## 2 Literature Review
### Demand Forecasting  
Probabilistic forecasting is a crucial tool for making informed decisions in the face of demand uncertainty. It encompasses two main forecasting modeling approaches: parametric and nonparametric. Parametric models specify a probability distribution and estimate its parameters, whereas nonparametric models eschew distributional assumptions and instead build empirical distributions, often focusing on quantiles.  
[PERSON] et al. [4] develop a comprehensive framework for predicting the distributions and performance evaluation metrics by employing Poisson and negative binomial distributions for inventory management. Similarly, [PERSON] et al. [2] concentrate on parametric methods utilizing Gaussian and negative binomial distributions to handle multi-series customer demand. [PERSON] et al. [5] propose an integrated approach that combines parametric and nonparametric models for probabilistic forecasting within electric power markets.  
Recent advancements in forecasting methodologies have seen a growing emphasis on nonparametric probabilistic techniques. Notably, the works of [PERSON] et al. [6], [PERSON] et al. [7], [PERSON] et al. [8], [PERSON] et al. [9], and [PERSON] et al. [10] have undersocored the robustness of these approaches, particularly in their ability to handle data that do not conform to predefined distributions. This flexibility renders nonparametric methods especially valuable in real-world forecasting scenarios. However, when data exhibit clear adherence to specific distributions or when strong theoretical foundations support particular distributional assumptions, parametric models can offer more nuanced insights into future uncertainties, thereby enhancing decision-making processes in areas such as inventory management.  
In the context of bike-sharing systems, while demand forecasting has been extensively studied, there remains a notable gap in research focusing on probabilistic forecasting methods. [PERSON] et al. [11] introduce a multi-categorical probabilistic approach for short-term demand prediction using Gaussian Mixture Models and Markov chain models, discretizing demand into categorical states. In contrast, our research employs continuous probabilistic forecasting with DeepAR, aiming to capture longer-term demand uncertainty across multiple stations for inventory optimization. [PERSON] et al. [12] proposed a queuing theory-based probabilistic forecasting approach and introduced new scoring rules for evaluating predictions. While their work centered on station-level forecasts, our study adopts a system-wide approach to simultaneously model demand patterns across multiple stations. More recently, [PERSON] et al. [13] developed a variational Poisson recurrent neural network for demand forecasting and inventory management, highlighting discrepancies between predictive and prescriptive metrics. [PERSON] et al. [3] utilized the DeepAR approach for probabilistic demand forecasting in bike-sharing systems, aiming to improve inventory rebalancing by accurately predicting station-level bike demand. Building upon these contributions, our research leverages DeepAR for system-wide probabilistic forecasting, specifically tailored to optimize rebalancing strategies across multiple stations, thus addressing the need for a comprehensive approach in bike-sharing demand prediction and inventory management.