## 1 Introduction  
The manufacturing industry is facing a great deal of pressure with regard to saving energy and reducing emissions, since it is an energy-intensive industry. In 2020, its energy consumption reached 1443.1 trillion British thermal units (Btu), of which electricity consumption was 200.7 trillion Btu [1]. In the same year, energy-related carbon dioxide emissions by the manufacturing industry reached 1057 million metric tons [2]. Studying energy-efficiency scheduling under electricity cost is of great significance for manufacturing firms to improve their energy efficiency. In a factory, machinery is the main energy-consuming unit [3, 4, 5, 6]. Reducing machine energy consumption economically and environmentally improves sustainable manufacturing. There are many potential energy-reduction approaches in a manufacturing plant, such as developing more energy-efficient machines and processes. In the production line, production planning and scheduling generally impact energy efficiency. Energy efficiency can also be increased by the suitable utilization of machines in the shop floor [6, 7, 8, 9, 10, 11, 12]. In the production process, the electricity consumption of each job is sometimes different, indicating that an operation with high electricity consumption is arranged for low processing time jobs, and an operation with low electricity consumption is arranged for high processing time jobs, so that the electricity cost can be decreased [5; 6; 13]. Paying too much attention to energy cost as an outstanding optimization objective may cause a long makespan and imbalanced workload on machines, as well as lead to bottlenecks, late deliveries, and even untimely machine failure [14; 15; 16; 17]. Production scheduling is explicitly important in a modern manufactory, consisting of planning and sequencing jobs into machines, since mass production heavily relies on a large number of machines working in parallel. Parallel machines can process several jobs simultaneously without affecting each other. Parallel machine scheduling is defined as sequencing and assigning jobs into machines when similar types of machines are available and jobs can be scheduled in these machines. A variety of sequencing and/or processing restrictions often exist when decision makers try to minimize some related objective functions [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]; however, most enterprises still use advanced machines running alongside outdated ones. In contrast to the old ones, modern machines are usually adjusted to work at a high speed and save energy. Speeding up old machines to operate as fast as modern ones results in them consuming more electric power and releasing more pollutants [25]. Sequencing and assigning jobs to a particular machine that can process different jobs at different production rates is considered to be the parallel-machine-scheduling problem. Solutions or algorithms constructed to schedule parallel machines are important because, when implemented in all sorts of parallel-machine problems, they can obtain good solutions for the overall production-planning process in a factory [18; 19]. Although most scheduling problems consider production efficiency, cost, and quality as optimization problems, considering the costs of energy and gas emissions--known as the green scheduling problem--has attracted the attention of researchers [9; 10; 11; 14; 25]. In addition, various constraints have been addressed in parallel-machine-scheduling problems, such as setup time, machine-available time, ready time, release date, due date, and delivery time [14; 26]. In this study, the due-date constraint was considered with makespan control, since late-delivery cost and production-overhead cost per working hour may be a thorny issue for entrepreneurs. Therefore, job priority and makespan control should be seriously considered for production planning as well.