Additive manufacturing multi-target collaborative optimization method fusing direction selection and multi-machine scheduling

Abstract

The invention provides an additive manufacturing multi-target collaborative optimization method fusing direction selection and multi-machine scheduling, and belongs to the technical field of precision forming. Comprising the following steps: 1) a direction-boxing joint optimization step: based on a candidate construction direction of a part, deciding a target construction direction of the part, and generating an operation boxing scheme by taking minimization of a support volume ratio and maximization of a space utilization rate as targets; and 2) a multi-machine multi-target scheduling step: distributing the operation generated in the above step to heterogeneous additive manufacturing equipment, and generating a scheduling scheme by taking the minimization of the total completion time of the system and the unit volume cost of the system as targets. And collaborative optimization is realized through an NSGA-II algorithm improved by an improved genetic algorithm and a dynamic ideal point method. The problems that in the prior art, only a single-machine scene or isomorphic equipment is considered, equipment isomerism is not considered, the algorithm solution space searching efficiency is low, and Pareto solution set distribution is uneven are solved.

Description

Technical Field

[0001] This invention relates to the field of precision forming technology, and in particular to a multi-objective collaborative optimization method for additive manufacturing that integrates direction selection and multi-machine scheduling. Background Technology

[0002] Additive manufacturing (AM) is widely used in aerospace, medical devices, and other fields due to its ability to produce complex structures without molds and its customizable production capabilities. However, its industrialization faces two major bottlenecks:

[0003] 1. Orientation selection issue: The construction orientation of the part on the printing platform directly affects the volume of the supporting structure and the space utilization rate. Improper orientation selection will lead to material waste and extended post-processing time.

[0004] 2. Multi-machine scheduling problem: When multiple heterogeneous devices (such as those with different laser powers and scanning speeds) are producing in parallel, it is necessary to simultaneously optimize time (maximum completion time) and cost (materials, energy consumption, and labor). Differences in device performance lead to a game-theoretic relationship between objectives (e.g., high-speed machines have higher costs, while low-speed machines have longer cycles), making it difficult for traditional scheduling models to achieve collaborative optimization.

[0005] There are currently two main technical solutions. One is single-machine scheduling that considers direction selection. This involves establishing a multi-objective model of "part direction selection + packing + delivery time" in a single-device scenario and using an improved distribution estimation algorithm (IEDA) to optimize direction and job sequencing. The drawback is that it is limited to a single device and cannot be extended to multi-machine parallel scenarios. It also does not consider device heterogeneity (such as cost / speed differences) and cannot resolve objective conflicts in multi-machine task allocation. The other is multi-objective scheduling of the same parallel machine. This involves constructing a dual-objective model of "direction selection + load balancing" for multiple homogeneous devices (with the same process parameters) and using an improved firefly algorithm to optimize job allocation. The drawback is that it ignores device heterogeneity (such as high-precision devices being expensive but fast), which leads to an imbalance between cost and time in actual production. Furthermore, the algorithm does not introduce a dynamic normalization mechanism, resulting in an uneven distribution of the Pareto solution set (HV index is 15% lower than NSGA-II-DIP).

[0006] In summary, in a multi-machine parallel production environment for additive manufacturing, the lack of coordinated optimization between part construction direction selection and multi-machine multi-objective scheduling leads to problems such as low equipment resource utilization, serious waste of support materials, and difficulty in balancing production cycle and manufacturing cost. Specifically, these problems manifest as follows:

[0007] 1. Conflict between orientation selection and packing constraints: Traditional scheduling models do not take the part construction orientation as a decision variable, resulting in the non-optimal placement orientation of parts on the printing platform, leading to redundant support structures (increasing material costs) and low space utilization (reducing the number of parts formed per batch), indirectly extending the production cycle.

[0008] 2. Imbalance between multi-machine heterogeneity and multi-objective game: The process parameters (such as scanning speed and layer thickness) of different additive manufacturing equipment vary significantly, but existing scheduling algorithms are difficult to optimize time (completion cycle) and cost (materials, energy consumption, labor) in a coordinated manner, and are prone to getting trapped in local optima, resulting in unbalanced equipment load or cost overruns.

[0009] 3. Insufficient solution space complexity and algorithm efficiency: The strong coupling between the direction selection variable (such as each part having its own direction) and the task assignment variable causes the solution space to expand exponentially. Traditional optimization methods cannot efficiently search for the global optimal solution and cannot guarantee the convergence and uniformity of the Pareto solution set. Summary of the Invention

[0010] This invention provides a multi-objective collaborative optimization method for additive manufacturing that integrates direction selection and multi-machine scheduling, in order to solve the above-mentioned problems existing in the current additive manufacturing scheduling scheme.

[0011] To address the aforementioned technical problems, embodiments of the present invention provide a multi-objective collaborative optimization method for additive manufacturing that integrates direction selection and multi-machine scheduling. This method includes at least the following:

[0012] Direction-packing joint optimization steps: Based on the candidate construction directions of the parts, the target construction direction of the parts is determined to minimize the support volume ratio and maximize the space utilization rate, thereby generating a packing plan; the support volume ratio is the ratio of the total support volume of all parts to the total space of all parts, and the space utilization rate is the ratio of the total product volume of all parts to the total space of all parts.

[0013] Multi-machine multi-objective scheduling steps: Assign the jobs generated in the above steps to heterogeneous additive manufacturing equipment, with the goal of minimizing the total system completion time and the system unit volume cost, and generate a scheduling scheme; the total system completion time is the maximum value of the sum of the processing times of all equipment, and the system unit volume cost is the ratio of the total cost of all jobs to the sum of the volumes of all jobs;

[0014] Furthermore, the direction-packing joint optimization steps and the multi-machine multi-objective scheduling steps are jointly optimized by the improved NSGA-II algorithm using the improved genetic algorithm and the dynamic ideal point method.

[0015] In the aforementioned method, the direction-packing joint optimization step is implemented based on an improved genetic algorithm (IGA) with a hybrid adaptive mechanism, wherein the improved genetic algorithm includes:

[0016] 1) Adaptive population management mechanism: based on the fitness standard deviation of the population in generation t. Dynamically adjust mutation probability ,when hour, To set a diversity threshold, increase the mutation probability to break through local optima; when At the same time, reduce the probability of mutation to protect superior models;

[0017] 2) Hybrid Heuristic Search Framework: Integrating the probabilistic jump characteristics of simulated annealing, a temperature decay-controlled annealing tournament selection mechanism is proposed, based on the standard deviation of population fitness. The temperature parameter is dynamically adjusted based on the number of iterations t. To achieve a balance between global exploration and local development;

[0018] 3) Evolutionary operator for direction awareness: Introducing a dynamic direction sensitivity mechanism, which dynamically adjusts direction sensitivity based on the number of iterations t. This allows the direction adjustment probability to adapt to the iterative process, achieving multi-stage control through nonlinear decay.

[0019] In the aforementioned method, temperature regulation and selection operations: the temperature parameters are updated each generation according to the annealing tournament selection mechanism controlled by temperature decay;

[0020] Simulated Annealing Tournament Selection:

[0021] 1) Individual participants will be randomly selected, subject to the tournament's scale;

[0022] 2) Select the individual with the highest fitness as the winner;

[0023] 3) Calculate the acceptance probability of the suboptimal solution using the following formula:

[0024]

[0025] For individual fitness, T is temperature. The number of participating individuals;

[0026] 4) Accept a random challenger to replace the winner with probability p.

[0027] The encoding and decoding methods described above are as follows:

[0028] The encoding scheme uses a dual-sequence structure for chromosome representation: the first sequence is the part arrangement order, consisting of n natural numbers, each value corresponding to a unique part ID, and their arrangement order determines the processing priority; the second sequence is the construction direction sequence, where each element is an integer from 1 to z, representing the construction direction number used by the corresponding part; when all parts enter the production queue synchronously, the part... Both its construction direction s and its encoding are based on natural numbers;

[0029] Decoding is the process of converting chromosome codes into actual processing schemes. A set of jobs is generated through a segmented function. By decoupling the job allocation and sorting logic, the job division is dynamically determined by resource constraints during decoding. During the decoding process, the algorithm traverses the part sequence and accumulates resource consumption. When the machine capacity threshold is exceeded, a new batch of jobs is automatically generated, thereby realizing the transformation from coding to scheduling scheme.

[0030] In the aforementioned method, the steps for solving the additive manufacturing multi-directional bin packing problem using the IGA algorithm include genetic operator operations, specifically:

[0031] 1) Improved sequential crossover:

[0032] a. Randomly select two cutting points, start and end;

[0033] b. Inheriting the segmented gene region from parent 1;

[0034] c. Fill in the remaining genes in the order of parent generation 2;

[0035] 2) Two points of intersection in direction:

[0036] a. Randomly select intersection points point1 and point2;

[0037] b. The offspring inherits the [0,point1) and [point2,end) segments of the parent generation 1;

[0038] c. Inherit the [point1,point2) segment direction from parent 2;

[0039] 3) Adaptive mutation:

[0040] a. Calculate the dynamic mutation rate based on the adaptive population management mechanism;

[0041] b. Sequence Mutation: Randomly select 3 positions and perform a circular exchange;

[0042] c. Directional mutation: Select an available direction other than the current direction for mutation.

[0043] In the aforementioned method, the steps for solving the additive manufacturing multi-directional bin packing problem using the IGA algorithm also include local search optimization, population update, and elite retention:

[0044] Local search optimization includes:

[0045] 1) Exchange Mutation: For each part, attempt to exchange its position with adjacent parts;

[0046] 2) Orientation optimization: Try other available machining directions for each part;

[0047] 3) Optimal solution retention: The improved optimal individual is directly retained to the next generation.

[0048] Population renewal and elite retention include:

[0049] During the iteration process, the newly generated offspring population is first merged with elite individuals. Then, a truncation selection mechanism is adopted to sort individuals from high to low fitness values ​​and retain the individuals with the highest fitness. Then, neighborhood search-based reinforcement optimization is implemented on the current best individual until the maximum number of iterations is reached.

[0050] In the aforementioned method, the multi-machine multi-objective scheduling step is based on the NSGA-II algorithm improved by the dynamic ideal point method to solve the multi-machine parallel processing scheduling multi-objective optimization model. While retaining the classic NSGA-II non-dominated sorting and crowding distance mechanism, the fitness evaluation strategy is reconstructed by introducing the dynamic ideal point method to more accurately guide the population to converge toward the Pareto optimal frontier.

[0051] In the aforementioned method, the algorithm includes:

[0052] 1) Construct a population objective value matrix. After each individual completes the objective function evaluation, it will obtain its specific performance value on different optimization objectives. Based on this, any solution individual X in the population set... i All can be represented as eigenvectors composed of multidimensional objective function values: ,in, That is the maximum completion time. It is the processing cost per unit volume. Then, a decision matrix for a multi-objective optimization problem is constructed. The first individual is selected from the population in sequence, and its list of objective values ​​is read. The read values ​​are used as the first row of the matrix, and each column corresponds to an optimization objective. The above process is repeated to extract the objective values ​​of all individuals in the population in sequence and arrange them in order row by row. If the population contains N individuals and the number of optimization objectives is M, then an N-row M-column matrix is ​​generated.

[0053] 2) Determine the ideal point, which is a vector consisting of the minimum total completion time of the system and the minimum unit volume cost of the system in the current population, and is dynamically updated with iteration;

[0054] 3) Normalize the objective function and calculate the dynamic range and normalization weights for each objective;

[0055] 4) Calculate the weighted Euclidean distance between each individual and the ideal point, and combine the results of non-dominated ranking and crowding calculation to determine the individual fitness to guide population evolution.

[0056] This invention addresses the problems in existing technologies that only consider single-machine scenarios or homogeneous devices, neglecting device heterogeneity, low efficiency in algorithm solution space search, and uneven distribution of Pareto solution sets. It proposes a multi-objective collaborative optimization method that integrates direction selection and heterogeneous device scheduling. Combining a direction selection-binning joint optimization module and a multi-machine scheduling dynamic decision module, collaborative optimization is achieved through an improved genetic algorithm (IGA) and a dynamically optimized NSGA-II algorithm (NSGA-II-DIP). Furthermore, the above-mentioned solution of this invention also includes at least the following beneficial effects:

[0057] 1) Reduce support material waste: Through orientation-packing joint optimization, reduce support volume ratio and improve space utilization;

[0058] 2) Balance equipment load: Consider the heterogeneity of equipment, coordinate and optimize time and cost to avoid equipment load imbalance or cost overrun;

[0059] 3) Improved solution efficiency: The improved algorithm can efficiently search for the global optimal solution, ensuring the convergence and uniform distribution of the Pareto solution set, and is suitable for large-scale production scenarios. Attached Figure Description

[0060] Figure 1 This is a flowchart of the improved NSGA-II algorithm;

[0061] Figure 2 This is a schematic diagram of the improved sequential cross (OX);

[0062] Figure 3 This is a diagram showing the intersection of two points in a direction;

[0063] Figure 4 This is a schematic diagram of adaptive variation. Detailed Implementation

[0064] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.

[0065] In the following description, certain specific details are set forth for the purpose of illustrating various disclosed embodiments in order to provide a thorough understanding of the various disclosed embodiments. However, those skilled in the art will recognize that embodiments may be practiced without one or more of these specific details. In other instances, well-known apparatuses, structures, and techniques associated with this application may not have been shown or described in detail to avoid unnecessarily obscuring the description of the embodiments.

[0066] Throughout this specification, references to "an embodiment" or "an embodiment" indicate that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. Therefore, the appearance of "in an embodiment" or "an embodiment" in various places throughout the specification does not necessarily refer to the same embodiment. Furthermore, a particular feature, structure, or characteristic may be combined in any manner in one or more embodiments.

[0067] In the following description, in order to clearly demonstrate the structure and working method of the present invention, a number of directional terms will be used. However, terms such as "front", "back", "left", "right", "outside", "inside", "outward", "inward", "up", and "down" should be understood as convenient terms and not as limiting terms.

[0068] In this embodiment, a multi-objective collaborative optimization method for additive manufacturing that integrates direction selection and multi-machine scheduling is provided, including:

[0069] Additive manufacturing part orientation selection and packing collaborative optimization steps: Based on the candidate construction orientations of the parts, the target construction orientation of the parts is determined to minimize the support volume ratio and maximize the space utilization rate, and a packing scheme is generated; the support volume ratio is the ratio of the total support volume of all parts to the total space of all parts, and the space utilization rate is the ratio of the total product volume of all parts to the total space of all parts.

[0070] Multi-machine multi-objective scheduling steps: Assign the jobs generated in the above steps to heterogeneous additive manufacturing equipment, and generate a scheduling scheme with the goal of minimizing the total system completion time and the system unit volume cost; the total system completion time is the maximum value of the sum of the processing times of all equipment, and the system unit volume cost is the ratio of the total cost of all jobs to the sum of the volumes of all jobs.

[0071] Specifically as follows:

[0072] I. Collaborative Optimization of Additive Manufacturing Part Direction Selection and Packaging:

[0073] 1. For the problem of co-optimization of orientation selection and packing of additive manufacturing parts, the following model assumptions are set:

[0074] (1) All parts are processed using the same type of metal powder material, and the material consumption of the support structure is linearly related to the support volume;

[0075] (2) During the packing process, the parts shall not contact each other and nested layout is prohibited. The projected boundaries of the parts shall not overlap within the processing area.

[0076] (3) The pre-set candidate construction directions for each part have all passed the process feasibility verification, and the support volume calculation depends only on the direction selection results;

[0077] (4) The multi-machine system has the same spatial parameters and geometric constraints, including the height of the processing bay, the upper limit of the projected area and the upper limit of the space;

[0078] (5) Each operation consists of several parts, and the operation packing must simultaneously meet the constraints of cumulative projection surface value, total occupied volume and maximum height.

[0079] (6) The mapping relationship between jobs and equipment is a single job single machine exclusive mode, and multiple jobs cannot be processed concurrently on the same equipment;

[0080] (7) All parts are waiting in the system at the beginning, and all machines are empty at the beginning;

[0081] (8) The equipment is initially idle, the operation process is continuous and uninterrupted, and the equipment resources are released immediately after completion.

[0082] 2. Explanation of variable symbols:

[0083] No. One part,

[0084] : No. One assignment,

[0085] The first part Alternative construction directions,

[0086] :Component In direction The length below;

[0087] :Component Width in direction s;

[0088] :Component The height in the direction s;

[0089] :Component Product volume;

[0090] :Component Support volume in direction s;

[0091] : The maximum projected area of ​​the processing area of ​​a single machine;

[0092] Space limitations of a single machine;

[0093] Height limits for the processing bay of a single machine;

[0094]

[0095]

[0096] Component The projected area in direction s is:

[0097]

[0098] Component The volume occupied by the processing compartment in direction s is:

[0099]

[0100] No. Maximum height of parts within a single operation for:

[0101]

[0102] No. The remaining projected area of ​​each task for:

[0103]

[0104] No. Remaining space for each task for:

[0105]

[0106] No. The sum of the support volumes of each component within the operation for:

[0107]

[0108] The support volume ratio is:

[0109]

[0110] in, It is the sum of the supporting volumes of the parts being constructed. It is the sum of all constructed spaces.

[0111] Space utilization rate:

[0112]

[0113] in, It is the sum of the product volumes of all the parts being built.

[0114] The goal is to minimize the volume ratio and maximize the space utilization rate. Therefore, the optimization objective is:

[0115]

[0116] The additive manufacturing part orientation selection and packing co-optimization problem, with the optimization objectives of maximizing space utilization and minimizing support volume, can be described by the following model:

[0117]

[0118]

[0119]

[0120]

[0121] Equation (1-10) defines the objective function aimed at minimizing the support volume ratio and maximizing space utilization; Equation (1-11) stipulates that each part must select a single direction from its alternative construction directions during processing; Equation (1-12) restricts all parts to be assigned to only one job and prohibits repeated processing operations; Equation (1-13) requires that for any job, the sum of the projected areas of the parts must be less than or equal to the projected area of ​​the job, and the sum of the volumes occupied by the parts ( The workspace must be less than or equal to the workspace. The maximum height of the parts must be less than or equal to the working height. If this is violated, the parts must be reassigned, the direction must be changed, or the operation must be split up.

[0122] 3. Improved genetic algorithm design with hybrid adaptive mechanism:

[0123] Traditional genetic algorithms are suitable for general combinatorial optimization problems. However, the multi-directional bin packing problem in additive manufacturing has characteristics such as strict three-dimensional spatial constraints, sensitivity to direction selection, and discretization of the solution space. The fixed parameter mechanism and general crossover and mutation strategy of traditional genetic algorithms are difficult to effectively coordinate the coupling relationship between bin packing order optimization and direction selection, resulting in limited solution quality and convergence efficiency. In view of the process constraints of the multi-directional bin packing problem, an improved genetic algorithm (IGA) with a hybrid adaptive mechanism is proposed, which improves the basic genetic algorithm in the following three aspects:

[0124] 3.1 Adaptive Population Management Mechanism:

[0125] In traditional genetic algorithms, the mutation probability, as a fixed hyperparameter, has significant limitations: when the population diversity is high, an excessively high mutation rate can destroy superior gene fragments; while when the population is trapped in a local optimum, a fixed mutation rate is insufficient to provide enough perturbation to escape stagnation. This contradiction between the static setting of parameters and the dynamic requirements of evolution severely restricts the algorithm's efficiency in exploring the solution space of the multi-directional bin packing problem.

[0126] Let the standard deviation of fitness of the population in generation t be . It represents the degree of population diversity:

[0127]

[0128] Adaptive mutation probability based on diversity monitoring Defined as:

[0129]

[0130] In the formula In this context, N represents the population size. For individuals (i.e., parts) The fitness of ) For the average fitness of the population, the formula is... middle, As the baseline mutation probability, To adjust the amplitude, This is the diversity threshold.

[0131] This mechanism has a dual regulatory characteristic, during the exploration enhancement phase: when... When the exponent term approaches 1, By increasing the mutation rate, local optima can be broken. Development and maintenance phase: when When, the mutation rate follows The increase decreases exponentially, reaching a minimum of [missing value]. This protects superior patterns from being destroyed. Compared to a fixed mutation rate strategy, this method, when population diversity is insufficient, protects them by... Injecting perturbations during population growth, when population dispersion is too high, through... The descent maintains convergence, achieving a dynamic balance between global search and local optimization.

[0132] 3.2 Hybrid Heuristic Search Framework:

[0133] Traditional genetic algorithms rely solely on fitness ranking for selection, which can lead to premature loss of population diversity and local optima in multi-directional bin packing problems. This study proposes a temperature-decay-controlled annealing tournament selection mechanism, incorporating the probabilistic jump characteristics of simulated annealing. The temperature decay parameter plays a crucial role in balancing global exploration and local exploitation; reasonable dynamic temperature control improves the algorithm's convergence efficiency. Higher temperatures increase the probability of accepting poorer choices, enhancing the algorithm's ability to explore the solution space, but slowing down convergence. Lower temperatures increase selection pressure, improving algorithm development efficiency but increasing the risk of local optima. Traditional simulated annealing algorithms employ linear temperature decay strategies, which are ill-suited to the non-uniform nature of the solution space in multi-directional bin packing problems. Therefore, by improving the temperature update mechanism, the decay rate is dynamically adjusted according to population diversity. The temperature parameter for generation t is defined as:

[0134]

[0135] In the formula middle, =1.0 is the initial temperature. The standard deviation of population fitness is given by equation 1-14. For diversity threshold, This is the maximum number of iterations. The exponential decay model exhibits a dual adaptive property: when population diversity... When, the temperature decay is accelerated (the value of the exp term increases) to enhance local development; when At this time, the temperature drop is slowed down (the exp value decreases) to maintain global exploration.

[0136] 3.3 Evolutionary Operators for Direction Awareness:

[0137] In direction-aware neighborhood operations, the direction adjustment probability has a crucial impact on the efficiency of solution space exploration. Traditional neighborhood search employs a fixed direction adjustment strategy, which struggles to balance the dynamic demands of global exploration and local development. While an excessively high direction adjustment probability enhances direction diversity, it can lead to oscillations in the later stages of optimization; conversely, an excessively low probability results in premature loss of potential optimization directions, leading to local convergence. However, standard neighborhood operation methods often set the direction adjustment probability to a fixed value, which is clearly unsuitable for multi-stage optimization. Therefore, a dynamic direction sensitivity mechanism is introduced, allowing the direction adjustment probability to adaptively change with the iteration process. The direction sensitivity for the t-th iteration is defined as follows: for:

[0138]

[0139] In the formula middle, For maximum directional sensitivity, Minimum directional sensitivity, The sensitivity attenuation coefficient, This represents the maximum number of iterations. The dynamic orientation sensitivity mechanism achieves multi-stage regulation through nonlinear decay, for example: when , High-frequency directional adjustment improves solution space coverage, when , Dynamically adjust the search focus and gradually reduce the search frequency. , Low sensitivity maintains a stable direction and accelerates convergence.

[0140] In this invention, as an optional embodiment, improvements are made to the encoding and decoding methods:

[0141] The natural number encoding method proposed in this embodiment is specifically designed for the characteristics of additive manufacturing scheduling problems. Compared to the binary encoding used by genetic algorithms, this method achieves collaborative encoding of part allocation, job sorting, and build direction selection through a multi-dimensional natural number sequence. In multi-machine scheduling scenarios that consider part build direction, the processing of each part requires simultaneously determining its position in the job sequence, its job batch, and its spatial orientation parameters. Traditional matrix encoding can lead to dimensionality mismatch problems due to the dynamic changes in the number of jobs.

[0142] This encoding scheme uses a dual-sequence structure for chromosome representation: the first sequence is the part arrangement order, consisting of n natural numbers, each value corresponding to a unique part ID, and their arrangement order determines the processing priority; the second sequence is the construction direction sequence, where each element is an integer from 1 to z, representing the construction direction number used by the corresponding part. When all parts enter the production queue synchronously, the part... Both the component and its construction direction s are encoded using natural numbers. Specifically: Let the set of parts to be processed be... Each part The set of candidate construction directions is Chromosomes are defined as binary pairs:

[0143]

[0144] in, This represents the priority order of part processing; arrive This represents the n parts in the arrangement order; arrive This indicates the construction direction of the corresponding part.

[0145] For example, when there are 5 parts that need to be added to the queue, the natural number encoding method is as follows:

[0146]

[0147] The decoding process generates a set of jobs through a segmentation function:

[0148]

[0149]

[0150] The dividing point , Generated by recursion:

[0151]

[0152] in, The direction of assignment k; : The dividing point of job k, indicating the starting position of the job; K: The index of the last job, indicating the job with the construction direction n.

[0153] Equation (1-22) states that for any given task, the sum of the projected areas of the parts must be less than or equal to the projected area of ​​the task. The total volume occupied by each part must be less than or equal to the working space, and the maximum height of each part must be less than or equal to the working height. If any of these conditions are violated, a split point is created, and the part is transferred to the next job. In the previous example, part 4 in direction 2 and part 5 in direction 1 form one job, and part 1 in direction 3 cannot be added to this job. Therefore, part 1 in direction 3, part 3 in direction 1, and part 2 in direction 2 form another job. The processing plan can be represented as follows:

[0154]

[0155] This mathematical model uses dual-sequence encoding to map the solution space to the search space. I controls the part processing sequence and job division, while S controls the selection of process parameters; together, they constitute the genotype of the solution vector. The decoding process Φ(χ) transforms the genotype into a dominant scheduling scheme.

[0156] This encoding method decouples job allocation and sorting logic, with job division dynamically determined by resource constraints during decoding, avoiding the need to pre-fix the number of jobs. During decoding, the algorithm traverses the part sequence and accumulates resource consumption. When the machine capacity threshold is exceeded, a new batch of jobs is automatically generated, thus realizing the transformation from encoding to scheduling scheme.

[0157] In summary, all individuals in the algorithm can generate codes according to the specified encoding rules, and each encoded sequence can be restored using the corresponding decoding method. After correct decoding, each sequence corresponds to a unique processing implementation scheme. This encoding method ensures both the traversal of the search space and the validity of feasible solutions.

[0158] 4. The steps for solving the additive manufacturing multi-directional packing problem using the IGA algorithm are as follows:

[0159] 4.1 Algorithm parameter initialization: Set population size, maximum number of iterations, basic crossover rate, basic mutation rate, tournament size, number of elites retained, and initial temperature.

[0160] 4.2 Population Initialization and Fitness Calculation:

[0161] The initial population is generated randomly: each individual consists of the order of parts and the corresponding processing direction. The order of parts is generated by random shuffling, and the processing direction is randomly selected from the range of parts directions.

[0162] Calculate the initial fitness: Calculate the difference between the space utilization rate and the support consumption rate of each individual according to Equation (1-9), and then calculate the fitness.

[0163] Record the initial optimal solution: Sort the population in descending order of fitness and retain elite individuals.

[0164] 4.3 Temperature Adjustment and Selection Operation:

[0165] Temperature decay: The temperature parameters are updated according to formula (1-16) for each generation.

[0166] Simulated Annealing Tournament Selection:

[0167] 1) Individual participants will be randomly selected, subject to the tournament's scale;

[0168] 2) Select the individual with the highest fitness as the winner;

[0169] 3) Calculate the acceptance probability of the suboptimal solution according to equation (1-24):

[0170]

[0171] For individual fitness, T is temperature. The number of participating individuals;

[0172] 4) Accept a random challenger to replace the winner with probability p.

[0173] 4.4 Genetic Operator Operations:

[0174] The improved sequential cross (OX) diagram is shown below. Figure 2 As shown:

[0175] 1) Randomly select two cutting points, start and end.

[0176] 2) Inherited from the cutting interval gene segment of parent generation 1.

[0177] 3) Fill in the remaining genes in the order of parent generation 2.

[0178] A diagram showing the intersection of two points in a direction is shown below. Figure 3 As shown:

[0179] 1) Randomly select intersection points point1 and point2.

[0180] 2) The offspring inherits the [0,point1) and [point2,end) segments of the parent generation 1.

[0181] 3) Inherit the [point1,point2) segment direction from parent generation 2.

[0182] Adaptive mutation, such as Figure 4 As shown:

[0183] 1) According to the formula Calculate the dynamic variation rate.

[0184] 2) Sequence Mutation: Randomly select 3 positions and perform a circular exchange.

[0185] 3) Directional mutation: Select an available direction other than the current direction for mutation.

[0186] 4.5 Local search optimization:

[0187] 1. Exchange Mutation: For each part, try to exchange its position with adjacent parts.

[0188] 2. Orientation optimization: Try other available machining directions for each part.

[0189] 3. Optimal solution retention: The improved optimal individual is directly retained to the next generation.

[0190] 4.6 Population Renewal and Elite Retention:

[0191] During the iteration process, the newly generated offspring population is first merged with elite individuals. Then, a truncation selection mechanism is adopted to sort individuals from high to low fitness values ​​and retain the individuals with the highest fitness. Then, neighborhood search-based reinforcement optimization is implemented on the current best individual until the maximum number of iterations is reached.

[0192] II. Optimization of Multi-Objective Additive Manufacturing Parallel Processing Scheduling:

[0193] 1. For the multi-objective additive manufacturing multi-machine parallel processing scheduling optimization problem, the following model assumptions are set:

[0194] (1) All operations are processed using the same type of metal powder material, and the material cost per unit volume is only related to the material utilization rate of the machine;

[0195] (2) All machines have the same layer thickness when layering, the layering time is only related to the interlayer dwell time, and the volume processing time is only related to the scanning speed;

[0196] (3) The machine operating cost and the inert gas cost are constant and are not linked to fluctuations in electricity and gas prices;

[0197] (4) Machine setup time and parts disassembly time are not related to worker skill level, and worker costs do not affect machine setup time and parts disassembly time;

[0198] (5) All jobs are waiting in the system at the beginning, and all machines are empty at the beginning;

[0199] (6) The equipment is initially idle, the operation process is continuous and uninterrupted, and the equipment resources are released immediately after completion.

[0200] 2. Explanation of variable symbols:

[0201] : No. One assignment, ;

[0202] : No. One machine, ;

[0203] : No. One optimization objective, ;

[0204] : No. The volume of each task;

[0205] : No. The height of each task;

[0206] : No. Processing time per unit volume of each machine;

[0207] : No. Unit stratification time for each machine;

[0208] : No. The operating cost of each machine;

[0209] : No. The setup time for each machine;

[0210] : No. The time required for disassembling and cleaning parts of each machine;

[0211] : No. The labor cost of each machine;

[0212] : No. Material cost per unit volume of the machine;

[0213]

[0214] Operation In the machine Volume processing time for:

[0215]

[0216] Operation In the machine Layered processing time for:

[0217]

[0218] Operation In the machine Total processing time for:

[0219]

[0220] Operation In the machine The machine operating cost is:

[0221]

[0222] Operation In the machine The labor cost is:

[0223]

[0224] The cost of labor per unit of time;

[0225] Operation In the machine The material cost is:

[0226]

[0227] Operation In the machine Total cost for:

[0228]

[0229] Total system completion time t for:

[0230]

[0231] System unit volume cost C for:

[0232]

[0233] 3. Model Establishment

[0234] This embodiment constructs a dual-objective optimization model for different multi-machine parallel processing scenarios in additive manufacturing. This model simultaneously pursues the maximization of production system efficiency and the minimization of cost, including the two core objectives of the shortest system manufacturing cycle and the lowest unit volume processing cost. Its mathematical expression framework is shown below:

[0235] Minimize the total system completion time t :

[0236]

[0237] Minimize the system unit volume cost C :

[0238]

[0239] Task assignments must be unique:

[0240]

[0241] Homework assignments must not be interrupted or taken over.

[0242]

[0243] Equation (2-10) represents the objective function of minimizing the total system completion time; Equation (2-11) represents the objective function of minimizing the unit volume cost; Equation (2-12) indicates that a job can only be processed on one machine and cannot be processed repeatedly; Equation (2-13) indicates that for any given machine, only one job can be produced at a time, and production cannot be interrupted or preempted. If any violation occurs, jobs and machines must be reallocated, and the production plan must be redesigned. and : represent assignments respectively and homework The start time; and : represent assignments respectively and homework In the machine Processing time; Binary decision variables, representing the job Are you doing homework? Previously completed; if the value is 1, it means... exist Completed beforehand, otherwise 0. M: A large constant, typically used to ensure the correctness of constraints (similar to a penalty term), when the task... Use this constant when completing before k´ to ensure that time constraints are not violated. and Binary decision variables, representing the job and homework Is it assigned to a machine? If it is 1, it indicates an assignment. and homework All were assigned to the machine .

[0244] 4. Design of NSGA-II Optimization Algorithm Improved by Dynamic Ideal Point Method

[0245] The ideal point method is a multi-objective decision-making method based on the geometric relationship of the objective space. Its core idea is to evaluate the merits of a solution by quantifying the spatial distance between feasible solutions and ideal reference points. This method uses the theoretical optimal value of each optimization objective as a benchmark, constructs a virtual ideal reference point, and makes a comprehensive evaluation by calculating the relative distance between feasible solutions and extreme points.

[0246] This embodiment employs an improved NSGA-II algorithm (NSGA-II-DIP) using Dynamic Ideal Points (DIP) to solve a multi-machine parallel processing scheduling multi-objective optimization model. While retaining the classic NSGA-II non-dominated sorting and crowding distance mechanisms, the algorithm introduces the Dynamic Ideal Points method to reconstruct the fitness evaluation strategy, thereby more accurately guiding the population towards the Pareto optimal front. The proposed algorithm flow is as follows: Figure 1 As shown.

[0247] In an optional embodiment of the present invention, a normalized fitness function based on a dynamic ideal point is constructed:

[0248] Guided by the dual optimization objectives of the additive manufacturing scheduling model established in this embodiment—minimizing the overall processing cycle and reducing the unit volume production cost—this approach addresses the issue of the two different optimization objectives having different dimensions (h and RMB / cm²). 3Therefore, it is necessary to eliminate the influence of dimensions by scaling different objectives to a unified dimensionless interval, making the numerical differences between objectives comparable. Furthermore, since the numerical fluctuation ranges of time and cost objectives differ and the differences are relatively large, normalization can also eliminate numerical discrepancies. Then, by defining an ideal point and calculating weighted Euclidean distance, multiple conflicting objectives are optimized simultaneously. A hybrid selection strategy is also incorporated to ensure that the algorithm always searches towards the historical optimum.

[0249] (1) Construct the population target value matrix:

[0250] After each individual completes the objective function evaluation, it obtains its specific performance values ​​on different optimization objectives. Based on this, any solution individual X in the population set... i All of these can be represented as eigenvectors composed of multidimensional objective function values. For example... .in, That is the maximum completion time. This refers to the processing cost per unit volume. Then, a decision matrix for a multi-objective optimization problem is constructed. The first individual is selected sequentially from the population, and its list of objective values ​​is read. These read values ​​are used as the first row of the matrix, with each column corresponding to an optimization objective. This process is repeated to extract the objective values ​​of all individuals in the population and arrange them sequentially row by row. If the population contains N... There are M individuals, and the objective is to optimize for M. If there are 1, then an N is generated. Line M A matrix of columns.

[0251] (2) Ideal point method:

[0252] The ideal point is the minimum objective value of all solutions in the current population, which is the theoretically optimal direction for each objective.

[0253]

[0254] For example, the current population's minimum time value ( The minimum cost is 100 hours. The price is 50 RMB / cm³. =[100,50].

[0255] Dynamically updating the ideal point: As the algorithm iterates and continuously optimizes, the new offspring population generated in each iteration may contain better solutions. The algorithm improves the ideal point parameters by capturing the target values ​​corresponding to these solutions in real time. This dynamic adjustment strategy can gradually correct the search direction as the population evolves, allowing the position of the ideal point to continuously converge towards the theoretical optimum as the solution set is optimized, thereby achieving an effect that accurately approximates the real Pareto front.

[0256] (3) Objective function normalization:

[0257] For each objective function (total processing time, cost per unit volume) in the current population, calculate the difference between its maximum and minimum values, and use this difference as the dynamic range of that objective.

[0258]

[0259] Mode middle, and Indicates the k-th element in the current population The maximum and minimum values ​​of each objective (e.g., time and cost). To minimize the value, prevent the denominator from being 0.

[0260] Take the reciprocal of the range of each target to obtain the normalized weights:

[0261]

[0262] For example, if the total processing time ranges from 200 to 100, then the weight is 1 / 100 = 0.01. If the unit volume cost ranges from 100 to 50, then the weight is 1 / 50 = 0.02.

[0263] (4) Calculate the weighted distance:

[0264] First, calculate the weighted bias, and then calculate the target value for each solution. Calculate its deviation from the ideal point and multiply it by the weight:

[0265]

[0266] Then calculate the weighted Euclidean distance (i.e., the vector length):

[0267]

[0268] (5) Non-dominated sorting:

[0269] Let p be the solution. and q The target values ​​are respectively and If and only if:

[0270]

[0271] Let p dominate q If the solution is p Non-dominated solution q And solve q It also does not dominate the solution p Then p is called and q They do not dominate each other. For example: if the solution p The time is 150 hours, the cost is 70 yuan / cm³, and the solution is q. The time is 160 hours, the cost is 60 yuan / cm³, p The timing is better, but q If the cost is better, then p and q They do not control each other.

[0272] For each solution p Define Domination Count (Number of times dominated by other solutions) and the dominating set (Solve p) The set of all dominated solutions. The set containing all non-dominated solutions (solutions with a dominance count of 0) is defined as the first frontier F1. .

[0273] Then from the current forefront of F1 Start by traversing its dominant set. will be p Decrease the count of the dominant solution by 1:

[0274]

[0275] Add the solution whose dominance count drops to 0 to the next frontier:

[0276]

[0277] Repeat until all solutions are assigned to a certain frontier.

[0278] (6) Crowding Calculation:

[0279] Crowding density is an indicator of the sparsity of the distribution of solutions within the same non-dominated front. Its core purpose is to ensure that the population is evenly distributed in the target space while preserving the quality of the solutions, thus avoiding excessive clustering.

[0280] First, the crowding degree of each solution within the frontier is initialized to 0. Then, all solutions within the frontier are sorted in ascending order according to the value of each objective function. Furthermore, the crowding degree of the extreme solutions (minimum and maximum values) in each objective direction is set to infinity to maintain the distribution range of the population. Then, the crowding degree of intermediate solutions is calculated:

[0281] For target k Calculate the crowding contribution of non-boundary solutions. :

[0282] For the target k The One solution :

[0283]

[0284] in, It is the current frontier target k The maximum and minimum values, It is a local minimum value to prevent the denominator from being zero.

[0285] untie Total congestion C ( The sum of contributions to all objectives is accumulated.

[0286]

[0287] Finally, to balance convergence and diversity, a weighted combination of convergence distance and distribution crowding is used to determine the fitness of individual solutions in the population:

[0288]

[0289] In equation (2-23), α It is the diversity weight, β These are convergence weights. It is a weighted Euclidean distance. Furthermore, since higher fitness is better, a smaller distance is better, and taking a negative sign makes the score positive.

[0290] By combining non-dominated ordination and crowding calculations, individual fitness is determined to guide population evolution.

[0291] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A multi-objective collaborative optimization method for additive manufacturing that integrates direction selection and multi-machine scheduling, characterized in that, At least including: Direction-packing joint optimization steps: Based on the candidate construction directions of the parts, the target construction direction of the parts is determined to minimize the support volume ratio and maximize the space utilization rate, and a packing plan is generated. The support volume ratio is the ratio of the total support volume of all parts to the total space of all parts, and the space utilization ratio is the ratio of the total product volume of all parts to the total space of all parts. Multi-machine multi-objective scheduling steps: Assign the jobs generated in the above steps to heterogeneous additive manufacturing equipment, with the goal of minimizing the total system completion time and system unit volume cost, and generate a scheduling scheme; The total completion time of the system is the maximum value of the total processing time of all equipment, and the unit volume cost of the system is the ratio of the total cost of all operations to the total volume of all operations. Furthermore, the direction-packing joint optimization step and the multi-machine multi-objective scheduling step are jointly optimized by the improved NSGA-II algorithm using the improved genetic algorithm and the dynamic ideal point method; The improved genetic algorithm is an improved genetic algorithm with a hybrid adaptive mechanism; The multi-machine multi-objective scheduling step is based on the NSGA-II algorithm improved by the dynamic ideal point method to solve the multi-machine parallel processing scheduling multi-objective optimization model. While retaining the classic NSGA-II non-dominated sorting and crowding distance mechanism, the fitness evaluation strategy is reconstructed by introducing the dynamic ideal point method to more accurately guide the population to converge toward the Pareto optimal frontier.

2. The method according to claim 1, characterized in that, The direction-packing joint optimization step is implemented based on an improved genetic algorithm with a hybrid adaptive mechanism, the improved genetic algorithm including: a. Adaptive population management mechanism: based on the standard deviation of the population's fitness in generation t. Dynamically adjust mutation probability ,when hour, To set a diversity threshold, increase the mutation probability to break through local optima; when At the same time, reduce the probability of mutation to protect superior models; b. Hybrid Heuristic Search Framework: Integrating the probabilistic jump characteristics of simulated annealing, a temperature decay-controlled annealing tournament selection mechanism is proposed, based on the standard deviation of population fitness. The temperature parameter is dynamically adjusted based on the number of iterations t. To achieve a balance between global exploration and local development; c. Evolutionary operator for direction awareness: Introducing a dynamic direction sensitivity mechanism, dynamically adjusting direction sensitivity based on the number of iterations t. This allows the direction adjustment probability to adapt to the iterative process, achieving multi-stage control through nonlinear decay.

3. The method according to claim 2, characterized in that, Temperature regulation and selection operation: The temperature parameters are updated each generation by an annealing tournament selection mechanism controlled by temperature decay. Simulated Annealing Tournament Selection: a. Participants will be randomly selected, subject to the tournament's scale; b. Select the individual with the highest fitness as the winner; c. Calculate the acceptance probability of the suboptimal solution using the following formula: ； For individual fitness, T is temperature. The number of participating individuals; d. Accept a random challenger to replace the winner with probability p.

4. The method according to claim 2, characterized in that, The encoding and decoding methods are as follows: The encoding scheme uses a dual-sequence structure for chromosome representation: the first sequence is the part arrangement order, consisting of n natural numbers, each value corresponding to a unique part ID, and their arrangement order determines the processing priority; the second sequence is the construction direction sequence, where each element is an integer from 1 to z, representing the construction direction number used by the corresponding part; when all parts enter the production queue synchronously, the part... Both its construction direction s and its encoding are based on natural numbers; Decoding is the process of converting chromosome codes into actual processing schemes. A set of jobs is generated through a segmented function. By decoupling the job allocation and sorting logic, the job division is dynamically determined by resource constraints during decoding. During the decoding process, the algorithm traverses the part sequence and accumulates resource consumption. When the machine capacity threshold is exceeded, a new batch of jobs is automatically generated, thereby realizing the transformation from coding to scheduling scheme.

5. The method according to claim 2, characterized in that, The steps for solving the additive manufacturing multi-directional bin packing problem using the IGA algorithm include genetic operator operations, specifically: A. Improved sequential crossover: a. Randomly select two cutting points, start and end; b. Inheriting the segmented gene region from parent 1; c. Fill in the remaining genes in the order of parent generation 2; B. Two points intersect in the direction: a. Randomly select intersection points point1 and point2; b. The offspring inherits the [0,point1) and [point2,end) segments of the parent generation 1; c. Inherit the [point1,point2) segment direction from parent 2; C. Adaptive mutation: a. Calculate the dynamic mutation rate based on the adaptive population management mechanism; b. Sequence Mutation: Randomly select 3 positions and perform a circular exchange; c. Directional mutation: Select an available direction other than the current direction for mutation.

6. The method according to claim 2, characterized in that, The steps for solving the additive manufacturing multi-directional bin packing problem using the IGA algorithm also include local search optimization: a. Exchange Mutation: For each part, try to exchange its position with adjacent parts; b. Direction optimization: Try other available machining directions for each part; c. Optimal solution retention: The improved optimal individual is directly retained to the next generation.

7. The method according to claim 2, characterized in that, The steps for solving the additive manufacturing multi-directional bin packing problem using the IGA algorithm also include population update and elite retention: during the iteration process, the newly generated offspring population is first merged with elite individuals, and then a truncation selection mechanism is adopted to sort individuals from high to low fitness values ​​and retain the individuals with the highest fitness. Then, neighborhood search-based reinforcement optimization is implemented on the current best individual until the maximum number of iterations is reached.

8. The method according to claim 1, characterized in that, The algorithm includes: a. Construct a population objective value matrix. After each individual completes the objective function evaluation, it will obtain its specific performance value on different optimization objectives. Based on this, any solution individual X in the population set... i All can be represented as eigenvectors composed of multidimensional objective function values: ,in, That is the maximum completion time. It is the processing cost per unit volume. Then, a decision matrix for a multi-objective optimization problem is constructed. The first individual is selected from the population in sequence, and its list of objective values ​​is read. The read values ​​are used as the first row of the matrix, and each column corresponds to an optimization objective. The above process is repeated to extract the objective values ​​of all individuals in the population in sequence and arrange them in order row by row. If the population contains N individuals and the number of optimization objectives is M, then an N-row M-column matrix is ​​generated. b. Determine the ideal point, which is a vector consisting of the minimum total completion time of the system and the minimum unit volume cost of the system in the current population, and is dynamically updated with iteration; c. Normalize the objective function and calculate the dynamic range and normalization weights for each objective. d. Calculate the weighted Euclidean distance between each individual and the ideal point, and combine the results of non-dominated ranking and crowding calculation to determine the individual fitness to guide population evolution.

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