulation study we investigate the performance of the proposed safety stock variants.     

Forecasting sales of slow and fast moving inventories

Traditional computerised inventory control systems usually rely on exponential smoothing to forecast the demand for fast moving inventories. Practices in relation to slow moving inventories are more varied, but the Croston method is often used. It is an adaptation of exponential smoothing that (1) incorporates a Bernoulli process to capture the sporadic nature of demand and (2) allows the average variability to change over time. The Croston approach is critically appraised in this paper. Corrections are made to underlying theory and modifications are proposed to overcome certain implementation difficulties. A parametric bootstrap approach is outlined that integrates demand forecasting with inventory control. The approach is illustrated on real demand data for car parts.     

Spare parts management: Linking distributional assumptions to demand classification

Spare parts are known to be associated with intermittent demand patterns and such patterns cause considerable problems with regards to forecasting and stock control due to their compound nature that renders the normality assumption invalid. Compound distributions have been used to model intermittent demand patterns; there is however a lack of theoretical analysis and little relevant empirical evidence in support of these distributions. In this paper, we conduct a detailed empirical investigation on the goodness of fit of various compound Poisson distributions and we develop a distribution-based demand classification scheme the validity of which is also assessed in empirical terms. Our empirical investigation provides evidence in support of certain demand distributions and the work described in this paper should facilitate the task of selecting such distributions in a real world spare parts inventory context. An extensive discussion on parameter estimation related difficulties in this area is also provided.     

Forecasting for the ordering and stock-holding of spare parts

A modern military organization like the UK's Royal Air Force is dependent on readily available spare parts for in-service aircraft in order to maximize operational capability. A large proportion of spare parts are known to have an intermittent or slow-moving demand pattern, presenting particular problems as far as forecasting and inventory control are concerned. In this paper, we use extensive demand and replenishment lead-time data to assess the practical value of forecasting models put forward in the literature for addressing these problems. We use an analytical method for classifying the consumable inventory into smooth, irregular, slow-moving and intermittent demand patterns. Recent forecasting developments are compared against more commonly used methods across the identified demand patterns. One recently developed method, a modification to Croston's method referred to as the approximation method, is observed to provide significant reductions in the value of the stock-holdings required to attain a specified service level for all demand patterns.     

Customer waiting times in an (R,S) inventory system with compound Poisson demand

Besides service level and mean physical stock, customer waiting time is an important performance characteristic for an inventory system. In this paper we discuss the calculation of this waiting time in case a periodic review control policy with order-up-to-levelS is used and customers arrive according to a Poisson process. For the case of Gamma distributed demand per customer, we obtain (approximate) expressions for the waiting time characteristics. The approach clearly differs from the traditional approaches. It can also be used to obtain other performance characteristics such as the mean physical stock and the service level.     

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.     

Potential inventory cost reductions using advanced time series forecasting techniques

This paper compares demand forecasts computed using the time series forecasting techniques of vector autoregression (VAR) and Bayesian VAR (BVAR) with forecasts computed using exponential smoothing and seasonal decomposition. These forecasts for three demand data series were used to determine three inventory management policies for each time series. The inventory costs associated with each of these policies were used as a further basis for comparison of the forecasting techniques. The results show that the BVAR technique, which uses mixed estimation, is particularly useful in reducing inventory costs in cases where the limited historical data offer little useful information for forecasting. The BVAR technique was effective in improving forecast accuracy and reducing inventory costs in two of the three cases tested. In the third case, unrestricted VAR and exponential smoothing produced the lowest experimental forecast errors and computed inventory costs. Furthermore, this research illustrates that improvements in demand forecasting can provide better cost reductions than relying on stochastic inventory models to provide cost reductions.     

Inventory rationing via drop-shipping in Internet retailing: A sensitivity analysis

In this paper, we study a threshold level inventory rationing policy that is of interest to e-tailers, operating in a business to consumer (B2C) environment and selling non-perishable, made-to-stock items such as books, CDs, consumer electronics, and body and bath products. A Monte Carlo simulation model is developed to examine this policy when the demand process is stochastic, lead-time is stochastic, and the e-tailer uses ‘drop-shipping’ as an order fulfillment option. The methodology presented, which includes computer simulation and a full factorial experimental design, permits understanding of the complexity of the decision-making environment and implications of different sources of uncertainty (e.g. demand variability and lead-time variability) on a profit-maximizing threshold level of inventory, a stock level below which low margin orders are drop-shipped directly from the e-tailer’s supplier rather than fulfilled from internal stock.     

Using an Expert System to Monitor an Automatic Stock Control System

The theory of the control of stock is well understood, and a large number of firms use automatic stock control and ordering techniques as a major part of their inventory control systems. However, the effectiveness of such automatic stock control systems depends on the correct selection of system parameters for each item in the inventory and on the accurate reporting of information. It is not unusual for errors to be made in both the selection of parameters and the entry of data, and regular failures of either can cause severe problems in stock control. It is generally difficult for a human stock controller to discover errors quickly because of the large number of items that is usually found in a company inventory. This paper describes the use of an expert system to examine stock records and diagnose anomalies in the data provided or parameters applied to individual items in a stock control system. The expert system ‘reasons’ from symptoms such as reports from the forecasting system or high stock levels, to defects in the stock model used or data-collection processes, and from there to possible remedial action. Results are presented for runs of the expert system on a simulated stock control system.