Aircraft system manufacturing process optimization method based on improved SPEA2 algorithm

Abstract

The invention discloses an aircraft system manufacturing process optimization method based on an improved SPEA2 algorithm, and relates to the field of aviation part manufacturing. The method comprises the steps of firstly constructing a manufacturing process mathematical model; an improved SPEA2 algorithm is used for optimizing the mathematical model, the improved SPEA2 algorithm is a neighborhood search method for adding solution region density on the basis of the SPEA2 algorithm, and global search and local search can be organically combined; and finally, the aircraft system manufacturing process with the shortest manufacturing period and the lowest resource consumption is obtained through decoding. According to the method, the aircraft system manufacturing process corresponding to the mathematical model is mapped to the SPEA2 algorithm through encoding and decoding, so that optimization of the aircraft system manufacturing process is realized, the aircraft system manufacturing time is saved, and the manufacturing cost is reduced; and by using the improved SPEA2 algorithm, the diversity of the population is effectively ensured, and the convergence speed of the SPEA2 algorithm can be accelerated.

Description

Technical Field

[0001] The invention relates to the field of aviation component manufacturing, and in particular to an aircraft system manufacturing process optimization method based on an improved SPEA2 algorithm. Background Art

[0003] At present, it is still difficult to make a more scientific and reasonable plan for the large number of parallel and intertwined manufacturing processes that objectively exist in the manufacturing process of aircraft. The aircraft system manufacturing process often has typical problems such as long cycle and low efficiency. On the one hand, due to the strict requirements of aircraft integration verification, there is a complex coupling relationship between the manufacturing sequence of various components, which is manifested as complex timing constraints between manufacturing processes. On the other hand, even for the same aircraft component manufacturing, different process planning will lead to differences in the type and quantity of manufacturing resources. At the same time, the use of various manufacturing resources is often subject to various restrictions in the actual manufacturing process. Therefore, before the aircraft is officially put into manufacturing, it is the key to solve the complex problems of coupling relationship and actual constraints in the current aircraft system manufacturing process to deeply sort out the logical relationship between the manufacturing of various components and clarify the actual constraints on the consumption of manufacturing resources under different process planning.

[0004] Among various algorithms, SPEA2 (Strength Pareto Evolutionary Algorithm 2) is a classic multi-objective evolutionary algorithm. The core of SPEA2 lies in its unique fitness function, which combines the degree to which an individual is dominated and its ability to dominate other individuals to evaluate its fitness. In addition, the algorithm uses a fixed-size external archive to save the current best non-dominated solution, and continuously generates new solutions through selection, crossover, and mutation operations. SPEA2 has the advantages of strong search capability, relatively balanced convergence and diversity, and easy implementation and tuning. It has good results when facing low-complexity parallel process optimization problems.

[0005] However, when dealing with complex aircraft system manufacturing process optimization problems, there are problems such as slow algorithm convergence, poor population distribution diversity, and easy to fall into local optimality. The improved SPEA2 algorithm can effectively make up for the shortcomings of the traditional SPEA2 algorithm in dealing with aircraft system manufacturing process optimization problems by introducing a neighborhood search method based on solution area density, so as to adapt to the actual needs of aircraft system manufacturing process optimization. Summary of the invention

[0006] In order to solve the above technical problems, the purpose of the present invention is to provide an aircraft system manufacturing process optimization method based on an improved SPEA2 algorithm. First, a mathematical model of the manufacturing process is constructed. Secondly, the improved SPEA2 algorithm is used to optimize the mathematical model. The improved SPEA2 algorithm is a neighborhood search method that increases the density of the solution area on the basis of the SPEA2 algorithm, and can organically combine the global search with the local search. Finally, the aircraft system manufacturing process with the shortest manufacturing cycle and the lowest resource consumption is obtained through decoding. The present invention maps the aircraft system manufacturing process corresponding to the mathematical model to the SPEA2 algorithm through encoding and decoding, thereby realizing the optimization of the aircraft system manufacturing process, saving the aircraft system manufacturing time, and reducing the manufacturing cost. The improved SPEA2 algorithm is used to effectively ensure the diversity of the population, and can accelerate the convergence speed of the SPEA2 algorithm. Figure 1 As shown,

[0007] The technical solution process of the present invention is specifically implemented according to the following steps:

[0008] Step 1: Construct a mathematical model of the aircraft system manufacturing process:

[0009] In order to optimize the manufacturing cycle and resource consumption of the aircraft manufacturing process, when establishing a mathematical model, it is necessary to consider the interaction and influence of timing constraints, resource constraints, and different manufacturing methods in the same manufacturing process on the aircraft manufacturing process;

[0010] S1.1, build a mathematical model of timing constraints:

[0011] First, sort out the timing constraints:

[0012] In order to clearly describe various constraints and accurately express the optimization goals of shortest manufacturing cycle and least resource consumption, the aircraft manufacturing process includes J processes, where process i and process j are any two manufacturing processes in J processes; the start time of process i is S i , the end time is F i , the duration of process i is d i ; The start time of process j is S j , the end time is F j , the duration of process j is d j , the weight between process i and process j is l ij , the weight between process j and process i is l ji ;

[0013] The generalized precedence relations of the timing constraints of the aircraft system manufacturing process are divided into four types: Start-Start (SS), Finish-Start (FS), Start-Finish (SF) and Finish-Finish (FF). The four types of process relations are also accompanied by minimum and maximum time delays.

[0014] like Figure 2 As shown:

[0015] exist Figure 2 (a) In the SS relationship, a directed link is derived from the left side of process i and introduced from the left side of process j. The number of pairs on the directed link is Indicates that process j should be started after process i arrive This period of time began, are the minimum and maximum lags under the SS relation, respectively;

[0016] exist Figure 2 (b) In the FS relationship, the directed connection is derived from the right side of process i and introduced from the left side of process j. The number of pairs on the directed connection Indicates that process j should be completed after process i arrive This period of time began, are the minimum and maximum lags under the FS relationship, respectively;

[0017] exist Figure 2 (c) In the SF relationship, a directed link is drawn from the left side of process i and introduced from the right side of process j. The number of pairs on the directed link is Indicates that process j should be started after process i arrive This period of time ends, are the minimum and maximum lags under the SF relation, respectively;

[0018] exist Figure 2 (d) In the FF relationship, the directed connection is derived from the right side of process i and introduced from the right side of process j. The number of pairs on the directed connection Indicates that process j should be completed after process i arrive This period of time ends, are the minimum and maximum lags under the FF relationship respectively;

[0019] Then, the timing constraint model is determined: the end time of the process is represented by the start time plus the manufacturing cycle, and the generalized priority relationship between process i and process j is represented and uniformly converted into formulas (1) to (8):

[0020] For SS relationship, the start time of process j is within the start time of process i. back, Before, therefore need to meet The relationship between process i and process j is transformed into the following formula:

[0021]

[0022] For FS relationship, the start time of process j is within the start time of process i. after, Before, therefore need to meet The relationship between process i and process j is transformed into the following formula:

[0023]

[0024] For SF relationship, the start time of process j is within the start time of process i. after, Before, therefore need to meet The relationship between process i and process j is transformed into the following formula:

[0025]

[0026] For FF relationship, the start time of process j is within the start time of process i. after, Before, therefore need to meet The relationship between process i and process j is transformed into the following formula:

[0027]

[0028] S1.2, construct a mathematical model of manufacturing method constraints:

[0029] Among the J processes in the aircraft manufacturing process, any process j has M j The selection method of the manufacturing method of the aircraft manufacturing process is shown in formula (9);

[0030]

[0031] Among them, x jm represents the manufacturing method selected by process j, x jm When the value is 1, it means that process j selects manufacturing method m; jm When the value is 0, it means that process j does not select manufacturing method m;

[0032] Due to M jEach manufacturing process can only choose one manufacturing method, and multiple manufacturing methods cannot be used at the same time. jm The constraints that need to be met are shown in formula (10):

[0033]

[0034] S1.3, construct a mathematical model of resource constraints:

[0035] For updateable resources: for The set of processes being executed at all times. The completion of the aircraft system manufacturing process requires consumption of three renewable resources. When process j is executed under manufacturing mode m, the number of processes per day is The demand for this resource is o jmk , No. The daily supply of renewable resources is Since the use of renewable resources at any time cannot exceed the maximum supply, the mathematical model of the aircraft system manufacturing process must satisfy the constraints shown in formula (11):

[0036]

[0037] For non-renewable resources: The aircraft system manufacturing process needs to consume W non-renewable resources to complete. When the jth process is executed under manufacturing mode m, the jth process consumes W non-renewable resources every day. w The demand for non-renewable resources is o jmw , No. w The supply of non-renewable resources is O w Since the use of non-renewable resources cannot exceed the maximum supply during the entire process of aircraft system manufacturing execution, the constraint conditions shown in formula (12) must be met:

[0038]

[0039] When the objective function f1 conforms to the following formula (13), the optimized aircraft manufacturing cycle is the shortest;

[0040] f1=min max(F j )(j=1,2,…,J) (13)

[0041] When the aircraft manufacturing cycle is the shortest, when the objective function f2 meets the following formula (14), the resource consumption in the aircraft manufacturing process is the lowest;

[0042]

[0043] Among them, the unit price of the kth renewable resource is C k , the unit price of the lth non-renewable resource is Cl ;

[0044] S1.4, finally, while comprehensively considering the timing constraints, resource constraints and manufacturing method constraints, the aircraft system manufacturing process optimization mathematical model with the shortest manufacturing cycle and the lowest resource consumption as the optimization goals is established as follows:

[0045] f1=min max(F j )(j=1,2,…,J)

[0046]

[0047] Step 2: Use the improved SPEA2 algorithm to optimize the mathematical model:

[0048] The improved SPEA2 algorithm adds a neighborhood search method for solution area density based on the original SPEA2 algorithm, and organically combines global search with local search.

[0049] S2.1, encoding and decoding of SPEA2 algorithm;

[0050] S2.2, population initialization: Generate an initial population P0 of size N in a random manner, create an archive set Q0 of size η, and set algorithm parameters, including the current number of iterations t, the maximum number of iterations T, and the crossover probability Λ c , mutation probability Λ m ;

[0051] S2.3, calculate individual fitness function;

[0052] S2.4, environmental selection;

[0053] S2.5, determine whether to terminate the program.

[0054] Furthermore, the encoding and decoding process of the SPEA2 algorithm is as follows:

[0055] Chromosome coding: Using the three-dimensional coding method of priority, the three-dimensional coding structure of chromosomes is divided into three dimensions: priority, manufacturing method, and time lag coefficient of the aircraft system manufacturing process; Figure 3 As shown in the figure, the first dimension is the priority of process j, and its value range is between [0,1]. The larger the value, the higher the priority. The second dimension is the manufacturing method of process j, and its value range is [0,M j ) is an integer in the range of [0,1]; the third dimension is the time lag coefficient of process j, which is a real number in the range of [0,1], indicating the proportion of the interval time between process j and the previous process of process j to the maximum and minimum time lag difference; the priority, manufacturing method and time lag coefficient of the J process of aircraft system manufacturing are obtained through chromosome three-dimensional coding;

[0056] Chromosome decoding: V a =(v1,v2,...v a ) represents a compatible linked list with a process, where 1≤a≤J, H a =(h1,h2,...h a ) represents the sequential list of manufacturing methods corresponding to the compatible linked list, Z a =(z1,z2,...z a ) represents the sequential list of time delay coefficients corresponding to the compatible linked list;

[0057] S2.1.1, traverse all unsorted processes, when all the immediate predecessor processes of process a+1 are already in the compatible linked list V a Then process a+1 is placed into the qualified process set U;

[0058] S2.1.2, traverse the qualified process set U, select the process a+1 with the highest priority in the qualified process set U and put it into the compatible linked list, update the compatible linked list at the same time, and clear the qualified process set U in sequence;

[0059] S2.1.3, repeat S2.1.1 and S2.1.2 above until the compatible linked list contains J processes;

[0060] S2.1.4, after obtaining the complete compatible list, directly obtain H according to the chromosome decoding a and Z a ;

[0061] S2.1.5, according to the three Tables V a , H a and Z a , through the serial scheduling method, an aircraft system manufacturing plan is generated, and the generated aircraft system manufacturing plan includes the start time and end time of each process and the manufacturing method in which each process is executed;

[0062] Further, the individual fitness function calculation method is as follows:

[0063] P t represents the population after iteration t, Q t represents the archive set of iteration t times, the number of individuals dominated by individual α is β, and the intensity of the number of individuals dominated by individual α is

[0064]

[0065] In the formula, |.| represents the size of the set; It means that α dominates β;

[0066] The rough fitness of individual α is R(α). R(α) not only considers the information of the non-dominated set, but also considers the information of the dominant individual α. The solution formula of R(α) is as follows:

[0067]

[0068] The smaller the R(α) value, the more individuals are dominated by individual α. When R(α) = 0, individual α is a non-dominated individual. When multiple individuals have the same rough fitness, in order to reflect the differences between different individuals, the crowding degree D(α) between individual α and other individuals with the same rough fitness is defined, and the calculation method is as follows:

[0069] First, the k-nearest neighbor method is used to evaluate the density between individuals and calculate the relationship between individual α and population P t and archive set Q t The distance z between other individuals in j (α), in ascending order:

[0070] z1(α)≤z2(α)≤…≤z n (α) (18)

[0071] Where n = N + η;

[0072] Then, select the kth distance among the above n distances as [] indicates rounding operation;

[0073] The density D(α) of individual α is:

[0074]

[0075] Where, 0 <D(α)<1,k≤n;

[0076] Finally, the exact fitness Ψ(α) of individual α is defined as:

[0077] Ψ(i)=R(α)+D(α) (20)

[0078] In summary, by t and archive set Q t All individuals in the model define the intensity function S(α) and the density function D(α), and finally obtain the fitness of each individual ψ(α);

[0079] Further, the environment selection method is as follows:

[0080] In the iterative process of the previous step S2.3, suitable individuals are selected through environmental selection to enter the archive set Q t, non-dominated individuals are preferentially selected to enter the archive set. When the number of non-dominated individuals is smaller than the size of the archive set, the dominated individuals with the smallest fitness are selected in turn to enter the archive set. This method of environmental selection ensures the diversity of the population and avoids the result from falling into the local optimum.

[0081] In the process of environmental selection, individuals with exact fitness values ​​Ψ(α) smaller than 1 are preferentially stored in the archive set, i.e.

[0082] Q t+1 ={α|α∈P t +Q t ∩Ψ(α)<1} (21)

[0083] When Q t+1 The number of individuals in is less than the size of the archive set η, that is, Q t+1 <η, then in the previous generation P t and Q t Select (η-|Q t+1 |) excellent individuals with small fitness values ​​enter Q t+1 In; when Q t+1 >η, use the pruning process to reduce Q t+1 The number of individuals in the pruning process is as follows:

[0084] First, the distance between any two bodies in the archive set is calculated to generate a distance table;

[0085] Next, select the two points ε and γ with the smallest distance from the distance table;

[0086] Then, select the second nearest neighbor of ε and γ, that is, the point with the second smallest distance to ε and γ, compare the distances of the second nearest neighbor, and delete the point with the smallest distance to the second nearest neighbor among ε and γ.

[0087] Further, the method for determining whether to terminate the program is as follows:

[0088] If t ≥ T, then Q t+1 All non-dominated individuals in are returned as results;

[0089] On the contrary, if t<T, continue with the following steps:

[0090] S2.5.1, for archive set Q t To update:

[0091] First, the function value of the objective function is normalized:

[0092] In order to avoid the large difference between the objective function f1 and the objective function f2 and thus affect the optimization results, the following formula is used for processing:

[0093]

[0094] Where φ represents one of the objective functions f1 and f2, f t,φ is the function value to be normalized in the objective function φ of the tth generation in the archive set, and f t,φ The maximum and minimum values ​​in f' t,φ is the normalized objective function φ;

[0095] Secondly, the normalized objective function is divided into solution areas and assigned to the target region:

[0096] Solution region division: The value range of the objective function φ is [0,1], the coordinates of the extreme point in the solution space are (1,1), and a curve is constructed with the extreme point as the center and the objective function range as the radius. Rays are drawn from the extreme point to the equal division points of the curve to divide the solution space into r solution regions with equal angles; Figure 4 As shown, the solution area B v (1≤v≤r) corresponds to the angle θ v (1≤v≤r) are equal;

[0097] Solution region attribution determination: For individual α in the archive set, the solution region to which individual α belongs is determined by calculating the angle between individual α and the boundary of the solution space. The specific method is:

[0098]

[0099] Among them, B v represents the solution region to which it belongs, f′ t,1 represents the normalized t-th generation objective function f1, f′ t,2 represents the normalized t-th generation objective function f2, Indicates rounding down;

[0100] Then, the solution region is classified: the sparse solution region and the crowded solution region in the archive set are determined by the number of solutions in the solution region; the upper bound of the number of solutions in the sparse solution region is defined as BN min , the number of solutions in the crowded solution region is lower bounded by BN max , the number of individuals is greater than 0 and less than BN min The solution region is defined as the sparse solution region, and the solution in the sparse solution region is defined as a sparse solution. Similarly, the number of individuals greater than BN max The solution region less than η is defined as the crowded solution region, and the solutions in the crowded solution region are defined as crowded solutions;

[0101] Next, the neighborhood solution is generated:

[0102] In the sparse solution region: For each sparse solution μ represents the number of decision variables, is a sparse solution The ξ-th decision variable, 1≤ξ≤μ; the neighborhood interval Divide into ρ equal parts, and use the neighborhood search strategy to generate neighborhood solutions {y1,y2,...,y σ}(1≤σ≤ρ), the generation method is as follows:

[0103]

[0104] Where τ is the neighborhood radius, y σ,ξ is the neighborhood solution y σ The ξ-th decision variable of ;

[0105] In the crowded solution region: For each crowded solution Randomly select three solutions Υ=(Υ1, Υ2, ..., Υ μ ), e=(e1,e2,...,e μ ) and Φ=(Φ1,Φ2,...,Φ μ ) perturbs the neighborhood of the crowded solution x, and obtains two neighborhood solutions after perturbation

[0106]

[0107] in, It is a congestion solution The Θth decision variable of It is a congestion solution The Θth decision variable of It is a congestion solution The Θth decision variable, Υ Θ , e Θ , Φ Θ are the Θth decision variables of the crowded solutions Υ, e, Φ, respectively, and Δ is the disturbance factor;

[0108] Finally, the solution area is processed:

[0109] For each sparse solution region, the sparse solutions and neighborhood solutions in the solution region are merged to form a new population P′. The solutions with the lowest rough fitness values ​​are selected in turn from the population. If the rough fitness values ​​are the same, the solutions with the smallest density D(α) are selected in turn according to the k-nearest neighbor method until the number of solutions reaches the number boundary of the sparse solution region, which is BN. min ; For each crowded area, the crowded solutions and neighborhood solutions in the solution area are merged to form a new population P″. The solutions with the lowest rough fitness R(α) are selected in turn from the population. If the rough fitness is the same, the solutions with the smallest density D(α) are selected in turn according to the k-nearest neighbor method until the number of solutions reaches the number of crowded solution areas, which is BN.max ;

[0110] To ensure that the size of the archive set remains unchanged, if the size of the archive set is less than η after the solution area processing, individuals with high fitness are selected from the solutions deleted in the solution area processing results to enter the archive set until the size of the archive set reaches η. If the size of the archive set is greater than η after the solution area processing, the archive set is reduced through the same pruning process as in S2.4 until the size of the archive set reaches η.

[0111] At this point, the archive set has been updated;

[0112] S2.5.2, Individual Selection:

[0113] Use the binary tournament method to extract the updated archive set Q t Select individuals from the mating pool G t , from the archive set Q t Two individuals are randomly selected, and the one with higher fitness among the two individuals enters the mating pool G t , the individual with lower fitness among the two individuals is put back into the archive set Q t , re-execute the selection operation of the binary tournament method until the mating pool G t The size reaches the population size N;

[0114] S2.5.3, for mating pool G t The individuals in the crossover and mutation:

[0115] Crossover operation: According to the crossover probability Λ c , for the mating pool G t The individuals in are cross-operated with each other; Figure 5 As shown, for any two individuals g1 and g2, two integers q1, q2∈[1,J] are randomly generated; the genes q1 and q2 in the chromosome of individual g1 are kept in their original positions, and the genes with unchanged positions are copied to the offspring individual c1. Except for the genes with unchanged positions in the chromosome of individual g1, the other genes are the genes in the corresponding positions of g2; the genes q1 and q2 in the chromosome of parent individual g2 are kept in their original positions and copied to the offspring individual c2. Except for the genes with unchanged positions in the chromosome of individual g2, the other remaining genes are the genes in the corresponding positions of parent g1;

[0116] Mutation operation: according to the mutation probability Λ m , for the mating pool G t The individual g3 in the mutated state is mutated to obtain the offspring individual c3; Figure 6 As shown, a mutation probability Λ is set to determine whether each gene on the chromosome mutates g , for each gene in the parent individual g3, randomly generate a random number ο∈[0,1], if ο<Λg For the priority of the gene corresponding to the random number, a random real number in the interval [0,1] is used to replace the original gene. For the method of making the gene corresponding to the random number, a random real number in the interval [0,M J ) replaces the original gene, and for the lag coefficient of the gene corresponding to the random number, a random real number in the interval [0,1] is used to replace the original gene;

[0117] S2.5.4, after completing crossover and mutation, the next generation population P is formed t+1 , let the number of iterations be t+1, go to S2.3, and continue the loop until t≥T, then Q t+1 All non-dominated individuals in are returned as the results.

[0118] Beneficial effects of the present invention:

[0119] 1. The manufacturing process optimization method of the present invention involves multiple constraints on timing constraints, resource constraints and manufacturing method selection, achieving the dual goals of minimizing the cycle and optimizing resource consumption of the aircraft system manufacturing process;

[0120] 2. The present invention comprehensively considers the actual needs and complex situations in the aircraft manufacturing process, so that the improved SPEA2 algorithm can more comprehensively solve practical problems in the optimization of aircraft system manufacturing processes;

[0121] 3. The improved SPEA algorithm adopts a neighborhood search strategy based on the density of the solution area, which realizes the organic combination of global search and local search, effectively ensures the diversity of the population, and enables the final solution set to perform well in terms of diversity and uniformity. In addition, due to the enhanced search effect, the convergence speed of the SPEA algorithm can be accelerated. BRIEF DESCRIPTION OF THE DRAWINGS

[0122] Figure 1 It is a flow chart of the present invention;

[0123] Figure 2 It is a directed connection graph of four generalized priority relations and the corresponding process relations;

[0124] Figure 2 Among them, (a) is SS relationship, (b) FS relationship, (c) SF relationship, and (d) FF relationship;

[0125] Figure 3 This is a schematic diagram of chromosome coding;

[0126] Figure 4 Schematic diagram of the partitioning method of the solution space;

[0127] Figure 5 This is a schematic diagram of chromosome crossing over;

[0128] Figure 6 This is a schematic diagram of chromosome mutation;

[0129] Figure 7 A timing constraint diagram for the component manufacturing process;

[0130] Figure 8 Optimization result diagram for improved SPEA2;

[0131] Fig. 9 Gantt chart for HUDS. DETAILED DESCRIPTION

[0132] The technical solution of the present invention is described in detail below in conjunction with the accompanying drawings.

[0133] The present invention provides an aircraft system manufacturing process optimization method based on improved SPEA2, which achieves the purpose of shortening the construction period and reducing resource consumption by planning the start and end time of each aircraft process. A neighborhood search method based on solution area density is adopted to re-search the area with too small or too large solution area density. The manufacturing process of the airborne head-up display system (HUDS) is now optimized as an example. Figure 7 To implement the timing constraints of the manufacturing process of each component, it can be seen from the figure that HUDS is decomposed into three subsystems, namely the data processing subsystem (DPS), the control subsystem (CS), and the display subsystem (DS). The numbered boxes in the figure represent a process, and the numbers in the boxes represent the process numbers. The arrows between the process blocks represent the generalized priority relationship between the processes, and the number pairs on the arrows represent the minimum and maximum time delays. For example, for the two processes numbered 1 and 2 in the figure, the arrows are drawn from the left side of box 1 and introduced from the left side of box 2, indicating that there is an SS relationship between the two processes. The number pair on the arrow (3,5) indicates that the minimum time delay is 3 days and the maximum time delay is 5 days. For another example, for the two processes numbered 5 and 9 in the figure, the arrows are drawn from the right side of box 5 and introduced from the left side of box 9, indicating that there is an FS relationship between the two processes. The number pair on the arrow (2,4) indicates that the minimum time delay is 2 days and the maximum time delay is 4 days. From Figure 7It can also be seen that the number of timing constraints within each subsystem is large, and the number of timing constraints between subsystems is small. In addition, Table 1 shows the consumption of different types of resources by each component of the airborne head-up display system (HUDS) under different manufacturing methods. It can be seen from Table 1 that each process has two manufacturing methods, and generally speaking, the first manufacturing method requires more time but less resource requirements, while the second manufacturing method requires less time but more resource requirements. When both time and resources are constrained, it is necessary to reasonably select the manufacturing method for each process to achieve the completion of HUDS manufacturing in a shorter time with less resource consumption. In addition, it can be seen from Table 1 that the number of renewable resources is 3, including technicians, display equipment, and manufacturing equipment; the number of non-renewable resources is 3, including peripheral components, core components, and hardware materials. Therefore, the manufacturing process of these 30 components is planned to obtain the shortest manufacturing cycle and the lowest resource consumption.

[0134] Table 1 Different manufacturing methods of HUDS component-level manufacturing process

[0135]

[0136]

[0137]

[0138] The following are the specific implementation steps of the present invention:

[0139] Step 1: Construct a mathematical model of the aircraft system manufacturing process:

[0140] S1.1, build a mathematical model of timing constraints:

[0141] The aircraft manufacturing process is defined as J processes, J = 30, where process i and process j are any two manufacturing processes in J processes; the start time of process i is S i , the end time is F i , the duration of process i is d i ; The start time of process j is S j , the end time is F j , the duration of process j is d j , the weight between process i and process j is l ij , the weight between process j and process i is l ji; The generalized precedence relations of the timing constraints of the aircraft system manufacturing process are divided into four types: Start-Start (SS), Finish-Start (FS), Start-Finish (SF) and Finish-Finish (FF). The four types of process relations are also accompanied by minimum and maximum time delays;

[0142] The end time of the process is expressed as the start time plus the manufacturing cycle, and the generalized priority relationship between process i and process j is expressed and uniformly converted as shown below:

[0143] S i +l ij ≤S j ,

[0144]

[0145] S1.2, construct a mathematical model of manufacturing method constraints:

[0146] Among the J processes in the aircraft manufacturing process, any process j has M j Manufacturing method, M j =2; The selection method of the manufacturing process of the aircraft manufacturing process is as follows:

[0147]

[0148] Among them, x jm represents the manufacturing method selected by process j, x jm When the value is 1, it means that process j selects manufacturing method m; jm When the value is 0, it means that process j does not select manufacturing method m;

[0149] Due to M j Each manufacturing process can only choose one manufacturing method, and multiple manufacturing methods cannot be used at the same time. jm The constraints that need to be met are shown in the formula:

[0150]

[0151] S1.3, construct a mathematical model of resource constraints:

[0152] For updateable resources: for The set of processes being executed at the moment. The aircraft system manufacturing process needs to consume Ξ kinds of renewable resources to complete, Ξ = 3. When process j is executed under manufacturing mode m, the number of renewable resources per day is The demand for this resource is o jmk , No. The daily supply of renewable resources is Since the use of renewable resources at any time cannot exceed the maximum supply, the mathematical model of the aircraft system manufacturing process must meet the following constraints:

[0153]

[0154] For non-renewable resources: The aircraft system manufacturing process needs to consume W non-renewable resources to complete, W = 3, and the jth process is executed under manufacturing mode m. w The demand for non-renewable resources is o jmw , No. w The supply of non-renewable resources is O w , since the use of non-renewable resources cannot exceed the maximum supply during the entire process of aircraft system manufacturing execution, the following constraints must be met:

[0155]

[0156] When the objective function f1 conforms to the following formula, the optimized aircraft manufacturing cycle is the shortest;

[0157] f1=minmax(F j )(j=1,2,…,J)

[0158] When the aircraft manufacturing cycle is the shortest, when the objective function f2 meets the following formula, the resource consumption in the aircraft manufacturing process is the lowest;

[0159]

[0160] Among them, the unit price of the kth renewable resource is C k , the unit price of the lth non-renewable resource is C l ;

[0161] Therefore, the mathematical model for aircraft system manufacturing process optimization is as follows:

[0162] f1=minmax(F j )(j=1,2,…,J)

[0163]

[0164] Step 2: Use the improved SPEA2 algorithm to optimize the mathematical model:

[0165] The improved SPEA2 algorithm adds a neighborhood search method for solution area density based on the original SPEA2 algorithm, and organically combines global search with local search.

[0166] S2.1, encoding and decoding of SPEA2 algorithm:

[0167] The chromosome coding adopts the three-dimensional coding method of priority. The three-dimensional coding structure of the chromosome is the priority (Priority), manufacturing method (Pattern) and time lag coefficient (Timelag) of the aircraft system manufacturing process. The first dimension is the priority of process j, and the value range is between [0,1]. The larger the value, the higher the priority. The second dimension is the manufacturing method of process j, and the value range is [0,M j ) is an integer in the range of [0,1]; the third dimension is the time lag coefficient of process j, which is a real number in the range of [0,1], indicating the proportion of the interval time between process j and the previous process of process j to the maximum and minimum time lag difference; the priority, manufacturing method and time lag coefficient of the J process of aircraft system manufacturing are obtained through chromosome three-dimensional coding;

[0168] Chromosome decoding: V a =(v1,v2,...v a ) represents a compatible linked list with a process, where 1≤a≤J, H a =(h1,h2,...h a ) represents the sequential list of manufacturing methods corresponding to the compatible linked list, Z a =(z1,z2,...z a ) represents the sequential list of time delay coefficients corresponding to the compatible linked list;

[0169] S2.1.1, traverse all unsorted processes, when all the immediate predecessor processes of process a+1 are already in the compatible linked list V a Then process a+1 is placed into the qualified process set U;

[0170] S2.1.2, traverse the qualified process set U, select the process a+1 with the highest priority in the qualified process set U and put it into the compatible linked list, update the compatible linked list at the same time, and clear the qualified process set U in sequence;

[0171] S2.1.3, repeat S2.1.1 and S2.1.2 above until the compatible linked list contains J processes;

[0172] S2.1.4, after obtaining the complete compatible list, directly obtain H according to the chromosome decoding a and Z a ;

[0173] S2.1.5, according to the three Tables V a , H a and Z a , through the serial scheduling method, an aircraft system manufacturing plan is generated, and the generated aircraft system manufacturing plan includes the start time and end time of each process and the manufacturing method in which each process is executed;

[0174] S2.2, population initialization:

[0175] The initial population P0 of size N = 100 is generated randomly, and the archive set Q0 of size η = 50 is created at the same time. The current iteration number t = 0, the maximum iteration number T = 200, and the crossover probability Λ c =0.9, mutation probability Λ m =0.1;

[0176] S2.3, calculate the individual fitness function:

[0177] P t represents the population after iteration t, Q t represents the archive set of iteration t times, the number of individuals dominated by individual α is β, and the intensity of the number of individuals dominated by individual α is

[0178]

[0179] In the formula, |.| represents the size of the set; It means that α dominates β;

[0180] The rough fitness of individual α is R(α), and the solution formula of R(α) is as follows:

[0181]

[0182] When multiple individuals have the same rough fitness, in order to reflect the differences between different individuals, the crowding degree D(α) between individual α and other individuals with the same rough fitness is defined, and the calculation method is as follows:

[0183] First, the k-nearest neighbor method is used to evaluate the density between individuals and calculate the relationship between individual α and population P t and archive set Q t The distance z between other individuals in j (α), in ascending order:

[0184] z1(α)≤z2(α)≤…≤z n (α)

[0185] Where n = N + η;

[0186] Then, select the kth distance among the above n distances as [] indicates rounding operation;

[0187] The density D(α) of individual α is:

[0188]

[0189] Where, 0 <D(α)<1,k≤n;

[0190] Finally, the exact fitness Ψ(α) of individual α is defined as:

[0191] Ψ(i)=R(α)+D(α)

[0192] In summary, by t and archive set Q t All individuals in the model define the intensity function S(α) and the density function D(α), and finally obtain the fitness ψ(α) of each individual;

[0193] S2.4, Environment Selection:

[0194] In the iterative process of the previous step S2.3, suitable individuals are selected through environmental selection to enter the archive set Q t , non-dominated individuals are preferentially selected to enter the archive set. When the number of non-dominated individuals is smaller than the size of the archive set, the dominated individuals with the smallest fitness are selected in turn to enter the archive set. This method of environmental selection ensures the diversity of the population and avoids the result from falling into the local optimum.

[0195] In the process of environmental selection, individuals with exact fitness values ​​Ψ(α) smaller than 1 are preferentially stored in the archive set, i.e.

[0196] Q t+1 ={α|α∈P t +Q t ∩Ψ(α)<1}

[0197] When Q t+1 The number of individuals in is less than the size of the archive set η, that is, Q t+1 <η, then in the previous generation P t and Q t Select (η-|Q t+1 |) excellent individuals with small fitness values ​​enter Q t+1 In; when Q t+1 >η, use the pruning process to reduce Q t+1 The number of individuals in the pruning process is as follows:

[0198] First, the distance between any two bodies in the archive set is calculated to generate a distance table; then, the two points ε and γ with the smallest distance are selected from the distance table; then, the second nearest neighbor points of ε and γ, that is, the points with the second smallest distance to ε and γ, are selected, and the distances of the second nearest neighbor points are compared, and the points with the smallest distance to the second nearest neighbor points in the two points of ε and γ are deleted;

[0199] S2.5, determine whether to terminate the program:

[0200] If t ≥ T, then Q t+1All non-dominated individuals in are returned as the result; otherwise, if t<T, continue with the following steps:

[0201] S2.5.1, for archive set Q t To update:

[0202] First, the function value of the objective function is normalized by the following formula:

[0203]

[0204] Where φ represents one of the objective functions f1 and f2, f t,φ is the function value to be normalized in the objective function φ of the tth generation in the archive set, and f t,φ The maximum and minimum values ​​in f' t,φ is the normalized objective function φ;

[0205] Secondly, the normalized objective function is divided into solution areas and assigned to the target region:

[0206] Solution region division: The value range of the objective function φ is [0,1], the coordinates of the extreme point in the solution space are (1,1), a curve is constructed with the extreme point as the center and the objective function range as the radius, and rays are drawn from the extreme point to the equally divided points of the curve to divide the solution space into r = 10 solution regions with equal angles;

[0207] Solution region attribution determination: For individual α in the archive set, the solution region to which individual α belongs is determined by calculating the angle between individual α and the boundary of the solution space. The specific method is:

[0208]

[0209] Among them, B v represents the solution region to which it belongs, f′ t,1 represents the normalized t-th generation objective function f1, f′ t,2 represents the normalized t-th generation objective function f2, Indicates rounding down;

[0210] Then, the solution region is classified: the sparse solution region and the crowded solution region in the archive set are determined by the number of solutions in the solution region; the upper bound of the number of solutions in the sparse solution region is defined as BN min The value is 4, and the number of solutions in the crowded solution region is lower bounded by BN max The value is 8, which means the number of individuals is greater than 0 and less than BN. min The solution region is defined as the sparse solution region, and the solution in the sparse solution region is defined as a sparse solution. Similarly, the number of individuals greater than BN maxThe solution region less than η is defined as the crowded solution region, and the solutions in the crowded solution region are defined as crowded solutions;

[0211] Next, the neighborhood solution is generated:

[0212] In the sparse solution region: For each sparse solution μ represents the number of decision variables, which is 3, namely the priority, execution mode and lag coefficient of the process. is a sparse solution The ξ-th decision variable, 1≤ξ≤μ; the neighborhood interval It is divided into ρ equal parts, ρ is 6, and a neighborhood search strategy is used to generate neighborhood solutions {y1,y2,...,y σ}(1≤σ≤ρ), the generation method is as follows:

[0213]

[0214] Among them, τ is the neighborhood radius, where the neighborhood radius of the priority dimension is 0.1, the neighborhood radius of the execution mode dimension is 1, and the neighborhood radius of the lag coefficient dimension is 0.1. σ,ξ is the neighborhood solution y σ The ξ-th decision variable of ;

[0215] In the crowded solution region: For each crowded solution Randomly select three solutions Υ=(Υ1, Υ2, ..., Υ μ ), e=(e1,e2,...,e μ ) and Φ=(Φ1,Φ2,...,Φ μ ) perturbs the neighborhood of the crowded solution x, and obtains two neighborhood solutions after perturbation

[0216]

[0217] in, It is a congestion solution The Θth decision variable of It is a congestion solution The Θth decision variable of It is a congestion solution The Θth decision variable, Υ Θ , e Θ , Φ Θ are the Θth decision variables of the crowded solutions Υ, e, Φ respectively, Δ is the disturbance factor, and its value is 0.03;

[0218] Finally, the solution area is processed:

[0219] For each sparse solution region, the sparse solutions and neighborhood solutions in the solution region are merged to form a new population P′. The solutions with the lowest rough fitness values ​​are selected in turn from the population. If the rough fitness values ​​are the same, the solutions with the smallest density D(α) are selected in turn according to the k-nearest neighbor method until the number of solutions reaches the number boundary of the sparse solution region, which is BN. min ; For each crowded area, the crowded solutions and neighborhood solutions in the solution area are merged to form a new population P″. The solutions with the lowest rough fitness R(α) are selected in turn from the population. If the rough fitness is the same, the solutions with the smallest density D(α) are selected in turn according to the k-nearest neighbor method until the number of solutions reaches the number of crowded solution areas, which is BN. max ;

[0220] To ensure that the size of the archive set remains unchanged, if the size of the archive set is less than η after the solution area processing, individuals with high fitness are selected from the solutions deleted in the solution area processing results to enter the archive set until the size of the archive set reaches η. If the size of the archive set is greater than η after the solution area processing, the archive set is reduced through the same pruning process as in S2.4 until the size of the archive set reaches η.

[0221] At this point, the archive set has been updated;

[0222] S2.5.2, Select:

[0223] Use the binary tournament method to extract the updated archive set Q t Select individuals from the mating pool G t , from the archive set Q t Two individuals are randomly selected, and the one with higher fitness among the two individuals enters the mating pool G t , the individual with lower fitness among the two individuals is put back into the archive set Q t , re-execute the selection operation of the binary tournament method until the mating pool G t The size reaches the population size N;

[0224] S2.5.3, for mating pool G t The individuals in the crossover and mutation:

[0225] Crossover operation: According to the crossover probability Λ c , for the mating pool G tThe individuals in the crossover operation are performed two by two; for any two individuals g1 and g2, two integers q1, q2∈[1,J] are randomly generated; the genes q1 and q2 in the chromosome of individual g1 are kept unchanged in their original positions, and the genes with unchanged positions are copied to the offspring individual c1. Except for the genes with unchanged positions in the chromosome of individual g1, the other genes are the genes with corresponding positions in g2; the genes q1 and q2 in the chromosome of parent individual g2 are kept unchanged in their original positions, and copied to the offspring individual c2. Except for the genes with unchanged positions in the chromosome of individual g2, the other remaining genes are the genes with corresponding positions in the parent g1;

[0226] Mutation operation: according to the mutation probability Λ m , for the mating pool G t The individual g3 in the chromosome is mutated to obtain the offspring individual c3; a mutation probability Λ is set to determine whether each gene on the chromosome mutates g , for each gene in the parent individual g3, randomly generate a random number ο∈[0,1], if ο<Λ g For the priority of the gene corresponding to the random number, a random real number in the interval [0,1] is used to replace the original gene. For the method of making the gene corresponding to the random number, a random real number in the interval [0,M J ) replaces the original gene, and for the lag coefficient of the gene corresponding to the random number, a random real number in the interval [0,1] is used to replace the original gene;

[0227] S2.5.4, after completing crossover and mutation, the next generation population P is formed t+1 , let the number of iterations be t+1, go to S2.3, and continue the loop until t≥T, then Q t+1 All non-dominated individuals in are returned as results.

[0228] The group Pareto solution set obtained by optimizing and solving this example using the improved SPEA2 proposed in this paper is as follows: Figure 8 As shown in the figure, each point represents a Pareto optimal solution. The horizontal axis represents the manufacturing time of the airborne head-up display system component level, and the vertical axis represents the resource consumption of the system. For each Pareto solution, we can determine its process sequence and manufacturing method through model calculation. From the distribution of solutions, it can be seen that in the actual optimization process, the improved SPEA2 proposed in this paper has a better solution set than the original SPEA2 algorithm, with more solutions, more uniform distribution of solutions, and better diversity and convergence. It proves that the improved SPEA2 can obtain better solutions than the original SPEA2, and also proves that the proposed method can be effectively applied to the optimization of aircraft system manufacturing processes.

[0229] In addition, if Fig. 9The HUDS (Head-up Display System, HUDS) Gantt chart shows the feasible detailed scheduling plan corresponding to the solution with a manufacturing cycle of 97 days and a resource consumption of 66,861 yuan in the Pareto chart, which is expressed in the form of a Gantt chart. The horizontal axis in the figure represents the number of days the plan is executed, and the vertical axis represents the manufacturing status of 30 components. Each rectangle represents the manufacturing process of a component being executed, where orange indicates that the manufacturing process is executed using manufacturing method 1, and blue indicates that the manufacturing process is executed using manufacturing method 2. It can be seen from the figure that this aspect can reasonably arrange the start and end times of 30 processes under the conditions of meeting timing constraints and resource constraints, and can reasonably choose between the two manufacturing modes to achieve the purpose of shortening the manufacturing cycle and reducing resource consumption.

Claims

1. An aircraft system manufacturing process optimization method based on an improved SPEA2 algorithm, characterized in that: The manufacturing process optimization method comprises the following steps: Step 1: Construct a mathematical model of the aircraft system manufacturing process; Step 2: Optimize the mathematical model using the improved SPEA2 algorithm; Step three: Optimize the aircraft system manufacturing process based on the mathematical model.

2. The aircraft system manufacturing process optimization method based on the improved SPEA2 algorithm according to claim 1 is characterized in that: In step 1, the steps of constructing the mathematical model are as follows: S1.1, build a mathematical model of timing constraints: First, the timing constraints are sorted out: the aircraft manufacturing process includes J processes, where process i and process j are any two manufacturing processes in J processes; the start time of process i is S i , the end time is F i , the duration of process i is d i ; The start time of process j is S j , the end time is F j , the duration of process j is d j , the weight between process i and process j is l ij , the weight between process j and process i is l ji ; The generalized precedence relations of the timing constraints of the aircraft system manufacturing process are divided into four types: start-start, end-start, start-end and end-end. The four types of process relations are also accompanied by minimum and maximum time delays. In the SS relationship, the directed connection is derived from the left side of process i and introduced from the left side of process j. Indicates that process j should be started after process i arrive This period of time began, are the minimum and maximum lags under the SS relation, respectively; In the FS relationship, the directed connection is derived from the right side of process i and introduced from the left side of process j. Indicates that process j should be completed after process i arrive This period of time began, are the minimum and maximum lags under the FS relationship, respectively; In the SF relationship, the directed connection is derived from the left side of process i and introduced from the right side of process j. Indicates that process j should be started after process i arrive This period of time ends, are the minimum and maximum lags under the SF relation, respectively; In the FF relationship, the directed connection is derived from the right side of process i and introduced from the right side of process j. Indicates that process j should be completed after process i arrive This period of time ends, are the minimum and maximum lags under the FF relationship respectively; Then, the timing constraint model is determined: the end time of the process is represented by the start time plus the manufacturing cycle, and the generalized priority relationship between process i and process j is represented and uniformly converted into formulas (1) to (8): For SS relationship, the start time of process j is within the start time of process i. back, Before, therefore need to meet The relationship between process i and process j is transformed into the following formula: For FS relationship, the start time of process j is within the start time of process i. after, Before, therefore need to meet The relationship between process i and process j is transformed into the following formula: For SF relationship, the start time of process j is within the start time of process i. after, Before, therefore need to meet The relationship between process i and process j is transformed into the following formula: For FF relationship, the start time of process j is within the start time of process i. after, Before, therefore need to meet The relationship between process i and process j is transformed into the following formula: S1.2, construct a mathematical model of manufacturing method constraints: Among the J processes in the aircraft manufacturing process, any process j has M j The selection method of the manufacturing method of the aircraft manufacturing process is shown in formula (9); Among them, x jm represents the manufacturing method selected by process j, x jm When the value is 1, it means that process j selects manufacturing method m; jm When the value is 0, it means that process j does not select manufacturing method m; Due to M j Each manufacturing process can only choose one manufacturing method, and multiple manufacturing methods cannot be used at the same time. jm The constraints that need to be met are shown in formula (10): S1.3, construct a mathematical model of resource constraints: For updateable resources: for The set of processes being executed at all times. The completion of the aircraft system manufacturing process requires consumption of three renewable resources. When process j is executed under manufacturing mode m, the number of processes per day is The demand for this resource is o jmk , No. The daily supply of renewable resources is Since the use of renewable resources at any time cannot exceed the maximum supply, the mathematical model of the aircraft system manufacturing process must satisfy the constraints shown in formula (11): For non-renewable resources: The aircraft system manufacturing process needs to consume W non-renewable resources to complete. When the jth process is executed under manufacturing mode m, the daily demand for the wth non-renewable resource is o. jmw , the supply of the wth non-renewable resource is O w Since the use of non-renewable resources cannot exceed the maximum supply during the entire process of aircraft system manufacturing execution, the constraint conditions shown in formula (12) must be met: When the objective function f1 conforms to the following formula (13), the optimized aircraft manufacturing cycle is the shortest; f1=min max(F j )(j=1,2,…,J) (13) When the aircraft manufacturing cycle is the shortest, when the objective function f2 meets the following formula (14), the resource consumption in the aircraft manufacturing process is the lowest; Among them, the unit price of the kth renewable resource is C k , the unit price of the lth non-renewable resource is C l ; S1.4, finally, while comprehensively considering the timing constraints, resource constraints and manufacturing method constraints, the aircraft system manufacturing process optimization mathematical model with the shortest manufacturing cycle and the lowest resource consumption as the optimization goals is established as follows: f1=minmax(F j )(j=1,2,…,J) 3. The aircraft system manufacturing process optimization method based on the improved SPEA2 algorithm according to claim 1 is characterized in that: In step 2, the optimization method of the mathematical model is as follows: S2.1, encoding and decoding of SPEA2 algorithm; S2.2, population initialization: Generate an initial population P0 of size N in a random manner, create an archive set Q0 of size η, and set algorithm parameters, including the current number of iterations t, the maximum number of iterations T, and the crossover probability Λ c , mutation probability Λ m ; S2.3, calculate individual fitness function; S2.4, environmental selection; S2.5, determine whether to terminate the program.

4. The method for optimizing aircraft system manufacturing process based on improved SPEA2 algorithm according to claim 3, characterized in that: The encoding and decoding process of the SPEA2 algorithm is as follows: Chromosome coding: The three-dimensional coding method of priority is adopted. The three-dimensional coding structure of chromosome is divided into three dimensions: priority, manufacturing method and time lag coefficient of aircraft system manufacturing process. The first dimension is the priority of process j, and the value range is between [0,1]. The larger the value, the higher the priority. The second dimension is the manufacturing method of process j, and the value range is [0,M j ) is an integer in the range of [0,1]; the third dimension is the time lag coefficient of process j, which is a real number in the range of [0,1], indicating the proportion of the interval time between process j and the previous process of process j to the maximum and minimum time lag difference; the priority, manufacturing method and time lag coefficient of the J process of aircraft system manufacturing are obtained through chromosome three-dimensional coding; Chromosome decoding: V a =(v1,v2,...v a ) represents a compatible linked list with a process, where 1≤a≤J, H a =(h1,h2,...h a ) represents the sequential list of manufacturing methods corresponding to the compatible linked list, Z a =(z1,z2,...z a ) represents the sequential list of time delay coefficients corresponding to the compatible linked list; S2.1.1, traverse all unsorted processes, when all the immediate predecessor processes of process a+1 are already in the compatible linked list V a Then process a+1 is placed into the qualified process set U; S2.1.2, traverse the qualified process set U, select the process a+1 with the highest priority in the qualified process set U and put it into the compatible linked list, update the compatible linked list at the same time, and clear the qualified process set U in sequence; S2.1.3, repeat S2.1.1 and S2.1.2 above until the compatible linked list contains J processes; S2.1.4, after obtaining the complete compatible list, directly obtain H according to the chromosome decoding a and Z a ; S2.1.5, according to the three Tables V a , H a and Z a , through the serial scheduling method, an aircraft system manufacturing plan is generated, and the generated aircraft system manufacturing plan includes the start time and end time of each process and the manufacturing method in which each process is executed.

5. The method for optimizing aircraft system manufacturing process based on improved SPEA2 algorithm according to claim 3, characterized in that: The individual fitness function calculation method is as follows: P t represents the population after iteration t, Q t represents the archive set of iteration t times, the number of individuals dominated by individual α is β, and the intensity of the number of individuals dominated by individual α is In the formula, |.| represents the size of the set; α<β means α dominates β; The rough fitness of individual α is R(α). R(α) not only considers the information of the non-dominated set, but also considers the information of the dominant individual α. The solution formula of R(α) is as follows: The smaller the R(α) value, the more individuals are dominated by individual α. When R(α) = 0, individual α is a non-dominated individual. When multiple individuals have the same rough fitness, in order to reflect the differences between different individuals, the crowding degree D(α) between individual α and other individuals with the same rough fitness is defined, and the calculation method is as follows: First, the k-nearest neighbor method is used to evaluate the density between individuals and calculate the relationship between individual α and population P t and archive set Q t The distance z between other individuals in j (α), in ascending order: z1(a)≤z2(a)≤…≤z n (a) (18) Where n = N + η; Then, select the kth distance among the above n distances as [] indicates rounding operation; The density D(α) of individual α is: Where, 0 <D(α)<1,k≤n; Finally, the exact fitness Ψ(α) of individual α is defined as: Ψ(i)=R(α)+D(α) (20) In summary, by t and archive set Q t The intensity function S(α) and density function D(α) are defined for all individuals in the , and finally the fitness ψ(α) of each individual is obtained.

6. The method for optimizing aircraft system manufacturing process based on improved SPEA2 algorithm according to claim 3, characterized in that: The environment selection method is as follows: Select suitable individuals through environmental selection and enter the archive set Q t , non-dominated individuals are preferentially selected to enter the archive set. When the number of non-dominated individuals is smaller than the size of the archive set, the dominated individuals with the smallest fitness are selected in turn to enter the archive set; In the process of environmental selection, individuals with exact fitness values ​​Ψ(α) smaller than 1 are preferentially stored in the archive set, i.e. Q t+1 ={α|α∈P t +Q t ∩Ψ(α)<1} (21) When Q t+1 The number of individuals in is less than the size of the archive set η, that is, Q t+1 <η, then in the previous generation P t and Q t Select (η-|Q t+1 ) excellent individuals with small fitness values ​​enter Q t+1 In; when Q t+1 >η, use the pruning process to reduce Q t+1 The number of individuals in the pruning process is as follows: First, the distance between any two bodies in the archive set is calculated to generate a distance table; Next, select the two points ε and γ with the smallest distance from the distance table; Then, select the second nearest neighbor points of ε and γ, that is, the points with the second smallest distance to ε and γ, compare the distances to the second nearest neighbor points, and delete the points of ε and γ with the smallest distance to the second nearest neighbor points.

7. The method for optimizing aircraft system manufacturing process based on improved SPEA2 algorithm according to claim 3, characterized in that: The determination method of the termination procedure is as follows: If t ≥ T, then Q t+1 All non-dominated individuals in are returned as results; On the contrary, if t<T, continue with the following steps: S2.5.1, for archive set Q t To update: First, the function value of the objective function is normalized: In order to avoid the large difference between the objective function f1 and the objective function f2 and affect the optimization results, the following formula is used for processing: Where φ represents one of the objective functions f1 and f2, f t,φ is the function value to be normalized in the objective function φ of the tth generation in the archive set, and f t,φ The maximum and minimum values ​​in f′ t,φ is the normalized objective function φ; Secondly, the normalized objective function is divided into solution areas and assigned to the target region: Solution region division: The value range of the objective function φ is [0,1], the coordinates of the extreme point in the solution space are (1,1), and a curve is constructed with the extreme point as the center and the objective function range as the radius. Rays are drawn from the extreme point to the equally divided points of the curve to divide the solution space into r solution regions with equal angles; Solution region B v The corresponding angle θ v Equal, where 1≤v≤r; Solution region attribution determination: For individual α in the archive set, the solution region to which individual α belongs is determined by calculating the angle between individual α and the boundary of the solution space. The specific method is: Among them, B v represents the solution region to which it belongs, f′ t,1 represents the normalized t-th generation objective function f1, f′ t,2 represents the normalized t-th generation objective function f2, Indicates rounding down; Then, the solution region is classified: the sparse solution region and the crowded solution region in the archive set are determined by the number of solutions in the solution region; the upper bound of the number of solutions in the sparse solution region is defined as BN min , the number of solutions in the crowded solution region is lower bounded by BN max , the number of individuals is greater than 0 and less than BN min The solution region is defined as the sparse solution region, and the solution in the sparse solution region is defined as a sparse solution. Similarly, the number of individuals greater than BN max The solution region less than η is defined as the crowded solution region, and the solutions in the crowded solution region are defined as crowded solutions; Next, the neighborhood solution is generated: In the sparse solution region: For each sparse solution μ represents the number of decision variables, is a sparse solution The ξ-th decision variable, 1≤ξ≤μ; the neighborhood interval Divide into ρ equal parts, and use the neighborhood search strategy to generate neighborhood solutions {y1,y2,...,y σ }(1≤σ≤ρ), the generation method is as follows: Where τ is the neighborhood radius, y σ,ξ is the neighborhood solution y σ The ξ-th decision variable of ; In the crowded solution region: For each crowded solution Randomly select three solutions Υ=(Υ1, Υ2, ..., Υ μ ), e=(e1,e2,...,e μ ) and Φ=(Φ1,Φ2,...,Φ μ ) perturbs the neighborhood of the crowded solution x, and obtains two neighborhood solutions after perturbation in, It is a congestion solution The Θth decision variable of It is a congestion solution The Θth decision variable of It is a congestion solution The Θth decision variable, Υ Θ , e Θ , Φ Θ are the Θth decision variables of the crowded solutions Υ, e, Φ, respectively, and Δ is the disturbance factor; Finally, the solution area is processed: For each sparse solution region, the sparse solutions and neighborhood solutions in the solution region are merged to form a new population P′. The solutions with the lowest rough fitness values ​​are selected in turn from the population. If the rough fitness values ​​are the same, the solutions with the smallest density D(α) are selected in turn according to the k-nearest neighbor method until the number of solutions reaches the number boundary of the sparse solution region, which is BN. min ; For each crowded area, the crowded solutions and neighborhood solutions in the solution area are merged to form a new population P″. The solutions with the lowest rough fitness R(α) are selected in turn from the population. If the rough fitness is the same, the solutions with the smallest density D(α) are selected in turn according to the k-nearest neighbor method until the number of solutions reaches the number of crowded solution areas, which is BN. max ; To ensure that the size of the archive set remains unchanged, if the size of the archive set is less than η after the solution area processing, individuals with high fitness are selected from the solutions deleted in the solution area processing results to enter the archive set until the size of the archive set reaches η. If the size of the archive set is greater than η after the solution area processing, the archive set is reduced through the same pruning process as in S2.4 until the size of the archive set reaches η. At this point, the archive set has been updated; S2.5.2, Individual Selection: Use the binary tournament method to extract the updated archive set Q t Select individuals from the mating pool G t , from the archive set Q t Two individuals are randomly selected, and the one with higher fitness enters the mating pool G t , the individual with lower fitness among the two individuals is put back into the archive set Q t , re-execute the selection operation of the binary tournament method until the mating pool G t The size reaches the population size N; S2.5.3, for mating pool G t The individuals in the crossover and mutation: S2.5.4, after completing crossover and mutation, the next generation population P is formed t+1 , let the number of iterations be t+1, go to S2.3, and continue the loop until t≥T, then Q t+1 All non-dominated individuals in are returned as results.

8. The aircraft system manufacturing process optimization method based on the improved SPEA2 algorithm according to claim 7 is characterized in that: In S2.5.3, the specific method of the crossover operation is as follows: According to the crossover probability Λ c , for the mating pool G t The individuals in the set are crossovered in pairs; for any two individuals g1 and g2, two integers q1, q2∈[1,J] are randomly generated; the genes q1 and q2 in the chromosome of individual g1 are kept in their original positions, and the genes with unchanged positions are copied to the offspring individual c1. Except for the genes with unchanged positions in the chromosome of individual g1, the other genes are the genes in the corresponding positions of g2; the genes q1 and q2 in the chromosome of parent individual g2 are kept in their original positions and copied to the offspring individual c2. Except for the genes with unchanged positions in the chromosome of individual g2, the other remaining genes are the genes in the corresponding positions of parent g1.

9. The aircraft system manufacturing process optimization method based on the improved SPEA2 algorithm according to claim 7 is characterized in that: In S2.5.3, the specific method of the mutation operation is as follows: According to the mutation probability Λ m , for the mating pool G t The individual g3 in the chromosome is mutated to obtain the offspring individual c3; a mutation probability Λ is set to determine whether each gene on the chromosome mutates g , for each gene in the parent individual g3, randomly generate a random number ο∈[0,1], if ο<Λ g For the priority of the gene corresponding to the random number, a random real number in the interval [0,1] is used to replace the original gene. For the method of making the gene corresponding to the random number, a random real number in the interval [0,M J ) replaces the original gene, and for the lag coefficient of the gene corresponding to the random number, a random real number in the interval [0,1] is used to replace the original gene.

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