Multi-stage hybrid flow shop scheduling problem with lag, unloading, and transportation times

This study aims to address a variant of the hybrid flow shop problem by simultaneously integrating lag times, unloading times, and transportation times, with the goal of minimizing the maximum completion time, or makespan. With applications in image processing, manufacturing, and industrial environments, this problem presents significant theoretical challenges, being classified as NP-hard. Notably, the problem demonstrates a notable symmetry property, resulting in a symmetric problem formulation where both the scheduling problem and its symmetric counterpart share the same optimal solution. To improve solution quality, all proposed procedures are extended to the symmetric problem. This research pioneers the consideration of the hybrid flow shop scheduling problem with simultaneous attention to lag, unloading, and transportation times, building upon a comprehensive review of existing literature. A two-phase heuristic is introduced as a solution to this complex problem, involving iterative solving of parallel machine scheduling problems. This approach decomposes the problem into manageable sub-problems, facilitating focused and efficient resolution. The efficient solving of sub-problems using the developed heuristic yields satisfactory near-optimal solutions. Additionally, two new lower bounds are proposed, derived from estimating minimum idle time within each stage via solving a polynomial parallel machine problem aimed at minimizing total flow time. These lower bounds serve to evaluate the performance of the developed two-phase heuristic, over measuring the relative gap. Extensive experimental studies on benchmark test problems of varying sizes demonstrate the effectiveness of the proposed approaches. All test problems are efficiently solved within reasonable timeframes, indicating practicality and efficiency. The proposed methods exhibit an average computational time of 8.93 seconds and an average gap of 2.75%. These computational results underscore the efficacy and potential applicability of the proposed approaches in real-world scenarios, providing valuable insights and paving the way for further research and practical implementations in hybrid flow shop scheduling.