Production line feeding control method based on max-plus algebra and model predictive control

Abstract

The application relates to the technical field of production line feeding control, and proposes a production line feeding control method based on a max-plus algebra and model predictive control, which comprises the following steps: establishing a mathematical model of a production line based on a max-plus algebra; defining an objective function, wherein the objective function comprises a tracking accuracy cost and a system overall performance cost; setting a prediction time domain and a control time domain of the model predictive control, and initializing a feeding time of a workpiece and a reference output sequence; at a current control time, predicting output sequences of multiple future workpieces based on a current state of the production line and the mathematical model; solving an optimization problem based on the objective function to determine a feeding sequence of the multiple future workpieces, wherein the optimization problem is converted into a mixed integer linear programming problem, and a solver is used to solve the optimization problem; and executing a first feeding time from the feeding sequence, and updating the current control time.

Description

Technical Field

[0001] This invention relates to the field of production line feeding control technology, and in particular to a production line feeding control method based on maximum algebra and model predictive control. Background Technology

[0002] Existing production line material control methods mainly rely on empirical rules, fixed-cycle material feeding, or simple scheduling algorithms, but these methods have many shortcomings in practical applications. These methods struggle to accurately coordinate production cycle time with work-in-process flow, easily leading to waiting or backlog in some processes, thus affecting overall output stability. Simultaneously, material feeding decisions lack foresight and predictive capabilities, failing to effectively respond to order changes or production fluctuations. Premature material feeding results in excessive work-in-process inventory, increasing inventory costs, while delayed material feeding may cause production delays, affecting delivery timeliness and potentially incurring default or penalty costs. Furthermore, traditional methods typically consider only a single objective, lacking the ability to comprehensively optimize output, inventory costs, and delivery timeliness, leading to decreased overall production efficiency. They also exhibit poor adaptability to adjustments in production line structure or processes, making it difficult to cope with changes in production conditions and easily creating bottlenecks or wasting resources. Therefore, existing material feeding control methods are insufficient to meet the requirements of modern production lines for efficient, stable, and low-cost operation. Summary of the Invention

[0003] To address the aforementioned shortcomings, the present invention aims to propose a production line feeding control method based on maximum algebra and model predictive control. This method utilizes maximum algebra to accurately model the production line and combines it with the rolling optimization mechanism of model predictive control to achieve dynamic and forward-looking optimization of the feeding strategy. This ensures that output and demand are matched, and through comprehensive optimization of multi-dimensional performance indicators, it synergistically improves the production line's resource utilization, work-in-process inventory control, and on-time delivery, thereby comprehensively enhancing the overall operating efficiency, stability, and adaptability of the production line.

[0004] To achieve this objective, the present invention adopts the following technical solution:

[0005] The production line feeding control method based on maximum algebra and model predictive control includes the following steps:

[0006] S1: A mathematical model of the production line is established based on maximal addition algebra, wherein the mathematical model includes multiple workstations and buffers, and the mathematical model describes the processing sequence, waiting relationship and cumulative delay of workpieces between the workstations;

[0007] S2: Define the objective function, which consists of tracking accuracy cost and overall system performance cost. The tracking accuracy cost is used to ensure that the production line output matches the planned demand, and the overall system performance cost is used to optimize the production line's resource utilization, work-in-process inventory, and operating efficiency.

[0008] S3: Set the prediction time domain and control time domain of model predictive control, and initialize the workpiece feeding time and reference output sequence;

[0009] S4: At the current control moment, based on the current state of the production line and the mathematical model, predict the output sequence of multiple workpieces in the future;

[0010] Based on the objective function, an optimization problem is solved to determine the material feeding sequence of multiple future workpieces, wherein the optimization problem is transformed into a mixed integer linear programming problem and solved using a solver;

[0011] Execute the first feeding moment from the feeding sequence and update the current control moment;

[0012] S5: Repeat step S4 until the feeding sequence of all workpieces is solved and executed.

[0013] Preferably, the mathematical model includes serial units, assembly units, disassembly units, and buffer units.

[0014] Preferably, in the serial unit, for any workstation Start processing the first Start time of machining of each workpiece This includes the following steps:

[0015] S111: Obtain the preceding workstation For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates the preceding workstation Start processing the first The moment of each workpiece;

[0016] S112: By acquiring downstream workstations For the Start time of machining of each workpiece Determine the workstation Is there at least one available slot to receive the first... One workpiece; among which For workstations The capacity, when season ,in For the zero element in maximal additive algebra;

[0017] S113: Combine the processing completion time obtained in step S111 with the time obtained in step S112. In maximal addition algebra The calculation is performed, and the maximum value between the two is taken as the workstation. Start processing the first Start time of machining of each workpiece The following relation is satisfied: .

[0018] Preferably, in the assembly unit, for the assembly station Start processing the first Start time of machining of each workpiece This includes the following steps:

[0019] S121: Obtain all preceding supply stations For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates workstation Start processing the first The moment of each workpiece;

[0020] S122: By acquiring downstream workstations For the Start time of machining of each workpiece Determine the assembly station Is there at least one available slot to receive the first... One workpiece, of which For workstations The capacity, when season ,in For the zero element in maximal additive algebra;

[0021] S123: Combine the processing completion times of all preceding supply stations obtained in step S121 with the processing start times of the downstream stations obtained in step S122. In maximal addition algebra The calculation is performed, and the maximum value at all times is taken as the assembly station value. Start processing the first Start time of machining of each workpiece The following relation is satisfied:

[0022] .

[0023] Preferably, in the disassembly unit, for the disassembly station Start processing the first Start time of machining of each workpiece This includes the following steps:

[0024] S131: Obtain the preceding workstation For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates the preceding workstation Start processing the first The moment of each workpiece;

[0025] S132: By acquiring all subsequent workstations For the Start time of machining of each workpiece Determine the disassembly station Is there at least one available slot to receive the first... One workpiece, of which For workstations The capacity, when season ,in For the zero element in maximal additive algebra;

[0026] S133: Perform a maximum addition algebraic operation on the processing completion time obtained in step S131 and the start times of all subsequent workstations obtained in step S132. The calculation is performed, and the maximum value at all times is taken as the disassembly station value. Start processing the first Start time of machining of each workpiece The following relation is satisfied:

[0027] .

[0028] Preferably, in the buffer unit, the buffer unit is abstracted as a workstation with capacity and processing time, and the calculation of the start processing time of the buffer unit adopts the same recursive logic as the serial unit, assembly unit or disassembly unit.

[0029] The processing time of the buffer unit is set according to its function: when only capacity constraints are reflected, the processing time of the buffer unit is zero; when undertaking process or cycle adjustment functions, the processing time of the buffer unit is the specified time for the workpiece to stay.

[0030] Preferably, in step S2, the objective function Satisfying the relation:

[0031] ;

[0032] in, Indicates the cost of tracking accuracy. Indicates the overall performance cost of the system. This represents a weighting factor used to balance the relative importance of tracking accuracy cost and overall system performance cost.

[0033] The cost of tracking accuracy It can be divided into and :

[0034] ;

[0035] ;

[0036] in, It is in the Time prediction The system output time at that moment. yes The reference output time of the time. For prediction in the time domain;

[0037] The overall performance cost of the system It can be divided into , , , and ,in:

[0038] ;

[0039] ;

[0040] ;

[0041] ;

[0042] ;

[0043] in, and They are respectively in the 1st Time prediction Workstation The input and output times, For workstations Processing time, and These represent the sets of all workstations and all buffer units, respectively.

[0044] Preferably, step S3 includes:

[0045] Define the total quantity to be processed Prediction time domain and control time domain Given the initial workpiece feeding time and reference output sequence ;

[0046] Step S4 includes:

[0047] During control time Define the feeding sequence The feeding time outside the control time domain remains unchanged, satisfying ,in ;

[0048] The feeding sequence Substituting into the mathematical model in step S1, the corresponding predicted output sequence is calculated. ;

[0049] Reference output sequence Feeding sequence and predicted output sequence Substituting into the objective function defined in step S2, a result is formed with An optimization problem with 10 decision variables;

[0050] The optimization problem is transformed into a mixed-integer linear programming problem, and solved using a solver to obtain the optimal feeding sequence. ;

[0051] The rolling execution control strategy executes only the optimal feeding sequence. The first feeding moment in the process ;

[0052] Update control time .

[0053] Preferably, transforming the optimization problem into a mixed-integer linear programming problem includes:

[0054] Using the Big-M method to study maximal operators in maximal additive algebra. Linearization is performed by introducing binary decision variables and sufficiently large constants, which is equivalent to a set of linear constraints.

[0055] Constraints are imposed on the decision variables, including monotonicity constraints and non-negativity constraints. Monotonicity constraints are used to ensure the time monotonicity of the feeding sequence, satisfying... The non-negativity constraint is used to ensure that the material is non-negative at all feeding times, satisfying... ;

[0056] A mixed-integer linear programming solver was selected to solve the transformed problem.

[0057] One of the above technical solutions has the following advantages or beneficial effects:

[0058] This invention introduces a mathematical model of the production line using maximal algebra, which accurately characterizes the temporal dependencies and flow constraints of workpieces at each workstation, thus overcoming the structural limitations of traditional empirical rules in coordinating production cycle time and work-in-process flow. Building upon this, by defining an objective function that includes tracking accuracy cost and overall system performance cost, it incorporates multi-dimensional indicators such as output matching, resource utilization, and work-in-process inventory control into a unified optimization framework, compensating for the shortcomings of existing methods in comprehensive optimization capabilities. Furthermore, by leveraging a rolling optimization mechanism based on model predictive control, it can predict future output based on real-time conditions and dynamically adjust material feeding strategies, thereby achieving proactive adaptation to order changes and production fluctuations at the operational level. Finally, by transforming the optimization problem into mixed-integer linear programming and solving it efficiently, it effectively coordinates multiple conflicting objectives while ensuring real-time performance. In summary, through the synergistic effect of multiple steps, this invention can significantly improve the comprehensive performance of the production line in terms of output stability, equipment utilization, inventory control, and on-time delivery without relying on fixed cycle time or empirical scheduling, while also enhancing its robustness and adaptability in response to changes in production conditions. Attached Figure Description

[0059] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0060] Figure 1 This is a flowchart of the production line feeding control method based on maximum addition algebra and model predictive control provided in the embodiments of the present invention;

[0061] Figure 2 This is a simplified serial unit diagram of the production line feeding control method based on maximum addition algebra and model predictive control provided in an embodiment of the present invention.

[0062] Figure 3 This is a detailed schematic diagram of the serial unit of the production line feeding control method based on maximum addition algebra and model predictive control provided in the embodiments of the present invention;

[0063] Figure 4 This is a schematic diagram of an assembly unit for a production line feeding control method based on maximum addition algebra and model predictive control provided in an embodiment of the present invention.

[0064] Figure 5 This is a schematic diagram of the disassembly unit of the production line feeding control method based on maximum addition algebra and model predictive control provided in an embodiment of the present invention;

[0065] Figure 6This is a schematic diagram of a mobile phone product repair and refurbishment line provided in an embodiment of the present invention;

[0066] Figure 7 This is a schematic diagram of a production line case of the production line feeding control method based on maximum addition algebra and model predictive control provided in the embodiments of the present invention;

[0067] Figure 8 This is a schematic diagram of case data for a production line feeding control method based on maximum algebra and model predictive control provided in an embodiment of the present invention. Detailed Implementation

[0068] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0069] In this invention, the terms "comprising," "including," or any other variations thereof are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0070] Production line feeding control methods based on maximum algebra and model predictive control, such as Figure 1 As shown, it includes the following steps:

[0071] S1: A mathematical model of the production line is established based on maximal addition algebra, wherein the mathematical model includes multiple workstations and buffers, and the mathematical model describes the processing sequence, waiting relationship and cumulative delay of workpieces between the workstations;

[0072] It should be noted that the Max-Plus Algebra is a type of algebra defined on an extension of the real number set. An algebraic system on the π- ... Composition, for any real number , , Maximal addition algebra defines the maximal operator. sum operator The operation is as follows:

[0073] addition: ;

[0074] multiplication: ;

[0075] Accumulation: ;

[0076] Multiplication: ;

[0077] In particular, elements It is an operation The zero element (also called the neutral element) is because: , ,also, It is an operation The absorption element, because: , ,element It is an operation The unit element (neutral element) is because: , A workstation (e.g., a machine tool, assembly station) is a processing unit in a production line used to perform processing operations on workpieces; a buffer zone (e.g., a temporary storage area) is an area used to store workpieces and balance the production cycle time. In this step, maximum addition algebra is used to precisely describe the start time of workpiece processing between workstations through mathematical expressions, thereby characterizing the processing sequence (e.g., workpieces must pass through workstations in sequence), waiting relationships (e.g., workpieces must wait for the preceding workstation to complete processing), and cumulative delays (e.g., overall delays caused by bottleneck workstations).

[0078] Understandably, the purpose of step S1 is to establish a mathematical model of the production line using maximal algebra. Maximal algebra can effectively describe the temporal relationships in the production process, such as the processing sequence of workpieces at different workstations, waiting time in buffer zones, and potential cumulative delays. By establishing such a model, the flow path and timing of each workpiece on the production line can be clearly understood. For example, the model can help identify which workstations are prone to becoming production bottlenecks, or under what circumstances workpieces will accumulate in buffer zones. This mathematical model provides a clear time framework for the production process, offering the necessary information for further optimization control.

[0079] Specifically, first, all workstations and buffer zones on the production line are determined; for example, a production line might have 3 workstations and 2 buffer zones. Then, the processing time for each workstation is defined, such as 5 minutes for workstation 1, 10 minutes for workstation 2, and 7 minutes for workstation 3. Next, maximal algebraic operators are used to describe the flow of workpieces between workstations. For example, the start processing time of a workpiece at workstation 1 can be represented as... (Feeding time), the start time of processing at station 2. The completion time of workstation 1 and the state of the buffer are jointly determined, i.e. ,in and These represent the processing times for station 1 and station 2, respectively. In this way, a mathematical model of the entire production line can be constructed, describing the processing sequence and timing of each workpiece at each station. Different implementation methods, such as considering different numbers of stations and buffer zones, or different processing times, can be achieved by adjusting the model parameters.

[0080] S2: Define the objective function, which consists of tracking accuracy cost and overall system performance cost. The tracking accuracy cost is used to ensure that the production line output matches the planned demand, and the overall system performance cost is used to optimize the production line's resource utilization, work-in-process inventory, and operating efficiency.

[0081] It's important to note that tracking accuracy cost is used to ensure consistency between actual output and planned demand. It assesses the accuracy of production plan execution by quantifying the deviation between the predicted output time and the reference time. Overall system performance cost, on the other hand, comprehensively considers various production line operating indicators, including equipment utilization, work-in-process inventory levels, and overall production efficiency. It evaluates the comprehensive performance of the production system through multi-dimensional indicators. Weighting coefficients are used to balance the relative importance of these two types of costs, adjusting the optimization focus according to specific production needs.

[0082] Understandably, step S2 achieves multi-objective optimization of production line material input control by constructing a dual cost structure that includes tracking accuracy and system performance. Tracking accuracy cost ensures the match between output and plan, reducing delivery deviations; overall system performance cost optimizes resource allocation, improves production efficiency, and reduces operating costs. By adjusting the weighting coefficients, the optimization focus can be flexibly adjusted according to actual production needs, ensuring both the execution of the production plan and improving the overall performance of the production system, overcoming the limitations of traditional single-objective optimization methods.

[0083] S3: Set the prediction time domain and control time domain of model predictive control, and initialize the workpiece feeding time and reference output sequence;

[0084] It's important to note that the prediction time domain refers to the time range used in model predictive control to predict future workpiece output. For example, if the prediction time domain is set to 5, the control system will predict the output of the next 5 workpieces. The control time domain refers to the number of feed times actually adjusted in each optimization calculation. The initial feed time for workpieces is usually set according to the production plan; for example, the feed time for the first workpiece can be set to time 0. The reference output sequence is the expected output time, used to guide the production process. For example, the reference output sequence can be an array containing the planned output time for each workpiece, such as... .

[0085] Understandably, step S3 provides the necessary parameters and initial conditions for the implementation of model predictive control. The settings of the prediction time domain and control time domain determine how the control system predicts and adjusts within the time frame. Initializing the feeding time provides a starting point for the optimization process, while the reference output sequence clarifies the production target. For example, setting the prediction time domain to 5 means that the system will consider the output of the next 5 workpieces, and setting the control time domain to 3 means that each optimization will adjust the feeding time of the next 3 workpieces. The effect is that the control system can adjust the feeding strategy in a timely manner based on the prediction of future production conditions to get as close as possible to the production target set by the reference output sequence.

[0086] S4: At the current control moment, based on the current state of the production line and the mathematical model, predict the output sequence of multiple workpieces in the future;

[0087] Based on the objective function, an optimization problem is solved to determine the material feeding sequence of multiple future workpieces, wherein the optimization problem is transformed into a mixed integer linear programming problem and solved using a solver;

[0088] Execute the first feeding moment from the feeding sequence and update the current control moment;

[0089] It should be noted that in step S4, the current state of the production line includes information such as the number of workpieces currently being processed at each workstation and the number of workpieces in the buffer. For example, workstation 1 is currently processing the second workpiece, and there is one workpiece in the buffer waiting for the next processing step. The mathematical model, previously established based on maximal addition algebra, is used to describe the flow pattern of the workpieces. Predicting the output sequence of multiple workpieces in the future is obtained by simulating the production process under the current state. When solving the optimization problem, the objective function and constraints need to be transformed into the form of a mixed-integer linear programming problem so that a solver (such as Gurobi or CPLEX) can be used for calculation. After executing the first feeding moment from the feeding sequence, the current control moment needs to be updated for the next round of prediction and optimization.

[0090] Understandably, the purpose of step S4 is to predict future production conditions and calculate the optimal material feeding strategy based on the current production state and mathematical model using model predictive control. By predicting the future output sequence of workpieces, it is possible to assess whether the current production state can meet the requirements of the production plan. Solving the optimization problem based on the objective function allows for finding the optimal solution among multiple possible material feeding schemes, thereby optimizing the production process. Executing the first material feeding moment and updating the control moment ensures that the optimization process can continue, adjusting the material feeding strategy in real time to cope with changes in the production process. This allows the production process to dynamically adapt to actual conditions, continuously optimizing the material feeding moment, thereby improving production efficiency and product quality.

[0091] S5: Repeat step S4 until the feeding sequence of all workpieces is solved and executed.

[0092] Preferably, the mathematical model includes serial units, assembly units, disassembly units, and buffer units.

[0093] It's important to note that in maximal algebra modeling, a serial unit refers to a unit structure where workpieces are processed sequentially through multiple workstations. For example, in a machining workshop, a workpiece is first turned on a lathe (workstation 1), then milled on a milling machine (workstation 2), and finally ground on a grinding machine (workstation 3). This is a typical serial unit structure. Each workstation completes a specific processing task, and the workpiece flows through each workstation in a predetermined order. An assembly unit refers to a production structure that brings together multiple components at one workstation for assembly. For example, in an electronic product assembly line, multiple components such as motherboards, chips, and displays are assembled into complete electronic products at assembly workstations. A disassembly unit, on the other hand, is the opposite of an assembly unit; it disassembles a complete product or component into multiple parts at a certain workstation. For example, in an automotive scrapping line, scrapped cars are disassembled into parts such as engines, frames, and tires at disassembly workstations. A buffer unit is mainly used to store workpieces and has a certain capacity limit; it can be a temporary storage area or warehouse in the production line.

[0094] Understandably, by clearly defining the components of the mathematical model—that is, decomposing the production line into serial units, assembly units, disassembly units, and buffer units—the production process can be described and optimized more accurately. Identifying and modeling these basic units allows for a more detailed depiction of the workpiece flow path and timing within the production line. For example, for serial units, the processing sequence and waiting time of workpieces at each station can be accurately calculated; for assembly units, the convergence timing of multiple components can be optimized; for disassembly units, disassembly time can be rationally scheduled; and for buffer units, the storage and transfer of workpieces can be effectively managed. Accurate modeling of these units helps improve the overall production line's efficiency and stability, reduces work-in-process inventory and delivery delay risks, and makes the production process smoother and more efficient.

[0095] Specifically, in the implementation process, the first step is to perform a structural analysis of the production line to identify sequential units, assembly units, disassembly units, and buffer units. For example, in an automobile manufacturing plant, the body welding workshop may contain multiple sequential units (different welding stations sequentially weld different parts of the body), the assembly workshop contains assembly units (assembling components such as engines and transmissions onto the body), and the cooling area after the painting workshop can be considered a buffer unit (temporarily storing painted body parts while they cool). For each identified unit, a corresponding maximal algebraic model is established based on its type and function. For sequential units, maximal algebraic operators are used to describe the flow sequence and time of workpieces at each station; for assembly units, the synchronous arrival and assembly time of multiple components are considered; for disassembly units, the disassembly sequence and timing are considered; and for buffer units, corresponding parameters are set according to their capacity and function. In this way, an accurate mathematical model of the entire production line can be constructed. Combinations of different implementation methods, such as for different types of production lines (discrete manufacturing, process industries, etc.) or different production needs (mass production, small-batch customization, etc.), allow for flexible adjustments to the modeling strategies and parameter settings of each unit to achieve the best optimization results. For example, in mass production, the cycle time consistency of serial units can be optimized; in small-batch customization, the flexibility of assembly and disassembly units can be enhanced through modeling.

[0096] Preferably, such as Figure 2 and Figure 3 ( Figure 3 As shown in the diagram (with station capacities of 2, 3, and 4 respectively), in the serial unit, for any station... Start processing the first Start time of machining of each workpiece This includes the following steps:

[0097] S111: Obtain the preceding workstation For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates the preceding workstation Start processing the first The moment of each workpiece;

[0098] S112: By acquiring downstream workstations For the Start time of machining of each workpiece Determine the workstation Is there at least one available slot to receive the first... One workpiece; among which For workstations The capacity, when season ,in For the zero element in maximal additive algebra;

[0099] S113: Combine the processing completion time obtained in step S111 with the time obtained in step S112. In maximal addition algebra The calculation is performed, and the maximum value between the two is taken as the workstation. Start processing the first Start time of machining of each workpiece The following relation is satisfied: .

[0100] It should be noted that in the modeling process of the serial unit, the preceding station... Refers to the current workstation The previous processing station is responsible for processing the workpiece using the previous process. Processing completion time. Indicates the workpiece is in the preceding station. The moment when processing is completed, among which Workstation Start processing the first The moment of each workpiece Workstation Processing time, This represents the addition operation in maximal additive algebra, i.e. Downstream workstations This is the current workstation. The next processing station, which is related to the first Start time of machining of each workpiece Used to determine the current workstation Is there an available slot to receive a new workpiece? When, indicate the current workstation In the initial processing state, at this time let ( (The zero element in maximal additive algebra, i.e., negative infinity), indicates that there are no workpieces at the downstream station. Waiting. In maximal addition algebra... The calculation takes the maximum value between the two moments to ensure the workpiece is at the current workstation. Start of processing time The conditions must be met: the previous workstation has completed its processing and there is an available workstation in the current workstation.

[0101] Understandably, the purpose of a serial unit is to accurately describe the flow pattern of workpieces within the unit, ensuring a reasonable processing sequence and timing between workpieces at each station. By acquiring the completion time of the preceding station and the idle status of the downstream station, and utilizing the algebraic rules of maximum addition, the start time of processing at the current station can be accurately calculated. By modeling the time dependencies in the production process, the serial unit ensures smooth flow of workpieces between stations, avoiding waiting or backlogs caused by improper process connections. This improves the timeliness and resource utilization of the production process, reduces work-in-process inventory backlog, and provides a precise time basis for subsequent optimization control.

[0102] For example, consider a serial unit with 3 stations, where the processing time and capacity of each station are as follows: Station Processing time minutes, capacity Workstation Processing time minutes, capacity Workstation Processing time minutes, capacity Assume the feeding time of workpiece 1 is... minutes, that is According to the rules of operation for maximal addition algebra:

[0103] workstation The time to complete the machining of workpiece 1 is minutes, workstation The moment when workpiece 1 begins to be processed It is determined by the following two conditions:

[0104] Preceding workstation Processing completion time: 2 minutes;

[0105] Downstream workstations The start time of machining for workpiece 1-1=0 (Negative infinity), therefore, minutes, workstation The time to complete the machining of workpiece 1 is minute;

[0106] workstation The moment when workpiece 1 begins to be processed It is determined by the following two conditions:

[0107] Preceding workstation Processing completion time: 5 minutes;

[0108] Since there are no downstream workstations, there is no need to consider the availability of downstream workstations. minutes, workstation The time to complete the machining of workpiece 1 is minute.

[0109] For workpiece 2: workstation The moment when workpiece 2 begins to be processed It is determined by the following two conditions:

[0110] Preceding workstation The (non-existent) processing completion time is Downstream workstations The start time of machining for workpiece 2-1=1 minutes, therefore, minutes, workstation The time to complete the machining of workpiece 2 is minute.

[0111] workstation The moment when workpiece 2 begins to be processed It is determined by the following two conditions:

[0112] Preceding workstation Processing completion time: 4 minutes, downstream station The start time of machining for workpiece 2-1=1 minutes, therefore minutes, workstation The time to complete the machining of workpiece 2 is minute.

[0113] workstation The moment when workpiece 2 begins to be processed It is determined by the following two conditions:

[0114] Preceding workstation The processing completion time is 8 minutes. Since there is no downstream workstation, the vacancy status of the downstream workstation does not need to be considered. minutes, workstation The time to complete the machining of workpiece 2 is minute.

[0115] Preferably, such as Figure 4 As shown, in the assembly unit, for the assembly station Start processing the first Start time of machining of each workpiece This includes the following steps:

[0116] S121: Obtain all preceding supply stations For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates workstation Start processing the first The moment of each workpiece;

[0117] S122: By acquiring downstream workstations For the Start time of machining of each workpiece Determine the assembly station Is there at least one available slot to receive the first... One workpiece, of which For workstations The capacity, when season ,in For the zero element in maximal additive algebra;

[0118] S123: Combine the processing completion times of all preceding supply stations obtained in step S121 with the processing start times of the downstream stations obtained in step S122. In maximal addition algebra The calculation is performed, and the maximum value at all times is taken as the assembly station value. Start processing the first Start time of machining of each workpiece The following relation is satisfied:

[0119] .

[0120] It should be noted that during the modeling process of the assembly unit, the preceding supply station... It refers to the assembly station. Multiple upstream workstations supplying components; for example, in automobile manufacturing, upstream supply workstations might include engine assembly workstations, transmission assembly workstations, and body welding workstations, each responsible for supplying different components required by the assembly workstations. (Processing completion time) Indicates the first The preceding supply station completed the first... The processing time of each workpiece, among which Workstation Start processing the first The moment of each workpiece Workstation Processing time, This represents the addition operation in maximal additive algebra, i.e. Downstream workstations It is an assembly station The next workstation is used for further processing or storing assembled workpieces. Start of processing time. Used to determine assembly station Are there enough empty spaces to receive new workpieces? At that time, explain the assembly station In the initial processing state, at this time let ( (The zero element in maximal additive algebra, i.e., negative infinity), indicates that there are no workpieces at the downstream station. Waiting. In maximal addition algebra... The calculation takes the maximum value across all time points to ensure the assembly station... Processing of the next step only begins when all preceding supply stations have completed their processing and there is an available slot at the current station. One workpiece.

[0121] Understandably, the purpose of an assembly unit is to accurately describe the assembly time of the workpieces within the unit, ensuring that all components arrive at the assembly station synchronously and are assembled. By acquiring the completion times of all preceding supply stations and the availability status of downstream stations, and utilizing the algebraic rules of maximum addition, the start time of the assembly station can be accurately calculated. By modeling the time dependencies in the assembly process, it is ensured that the assembly station only begins assembly when all components are ready and it has available space. This avoids delays caused by untimely component supply or congestion at the assembly station, improving the synchronization and efficiency of the assembly process, reducing work-in-process inventory backlog, and providing a precise time basis for subsequent optimization control.

[0122] For example, consider an assembly unit that includes two preceding supply stations. and and an assembly station The processing time and capacity of each workstation are as follows: Workstation Processing time minutes, capacity Workstation Processing time minutes, capacity Assembly station capacity Assume the feeding time of workpiece 1 is... minutes, that is and (Assuming two supply stations feed materials simultaneously), according to the rules of maximal addition algebra, the stations... The time to complete the machining of workpiece 1 is minutes, workstation The time to complete the machining of workpiece 1 is Minutes. Assembly station. The moment when workpiece 1 begins to be processed It is determined by the completion time of all preceding supply stations and the availability status of downstream stations. Assume the downstream station... The start time of machining for workpiece 1-1=0 (Negative infinity), therefore, Minutes, assembly station The time to complete the machining of workpiece 1 is minutes (assuming assembly station) Processing time (minutes). For workpiece 2, station The moment when workpiece 2 begins to be processed From the preceding workstation (Non-existent) processing completion time and downstream workstations The start time of machining for workpiece 2-1=1 Decision, therefore Minutes. Workstation The time to complete the machining of workpiece 2 is minutes, workstation The moment when workpiece 2 begins to be processed From the preceding workstation (Non-existent) processing completion time and downstream workstations The start time of machining for workpiece 2-1=1 The decision is made in minutes, therefore Minutes. Workstation The time to complete the machining of workpiece 2 is Minutes, assembly station The moment when workpiece 2 begins to be processed The processing completion times of all preceding supply stations (2 minutes and 6 minutes) and downstream stations are determined by... The start time of machining for workpiece 2-1=1 The decision is made in minutes, therefore Minutes, assembly station The time to complete the machining of workpiece 2 is Minutes. This example demonstrates that the start time of processing at the assembly station strictly follows the algebraic operation rules of maxima, ensuring the synchronous arrival and smooth assembly of all parts at the assembly station, and avoiding delays caused by untimely parts supply or congestion at the assembly station.

[0123] Preferably, such as Figure 5 As shown, in the disassembly unit, for the disassembly station Start processing the first Start time of machining of each workpiece This includes the following steps:

[0124] S131: Obtain the preceding workstation For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates the preceding workstation Start processing the first The moment of each workpiece;

[0125] S132: By acquiring all subsequent workstations For the Start time of machining of each workpiece Determine the disassembly station Is there at least one available slot to receive the first... One workpiece, of which For workstations The capacity, when season ,in For the zero element in maximal additive algebra;

[0126] S133: Perform a maximum addition algebraic operation on the processing completion time obtained in step S131 and the start times of all subsequent workstations obtained in step S132. The calculation is performed, and the maximum value at all times is taken as the disassembly station value. Start processing the first Start time of machining of each workpiece The following relation is satisfied:

[0127] .

[0128] It should be noted that during the modeling process of the disassembly unit, the preceding workstations... It refers to the disassembly station. This provides an upstream station for the workpieces to be disassembled; for example, in an electronics recycling line, the preceding station might be responsible for transporting waste electronics. (Processing completion time) This indicates that the preceding workstation has completed the first... The processing (or transportation) time of each workpiece, among which Workstation Start processing the first The moment of each workpiece Workstation Processing time, This represents the addition operation in maximal additive algebra, i.e. Subsequent workstations It is a disassembly station The downstream station is used for further processing of the disassembled parts. Start of processing time. Used to determine the disassembly station Are there enough empty spaces to receive new workpieces? At that time, explain the disassembly station. In the initial processing state, at this time let ( For the zero element in maximal additive algebra (i.e., negative infinity), it indicates that there is no workpiece at the subsequent station. Waiting. In maximal addition algebra... The calculation takes the maximum value across all time points to ensure the disassembly station... The next workpiece will only begin processing after the previous workpiece has been completed and there is available space at the current workstation. One workpiece.

[0129] Understandably, by obtaining the completion time of the preceding station and the availability status of the subsequent station, and using the operational rules of maximal addition algebra, the start time of the disassembly station can be accurately calculated. By modeling the time dependencies in the disassembly process, it can be ensured that the disassembly station only starts disassembly when the workpiece to be disassembled arrives and it has an available space. This avoids delays caused by untimely workpiece supply or congestion at the disassembly station, improves the synchronization and efficiency of the disassembly process, reduces the backlog of work-in-process inventory, and provides a precise time basis for subsequent optimization control.

[0130] For example, consider a disassembly unit with a preceding station. and two subsequent workstations and and a dismantling station The parameters for each workstation are as follows: Workstation Processing time minutes, capacity Disassembly station capacity Subsequent workstations Processing time minutes, capacity Subsequent workstations Processing time minutes, capacity Assume the feeding time of workpiece 1 is... minutes, that is According to the operational rules of maximal addition algebra, the workstation The time to complete the machining of workpiece 1 is Minutes, disassembly station The moment when workpiece 1 begins to be processed From the preceding workstation The processing completion time is 5 minutes and subsequent workstations and The start time of machining for workpiece 1-1=0 (Negative infinity) determines, therefore Minutes. Disassembly station. The time to complete the machining of workpiece 1 is minutes (assuming disassembly station) Processing time (minutes). For workpiece 2, station The moment when workpiece 2 begins to be processed From the preceding workstation (Non-existent) processing completion time and subsequent workstations The start time of machining for workpiece 2-1=1 The decision is made in minutes, therefore Minutes. Workstation The time to complete the machining of workpiece 2 is Minutes. Disassembly station. The moment when workpiece 2 begins to be processed From the preceding workstation The processing completion time is 10 minutes, and the subsequent workstations The start time of machining for workpiece 2-1=1 Minutes (hypothetically), subsequent workstations The start time of machining for workpiece 2-1=1 The decision is made in minutes (hypothetically), therefore Minutes. Disassembly station. The time to complete the machining of workpiece 2 is Minutes. This example demonstrates that strictly adhering to the algebraic operation rules of the disassembly station at the start of processing ensures the timely arrival and smooth disassembly of the workpieces, avoiding delays caused by untimely workpiece supply or congestion at the disassembly station.

[0131] Preferably, in the buffer unit, the buffer unit is abstracted as a workstation with capacity and processing time, and the calculation of the start processing time of the buffer unit adopts the same recursive logic as the serial unit, assembly unit or disassembly unit.

[0132] The processing time of the buffer unit is set according to its function: when only capacity constraints are reflected, the processing time of the buffer unit is zero; when undertaking process or cycle adjustment functions, the processing time of the buffer unit is the specified time for the workpiece to stay.

[0133] It's important to note that in the modeling of buffer units, the buffer unit is abstracted as a workstation with capacity and processing time. This abstraction ensures that the modeling logic of buffer units remains consistent with that of serial, assembly, or disassembly units. The capacity of a buffer unit determines the number of workpieces it can hold, while the processing time is set according to the specific function of the buffer unit. For example, in a buffer unit used only for temporary workpiece storage, the processing time can be set to zero, as the workpieces can leave without any processing after entering the buffer. However, when the buffer unit performs process or cycle time regulation functions, such as in cooling, drying, or fixed-cycle conveying equipment, the workpieces need to stay in the buffer unit for a certain period of time to complete specific process steps. In this case, the processing time is set to the specified time the workpieces need to stay. The recursive logic in the maximal algebra, by considering the capacity and processing time of the buffer unit, ensures that the timing of workpieces entering and leaving the buffer unit conforms to the requirements of the production process. In this way, the model can accurately reflect the actual role of the buffer unit in the production line, thereby improving the predictability and controllability of the production process and reducing production delays or resource waste caused by improper buffer unit management.

[0134] like Figure 6 As shown, the disassembly of a mobile phone product repair and refurbishment production line includes two system inputs. and Two system outputs and Multiple workstations (where the buffer) Unified representation as Each workstation has a capacity and processing time The production line is broken down into four basic types: serial units, assembly units, disassembly units, and buffer units. Serial units contain a sequence of workstations. , , and Assembly units are embodied in workstations. and Joint supply (Right now The disassembly unit is manifested as a workstation. Decomposed into and (Right now The buffer unit then passes through the workstation. Unified modeling. Based on maximum addition algebra, the start time of each station can be accurately represented by the following formula, thus fully describing the timing dependencies and work-in-process flow of the production line:

[0135] ；

[0136] ；

[0137] ；

[0138] ；

[0139] ；

[0140] ；

[0141] ；

[0142] ；

[0143] ；

[0144] ；

[0145] ；

[0146] ；

[0147] ；

[0148] This disassembly example demonstrates the practical application of production line modeling within a unified framework, providing an accurate mathematical foundation for predictive control using mathematical models.

[0149] Preferably, in step S2, the objective function Satisfying the relation:

[0150] ;

[0151] in, Indicates the cost of tracking accuracy. Indicates the overall performance cost of the system. This represents a weighting factor used to balance the relative importance of tracking accuracy cost and overall system performance cost.

[0152] The cost of tracking accuracy It can be divided into and :

[0153] ;

[0154] ;

[0155] in, It is in the Time prediction The system output time at that moment. yes The reference output time of the time. For prediction in the time domain;

[0156] The overall performance cost of the system It can be divided into , , , and ,in:

[0157] ;

[0158] ;

[0159] ;

[0160] ;

[0161] ;

[0162] in, and They are respectively in the 1st Time prediction Workstation The input and output times, For workstations Processing time, and These represent the sets of all workstations and all buffer units, respectively.

[0163] It should be noted that the cost of tracking accuracy Used to measure the degree of match between production line output and planned demand, including and Two sub-items: By calculating the predicted output time Compared to the reference output time The sum of absolute deviations penalizes both early and late completion, making it suitable for scenarios that require balancing inventory costs and delay risks. By calculating the sum of positive deviations between the predicted output time and the reference time, only the delayed completion is penalized, making it suitable for scenarios that emphasize on-time delivery. Overall system performance cost. This is used to optimize resource utilization and operational efficiency of the production line, and includes five sub-items: By calculating the reciprocal of the throughput in the predicted time domain and taking its negative value, low throughput is penalized in order to maintain high output. By calculating the sum of the difference between the predicted output and the input time of each workstation and subtracting the processing time, the workstation congestion status is penalized to reduce resource consumption. By calculating the sum of the difference between the consecutive predicted input times of each workstation and the processing time, the starvation state of the workstation is penalized to reduce idle time. By calculating the negative value of the sum of the theoretical processing times of each workstation, low equipment utilization is penalized to improve overall efficiency. By calculating the sum of the differences between the predicted output and input times of each buffer unit, high inventory levels in the buffer are penalized to reduce work-in-process dwell time. Weighting coefficients. Used to adjust the relative weights of tracking accuracy and system performance in the objective function, for example when When the value is large, more emphasis is placed on system performance optimization; when the value is small, more emphasis is placed on output tracking accuracy. These cost items are calculated based on the predicted output and input time of the maximal algebraic model, providing quantitative indicators for the optimization problem.

[0164] Understandably, by defining a comprehensive objective function, the multi-objective optimization problem of production line material feeding control is structured, tracking accuracy and cost to ensure output matches the plan, reducing delivery deviations, optimizing overall system performance and cost, resource utilization and operational efficiency, and avoiding production bottlenecks and waste. By balancing the priorities of both through weighted coefficients, the system can flexibly adjust the optimization focus according to actual needs. This design overcomes the shortcomings of traditional single-objective optimization methods, transforming production management objectives into calculable cost items through mathematical form, providing a clear optimization direction for model predictive control, thereby improving the overall performance of the production line while ensuring on-time delivery.

[0165] Preferably, step S3 includes:

[0166] Define the total quantity to be processed Prediction time domain and control time domain Given the initial workpiece feeding time and reference output sequence ;

[0167] Step S4 includes:

[0168] During control time Define the feeding sequence The feeding time outside the control time domain remains unchanged, satisfying ,in ;

[0169] The feeding sequence Substituting into the mathematical model in step S1, the corresponding predicted output sequence is calculated. ;

[0170] Reference output sequence Feeding sequence and predicted output sequence Substituting into the objective function defined in step S2, a result is formed with An optimization problem with 10 decision variables;

[0171] The optimization problem is transformed into a mixed-integer linear programming problem, and solved using a solver to obtain the optimal feeding sequence. ;

[0172] The rolling execution control strategy executes only the optimal feeding sequence. The first feeding moment in the process ;

[0173] Update control time .

[0174] It should be noted that the total number of processes This refers to the total number of workpieces planned for production on the production line, in the forecast time domain. In model predictive control, this refers to the time range used to predict future workpiece output, typically expressed in terms of workpiece quantity or time units; it is the control time domain. This refers to the number of feeding times actually adjusted in each optimization calculation, which is usually less than or equal to the feeding time of the initial workpiece in the prediction time domain. This is the start time of processing the first workpiece, referencing the output sequence. This refers to the planned output time for each workpiece, used to guide the production process. (Material feeding sequence) It is the current control moment. next future The feeding time schedule for each workpiece. The feeding time outside the control time domain remains unchanged, that is, for... , This ensures that the scope of optimization calculations is limited and controllable, and that the predicted output sequence is obtained by substituting the feeding sequence into the mathematical model. The future production process is simulated using a maximum addition algebraic model. The objective function combines the reference output sequence, the feeding sequence, and the predicted output sequence to form an optimization problem aimed at minimizing costs and penalties while meeting production targets. The optimization problem is then transformed into a mixed-integer linear programming (MILP) problem. This step uses mathematical transformations to enable the problem to be efficiently solved by existing solvers, such as Gurobi or CPLEX, which will be used to compute the optimal feeding sequence. The rolling execution control strategy means that after each optimization, only the first feeding time is executed, then the control time is updated, and the optimization calculation is performed again, ensuring the real-time performance and dynamic adaptability of the control process.

[0175] Preferably, transforming the optimization problem into a mixed-integer linear programming problem includes:

[0176] Using the Big-M method to study maximal operators in maximal additive algebra. Linearization is performed by introducing binary decision variables and sufficiently large constants, which is equivalent to a set of linear constraints.

[0177] Constraints are imposed on the decision variables, including monotonicity constraints and non-negativity constraints. Monotonicity constraints are used to ensure the time monotonicity of the feeding sequence, satisfying... The non-negativity constraint is used to ensure that the material is non-negative at all feeding times, satisfying... ;

[0178] A mixed-integer linear programming solver was selected to solve the transformed problem.

[0179] It should be noted that the maximal operator in maximal addition algebra... In optimization problems, this manifests as the operation of finding the maximum value, which is difficult to handle directly in traditional linear programming. To address this issue, the Big-M method is used for linearization. The Big-M method introduces a binary decision variable and a sufficiently large constant. This transforms the operation of finding the maximum value into a set of linear constraints. For example, for the expression... This can be achieved by introducing binary variables. and constant Construct the following constraints: , , , , Furthermore, imposing constraints on decision variables is a crucial step in ensuring the feasibility and rationality of the solution. Monotonicity constraints are one such constraint. Ensure the feeding sequence time is monotonically increasing, meaning the feeding time of subsequent workpieces must not be earlier than the feeding time of the previous workpiece, which conforms to the logical order of actual production. Non-negativity constraint. This ensures that all material feeding times are non-negative, avoiding unrealistic negative-time feeding plans. These constraints are integrated into the optimization problem through mathematical expressions, forming a complete MILP problem together with the objective function. Finally, mixed-integer linear programming solvers (such as Gurobi and CPLEX) are used to solve the transformed problem. These solvers employ advanced algorithms (such as branch and bound and cutting plane methods) to find the optimal solution that satisfies all constraints and minimizes the objective function. In this way, the optimization problem can be effectively solved within a reasonable time, providing real-time and feasible decision support for production line material feeding control.

[0180] One embodiment of the present invention relates to the feeding control of a simple serial production line, such as... Figure 7 and 8 As shown, the production line consists of two workstations, each with a capacity of 1 and processing times of 3 seconds and 4 seconds, respectively. A mathematical model is established using maximal addition algebra, and the start time of processing at each workstation is expressed by formulas (17) to (19):

[0181] , , ;

[0182] in For the feeding time, To determine the output time, during implementation, define the processing quantity. Prediction time domain Control Time Domain Initialize workpiece feeding time Seconds and reference output sequence During control time Define the feeding sequence ,in To reflect that time outside the control time domain remains unchanged; Substitute into the mathematical model to calculate the predicted output sequence The result is as follows Figure 8 As shown. For example , , ,and then , , Similar derivation and ,based on Construct the objective function (Including tracking accuracy costs and overall system performance costs), among which have 1 decision variable ( and The optimal feeding sequence can be solved using a mixed-integer linear programming solver (such as Gurobi). Then, the execution continues in a rolling fashion, executing only the specified number of times. The first feeding moment in the process and update control timing This process is repeated until all workpieces have been fed. This embodiment demonstrates how to dynamically optimize the feeding strategy through maximum algebraic modeling and model predictive control, thereby reducing work-in-process inventory and delay risks while ensuring on-time delivery.

[0183] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "illustrative embodiment," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0184] Although embodiments of the invention have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims

1. A production line material feeding control method based on maximum addition algebra and model predictive control, characterized in that, Includes the following steps: S1: A mathematical model of the production line is established based on maximal addition algebra, wherein the mathematical model includes multiple workstations and buffers, and the mathematical model describes the processing sequence, waiting relationship and cumulative delay of workpieces between the workstations; S2: Define the objective function, which consists of tracking accuracy cost and overall system performance cost. The tracking accuracy cost is used to ensure that the production line output matches the planned demand, and the overall system performance cost is used to optimize the production line's resource utilization, work-in-process inventory, and operating efficiency. S3: Set the prediction time domain and control time domain of model predictive control, and initialize the workpiece feeding time and reference output sequence; S4: At the current control moment, based on the current state of the production line and the mathematical model, predict the output sequence of multiple workpieces in the future; Based on the objective function, an optimization problem is solved to determine the material feeding sequence of multiple future workpieces, wherein the optimization problem is transformed into a mixed integer linear programming problem and solved using a solver; Execute the first feeding moment from the feeding sequence and update the current control moment; S5: Repeat step S4 until the feeding sequence of all workpieces is solved and executed. The mathematical model includes serial units, assembly units, disassembly units, and buffer units; In the serial unit, for any workstation Start processing the first Start time of machining of each workpiece This includes the following steps: S111: Obtain the preceding workstation For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates the preceding workstation Start processing the first The moment of each workpiece; S112: By acquiring downstream workstations For the Start time of machining of each workpiece Determine the workstation Is there at least one available slot to receive the first... One workpiece; among which For workstations The capacity, when season ,in For the zero element in maximal additive algebra; S113: Combine the processing completion time obtained in step S111 with the time obtained in step S112. In maximal addition algebra The calculation is performed, and the maximum value between the two is taken as the workstation. Start processing the first Start time of machining of each workpiece The following relation is satisfied: .

2. The production line feeding control method based on maximum addition algebra and model predictive control according to claim 1, characterized in that, In the assembly unit, for the assembly station Start processing the first Start time of machining of each workpiece This includes the following steps: S121: Obtain all preceding supply stations For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates workstation Start processing the first The moment of each workpiece; S122: By acquiring downstream workstations For the Start time of machining of each workpiece Determine the assembly station Is there at least one available slot to receive the first... One workpiece, of which For workstations The capacity, when season ,in For the zero element in maximal additive algebra; S123: Combine the processing completion times of all preceding supply stations obtained in step S121 with the processing start times of the downstream stations obtained in step S122. In maximal addition algebra The calculation is performed, and the maximum value at all times is taken as the assembly station value. Start processing the first Start time of machining of each workpiece The following relation is satisfied: 。 3. The production line feeding control method based on maximum addition algebra and model predictive control according to claim 1, characterized in that, In the disassembly unit, for the disassembly station Start processing the first Start time of machining of each workpiece This includes the following steps: S131: Obtain the preceding workstation For the The completion time of machining each workpiece is expressed as: ,in For workstations Processing time, Indicates the preceding workstation Start processing the first The moment of each workpiece; S132: By acquiring all subsequent workstations For the Start time of machining of each workpiece Determine the disassembly station Is there at least one available slot to receive the first... One workpiece, of which For workstations The capacity, when season ,in For the zero element in maximal additive algebra; S133: Perform a maximum addition algebraic operation on the processing completion time obtained in step S131 and the start times of all subsequent workstations obtained in step S132. The calculation is performed, and the maximum value at all times is taken as the disassembly station value. Start processing the first Start time of machining of each workpiece The following relation is satisfied: 。 4. The production line feeding control method based on maximum addition algebra and model predictive control according to claim 1, characterized in that, In the buffer unit, the buffer unit is abstracted as a workstation with capacity and processing time. The calculation of the start processing time of the buffer unit adopts the same recursive logic as the serial unit, assembly unit or disassembly unit. The processing time of the buffer unit is set according to its function: when only capacity constraints are reflected, the processing time of the buffer unit is zero; when undertaking process or cycle adjustment functions, the processing time of the buffer unit is the specified time for the workpiece to stay.

5. The production line feeding control method based on maximum addition algebra and model predictive control according to claim 1, characterized in that, In step S2, the objective function Satisfying the relation: ； in, Indicates the cost of tracking accuracy. Indicates the overall performance cost of the system. This represents a weighting factor used to balance the relative importance of tracking accuracy cost and overall system performance cost. The cost of tracking accuracy It can be divided into and : ； ； in, It is in the Time prediction The system output time at that moment. yes The reference output time of the time. For prediction in the time domain; The overall performance cost of the system It can be divided into , , , and ,in: ； ； ； ； ； in, and They are respectively in the 1st Time prediction Workstation The input and output times, For workstations Processing time, and These represent the sets of all workstations and all buffer units, respectively.

6. The production line feeding control method based on maximum addition algebra and model predictive control according to claim 1, characterized in that, Step S3 includes: Define the total quantity to be processed Prediction time domain and control time domain Given the initial workpiece feeding time and reference output sequence ; Step S4 includes: During control time Define the feeding sequence The feeding time outside the control time domain remains unchanged, satisfying ,in ; The feeding sequence Substituting into the mathematical model in step S1, the corresponding predicted output sequence is calculated. ; Reference output sequence Feeding sequence and predicted output sequence Substituting into the objective function defined in step S2, a result is formed with An optimization problem with 10 decision variables; The optimization problem is transformed into a mixed-integer linear programming problem, and solved using a solver to obtain the optimal feeding sequence. ; The rolling execution control strategy executes only the optimal feeding sequence. The first feeding moment in the process ; Update control time .

7. The production line feeding control method based on maximum addition algebra and model predictive control according to claim 6, characterized in that, Transforming the optimization problem into a mixed-integer linear programming problem includes: Using the Big-M method to study maximal operators in maximal additive algebra. Linearization is performed by introducing binary decision variables and sufficiently large constants, which is equivalent to a set of linear constraints. Constraints are imposed on the decision variables, including monotonicity constraints and non-negativity constraints. Monotonicity constraints are used to ensure the time monotonicity of the feeding sequence, satisfying... The non-negativity constraint is used to ensure that the material is non-negative at all feeding times, satisfying... ; A mixed-integer linear programming solver was selected to solve the transformed problem.

Images