# Modified Migrating Birds Optimization for Energy-Aware Flexible Job Shop Scheduling Problem
## 1 Introduction
[14] presented a mathematical model of a flexible job shop, considering energy consumption and preventive maintenance and proposed a non-dominated sorting genetic algorithm II (NSGA-II) to optimize the maximum completion time, the total processing energy cost and the total maintenance energy cost. [PERSON] et al. [15] introduced an integrated green flexible job shop scheduling problem with the consideration of crane transportation. A hybrid algorithm, called as GA-GSO-GTHS, was developed based on a genetic algorithm (GA), a glowworm swarm optimization (GSO) algorithm and a green transport heuristic strategy (GTHS). [PERSON] et al. [16] built a new mathematical model of the flexible job shop under a time of use strategy to minimize the makespan and electricity consumption cost. A hybrid algorithm based on the biogeography-based optimization algorithm and variable neighborhood search was proposed to solve the bi-objective problem. [PERSON] et al. [17] studied an energy-saving flexible job shop scheduling problem to minimize the makespan and the total energy consumption. A modified shuffled frog-leaping algorithm (SFLA) was employed to solve the model. [PERSON] et al. [18] proposed a discrete water wave optimization algorithm to solve an energy-conscious FJSP with various machining speeds. With regards to the above literature, the relevant research on the energy-aware FJSP is still in the initial stage of exploration. Extensive work is yet to be carried out to narrow the gap between theoretical research and practical production. In the practical production, each machine can be operated by different workers belonging to an eligible worker set. In other words, for each machine, an appropriate worker needs to be chosen from the eligible worker set before processing jobs on it. In this case, the processing time of each job is dependent on both the assigned machine and the selected worker. Therefore, the worker can be also viewed as a kind of limited resource to be scheduled in the workshop. Such a problem is always categorized as a dual-resource constrained flexible job shop scheduling problem (DRCFJSP) [19]. However, in the previous research, the environmental metrics were not considered in the DRCFJSP. Nowadays, with increasing wage costs, the rational use of limited workershas become more and more important. Therefore, the integration of energy-aware scheduling and dual-resource constraints in the FJSP, named the energy-aware dual-resource constrained flexible job shop scheduling problem (EDRCFJSP), is a new issue which deserves to be studied. To the best of the authors' knowledge, there are few studies related to the EDRCFJSP that are more complex and that require more hard work to solve than the traditional FJSP [20]. Thus, an effective scheduling algorithm is highly desirable for the solving of the problem under study. In recent years, swarm intelligence algorithms have been developed and widely used for solving various optimization problems [21; 22; 23; 24; 25; 26; 27; 28]. In this paper, a swarm intelligence algorithm, namely the migrating birds optimization (MBO), is introduced to deal with the addressed problem, which was originally proposed for quadratic assignment problems [29]. Since then, the MBO algorithm has been successfully applied to various optimization problems, such as the production scheduling problem [30], closed loop layout [31], knapsack problem [32], travelling salesman problem [33] and task allocation problem [34]. However, as far as the we know, there is no application of the MBO algorithm on the EDRCFJSP. In addition, the MBO is a kind of neighborhood search algorithm, which makes it easy to adopt for the production scheduling problem. Thus, according to the characteristics of the addressed problem, we present a modified MBO algorithm (MMBO) for solving the EDRCFJSP. Some effective technologies are included alongside the original MBO, such as population initialization, problem-based neighborhood structures, local search strategy and an age-based re-initialization mechanism.