# Energy-Efficient Hybrid Flow-Shop Scheduling under Time-of-Use and Ladder Electricity Tariffs
## 1 Introduction
Two multi-objective MIP formulations are given for the HFS scheduling problem considering variable production speeds to reduce energy consumption at the expense of longer processing times [28]. Energy costs can be reduced not only by variable speeds but also by taking advantage of fluctuating TOU energy prices. To solve the problem, eps-constraint method is used. A numerical case study showed that energy costs can be enormously reduced by just a few delays in delivery. Taking into account the limitations of TOU electricity, researchers combined the characteristics of TOU electricity prices and the characteristics of the ladder electricity price, to explore a more effective means of reducing energy consumption [29]. Rising the electricity price of each step under the ladder electricity price scheme encourages users to reduce electricity consumption efficiently [30]. Compared with the traditional scheduling problem, the green workshop scheduling problem is trickier and is more challenging to resolve, and the research has greater practical significance and applied value. At the same time, a broad range of energy price control schemes and mechanisms exist, the most common of which is the TOU pricing policy [31]. TOU electricity price mechanism is an important mechanism arrangement that guides the power user to utilize energy based on the electricity zone and ensures the power system's safety and stable economic operation [32]. Many scholars have conducted a certain study on such issues. Right-shift is a common method, proposed in the energy-aware scheduling problem under the TOU environment [33]. The studies prove that reducing energy consumption by scheduling the flow shop problem is useful and effective [34]. At the same time, research on the impact of TOU price strategy on flow shop scheduling problems and the scheduling method of adjusting to TOU price has also been discussed [35]. Considering a multi-item dynamic lot sizing strategy with production and warehouse capacities with different storage allocation policies, a mathematical model of the related problem was detailed, and dominance properties were presented separately for each storage allocation policy [36]. Based on no-shortage policy, the proposed algorithm utilized transfer of lots between consecutive periods. Besides, the algorithm is also applied to a shop floor planning problem. With the total energy consumption limits in the specified time intervals, robust production schedules were designed to guarantee that the energy consumption limits are not violated for the given set of uncertainty scenarios [37]. Considering scheduling with energy constraints and robust scheduling, the paper focused on constructing pro-active production schedules for one machine that guarantee compliance with the contracted energy consumption limits if the operations' start times are delayed within a predetermined range. Notions of latest start times and right-shift schedules were defined in a pseudo-polynomial algorithm. And the algorithm can be incorporated into a wide variety of methods for solving the Robust Scheduling with Energy Consumptions Limits problem (RSECLP) that are based on searching the space of the permutations of the operations. Ladder electricity pricing is the process of dividing electricity consumption into several steps, and each step is carried out at a different price. It can reduce energy usage and encourage improving energy utilization efficiency by increasing the electricity price. However, the ladder electricity price cannot effectively reduce the power grid pressure for the power grid. The TOU energy pricing may efficiently reduce peaks and fill troughs, ensuring the power grid's safety and stability while achieving a smooth load curve. However, they cannot encourage the users' power saving, and even users have increased electricity at the time of load, thereby causing waste of resources [38]. Due to the limitations of TOU, the combination of TOU and ladder pricing is applied to industries having high energy consumption.