The multiple travelling salesman problem (MTSP) extends the classical travelling salesman problem (TSP) by involving multiple salesman in the solution. MTSP has found widespread applications in various domains, such as transportation, robotics, and networking. Despite extensive research on MTSP and its variants, there has been limited attention given to the open close multiple travelling salesman problem (OCMTSP) and its variants in the literature. To the best of the author's knowledge, only one study has addressed OCMTSP, introducing an exact algorithm designed for optimal solutions. However, the efficiency of this existing algorithm diminishes for larger instances due to computational complexity. Therefore, there is a crucial need for a high-level metaheuristic to provide optimal/best solutions within a reasonable timeframe. Addressing this gap, this study proposes a first meta-heuristic called multi-chromosome-based Genetic Algorithm (GA) for solving OCMTSP. The effectiveness of the developed algorithm is demonstrated through a comparative study on distinct asymmetric benchmark instances sourced from the TSPLIB dataset. Additionally, results from comprehensive experiments conducted on 90 OCMTSP symmetric instances, generated from the renowned TSPLIB benchmark dataset, highlight the efficiency of the proposed GA in addressing the OCMTSP. Notably, the proposed multi-chromosome-based GA stands out as the top-performing approach in terms of overall performance. Further, solutions to symmetric TSPLIB benchmark instances are also reported, which will be used as a basis for future studies.

In reality, the machines may interrupt because of the nature of deterioration of the machines. Thus, it is inevitable to perform maintenance alongside production planning. The preventive maintenance is a schedule of strategic operations that are performed prior to the failure occurring, to retain the system operating at the preferred level of consistency. Thus, preventive maintenance plays a significant role in flow shop scheduling models. With its practical significance, this study addresses a practical three-machine n jobs flow shop-scheduling problem (FSSP) in which machine specific preventive maintenance, where each machine is given with a maintenance schedule is considered. In addition, a practical ordered precedence constraint in which some set of jobs has to process in the specified order irrespective of their processing times is also considered. The problem’s goal is to establish the optimal job sequence and preventive maintenance such that the overall cost of tardiness and preventive maintenance is as minimum as possible. An efficient heuristic approach is designed to tackle the present model, resulting in total cost savings. A comparative analysis is not conducted due to absence of studies on the current problem in the literature. However, Computational experiments are carried out on some test instances and results are reported. The reported results may be useful for future studies.

Job precedence can often be seen in various manufacturing process scenarios. For instance, in the context of flow shop scheduling, certain jobs must be processed before a specific job may be executed. Formally, this scenario is known as precedence constraint, which influences the optimal job sequence. Because of this practical significance, in this study, a two-machine flow shop scheduling problem in which transportation times, breakdown time, and weighted jobs are considered. In addition to that, an ordered precedence constraint is considered that ensures a successor job cannot start on any machine before its predecessor job has been done on all machines. This is the first study that deals with flow shop scheduling problems with transportation times, breakdown time, job weights, and precedence constraints altogether, to the best of the author’s knowledge. To solve this problem, a simple and efficient solution methodology is developed that assures optimal or near-optimal solutions effectively. The developed algorithm is tested on various test instances and results are reported, which will be useful for future comparative studies.