A Lagrangian relaxation based approach for the capacitated lot sizing problem in closed-loop supply chain

This paper investigates the capacitated lot sizing problem in closed‐loop supply chain considering setup costs, product returns, and remanufacturing. We formulate the problem as a mixed integer program and propose a Lagrangian relaxation‐based solution approach. The resulting Lagrangian subproblems are then solved by polynomial time algorithms. Compared to existing solution methods in the literature, our Lagrangian relaxation based approach is advantageous in that it naturally provides a lower bound on the optimal objective function value, which allows us to assess the quality of solutions found. Numerical experiments using synthesized data demonstrate that our approach can find quality solutions efficiently.