Fight inventory shrinkage: Simultaneous learning of inventory level and shrinkage rate

In 2020, inventory shrinkage eroded $61.7 billion profit in the U.S. retail industry. Unfortunately, fighting inventory shrinkage to protect retailers' already slim profits is challenging due to unknown shrinkage rates and invisible inventory levels. While the latter has been studied in the literature, the former has not. To deal with this challenge, we introduce two new features to the Bayesian inventory models: (1) interleaving customer and theft arrival processes that contribute to actual sales and shrinkages, respectively, and (2) learning of both inventory level and shrinkage rate. We first derive the learning formulae using the triple-censored sales data (invisible lost sales, shrinkages, and "lost shrinkages") and then use them to construct a POMDP (partially observable Markov decision process) model for making inventory and loss prevention decisions. For a different level of information deficiency, we analyze the model property and design heuristic order policies to capture the benefit of learning. Through a numerical study, we show that our estimated shrinkage rate converges quickly and monotonically to the actual value. For products with high shrinkage rates (5-12%), our heuristic policy can help seize 82-94% of the ideal profit retailers could earn under full information. We note that feature (1) of our model is crucial. It not only reflects the actual arrival order but also allows us to learn the unknown shrinkage rate, which, in turn, can prevent serious underordering and vicious inventory cycles and can increase the profit by 108% in some cases. Our approach thus enables both effective inventory management and early identification of ineffective loss prevention strategies, reducing shrinkage, and increasing sales and profit.