Inventory estimation from transactions via hidden Markov models

Abstract

Our work solves the problem of inventory tracking in the retail industry using Hidden Markov Models. It has been observed that inventory records are extremely inaccurate in practice (cf. [1{4]). Reasons for this inaccuracy are item losses due to item theft, mishandling, etc. which are unaccounted. Even more important are the lost sales due to lack of items on the shelf, called stockout losses. In several industries, stockout is responsible for billions of dollars of lost sales each year (cf. [4]). In [5], it is estimated that 4% of annual sales are lost due to stockout, across a range of industries. Traditional approaches toward solving the inventory problem have been geared toward designing better inventory management practices, to reduce or account for stock uncertainity. However, such strategies have had limited success in overcoming the effects of inaccurate inventory (cf. [1]). Thus, inventory tracking remains an important unsolved problem. The work done in this thesis is a step toward solving this problem. Our solution follows a novel approach of estimating inventory using accurately available point-of-sales data. A similar approach has been seen in other recent work such as [1, 6, 7]. Our key idea is that when the item is in stockout, no sales are recorded. Thus, by looking at the sequence of sales as a time-series, we can guess the period when stockout has occured. In our work, we nd that under appropriate assumptions, exact stock recovery is possible for all time. To represent the evolution of inventory in a retail store, we use a Hidden Markov Model (HMM), along the lines of [6]. In the latter work, the authors have shown that an HMM-based framework, with Gibbs sampling for estimation, manages to recover stock well in practice. However, their methods are computationally expensive and do not possess any theoretical guarantees. In our work, we introduce a slightly dierent HMM to represent the inventory process, which we call the Sales-Refills model. For this model, we are able to determine inventory level for all times, given enough data. Moreover, our recovery algorithms are easy to implement and computationally fast. We also derive sample complexity bounds which show that our methods are statistically ecient. Our work also solves a related problem viz. accurate demand forecasting in presence of unobservable lost sales (cf. [8{10]). The naive approach of computing a time-averaged sales rate underestimates the demand, as stockout may cause interested customers to leave without purchasing any items (cf. [8, 9]). By modelling the retail process explicitly in terms of sales and refills, our model achieves a natural decoupling of the true demand from other parameters. By explicitly determining instants where stock is empty, we obtain a consistent estimate of the demand. Our work also has consequences for HMM learning. In this thesis, we propose an HMM model which is learnable using simple and highly ecient algorithms. This is not a usual property of HMMs; indeed several problems on HMMs are known to be hard (cf. [11{13]). The learnability of our HMM can be considered a consequence of the following property: We have a few parameters which vary over a finite range, and for each value of the parameters we can identify a signature property of the observation sequence. For the Sales-Refills model, the signature property is the location of longer inter-sale intervals in the observation sequence. This simple idea may lead to practically useful HMMs, as exemplied by our work.