  We consider the capacitated three-level lot-sizing and replenishment problem
with a distribution structure (3LSPD-C), recently proposed in the literature.
In 3LSPD-C, a single production plant delivers items to the warehouses from
where they are distributed to their corresponding retailers. There is a
capacity on the total amount the plant can produce in each period, whereas
there are no capacities on transportation. The goal of the problem consists in
determining an integrated three-echelon production and distribution plan
minimizing the total cost, which is composed of fixed costs for production and
transportation setups as well as variable inventory holding costs.
Additionally, we consider a generalization of the problem which also
establishes storage capacities on the warehouses and/or retailers, given the
importance of such characteristics in practical industrial and commercial
environments. Such extension is denoted generalized capacitated three-level
lot-sizing and replenishment problem with a distribution structure (G3LSPD-C).
We propose a hybrid mixed integer programming (MIP) heuristic consisting of a
relax-and-fix approach to generate initial feasible solutions and a
fix-and-optimize improvement procedure grounded on varying-size neighborhoods
to obtain high-quality solutions. Computational experiments are performed to
analyze the potential cost reductions achieved using the new hybrid heuristic
when compared with a state-of-the-art MIP formulation. The results show that
the proposed hybrid heuristic can match or improve the solution quality
obtained by the MIP formulation for the majority of the 3LSPD-C instances.
Additionally, such superior behavior remains valid when the approaches are
applied to the more general G3LSPD-C. Furthermore, we investigate the economic
impacts of the storage capacities and how they affect the performance of our
newly proposed hybrid heuristic.
