## 6 Conclusions and Future Research  
In this paper, a novel metaheuristic approach based on scatter search was developed to provide efficient, non-dominated solutions to a bi-objective, dynamic, multi-processor open-shop scheduling problem (Bi-DMOSP). This problem is frequently encountered in maintenance and healthcare diagnostic applications, and therefore, providing efficient solutions is particularly important to effectively manage these systems. The two objectives considered in this paper were the minimization of the makespan and the minimization of the mean weighted flow time (MWFT). The former objective targets maximizing the utilization of the available machines, while the latter aims to improve customer satisfaction. Since the studied problem is known to be NP-hard for both objectives, their simultaneous consideration represents a challenge in designing efficient algorithmic approaches. This paper adopted a multi-objective scatter-search (MOSS) approach combined with an efficient novel bi-objective tabu search for local improvements. It is intended to reduce the computational burden caused by the large size of the solution set and the redundant  
Figure 11: Main effect plots for \(\Delta\overline{HV\%}\). Figure 12: Interaction plots for \(\Delta\overline{HV\%}\). local search moves found in a formerly developed metaheuristic based on NSGA-II. The developed metaheuristic in this paper utilizes a small set of reference solutions to generate a new set of test solutions in every iteration, combined with an efficient bi-objective tabu search that targets improving both objectives. With an efficient updating mechanism for the reference and non-dominated solutions throughout iterations, the developed metaheuristic can effectively achieve the stated quest of finding efficient non-dominated solutions. Computational experiments were conducted on randomly generated small and large instances of the studied Bi-DMOSP. Thirty small-sized benchmark instances were used to fine-tune the metaheuristic parameters and compare their performance against the formerly developed NSGA-II and an exact algorithm based on the \(\varepsilon\)-constraint method. Three performance metrics were used in the comparisons: the number of generated non-dominated solutions, the total gravitational distance, and the percentage hypervolume deviation. The last two performance metrics were developed in this paper to represent deviations from the known optimal Pareto front, obtained earlier using the \(\varepsilon\)-constraint method. Computational results revealed the superiority of the developed MOSS compared to NSGA-II, resulting in better metrics in 23 instances versus only three instances for NSGA-II. Meanwhile, at a 95% confidence level, the developed MOSS's overall average performance was significantly better. Furthermore, the average percentage hypervolume deviation of the developed MOSS was found to be only 1.83% versus 2.74% for NSGA-II, which reflects its capability of generating non-dominated solutions that are very close to the optimal Pareto front and its superiority over NSGA-II. The large-sized instances are randomly generated at different levels of selected problem parameters representing its structure. These parameters include the number of workstations, the number of jobs, the loading level (percentage of workstations needed for a job), and two parameters representing the job release and machine-ready times. Computational results reveal that the difference between the developed MOSS and NSGA-II in terms of their percentage hypervolumes is mainly affected by the first three parameters. It was found that the developed MOSS can produce competitive results in cases of low density, which are measured by the loading level, combined with the number of jobs. Meanwhile, this was reversed in the cases with large densities. The studied Bi-DMOSP is, undoubtedly, an important problem in maintenance and healthcare applications. Providing efficient Pareto front solutions to this problem is of utmost importance for efficiently managing such systems.