Estimation of Consumer Demand with Stock--out Based
Substitution: An Application to Vending Machine Products
R. Anupindi, M. Dada and S. Gupta
*** Abstract ****
In this paper we develop a model of customer arrivals and choice
between goods that allows for possible product substitution and lost
sales when a customer faces a stock-out at retail. The model is
formulated in the context of retail vending, an industry that accounts
for a sizable part of the retail sales of many consumer products. We
focus on the problem of estimating core demand rates and substitution
rates between items in the assortment carried, based on information
available from two different kinds of inventory tracking systems. In
the best case scenario of a perpetual inventory system in which times
of stock-out occurrence and cumulative sales of all goods at these
times are observed, we derive Maximum Likelihood Estimates (MLEs) of
the demand parameters and show that they are especially simple and
intuitive. However, state-of-the-art inventory systems in retail
vending provide only periodic data, i.e., data in which times of
stock-out occurrence are unobserved or ``missing''. For these data we
show how the Expectation-Maximization (E-M) algorithm may be employed
to obtain the MLEs by treating the stock-out times as missing
data. The model is applied to daily sales and stock data from beverage
vending machines in a mid-Western US city.