Dynamic Pricing and Inventory Model with Setup Costs, Capacity Constraints, and Lead Times

This paper investigates the joint pricing and inventory decision problem in a periodic review system with a positive lead time, supply capacity constraints, and fixed setup costs. To address the computational complexity associated with dynamic programming, we propose a heuristic algorithm that effectively reduces the dimensions of the state space. Our approach adopts a myopic pricing policy, maximizing one-stage profit as the pricing strategy, and approximating the myopic expected demand as a linear function of the net inventory. Furthermore, we introduce a modified policy that leverages the capacity and fixed setup cost characteristics by proving a specific concavity property of the expected discounted profit function. The proposed policy exhibits a band structure for the optimal inventory strategy: when the price-deflated inventory position exceeds an upper bound, no orders are placed; when the inventory position falls below a lower bound, orders are placed up to capacity. The width of the band structure is at least one capacity unit. Numerical studies demonstrate the strong performance of the heuristic algorithm and the modified policy, significantly reducing computational complexity. Notably, higher capacity levels lead to smaller profit gaps. The methodology presented in this study offers practical guidance for decision-making in joint pricing and inventory management, considering lead time, capacity, and setup costs.