Logic-based Benders decomposition methods for the distributed permutation flow shop scheduling problem with production and transportation cost

Distributed manufacturing mode can significantly enhance production flexibility and efficiency. Considering that factories and customers in distributed manufacturing environments may be geographically dispersed, address a distributed permutation flow shop scheduling problem (DPFSP) with direct transportation different cost of production and transportation while the goal is to minimize of weighted sum cost makespan (DPFSP-PTM). First, we formulate two mixed-integer linear programming (MILP) models and constraint programming (CP) model to optimize the objective simultaneously. Then, by decomposing DPFSPPTM into an order assignment master problem (AMP) and a series of scheduling subproblems (SSPs), develop two exact methods based on logic-based Benders decomposition (LBBD) and Branch-and-Check (BCH). To accelerate convergence, we propose three strong SSP relaxations based on the single-machine bottleneck enhance the MILP models and AMP. Additionally, we introduce an initial solution generated by the iterated greedy (IG) algorithm to warm-start the LBBD. Finally, we demonstrate the effectiveness of the proposed methods in achieving competitive average optimality gaps and lower bounds across both small-scale large-scale instances.