# Energy-Efficient Hybrid Flow-Shop Scheduling under Time-of-Use and Ladder Electricity Tariffs
## 1 Introduction
The ladder electricity price refers to the division of electricity consumption into several steps, in which each step is performed at different electricity prices. By combining the advantages of the two electricity price mechanisms, the government can reduce energy consumption while achieving the goal of \"shaving peaks and filling valleys\". In order to reduce energy consumption in the flow-shop problem, [PERSON] [39] established a mathematical model considering TOU and ladder prices. Other researchers, such as [PERSON] et al. [40] evaluated an integrated scheduling problem from iron-steel manufacturing under TOU electricity tariffs. An improved Strength Pareto Evolutionary Algorithm (SPEA2) based on the relationship propagation chain was customized for the problem. To the best of our knowledge, most studies have focused on improving algorithms and production processes [41]. Since electricity price mechanism has a great impact on energy consumption it is also worth researching the production process under the electricity price mechanism [42]. Nonetheless, the initiatives mentioned above offer a starting point for investigating the energy-efficient design of HFSP under TOU and ladder power rates. Together, these studies provide important insights into applying the HFSP to reduce energy consumption. A significant amount of existing research has explored the application of algorithms, including new heuristic algorithms. However, these studies ignore the advantage of constructive algorithms in solving targeted problems. To reduce energy consumption, the general TOU electricity price policy has been applied for a number of years, and many of its aspects have been studied. In addition, research on TOU and ladder electricity price schemes have received little attention. Furthermore, the application of renewable energy to reduce carbon emissions in production dispatch scenarios has been studied. However, few studies exist on the efficient use of renewable energy during production. The optimization method for solving HFS problems includes exact, constructive, and heuristic algorithms. The first step in solving HFS problems will be to apply exact algorithms, such as branch and bound methods [43] and dynamic programming [44]. On the other hand, exact algorithms cannot solve large-scale HFS problems due to the tremendous amount of computation required. Recent years have seen the development of heuristic and constructive algorithms to solve HFS problems. Constructive algorithms use a design rule to construct the solution. For instance, [PERSON], Enscore & Ham (NEH) proposed a constructive heuristic algorithm. improved to be effective in flow shop scheduling. The basic idea of NEH algorithms is giving the priority of the workpiece based on the processingtime [45]. Heuristic algorithms including genetic algorithm (GA) [46], Simulated Annealing (SA) [47], Tabu Search (TS) [48], Particle Swarm Optimization (PSO) [49], etc. As a well-known scheduling problem, HFSP has substantial research value and has been demonstrated to be an NP-Hard problem. Furthermore, considering that many problems in the real world is usually consisting of multiple targets which may conflict with each other. Researchers have proposed many multi-objective optimization algorithms to find a Pareto solution, such as Non dominated sorting genetic algorithm-II (NSGA-II) [50], and multi-objective evolutionary algorithm based on decomposition (MOEA/D) [51]. [PERSON] et al. [52] proposed an energy-efficient distributed permutation flow-shop inverse scheduling problem to simultaneously minimize adjustment and energy consumption. This model contained realistic constraints, controllable processing times, and energy consumption factors. We developed a hybrid collaborative algorithm that combines a cooperative search scheme to effectively address the problem.