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As a significant branch of production scheduling problem, the Flexibility Job Shop Scheduling Problem (FJSP) is a typical NP-hard problem. Most conventional flexible workshop scheduling primarily focuses on performance aspects involving production efficiency such as time and quality. In recent years, due to increased energy costs and environmental pollution, 'low-carbon scheduling' has garnered attention as a new scheduling paradigm among scholars and engineers. This paper investigates a low-carbon flexible Job shop scheduling problem, proposing a Grey Wolf Optimization algorithm (SC-GWO), aiming to minimize the sum of carbon emission costs and makespan costs. This algorithm employs the Grey Wolf Algorithm (GWO) as the fundamental optimization method, adaptively choosing between global and local searches based on the dispersion degree of individuals. Firstly, integrating the Sine Cosine Algorithm (SCA), the sinusoidal cosine search mechanism is applied to GWO to enhance its local search capability. Secondly, a new leader selection mechanism is introduced to prevent leaders from falling into local optima, thus improving the algorithm's global exploration capability. Utilizing a nonlinear convergence factor strategy controls the global exploration and local exploitation capabilities in different algorithm stages, enhancing optimization accuracy and accelerating convergence, achieving a dynamic balance between the two. Finally, validation of the SC-GWO algorithm's ability to solve low-carbon scheduling problems in flexible job shop scheduling instances is conducted. Experimental results demonstrate the superior performance of SC-GWO in solving low-carbon flexible workshop scheduling instances. Comparative experiments against four other advanced algorithms on 22 classic benchmark test functions confirm SC-GWO's better convergence. Through standard test functions like Bandimarte instances applied to solve FJSP, experimental results showcase the excellent optimization performance of SC-GWO. Compared to HGWO and GWO, the makespan time is reduced by 22.25% and 39.27%, respectively. The proposed SC-GWO algorithm demonstrates favorable solving effects on flexible job shop scheduling instances, meeting actual production scheduling needs. [ABSTRACT FROM AUTHOR]

Multi-objective particle swarm optimization algorithms encounter significant challenges when tackling many-objective optimization problems. This is mainly because of the imbalance between convergence and diversity that occurs when increasing the selection pressure. In this paper, a novel adaptive MOPSO (ANMPSO) algorithm based on R2 contribution and adaptive method is developed to improve the performance of MOPSO. First, a new global best solutions selection mechanism with R2 contribution is introduced to select leaders with better diversity and convergence. Second, to obtain a uniform distribution of particles, an adaptive method is used to guide the flight of particles. Third, a re-initialization strategy is proposed to prevent particles from trapping into local optima. Empirical studies on a large number (64 in total) of problem instances have demonstrated that ANMPSO performs well in terms of inverted generational distance and hyper-volume metrics. Experimental studies on the practical application have also revealed that ANMPSO could effectively solve problems in the real world. [ABSTRACT FROM AUTHOR]

This paper describes a dynamic group-based differential evolution (GDE) algorithm for global optimization problems. The GDE algorithm provides a generalized evolution process based on two mutation operations to enhance search capability. Initially, all individuals in the population are grouped into a superior group and an inferior group based on their fitness values. The two groups perform different mutation operations. The local mutation model is applied to individuals with better fitness values, i.e., in the superior group, to search for better solutions near the current best position. The global mutation model is applied to the inferior group, which is composed of individuals with lower fitness values, to search for potential solutions. Subsequently, the GDE algorithm employs crossover and selection operations to produce offspring for the next generation. In this paper, an adaptive tuning strategy based on the well-known 1/5th rule is used to dynamically reassign the group size. It is thus helpful to trade off between the exploration ability and the exploitation ability. To validate the performance of the GDE algorithm, 13 numerical benchmark functions are tested. The simulation results indicate that the approach is effective and efficient. [ABSTRACT FROM AUTHOR]