## 1 Introduction
The first one is the minimization of the makespan (\(C_{max}\)), which is the maximum time at which all jobs will be completed. The makespan is equivalent to the maximization of the utilization of the available machines. The second objective is the minimization of the mean weighted flow time (MWFT). The MWFT is calculated as the weighted average of all jobs' flow times, for which the weights are the priority values assigned to the jobs. The flow time of a job is the difference between its completion time and its release time. The minimization of the MWFT improves customer satisfaction levels. We follow an a posteriori approach [5] to solving the Bi-DMOSP, meaning that a set of non-dominated solutions will be generated and delivered to the decision-maker so that they can choose one to implement. The considered problem is NP-hard for both objectives since it is a generalization of the traditional static, deterministic open-shop scheduling problem (OSP). In the OSP, all machines are not busy at the beginning of the schedule, at which time all jobs are released, and the system consists of single-machine workstations. For more than two machines, the makespan minimization OSP is known to be NP-hard [6], as it is for the MWFT minimization OSP [7]. Therefore, an efficient metaheuristic is needed to generate non-dominated solutions that coincide with or are very close to the optimal Pareto front solutions. This process needs to be done in a practically acceptable computational time. Despite its frequent occurrence in real-life applications, the literature has rarely addressed the multi-objective DMOSP. [PERSON] and [PERSON] [8] studied a special case of four workstations, each containing parallel identical machines. This DMOSP special case is encountered in chip sorting in light-emitting diode (LED) manufacturing. They considered the objectives of the minimization of the makespan and the minimization of total weighted tardiness simultaneously. They proposed two simulated annealing approaches and pro vided minor computational experiments for them. Recently, [PERSON] [3] developed a mixed-integer linear programming model and an exact algorithm based on the \(\varepsilon\)-constraint method [5] for the studied Bi-DMOSP. That algorithm can generate optimal Pareto front solutions for small-sized instances; nevertheless, its computational time is prohibitive for intermediate and large-sized instances. Consequently, two metaheuristic approaches that can provide efficient, non-dominated solutions in practically acceptable computational times for large-sized instances were developed by [PERSON] [4]. The first metaheuristic is an evolutionary algorithm based on NSGA-II [9]. Meanwhile, the second metaheuristic is a swarm-intelligence one based on the multi-objective gray wolf optimizer (MOGWO) [10]. Both metaheuristics are population-based and represent two different lines of research. They are both hybridized with a simulated annealing local search heuristic. The computational results in Abdelmaguid [4] show that NSGA-II outperforms MOGWO. This paper contributes to the literature as follows:  
* It develops a new solution representation scheme (chromosome structure) and a recombination operator that can be used by metaheuristics in solving multiprocessor scheduling problems. This scheme is a tight representation of semi-active schedules that eliminates redundancy, which is a common problem encountered with permutations-based representations;
* It develops a novel, multi-objective scatter-search algorithm (MOSS) for the Bi-DMOSP. The proposed MOSS is a population-based metaheuristic that differs from the recently developed metaheuristics in [4] since it has a population-control mechanism. This mechanism allows a solution in the reproductive population set, called the reference set, only if it satisfies specific solution-quality and diversity criteria.