The Green Flexible Job-Shop Scheduling Problem Considering Cost, Carbon Emissions, and Customer Satisfaction under Time-of-Use Electricity Pricing

: Exploration of the green flexible job-shop scheduling problem is essential for enterprises aiming for sustainable practices, including energy conservation, emissions reduction, and enhanced economic and social benefits. While existing research has predominantly focused on carbon emissions or energy consumption as green scheduling objectives, this paper addresses the broader scope by incorporating the impact of variable energy prices on energy cost. Through the introduction of an energy cost model based on time-of-use electricity pricing, the study formulates a multi-objective optimization model for green flexible job-shop scheduling. The objectives include minimizing cost, reducing carbon emissions, and maximizing customer satisfaction. To prevent premature convergence and maintain population diversity, an enhanced genetic algorithm is employed for solving. The validation of the algorithm’s effectiveness is demonstrated through specific examples, providing decision results for optimal scheduling under various weight combinations. The research outcomes hold substantial practical value as they can significantly reduce energy expenses, lower carbon emissions, and elevate customer satisfaction while safeguarding production efficiency. This contributes to enhancing the market competitiveness and green brand image of businesses.


Introduction
The flexible job-shop scheduling problem (FJSP) is essentially about efficiently managing a sequence of jobs using a specific number of machines [1].Each job comprises multiple operations, and each operation can be executed on several machines [2].The primary optimization goals in traditional FJSP are time-related, encompassing metrics like maximum completion time [3], maximum tardiness [4], total tardiness [5], mean tardiness [6], maximum lateness [7], total idle time [8], total flow time [4], maximum workload [9], total workload [10], and jobs' number-related aspects, such as the number of late jobs [11].Previous research has consistently underscored production efficiency in the traditional FJSP.
Improving the energy efficiency of products is a basic condition for the healthy development of economic activity and for sustainable development [12].Consequently, the green flexible job-shop scheduling problem (GFJSP) has garnered widespread attention [13].Researchers like Lei et al. [14] emphasize minimizing total energy consumption while considering workload balance, reflecting a strong emphasis on goals related to energy and analyzing conflicts between these objectives.Nouiri et al. [15] proposed a rescheduling approach to minimize make-span and reduce overall energy consumption.Considering the uncertainty of job processing times, Wu et al. [16] studied FJSP considering deterioration effects and energy consumption simultaneously.Zhang et al. [17] presented a model for low-carbon scheduling of the flexible job shop, considering make-span, machine workload, and carbon emission factors.Moreover, Zhu et al. [18] developed a scheduling model to minimize both make-span and carbon emissions by considering the impact of workers' learning abilities on job processing times and energy consumption.
Rising energy costs play a substantial role in the overall increase in manufacturing production cost.Moreover, small businesses tend to adopt innovations that promote sustainable development, with a specific focus on frugal innovations prioritizing cost reduction [19].Consequently, streamlining the energy cost of production systems can enhance overall cost-effectiveness, aligning with broader sustainability objectives.Accounting for changes in energy prices throughout the day, the energy consumption cost of a single machine was optimized [20,21].Zhang et al. [22] considered the time-indexed integer flow shop scheduling problem, aiming to minimize both the total electrical cost and the carbon footprint.Che et al. [23] proposed a continuous-time mixed-integer linear programming model to allocate jobs to different time slots with the aim of minimizing total electricity cost.Geng et al. [24] introduced an improved multi-objective ant-lion optimization algorithm to minimize make-span and energy consumption costs in hybrid flow shop scheduling.Schulz et al. [25] considered a hybrid flow shop scheduling problem, aiming to minimize both energy cost and total tardiness.Chen et al. [26] developed an improved non-dominated sorting genetic algorithm II for the hybrid flow shop scheduling problem under time-of-use.Wang et al. [27] addressed the two-machine permutation flow shop scheduling problem with the objective of minimizing the total electrical cost under time-of-use electricity pricing.
As shown in Table 1, existing studies predominantly optimize energy consumption or carbon emissions alongside traditional production metrics in GFJSPs.Limited attention is given to energy consumption cost in GFJSPs, with studies mostly focused on simple systems such as single machines or flow shops.Consequently, further research is essential to address this issue in complex GFJSPs.Customer importance is a factor that reflects the priority of different customers in a FJSP.Considering customer importance can help to improve customer satisfaction, as well as to increase competitiveness [28].However, very few existing studies have considered customer importance and order delivery time in scheduling to maximize customer satisfaction, and only Shi et al. minimized customer dissatisfaction using fuzzy delivery time [29], and Álvarez-Gil et al.Higher-priority customer requests can be prioritized in scheduling while balancing the workload of resources to avoid saturated or underutilized machines [30].Therefore, this paper proposes a multiobjective scheduling optimization model for GFJSP, aiming to minimize costs (including energy consumption cost and order delay cost) under a time-of-use electricity price, while minimizing carbon emissions and maximizing customer satisfaction.The introduction of order delay cost comprehensively considers the production process, ensuring a holistic optimization of production scheduling for increased profitability, taking into account both energy consumption cost and order delay cost due to product delays during periods of low electricity prices.
Many different approaches have been applied to solve the problem due to the difficult nature of FJSP.Some of the very recent approaches include biogeography-based optimization [31], the firefly algorithm [32], heuristics [33], invasive weed optimization [34], and differential evolution [35].However, the genetic algorithm (GA) remains the most frequently used algorithm for the FJSP [36] and has proven to be one of the most effective evolutionary techniques for solving FJSPs [37].Hence, considering the multi-objective nature of the GFJSP constructed in this paper, an improved genetic algorithm (IGA) has been designed to solve the model, enabling better scheduling decisions compared to empirical solutions under different weight combinations.The description of the flexible job-shop scheduling problem is as follows: There are n types of workpieces can be processed on multiple different machines, and the machining time varies with the machine.Under the time-of-use electricity price policy, a period is divided into t periods T = {T 1 , T 2 , . . . ,T t }, and the electricity price for each period T j ∈ T is e j , as shown in Figure 1.
GFJSP Under Time-of-Use (in this paper) YES YES YES YES Many different approaches have been applied to solve the problem due to the difficult nature of FJSP.Some of the very recent approaches include biogeography-based optimization [31], the firefly algorithm [32], heuristics [33], invasive weed optimization [34], and differential evolution [35].However, the genetic algorithm (GA) remains the most frequently used algorithm for the FJSP [36] and has proven to be one of the most effective evolutionary techniques for solving FJSPs [37].Hence, considering the multi-objective nature of the GFJSP constructed in this paper, an improved genetic algorithm (IGA) has been designed to solve the model, enabling better scheduling decisions compared to empirical solutions under different weight combinations.If different machines are selected for the operation, the required machining time and energy consumption will vary, leading to different carbon emissions.Additionally, selecting different periods results in varying electricity prices and, consequently, different energy costs.Constrained by the machine capacity and the order delivery date, although the energy cost can be reduced by processing during off-peak periods, the order delay cost If different machines are selected for the operation, the required machining time and energy consumption will vary, leading to different carbon emissions.Additionally, selecting different periods results in varying electricity prices and, consequently, different energy costs.Constrained by the machine capacity and the order delivery date, although the energy cost can be reduced by processing during off-peak periods, the order delay cost will also increase.Furthermore, when dealing with significant customers, failing to prioritize meeting their supply demands may have negative repercussions on future orders.

Modeling the Multi-Objective GFJSP Model
We made the following assumptions: (a) each workpiece is machined independently of each other, and the workpieces have no priority; (b) each workpiece can be machined at time t = 0, and each machine is available at time t = 0; (c) there are several machines available for each operation, but each operation can be machined by only one machine; (d) each workpiece must be machined in a specific sequence of machining operations, and the next operation can be machined only after the completion of the previous operation; (e) each machine can machine only one operation at a time at the same moment; (f) the die change time is shorter than the machining time and is not taken into account.

Symbol Explanation
Actual delivery time (unit: days) x ∈ {1,         The importance level of customer H c has a range from 0 to 1; the larger the value, the higher the importance level.(1) Within the machining process, machines typically operate in two primary states: machining and idle.Despite high-energy-consumption machines exhibiting significant power requirements during start-up and shutdown, these periods are notably brief.Thus, the study omits the consideration of energy consumption related to switching on and off, focusing instead on incorporating idle energy consumption.In summary, the overall energy consumption encompasses both machining and idle energy consumption.It is essential to note that energy consumption varies depending on the specific machining operation being performed on the same machine.Furthermore, variations arise when the same operation is processed on different machines.In light of the time-of-use electricity price, even if energy consumption and the machine remain constant, the energy cost will differ based on variations in the machining period.
To summarize, the energy cost can be expressed as the sum of two components: the machining energy cost and the idle energy cost.Therefore, the energy cost model f ec (c) can be formulated as: Constrained by the machine capacity and the order delivery date, the pursuit of minimizing the energy cost may lead to delayed delivery of the order, thereby increasing the order delay cost.It can be expressed as the product of the number of delayed workpieces late yc xk , the delay time x − y, and the unit delay cost p ′ kc (unit: CNY/workpiece).Therefore, the order delay cost f dc (c) can be formulated as: Therefore, to maximize profits, the sum of the energy and order delay cost should be minimized rather than simply reducing one of them.In summary, the cost model f (c) can be expressed as: (2) Moreover, carbon emissions constitute a critical metric for the GFJSP.Carbon emissions can be assessed by employing the carbon emission coefficient of electricity ∂ (kg CO 2 /kWh).It can be made available through the local government or energy agency.Therefore, the carbon emissions model f (e) can be expressed as: (3) Regarding a specific order for a particular customer, if no delay has occurred, the customer satisfaction is 1.Otherwise, customer satisfaction can be expressed as the product of "1 − do ykc ", "1 − q c ", and "1 − dt ykc ".Therefore, the customer satisfaction model can be formulated as: In summary, the objectives for the GFJSP model under the time-of-use electricity price are as follows: where Equations ( 6)-( 8) represent minimizing the cost and the carbon emissions and maximizing customer satisfaction, respectively.Thus, the multi-objective GFJSP model of GFJSP under the time-of-use electricity price can be expressed as Equation (9).w 1 , w 2 , and w 3 represent the weight coefficient, and f ′ (c), f ′ (e), and f ′ (q) are the normalized objective functions.
The constraints of the multi-objective GFJSP model of a GFJSP under the time-of-use electricity price are as follows: ) ) Equation (10) represents the intermediate variable g xkc , which denotes the sum of all orders o ykc submitted on the day x.If the workpiece J k for a customer H c with a delivery period y is supplied on the day x, x ′yk xc is set to 1, and thus the supply quantity is the sum of o ykc .Equation (11) represents the intermediate variable wip jk xs , which can be expressed as the residual of semi-finished workpieces from the previous time slot, plus the workpiece in the current time slot, minus the quantity produced in the current time slot for the next operation.Equation (12) indicates that when calculating the final workpiece of a workpiece's last operation, the workpiece of the next operation in this time slot is not considered.Equation (13) states that the workpiece quantity of the next operation cannot exceed the quantity of semi-finished workpieces from the previous operation.Equation (14) specifies that at the beginning of the first day, all semi-finished workpiece quantities are zero.Equation (15) ensures that starting from the second day, at the beginning of the workpiece, all semifinished workpiece quantities are the carryover from the previous day's output.Workpieces completed in the last operation do not carry over.Equation (16) describes the finished workpieces product xk at the end of each day, with the semi-finished workpieces after the completion of the last operation being treated as finished workpieces.Equation ( 17) denotes the inventory of finished workpieces stock xk , which is the sum of the inventory from the previous day plus today's finished workpieces minus the quantity allocated to customers today.Equation (18) specifies that the inventory is zero at the beginning.Equation ( 19) ensures that supply is not possible when the workpiece is insufficient.Equation (20) represents the intermediate variable late yc xk .When x ′yk xc = 0, indicating no orders were provided on day x with a delivery period of y, then late yc xk = 0.When x ′yk xc = 1 and x = y, indicating that orders provided on the day x have a delivery period of x, and thus no delay occurred, late yc xk = 0. Otherwise, the quantity of delayed orders is o ykc .Equation (21) ensures that the order cannot be delivered before the delivery date.Equation (22) stipulates that all orders must be delivered.Equation (23) states that one machine M i at a time T j can only produce one type of workpiece.Equation (24) indicates that each batch of orders can only be submitted once.

Model Solving Based on an Improved Genetic Algorithm
To address the characteristics of the multi-objective GFJSP model, we propose an improved genetic algorithm.It utilizes a four-layer encoding scheme, incorporating the operation code, the machine code, the time code, and the allocation code for chromosome encoding.The initialization strategy combines global search and local search, utilizing a combination of dominant and recessive individuals selected through the fitness function.Dominant individuals are added to the dominant set, and a new population is formed by combining dominant individuals with a population selected through roulette wheel selection.This ensures the preservation of dominant individuals and enhances the quality of the new population.In the crossover and mutation stages, operations are performed on the operation code, machine code, time code, and allocation code separately, promoting diversity in the offspring population and preventing the generation of undesirable genes.Finally, the optimal individual is obtained by selecting from the dominant set.The specific flow of the improved genetic algorithm is shown in Figure 2.
The flow of the improved genetic algorithm.

Chromosome Encoding
Currently, for the FJSP, a common approach involves a two-layer encoding method, comprising the machine assignment code and operation assignment code.However, this paper focuses on scenarios involving large-scale scheduling and order-based delivery.Recognizing the inadequacy of two-layer encoding in meeting the requirements of the addressed problem, a four-layer encoding method is introduced, encompassing machine

Chromosome Encoding
Currently, for the FJSP, a common approach involves a two-layer encoding method, comprising the machine assignment code and operation assignment code.However, this paper focuses on scenarios involving large-scale scheduling and order-based delivery.Recognizing the inadequacy of two-layer encoding in meeting the requirements of the addressed problem, a four-layer encoding method is introduced, encompassing machine assignment code, operation assignment code, time assignment code, and order assignment code.The genes of the operation assignment code correspond to operation numbers.Genes in the machine assignment code are linked to gene positions in the operation assignment code, representing machine numbers and indicating the machine selected for the operation at the respective position in the operation assignment code.To enhance the utilization of time-based pricing policies in large-scale scheduling, a modeling approach is introduced where machines batch-process one operation per hour, and scheduling is conducted based on these hourly batches.Consequently, time assignment codes are introduced to determine the quantity of batches (1 h per batch) for assigning each operation to each machine.Genes within the order assignment code represent distinct order numbers for different customers.The order number appearing in the leftmost position signifies that the order that should be fulfilled first for the respective customer, serving their needs promptly.As an example of a specific scheme, the encoding rules are illustrated in Figure 3.In this example, the order assignment code indicates the prioritization of the fulfillment of Customer 1's Order 1, and the operation assignment code, machine assignment code, and time assignment code signify that the machine M 2 processes two batches of the operation O 21 (2 h) first, and so on.

Initial Population
There are two common ways to initialize populations: a global search based on feasible solutions and a local search that narrows down the feasible solution space based on the actual situation.The advantage of a global search is that it can explore all feasible solutions, ensuring genetic diversity.However, it has the disadvantage of a large feasible solution base, making the search challenging and resulting in lower population quality.A local search, on the other hand, improves search efficiency by narrowing the search range and generally leads to higher-quality feasible solutions within the narrowed range.However, it lacks genetic diversity and may lead to local optima.Thus, this study employs a combined approach of global and local searches for population initialization to ensure both the quality of the initial population and genetic diversity.
The local search strategy includes: (1) Order Assignment Code: (a) prioritize the allocation of orders with earlier delivery deadlines and (b) prioritize the allocation of orders from more important customers.
(2) Operation Assignment Code: Initially, arrange the first operation of each workpiece into a column and shuffle them randomly.Allocate the first n bits (n kinds of workpieces) to the operation code.Shuffle the second operation of each workpiece and continue the allocation to the operation assignment code in the same order.Repeat this process until all operations have been allocated.When the allocation reaches the maximum length of the operation assignment code, start the allocation again from the first operation of each workpiece.
(3) Machine Assignment Code: (a) prioritize selecting machines with lower energy

Initial Population
There are two common ways to initialize populations: a global search based on feasible solutions and a local search that narrows down the feasible solution space based on the actual situation.The advantage of a global search is that it can explore all feasible solutions, ensuring genetic diversity.However, it has the disadvantage of a large feasible solution base, making the search challenging and resulting in lower population quality.A local search, on the other hand, improves search efficiency by narrowing the search range and generally leads to higher-quality feasible solutions within the narrowed range.However, it lacks genetic diversity and may lead to local optima.Thus, this study employs a combined approach of global and local searches for population initialization to ensure both the quality of the initial population and genetic diversity.
The local search strategy includes: (1) Order Assignment Code: (a) prioritize the allocation of orders with earlier delivery deadlines and (b) prioritize the allocation of orders from more important customers.
(2) Operation Assignment Code: Initially, arrange the first operation of each workpiece into a column and shuffle them randomly.Allocate the first n bits (n kinds of workpieces) to the operation code.Shuffle the second operation of each workpiece and continue the allocation to the operation assignment code in the same order.Repeat this process until all operations have been allocated.When the allocation reaches the maximum length of the operation assignment code, start the allocation again from the first operation of each workpiece.
(3) Machine Assignment Code: (a) prioritize selecting machines with lower energy consumption for each operation and (b) prioritize selecting machines with a higher production capacity for each operation.
This paper defines the local search strategies formed by the (a) strategy for the allocation code, the strategy for the operation code, and the (a) strategy for the machine code as Strategy A; the strategies formed by the (b) strategy for the allocation code, the strategy for the operation code, and the (a) strategy for the machine code as Strategy B; the strategies formed by the (a) strategy for the allocation code, the strategy for the operation code, and the (b) strategy for the machine code as Strategy C; and the strategies formed by the (b) strategy for the allocation code, the strategy for the operation code, and the (b) strategy for the machine code as Strategy D. In the initial population, individuals generated using Strategies A, B, C, and D each account for 10% of the total population, while the global search strategy accounts for 60%.

Fitness Calculation
The objectives of the GFJSP model are to minimize cost, and carbon emissions, and maximize customer satisfaction.After normalizing each objective, we obtain f ′ (c), f ′ (e), and f ′ (q), each assigned a weight, w 1 , w 2 , and w 3 , respectively.Thus, we define as the fitness function.

Selection of Operations
This paper utilizes the roulette wheel method to select individuals for the new population.The optimal individual is also selected and added to the new population, ensuring that the population, after selection, contains dominant individuals.

Crossover Operation
Crossover includes the operations on operation assignment code, machine assignment code, time assignment code, and order assignment code.
(1) Operation Assignment Code Crossover: Step 1: Randomly select two chromosomes as parents, denoted Parent 1 and Parent 2 , based on the crossover probability.Randomly choose two crossover gene positions before the smaller gene boundary point of the two parents.
Step 2: Exchange the genes between the two crossover gene positions of the parents to form offspring chromosomes Child 1 and Child 2 .
Step 3: Gene exchange will change the occurrence of each gene in the chromosome, disrupting the maximization principle followed during encoding.Therefore, perform a gene check on the offspring chromosomes after gene exchange.As shown in step 3 of Figure 2, Child 1 has an additional "12" gene and a "22" gene and is missing one "21" gene and one "11" gene.
Step 4: Based on the results detected in step 3, replace one "12" gene outside the exchange region of Child 1 with a "21" gene.Similarly, complete all gene replacements.The crossover steps are shown in Figure 4.
(2) Machine Assignment Code Crossover: If the Machine Assignment code crossover does not encounter gene conflict issues, only steps 1 and 2 of the operation assignment code crossover need to be completed.Since the gene positions of the machine assignment code correspond to those of the operation assignment code, the two crossover gene positions chosen randomly during the operation assignment code crossover will be the same for the machine assignment code crossover.
(3) Time Assignment Code Crossover: The time assignment code crossover is the same as the machine assignment code crossover.
(4) Order Assignment Code Crossover: Since each gene in the order assignment code is unique, gene checking is required during crossover, following the same steps as the operation assignment code crossover.However, since the gene positions of the order assignment code are not relative to the gene positions of the operation assignment code, two new crossover gene positions need to be randomly selected.Additionally, since the genes in the order assignment code represent customer orders, and all orders from customers need to be satisfied, there is no gene boundary point.When selecting crossover gene positions, the gene positions can be randomly chosen anywhere within the entire chromosome.
ment code, time assignment code, and order assignment code.
(1) Operation Assignment Code Crossover: Step 1: Randomly select two chromosomes as parents, denoted Parent1 and Parent2, based on the crossover probability.Randomly choose two crossover gene positions before the smaller gene boundary point of the two parents.
Step 2: Exchange the genes between the two crossover gene positions of the parents to form offspring chromosomes Child1 and Child2.
Step 3: Gene exchange will change the occurrence of each gene in the chromosome, disrupting the maximization principle followed during encoding.Therefore, perform a gene check on the offspring chromosomes after gene exchange.As shown in step 3 of Figure 2, Child1 has an additional "12" gene and a "22" gene and is missing one "21" gene and one "11" gene.
Step 4: Based on the results detected in step 3, replace one "12" gene outside the exchange region of Child1 with a "21" gene.Similarly, complete all gene replacements.The crossover steps are shown in Figure 4. 31  Step 1： Step 2： Step 3：

Child2
Step 4： (2) Machine Assignment Code Crossover: If the Machine Assignment code crossover does not encounter gene conflict issues, only steps 1 and 2 of the operation assignment code crossover need to be completed.Since the gene positions of the machine assignment code correspond to those of the operation assignment code, the two crossover gene positions chosen randomly during the operation assignment code crossover will be the same for the machine assignment code crossover.
(3) Time Assignment Code Crossover: The time assignment code crossover is the same as the machine assignment code crossover.
(4) Order Assignment Code Crossover: Since each gene in the order assignment code is unique, gene checking is required during crossover, following the same steps as the operation assignment code crossover.However, since the gene positions of the order

Mutation Operation
Mutation includes operation assignment code mutation, machine assignment code mutation, time assignment code mutation, and order assignment code mutation.The processes for operation assignment code mutation, time assignment code mutation, and machine assignment code mutation are the same.Based on the mutation probability, a chromosome is randomly selected, and two crossover gene positions are randomly chosen before the gene boundary.The genes at these two positions are then exchanged.The difference in order assignment code mutation lies in the fact that two mutation gene positions need to be reselected, and there is no restriction on gene boundary points for the selection of mutation positions.

Description of the Flexible Job-Shop
This paper conducted application validation in the stamping workshop of a prominent home appliance manufacturing enterprise in Shandong Province, China.One kilowatt-hour of electricity saved is equivalent to reducing 0.997 kg of carbon dioxide emissions in this area [38], that is, ∂ = 0.997.The chosen stamping workshop is equipped with six machines, tasked with processing six distinct types of stamped parts.These parts are supplied to five different customers.Additionally, each type of stamped part has varying batch sizes, and customer demands exhibit diversity.Furthermore, for the same type of stamped part, the identical operation is processed in a non-continuous manner.Consequently, the scheduling problem is intricate.
The stamping workshop operates for eight hours a day (8:00-12:00 and 14:00-18:00).Dispatchers formulate dispatch plans on a unit-time basis, signifying that each machine undergoes the stamping process for the same type of part within one hour.Detailed data for the stamped parts and machines are presented in Table 3, and Table 4 provides information about the orders involved in scheduling, including the types and quantities of stamped parts ordered by a customer, delivery deadlines, compensation required in case of delays, and the importance of the customer.Table 5 displays the idle energy consumption for each machine per unit of time.According to the "Notice on Relevant Matters Concerning the Time-of-Use Electricity Pricing Policy for Industrial and Commercial Enterprises in Shandong Province" and other related documents issued by the Development and Reform Commission of Shandong Province, the industrial electricity consumption in Shandong Province follows a time-of-use electricity pricing policy.The period from 8:00 A.M. to 6:00 P.M. during the autumn season (September to November) is divided into different time-of-use periods, including Standard Periods (08:00-10:00 and 15:00-16:00), Off-Peak Periods (10:00-11:00 and 14:00-15:00), the Deep Off-Peak Period (11:00-14:00), the Peak Period (16:00-17:00), and the High Peak Period (17:00-18:00).Based on the "State Grid Shandong Electric Power Company's Agency Power Purchase Price in November 2023", the time-of-use electricity pricing function during working hours can be expressed as follows (unit: CNY/(kW•h)):

Single-Objective Solution
In this study, the case presented in Section 3.1 is used as the experimental subject, and IGA is employed to solve single-objective problems.The weight ratios assigned to cost, carbon emissions, and customer satisfaction are set as [1, 0, 0], [0, 1, 0], and [0, 0, 1], respectively, indicating that each solving instance only considers a single objective.The results of IGA in solving single-objective problems are illustrated in Figure 5.

Single-Objective Solution
In this study, the case presented in Section 3.1 is used as the experimental subject, and IGA is employed to solve single-objective problems.The weight ratios assigned to cost, carbon emissions, and customer satisfaction are set as [1, 0, 0], [0, 1, 0], and [0, 0, 1], respectively, indicating that each solving instance only considers a single objective.The results of IGA in solving single-objective problems are illustrated in Figure 5.In Figure 6a, we can clearly observe the cost situation under [1, 0, 0], [0, 1, 0], and [0, 0, 1].The results show that scheduling with cost as the primary objective significantly reduces cost compared to scheduling with carbon emissions and customer satisfaction as objectives.This highlights that when using IGA for scheduling optimization, prioritizing cost as the main goal is effective in lowering the cost.Figure 6b illustrates the carbon emissions under [1, 0, 0], [0, 1, 0], and [0, 0, 1].Noticeably, scheduling with carbon emissions as the objective demonstrates lower carbon emissions relative to schedules with cost and customer satisfaction as objectives.This suggests that employing IGA for optimization can significantly reduce carbon emissions, contributing to environmentally friendly operations.Figure 6c displays customer satisfaction under [1, 0, 0], [0, 1, 0], and [0, 0, 1].It is noteworthy that scheduling with customer satisfaction as the objective excels in customer In Figure 6a, we can clearly observe the cost situation under [1, 0, 0], [0, 1, 0], and [0, 0, 1].The results show that scheduling with cost as the primary objective significantly reduces cost compared to scheduling with carbon emissions and customer satisfaction as objectives.This highlights that when using IGA for scheduling optimization, prioritizing cost as the main goal is effective in lowering the cost.Figure 6b illustrates the carbon emissions under [1, 0, 0], [0, 1, 0], and [0, 0, 1].Noticeably, scheduling with car-bon emissions as the objective demonstrates lower carbon emissions relative to schedules with cost and customer satisfaction as objectives.This suggests that employing IGA for optimization can significantly reduce carbon emissions, contributing to environmentally friendly operations.Figure 6c displays customer satisfaction under [1, 0, 0], [0, 1, 0], and [0, 0, 1].It is noteworthy that scheduling with customer satisfaction as the objective excels in customer satisfaction, significantly surpassing schedules with cost and carbon emissions as objectives.This emphasizes that optimizing scheduling through IGA can effectively enhance customer satisfaction, thereby boosting the competitiveness and reputation of the enterprise.In summary, by balancing the objectives of cost, carbon emissions, and customer satisfaction, IGA demonstrates reliability in optimizing scheduling problems.This also validates the model's reliability, providing robust support for achieving comprehensive and efficient operational management.Perform 10 runs each for the traditional GA and the IGA separately.Evaluate the algorithms based on three indicators: the optimal solution, the worst solution, and the average deviation from 10 consecutive runs, as shown in Table 6.A comparison of the results obtained by the two intelligent algorithms reveals that the IGA outperforms the traditional GA in terms of the optimal solution, the worst solution, and the average deviation when solving three single-objective problems.Therefore, the IGA proposed in this paper demonstrates stronger optimization capabilities in solving single-objective problems.Perform 10 runs each for the traditional GA and the IGA separately.Evaluate the algorithms based on three indicators: the optimal solution, the worst solution, and the average deviation from 10 consecutive runs, as shown in Table 6.A comparison of the results obtained by the two intelligent algorithms reveals that the IGA outperforms the traditional GA in terms of the optimal solution, the worst solution, and the average deviation when solving three single-objective problems.Therefore, the IGA proposed in this paper demonstrates stronger optimization capabilities in solving single-objective problems.

Multi-Objective Optimization
Using the case in Section 3.1 as the experimental object, GA and IGA are, respectively, applied to solve multi-objective problems.Due to the different dimensions of the three objectives, cost, carbon emissions, and customer satisfaction, normalization is required.After assigning weights of 1/3, 1/3, and 1/3 to the three objectives, the optimal scheduling solution for multi-objectives can be obtained.The convergence curves of the two algorithms in solving multi-objective problems are shown in Figure 7.It can be observed that GA starts to find the optimal solution around the 248th generation, but the algorithm becomes stuck in a local optimum.IGA, on the other hand, obtains the optimal solution at the 134th generation.In terms of convergence speed and optimization effectiveness, the improved genetic algorithm demonstrates greater advantages in solving multi-objective problems.objectives, cost, carbon emissions, and customer satisfaction, normalization is required.
After assigning weights of 1/3, 1/3, and 1/3 to the three objectives, the optimal scheduling solution for multi-objectives can be obtained.The convergence curves of the two algorithms in solving multi-objective problems are shown in Figure 7.It can be observed that GA starts to find the optimal solution around the 248th generation, but the algorithm becomes stuck in a local optimum.IGA, on the other hand, obtains the optimal solution at the 134th generation.In terms of convergence speed and optimization effectiveness, the improved genetic algorithm demonstrates greater advantages in solving multi-objective problems.To further compare the solving effectiveness of the two algorithms, each algorithm is run 20 times, and the scheduling solutions from the 20 runs are retained.After normalization, the model's objective values are calculated, and the results are shown in Figure 8.The performance analysis of the algorithms is presented in Table 7. From Table 7, it is observed that the IGA, when solving multi-objective problems, outperforms the traditional GA in terms of the mean, optimal, and worst values of the overall objective.The IGA exhibits stronger solution diversity, leading to a slightly higher average deviation compared to the traditional GA.Considering all indicators comprehensively, the IGA demonstrates greater advantages in solving multi-objective problems.To further compare the solving effectiveness of the two algorithms, each algorithm is run 20 times, and the scheduling solutions from the 20 runs are retained.After normalization, the model's objective values are calculated, and the results are shown in Figure 8.The performance analysis of the algorithms is presented in Table 7. From Table 7, it is observed that the IGA, when solving multi-objective problems, outperforms the traditional GA in terms of the mean, optimal, and worst values of the overall objective.The IGA exhibits stronger solution diversity, leading to a slightly higher average deviation compared to the traditional GA.Considering all indicators comprehensively, the IGA demonstrates greater advantages in solving multi-objective problems.
The performance analysis of the algorithms is presented in Table 7. From Table 7, it is observed that the when solving multi-objective problems, outperforms the traditional GA in terms of the mean, optimal, and worst values of the overall objective.The IGA exhibits stronger solution diversity, leading to a slightly higher average deviation compared to the traditional GA.Considering all indicators comprehensively, the IGA demonstrates greater advantages in solving multi-objective problems.

Results and Discussions
The simplex grid method is an effective tool for multi-objective analysis.Its calculations are simple, allowing for a reasonable and uniform valuation of each weight.By defining the number of grids, it can meet the precision requirements for weights.In this study, a five-order simplex grid table with a weight precision of 0.2 was selected, as illustrated in Figure 9.

Results and Discussions
The simplex grid method is an effective tool for multi-objective analysis.Its calculations are simple, allowing for a reasonable and uniform valuation of each weight.By defining the number of grids, it can meet the precision requirements for weights.In this study, a five-order simplex grid table with a weight precision of 0.2 was selected, as illustrated in Figure 9.This study employs the improved genetic algorithm designed in Section 3 to solve the proposed model.The algorithm parameters are set as follows: the initial population size is 800, the crossover probability is 0.6, mutation probability is 0.02, and the maximum evolution iterations are set to 400.The improved algorithm is implemented using Python 3.7, and computations are performed on an Intel ® Core (TM) i5_6300HQ CPU @ 2.80 GHz with 8 GB of random-access memory, running on the Windows 10 operating system.The computational results are presented in Table 8.This study employs the improved genetic algorithm designed in Section 3 to solve the proposed model.The algorithm parameters are set as follows: the initial population size is 800, the crossover probability is 0.6, mutation probability is 0.02, and the maximum evolution iterations are set to 400.The improved algorithm is implemented using Python 3.7, and computations are performed on an Intel ® Core (TM) i5_6300HQ CPU @ 2.80 GHz with 8 GB of random-access memory, running on the Windows 10 operating system.The computational results are presented in Table 8.
For the scheduling results in Table 8, we observe a cost range of CNY 1700.94 to 7433.04, a carbon emission range of 1194.39 to 1815.34 kg, and a customer satisfaction range of 5.64 to 13.30.The cost varies by CNY 5732.10,carbon emissions differ by 600.94 kg, and satisfaction varies by 7.66.This indicates significant fluctuations in cost, carbon emissions, and customer satisfaction with changing weights.The article defines four scheduling modes, Profitoriented, Energy-saving, Customer-oriented, and Balanced, with weight configurations for minimizing cost, carbon emissions, and maximizing customer satisfaction of [1, 0, 0], [0, 1, 0], [0, 0, 1], and [1/3, 1/3, 1/3], respectively.Figure 10 illustrates the cost, carbon emissions, and customer satisfaction under different modes.
The balanced model emerges as a strategic compromise, aiming to harmonize the key objectives of cost minimization, carbon emission reduction, and customer satisfaction.This approach provides a nuanced and well-rounded solution compared to the singular objectives of the other three models.In terms of cost, the balanced model strikes a middle ground.While its cost is slightly higher than the profit-oriented model, it significantly outperforms the energy-saving model by saving approximately CNY 4863.92.This financial equilibrium positions the balanced model as a prudent choice for businesses seeking cost optimization without compromising excessively on other objectives.Carbon emissions present another pivotal dimension.The balanced model, with 1368.68 kg of emissions, achieves a delicate balance.It outperforms the energy-saving model by reducing emissions by 390.65 kg, showcasing a conscientious approach towards environmental sustainability.Simultaneously, it remains competitive when compared to the profit-oriented and customer-centric models, avoiding the excessive emissions seen in the profit-oriented model.Crucially, customer satisfaction under the balanced model experiences a remarkable improvement of 92.55% compared to the energy-saving model.This enhancement is substantial and positions the balanced model as an attractive choice for businesses aiming to enhance customer loyalty and repeat business without neglecting financial considerations and environmental responsibilities.When comparing the balanced model to the profit-oriented model, the slight increase in cost is justified by the notable improvements in both carbon emission reduction and customer satisfaction.The balanced model offers a superior alternative for businesses that seek a more holistic and sustainable approach to their operations.In contrast to the energy-saving model, the balanced model presents a more financially viable option while still making commendable strides in emission reduction and customer satisfaction.The balanced model, therefore, stands out as a pragmatic compromise, providing businesses with a versatile strategy that considers the triad of cost, environmental impact, and customer centricity.This strategic alignment positions the balanced model as a compelling choice for companies aiming to achieve comprehensive optimization across multiple objectives.Figure 11 shows the Gantt chart under this model, with the order priority allocation sequence H 3 J 2 (the customer H 3 's order for product J 2 ), and H 4 J 4 .Figure 12 illustrates the Gantt chart under this model, with the order priority allocation sequence and H 4 J 4 .Figure 13 illustrates the Gantt chart under this model, with the order priority allocation sequence and H 4 J 4 .Figure 14 illustrates the Gantt chart under this model, with the order priority allocation sequence and H 4 J 4 .For the scheduling results in Table 8, we observe a cost range of CNY 1700.94 to 7433.04, a carbon emission range of 1194.39 to 1815.34 kg, and a customer satisfaction range of 5.64 to 13.30.The cost varies by CNY 5732.10,carbon emissions differ by 600.94 kg, and satisfaction varies by 7.66.This indicates significant fluctuations in cost, carbon emissions, and customer satisfaction with changing weights.The article defines four scheduling modes, Profit-oriented, Energy-saving, Customer-oriented, and Balanced, with weight configurations for minimizing cost, carbon emissions, and maximizing customer satisfaction of [1, 0, 0], [0, 1, 0], [0, 0, 1], and [1/3, 1/3, 1/3], respectively.Figure 10 illustrates the cost, carbon emissions, and customer satisfaction under different modes.The balanced model emerges as a strategic compromise, aiming to harmonize the key objectives of cost minimization, carbon emission reduction, and customer satisfaction.This approach provides a nuanced and well-rounded solution compared to the singular objectives of the other three models.In terms of cost, the balanced model strikes a middle ground.While its cost is slightly higher than the profit-oriented model, it significantly outperforms the energy-saving model by saving approximately CNY 4863.92.This financial equilibrium positions the balanced model as a prudent choice for businesses seeking cost optimization without compromising excessively on other objectives.Carbon emissions present another pivotal dimension.The balanced model, with 1368.68 kg of emissions, achieves a delicate balance.It outperforms the energy-saving model by reducing emissions by 390.65 kg, showcasing a conscientious approach towards environmental sustainability.Simultaneously, it remains competitive when compared to the profit-oriented and customer-centric models, avoiding the excessive emissions seen in the profit-oriented model.Crucially, customer satisfaction under the balanced model experiences a remarkable improvement of 92.55% compared to the energy-saving model.This enhancement is H3J2, H5J2, H2J2, H3J3, H1J4, H4J1, H1J6, H1J3, H4J5, H5J6, H1J1, H5J3, H2J5, and H4J4. Figure 13 illustrates the Gantt chart under this model, with the order priority allocation sequence H5J2, H3J2, H3J3, H2J1, H2J2, H1J4, H5J3, H1J3, H2J5, H1J6, H1J1, H4J1, H5J6, H4J5, and H4J4. Figure 14 illustrates the Gantt chart under this model, with the order priority allocation sequence H5J2, H3J2, H2J1, H1J4, H2J2, H3J3, H1J1, H5J3, H1J3, H2J5, H4J1, H5J6, H1J6, H4J5, and H4J4.trates the Gantt chart under this model, with the order priority allocation sequence H5J2, H3J2, H3J3, H2J1, H2J2, H1J4, H5J3, H1J3, H2J5, H1J6, H1J1, H4J1, H5J6, H4J5, and H4J4. Figure 14 illustrates the Gantt chart under this model, with the order priority allocation sequence H5J2, H3J2, H2J1, H1J4, H2J2, H3J3, H1J1, H5J3, H1J3, H2J5, H4J1, H5J6, H1J6, H4J5, and H4J4.The profit-oriented model inadvertently leads to a surge in carbon emissions due to high-energy-consumption machines deployed to meet demanding production schedules.Conversely, the energy-saving model, despite showcasing a commendable commitment to lower carbon emissions, incurs higher costs.Similarly, the customer-centric model, prioritizing customer satisfaction, relies on high-energy-consumption machines operating at maximum capacity, contributing to heightened carbon emissions.The balanced model offers a comprehensive solution that addresses the inherent limitations of the above models.It provides a strategic advantage by coordinating the synergies between minimizing cost and carbon emissions and maximizing customer satisfaction to achieve sustainable growth.This innovative approach incorporates energy price fluctuations into scheduling strategies, enabling a multi-objective optimization model that considers cost, carbon emissions, and customer satisfaction.Companies can effectively control energy consumption, reduce operating cost, minimize environmental impacts, and improve customer satisfaction.At the societal level, this research contributes to sustainable development goals by promoting low-carbon production.
Contrasting with prior studies, such as those by Jiang et al. [39] and Zhang et al. [40] that prioritized minimizing completion times without regard for energy consumption implications, this approach falls short in fostering environmental sustainability.Although Jiang et al. [41] delved into energy-efficient scheduling to curtail energy expenditure, the omission of time-of-use electricity rates and customer satisfaction considerations from The profit-oriented model inadvertently leads to a surge in carbon emissions due to high-energy-consumption machines deployed to meet demanding production schedules.Conversely, the energy-saving model, despite showcasing a commendable commitment to lower carbon emissions, incurs higher costs.Similarly, the customer-centric model, prioritizing customer satisfaction, relies on high-energy-consumption machines operating at maximum capacity, contributing to heightened carbon emissions.The balanced model offers a comprehensive solution that addresses the inherent limitations of the above models.It provides a strategic advantage by coordinating the synergies between minimizing cost and carbon emissions and maximizing customer satisfaction to achieve sustainable growth.This innovative approach incorporates energy price fluctuations into scheduling strategies, enabling a multi-objective optimization model that considers cost, carbon emissions, and customer satisfaction.Companies can effectively control energy consumption, reduce operating cost, minimize environmental impacts, and improve customer satisfaction.At the societal level, this research contributes to sustainable development goals by promoting low-carbon production.
Contrasting with prior studies, such as those by Jiang et al. [39] and Zhang et al. [40] that prioritized minimizing completion times without regard for energy consumption implications, this approach falls short in fostering environmental sustainability.Although Jiang et al. [41] delved into energy-efficient scheduling to curtail energy expenditure, the omission of time-of-use electricity rates and customer satisfaction considerations from their analysis precludes a holistic embrace of enterprise sustainability.The GFJSP designed in this study contributes to the overall sustainability of the enterprise by harmonizing economic performance with ecological responsibility and consumer satisfaction.Of course, this study needs to be further extended in the future.Industry 4.0 (I4.0) technologies provide favorable conditions for sustainable development [42].For example, dynamic layout planning based on I4.0 reduces costs, improves society, and protects the environment [43].Consider the combination of collaborative robot allocation and job-shop scheduling to minimize cost and manufacturing spans [44].Therefore, our work can be extended by I4.0 technologies, designing an adaptive, green, flexible job-shop scheduling system by combining I4.0 technologies such as Internet of Things, collaborative robots, and augmented reality [45].This system enables collaborative multi-robot-machine scheduling, responding in real-time to market changes and energy price fluctuations.It automatically adjusts scheduling strategies to optimize energy efficiency and reduce carbon emissions.

Conclusions
The global shift towards sustainable practices has underscored the need for manufacturing systems that prioritize energy efficiency and emission reduction.Within this context, the Green Flexible Job-shop Scheduling Problem has emerged as a critical area for research and innovation.While existing studies have made significant strides in addressing carbon emissions and energy consumption, there is a notable gap in considering the dynamic impact of variable energy prices on the overall energy cost within the scheduling framework.This paper addresses this gap by introducing a comprehensive energy cost model based on time-of-use electricity pricing.Simultaneously, a multi-objective optimization model is developed to strike a balance between minimizing cost and carbon emissions and maximizing customer satisfaction.By achieving a balance between economic and environmental concerns and customer satisfaction, this study not only helps businesses more effectively reduce operational costs but also mitigates environmental impacts and enhances customer satisfaction.Furthermore, it contributes to advancing the three pillars of sustainability in societal construction-environment, society, and economy.
I4.0 technologies, particularly collaborative robots and augmented reality, have been successfully applied to job-shop problems.Collaborative multi robot-machine scheduling can help improve productivity [46].Augmented reality can reduce error rates and increases productivity by providing real-time information and visual guidance [47].Therefore, future GFJSP studies should focus on collaborative multi-robot-machine scheduling, with due consideration given to the energy consumption of robots.And new technologies should be integrated to design more adaptable and intelligent production systems that optimize energy efficiency, reduce carbon emissions, and support sustainable development.

Figure 1 .
Figure 1.The time-of-use electricity price model.

Figure 2 .
Figure 2. The flow of the improved genetic algorithm.

Figure 4 .
Figure 4. Example of the operation code crossover.

Figure 4 .
Figure 4. Example of the operation code crossover.

Sustainability 2024 ,
16, x FOR PEER REVIEW 15 of 23 validates the model's reliability, providing robust support for achieving comprehensive and efficient operational management.

Figure 6 .
Figure 6.Comparison of (a) cost, (b) carbon emissions, and (c) customer satisfaction under the single objective.

Figure 6 .
Figure 6.Comparison of (a) cost, (b) carbon emissions, and (c) customer satisfaction under the single objective.

Figure 7 .
Figure 7. Convergence curves of two algorithms for solving multi-objective problems.

Figure 7 .
Figure 7. Convergence curves of two algorithms for solving multi-objective problems.

Figure 8 .
Figure 8.Comparison of operation results.

Figure 8 .
Figure 8.Comparison of operation results.

Figure 10 .
Figure 10.Comparison of (a) the cost, (b) the carbon emission, and (c) the customer satisfaction in different models.

Figure 10 .
Figure 10.Comparison of (a) the cost, (b) the carbon emission, and (c) the customer satisfaction in different models.

Figure 11 .
Figure 11.Gantt chart for the profit-oriented model.

Figure 12 .
Figure 12.Gantt chart for the energy-saving model.

Figure 11 .
Figure 11.Gantt chart for the profit-oriented model.

Figure 11 .
Figure 11.Gantt chart for the profit-oriented model.

Figure 12 .
Figure 12.Gantt chart for the energy-saving model.

Figure 13 .
Figure 13.Gantt chart for the customer-centric model.

Figure 13 .
Figure 13.Gantt chart for the customer-centric model.

Figure 14 .
Figure 14.Gantt chart for the balanced model.

Figure 14 .
Figure 14.Gantt chart for the balanced model.

Table 1 .
Comparison of research status.

Table 2 .
Explanation of the symbols. 2 2 2 • • • h}. e j Electricity price at time T j .∂ Conversion coefficient between electrical energy consumption and carbon emissions.w iks Energy consumption of machine M i processes operation O ks of the workpiece J k in one hour.w ′ CNY/workpiece) due to customer H c not delivering workpiece J k on time.capacity sik The capacity of operation O ks of machine M i machining workpiece J k .o ykc Customer H c orders quantity of workpiece J k with delivery date y. q c i Idle energy consumption of machine M i in one hour.p k Profit (CNY/workpiece) of the workpiece J k .p ′ kc Penalty ( stock xk On day x, the inventory of finished workpiece J k .gxkcThequantity of the workpiece J k supplied to the customer H c on day x, including all overdue orders for delivery on day x.late yc xk On day x, provide the customer H c with the quantity of overdue the workpiece J k , which should have been submitted on day y.On day x and time T j , the quantity of semi-finished workpieces in the operation O ks of the workpiece J k .productxkOnday x, the output of the workpiece J k .numkcThenumber of batch deliveries of the workpiece J k ordered by the Customer H c .do ykc For all overdue orders, the normalized value of the order quantity o ykc , equal to(o ykc − o min )/(o max − o min ).dtykcFor all overdue orders, the normalized value of the order overdue time t ykc , equal to (t ykc − t min )/(t max − t min ).

Table 3 .
Stamped parts and machines.

Table 4 .
Details about the order.

Table 5 .
Machine ideal energy consumption (in one hour).

Table 6 .
Performance analysis of algorithms for solving single-objective problems.

Table 6 .
Performance analysis of algorithms for solving single-objective problems.

Table 7 .
Performance analysis of algorithms for solving multi-objective problems.

Table 7 .
Performance analysis of algorithms for solving multi-objective problems.

Table 8 .
Based on the computation results of the five-order simplex grid.

Table 8 .
Based on the computation results of the five-order simplex grid.