Hot rolling turn scheduling using Lagrangian relaxation

In this paper, we address the problem of scheduling n slabs belonging to F rolling turns on m identical/unrelated parallel machines with slab´s chain precedence constraints and the objective of minimizing the total weighted completion time. Two subgradient method-based Lagrangian relaxation algorithms are developed for solving the two models by using different decomposing strategies accordingly. When all rolling machines are identical, the problem is relaxed and decomposed into multiple independent turn-level subproblems, each can be solved to optimality by a polynomial algorithm with complexity of O(T) where T is the scheduling horizon. When the rolling machines are unrelated, a novel decomposition is proposed based on machine-level separating strategy. Each machine-level subproblem is optimally solved by combining the dynamic programming with the WSPT (weighted shortest processing time) rule. Numerical results show that the proposed methods can generate near-optimal schedules in a timely fashion that the average duality gaps are lower than 5% for the identical case, and less than 0.5% for the unrelated case.