Two-Stage Hybrid Optimization Algorithm for Silicon Single Crystal Batch Scheduling Problem under Fuzzy Processing Time

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Introduction
As an important basic semiconductor material, silicon single crystal is widely used in semiconductor integrated circuit chips and other felds.With the rapid development of the information industry, several types of electronic equipment are increasing, and the replacement is rapid, resulting in a sharp increase in the number and types of semiconductor chips.China's industrial demand for silicon single crystals accounts for about one-third of the global total demand, but its production capacity lags far behind that of developed countries and mainly depends on import supply.To break the import restriction, get rid of the industrial monopoly and promote the development of silicon single crystal production in the direction of mass production and industrialization; China has carried out a series of scientifc and technological reform policies [1].In view of the demand for large-scale and multiple variety orders, how to improve production efciency and reduce production costs is the core for semiconductor manufacturing enterprises to improve their competitiveness and the key to further realize scale and industrialization.
Te Czochralski method is the main technical method for growing silicon single crystals.Its manufacturing process for growing silicon single crystals is complex and changeable, with many complex growth processes and long production cycles.It is recognized as one of the most complex manufacturing systems [2].To break the bottleneck of foreign monopoly technology and have its own core technology, many domestic scholars are committed to improving the growth quality of silicon single crystals [3][4][5].In recent years, many enterprises have conquered large-scale silicon single crystals and entered the preliminary mass production stage.Foreign silicon single crystal production enterprises developed earlier than domestic silicon single crystal enterprises.To carry out large-scale production and improve production efciency, foreign silicon single crystal production enterprises have begun to study and practice the issue of silicon single crystal order allocation, shorten the production cycle, improve production efciency, and reduce production costs through reasonable production scheduling [6,7].However, domestic silicon single crystal enterprises still use manual scheduling methods for production planning.Terefore, it has become a major research hotspot to study advanced and practical algorithms to solve the production scheduling problem for domestic production enterprises.
However, the infuence of uncertain processing time will cause a huge social and economic impact on enterprise production, which has attracted more and more researchers' attention [8][9][10][11].For example, Sakaw and Mori [8] frst proposed a triangular fuzzy number ranking criterion to compare the size of fuzzy numbers and applied this criterion to solve the fuzzy job shop scheduling problem.Engin and Yilmaz [9] used a fuzzy processing time and a fuzzy due date to formulate the multiobjective hybrid fow-shop scheduling with multiprocessor tasks problem.Alharbi and El-Wahed Khalifa [10] adopted pentagonal fuzzy numbers to represent the processing time to solve the fow-shop scheduling problem under fuzzy environment.Chen [11] used triangular fuzzy number to represent the uncertainty of job processing time to minimize the maximum fuzzy completion time for the fuzzy distributed fexible job shop scheduling problem.Te silicon single crystal production batch scheduling problem is a constrained combinatorial optimization problem, which belongs to NP-hard problem; it is difcult to solve it by traditional optimization methods.In recent years, most researchers use intelligent optimization algorithm to solve the NP-hard problem and have achieved good results.For example, Lei et al. [12] improved the coding method and proposed a hybrid frog leaping algorithm (SFLA) to solve the green fexible job shop scheduling problem with the objective function of minimizing workload and total energy consumption.Lei and Yang [13] took the energy consumption constraint as the objective function in the frst stage, and then sent the solution set obtained in the frst stage to the second stage.In the second stage, a certain energy consumption value was taken as the constraint for fne solution to obtain a scheduling scheme meeting the energy consumption constraint.Guo and Zhong [14] put forward many requirements on the scheduling model of remanufacturing jobs, established a four objective scheduling model, and introduced multiple population coevolution based on artifcial fsh swarm algorithm to solve the model.Li et al. [15] proposed a new improved artifcial immune algorithm (IAIS) for the second type of fuzzy fexible job shop scheduling problem under the fuzzy processing time.Rui and Gong [16] adopted the improved decomposition based multiobjective evolutionary algorithm (IMOEA/D) and designed a variable neighborhood search combining fve local search strategies to optimize the maximum fuzzy completion time.Tere is no related literature about the existing fuzzy scheduling research on the batch scheduling problem of silicon single crystal production.
To sum up, the production scheduling problem generally focuses on two aspects, that is, how to establish an appropriate model to describe the actual scheduling and how to solve the model to fnd the best feasible solution that meets the objective.In view of the solution ideas and methods of above scheduling problems, this paper establishes a minimum fuzzy maximum completion time optimization model for silicon single crystal production batch scheduling problem and proposes an intelligent optimization algorithm to solve the model.

Silicon Single Crystal Batch Scheduling
Problem with Multiple Size under Fuzzy Processing Time

Te Model of the Silicon Single Crystal Batch Scheduling
Problem.Tis paper studies the optimal scheduling according to the production equipment matching situation and production task demand of a silicon single crystal production plant.Among them, the corresponding thermal feld dimensions of the single crystal furnace equipped by the plant are 22-inch thermal feld, 24-inch thermal feld, 28inch thermal feld, and 32-inch thermal feld.Te types of silicon single crystal required for the order task are 6-inch, 8-inch, 10-inch, and 12-inch.According to the production conditions, the types of silicon single crystal that can be grown by diferent types of single crystal furnaces are shown in Table 1.Where "Yes" means that this type of silicon single crystal can be grown by this type of single crystal furnace and "No" means that this type of silicon single crystal cannot be grown by this type of single crystal furnace.
Because the production process is often accompanied by many uncertainties, the processing time of each silicon single crystal can only be determined as a rough range.Terefore, this article uses triangular fuzzy number (TFN) to represent the processing time of each process with TFN = (t 1 , t 2 , t 3 ).Te membership function graph is shown in Figure 1.Te abscissa represents the processing time, and the ordinate represents the membership value.Te closer the membership value is to 1, the higher the degree of belonging to the fuzzy set is.Among them, t 1 closest to the origin represents the most optimistic processing time, t 2 that in the middle represents the most probable processing time, and t 3 farthest from the origin represents the most conservative processing time.Te membership expression of TFN is shown in formula (1) [17] and that will have several special cases as shown in formula (2)-(4). (2 Because the processing time is expressed in TFN, the fuzzy completion time C i is also TFN, expressed as C i � (c 1 , c 2 , c 3 ); c 1 , c 2 , and c 3 represent the most optimistic completion time (i.e., the lower bound of C i ), the most likely completion time, and the most conservative completion time (i.e., the upper bound of C i ) of the i-th size order job, respectively.
Te model of the scheduling problem in this paper is described as follows.Te order quantity of L-type silicon wafers with diferent sizes is converted into N jobs and allocated to M single crystal furnaces with diferent types.Each job has multiple processes, and each process can only have one optional single crystal furnace.Te processing sequence of the processes is fxed, and the performance of each single crystal furnace equipment is diferent.Te purpose of scheduling is to determine the jobs allocated on diferent types of single crystal furnace equipment, so as to minimize the maximum fuzzy completion time of the whole production system.Te following constraints must be met [18]: (1) Te same job can only be processed on one equipment at any time (2) Only one job can be processed by the same equipment at any time (3) Te single crystal furnace cannot be interrupted during the production process, but can wait at a specifc process (4) Te same job can only be processed in the next operation after the previous operation is completed (5) Assuming that the failure factor is not considered, all single crystal furnaces are able to work normally Te symbols and defnitions involved in this study are shown in Table 2.
According to the symbol defnition, the objective function in this paper can be expressed as follows: Objective function: Subject to: Mathematical Problems in Engineering where equation ( 5) represents the objective function.Equations ( 6) and ( 7) represent the calculation of fuzzy completion time.Equations ( 8)-( 10) are equality constraints, where equation (8) indicates that each job can only be processed by one furnace, equation (9) indicates that each furnace can only process one job at the same time, and equation (10) indicates that the fuzzy starting time of the next batch is equal to the fuzzy completion time of the previous batch plus the time to replace the quartz crucible, and the replacement time is 2 h.Equations ( 11)-( 13) are inequality constraints, where equation (11) indicates that the actual production quantity must be greater than the order required quantity.Equation (12) indicates that the starting time variables and running time variables must be greater than or equal to 0. Equation ( 13) represents the upper and lower limits of the number of the i-th size silicon single crystal produced on the s-type single crystal furnace, and Ceiling means rounding up.

Fuzzy Number and Its Operation.
When calculating the objective function corresponding to diferent individuals or solutions of the fuzzy production scheduling problem, it is necessary to add, subtract, multiply, divide, and take the largest of the fuzzy number (the processing time).At the same time, when evaluating the advantages and disadvantages of diferent solutions, it is necessary to carry out comparative operations.Te defnition for two triangular fuzzy numbers, TEN s � (s 1 , s 2 , s 3 ) and  t � (t 1 , t 2 , t 3 ), is as follows: (1) Addition, subtraction, and multiplication operations: (2) Comparison operations: for triangular fuzzy numbers  s � (s 1 , s 2 , s 3 ) and  t � (t 1 , t 2 , t 3 ), the following criteria are usually used for comparison [13]: Te k-th job is operated on the j-th machine U i,j Te number of the i-th size silicon single crystal grown on the j-th machine  t i,j Fuzzy processing time of the i-th size silicon single crystal on the j-th machine C i Fuzzy completion time of the i-th size silicon single crystal order  T END Fuzzy maximum completion time  st i,j,z Fuzzy starting time of the i-th size silicon single crystal in z-batch on the j-th machine  et i,j,z Fuzzy completion time of the i-th size silicon single crystal in z-batch on the j-th machine 4 Mathematical Problems in Engineering (3) Larger operations: at present, the following two methods are commonly used to calculate the larger operation of triangular fuzzy numbers, namely, the S method and the L method [8].Te defnitions are as follows: S method defnition: L method defnition: Taking  s and  t as examples, the results, respectively, obtained by S method and L method for fuzzy larger operation are shown in Figure 2.
According to the two operations above, the result obtained by L method is  s or  t and the result obtained by S method is the set combination of two triangular fuzzy numbers.Terefore, this paper adopts the L method as larger operation of triangular fuzzy numbers.

Framework of a Two-Stage Hybrid
Optimization Algorithm (TSHOA) 3.1.Encoding and Decoding.According to the problem description of silicon single crystal batch scheduling problem with multiple size under fuzzy processing time, the fnal scheduling result is composed of two parts.Te frst part is how to allocate the order numbers of silicon single crystal rod for diferent types of single crystal furnaces, and the other part is how to allocate the batch plans for produced order quantity on the single crystal furnaces.Terefore, algorithm coding consists of two stages.

Te First Stage.
Te coding in the frst stage is represented by a vector composed of the number of silicon single crystal with diferent sizes produced by diferent types of single crystal furnaces as shown in Table 1.Te vector dimension is 10 dimensions, and the vector composition form is shown in the following formula: where x i1 represents the number of 6-inch silicon single crystal produced by the single crystal furnace corresponding to the 22-inch thermal feld.x i2 represents the number of 8-inch silicon single crystal produced by the single crystal furnace corresponding to the 22-inch thermal feld.x i3 represents the number of 10-inch silicon single crystal produced by the single crystal furnace corresponding to the 22-inch thermal feld.x i4 represents the number of 6-inch silicon single crystal produced by the single crystal furnace corresponding to the 24-inch thermal feld, and the following variables meaning can be analogized in the same way.
According to the variable solution results of the frst stage, it is analyzed and judged whether it is necessary to enter the second stage.To describe the assignments more clearly, a visual representation of variable assignments is presented in Table 3.When the sum of the order quantity exceeds the total number of single crystal furnaces, it is necessary to enter the second stage to obtain the batch allocation scheme to solve the objective function.Otherwise, it means that one batch can complete the production.Te maximum fuzzy processing time is taken as the fuzzy completion time of this type of single crystal furnace, and the fnal completion time is the maximum fuzzy completion time of all types of single crystal furnaces, that is, in Table 3, the total completion time of each column of variables to complete the production is summed, and the fuzzy completion time of the column with the maximum completion time is the maximum fuzzy completion time of the total factory.For example, if x i1 + x i2 + x i3 = 10 and M 1 = 5, the number of 22-inch thermal feld single crystal furnaces is 5, and it is necessary to enter the second stage at this time.

Te Second Stage.
Suppose the frst stage solution result is x i1 � 4, x i2 � 3, and x i3 � 3, that is, the result of the frst stage solution is that 6-inch silicon single crystal rods should be produced 4, 8-inch and 10-inch silicon single crystal rods should be produced 10, respectively; all types of this silicon single crystal rods need to be produced on a 22inch single crystal furnace.Because the processing time from the same type of single crystal furnace that processes different sizes of silicon single crystals is diferent, so the main purpose of the second stage is how to distribute the production process, so that the total completion time of the distribution results is the shortest.Te dimension of the  [19].Te algorithm has the characteristics of memorizing individual optimal solutions and sharing information within the population, that is, optimizing the solutions to the problem through cooperation and competition among individuals in the population.Te mutation operation is the core of the DE algorithm, which afects the performance of the algorithm to a large extent.Te fve types of typical mutation strategies commonly used are as follows: where r1, r2, r3, r4, and r5 are integers that are diferent from each other, representing diferent individuals in the population; j denotes the dimension of individuals in the population; x G i,j denotes the parent individual of the G-th generation; xbest G j denotes the optimal individual of the G-th generation, and F is the variation factor.
According to formula (20), in the standard DE, the mutation vector v G+1 i,j , basis vector x G i,j , and diference vector x G r1,j − x G r2,j are all taken from the same dimension, then for the entire population, the mutation, crossover, or selection operations between each dimension are mutually independent, that is, the evolutionary process between each dimension is unrelated.If a certain dimension or several dimensions in the population cause the algorithm to fall into a precocious state due to the high degree of aggregation of individuals, it is difcult to jump out of the local optimal solution by its own dimension alone.Terefore, this paper introduces the idea of diferent-dimensional mutation proposed in the literature [20], and at the same time, this paper adopts a variety of mutation strategies.Te specifc expression is as follows [21]: Mutation strategy 1: Mutation strategy 2: (26) Mutation strategy 3: where Np represents the number of individuals in the population, i ≠ r1 ≠ r2 ≠ r3, and j ≠ n ≠ m.It can be seen from formulas ( 25)-( 27) that the whole population frst performs a random global search mutation strategy, such as formula (25).Ten, the mutation strategy is transformed, and the global best individual is used as the base vector for local optimization, as shown in formula (26).Finally, a different-dimensional mutation strategy such as formula ( 27) is adopted.When the evolutionary solution of one-dimension falls into a local optimum, the particles of other dimensions can be used to jump out of the current local optimum to avoid the algorithm from entering a premature state.Tis multistrategy diferent-dimensional mutation method combines the advantages of each mutation strategy, efectively balancing the exploration and exploitation capabilities of the algorithm.
It can be seen from the mutation strategy that the mutation factor F has a greater impact on the performance of the algorithm.In the early stage of the algorithm, mutation factor F is a larger value, which improves the global search ability of the algorithm; in the later stage of the algorithm, a smaller mutation factor F can make the algorithm more refned to improve the local search ability of the algorithm.Terefore, dynamically adjusting the value of the variation factor F is crucial to the balance the exploration ability and optimization ability of the algorithm.Terefore, this article proposes a new dynamic adjustment variation factor, which determines the variation factor of the current individual according to the ftness value ranking of the individual in population.Te specifc expression is shown in the following formulas: where F G max is the maximum variation factor of the G-th generation population, which is generated by formula (29) [22], F max and F min , respectively, represent the maximum and minimum values of the variation factor, Gen and G max , respectively, represent the current algebra and the maximum algebra, and s(i) represent the ranking of individual i in the entire population Np, which is sorted according to the individual ftness value.In this way, each individual in the population has an independent variation factor, and selects the desired search method according to its own characteristics, that is, if the variation factor is large, it means that the ranking is high in the entire population, and the corresponding individual ftness value is large, and the individual should focus on the global search, otherwise, the individual should focus on the local fne search.

Crossover Improvements. Te crossover operation recombines the mutant individual generated by the mutation operation with the original individual according to the crossover probability CR to generate a new individual.
Te traditional crossover operation is obtained by synthesizing the mutant individual and the original individual, but the conventional crossover operation has a slow evolution speed and produces better results.Te probability of the individual is small, so according to the idea of crossover operation between the optimal individual and the original individual in the improved particle swarm algorithm [23], the improved crossover operation is shown in the following formula [21]: where CR is adaptively adjusted by formula (31), xbest G is the optimal individual of the G-th generation, and the new individual is synthesized from the optimal individual and the mutant individual.In the early stage of the algorithm, the mutation individual v G+1 i,j accounts for a large proportion of the newly generated crossover individuals, which is helpful for the global optimization of the algorithm.In the later stage of the algorithm, the optimal individual xbest G j accounts for a large proportion of the newly generated crossover individuals, which is helpful for the local optimization of the algorithm.

Selection Improvements.
Te individual obtained by the crossover operation and the original individual will retain the individual with good ftness value to the next generation, so as to ensure that the algorithm is constantly approaching the optimal solution.Te traditional selection operation ignores the excellent characteristics of the mutant individual.Tis paper adopts the selection strategy idea of the traditional GA algorithm [24], the generation of new individuals is selected among the original individuals, mutant individuals, and crossover individuals.Take the minimum problem as an example, as shown in the following formula [21]: 3.3.Variable Neighborhood Search Algorithm.Te variable neighborhood search (VNS) algorithm [25] is an improved local search algorithm.It is mainly composed of two parts, namely, neighborhood action and variable neighborhood descent.Te neighborhood action can generate multiple diferent neighborhoods of the current solution, and the variable neighborhood descent can obtain the optimal value by searching alternately with the neighborhood search structure composed of diferent actions.Te basic steps of the VNS algorithm are as follows: Step 1: we generate the initial solution x according to the upper and lower limits of the variables and set the number of neighborhood search structures k max , the maximum number of searches d max , and the maximum number of iterations G max .
Step 2: we randomly select a feasible solution x ′ according to the k-th neighborhood structure N k (x) of the current solution x and then search for x ′ in the frst neighborhood search structure; at this time, k = 1, d = 1, and g � 1.
Step 3: if the ftness value of the newly generated solution x ″ is better than the current solution, update the optimal solution to x ″ , we continue to use the current neighborhood search structure to search, otherwise, let It shows that the neighborhood search structure has been unable to obtain a better solution, that is, the neighborhood search structure has been obtained the local optimal solution of the neighborhood, let k = k + 1.
Step 4: if k ≤ k max is not reached, we go to step 3, otherwise, we go to step 5.
Step 5: let g � g + 1; if g ≤ G max , we go to step 2; otherwise, we end the iteration and output the optimal solution.
Tis article mainly adopts the following three neighborhood structures: Neighborhood search structure, N 1 : we randomly select two processes with diferent process codes in the processes to exchange.Neighborhood search structure, N 2 : we randomly select a process in the processes and exchange this process with its subsequent process, if the process is the same, we reselect it backward until the process is diferent, if this process is the last process in the processes, then we rerandomize.Neighborhood search structure N 3 : we randomly select a process in the processes and exchange this process with its previous process; if the process is the same, we reselect it forward until the process is diferent; if this process is the frst in the processes, then we rerandomize.
3.4.Te TSHOA Algorithm.In this article, a two-stage optimization algorithm (TSHOA) is used to solve the fuzzy silicon single crystal scheduling problem under uncertain time with multiple sizes.First, the silicon wafer orders are converted into the number of silicon single crystal rods after obtaining orders with diferent types of single crystal furnaces and silicon wafers of diferent sizes, and the IDE algorithm is used to obtain the allocation of silicon single crystal rods of diferent sizes produced by diferent types of single crystal furnaces to complete the frst stage of problem solving.Ten, the VNS algorithm is used to solve the second stage, and the batch allocation solution are carried out for the silicon single crystal rods that need to be produced in multiple batches, so as to minimize the fnal fuzzy completion time.
Te TSHOA proposed in this article combines the advantages of the DE algorithm and the VNS algorithm.Te DE algorithm has fast convergence performance, simple parameters, and easy operation, but the algorithm is very easy to fall into local optimization.Te VNS algorithm is an improved local search algorithm, which has strong local search ability and can efectively help the DE algorithm jump out of the local optimum.Terefore, the proposed algorithm has better global and local search capabilities and has high solution accuracy.But because of the variable neighborhood search feature of the VNS algorithm, when the problem scale is large, the space of the neighborhood solution will also grow, which will consume a lot of search time.Specifcally, the steps are as follows: Step 1: we set the initialization parameters of the algorithm, the population size Np, the maximum iteration termination number G max , the value range of the independent variable, and let the initial iteration number Gen � 1.
Step 2: we initialize the population.Te frst stage initialization is performed.Te initial population is obtained by random initialization according to the upper and lower limits of the variables as shown in formula (17).
Step 3: we calculate and analyze whether it is necessary to enter the second stage according to the individual results.As shown in Table 3, when the sum of the corresponding columns of single crystal furnaces with diferent thermal felds exceeds the number of single crystal furnaces, enter the second stage to fnd out the allocation plan and obtain the objective function.If entering the second stage, use the neighborhood search Mathematical Problems in Engineering algorithm under the three criteria to schedule and optimize the process.If it does not exceed, it means that the production can be completed in one batch, and the longest fuzzy processing time is regarded as the fuzzy completion time of this type of single crystal furnace.
Step 4: judging Gen ≤ G max , if satisfed, let Gen � Gen + 1, otherwise, output the optimal allocation plan for silicon single crystal production according to the result with the smallest fuzzy completion time.
Step 5: we calculate the ftness value according to the ftness value to obtain the optimal individual xbest and the ranking of the ftness value of the individual in the whole population.
Step 6: we obtain the adaptive mutation factor according to the ranking of ftness values and perform mutation operation according to formulas ( 25)-(28).
Step 7: we perform the crossover operation according to formula (30).
Step 8: we perform the selection operation according to formula (32), select individuals with lower ftness values to form a new population, and then go to step 3.

Simulation Test
Te research in this article is based on the order task of a single crystal silicon manufacturer.Te size of the silicon single crystal rods, charging amount of single crystal furnace and the corresponding processes production time are shown in Table 4. Te scheduling results obtained by the proposed THSOA algorithm are compared to the results obtained by the manual scheduling.All simulations are implemented in MATLAB R2016a, Intel(R) Core ( ™ ) i7-7700 CPU @3.60 GHz processor with 8.00 GB memory programming.Charging fuzzy processing time 4.1.Parameter Settings.In the frst stage, the IDE algorithm parameters are set as follows: the maximum variation factor is Fmax = 1, the minimum variation factor is Fmin = 0.1, crossover probability, the maximum crossover probability is CR max � 0.9, the minimum crossover probability is CR min � 0.1, the population size is Np � 60, and the maximum number of iterations is 300.Te VNS algorithm is used in the second stage, and the population size Np � 50, the maximum number of iterations is 100, the maximum number of searches is 10 times, and three kinds of neighborhood search structures in Section 3.3 are used.

Experimental Results' Comparison and Analysis.
According to the order demand, the objective of this study is that the fuzzy completion time is the shortest.Figure 3 is the objective convergence curve, the abscissa represents the number of iterations times, and the ordinate represents the convergent completion time.It can be seen from Figure 3 that the TSHOA has reached convergence nearly 80 iterative times and its convergence speed is fast, so the solution efciency can be greatly improved in the process of solving the problem.
At present, the single crystal furnace of larger thermal felds is used to produce the corresponding size silicon single crystal rods by using the manual scheduling method.And the order allocation results obtained by the proposed algorithm and manual scheduling method are shown in Tables 5 and 6.When the quantity of silicon single crystals with diferent sizes is determined, it will be more wasteful to use manual methods, and the use of optimization-based algorithm will not only meet the production needs but also try to avoid resource waste.For example, it can be seen from Tables 5 and 6 that if the 10 Mathematical Problems in Engineering output quantity of a 6-inch silicon single crystal is 2000 kg, the optimal production result will also produce 2100 kg obtained by the manual method production, while only 2010 kg will be produced based on the optimization-based method, which will reduce the waste of 90 kg output resources.
Te allocation results obtained by the TSHOA algorithm and the manual scheduling method are given in Tables 7 and  8, and the production batches are allocated in silicon single crystal furnace of diferent thermal felds size, among them, 1, 2, 3, and 4 stand for numerical index, which means silicon single crystal rods of 6 inch, 8 inch, 10 inch, and 12 inch, respectively.
From the batch allocation results of Tables 7 and 8, the solution results obtained by the optimization-based algorithm can be used for production in a 22-inch single crystal furnace.However, if the production cycle is shortened through the manual scheduling method, a large size silicon single crystal furnace can be used to produce size silicon single crystal rods to improve production efciency, but it will undoubtedly increase the burden of production energy consumption, material consumption, etc. Table 8 shows that the 22-inch silicon single crystal furnace is not used, which undoubtedly increases the waste of resources.From the batch allocation results of Tables 7 and 8 and the fuzzy processing times of Table 4, the fuzzy completion time   of every size single crystal furnace can be obtained.Te maximum fuzzy completion time obtained by the TSHOA and manual scheduling method is (273.9,290.1, 312) and (384.4,407.6, 438.8), respectively.So, the proposed TSHOA algorithm can greatly reduce the processing time, improve production efciency, and thus reduce production cost.Figures 4-6 are the fuzzy scheduling Gantt charts obtained by the single crystal furnaces corresponding to the 22inch thermal feld, the 24-inch thermal feld, and the 28-inch thermal feld, respectively.Te lower triangle represents the fuzzy opening time of the process, and the upper triangle represents the fuzzy completion time of the process.Tere are three subscripts on the triangle, which represent the batch of processing, the size of processing, and the current process.

Figure 4 :
Figure 4: Fuzzy scheduling Gantt chart obtained by the 22-inch thermal feld single crystal furnace.

Table 1 :
Corresponding production type representation.

Table 2 :
Symbol defnition.Te order quantity of the i-th size order silicon single crystal D i Actual production quantity of the i-th size order silicon single crystal o k,j (19)lems in Engineering variables in the second stage is determined by the summation of the corresponding columns in the frst stage.For the convenience of recording, the variables in the second stage are composed of s (s � 4, it represents 4 diferent types of single crystal furnaces) cell arrays.Suppose that, each array ceils consist of 1, 2, 3, and 4, which represent 6-inch, 8inch, 10-inch, and 12-inch silicon single crystal rods, respectively.Terefore, each digit in the coding string of the second stage represents a job number, respectively.For the same job number, the number of occurrences represents the number of the processing of the same size silicon single crystal rod.And the second stage variable encoding representation is shown in equation(18).Take Table3as example, in the frst stage, the frst column corresponds to the column summation x i1 + x i2 + x i3 � 10, then the variable dimension is 10 dimensions.Assume that there are 5 single crystal furnaces corresponding to the 22-inch thermal feld, i.e., if 10 silicon single crystal rods need to be produced, the initial variables are randomly generated as shown in formula(19).�p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 where p represents the second stage variable representation.p 1 represents the cell array variable representation of the frst type of single crystal furnace, and the s type of single crystal furnace models are arranged in order.It can be seen from formula(19)that the p 1 code length is 10 in which p 1 � 2 means that the frst 8-inch silicon single crystal rod is produced on the M 1,1 , p 2 � 3 means that the frst 10-inch silicon single crystal rod is produced on the M 1,2 , p 3 � 2 means that the second 8-inch silicon single crystal rod is produced on the M 1,3 , and the following is the same, but p 6 � 3 does not represent the sixth single crystal furnace, but means that the second 10-inch silicon single crystal rod is produced on the frst completed single crystal furnace in the frst batch of 5 sets single crystal furnaces; it can be noted for the virtual sixth single crystal furnace.By analogy, the fuzzy completion time of the last single crystal furnace in the last batch can be obtained by solving the variables.Similarly, the fuzzy completion time of diferent types of single crystal furnaces can be obtained.Te maximum fuzzy completion time in s type of single crystal furnace is obtained as the objective function through L method comparison criterion.
~Figure 2: Larger operation of triangular fuzzy number.Mathematical

Table 3 :
Te frst stage encoding variable representation.

Table 4 :
Fuzzy processing time of each process.

Table 5 :
Te allocation results obtained by the TSHOA scheduling.

Table 6 :
Te allocation results obtained by the manual scheduling.

Table 7 :
Te production batch allocation results obtained by the TSHOA scheduling.

Table 8 :
Te production batch allocation results obtained by the manual scheduling.