Estimating Life Cycle Sales of Technology Products with Frequent Repeat Purchases: A Fractional Calculus-Based Approach

Abstract

Accurately predicting the sales trajectory of a product is critically important for firms’ medium- and long-term planning. However, reliable sales prediction models are very difficult to find when repeat purchases or subscription renewals account for a large proportion of product sales, which is often the case for technological products. This study introduces a new sales growth model, the generalized diffusion model with repeat purchases (GDMR), to address this problem. The GDMR draws upon a branch of mathematics called fractional calculus and formulates a product’s sales growth rate using a novel noninteger-order integral equation. Compared with benchmark methods, the GDMR is simple and easy to implement, is suitable for a wide variety of products, and predicts better than benchmark models such as time series and machine learning models. Furthermore, the GDMR can reliably recover a product’s progress of adoptions even when only sales data are available. Because of these important advantages, the GDMR can help firms better understand their products’ market positions and, subsequently, make more informed decisions in production and inventory planning, transportation and logistics, and sales and marketing, thus improving the effectiveness and efficiency of their business operations.