Research on Two-Stage Joint Optimization Problem of Green Manufacturing and Maintenance for Semiconductor Wafer

*is paper proposes a two-stage joint optimization problem of green manufacturing and maintenance for semiconductor wafer (TSGMM-SW) considering manufacturing stage, inspection, and repair stage simultaneously, which is a typical NP-hard problem with practical research significance and value. Aiming at this problem, a green scheduling model with the objective of minimizing makespan, total carbon emissions, and total preventive maintenance (PM) costs is constructed, and an improved hybrid multiobjective multiverse optimization (IHMMVO) algorithm is proposed in this paper. *e joint optimization of green manufacturing and maintenance is realized by designing synchronous scheduling and maintenance strategy for wafer manufacturing and equipment PM.*e diversity of the population is expanded and the optimization performance of IHMMVO is improved by designing the initial population fusion strategy and subpopulation evolution strategy. In the experimental phase, we perform the simulation experiments of 900 test cases randomly generated from 90 parameter combinations. *e IHMMVO algorithm is compared with other existing algorithms to verify the effectiveness and feasibility for TSGMM-SW.


Introduction
e concept of industry 4.0 refers to the fourth industrial revolution to realize intelligent manufacturing through the Internet of things, big data analysis, and other technical means, which is led by intelligent factories, intelligent production, and intelligent logistics. It is bound to have new higher requirements for the high-tech electronic information industry. As a pillar industry of the national economy, semiconductor industry has a very important strategic position. Wafer fabrication is one of the most complex processes in semiconductor manufacturing system. It needs to form the required circuit layer on wafer surface through various physical and chemical operations such as oxidation, deposition, injection, sputtering, photolithography, and cleaning. Its essence is reentry. Wafers may repeatedly visit some equipment at different processing stations. e manufacturing process can be viewed as the reentrant hybrid flow shop scheduling (RHFS) problem. e general job shop scheduling problem (JSP) is that there are multiple machines with different functions in the processing system. e jobs to be processed include multiple operations. Each operation is processed on one machine, and the processing routes of jobs are different from each other. ere are sequence constraints between operations belonging to the same job. RHFS problem is that there are multiple processing stations in the processing system. All jobs have the same process constraints among each station and need to reentry the stations several times in the same order. At least one station has more than one machine. Jobs can be processed by any machine on the corresponding station. e RHFS problem is large in scale and high in complexity, and it belongs to the NP-hard problem [1]. erefore, there is practical value and significance of designing a new intelligent optimization algorithm to solve it. e development of global economy has also brought many environmental problems, among which greenhouse gas emissions, especially carbon dioxide emissions, have attracted the attention of scholars at home and abroad. Manufacturing industry is one of the main sources of carbon dioxide emissions. It is imperative to achieve energy saving and emission reduction in manufacturing industry and develop sustainable manufacturing. Once the semiconductor wafer manufacturing factories are established, they run 24 hours a day and 365 days a year. Machines will produce a degradation effect due to long-term continuous production, which will prolong processing time of jobs and increase machine energy consumption, resulting in a corresponding increase in carbon emissions. In addition, the equipment of semiconductor wafer manufacturing factories is generally expensive. If there is a breakdown for maintenance, it is difficult to find alternative processing equipment quickly, which will delay the production process of whole semiconductor production line and cause enormous economic losses for enterprises. erefore, it is necessary to perform PM on the equipment of semiconductor wafer manufacturing factories. After PM, machines can restore its processing ability to a certain extent, reduce the frequency of random breakdown, shorten processing time, and reduce energy consumption and carbon emissions of machines so as to make some contribution to achieve green production.
Inspired by the idea of multiverse theory, Mirjalili et al. [2] designed a new heuristic algorithm, multiverse optimizer (MVO) algorithm. e algorithm involves fewer parameters, simpler structure, and stronger search ability. It has been applied in parameter optimization [3], image processing [4], and flexible job shop scheduling [5]. However, its application in RHFS problem is few.
is paper carries out the research on TSGMM-SW, which only considers PM in the manufacturing stage. e contributions of this paper are as follows: (1) e TSGMM-SW model is proposed for the first time, which is more in line with the actual situation of semiconductor wafer manufacturing line and can achieve the balance among makespan, total carbon emissions, and total PM costs. (2) IHMMVO algorithm was designed for the first time to solve the problem, which is a combination of NSGA-II and MVO algorithm. e improved operations of the algorithm mainly include synchronous scheduling and maintenance strategy, initial population fusion strategy, subpopulation evolution strategy, etc. e remainder of this paper is organized as follows. e related research literature is introduced in Section 2. e description and model of TSGMM-SW are introduced in Section 3. Section 4 states the IHMMVO algorithm in detail. Section 5 presents experimental results and analysis. Section 6 is a discussion to explain the main limitations and implications of the study. Section 7 provides the conclusions and future work directions.

Literature Review
In recent years, experts and scholars at home and abroad are also continually conducting related discussion and research in the field of semiconductor manufacturing. Kim and Lee [6] used the input, output, and production quantity as the scheduling coordination factors to update manufacturing lead time and available WIP (work in process) level, which realized the synchronization of semiconductor production plan and scheduling decision. Bitar et al. [7] started the research on unrelated parallel machine resource scheduling problem in semiconductor lithography workshop. Jamrus et al. [8] solved the flexible job shop scheduling problem in semiconductor manufacturing under uncertain processing time, with the objective of optimizing makespan. Jia et al. [9] addressed the problem on reentrant batch-processing machines with incompatible job families in the semiconductor wafer fabrication system. Hur et al. [10] studied the machine setting, batch allocation, and batch sequence problems in the semiconductor manufacturing system, which had certain practical significance. Foumani et al. [11][12][13][14] started the research on scheduling, production and inspection of robotic cells in the flow shop environment. It is well known that robotic cells are specifically designed for semiconductor wafer fabrication. Tirkel [15] proposed an approach for establishing a quantitative model to design the efficiency of in-line inspection in semiconductor wafer. Ulrich et al. [16] presented an automated and cycle time optimized path planning algorithm for robot-based inspection systems. To the best of our knowledge, most of the research on the semiconductor manufacturing is limited to the study of its manufacturing stage in the current literature, ignoring its inspection and repair stage, which has a certain degree of limitations. In the actual wafer processing process, wafers are manufactured layer by layer. e defects caused by a manufacturing layer will be covered by a new layer because they are not inspected and repaired in time [17]. As a result, the final wafer testing phase cannot track the error link, so the wafer cannot be repaired in an effective time, which increases the proportion of defective products. e specific approaches and optimization objectives of the above typical literature are shown in Table 1.
In recent years, heuristic algorithms have played a greater advantage in solving RHFS combinatorial optimization problems and have shown strong competitiveness. Mousavi et al. [18] studied a biobjective hybrid flow shop scheduling problem, in which they considered reentrant lines, setup times, and position-dependent learning effects. Shen et al. [19] and Ying et al. [20] solved the reentrant hybrid flow shop scheduling problem with the makespan and the total tardiness criteria. Zhang and Chen [21] addressed a production scheduling problem originating from a real rotor workshop. Given its specific characteristics, the problem was formulated as a reentrant hybrid flow shop scheduling problem with machine eligibility constraints. Sangsawang et al. [22] designed a two-stage RHFS problem with blocking constraints, aiming at minimizing makespan. Chamnanlor et al. [23] studied the RHFS problem with time window constraints, which often occurred in the 2 Mathematical Problems in Engineering manufacturing system producing the slider parts of hard disk drives. e above literature provides a good reference for the research of RHFS problem, but most of them involve only one stage of RHFS problem. e specific approaches and optimization objectives of the above typical literature are shown in Table 2.
ere have been many studies on green manufacturing. Yin et al. [24] established a low-carbon mathematical scheduling model with machine speed as an independent decision variable and designed the corresponding encoding and decoding method and crossover mutation operator. e validity of the model and method for low-carbon job shop scheduling problem was illustrated by an engineering example. Liu et al. [25] proposed a method for calculating carbon footprint of products and a multiobjective model for minimizing production cycle and carbon footprint, which can effectively provide carbon labels for future life cycle products of manufacturing enterprises. Sihag and Sangwan [26] established a multiobjective mathematical model for optimizing carbon emissions and processing time in turneries, which could select the best cutting parameters for manufacturing enterprises, reduce carbon emissions, and improve processing efficiency. For single-machine scheduling and multivehicle routing problems, Wang et al. [27] designed a mathematical programming model with the objective of minimizing total carbon emissions. Experiments on an enterprise case and 20 simulation cases verified the effectiveness of the proposed algorithm in guiding green manufacturing. Foumani and Smith-Miles [28] put forward a mixed integer linear model aiming at optimizing production time and carbon emissions on the basis of considering three common carbon emission reduction policies of flow shop. He considered the impact of carbon emissions reduction policy on the economic competitiveness of manufacturing industry from the perspective of environmental policymakers. Wang and Feng [29] constructed the green degree evaluation system of equipment manufacturing industry and also proposed policy recommendations for the green degree of China's equipment manufacturing industry from the perspectives of improving relevant policies, technological upgrading and environmental protection input, applying legal policy systems, and increasing financial support. Zhang et al. [30] provided a general model and overall framework of green manufacturing for industrial enterprises. According to the needs and conditions, enterprises could construct and formulate a clear plan for the implementation of green manufacturing that conforms to their actual situation of enterprises. e specific approaches and optimization objectives of the above typical literature are shown in Table 3.
In a literature about PM research, Liu et al. [31] proposed a comprehensive decision-making model, which considered the coordination of preventive maintenance and single machine scheduling. e problem of maintenance shortage or overage caused by only considering PM was avoided. Huang et al. [32] designed cluster particle swarm optimization algorithm to shorten the manufacturing cycle and effectively reduce energy consumption aiming at the problem of cleaner production by adding deteriorating maintenance operations to flow shop with two-stage multiprocessors. Ladj et al. [33] addressed the PM problem of a single multifunctional machine based on health management. Xiao et al. [34] built a joint optimization model with the objective of minimizing the total costs, including production costs, PM costs, maintenance costs of unexpected breakdown, and delay costs. Optimal PM interval and allocation strategy for each job can be obtained by solving this problem. Golpîra and Tirkolaee [35] introduced an original concept entitled "stable maintenance tasks scheduling," which incorporated robust optimization concept into the maintenance and repair tasks scheduling problem. Goli et al. [36] addressed a novel robust flow shop scheduling problem with outsourcing option where jobs could be either scheduled for inside or outsourced to one of the available subcontractors. Capacity limitation for inside resource, just-in-time delivery policy, and uncertain processing time were the key assumptions of the proposed model. e specific approaches and optimization objectives of the above typical literature are shown in Table 4.
In summary, the current literature rarely uses MVO algorithm to solve the scheduling optimization integration problem, and it also rarely optimized makespan, total PM costs, and total carbon emissions simultaneously. In addition, Jia et al. [9] Improved combined scheduling algorithm Minimize total weight tardiness Hur et al. [10] Greedy random adaptive search algorithm Maximize weighted throughput during the 2-5 day planning period Foumani et al. [11][12][13][14] eoretical analysis, theorem proving and heuristic method Optimize the scheduling, production and inspection problems of robotic cells in flow shop environment Tirkel [15] Quantitative model and sensitivity analysis Modelling the efficiency of in-line inspection Ulrich et al. [16] Probabilistic roadmap method applied on all measurement poses and A * search algorithm Finding the optimal path between measurement poses most of them study the RHFS problem about one stage. e research of this paper fills the gap. e TSGMM-SW model established in this paper studies the two-stage reentrant hybrid flow shop scheduling problem in wafer manufacturing process. at is, when wafers are processed one layer through all stations in manufacturing stage, they enter into the inspection and repair stage immediately. e wafers undergo inspection and repair in time to improve the qualification rate. In addition, the model can achieve the balance among makespan, total PM costs, and total carbon emissions.

Problem Description and Model Establishment
3.1. Description of TSGMM-SW. TSGMM-SW consists of two stages: the first stage is the manufacturing stage of semiconductor wafers and the second stage is the inspection and repair stage, in which each stage is an independent RHFS problem. e problem can be described as that n jobs need to be processed. job i Among them, needs to processed L i layers (re-entry number). Besides, the re-entry number of each job can be different. ere are s stations in the first stage, and station l has m l (m l ≥ 1) parallel machines that can be chosen. Besides, at least one station has more than one parallel machine. In the second stage, there are two stations, one is the inspection station and the other is the repair station, and each job needs to reenter L i ′ times. All of the jobs pass through each station in the same sequence of processing constraints, and each job is processed L i (s + 2L i ′ ) times in total [17]. It is assumed that there is infinite capacity of buffer between continuous processing stations. e processing process is uninterrupted. Each wafer can restart processing Table 3: Typical literature about green manufacturing research.

Literature
Approaches Objectives Yin et al. [24] Multiobjective genetic algorithm based on simplex lattice Optimize productivity, energy efficiency, and noise reduction Liu et al. [25] Improved fruit fly algorithm Minimize makespan and energy consumption Sihag and Sangwan [26] Multiobjective genetic algorithm Minimize makespan and carbon emissions Wang et al. [27] Hybrid tabu search algorithm Minimize carbon emissions Foumani and Smith-Miles [28] CPLEX Minimize makespan and carbon emissions Wang and Feng [29] Improved particle swarm algorithm Measure the green degree Zhang et al. [30] System engineering method Improve the green rate of manufacturing technology and manufacturing process and reduce the environmental impact rate of resources  [31] Genetic algorithm Minimize the production costs and tardiness costs Huang and Yu [32] Cluster particle swarm algorithm Minimize makespan Ladj et al. [33] Improved genetic algorithm Minimize the total interventions costs Xiao et al. [34] Stochastic key genetic algorithm Minimize the total costs Golpîra and Tirkolaee [35] CPLEX Minimize the total costs of the maintenance and maximize the stability/robustness of the schedule Goli et al. [36] CPLEX Minimize the total-weighted time and the total costs of outsourcing Ying et al. [20] Iterative greedy algorithm Minimize makespan and tardiness Zhang and Chen [21] Improved discrete differential evolution algorithm Minimize tardiness Sangsawang et al. [22] Combining adaptive genetic algorithm based on fuzzy logic controller with particle swarm optimization based on Cauchy distribution Minimize makespan Chamnanlor et al. [23] Hybrid optimization algorithm based on genetic algorithm and ant colony algorithm Minimize makespan 4 Mathematical Problems in Engineering the next layer after the inspection and repair stage. e PM fixed costs are same for all machines in each station. Figure 1 shows the schematic diagram of TSGMM-SW problem. Semiconductor wafer fabrication factories cannot keep machines in working state all the time; jobs to be processed have to wait for the process of the previous operation to be completed or the selected machine to be idle. e longer the waiting time is, the greater the change of the production environment will occur. e machine will produce degradation effect, and the actual processing time of jobs will gradually increase. e actual processing time of machines is calculated by the following equation: where T ijlk is the actual processing time of o ij , T ijlk is the standard processing time of o ij , t ω is the waiting time before processing o ij , and τ is degradation factor, assuming that the degradation factor of machines with the same serial number at each station is the same. In practical production applications, the two-parameter Weibull distribution tends to describe the failure rule of machines [37]. e failure rate function of machines is, where m > 0 and β > 0, m denotes shape parameters, and β denotes scale parameter. e reliability R of machines at runtime t is shown in the following equation [38]: where Z denotes the service age. According to equation (4), the service age when R reaches the threshold value is calculated.
where R 0 is the minimum reliability threshold value of machines. During the processing, as long as service age of machines is not less than Z 0 , the machine needs to perform PM at any time. After PM, the machines cannot completely restore to their original state. e initial service age of machines after PM is shown in the following equation: where Z k is the service age of machine k after PM, Z ′ is actual service age of machine k before PM, and pk is service age regression factor of machine k.  Mathematical Problems in Engineering materials, and the carbon emissions from the power consumption of equipment switches are not related to workshop scheduling. So, this paper only considers the carbon emissions generated from power consumption during processing and idling of equipment. Also, the carbon emissions generated from lubricants used by equipment are considered too. e relevant parameters in TSGMM-SW model are shown in Table 5. e objective functions and constraints are shown as below:

Model of TSGMM-SW
f2 � min(TCT), f3 � min(TCost), where Equations (6)-(8) are three objective functions. Equation (9) is the definition of makespan. Equation (10) is the definition of the total carbon emissions. Equation (11) is the definition of the total PM costs. Equations (12) and (13) indicate the total carbon emissions generated from machines processing in manufacturing stage and in inspection and repair stage, respectively. Similarly, equations (14) and (15) indicate the total carbon emissions generated from machines in idle state in manufacturing stage and in inspection and repair stage, respectively. Equations (16) and (17) indicate the total carbon emissions generated from the use of lubricants in manufacturing stage and in inspection and repair stage, respectively. Equation (18)

Model Complexity Analysis.
Under the premise that the model structure is determined, the computational complexity is related to the number of decision variables and constraints.
e TSGMM-SW model involves two stages, namely, the manufacturing stage and the inspection and repair stage. e number of its decision variables is s l�1 m l ( n i�1 L i + 1) + 2 l′�1 m l′ ( n i�1 L i ′ + 1). e number of constraints in the manufacturing stage is ns 2 . e number of constraints in the inspection and repair stage is 4n From the number of variables and constraints above, we can see that the TSGMM-SW model solves very complex intractable problem. [2]. MVO finds the solutions through the interaction between white holes, black holes, and wormholes. Each universe represents a viable solution to the problem. Each object in the universe represents the variables in each solution, and the expansion rate of universes represents the fitness value of the solution. Universes with higher expansion rates are considered to have white holes, and those with lower expansion rates are considered to have black holes. Particles in the universe move from white holes (sending) to black holes (accepting) through a medium called wormholes. Eventually, all matter in the universe reaches its optimum position by exploring the search space. In each iteration of the algorithm, the roulette method is used to select a white hole from universe individuals which are sorted by expansion rate. e population is described as
where d is the number of variables, n is the number of universes (the number of candidate solutions), and x j i is  the j-th dimension of the i-th universe, which is described as where U i represents the i-th universe, NI(U i ) represents normalized expansion rate of the i-th universe, x j k represents the j-th dimension of the k-th universe selected by the roulette method, and r 1 is a random number between [0, 1]. In order to increase the expansion rate of objects through wormholes, it is assumed that wormhole tunnels are always built between each universe and the optimal universe. e location of black holes in the optimal universe is updated by the following equation: where X j represents the j-th dimension of the current optimal universe, ub j and lb j are the upper and lower bounds of the j-th dimension respectively, r 2 , r 3 , and r 4 represent random numbers between [0, 1], respectively, WEP indicates the existence rate of wormholes, and it is calculated by equation (25), and TDR indicates the travel distance ratio of the object around the current optimal universe, and it is calculated by equation (26).
where WEP max represents the maximum value of WEP, WEP min represents the minimum value of WEP, l represents the current number of iterations, L represents the maximum number of iterations, and p represents mining accuracy of the algorithm.

IHMMVO Algorithm Flow.
e IHMMVO algorithm proposed in this paper adds fast nondominated sorting, crowding distance calculation, and elite strategy to solve the multiobjective optimization problem based on the singleobjective MVO algorithm. In addition, the improvements mainly include the following aspects: synchronous scheduling and maintenance strategy, initial population fusion strategy, and subpopulation evolution strategy. e flowchart is shown in Figure 2.

Encoding and Decoding.
e basic MVO algorithm is used to solve the single-objective optimization problem of continuous function. If the individual's position corresponds to the continuous vector value, it is impossible to update the processing sequence of jobs in the job shop scheduling problem. In this paper, the random key rule based on ascending order is used to map the position of individuals and the processing order of jobs. By changing the position information of individuals, the processing order of jobs is changed, so different scheduling schemes are constructed. For the manufacturing stage of TSGMM-SW, greedy strategy is adopted to decode, i.e., arrange the processing operation of each job in turn on the machine that can start processing it at the earliest time at each station.
is decoding method can quickly calculate the service age of each machine and make it easier to judge whether the machine needs PM or not. For the inspection and repair stage of TSGMM-SW problem, PM operation is not considered, and the service age of machines is not required to be calculated. e PS decoding method [20] is used.
at is, if the idle time is enough to process the operation, it will be inserted into the idle time gap for processing.
ere are three scheduling strategies for joint optimization of production scheduling and maintenance, namely, scheduling before maintenance [39], maintenance before scheduling [40], and synchronous scheduling and maintenance [41]. e scheduling before maintenance strategy needs to arrange the scheduling plan of jobs first and then insert the PM into the idle gap of the machine. If the idle gap cannot meet the PM time, you need to dynamically adjust the start processing time of jobs. is strategy involves rescheduling, which is more complex and time consuming. Maintenance before scheduling strategy sets the PM period to be unavailable and schedules the jobs in the remaining time. is strategy ignores the actual processing situation of machines in the factory and excessively relies on the machine PM period determined by historical experience. e synchronous scheduling and maintenance strategy makes up for the shortcomings of the above two strategies. According to the machine reliability threshold, the service age threshold of machines is determined. Every time when a job is arranged, it is judged whether the service age of machines reaches the threshold value. If it reaches the threshold value, PM will be executed. Otherwise, the jobs will continue to be arranged. e actual processing situation of machines will be fully considered, and no additional dynamic adjustment of jobs is required. erefore, the third synchronous scheduling and maintenance strategy is adopted in this paper. e specific process is as follows.
Step 1. For the operation O ij+1 which waits for processing, the machine that can start to process it at the earliest time is selected at the corresponding station. Judge Z t (service age of the selected machine k at the moment t); if Z t + T i,j+1,lk < Z 0 , machine k does not need perform PM and O ij+1 is processed directly on the machine k. Otherwise go to Step 2.
te is the maximum of S m and C ij . at is, PM is inserted into the idle time on machine k before starting O ij+1 , without affecting the start time S ij+1 . en, ts � te − tp lk and S ij+1 � te, and execution ends and goes to Step 5. Otherwise go to Step 4.
Step 4. If Z t � Z 0 , ts is the time when the machine can start processing O ij+1 , i.e., ts � S m . en, te � ts + tp lk . S ij+1 is the maximum of te and C ij , and execution ends and goes to Step 5.
Step 5. Record S * ij+1,k (the earliest start time of O ij+1 on machine k) after considering PM on the selected machine k. Determine the earliest start time of O ij+1 on the remaining machines at the corresponding station in turn. First, judge whether the other machines need to perform PM if they process O ij+1 . If there is no need, then S * ij+1,k * � S ij+1 . Otherwise, perform the calculation of S ij+1 according to Steps 1-4. Finally, the earliest start time of processing O ij+1 for all machines (considering PM) is obtained.
Step 6. According to the machine serial number order, select a machine that can start processing O ij+1 at the earliest time and assign it to process.
In order to describe synchronous scheduling and maintenance strategy more clearly, this paper gives a detailed description through a small-scale test example. e example includes four jobs; [1,2,4,3] is the encoding mode, that is, decoding the job 1, 2, 4, and 3 in turn. In this example, all machines belonging to the same station in the two stages have the same processing power, idle power, effective use time, and usage of lubricants. For the manufacturing stage, there are 3 stations, and the number of parallel machines at the same speed is 3, 2, and 2, respectively. e PM fixed costs are the same for all machines belonging to the same station, but the PM unit activity costs are set differently. Reliability thresholds of the machines having the same serial number at each station are the same. ere are two stations in the inspection and repair stage. One is the inspection station and the other is the repair station. ere are two parallel machines at the same speed in each station. e relevant parameters and values for the two stages are shown in Tables 6 and 7. In addition, the carbon emission factor of electric energy is 1.8742 × 10 − 7 kgCO 2 /J, and the carbon emission factor of lubricants is 469 kgCO 2 /m 3 [42] (1 L � 0.001 m 3 , 1 kw/h � 3.6 × 10 6 J). According to equation (4), the service age thresholds of machines for PM are [21,28,43] (unit: min). Figure 3 shows the scheduling Gantt chart for this example. Take the first machine at station 1 as an example. Before processing the third layer of Job1, the first and second layers of it and the first layer of Job3 and the second layer of Job2 have already been arranged to process at station 1. e machine is idle before processing the second layer of Job1, and the idle time is 10 min, so it is necessary to consider the influence of machine's degradation effect. e machine's degradation factor is 0.1, and the actual processing time of Job1 on the first machine at station 1 is T 2 111 � round(2 × 10 0.1 ) � 3 min. Similarly, the machine is idle before processing the second layer of Job2, the idle time is 3 min, and T 2 211 � round(4 × 3 0.1 ) � 4 min. After the second layer of Job2 is processed, the machine' service age is the total processing time on the machine Z ′ � 5 + 6+ 3 + 4 � 18 min. If the third layer processing of Job1 is performed, the machine is idle before processing, and the idle time is 14 min; then, T 3 111 � round(4 × 14 0.1 ) � 5 min. e machine's service age threshold is 21 min and PM time is 4 min because Z ′ + T 3 111 � 18 + 5 � 23 > 21 and the PM time is less than the idle time, so according to the scheduling and maintenance strategy, PM should be executed in the idle time of the machine before processing the third layer of Job1, without affecting the start time of it. e second machine and the third machine at the station 1 are also analyzed. If the third layer of Job 1 is processed on them, although PM is not required, the start times are same with the first machine. en, when decoding, according to machine serial order, the final decision is to process the job on the first machine. e first machine of station 2 and station 3 is similar to the above example; due to limited space, there are no more details. On the second machine of station 3, the processing time of Job2 and Job3 is 13 min and 3 min, respectively, for the first layer. e machine is idle before processing the second layer of Job4, the idle time is 16 min, and the machine's degradation factor is 0.15. en, the actual processing time is T 2 432 � round(5 × 16 0.15 ) � 8 min. Also, the machine is idle before processing the third layer of Job2, the idle time is 11 min, and the actual processing time is T 3 232 � round(3× 11 0.15 ) � 4 min. At this moment, the machine's service age is Z ′ � 13 + 3 + 8 + 4 � 28 min. Because the second machine's service age is 28 min and Z ′ � Z 0 , PM should be executed immediately after Job2 is processed. Finally, three objective values were obtained as follows: C max � 77 min, TCT � 0.3538 kgCO 2 , and TCost � 143 RMB.

Initialization Population Fusion
where pop2(i, :) is the position variable of the i-th individual in population pop2 and pop1(j, :), pop1(k, :), and pop1(p, :) are the position variables of the j-th, k-th, and p-th individuals which are randomly selected from pop1. In addition, i ≠ j, k, p. By enlarging the position variables of individuals, more job scheduling schemes can be obtained by using random key coding rules. Combine the populations pop1 and pop2, calculate the crowding distance, and perform fast nondominated sorting [43]. en, select the first N individuals to form the final initial population and get the initial external archive Archive which is composed of noninferior solution set.

Subpopulation Evolution Strategy.
Divide the whole population into two subpopulations pop x and pop y , each with N/2 universe individuals. Implement different evolutionary strategies for the two subpopulations. Among them, Levy flight perturbation mutation operation is performed for universe individuals in pop x , and NSGA-II operation is performed for universe individuals in pop y . ese operations can expand the search scope and the population's diversity, make the algorithm can jump out of local optimum in the process of evolution, and avoid premature convergence.
For subpopulation pop x , WEP increases gradually during the iteration of the basic MVO algorithm. In the early iteration, it is less than the random number r 2 , and the j-th dimension variable of the i-th universe remains unchanged according to equation (24). is can easily cause the algorithm to fall into the local optimal value. erefore, this paper introduces the Levy flight perturbation strategy to perturb the position variables of universe individuals in the early iteration and increase the diversity of the population. By introducing the adaptive weighted learning factor w, the offspring universe individuals can maintain the original position information and can also carry out Levy disturbance to a certain extent. e above operation increases the universe individuals' diversity and obtains a new offspring  Stations  1  2  3  1  2  3  1  2  3  1  2  1  2  Job1  5  6  4  2  6  8  4  3  6  2  1  2  1  Job2  5  4  13  4  6  7  5  2  3  2  0  1  2  Job3  6  4  3  2  5  7  6  2  0  1  0  2  1  Job4  9  5  8  6  2  5  6  3  4  1  2 2 1 where w is a weighted learning factor, Levy is Levy flight step, and β is a parameter between [1,2], and we set it to 1.5. u and v obey normal distribution. For subpopulation pop y , NSGA-II algorithm is used to generate new subpopulations new1 and new2 by LOX crossover [44] and swap mutation. Combine them to calculate the crowding distance and perform fast nondominated ranking; the first N/2 individuals are selected to obtain a new offspring population C2. Finally, the subpopulations C1 and C2 are merged to form the final initial population C at the beginning of the next iteration. e noninferior solution set of the population C is obtained, and Archive is updated.

Simulation Experiments
In order to verify the effectiveness of the proposed IHMMVO algorithm for solving TSGMM-SW, the experimental phase is divided into two parts. e first part selects MMVO, MSSA, and IHMMVO algorithm for comparison. Among them, MMVO is the basic algorithm of IHMMVO, and MSSA has been proved to have high convergence and good divisibility in solving multiobjective optimization problems in literature [45]. Besides, both of them have simple structure, strong search ability, and fast convergence ability in solving complex problems, so we select them as comparative algorithms. In order to further verify the validity of IHMMVO algorithm, the second part selects two canonical multiobjective evolutionary algorithms (MOEA)-NSGA-II algorithm [43] and MOPSO algorithm [46] for comparative analysis, which are classical and effective algorithms for solving multiobjective problems. In the experimental stage, in order to make the three objective functions in the same order of magnitude, the min-max normalization was carried out. In order to ensure the fairness of the comparison, experimental simulation environment is set to Windows 10 operating system, with Intel (R) Core (TM) i7-4770 CPU @3.40 GHz and 8 GB memory, and programming software is MATLAB R2017a.

Test Cases and Parameter Setting.
Since there is no dataset for solving TSGMM-SW in the published literature, the dataset setting method in [17] is referenced to generate test cases. e dataset includes 5 job number criteria, 2 reentry number criteria, 3 station number criteria, and 3 machine number criteria, a total of 5 * 2 * 3 * 3 � 90 parameter combinations. For each parameter combination, 10 test cases are generated randomly, including 90 * 10 � 900 test cases. e parameters and their values involved in the dataset are shown in Table 8.
e setting of algorithm parameters has an important influence on its performance. e IHMMVO algorithm designed in this paper involves four key parameters, namely, the maximum value of weighted learning factor w max , the minimum value of weighted learning factor w min , crossover probability P crossover , and mutation probability P mutation . e Taguchi method is used for experiments to study the solution results of various parameter combinations to the scheduling problem, and finally the optimal parameter combination is determined. e remaining parameters are selected based on experience. For the fairness of experiments, the same parameter involved in the rest of comparison algorithms is same with IHMMVO. e specific parameter settings are shown in Table 9.

Performance Test and Result Analysis.
In this paper SP, GD, and IGD are selected as evaluation indicators [44]. For the investigated problem, the Pareto optimal solutions of all test algorithms are regarded as the final Pareto optimal solutions. For TSGMM-SW, 10 test cases are generated randomly by each parameter combination. Simulation experiments are carried out with all the comparison algorithms. e average value is taken as the final result of the parameter combination, and the optimal values of each evaluation indicator are shown in bold. Table 10 shows the experimental result comparison of IHMMVO, MMVO, and MSSA. It can be seen that for SP indicator, the dominant proportion of IHMMVO is 51%, the dominant proportion of MMVO is 43%, and the dominant proportion of MSSA is 6%. For GD indicator, the dominant proportion of IHMMVO is 91%, the dominant proportion of MMVO is 9%, and the dominant proportion of MSSA is 0%. For IGD indicator, the dominant proportion of IHMMVO is 96%, the dominant proportion of MMVO is 4%, the and dominant proportion of MSSA is 0%. All of the above indicates the diversity and convergence of Pareto frontier solutions obtained by IHMMVO algorithm are superior to MMVO and MSSA algorithms; for the distribution of Pareto frontier solutions, IHMMVO is superior to MSSA algorithm, just slightly better than MMVO. Table 11 shows the experimental result comparison of IHMMVO, NSGA-II, and MOPSO. It can be seen that for GD and IGD indicators, IHMMVO algorithm is superior to NSGA-II and MOPSO algorithm and the dominant proportion is 100%, which indicates the diversity and convergence of Pareto frontier solutions obtained by IHMMVO algorithm are superior to NSGA-II and MOPSO algorithm. For SP indicator, the dominant proportion of IHMMVO is 69%, the dominant proportion of NSGA-II is 10%, and the dominant proportion of MOPSO is 21%. at is, for most of the test cases, IHMMVO is superior to other comparison algorithms in the distribution of Pareto frontier solutions. Tables 10 and 11 show the advantages and disadvantages of several comparison algorithms from a macro perspective. Table 12 judges whether there is a significant difference between the evaluation indicator of each comparison algorithm by performing Wilcoxon signed rank test on all experimental results. e values outside the brackets are the experimental results between IHMMVO, MMVO, and MSSA, and the values inside the brackets are the experimental results between IHMMVO, NSGA-II, and MOPSO. It can be concluded that there are significant differences between IHMMVO algorithm and the other four comparison algorithms for GD and IGD indicators. For SP indicator, IHMMVO algorithm has significant differences with other comparison algorithms except MMVO. rough the analysis of the above experimental results, it can be inferred that IHMMVO algorithm proposed in this paper can effectively solve TSGMM-SW problem and has certain competitiveness and advantages compared with other comparative algorithms. e Pareto frontiers of several algorithms for 10 * 2 * 5 * 5 case, 30 * 2 * 8 * 9 case, 40 * 3 * 5 * 10 case, and 50 * 2 * 6 * 11 case are clearly illustrated in Figure 4. We can find IHMMVO is closer to the optimal frontier.

Case Analysis.
In this paper, a 10 * 2 * 8 * 3 test case is selected for concrete analysis. Assume that there are 8 stations in the production line, including oxidation, deposition, injection, metallization, lithography, etching, polishing, and cleaning. e number of parallel machines on each station is 3, and the number of jobs for processing is 10. e processing level is 2. e reliability thresholds of the three machines at each station are [0.5, 0.4, 0.55]. For other parameters, refer to Table 8. Compare the Pareto optimal solution set and job scheduling schemes obtained by considering PM and without considering PM. e results are shown in Table 13. Among them, the values outside brackets are the experimental results considering PM, while the values inside brackets are the experimental results without considering PM. It can be seen that makespan and total carbon emissions can be optimized to some extent by taking PM operation into account in semiconductor production Mathematical Problems in Engineering 13 line. Besides, Figure 5 shows the scheduling Gantt chart of the third scheduling scheme 4-9-3-5-8-6-7-10-2-1 in Table 13. It can be found that most PM is executed in the idle period of machines, which has little impact on the start time of the next operation. But it can effectively alleviate the degradation effect of machines caused by long-term continuous production and reduce the makespan of jobs as a whole. To sum up in conclusion, PM is an important factor for semiconductor wafer fabrication enterprises to achieve cleaner production. e sensitivity analysis method is used to analyze the importance degree of reliability threshold R for the three optimization objectives of this paper. e three objective function values are the averages of all the solutions in optimal Pareto solution set. e sensitivity coefficient is calculated by the following equation: where S fR is the sensitivity of objective function value to factor variable R, Δf/f is the change rate of objective function value f, and ΔR/R is the change rate of factor variable R. Figure 6 shows the effect of ΔR/R on the optimization objectives. Figures 6(a)-6(c), respectively, represent the change trend of the three objectives values with the increase of ΔR/R. It can be seen that C max and TCT increase with the increase of ΔR/R, while TCost decreases. Figure 6(d) shows the change trend of sensitivity coefficient of objective function values. e change curve of C max and TCT basically coincides, and the latter is slightly larger than the former, indicating that the influence of ΔR/R on the two is almost the same. e change curve of TCost is obviously above the change curve of the former two, indicating that ΔR/R had greater influence on TCost than on C max and TCT.

Discussion
From the experimental results in Figure 4, it can be seen that in the multiobjective optimization problem, the optimization of one indicator will inevitably cause the        Cases  IHMMVO  NSGA-II  MOPSO  SP  GD  IGD  SP  GD  IGD  SP  GD  IGD  deterioration of at least one other indicator. e obtained Pareto solution and the corresponding scheduling scheme can be an effective reference tool to help managers make intelligent decisions based on the actual situation of the enterprise and the goals they are focusing on when considering the production efficiency, green production,  Cases  IHMMVO  NSGA-II  MOPSO  SP  GD  IGD  SP  GD  IGD  SP  GD  IGD        and economic benefit balance in the semiconductor wafer manufacturing environment. In this paper, IHMMVO algorithm is used to perform a comparative experiment on a test case of 10 * 2 * 8 * 3 scale, showing the optimization objective values of partial solutions obtained with and without considering PM operation. On the other hand, sensitivity analysis is performed to show how the change rate of R affects the three objective function values. It can be concluded that the change rate of R has little effect on C max and TCT, which is basically the same, while the effect on TCost is greater than the former two. Based on this, enterprise decision makers can set appropriate machine reliability threshold to reduce PM costs and improve economic benefits on the basis of no significant impact on C max and TCT indicators. In short, the research of this paper can provide a certain reference value for the future transition of semiconductor wafer manufacturing to a clean production mode while weighing production efficiency and economic benefits. However, this paper mainly focuses on the deterministic scheduling of semiconductor wafer manufacturing, without considering the uncertainty of the production process.

Conclusions and Future Work
In this paper, a two-stage joint optimization model of green manufacturing and maintenance for semiconductor wafers is constructed for the first time, and an IHMMVO algorithm is proposed to solve the problem. Initial population fusion strategy, subpopulation evolution strategy, and synchronous scheduling and maintenance strategy are designed in IHMMVO algorithm. e algorithm can jump out of local optimum and obtain better Pareto frontier solution set, which realizes the two-stage joint optimization of green manufacturing and PM. Enterprise decision makers can choose an optimal scheduling scheme according to the actual situation. e simulation results show that the proposed algorithm has certain advantages and competitiveness in solving TSGMM-SW. Finally, through the specific experimental results analysis of a simulation case in semiconductor wafer manufacturing stage, it can be seen that makespan and total carbon emission indicators can be optimized to a certain extent by considering PM operation on semiconductor production line under reasonable machine reliability threshold setting. It can make a contribution to green manufacturing of semiconductor wafer enterprises.
With the development of collaborative production mode, distributed manufacturing has become a trend. In the future, we will focus on the research of semiconductor wafer distributed green manufacturing scheduling problem considering equipment preventive maintenance. We will also pay more attention to the uncertain scheduling problem in semiconductor wafer production. In addition, with the pursuit of "things perception, things interconnection, and things intelligence" in manufacturing workshop, combining IoT, cloud computing, and big data technologies to build intelligent scheduling algorithms and scheduling rules libraries and combining deep learning and machine learning method to construct more efficient algorithms are also our future effort direction.

Data Availability
e data used to support the findings of this paper are from previously reported studies, which have been cited.   Mathematical Problems in Engineering