Reformulations to improve the Lagrangian relaxation approach for the capacitated multi-product dynamic lot sizing problem with batch ordering
研究了产能约束下多产品动态批量问题，提出三种混合整数线性规划模型并应用拉格朗日松弛，通过数值实验比较不同模型和算法以找到最小对偶间隙的求解方案。
In this work, we study the multi-product dynamic lot-sizing problem with capacity constraints and batch ordering. This problem arises in short to medium range production scheduling for several products over a finite number of periods to meet known demand. Each period has a capacity for placing orders, and every order for each product must have a fixed quantity, or batch size, though multiple orders can be placed for each product. We define three mixed-integer linear programming (MILP) models and apply Lagrangian relaxation to formulate the corresponding dual problems by relaxing the capacity constraints. The aim is to identify the dual problem that is the easiest to solve and provides the solution with the smallest duality gap. Subgradient optimisation is applied to solve the preferred Lagrangian dual model, which uses one of two heuristics to find good feasible solutions. We also show that the special case, where the batch sizes for all products are the same, can be modeled as a transportation problem. A set of numerical experiments is designed to compare the performance of the Lagrangian relaxation approach with a commercial MILP solver to identify the version of the subgradient algorithm and the MILP model that provide the best solutions.