# Energy-Efficient Hybrid Flow-Shop Scheduling under Time-of-Use and Ladder Electricity Tariffs
## 2 Problem Statement and Methods  
This article selects makespan and TEC as two major indicators for energy-efficient scheduling in the HFSP. Generally, the energy consumption of a machine in the scheduling process consists of 5 stages, Startup, Preparation, Work, Idle and Shutdown [57]. Two stages have been considered; one considers the total amount in an interval and the other considers the reference energy consumption for the production [58]. The energy consumption during the preparation phase is not included in the consideration as we focused on the manufacturing process. Meanwhile, the machine is not turned off until the last task is completed. According to the study, energy consumption is mainly considered in processing mode and in standby mode. The energy consumption of the production process is different from each machine. As discussed earlier, electricity charges are determined based on real-time energy rates and electricity prices during the day. In the TOU electricity price scheme, the time horizon is divided into \(i\) time periods and each period \(i\) has a duration of \(Ti\) with electricity price \(Pi\) per unit time, where the length of each \(Ti\) can be different. The government presses higher requirements for a high energy consumption industry than the enterprise generally. As we know, there are two types of ladder price, one is considering the total amount in an interval, and the other is the penalty electricity price which referring the average energy consumption for the production. Considering the uniqueness of different industries, the total electricity consumption is chosen in the paper. Additionally, the TOU and ladder electricity price have three periods in the paper, valley period, usual period, and peak period. Furthermore, it has more than one price in each period. For example, there is a two-step TOU and ladder electricity price in Figure 1. When the total electricity consumption exceeds the set electricity standard of the time, the excess part would apply the second stage price \(P_{i}^{2}\). The Chinese government in July 2012 nationwide (not covered in Xinjiang and Tibet) implemented a ladder electricity price. Considering the complex situation in the HSFP, the TOU and ladder price model of this article is constructed as a simplified version with a double-layer schematic diagram electricity price as follows in Figure 1. Assuming that there are two electrical prices in each period, The first gear is standard electricity, and the second gear is punitive electricity price. Figure 1: TOU and ladder price electricity price scheme. The ladder price is a type of inclining block rate (IBR) tariff, in which the energy purchase cost increases exponentially with the increase in consumption, as shown in Figure 2.  
The TOU and ladder electrical price proposed by the government is mainly used in the constraints of energy-intensive industries to achieve carbon neutrality. For comprehensive exceeding the benchmark energy consumption of the whole year, the part that exceeds the benchmark energy consumption is converted into electricity according to the equivalent value, and the electrical price of the ladder is performed according to the three steps. As an extended form of the classic flow shop scheduling problem, the HFSP supposes that there are J stages during processing. Meanwhile, at least one processing stage contains two or more identical parallel machines. This HFSP consists of a set of n jobs \(\{I_{1},2,\cdots,I_{n}\}\) and a set of m machines \(\{M_{1},M_{2},\cdots,M_{m}\}\). Each job and machine at time zero are available. Each machine can process one job at a time and each job can be processed on one machine at a time. Once a job starts processing on the machine, its speed cannot be changed. This paper does not consider the time interval of gear switch, the time involved in setting up machines, and the time required to breakdown machines. Each machine cannot be turned off completely until all of its duties have been completed.