Optimizing joint allocation and pricing policies for a one-warehouse multi-retailer system with lost sales

In this paper, we consider a joint allocation, inventory, and pricing problem with one warehouse and heterogeneous retailers operating under price-sensitive deterministic demand. Retailers are geographically dispersed and may differ in terms of operating costs, market potential, and price elasticity. Moreover, they are required to offer the same selling price in each period, and the unsatisfied demand is lost. Given that there is an initial non-replenishable inventory at the central warehouse that needs to be distributed over a finite planning horizon, the objective is to maximize the total profit by jointly deciding on the product allocation to retailers, product pricing, and the amount of sales which also determines the inventory level at the retailers and the warehouse. We derive structural properties of the optimal solution and introduce a mixed-integer quadratically constrained programming model to efficiently obtain the global optimal solution. Furthermore, we prove that when the initial inventory at the warehouse is sufficiently large, it is possible to solve the problem optimally in polynomial time. Additionally, we devise a heuristic that sequentially solves the problem by decomposing it with respect to the pricing and allocation decisions. The computational study carried out using instances with different sizes and market conditions indicates that the developed heuristic generates remarkably small optimality gaps. Finally, we provide managerial insights about the impact of the initial product quantity, retailer heterogeneity, and demand seasonality on the profit and pricing scheme.