کنترل موجودی با یک روش Croston اصلاح شده و توزیع ارلانگ

This paper considers an inventory control system, primarily for a finished goods inventory. The purpose is to create a procedure that can handle both fast-moving items with regular demand and slow-moving items. The suggested procedure should be easy to implement in a modern computerized ERP-system. Essentially, the system is a periodic review system built around a Croston forecasting procedure. An Erlang distribution is fitted to the observed data using the mean and variance of the forecasted demand rate. According to probabilities for stock shortages, derived from the probability distribution, the system decides if it is time to place a new order or not. The Croston forecasting method is theoretically more accurate than ordinary exponential smoothing for slow-moving items. However, it is not evident that a Croston forecasting procedure (with assumed Erlang distribution) outperforms ordinary exponential smoothing (with assumed normal distribution) applied in a “practical” inventory control system with varying demand, automatically generated replenishment, etc. Our simulation study shows that the system in focus will present fewer shortages at lower inventory levels than a system based on exponential smoothing and the normal distribution.

Jonsson (2000) and Jonsson and Mattsson (2002) present a survey of used techniques for production and inventory control in 400 Swedish companies. Their study shows that many companies involved in the manufacturing and supply environment perform poorly when it comes to updating parameters used for inventory control, e.g. the reorder points in the year 1999 were even more scarcely updated than in a similar study 5 years ago. The companies answered that most of the items reorder points were not even updated once a year. A forecasting module, which automatically updates safety stocks and reorder points, would in our opinion, decrease inventory levels and inventory shortages for most items with changing demand. We suppose that the low usage of automatic reorder point updates in practical applications is due to the difficulty for ordinary exponential smoothing to handle slow-moving items; and the difficulty in a practical application to decide which forecast interval to use (Segerstedt, 2000). This paper considers the problem of forecasting and managing an inventory consisting of articles that can be both slow moving and fast moving. For fast-moving items, many methods for inventory control have been developed. The method that is most commonly used is the Exponential smoothing forecasting procedure in combination with an assumption that the demand size for the items belongs to the normal distribution. This method of controlling inventories works quite well for the fast-moving items. However, problems arise when the demand becomes intermittent. Intermittent demand seems to appear at random and there are many periods, i.e. production days, weeks or even months that do not show any demand at all. Another problem is that when there is a demand during one period this demand can be greater than one unit. Another important issue to discuss is what statistical distribution to use for items with intermittent demand. The most common assumption is that the demands for these items are normally distributed. However, e.g. Burgin (1975) has investigated the applicability of the gamma distribution for inventory control. His findings were that the gamma distribution was better suited for representing the demand of different items than the normal distribution. This is due to the fact that the gamma distribution is defined only for non-negative values and that it need not be symmetrical all the time. Both these facts are important when discussing the demand of an item. The fact that the gamma distribution is not defined for negative values implies that there is no negative demand, which is a reasonable assumption to make. Of course in some cases negative demand might occur due to customers returning their products but that is another problem. Burgin also states that the symmetrical shape of the normal distribution is a disadvantage when representing intermittent demand and that the gamma distribution, which is not symmetrical, is more appropriate for the representation of intermittent demand.

From this work we dare to draw the conclusion that when the demand is intermittent an inventory control system based on the modified Croston procedure and the Erlang distribution will outperform a “normal system” using the exponential smoothing forecasting technique and normal distribution when it comes to reducing shortages. It can be seen from the simulation results of the artificial demand series that the reductions in shortage volumes are not due to higher inventory levels. (If we have demand every period modified Croston turns to normal exponential smoothing.) The improvement, we believe, is primarily based on the modified Croston procedure and not on the assumption of Erlang distribution. For a lead-time distribution, which is a sum of demand rates per day, the Erlang distribution then becomes close to the normal distribution (the central limit theorem). In our simulation study if we use normal or Erlang distribution probably has very small importance. Other researchers have also come to the conclusion that changing probability density function will not give significant improvements (see for example Silver et al., 1998). However we believe, that with a skewed distribution like in Fig. 2, a better estimate must give information to an improved probability calculation and therefore to a better inventory control. One way to improve even the Erlang estimate may be to incorporate more moments than only E{X} and E{X2}.(μ=E{X},σ2=E{X2}−μ2). We hope in the future to present a procedure for this, which can be implemented in computer based inventory systems. With modified Croston we estimate a forecasted demand rate for the next day; with a long lead time, the sum of many Erlang distributed demand rates will be an Erlang distributed demand during lead time (plus review interval) that, even with a better estimate, is very close to the normal distribution (the central limit theorem). Even if the measured demand rate is much skewed, the sum of many skewed demand rates will be normal. Maybe that will not be true in a practical application; the real demand during lead time may still be skewed, science the assumed independence between the different demands per day does not exist. More tests with real life data must be performed. The modified Croston technique perhaps can also be modified to measure the demand during lead time separately but still keep the continuous time scale and renew the forecast only when a withdrawal occurs. To prove the usefulness of the modified Croston procedure, the evaluation of the model must be extended to include simulations with non-stationary stochastic demand series, or even better, real demand. However, these simulations, with stationary stochastic demand, indicate that this model is worth further investigation.