## 1 Introduction  
Driven by the rapid development of global economy and civilization, there will be a consistent growth in energy consumption in the years ahead. According to a survey of the International Energy Agency (IEA), the world-wide demand for energy will increase by 37% by 2040 [1]. Non-renewable energy resources such as coal, oil, and gas are diminishing day-by-day, which is threatening the sustainable development of many countries. Meanwhile, greenhouse gas emissions generated from inappropriate usage of fossil fuels have taken a heavy toll on the global climate as well as the atmospheric environment [2]. Therefore, how to save energy and then improve the environment quality has become a pressing matter of the moment. As the backbone of many countries, the industrial sector consumes about half of the world's total energy and emits the most greenhouse gases [3; 4]. Hence, energy saving in the industry sector has priority in promoting sustainable economic development. As we all know, most of the energy is converted into the form of electricity that numerous industrial sectors use as their main energy [5; 6]. Nevertheless, electricity is hard to store effectively, and thus, must be produced and delivered to its customers at once [7]. In addition, the electricity demand is always uneven, which leads to an increase in generating cost owing to the utilization of backup power facilities during peak hours [8]. In order to maintain balance between electricity supply and demand, electricity providers usually implement demand-side management programs [9], which are an essential component of realizing the goals of a smart grid and rationalizing the allocation of power resources [10]. One of the demand-side management programs is time-of-use (TOU) electricity tariffs, which have been widely used around the world. Usually, a common TOU tariff scheme can be divided into three types of periods: off-peak, mid-peak, and on-peak periods. The basic nature of the TOU scheme is that the retail prices set by electricity providers vary hourly throughout the day according to the amount of electricity demands; when there is an increase in demand, the electricity cost goes up correspondingly, and vice versa [11]. The practice of TOU electricity tariffs not only provides significant opportunities for the industrial sector to enhance energy efficiency, but also avoids power rationing during on-peak periods, and improves the stability of the power grid [7]. Using low-energy equipment and improving the efficiency of production management are two important methods to save energy [12]. As a widely used production management method, scheduling can effectively control energy consumption [13], which brings a lower cost of operation. However, the studies about energy-saving scheduling are still limited [14]. Over recent years, energy-efficient scheduling problems have gradually aroused the attention of scholars. To achieve the goal of energy saving during the production process, some researchers have investigated the problems with various energy-efficient mechanisms to reduce electricity costs by minimizing overall energy consumption, such as speed-scaling [15; 16; 17; 18] and power-down [19; 20; 21], while others have studied the problems from the perspective of TOU electricity tariffs, which has become a frontier issue in this field. As for researching scheduling problems under TOU electricity tariffs, there has been a growing interest recently. Considering both production and energy efficiency, [PERSON] et al. [22] proposed an ant colony optimization meta-heuristic algorithm for hybrid flow shop scheduling problems under TOU electricity tariffs. [PERSON] et al. [12] studied a flow shop scheduling problem with production throughput constraints to minimize electricity cost and the carbon footprint simultaneously. [PERSON] et al. [23] presented a so called \"econological scheduling\" model for a speed-scaling multi-machine scheduling problem aimed to minimize the electricity cost and environmental impact. [PERSON] et al.