Closed-loop supply chains are complex systems as they involve the seamless backward and forward flow of products and information. With the advent of e-commerce and online shopping, there has been a growing interest in product returns and the associated impact on inventory variance and the bullwhip effect. In this paper, a novel four-echelon closed-loop supply chain model is presented, where base-stock replenishment policies are modelled by means of a proportional controller. A stochastic state-space model is implemented, initially to capture the supply chain dynamics while the model is analysed under stationarity conditions with the aid of a covariance matrix. This allows the bullwhip effect to be expressed as a function of replenishment policies and product return rates. Next, an optimisation method is introduced to study the impact of the Internet of Things on inventory variance and the bullwhip effect. The results show that the Internet of Things can reduce costs associated with inventory fluctuations and eliminate the bullwhip effect in closed-loop supply chains.

闭环供应链是一个复杂的系统，因为它们涉及产品和信息的无缝前后流动。随着电子商务和网上购物的出现，人们对产品退货及其对库存差异和牛鞭效应的相关影响越来越感兴趣。在本文中，提出了一种新颖的四梯队闭环供应链模型，其中基本库存补货政策是通过比例控制器建模的。实现了随机状态空间模型，最初是为了捕捉供应链动态，同时借助协方差矩阵在平稳条件下分析模型。这使得牛鞭效应可以表示为补货政策和产品退货率的函数。下一个，引入优化方法研究物联网对库存方差和牛鞭效应的影响。结果表明，物联网可以降低与库存波动相关的成本，消除闭环供应链中的牛鞭效应。