## 1 Introduction
[24] examined the unrelated parallel machine scheduling problem under TOU electricity tariffs to optimize the weighted sum of makespan and electricity cost. [PERSON] et al. [7] and [PERSON] et al. [25] addressed a similar parallel machine scheduling problem under TOU electricity tariffs to minimize the total electricity cost. The former developed a time-interval-based mixed-integer linear programming (MILP) model and a column generation heuristic algorithm. The latter improved the former model by providing a linear programming relaxation and a two-stage heuristic algorithm. Single machine scheduling problems are of great significance both in theory and practice. On one hand, there are many single machine scheduling problems in the real industrial environment. For example, a Computer Numerical Control (CNC for short) planer horizontal milling and boring machine can be regarded as a single machine. On the other hand, the research results and methods of single machine scheduling problems can provide reference for other scheduling problems, such as flow shop, job shop, and parallel machine scheduling problems. For single machine scheduling problems under TOU electricity tariffs, [PERSON] et al. [26] investigated a single-machine batch scheduling problem to minimize the makespan and the total energy costs simultaneously. Considering the TOU electricity tariffs and the power-down mechanism, [PERSON] et al. [27] proposed a model that enables the operations manager to determine the \"turning on\" time, \"turning off\" time, and idle time at machine level, leading to a significant reduction in electricity cost by avoiding on-peak periods. [PERSON] et al. [28] developed a mixed integer linear programming model and a genetic algorithm for the same problem, reducing electricity cost and greenhouse gas emissions effectively during peak time periods. Without considering a power-down mechanism, [PERSON] et al. [29] studied the single machine scheduling problem under TOUelectricity tariffs systematically to minimize the total electricity costs. They divided the problem into the two cases of uniform-speed and speed-scalable, in which a preemptive version and a non-preemptive version were investigated respectively. For the uniform-speed case with non-preemptive assumption (Problem U-pyr), they demonstrated that the problem is strongly non-deterministic polynomial-time hard (NP-Hard). Note that the Problem U-pyr is the same as our problem. Based on [PERSON] et al. [29], [PERSON] et al. [9] investigated a continuous-time MILP model for Problem U-pyr and developed a greedy insertion heuristic algorithm(GIH) that is the most classic method for this problem until now, according to our best knowledge. In their algorithm, the jobs are inserted into available periods with lower electricity prices in sequence, and the jobs with higher power consumption rates are mostly assigned to periods with lower electricity prices by traversing all non-full \"forward blocks\" and \"backward blocks\". However, [PERSON] et al. [29] did not establish a complete mathematical model for the single machine scheduling problem (Problem U-pyr) under TOU electricity tariffs, and their algorithm is only feasible in the condition that all the jobs have the same workload and the TOU tariffs follow a so-called pyramidal structure. Regrettably, the TOU electricity tariffs rarely follow a complete pyramidal structure in most provinces in China. To perfect the theory of [PERSON] et al. [29], [PERSON] et al. [9] developed a new model and algorithm, but their algorithm requires that all the jobs must traverse all non-full \"forward blocks\" and \"backward blocks\", causing a strong high-time complexity. Especially when the processing times of the jobs are relatively short, it usually takes several jobs to fill one period and generates uneven remaining idle times, which leads to an increase in the number of forward and backward blocks required to be calculated.