The green capacitated multi-item lot sizing problem with parallel machines

Carbon emissions related to energy consumptions from the manufacturing industry have become a substantial part of environmental burdens. To reduce carbon emissions, we introduce carbon emission constraints into the capacitated multi-item lot sizing problem with nonidentical parallel machines. The problem aims to satisfy customer demand for various items over the planning horizon, with an objective to minimize total costs without violating the capacity and carbon emission constraints. We formulate the problem with a mixed integer programming model and propose Lagrangian relaxation and column generation methods to improve lower bounds over the linear programming relaxation. Furthermore, we apply a heuristic named progressive selection to solve the problem and compare the heuristic with other state-of-the-art approaches in the literature. Computational results indicate that the progressive selection heuristic is computationally tractable and can obtain superior results under the same computational resources.