Determination of the Collaborative Optimal Lot Size for Product Replenishment in a Supermarket Chain Using Dynamic Programming Algorithms

Abstract

The primary objective of this study is to determine the optimal collaborative replenishment size for a food product category consisting of 18 subcategories in a modern-channel supermarket chain with 14 points of sale. To achieve this, an annual planning horizon divided into 12 monthly periods was considered, employing the dynamic programming algorithm proposed by Harvey Wagner and Thomson Whitin. This algorithm was informed by demand forecasts generated through robust prediction methodologies (Holt-Winter, Prophet, Polynomial Regression, and Random Forest) and the cost structures established by the supermarket chain. To assess the sensitivity of the model to variations in ordering, purchasing, and inventory holding costs, two additional scenarios were designed alongside the baseline scenario. The first scenario presents an adverse environment, with a sustained monthly increase of 10% in these costs throughout the entire planning horizon. The second scenario combines favorable and unfavorable conditions: during the first half of the planning period, a 5% reduction in each of these costs is applied, followed by a 5% increase in the second half. The results showed that, in the first scenario, the total cost of the inventory policy increased by an average of 3.44% per period compared to the baseline scenario. In contrast, the second scenario revealed a decrease of 0.4% per period compared to the same reference point. These findings demonstrate the capacity of the Wagner-Whitin mathematical model based on dynamic programming, reinforced with robust forecasting methodologies, to anticipate changes in ordering, purchasing, and inventory holding costs, thereby optimizing decision-making in replenishment management.