**5. Conclusions**

Along with the trend of technological advance, a rapid increase in product diversity and shortening of product life cycles have become important features for enterprises. This means that practitioners need to use limited historical demand to forecast the future, leading to the failure of traditional forecasting approaches. Fortunately, potential relationships between demand time series of multiple products can provide abundant information to improve forecasting accuracy. Therefore, we proposed an improved approach, whose core idea is utilizing the demand of associated products (including highly correlated ones or ones having significantly causality) as predictors to add in AR model to improve its prediction accuracy. Considering that time series may be short, we introduce two feasible schemes to simplify variables to avoid over-fitting. Then monthly procurement data from a Chinese grid company is used to test the forecasting ability of the proposed approaches, as well as four conventional time series approaches SES, AR, VAR, and FDM.

We adopt two types of indicators including forecasting errors and inventory performance to compare these approaches. There are three main findings. Firstly, our new approaches can perform better than those conventional approaches, especially when the product dimensionality is large. Among them, CII with the critical value equal to 0.513 has the best forecasting performance, enjoying the lowest forecasting errors, total inventory costs and inventory ratio, followed by GII with the critical value equal to 0.01. Secondly, Scheme (II) is more effective for the approaches based on associated relationships to achieve higher accuracy, compared with Scheme (I) because it can refine more information and set off the errors of different time series. Finally, it is vital to set a rational threshold condition to select proper variables, which can help to ge<sup>t</sup> ideal prediction results. The condition should make sure that most objects have enough proper variables to improve prediction and while the irrelative information is also as less as possible.

However, due to the limitations of the real dataset, we can only pick up one predictor from the associated relationships, so we cannot discuss the influence of the number of selected predictors on the forecasting accuracy. In the future, we can extend our research from the following aspects. Firstly, we can properly test the new approaches through a large wide of diverse datasets, and analyze the most proper number of predictors that should be added in final forecasting models. Secondly, try more methods to dig for predictors from proper variable sets, not limited to PCA. Then compare their prediction performance to find a better one. In addition, "order-up-to level" is the only inventory strategy used to do inventory simulations in this empirical study. Therefore, we can introduce other widely applied strategies into inventory simulations and explore whether the approaches based on associated relationships are appropriate for different inventory strategies.

**Author Contributions:** Conceptualization, M.L.; methodology, S.L. and S.Y.; software, S.L.; validation, S.Y.; formal analysis, S.Y. and S.L.; investigation, S.L.; resources, S.L.; data curation, S.L. and S.Y.; writing—original draft preparation, S.Y.; writing—review and editing, S.Y.; visualization, S.L. and S.Y.; supervision, M.L.; project administration, M.L.; funding acquisition, M.L.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
