The value of integrated tactical planning optimization in the lumber supply chain

This study investigates the integrated annual planning of harvesting, procurement, production, distribution, and sale activities in the lumber supply chain. The problem is formulated as a mixed integer programming (MIP) model in which the binary variables correspond to the harvesting schedule over the planning horizon. The proposed model is applied on a real-size case study, which leads to a large-scale MIP model that cannot be solved by commercial solvers in a reasonable time. Consequently, we propose a heuristic algorithm which iteratively updates the search step-size of the sub-gradient method in the Lagrangian Relaxation algorithm through obtaining a new lower-bound on the objective function value based on the most recent upper-bound. While improving significantly the convergence rate, this heuristic also guarantees the feasibility of the converged solution. Furthermore, in order to measure the value of integration, we compare the integrated model with the decoupled planning models currently implemented in the lumber industry. It is observed that, depending on the number of decoupled models, 11%-84% profit improvement can be achieved by considering an integrated model. Finally, the advantage of the proposed heuristic algorithm in finding high quality plans in 51%-77% less CPU time comparing to a commercial solver and the classical Lagrangian Relaxation algorithm is demonstrated through a set of real-size test instances.