Energy-Efficient Hybrid Flow-Shop Scheduling under Time-of-Use and Ladder Electricity Tariffs

Abstract

With the increasing influence of climate change, green development has become an important trend. Since manufacturing represents approximately one-half of total energy consumption, reducing the amount of energy consumed within this industry is imperative. This study provides a hybrid flow shop scheduling issue under a time-of-use and ladder electricity price system to reduce total energy consumption without compromising maximum completion time. An improved non-dominated sorting genetic algorithm II with some optimization strategies is proposed to solve the problem. First, an enhanced constructive heuristic algorithm is used to improve the quantity of initial solution in the initialization. Besides, an adaptive genetic operation is introduced, aiming to avoid the emergence of locally optimal solutions. In addition, the right-shift approach is developed to reduce the total energy consumption without affecting completion time. By maintaining the production efficiency and reducing the energy consumption cost by 4.33%. A trade-off proposal is made between productivity and sustainability in view of the calculation results.

1. Introduction

As a result of the increasing influence of climate change, green development has become an important trend. As a concept of economic growth and social development aiming at efficiency and sustainability, green manufacturing is an integral part of green development [1]. Green manufacturing involves the entire process of production, such as renewable energy power generation, saving raw materials, reducing energy consumption, green product redesign, and reducing pollution emissions [2]. According to a China Electricity Council (CEC) report, China’s total electricity consumption reached 7511 TWh in 2020, with year-over-year growth of 3.1%. Specifically, the electricity consumption of the secondary industry was 5.121 TWh, accounting for 68.2% of the total electricity consumption [3]. Green development considers the environmental impact and resource efficiency while ensuring the economic and social benefits of the enterprise, especially in China [4].

Traditional workshop scheduling problems usually consider a single economic indicator, while green and economic indicator are both considered green manufacturing [5]. Green indicators usually contain minimizing the consumption of resources and minimizing the impact on the environment [6]. Due to most of the production methods of chemical, steel, and other high-profile production methods being flowing operations, the similar processing production processes can be abstracted as a hybrid flow-shop scheduling problem (HFSP) [7]. In industrial production, the flow shop scheduling problem (FSP) is a general problem model and a hot spot in the academic field [8]. As an extended form, HFSP is a popularization of the traditional workshop scheduling problem, which has the comprehensive characteristics of assembly-line work and parallel machines [9].

In the basic flowshop scheduling problem (FSP), n jobs must be processed on m machines in the same order, that is, the operations of each job follow the same route. It is commonly assumed that there is no job preemption, and all jobs are ready to be processed at the beginning. Therefore, if an operation of a job has begun on a machine, it cannot be interrupted until it is finished. All these assumptions are also valid for the permutation flowshop scheduling problem (PFSP).

Considering the green PFSP that minimizes total flowtime and total energy consumption concurrently, a bi-objective mixed-integer linear programming (MILP) model and three effective heuristic algorithms were developed for the problem, employing a simple job-based speed-scaling strategy [10]. The study incorporated the energy concerns into the PFSP and analyzes the trade-off between total flowtime and energy consumption criteria in an efficient way to assist decision making in this environment.

Environmental issues in green FSPs were proposed to capture both operational and tactical planning decisions, from an environmental policy-maker perspective [11]. Test instances were randomly generated and the optimal solution sets were derived for each of three carbon reduction policies. The outcome of evaluation in most cases has shown that an Emissions Trading (ET) policy yields a stronger market-based policy.

Many scholars have the intensive study of the methods to reduce energy consumption for HFSP optimization in existing literature, including energy-saving measures [12] and energy saving strategies [13]. After sorting out a large number of relevant documents on energy requirements in the production process, Gahm et al. [14] put forward the concept of “Energy Saving Scheduling” (EES). They gave a research framework of EES, which classified the existing literature according to three dimensions of energy coverage, energy supply, and energy demand. Undoubtedly, it helped scholars understand this research problem more systematically [15]. So far, several studies have demonstrated that reducing energy consumption has become one of the hot spots in scheduling [16].

Singh et al. [17] systematically analyzed the manufacturing process and described the relationship of process variables in energy consumption by constructing a mathematical model, while the parameters was optimized through an artificial bee colony algorithm. Additionally, Saberi-Aliabad et al. [18] conducted research in an unrelated parallel machine environment to minimize costs and consume energy. Their study developed a few dominance rules and valid inequalities in the mixed-integer linear programming model.

As a typical scheduling problem, HFSP widely exists in the actual manufacturing process [19]. Several studies have considered energy consumption and emphasized the importance of energy conservation as an environmental issue [20]. Hasani and Hosseini [21] considered the production costs and energy consumption in a flexible flow shop scheduling problem. The problem was described as a numeric example with machine dependent processing stages. To find the optimal or close to optimal solution, a new approach was introduced based on the NSGA-II.

It is possible to reduce total energy consumption (TEC) during the manufacturing process through adjustments and technical upgrades [22]. Furthermore, the electricity price mechanism has played an important role in energy conservation. The time-of-use (TOU) price strategy is a very useful one in green scheduling, by adjusting processing time from peak periods to off-peak periods effectively [23]. TOU applications in many filed, such as Household, Logistics and Manufacturing.

In order to minimize electricity cost and earn the relevant incentive, residential demand response was studied through the scheduling of typical home appliances as per the electricity price [24]. A practical optimization model for a household to determine the optimal scheduling of home appliances under TOU electricity prices is formulated and a mixed integer nonlinear optimization model was built. A case study showed that a household is able to shift consumption in response to the varying prices and incentives, through which the consumer may realize an electricity cost saving of more than 25%.

To provide a comprehensive overview of the research that aim to achieve energy efficiency for sustainable and green ports, a systematic literature review was conducted to analyze operational strategies, technology usage and energy management systems [25]. As an operational strategy, peak shaving aimed to reduce the peak energy consumption of the port. A number of different methods using the load profile curves where (1) Power sharing: Using any stored energy in the case of peak energy demand periods; (2) Load shifting: Shifting the energy demand in the peak period to non-peak periods; (3) Load Shedding: Turning off non-critical loads during peak periods.

Aiming at minimizing the overall electricity related cost as well as the potential penalty cost due to the failure of timely fulfillment of target production, a methodology was proposed to help the manufacturing enterprises identify a cost-effective reservation capacity by optimal production scheduling under Critical Peak Pricing (CPP) program, a typical price-driven electricity demand response program [26]. A Mixed Integer Nonlinear Programming (MINLP) is used to formulate the problem and approximate method is used to obtain a near optimal solution. The results of the case study show that a cost reduction of 31.42% and 11.04% can be achieved for a billing cycle compared with the simple method.

To minimize total lateness cost of operations and total energy costs considering hourly energy price with energy selling back option under uncertain renewable energy generation, an energy management and operations planning problem for next generation green ports was addressed [27]. In this paper, energy management matches energy demand and supply considering different energy pricing schemes and bidirectional energy trading between energy sources and energy storage systems. A mixed integer linear programming model was suggested and the model finds optimal results in short computational times.

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. The ladder electricity price refers to the division of electricity consumption into several steps, in which each step is performed at different electricity prices. By combining the advantages of the two electricity price mechanisms, the government can reduce energy consumption while achieving the goal of “shaving peaks and filling valleys”. In order to reduce energy consumption in the flow-shop problem, Yan [39] established a mathematical model considering TOU and ladder prices. Other researchers, such as Cao et al. [40] evaluated an integrated scheduling problem from iron-steel manufacturing under TOU electricity tariffs. An improved Strength Pareto Evolutionary Algorithm (SPEA2) based on the relationship propagation chain was customized for the problem.

To the best of our knowledge, most studies have focused on improving algorithms and production processes [41]. Since electricity price mechanism has a great impact on energy consumption it is also worth researching the production process under the electricity price mechanism [42]. Nonetheless, the initiatives mentioned above offer a starting point for investigating the energy-efficient design of HFSP under TOU and ladder power rates.

Together, these studies provide important insights into applying the HFSP to reduce energy consumption. A significant amount of existing research has explored the application of algorithms, including new heuristic algorithms. However, these studies ignore the advantage of constructive algorithms in solving targeted problems. To reduce energy consumption, the general TOU electricity price policy has been applied for a number of years, and many of its aspects have been studied. In addition, research on TOU and ladder electricity price schemes have received little attention. Furthermore, the application of renewable energy to reduce carbon emissions in production dispatch scenarios has been studied. However, few studies exist on the efficient use of renewable energy during production.

The optimization method for solving HFS problems includes exact, constructive, and heuristic algorithms. The first step in solving HFS problems will be to apply exact algorithms, such as branch and bound methods [43] and dynamic programming [44]. On the other hand, exact algorithms cannot solve large-scale HFS problems due to the tremendous amount of computation required. Recent years have seen the development of heuristic and constructive algorithms to solve HFS problems. Constructive algorithms use a design rule to construct the solution. For instance, Nawaz, Enscore & Ham (NEH) proposed a constructive heuristic algorithm. improved to be effective in flow shop scheduling. The basic idea of NEH algorithms is giving the priority of the workpiece based on the processing time [45]. Heuristic algorithms including genetic algorithm (GA) [46], Simulated Annealing (SA) [47], Tabu Search (TS) [48], Particle Swarm Optimization (PSO) [49], etc.

As a well-known scheduling problem, HFSP has substantial research value and has been demonstrated to be an NP-Hard problem. Furthermore, considering that many problems in the real world is usually consisting of multiple targets which may conflict with each other. Researchers have proposed many multi-objective optimization algorithms to find a Pareto solution, such as Non dominated sorting genetic algorithm-II (NSGA-II) [50], and multi-objective evolutionary algorithm based on decomposition (MOEA/D) [51].

Wang et al. [52] proposed an energy-efficient distributed permutation flow-shop inverse scheduling problem to simultaneously minimize adjustment and energy consumption. This model contained realistic constraints, controllable processing times, and energy consumption factors. We developed a hybrid collaborative algorithm that combines a cooperative search scheme to effectively address the problem. To build a green manufacturing model and realize energy conservation and emission reduction, a distributed two-stage reentrant hybrid flow-shop bi-level scheduling model was established, which takes the makespan, total carbon emissions, and total energy consumption costs as the optimization objectives [53]. The optimal scheduling scheme is determined using the upper model to minimize the time between jobs. Specifically, the lower model aims to minimize the overall carbon emissions and the overall energy costs by allocating the power supply among distributed energy resources (DERs), energy storage systems (ESSs), and the primary grid. An improved hybrid salp swarm (SSA) and NSGA-III algorithm was proposed to solve this problem.

Furthermore, aiming to minimize the makespan and total energy cost in capital-intensive manufacturing industries, Park and Ham [54] addressed flexible job-shop scheduling under time-of-use pricing and scheduled downtime. First, an integer linear programming model was proposed. Second, a constraint programming model was discussed. Third, managerial insights were derived.

A model under a sequential-dependent single machine scheduling problem was proposed to minimize energy costs and reduce carbon footprint emissions [55]. A Low Energy-Carbon Cost Sequence Dependent Job Scheduling optimization model (LEC-SDJS) was developed. Comparing the model to classical scheduling optimization approaches shows the model to be useful.

Considering a mixture of job-shop and flow-shop production scheduling problems with a speed-scaling policy and no-idle time strategy, a multi-objective Q-learning-based hyper-heuristic with Bi-criteria selection (QHH-BS) was developed. This can result in the production of a set of high-quality Pareto frontier solutions [56]. The manufacturing sequence of job-shop and flow-shop goods was represented by this method, which employed a novel three-layer encoding.

Our study considers a HFSP depending on electricity price and energy-saving measures to reach the goal of minimization of total energy consumption without effect the maximum complete time. The proposed mathematical programming model focuses on the characteristic of HFSP and electricity price scheme, while right-shift approach is geared toward the Pareto solution through the adoption of an improved NSGA-II algorithm. However, the purpose of research is to integrate multiple algorithms for a new problem, without a large degree innovation for the existing algorithms. Thus, we can verify its feasibility based on a general multi-objective algorithm. For these reasons, we adopt NSGA-II as the original algorithm, which is proven to be a useful method to solve multi-objective problems.

The contributions of the paper can be summarized as follows.

(i)

A new scenario containing energy-intensive industries under punitive electricity price is considered.

(ii)

In response to problems, several low-carbon strategies are integrated into a multi-objective algorithm framework.

(iii)

Experimental results show that the proposed framework reduces carbon reduction effectively without affecting make span.

In this work, the background, research significance, article contribution, and article structure are described in Section 1. Furthermore, the research problem is described in Section 2. Section 3 introduces an improved NSGA-II algorithm with a right-shift procedure. Section 4 reports the experimental results and compares them with the original NSGA-II algorithm. Finally, some conclusions and future work are provided in Section 5.

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.

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 $\left\{J_1,J_2,···,J_n\right\}$ and a set of m machines $\left\{M_1,M_2,···,M_m\right\}$. 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. The notations which used in this paper are summarized below:

Indices:

$C_{max}$	Makespan
$TEC$	Total energy consumption
$i$	Index of processing stages, $i=1,2,···,n$
j	Index of job, $j=1,2,···,m$
k	Index of machines in the processing stages, k = 1, 2, ···, p
t	Index of time period for TOU

Variables:

$s_{ijk}$	Start time of job j on machine k in stage $i$
$e_{ijk}$	Completion time of job j on machine k in stage $i$
$\sigma \left(t\right)$	Processing time of job j on machine k in stage $i$
${\sigma }^v\left(t\right)$	TOU price function
$q_0$	Electricity price under TOU and $\mathrm{v}$th ladder stage
$q_0$	Electricity price within time [0, $t_1$)
$q_i$	Electricity price within time [$t_1$, $t_i$)
$q_i^v$	Electricity price of stage v within time [$t_1$, $t_l$), $v\in \left[1,V\right]$
$E_p$	power consumption for each machine during processing mode
$E_i$	power consumption for each machine during standby mode

Decision variables:

$$x_{ijk}=\left\{\begin{array}{c}1 job j in sequence on machine k in stage i \\ 0 otherwise\end{array}\right.$$

$$y_{jk}=\left\{\begin{array}{c}1 job j is assingned to machine k\\ 0 otherwise\end{array}\right.$$

$$Z_{ijk}^v=\left\{\begin{array}{c}1 electricity consumption of job j reach the standard of stage v \\ 0 otherwise\end{array}\right.$$

The mathematical formulation for the problem is as follows. Equations (1) and (2) are the multi-objective of the HFSP. Equation (1) represents the make span, which is the maximum completion time of all jobs. Equation (2) represents the TEC, which depends on the machine power and the energy price in each period.

$$f_1=\min C_{\max }$$

(1)

$$f_2=\min TEC$$

(2)

Equation (3) ensures that only one job can be processed simultaneously. Equation (4) guarantees that each job can be processed by only one machine at any stage.

$${\sum }_{i=1}^n{x}_{ijk}=1j=1,2,···,m; k=1,2, ···, p;$$

(3)

$${\sum }_{j=1}^m{y}_{jk}=1k=1,2, ···, p;$$

(4)

Equations (5)–(8) formulate the processing contiguity constraints. Equation (5) illustrates the relationship between the start time and the completion time of the process at the same stage. Equation (6) indicates that the next process of the same job cannot be started before the previous process is completed.

$$e_{ijk}=s_{ijk}+p_{ijk}i=1,2,···,n; j=1,2,···,m; k=1,2, ···, p;$$

(5)

$$e_{ijk}\le s_{i\left(j+1\right)k}i=1,2,···,n; j=1,2,···,m; k=1,2, ···, p;$$

(6)

Equation (7) indicates that the next job cannot be processed earlier than the previous one at the same stage. Equation (8) means that the lower-ranked job allocated on the same machine at the same stage can only be processed after the previous job is processed, M is enough for when the job in different ranks is not on the same machine.

$${\sum }_{i=1}^n{x}_{ijk}{s}_{ij}\le {\sum }_{i=1}^n{x}_{\left(j+1\right)k}{s}_{ij} j=1,2,···,m; k=1,2, ···, p;$$

(7)

$$\begin{gathered} {\sum }_{i=1}^n{x}_{ijk}{y}_{ik}{e}_{ij}\le {\sum }_{i=1}^n{x}_{ijk}{y}_{ik}{s}_{ij}+\left(1-{\sum }_{i=1}^n{x}_{ijk}{y}_{ik}{s}_{ij}\right)\times M \\ j=1,2,···,m; k=1,2, ···, p; \end{gathered}$$

(8)

Equations (9) and (10) indicate electricity prices during different processing times and electricity consumption levels. $\sigma \left(t\right)$ is the TOU electricity price function and ${\sigma }^v\left(t\right)$ represents the electricity price under TOU and the $v$th ladder stage.

$$\sigma \left(t\right)=\left\{\begin{array}{c}{q}_1 0\le t<t_1\\ q_2 t_1\le t<t_2\\ ···\\ q_l t_{l-1}\le t<t_l\end{array}\right.$$

(9)

$${\sigma }^v\left(t\right)=\left\{\begin{array}{c}{\mathrm{q}}_1^v 0\le t<t_1\\ {\mathrm{q}}_2^v t_1\le t<t_2\\ ···\\ {\mathrm{q}}_l^v t_{l-1}\le t<t_l\end{array}\right.$$

(10)

There are two modes power consumption for each machine during processing mode ($E_p$) and standby mode ($E_i$), and the total electricity consumption is calculated by step prices. Thus, the TEC could be calculated as follows in Equation (11).

$$TEC={\int }_0^{C_{max}}\left({\sum }_{i=1}^m{E}_p*{\sigma }^h\left(t\right)+{\sum }_{i=1}^m{E}_i*{\sigma }^h\left(t\right)\right)dt$$

(11)

3. The Scheduling Approach

In response to this problem, an improved non-dominated sorting genetic algorithm-II with some optimization strategies is explored. There are two stages to the problem. First, an optimization solution is found according to the principles of shortest time. Then, the right-shift strategy reduces the energy consumption under the TOU and ladder electricity price.

Based on the characteristic of multi-objective problems in HFSP. The initial population is generated by the NEH heuristic algorithm, and the object of TEC is optimized with the right shift approach. Considering the static control parameters may not be effective, a deterministic control method is adopted in crossover and mutation by using dynamic control parameters. The flowchart of the improved NSGA-II algorithm is showed in Figure 3.

3.1. Representation of the Chromosome

Considering the HFSP, this paper adopts a simple coding method according to the process. The chromosome represented the scheduling of each workpiece at each stage. In the process-based coding method, the genes on the chromosome represent the processing sequence of a certain process on the machine. Therefore, the solution to the scheduling problem could be obtained directly. A gene on a chromosome is used to indicate the selection of processing procedures, and a number at each of these gene points indicates the corresponding workpiece for processing. Additionally, the length of the chromosome corresponds to the number of workpieces to be processed.

In this way, Figure 4 shows that genes in the chromosome denote the orders assignment for operations from $j_1$ to $j_n$. The length of the chromosome is equal to the number of jobs to be processed and represents the scheduling of each workpiece at each stage. For example, the vector with six jobs [341526] means that the jobs are processed in the order 3, 4, 1, 5, 2 and 6.

3.2. Initialization

A minimal number of organisms are created in the initial population as part of NEH’s heuristic method to ensure good-quality individuals. The remaining individuals in the population are generated randomly, which ensures the quality and diversity of the initial population. Since the job sequence obtained by the NEH algorithm cannot provide the processing order information of the jobs on each machine in the HFSP, if it is regarded as a population individual, it must be decoded to obtain the job processing sequence information on each machine to calculate the fitness for selection. The renowned NEH technique is the most efficient heuristic for minimizing the makespan in the PFS problem. Moreover, it is also effective for the multi-objective HFSP considered in this paper, with the basic idea of giving priority to the workpiece based on the processing time. The first-in-first-out rule is used to allocate the parallel machines of the same process, which means that the workpiece completed first in the previous process can occupy the first idle parallel machine of the following process.

3.3. Right-Shift Procedure

This paper considers power consumption during processing and standby time under the TOU and ladders electricity policy. The interval between the start processing time and the end time is calculated in the first step. Next, each order’s processing time is generated to judge the TOU interval of the start time and completion time. After calculating and summarizing the processing time for each order, the gap time of the neighboring two workpieces is compared to determine whether the right shift strategy or speed scaling strategy.

Due to the condition of the decoding method, each workpiece is decoded according to the earliest start time. Therefore, all processes have no room to move left and can only move to the right. It could be seen from the adjustment principle that the adjustment of the next process period will affect the adjustment of the previous process period, so the right shift operation should follow the rule of backward to forward.

The right-shift strategy is applied to reduce the energy consumption after obtaining the processing sequence based on the principle of the shortest time. The completion time of the last workpiece of each procedure remains unchanged.

One problem is whether to move from the high electricity price range to the low electricity price range. When calculating the energy consumption, the first step is judging the processing time in each electricity price zone. The next step is to calculate the power consumption during processing and standby modes from the last stage to the first stage. At each stage, each operation’s starting and ending times are recorded.

The algorithm of the right-shift procedure in shown in Algorithm 1. In our algorithm, the computational complexity of the TEC heuristic is O(PMN²), where p is the number of stages, m is the number of procedures, and n represents the number of jobs. Although the number of stages is constant, it has the same computational complexity as NSGA-II.

Before the right-shift procedure, a job sequence is generated based on initial rules. Determine the space to move to the right in the core rule. First, the jobs in nonincreasing order of their completion time are arranged from the last stage to the first stage. $S_{i,j,k}$ is the start time of job $j$ at stage i in machine k and $S_{i+1,j}$ is the next stage i + 1 of job j. $S_{i,j+1,k}$ is the start time of the next job j + 1 of the same machine k in this stage $i$. By comparing the time of $S_{i,j,k}$, $S_{i+1,j}$ and $S_{i,j+1,k}$, we find out whether there is a space for right shift. By comparing the electricity price of $S_{i,j,k}$, $S_{i+1,j}$ and $S_{i,j+1,k}$, we found out whether it is worth to right shift. It determined the start time relying on the domination relationship and calculated the corresponding energy consumption from the last stage to the first stage. It supposes that the time required for each statement is the unit time required. The time consumption of an algorithm is the sum of the frequency of all statements in the algorithm. The algorithm’s complexity usually includes encoding, selection operator, and the improving approach for the problem. Usually, the improved algorithm has a smaller or the same calculation complicated while the origin algorithm has known time complexity.

Cao et al. [40] supply the complexity of the improved SPEA2 based on the hybrid adjustment strategy of the single cast stage. Given that there are G casts, $l_{max}$ production stages, K periods, and $N_i$ charges in a single cast $i$, the complexity of the abovementioned encoding, crossover, and mutation operator for SPEA2 are $O\left(\left(GK+max\left\{N_i\right\}\right)\right)l_{max}$, $O\left(\left(GK+max\left\{N_i\right\}\right)\right)l_{max}$, and $O\left(GK{l}_{max}\right)$ in the worst case. Therefore, the whole complexity of the improved algorithm is $O\left(\left(GK+max\left\{N_i\right\}\right)\right)l_{max}$.

We develop an ad-hoc heuristic method to address a multi-objective combinatorial optimization problem of scheduling jobs among multiple parallel computers while minimizing both the total energy consumption and the makespan under time-of-use electricity prices [41]. The first part of the Split-Greedy heuristic (SPH) consists of an improved and refined version of the constructive heuristic (CH). The second part, called Exchange Search (ES), is a novel local search procedure to improve the quality of the Pareto optimal solutions. The related parameter is from the article.

By implementing an additional auxiliary table that stores information on the scheduled jobs, the complexity of SPH is improved and is expressed as $O\left(\left|P_j\right|·lo{g}_2\left|P_j\right|+N·k^{max}+M·\left(\mathsf{\Omega }·lo{g}_2\mathsf{\Omega }+\left|P_j\right|·k^{max}+N\right)\right)$. While the complexity of ES can be more compactly expressed as $O\left(R·N·M·K^{max}·p_{max}^2·lo{g}_2\left(p_{max}\right)·lo{g}_2\mathsf{\Omega }\right)$. A consideration of the complexity of ES in the worst-case scenario indicates that the computation time is not promising.

Considering that the purpose of this paper is to integrate the low-carbon method with an origin multi-objective algorithm for a new problem, and the computational complexity of the right-shifting heuristics method is also O(MN2) as NSGA-II, the computational complex would not change. Therefore, without a considerable degree of innovation for the existing algorithms, the method proposed in this paper for reducing energy consumption will not increase the time complexity.

3.4. Select, Crossover, and Mutation

Initially, a random parent population $P_0$ is obtained through initialization and the individual’s fitness in $P_0$ is calculated through the program. First, the binary tournament selection, crossover, and mutation operators are used to create an offspring population $Q_0$ of size $N$. The gene strategy is adopted after the initial generation. The next-generation population is derived by assigning non-dominated grades and crowding distances based on comparing the current population with the best non-dominated solution. The $t$th generation of the proposed algorithm is described, as shown in Section 3.5. The non-dominated Sorted is shown in Algorithm 1.

The crossover operation uses the two-point crossover method. Two offspring chromosomes are generated by randomly selecting two points on the two-parent chromosomes and exchanging the gene fragments between these two positions. The mutation operation adopts the single-point mutation method, in which the value of a certain gene of the chromosome is changed to transform it into a similar solution for selection.

By dynamically adjusting the update range of population individuals as the evolutionary generation increases, the convergence of the population and the accuracy of the search direction are increased. The adaptive adjustment of the genetic parameters allows the adaptive genetic algorithm to achieve convergence while maintaining the diversity of the population.

Algorithm 1. Right-shift procedure.

Input: A complete schedule decoded from a job sequence.

Output: A modified schedule with less electric power cost.

1: for i = 1, 2, …, m do

2:   Arrange the jobs in non-increasing order of their completion time at stage i

3:   for j = 1, 2…, J do

4:      Set tmin, = Si,j,k, where Si,j,k the start time of job j at stage i. Set tmax = Min(Si,j+1, Si+1,j,k)—Pi,j,k, where Sj+1,k is the start time of job j + 1 at stage i, the schedule Si+1,j the start time of job j at stage i + 1.

5:      By comparing the time of Si,j,k, Si+1,j and Si,j+1,k, find out whether there was a space for right shift. By comparing the electricity price of Si,j,k, Si+1,j and Si,j+1,k, find out whether it was worth to right shift.

6:      Arrange the start time between [tmin, tmax], calculate the objective function values Makespan and TEC, determine the start time relying on domination relationship.

7:   end for

8: end for

To study the impact of the parameter in genetic algorithms, Hassanat et al. [59] developed some new deterministic methods of controlling crossover rates and the mutation rate, which dynamically increase or decrease the crossover and mutation rates relying on the evolutionary generations. It provides a guideline for our research.

Considering the adaptability of the multi-objective problem and the complexity of HFSP, the adaptive DHM/ILC method is adopted. The crossover rate is changed linearly according to the generations. To calculate mutation rates in DHM/ILC, the following equation is used:

$$P_c=\frac{g}{G}$$

(12)

$$C=P_c\times popsize$$

(13)

$P_c$ as the crossover probability, $g$ is the current genetic generation number, G is the total genetic generation number, C is the size of chromosomes which are selected to crossover. At the same time, Equations (14) and (15) change the mutation rate linearly according to the number of the current generation and the total number of generations. The following equation can be used to calculate mutation rates in ILM/DHC:

$$P_m=1-\left(\frac{g}{G}\right)$$

(14)

$$M=P_m\times popsize$$

(15)

$P_m$ as the crossover probability, $g$ is the current genetic generation number, $G$ is the total genetic generation number, $M$ is the number of chromosomes that need to be used for the crossover process.

3.5. Non-Domination Sorted and Elite Select

This section is showed as Figure 5. Firstly, the offspring $Q_t$ is produced by genetic operation from the initial parent $P_t$ (initial t = 0), while the population sizes of $P_t$ and $Q_t$ are both N. Then the operation merges $P_t$ and $Q_t$ into $R_t$, which has 2N individuals. The algorithm assumes that every individual in the population has two attributes: (1) nondomination rank (2) crowding distance. Crowd-comparison guides the algorithm towards producing a Pareto-optimal result at different stages of the algorithm. The partial order is defined as the lower nondomination rank. If two individuals have the same nondomination rank, the larger crowding distance has a priority.

According to the nondomination rank and the crowded distances of all individuals in $F_i$, the differences non-dominated solution sets $F_1$, $F_2$, …, $F_n$ are performed from the $R_t$. The new set $P_{t+1}$ is constructed from $F_i$ based on the nondomination rule until the scale of $P_{t+1}$ is N, as presented in Figure 4.

In the end, it will output the best solution $P_N$ found in all iterations. Since this is a multi-objective problem, there would be multiple solutions. The setting is to output the same number of solutions as the population size N.

4. Case Study

We evaluate the performance of the improved NSGA-II algorithm with right-shift procedure and the original NSGA-II algorithm in a real-world environment. The calculation results are analyzed and discussed clearly. The model formulations are coded and implemented in Visual Studio2017 on a PC with a 2.8 GHz Intel core i5-7200 processor and 8 GB RAM on a Windows 10 system.

4.1. Evaluation of the Example

The TOU price refers to the current policy in Tianjin as shown in

Table 1. Electricity market price for industry during the non-summertime in Tianjin, China. (Source: Tianjin Electricity Price, 2019; www.tj.sgcc.com.cn/) (accessed on 23 June 2022).

Tianjin’s Commercial TOU Pricing Standard
Type	High-Peak	Mid-Peak	Off-Peak
Period	8:00~11:00	7:00~8:00	21:00~7:00
	18:00~21:00	11:00~18:00
Price (CNY/kWh)	0.9914	0.6980	0.4186

. As the ladder price is not apply in Tianjin, we use the high energy consumption industry ladder electricity price referring to another industry province. Considering the price of each interval, the up-step price is set 20 percent beyond the origin price and total amount in an interval is set 800 kwh per day [39].

The processing time and energy usage of each process in a stamping workshop in Tianjin are examples of real-world scheduling problems. The stamping workshop is needed to process different types of stamping parts, different types of stamping parts contain different batches, and the same process of the same type of stamping parts adopts continuous processing. Therefore, one type of stamping part could be regarded as an order containing different batches, and the production scheduling problem of the workshop could be simplified as one process. Its data demonstrates some common characteristics of the manufacturing. The data has been pre-processed in order to protect some commercial confidentiality.

Considering the advantage of improvement strategies, a right-shift method is designed based on the improvement of the NSGA-II as shown at part 4.3. The population size is set to 200. The maximum evolution generation of the algorithm is the same 200 to ensure results are obtained. The mutation probability and the crossover rate are set to be 0.05 and 0.95 in original NSGA-II.

From the data simulation, the Pareto optimal frontier formed in the two scheduling target dimensions of the maximum completion time and the energy consumption cost for the green scheduling considering the TOU electricity price is determined.

In response to this problem, an improved non-dominated sorting genetic algorithm-II with some optimization strategies is explored. Two stages are proposed to the problem. First, an optimization solution is found according to the principles of shortest time. Then, the right-shift strategy is used to to reduce the energy consumption under the TOU and ladder electricity price.

For each stage, there are two machines respectively.

Table 2. Average processing time and power demand of each job.

Jobs	Average Processing Time (h)/Average Power Demanded (kW)
	Stage 1	Stage 2	Stage 3	Stage 4	Stage 5	Stage 6	Stage 7	Stage 8
1	1.5/63.5	1.6/32.4	0.8/94.5	0.9/33.4	2.8/84.5	2.1/51.3	0.7/52.4	1.0/72.3
2	0.9/78.9	1.6/35.0	0.7/83.6	2.6/44.3	2.4/73.5	2.1/49.4	2.4/32.5	1.2/72.1
3	1.5/68.5	1.2/45.2	1.6/84.5	1.4/33.8	2.8/84.3	0.7/58.6	0.8/44.7	1.9/76.8
4	1.3/66.9	1.6/45.2	1.1/84.2	2.6/44.3	1.4/74.3	2.2/65.3	0.7/51.5	1.2/80.5
5	0.9/67.1	2.7/42.0	0.8/93.5	1.1/43.3	0.7/62.8	2.9/46.8	1.4/54.5	0.9/66.1
6	2.1/64.1	1.5/45.8	2.3/84.1	1.4/34.8	2.1/77.3	0.6/57.9	1.8/52.8	1.6/82.3
7	1.5/78.2	1.6/37.4	1.6/35.2	2.6/37.3	0.9/70.6	2.4/51.9	2.4/64.3	1.0/72.0
8	0.5/75.1	1.5/60.6	0.9/46.5	1.7/45.8	0.6/84.3	1.9/60.0	2.8/44.5	2.9/72.3
9	1.9/65.2	1.5/69.8	1.7/42.0	2.1/55.3	0.7/63.5	0.9/52.1	0.7/40.8	1.1/80.5
10	1.1/58.5	0.9/73.1	1.5/35.8	1.1/48.1	2.2/73.9	0.8/60.6	2.4/60.3	2.1/72.1
11	1.5/69.8	0.9/66.7	2.2/33.8	1.8/48.5	2.8/74.5	1.5/60.3	1.7/51.5	1.1/82.3
12	0.9/66.7	1.5/58.3	1.6/47.4	1.2/43.5	1.2/84.9	2.2/61.0	1.4/54.5	1.2/55.5
13	1.5/78.4	2.2/33.8	1.7/36.5	2.3/84.1	1.4/63.8	2.7/68.5	1.8/42.8	2.3/72.1
14	1.5/56.3	1.6/47.4	0.9/46.3	1.0/85.5	2.8/74.5	1.7/50.6	0.8/50.0	2.0/76.0
15	1.2/78.8	1.5/38.3	2.9/94.0	2.2/33.3	1.2/82.9	1.2/50.3	1.6/50.5	1.2/82.8

shows the average processing time and power demanded of each workpiece on each machine whose data is measured from the machine.

Table 3. Power of standby of each stage’s machine.

Machine	${\mathit{M}}_1$	${\mathit{M}}_2$	${\mathit{M}}_3$	${\mathit{M}}_4$	${\mathit{M}}_5$	${\mathit{M}}_6$	${\mathit{M}}_7$	${\mathit{M}}_8$
Power (kW)	18.00	9.00	23.00	10.00	15.00	13.00	18.00	25.00

provides an overview of the standby power of each machine tool.

The algorithm runs 10 times in order to assess the randomness, and the Pareto frontiers of the two algorithms are shown in Figure 6. It can be seen that most of the solutions obtained using the improved NSGA-II algorithm with the right-shift strategy dominate the solutions obtained by the improved algorithms.

As shown in Figure 6, the X-axis represents the processing time and the Y-axis represents the TEC. It is obvious that the larger time has a lower TEC. The black points are the multi-objective solutions without adopting the right-shift method, and the red points are the multi-objective solutions when adopting the right-shift method. After adopting the right-shift method, the TEC can be reduced without changing the makespan.

At the same time, it can be seen that the Pareto frontier of the improved NSGA-II algorithm with the right-shift strategy is not fluent, and the right-shift strategy does not change the overall time, but only reduces the energy consumption.

A non-dominated solution of improved NSGA-II algorithm (f1 = 23.6 h, f2 = 6173.6 yuan) corresponding to the processing sequence 8-2-10-7-5-3-12-13-14-6-4-9-11-15-1 is chosen as an example. Figure 7 is a Gantt graph with right shift operation under TOU and ladder electricity price, and Figure 8 is a Gantt graph without right shift operation at the TOU and ladder electricity price (f1 = 23.6 h, f2 = 6453.08 yuan).

The are 16 machines (one stage has parallel machines) and 12 jobs in the HFSP. While there is an optimization solution, Job 8 will work first in the machine 1 (M1), and Job 2 work on the machine 2 (M1), Job 7 and Job 10 will start after the previous jobs finishing the process. As shown in Figure 7, X axial represents the processing time (hour in a day), while the Y axial represents the Machine index. While M1 and M2 are the parallel machines in the same stage, each line belonging to the machine represents the jobs processed above the machine, The legend on the right represents job Index, while each color represents the different job.

The comparison shows that there are many processes that have shifted the processing time. For example, workpiece 12 on machine 7 and workpieces 11 on machine 16 had been transferred from high electricity price to low price. By maintaining the production efficiency and reducing the energy consumption cost by 4.33%, a total of 279.48 yuan is saved. In the trend chart of the processing cycle, the total electricity consumption trend chart of all machines is shown in Figure 7, where TEC represents the total electricity consumption of all machines. The periods of high electricity prices are close to 10h and 20h, and the total electricity load of all machines decreases after the machine is moved to the right. Therefore, the electricity load has shifted significantly under the TOU electricity price, achieving the goal of “shaving peaks and filling valleys”, effectively reducing energy consumption costs and carbon emissions.

The above conclusions shows that the right shift operation is effective for processing workpiece allocation without changing the maximum complete time. Moving high-energy-consuming workpieces into the low-electricity price range can greatly reduce costs and energy consumption. To some extent, carbon emissions would also be reduced with the emission factor of electricity. It could be said that this optimization method is also effective for green production.

In this paper, we develop a model that combines time-of-use and ladder electricity tariffs for scheduling production. The traditional research about HFSP considers a single economic indicator and has performed good work to search the optimal jobs’ sequence. For instance, the NEH technique, which is based on calculating processing time, can solve in a short amount of time. According to the research results of adjusting the tradeoff between makespan and energy consumption in literature, it is suggested to apply the existing multi-objective algorithm or set innovation for the existing algorithms in this issue.

The choice of plans after scheduling is also worth studying. Since the solution obtained is a set of pareto optimal solution sets, the corresponding choices will be different for different processing purposes. Based on the right shift procedure to reduce energy consumption, manufacturers considered that the difference in processing efficiency and energy consumption will also lead to differences in program selection.

As shown in

Table 4. Comparison of optimal scheduling schemes.

	Makespan (h)	TEC (Yuan)	Total Carbon Emission (tCO2)
Efficient	23.1	7498.6	4.529
Economical	24.2	5335.9	3.222
Trade-off	23.6	6173.6	3.729

, the shortest completion time is 23.1 h, while the energy consumption cost is 7498.6 yuan. Although different scheduling schemes had different emphasis, they can consider the coordination of the two scheduling objectives of time, cost, and carbon emissions to some extent. In the actual production process, companies can choose different ones according to actual needs. Without special preferences, a plan is proposed to make economic benefit and ecological efficiency considering the relevant weights, which the completion time being 23.6h and the cost of energy consumption being 6173.6 yuan. As the emission factor of electricity is 0.604 tCO2/MWh.

4.2. Random Instances

The computational findings for evaluating the modified NSGA-II method for minimizing makespan and total energy consumption in the HFSP issue are described in this part. Comparing their performance with the well-known NSGA-II algorithm indicates where their performance stands. All the algorithms are coded in n Visual Studio2017 on a PC with a 2.8 GHz Intel core i5-7200 processor and 8 GB RAM in Windows 10 system.

We need to consider HFSP with various machines and job numbers to evaluate the provided approaches under different problem sizes. Therefore, we generate a set of instances of various sizes that resemble the industrial data.

Table 5. Summary of test data.

Factors	Values	Number of Levels
Number of jobs	10, 20, 50	3
Number of stages	3, 5, 8	3
Number of parallel machines	2, 4	2
Processing time (min)	Discrete uniform [10, 50]	1
Power of machine (kW)	Discrete uniform [5, 10]	1
Standby power of machine (kW)	1 (fixed)	1

identifies and lists the variables that impact the manufacturing system’s performance.

This section describes the experiments that have been conducted to compare the results of the improved NSGA-II (I-NSGA-II) and traditional NSGA-II. In this study, experiment case is identified as the form of “Number of jobs-Number of machines at each stage-Number of stages”. For example, “20-5-4” means a case of 20 jobs, 5 stages and 4 parallel machines in each stage.

To test the efficiency of the I-NSGA-II algorithm in the environments with different complexity, the test set of instances is classified into the three categories according to the number of jobs (10, 30, and 50). Each category has different stages and parallel machines. Identical factors were generated randomly for each test problem. Genetic factors are important parameters that affect the results.

It should be mentioned that algorithm parameters affect the performance of the compared algorithms. Pilot experiments were conducted under a set of potential parameter values to find the best combinations. To establish fair comparisons, as a standard parameter, the maximum population size maxPop, is selected as 200 to represent the standard parameter in all three algorithms.

In addition, the performance of NSGA-II is affected by the termination condition, i.e., the number of iterations or the maximum execution time (tmax). These settings take different values in different experiments. In addition, the population size was dynamically determined according to the scale of problem. The mutation probability and the crossover rate were set to be 0.05 and 0.95, respectively.

We ran the algorithm ten times, taking into account randomness for each instance, and recorded the minimum (Best) and average values (Avg) of performance metrics in

Table 6. Comparisons of algorithmic BEST performance.

n	Instance	NSGA-II		I-NSGA-II
Factors		Makespan (h)	TEC (kJ)	Makespan (h)	TEC (kJ)	dev (TEC)
10	10-3-2	3.02	113.949	2.90	97.144	14.75%
	10-5-2	5.01	249.721	4.87	215.836	13.57%
	10-8-2	6.28	385.581	6.28	385.581	10.98%
	10-3-4	2.12	104.948	2.05	66.321	36.81%
	10-5-4	3.51	234.965	3.51	149.124	36.53%
	10-8-4	4.98	381.096	4.98	260.402	31.67%
20	20-3-2	6.16	285.215	6.1	263.268	7.69%
	20-5-2	7.08	458.979	7.08	434.537	5.33%
	20-8-2	7.13	469.815	7.13	441.721	5.98%
	20-3-4	3.56	290.724	3.56	229.164	21.01%
	20-5-4	4.70	496.805	4.55	415.557	16.19%
	20-8-4	5.93	800.921	5.93	683.832	15.46%
50	50-3-2	13.57	403.096	13.57	389.828	3.29%
	50-5-2	14.51	674.565	14.50	634.58	5.93%
	50-8-2	15.73	1007.18	15.73	974.164	3.28%
	50-3-4	7.30	646.214	7.22	621.997	3.75%
	50-5-4	8.18	976.182	8.18	1115.527	3.05%
	50-8-4	9.58	1391.56	5.58	1351.92	2.85%

and

Table 7. Comparisons of algorithmic AVG performance.

n	Instance	NSGA-II		I-NSGA-II
Factors		Makespan (h)	TEC (kJ)	Makespan (h)	TEC (kJ)	dev (TEC)
10	10-3-2	3.02	120.340	3.04	109.641	8.89%
	10-5-2	5.01	258.162	5.01	230.943	10.54%
	10-8-2	6.62	400.963	6.62	362.007	9.72%
	10-3-4	2.12	104.948	2.13	77.275	28.16%
	10-5-4	3.81	244.529	3.64	165.203	32.44%
	10-8-4	5.12	388.937	5.12	274.012	29.55%
20	20-3-2	6.42	313.324	6.41	296.597	5.34%
	20-5-2	7.52	502.665	7.52	479.378	4.63%
	20-8-2	8.96	712.917	8.96	686.943	3.64%
	20-3-4	3.83	320.724	3.83	273.377	14.76%
	20-5-4	4.70	504.441	4.68	434.973	13.77%
	20-8-4	6.10	826.370	6.10	708.296	14.29%
50	50-3-2	14.01	440.8635	14.03	431.158	2.20%
	50-5-2	15.07	743.992	15.07	728.030	2.14%
	50-8-2	16.62	1133.487	16.62	1104.056	2.60%
	50-3-4	7.70	729.198	7.71	696.833	4.43%
	50-5-4	8.71	1152.805	8.71	1115.527	3.23%
	50-8-4	10.13	1674.597	10.13	1613.22	3.66%

. The overall comparison results of HPSO and NSGA-II on the benchmark instances. The deviation between the I-NSGA-II solutions and the NSGA-II solutions is output, which is defined as follows.

$$\mathrm{dev}\left(\mathrm{TEC}\right)=\frac{\mathrm{TEC}\left(\mathrm{NSGA}-\mathrm{II}\right)-\mathrm{TEC}\left(\mathrm{I}-\mathrm{NSGA}-\mathrm{II}\right)}{\mathrm{TEC}\left(\mathrm{NSGA}-\mathrm{II}\right)}$$

(16)

where TEC(NSGA-II) denotes the TEC objective value of the solutions obtained by algorithm NSGA-II.

It can be seen from the Table 6 and Table 7, if the processing time is too short without beyond a price interval, there will be no impact on energy consumption under TOU and ladder electricity price. In the case of electricity prices in multiple intervals, the increase of parallel machines reducing the processing time while increasing the energy consumption. However, the existence of parallel machines makes the gap between workpiece processing possible, thereby increasing the space for the right-shift movement.

Comparing the best solutions of the two algorithms, it can be found that the two algorithms both have their own advantages. For small-scale and medium-scale cases, the two target values of the best solutions obtained by the I-NSGA-II are both smaller than or equal to the corresponding target value of the traditional NSGA-II. It can be seen from Table 7 that the average solution of the I-NSGA-II is better than the average solution of the traditional algorithms.

Figure 9, Figure 10 and Figure 11 show the Pareto diagram of each instance. Through the graph, we can see the comparison results directly.

Figure 9 represent the approximate Pareto fronts found by I-NSGA-II and NSGA-II for 10 jobs. It can be seen from the figure that, on average, the Pareto solutions obtained by I-NSGA-II are better than those obtained by NSGA-II. For small-scale calculation examples where there is right-shift operation space, TEC can reach a significant decline, such as 10-3-4,10-5-4, and 10-8-4. Figure 10 shows the Pareto fronts found by the methods for 20-job instances. The results are similar with the small instances. For medium-scale calculation examples, the degree of improvement is reduced.

Figure 11 presents the Pareto fronts found by I-NSGA-II and NSGA-II for large instances with 50 jobs. In general, it may be concluded from the figures that the improved algorithm does not achieve ideal results, which shows that there are certain limitations. As the number of workpieces increases while machines used for processing have not increased, the space for right movement operations is reduced. Nevertheless, energy consumption can still be improved.

5. Conclusions and Future Work

From the perspective of consumption reduction and sustainable development, a HFSP which aiming to minimize the TEC without altering the completion time under TOU and ladder electricity tariffs is derived. This problem is addressed by an improved NSGA-II that integrates the low-carbon method with existing algorithms. First, the NEH heuristic algorithm is used to improve the quality of initial solution in the initialization and an adaptive genetic operation is introduced aiming to avoid the emergence of locally optimal solutions. Then the right-shift strategy is adapted to reduce TEC without affecting completion time. The result of the two algorithms has been compared by a real example which shows that total electricity consumption and carbon emission are reduced after adopting the improved algorithm. We propose some suggestions for the trade-off between maximum completion time and total electricity consumption. By maintaining the production efficiency and reducing the energy consumption cost by 4.33%, a total of 279.48 yuan is saved. The conclusions show that the right shift approach effectively processes workpiece allocation without changing the maximum completion time. Moving high-energy-consuming workpieces into the low-electricity price range can reduce costs and energy consumption. The electricity emission factor would also reduce carbon emissions to some extent. In this sense, this optimization method is also conducive to green production. Therefore, the electricity load has shifted significantly under the TOU electricity price, achieving the goal of “shaving peaks and filling valleys”, effectively reducing energy consumption costs and carbon emissions.

As a result of the coordination of the two scheduling objectives of time, cost, and carbon emissions, we construct a plan to balance the economic benefit and the ecological efficiency, taking into account the relevant weights.

In the future, the research method should be further improved, and the accuracy and calculation efficiency of the algorithm should be further improved. Besides, we will consider the relationship between energy consumption and carbon emissions. The environmental impact of renewable energy in the production process also needs to be concerned. Furthermore, the carbon footprint on cycle life is also worthy of attention.