Parallel Machine Scheduling with Additional Resources: A Lagrangian-Based Constraint Programming Approach

This study deals with an unrelated parallel machine scheduling problem with one additional resource type (e. g., machine operators). The objective is to minimize the total completion time. After giving the integer programming model of the problem, a Lagrangian relaxation problem (LRP) is introduced by relaxing the constraint set concerning the additional resource. A general subgradient optimization procedure is applied to a series of LRPs to maximize the lower bound for the original problem. To generate efficient upper bounds for the original problem, a constraint programming (CP) model is applied by taking the LRP solutions as input regarding the machine assignments. For the problem, a pure CP model is also developed to evaluate its performance. All the solution approaches are evaluated through a range of test problems. The initial computational results show that Lagrangian-based CP approach produces promising results especially for larger problem sizes.