پیش بینی و عملکرد موجودی در زنجیره تامین دو مرحله ای همراه با تقاضای (ARIMA (0،1،1: نظریه و تجزیه و تحلیل تجربی

The ARIMA(0,1,1) demand model has been analysed extensively by researchers and used widely by forecasting practitioners due to its attractive theoretical properties and empirical evidence in its support. However, no empirical investigations have been conducted in the academic literature to analyse demand forecasting and inventory performance under such a demand model. In this paper, we consider a supply chain formed by a manufacturer and a retailer facing an ARIMA(0,1,1) demand process. The relationship between the forecasting accuracy and inventory performance is analysed along with an investigation on the potential benefits of forecast information sharing between the retailer and the manufacturer. Results are obtained analytically but also empirically by means of experimentation with the sales data related to 329 Stock Keeping Units (SKUs) from a major European superstore. Our analysis contributes towards the development of the current state of knowledge in the areas of inventory forecasting and forecast information sharing and offers insights that should be valuable from the practitioner perspective.

The classical approach towards inventory forecasting considered by researchers and practitioners alike consists of selecting an accurate forecasting method that is subsequently used for stock control purposes. However, it should be noted that this approach tends to look at demand forecasting and inventory management as two independent stages without interactions, which may lead to a sub-optimal performance of the whole system (Syntetos et al., 2010). In fact, previous research has shown that forecast accuracy is to be distinguished from the performance of the forecasts when utility measures are employed, especially in an inventory management context where the interactions between forecasting and stock control are not yet fully understood (Syntetos et al., 2009b, Babai et al., 2010 and Ali et al., 2011). That is to say, forecast accuracy improvements do not necessarily imply inventory cost savings and/or a service level increase. Ali et al. (2011) have investigated the scale of inventory savings according to the degree of improvement in forecasting accuracy by analysing a two-stage supply chain where two information sharing strategies are considered. The first, termed as No Information Sharing (NIS), relates to not sharing any information with the downstream supply chain members. Under such a strategy, the upstream members base their forecasts on the orders received from the downstream members and no information sharing mechanisms are employed. The second strategy, termed as Forecast Information Sharing (FIS), relates to the supply chain members (say a retailer and the manufacturer) using the same forecasts to place orders. The research under concern was conducted assuming stationary demand processes (Auto-Regressive, AR(1); Moving Average, MA(1) and ARMA(1,1)) and it resulted in some very important findings: (i) there is a substantial forecast-accuracy related benefit resulting from FIS and the relevant gains depend on the demand process; (ii) the percentage reductions in inventory holdings and costs are generally less than the percentage gains in forecasting errors; (iii) the translation of the accuracy gains to inventory savings depends on the magnitude of the forecast accuracy improvement, regardless of the demand process. Non-stationary processes though were not considered. The (Auto-Regressive Integrated Moving Average) ARIMA(0,1,1) demand model in particular has been analysed extensively by researchers and used widely by forecasting practitioners due to its attractive theoretical properties and empirical evidence in its support. It is important to note that the optimal estimator for an ARIMA(0,1,1) model is the Simple Exponential Smoothing method that is widely used in practise. A well known result in the supply chain inventory forecasting literature (see Graves, 1999; Gilbert, 2005) shows that, for an order-up-to level (OUT) system, the orders resulting from a downstream stage facing an ARIMA(0,1,1) demand process follow also an ARIMA(0,1,1) process. Based on this mathematical relationship, various papers have argued that there is no benefit from any information sharing between downstream and upstream stages (e.g. Graves, 1999). However, it has recently been shown in the academic literature that orders faced by an upstream stage in the supply chain that follow an ARIMA(0,1,1) process may be generated from various demand processes at the downstream stage (including the ARIMA(0,1,1)) (Ali and Boylan, 2011). As such, it may not be possible for the upstream stage to infer the demand at the downstream stage rendering FIS a potentially very beneficial strategy that may result in reduced forecasting errors and inventory costs. Addressing the plausibility of this assumption constitutes the aim of this work. To the best of our knowledge, there have been no attempts in the academic literature to analyse empirically inventory systems with ARIMA(0,1,1) demand data. It is our aim to attempt to fill this gap. In more detail, the aim of this paper is twofold: (i) to analyse the scale of inventory savings according to the degree of improvement in forecasting accuracy for an ARIMA(0,1,1) demand process; and (ii) to assess the inventory benefits of information sharing under an ARIMA(0,1,1) structure. Our analysis is conducted analytically but also empirically by means of experimentation with the sales data related to 329 Stock Keeping Units (SKUs) from a major European superstore. The remainder of the paper is organised as follows: in the next section the background of our work is briefly discussed followed, in Section 3, by the presentation of the assumed supply chain structure and the theoretical analysis of forecasting and inventory performance. In Section 4 we analyse the scale of inventory savings according to the degree of improvement in forecasting accuracy along with the inventory benefits of information sharing in the supply chain. This is conducted by means of simulation on theoretically generated and empirical data. The paper concludes, in Section 5, with a discussion of our main findings and the natural next steps of research.

This paper extends the existing theory related to ARIMA(0,1,1) processes in two regards. First of all, it explores the benefits of sharing forecast information in a two-stage supply chain when demand follows such a process. Secondly, the relationship between MSE and inventory holdings/costs (for which analysis was previously limited only to ARMA processes) has been considered. We have validated our theoretical results for Mean Square Error via simulation and found them to closely match actual performance. We have also quantified the differences between theoretical and simulation results for inventory holdings. We have analysed a two-stage supply chain that consists of one retailer and one manufacturer where the retailer faces a non-stationary ARIMA(0,1,1) demand process. Two information sharing strategies have been considered in the analysis, namely, the NIS and the FIS strategies. Analytical expressions of the demand forecasts and the inventory levels have been derived for both strategies. The percentage reduction of the forecasting error and the inventory cost resulting from using the FIS instead of the NIS strategy have been investigated with respect to the smoothing constant α  and the lead-time L. We have first shown that the percentage reduction in the MSE and the inventory cost increases when the smoothing constant α  increases. The results also show that this percentage reduction is an increasing function of the lead-time. The reduction in the MSE and the inventory cost can go up to 84% and 70%, respectively, showing the substantial benefits that may be derived from sharing forecast information between the retailer and the manufacturer under the ARIMA(0,1,1) framework. This also confirms the findings reported in the academic literature for other demand processes. In addition, this investigation provides further evidence on the gap between forecasting accuracy and forecasting utility performance since the results show that the percentage reductions in inventory costs are generally less than the percentage gains in MSE. Through an empirical investigation based on the sales data related to 329 SKUs from a major European superstore, we have shown that the benefit resulting from using the FIS instead of the NIS strategy is substantial. This benefit is expressed in terms of a percentage reduction of the forecasting error that reaches 81% and a percentage reduction of the inventory cost that is equal to 68%. Please note that this is validated for ranges of estimated α values going up to 0.4. Various case studies have shown how supply chain collaborations may result in improved performance. For example Ireland and Crum (2006) suggest a 40% reduction in inventory costs. The study under concern investigates the benefits of sharing information in supply chains. Recently, Boone and Ganeshan (2008) specifically considered sharing forecasts in supply chains and reported a 40% reduction in supply chain costs in their study of two medium sized companies. The results of the above case studies clearly support the findings presented in this paper i.e. the benefits of sharing information in supply chain can be very large. How large these benefits may be for a certain organisation would depend on various factors. In this paper, we have looked at two of these factors: the smoothing constant and the lead-time. Further research is required to explore the effects of other factors e.g. cost structure and forecasting methods. Companies wishing to investigate how much savings there may be for their own supply chains by utilising FIS would need to simulate their own sales forecast and inventory costs. Such estimates on potential savings in costs would help companies make informed decisions about the investment required in IS/IT for such collaborations. Although the empirical results provided in this paper generally agree with our theoretical findings, it should also be noted that our empirical investigation has been based on a small dataset that corresponds to the downstream demand of a supermarket. An interesting avenue for further research could be to extend this empirical investigation by considering bigger datasets in multi-stage supply chains. Consideration of other ARIMA demand processes could also contribute towards the development of the current state of knowledge in the areas of inventory forecasting with non-stationary demand processes.