A Lagrangian Relaxation Based Heuristic for Integrated Lumber Supply Chain Tactical Planning

The problem investigated in this paper is focused on integrated tactical planning in the lumber supply chain which is featured as a divergent value chain. In this industry, raw materials (logs) are shipped from forest contractors to sawmills. Then the logs are sawn to finished lumbers and are distributed to the lumber market through different channels. A mixed integer programming (MIP) model is proposed to address harvesting, procurement, production, distribution, and sales decisions in an integrated scheme so as to maximize the total profit. Three decoupled models are also formulated representing, respectively, harvesting and procurement, production, sales and distribution. The proposed MIP model is hard to solve for real-life size instances. As a consequence, a Lagrangian Relaxation (LR) algorithm is developed to solve the proposed integrated model in a reasonable time with high quality results. In order to accelerate the LR algorithm and to obtain a feasible converged solution, a heuristic algorithm is proposed. The latter obtains a high quality lower bound in each iteration of the algorithm based on the most recent upper bound. The benefit of the integrated model is evaluated by comparing the revenue and cost of the integrated model and decoupled models in a realistic-size case study. It is found that substantial improvement can be reached by considering an integrated model. Finally, the solutions quality and the resolution time of the proposed LR heuristic algorithm are evaluated through comparison with a commercial solver, the classical LR, and a time-decomposition heuristic algorithm.