Permutation flow shop scheduling problems (PFSP) with makespan minimization are a standard benchmark in production scheduling, where solution quality depends on effective search in a discrete permutation space. This paper proposes HBFO, a hybrid discrete bacterial foraging optimization method that implements bacterial foraging dynamics through permutation-preserving operators and integrates intensification and diversification mechanisms. HBFO is evaluated on Taillard's 120 benchmark instances using 20 independent runs per instance. Performance is assessed using the best-run makespan, the relative percentage deviation (RPD) relative to best-known reference values, and run-to-run variability. Compared with recent advanced metaheuristics reported on the same benchmark families, HBFO achieves the best or tied-best best-run makespan in 115 of the 120 instances, with five instances where another method attains a lower best-run value. The RPD results show frequent exact matches to the reference values and predominantly low-percentage deviations otherwise. Configuration-level summaries further indicate that both average deviation and variability vary across Taillard instance families.

With a decentral and global economy, distributed scheduling problems are getting a lot of attention. This paper addresses a distributed flexible job shop scheduling problem (DFJSP) with minimizing makespan, in which three subproblems, namely operations sequencing, factory selection and machine selection must be determined. To solve the DFJSP, a novel mixed-integer linear programming (MILP) model is first developed, which can solve the small-scaled instances to optimality. Since the NP-hard characteristic of DFJSP, a hybrid algorithm (GA-VNS-CP) of genetic algorithm (GA), variable neighborhood search (VNS) and constraint programming (CP). Specifically, the GA-VNS-CP is divided into two stages. The first stage uses the hybrid meta-heuristic algorithms of GA and VNS (GA-VNS), and the VNS is designed to improve the local search ability of GA. In GA-VNS, the encoding only considers the factory selection and the operations sequencing problems, and the machine selection problem is determined by the decoding rule. Because the solution space may be limited by the decoding rule, the second stage uses the CP to extend the solution and further improve the solution. Numerical experiments based on benchmark instances are conducted to evaluate the effectiveness of the MILP model, VNS, CP and GA-VNS-CP. The experimental results show effectiveness of the MILP model, VNS and CP. Moreover, the GA-VNS-CP algorithm has better performance than traditional algorithms and improves 6 current best solutions for benchmark instances

Jaya algorithm has recently been proposed, which is simple and efficient meta-heuristic optimization technique and has received a great attention in the world of optimization. It has been successfully applied to some thermal, design and manufacturing associated optimization problems. This paper aims to analyze the performance of Jaya algorithm for permutation flow-shop scheduling problem which is a well-known NP-hard optimization problem. The objective is to minimize the makespan. First, to make Jaya algorithm adaptive to the problem, a random priority is allocated to each job in a permutation sequence. Second, a job priority vector is converted into job permutation vector by means of Largest Order Value (LOV) rule. An exhaustive comparative study along with statistical analysis is performed by comparing the results with public benchmarks and other competitive heuristics. The key feature of Jaya algorithm of simultaneous movement towards the best solution and going away from the worst solution enables it to avoid being trapped in the local optima. Furthermore, the uniqueness of Jaya algorithm compared with any other evolutionary based optimization technique is that it is totally independent of specific parameters. This substantially reduces the computation effort and numerical complexity. Computational results reveal that Jaya algorithm is efficient in most cases and has considerable potential for permutation flow-shop scheduling problems.

Dynamic Flexible Job Shop Scheduling (DFJSS) is a critical challenge in smart manufacturing due to dynamic job arrivals, machine breakdowns, routing flexibility, and multi-objective performance requirements. Conventional dispatching rules and metaheuristic algorithms often fail to provide adaptive real-time decisions. This paper proposes a Greedy Actor Neural Critic PPO based Reinforcement Learning (GANC-PPO-RL) framework to minimize makespan and total energy consumption. The problem is formulated as a Markov Decision Process, where machine availability, job progress, queue status, processing times, and energy levels define states, and job–machine assignments define actions. A multi-objective reward penalizes higher makespan and energy usage. The PPO algorithm ensures stable policy updates using a clipped objective. Results show makespan within ±2% of benchmarks for small instances and about 23% reduction for larger cases, with 16–20% lower makespan and 15–25% lower energy consumption compared to RL-QL.