Spares supply and replacement collaborative decision method for multi-modal system in multi-task scenario

Abstract

This invention relates to a collaborative decision-making method for spare parts supply and replacement in multimodal systems under multi-task scenarios. The method includes constructing a set of component performance levels and a set of system performance levels; obtaining the probability distribution of components and the system at different performance levels; defining multi-task modes; calculating task reliability; defining a component replacement state equation to update the component performance level at the start of each sub-task; defining decision variables and constraints; defining an objective function; designing and solving the basic dynamic programming equations; obtaining the optimal component replacement strategy for each maintenance stage; constructing an optimal strategy table; selecting the optimal initial spare parts carrying capacity; and forming the optimal decision. This achieves collaborative decision-making for spare parts supply and replacement in multimodal systems under multi-task scenarios. This invention, while ensuring the reliable operation of the multi-task sequence, rationally allocates maintenance resources and spare parts, avoiding resource waste and the risk of task failure.

Description

Technical Field

[0001] This invention relates to the fields of reliability engineering, maintenance decision optimization, and multimodal system management, and particularly to a collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios. Background Technology

[0002] In power systems, aerospace equipment, marine systems, and complex industrial systems, systems typically need to perform multiple tasks continuously throughout their entire lifecycle. Due to the complex operating environment, limited maintenance windows between tasks, and resource constraints such as maintenance time, maintenance costs, and spare parts availability, it is often impossible to fully repair all degraded components at every maintenance stage.

[0003] Existing selective maintenance research is mostly based on two-state systems (normal / failure) or single-task modes, making it difficult to accurately describe the characteristics of component performance degradation, coexistence of multiple performance levels, and continuous execution of multiple tasks in real-world systems. Furthermore, the initial spare parts inventory directly affects system task success rate and maintenance costs, but existing methods typically separate maintenance decisions from spare parts supply decisions, resulting in insufficient overall decision-making efficiency and reliability. In multimodal systems (MSS) under multi-task modes with varying sub-task times and intensities, existing selective maintenance strategies suffer from the following problems: (1) The problem of dynamic degradation modeling of component and system performance levels in multimodal systems; (2) The problem of quantifying the reliability of system tasks under multiple task sequences (different sub-tasks); (3) Decision problem of optimal maintenance behavior (selective replacement) during the interval between tasks under the constraint of limited spare parts resources; (4) Optimize the determination of the initial spare parts carrying capacity and clarify the relationship between the upper limit of the spare parts carrying capacity and the system reliability.

[0004] Therefore, it is necessary to propose a joint decision-making method for selective replacement and spare parts supply under spare parts constraints for multi-task modes and multi-modal systems. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a collaborative decision-making method for spare parts supply and replacement in multimodal systems under multi-task scenarios. This method solves the technical problems of traditional approaches, which lack overall resource planning for the entire task sequence, leading to overuse or underuse of maintenance resources and difficulty in optimizing the initial spare parts carrying capacity. This invention establishes a joint decision-making model for selective replacement and spare parts supply in multimodal systems under multi-task modes, designs a dynamic programming optimization algorithm, and achieves optimal matching between component replacement strategies and initial spare parts carrying capacity. Under the premise of ensuring reliable operation of the multi-task sequence, it rationally allocates maintenance resources and spare parts, avoiding resource waste and the risk of task failure.

[0006] To achieve the above technical objectives, the present invention provides the following technical solution: a collaborative decision-making method for spare parts supply and replacement of a multimodal system in a multi-task scenario, comprising the following steps: Construct a set of component performance levels, and obtain the transition rate matrix of the components and the probability distribution of the components at each performance level; Define the system structure function, construct the system performance level set, construct the general generating function of each component and the general generating function of the system, and map the probability distribution of each component at each performance level to the probability distribution of the system at each performance level; Define a multitasking mode; the multitasking mode includes a multitasking sequence consisting of several subtasks, and the parameters of each subtask in the multitasking sequence include system performance level requirements and duration; Define the component replacement state equation, and calculate the performance level of each component at the beginning of the current subtask based on the performance level of each component at the end of the previous subtask and the component replacement strategy. Calculate task reliability; the task reliability includes the sub-task reliability of each sub-task and the multi-task sequence reliability of the system, calculated based on the performance level of each component at the beginning of each sub-task and the probability distribution of the system at each performance level; Define the component replacement strategy and the initial spare parts carrying quantity as decision variables, define spare parts resource constraints and construct the feasible replacement set for each maintenance stage, and calculate the remaining spare parts quantity after each maintenance stage; Define an objective function that aims to maximize the reliability of the multi-task sequence while optimizing the component replacement strategy and the initial spare parts carrying capacity. Based on the defined objective function, design and solve the basic equations of dynamic programming to obtain the optimal component replacement strategy and the optimal multi-task sequence reliability of each sub-task in each maintenance stage, and construct the optimal strategy table. Different initial spare parts carrying quantities are set, and the optimal initial spare parts carrying quantity is selected based on the optimal strategy table. The optimal initial spare parts carrying quantity and the optimal component replacement strategy constitute the optimal decision, thereby realizing spare parts supply and replacement.

[0007] Optionally, the step of constructing a set of component performance levels and obtaining the transition rate matrix of the components and the probability distribution of the components at each performance level includes: The components are sorted by performance level from smallest to largest to construct a set of component performance levels, which is mathematically represented as follows: ; in Indicates components A set of component performance levels; Indicates components The Performance levels, sorted from smallest to largest. For components Total number of performance levels; Define a transition rate matrix to describe the state transition characteristics of each component. The transition rate matrix is ​​defined as follows: ; in, Indicates components The transition rate matrix; Indicates components By the performance level Degenerate to the first performance level The transfer rate and ; The probability distribution of each component at different performance levels is defined mathematically as follows: ; in Indicates components At any moment The probability distribution, Each represents a component At any moment In the first performance level The probability of and satisfying ; This represents the transpose operation; the probability distribution follows the Fokker-Planck equation, mathematically represented as follows: ; in, For components At any moment probability distribution The first derivative with respect to time represents the component. The rate at which the probability of being at each performance level changes over time.

[0008] Optionally, the definition of system structure functions, the construction of a system performance level set, and the construction of general generation functions for each component and a system-wide general generation function include: The system structure function maps the performance levels of each component to the overall system performance level. When the system is a series system, the system structure function is defined as follows: The system structure function, when the system is a parallel system, is defined as follows: ,in Represents the system structure function. Indicates at time From components , The system performance level is jointly determined by the performance levels of the components. , Representing time respectively part , Performance level, This indicates taking the minimum value of each performance level within the parentheses; The system performance levels are sorted from smallest to largest to construct a set of system performance levels, which is mathematically represented as follows: ; in, Represents the set of system performance levels; The system's first Performance levels, sorted from smallest to largest. The total number of system performance levels; The general generating functions for each component are constructed, and their mathematical representation is as follows: ; in, Indicates components General generating functions, , This represents the total number of components. Indicates components At any moment In the first performance level The probability, For components Total number of performance levels; Formal variables of general generating functions, Indicates components The Performance level; Indicates in Take 1 to Within range Perform cumulative calculations; Constructing general generation functions for the system The mathematical representation is as follows: ; in, Representation and system structure function The corresponding tensor product operation; Indicates components In the first performance level The probability, For components Total number of performance levels; Indicates in Take 1 to Within range Perform cumulative multiplication calculation; This indicates the use of system structure functions. Performance levels of each component After mapping to system performance level, with The power form represents the system's performance level; Indicates the system at time 10:00 In the first performance level The probability distribution; The system's first Performance level.

[0009] Optionally, the component replacement strategy is mathematically represented as follows: ; in, Indicates the maintenance phase Component replacement strategy; Indicates the maintenance phase Components Part replacement instructions The total number of components. , Indicates that during the maintenance phase Components Replace with brand new parts. This indicates that no replacement will be performed; The component replacement state equation is defined as follows: ; in, Indicates components In subtask Initial performance level; Indicates components In subtask Performance level at the end Indicates components The Performance level; subtasks The maintenance phase begins at the beginning. At the end, subtask The end is the maintenance phase. At the beginning.

[0010] Optionally, the reliability of the subtask is the sum of conditional probabilities that the system's performance level meets the system's performance level requirements at the end of the subtask, mathematically represented as follows: ; in, This indicates that all components of the system are in the subtask. The initial combination of performance levels is defined as follows: , Indicates components In subtask Initial performance level; Subtasks ; Subtasks The subtask reliability calculation function, For subtasks Subtask reliability; Subtasks When the system ends, it is in the first position. performance level The conditional probability, based on the system being in the first position. performance level probability distribution and subtasks Duration calculation, The total number of system performance levels; For indicator functions, This indicates that the condition is met. The value is 1 if the value is 1, otherwise the value is 0. For subtasks The system performance level requirements;

[0011] The reliability of the multi-task sequence is the product of the reliability of each sub-task, mathematically represented as follows: ; in, Indicates subtasks The initial multi-task sequence reliability calculation function, For subtasks Initial multi-task sequence reliability; Indicates subtasks The initial multi-task sequence The highest subtask number; operation Indicates to exist arrive Perform cumulative multiplication within the range; Subtasks The subtask reliability calculation function.

[0012] Optional, ; in, Indicates that the repair phase has been completed. The remaining amount of spare parts after the event. For dimension A vector of all 1s Used to calculate maintenance phase Spare parts consumption; The set of feasible replacements is defined as follows: ; in Indicates based on maintenance phase Remaining spare parts The obtained maintenance stage The set of feasible replacements; The remaining quantity of spare parts is calculated using the following equation: ; in, After the maintenance phase The remaining amount of spare parts.

[0013] Optionally, the objective function is defined as follows: ; in, Indicates the reliability of the system's multi-task sequence. This represents the combined performance levels of all system components at the start of the first subtask. This represents a multitasking sequence consisting of all subtasks in the system. Indicates the need to make Maximum initial spare parts carrying capacity Component replacement strategies for each maintenance stage , This is the maximum number for the maintenance phase.

[0014] Optionally, the step of designing and solving the basic equations of dynamic programming based on the defined objective function to obtain the optimal component replacement strategy for each maintenance stage and the optimal multi-task sequence reliability for each corresponding sub-task includes: The basic equations of dynamic programming are constructed and defined as follows: when When the fundamental equation of dynamic programming is the terminal stage equation, its mathematical representation is as follows: ; when At this point, the fundamental equations of dynamic programming are intermediate-stage equations, mathematically represented as follows: ; in, Subtasks The reliability of the optimal multi-task sequence at the end. This indicates the reliability of the optimal multi-task sequence in the system. This indicates that all components of the system are in the subtask. The combination of performance levels at the end is defined as , Indicates components In subtask Performance level at the end Indicates that the repair phase has been completed. The remaining amount of spare parts after the event. Indicates the maintenance phase Component replacement strategy Indicates based on maintenance phase Remaining spare parts The obtained maintenance stage The set of feasible replacements; Indicates in subtask At the end, based on the combined performance levels of all components of the current system. After the maintenance phase Remaining spare parts and subsequent subtask sequences For the reliability of subsequent optimal multi-task sequences Take the conditional expectation; This indicates that all components of the system are in the subtask. Performance level combination at the time of technology After the maintenance phase Remaining spare parts Maintenance phase Component replacement strategy The obtained system components in the subtask The initial performance level combination, Indicates components In subtask The state transition probability during the period For subtasks The duration; Indicates components In subtask From the initial performance level during the period Transfer to the performance level at the end The probability of the component Transition rate matrix The solution is obtained using the inverse Laplace transform, and the mathematical representation of the solution process is as follows: ; in, Indicates components The state transition matrix, whose elements correspond to components The transition probability between different performance levels; Indicates components Transition rate matrix Matrix exponent; Indicates the inverse Laplace transform; Represents the complex frequency domain variables of the Laplace transform; Representation and transition rate matrix Identity matrices of the same dimension; set up Iterate through all state combinations Solving the maintenance phase equations based on the terminal phase equations Optimal component replacement strategy and its corresponding subtasks Optimal multi-task sequence reliability at the end ; set up Iterate through all state combinations Solving the maintenance stage equations based on intermediate stage equations Optimal component replacement strategy and its corresponding subtasks Optimal multi-task sequence reliability at the end ; After the iteration is completed, the solution results of all iteration stages are sorted out and the optimal strategy table is constructed. The optimal strategy table takes the subtask number, the performance level combination at the end of the previous subtask of the component, and the remaining spare parts as input, and outputs the optimal component replacement strategy and the optimal multi-task sequence reliability of the current maintenance stage.

[0015] Optionally, setting different initial spare parts carrying quantities and selecting the optimal initial spare parts carrying quantity based on the optimal strategy table includes: Set different initial spare parts carrying quantities Input the optimal strategy table, combined with the initial performance levels of the components. Generate different initial spare parts carrying capacity The optimal multi-task sequence reliability of the corresponding subtask 1 ; Choose to optimize the reliability of the multi-task sequence. Maximum initial spare parts carrying capacity Optimal initial spare parts carrying quantity .

[0016] Optionally, if multiple optimal decisions exist, the decision with the lowest spare parts consumption should be selected first.

[0017] By employing the above technical solution, the present invention provides a collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios, which has at least the following beneficial effects: (1) This invention constructs a theoretical model of a multimodal system based on a time-homogeneous continuous-time Markov process and a universal generating function (UGF), thereby accurately depicting the relationship between component gradual degradation and system performance transfer and reducing simulation errors; (2) This invention aims to maximize the reliability of multi-task sequences and uses a dynamic programming optimization algorithm to achieve global coordination of maintenance resources, thereby improving reliability and ensuring the stable completion of high-demand sub-tasks; (3) By defining the optimal initial spare parts carrying quantity, the present invention achieves the optimal balance between reliability and cost, and avoids insufficient or redundant spare parts waste; (4) The optimal strategy table constructed by the present invention can be quickly queried directly through “state input → decision output” without complex calculation. It is suitable for engineering scenarios in multiple fields such as power, aviation, and shipbuilding. It is convenient to deploy and highly practical, providing a reliable and efficient technical solution for multi-modal system multi-task operation. Attached Figure Description

[0018] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a flowchart of the collaborative decision-making method for spare parts supply and replacement in a multi-modal system under a multi-task scenario, as described in this invention. Figure 2 This is a structural diagram of the fluid transport system in an embodiment of the present invention; Figure 3 This is a schematic diagram illustrating the relationship between time and performance level in the multi-tasking mode of the present invention; Figure 4 This is a schematic diagram illustrating the relationship between the initial spare parts carrying capacity and the reliability of the multi-task sequence in the simulation of this invention; Figure 5 This is a comparison chart of simulation results in single-task mode and multi-task mode in the simulation of this invention. Detailed Implementation

[0019] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. This will allow for a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects, and to facilitate its implementation.

[0020] Those skilled in the art will understand that all or part of the steps in the implementation of the methods of the embodiments can be implemented by a program instructing related hardware. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0021] Please refer to Figures 1-5 This illustration shows a specific implementation of the present embodiment. This embodiment obtains the probability distribution of components and the system at different performance levels, defines a multi-task mode, calculates the sub-task reliability of each sub-task and the multi-task sequence reliability of the system, defines the component replacement state equation, defines spare parts resource constraints and constructs feasible replacement sets for each maintenance stage, calculates the remaining spare parts after each maintenance stage, constructs an objective function, and then designs and solves the basic equation of dynamic programming to obtain the optimal component replacement strategy and the optimal initial spare parts carrying quantity. This achieves the optimal matching between maintenance behavior selection and initial spare parts quantity, and under the premise of ensuring the reliable operation of the multi-task sequence, rationally allocates maintenance resources and spare parts, avoiding resource waste and task failure risks.

[0022] Please refer to Figure 1 This embodiment proposes a collaborative decision-making method for spare parts supply and replacement in a multi-task scenario for a multimodal system. The method includes the following steps:

[0023] S1. Construct a set of component performance levels, and obtain the transition rate matrix of the components and the probability distribution of the components at each performance level.

[0024] As a preferred embodiment of step S1, the specific process includes: S11. Sort the performance levels of the components from smallest to largest, and construct a set of component performance levels. The mathematical representation of the set of component performance levels is as follows: ; in Indicates components A set of component performance levels; Indicates components The Performance levels, sorted from smallest to largest, among which For optimal performance (like new). The worst performance level (failure); For components The total number of performance levels. This represents the total number of components.

[0025] S12. The component performance degradation process is described using a continuous-time Markov process. A transition rate matrix is ​​defined to describe the state transition characteristics of each component. The transition rate matrix is ​​defined as follows: ; in, Indicates components The transition rate matrix; Indicates components By the performance level Degenerate to the first performance level The transfer rate and The transition rate matrix is ​​a lower triangular matrix, indicating that the component performance degradation process is unidirectional and irreversible.

[0026] S13. Define the probability distribution of each component at different performance levels, mathematically represented as follows: ; in Indicates components At any moment The probability distribution, Each represents a component At any moment In the first performance level The probability of and satisfying ; This indicates a special transposition; the probability distribution follows the Fokker-Planck equation, mathematically represented as follows: ; in, For components At any moment probability distribution The first derivative with respect to time represents the component. The rate at which the probability of being at each performance level changes over time. This equation describes the component. The dynamic evolution of the probability distribution over time indicates that the rate of change of the probability distribution of component performance level over time is determined by the transition rate matrix. With the current probability distribution Together, the probability distribution at each time step can be solved by combining the Laplace transform. The solution depends on the transition rate matrix parameters and the probability distribution at the initial time step.

[0027] S2. Define the system structure function, construct the system performance level set, construct the general generating function of each component and the general generating function of the system, and map the probability distribution of each component at each performance level to the probability distribution of the system at each performance level.

[0028] A multi-state system (MSS) comprises M independent components, each with a finite number of discrete performance levels. Performance degradation follows a time-homogeneous continuous-time Markov process. The system must execute a multi-task sequence consisting of several subtasks and is constrained by the initial spare parts carrying capacity. The performance level of a multi-state system is determined by both the component performance levels and the system architecture (series, parallel, or a combination of series and parallel), and is expressed through a system architecture function. Achieving performance transfer from the component layer to the system layer. As a preferred embodiment of step S2, the specific process includes:

[0029] S21. Construct a system structure function to realize energy transfer from the component layer to the system layer. The system structure function is defined as follows when the system is a series system: The system structure function, when the system is a parallel system, is defined as follows: ,in For system structure functions, Indicates at time From components , The system performance level is jointly determined by the performance levels of the components. , Representing time respectively part , Performance level, This indicates taking the minimum value of each performance level within the parentheses, used to characterize the performance transfer rule of the "bottleneck effect" in a series system. If the system has a complex series-parallel / parallel-series structure, the system performance level can be calculated by decomposing it layer by layer and combining it with the above system structure function.

[0030] S22. Sort the system performance levels from smallest to largest to construct a set of system performance levels. The mathematical representation of the set of system performance levels is as follows: ; in, Represents the set of system performance levels; The system's first Performance levels, sorted from smallest to largest, among which... For optimal performance, This represents the worst performance level. This represents the total number of performance levels in the system.

[0031] S23. A Universal Generating Function (UGF) is used to describe the probability distribution of the system performance level. The system's universal generating function is obtained through tensor product operations on the component UGFs. The mathematical representation is as follows: ; in, Indicates components The general generating function, the general generating function of each component, is mathematically represented as: , This represents the total number of components. For general generation operators, denote the relationship with system structure functions. The corresponding tensor product operation is used to pass the performance level and probability distribution of each component to the system layer, and finally obtain the general generating function of the system. Indicates that component 1 is at time In the first performance level The probability, This represents the total number of performance levels for component 1. Indicates components At any moment In the first performance level The probability, For components Total number of performance levels; Indicates the system at time 10:00 In the first performance level The probability distribution is the core foundation for subsequent calculations of subtask reliability and dynamic programming state transitions. Formal variables of the Universal Generating Function (UGF) have no actual physical meaning; they only carry performance level information through the form "power = performance level," achieving a joint representation of "probability (coefficient) + performance level (power)." This indicates the use of system structure functions. Performance levels of each component After mapping to system performance level, with The power form characterizes the system performance level; operation Indicates in Take 1 to Within range Perform cumulative multiplication calculations.

[0032] The system's general generating function As a core tool for mapping component performance probabilities to system performance probabilities, its probability solution process relies on structure functions. Corresponding general generation operator Finish.

[0033] Steps S1 and S2 respectively construct the component layer model and the system layer model, realizing the modeling of the multimodal system and forming a complete theoretical model of the multimodal system.

[0034] This invention constructs a theoretical model of a multimodal system based on a time-homogeneous continuous-time Markov process and a universal generating function (UGF), thereby accurately depicting the relationship between component gradual degradation and system performance transfer and reducing simulation errors.

[0035] S3, Define multi-tasking mode.

[0036] As a preferred embodiment of step S3, it specifically includes: The multitasking mode includes A multi-task sequence consisting of sub-tasks Each subtask , Number the subtasks. , Representing subtasks The system performance level requirements and duration must meet the following conditions: system performance is not less than [a certain value]. Only then can the task be completed. and All are constants.

[0037] The following section introduces the key variables that characterize the system state during the task process: (1) :part In subtask At the beginning (the first) The performance level at the end of each maintenance phase; (2) :part In subtask At the end (the first) Performance level at the start of each maintenance phase; (3) All system components in subtasks At the beginning (the first) The performance level combination at the end of each maintenance phase; (4) All system components in subtasks At the end (the first) The performance level combination at the start of each maintenance phase; (5) The initial state of the system ( The performance level combination at any given time is denoted as The initial maintenance phase is incorporated into the first decision-making phase, which is the initial state preparation before the system officially executes the sub-task (including determining the initial spare parts carrying quantity and deciding on the initial component replacement strategy). This is considered the first maintenance phase in the entire dynamic programming decision-making process. ), and its output system performance level combination Directly used as the first subtask ( Initial performance level combination at the beginning This achieves a seamless transition from "initial maintenance decision → subtask 1 start state," and incorporates the determination of the initial spare parts carrying quantity and the decision of the initial component replacement strategy into a unified dynamic programming optimization framework, rather than treating them as independent external parameters. Subtasks The maintenance phase begins at the beginning. At the end, subtask The end is the maintenance phase. At the beginning.

[0038] S4. Define the component replacement state equation as a constraint for updating the state of the replaced component. Based on the component's performance level at the end of the previous subtask and the component replacement strategy, calculate the component's performance level at the beginning of the current task.

[0039] The performance level after component replacement is determined by the component replacement strategy, if during the maintenance phase For components Replace ( ), then component The system has been restored to its optimal performance level. If not replaced ( If so, then the previous subtask is retained. Performance level at the end .

[0040] As a preferred embodiment of step S4, it specifically includes: The component replacement strategy is mathematically represented as follows: ; in, Indicates the maintenance phase Component replacement strategy; Indicates the maintenance phase Components Part replacement instructions The total number of components. , Indicates that during the maintenance phase Components Replace with brand new parts (replacing with brand new parts restores performance to the original level). ), This indicates that no replacement will be performed (to maintain the current performance level).

[0041] The state equation for component replacement is defined as follows: .

[0042] S5. Reliability of computational tasks.

[0043] As a preferred embodiment of step S5, the specific process includes: S51. Define the reliability of subtasks.

[0044] The reliability of a subtask is the sum of conditional probabilities that the system's performance level meets the system's performance level requirements at the end of the subtask, given that all system components are known during the subtask. The initial performance level combination is Subtask The system performance level requirements are The mathematical representation of the subtask reliability calculation process is as follows: ; in, Subtasks The subtask reliability calculation function, For subtasks Subtask reliability; Subtasks When the system ends, it is in the first position. performance level The conditional probability, based on the system at each time step obtained from the mapping in step S2, is the system in the first... performance level probability distribution Combined with subtask duration calculate; For the characteristic function, in the above formula This indicates that the condition is met. The value is 1 if it is true, otherwise the value is 0.

[0045] S52. Define the reliability of multi-task sequences.

[0046] The success of a multi-task sequence depends on the success of all its subtasks. Therefore, the reliability of a multi-task sequence is defined as the product of the reliabilities of each subtask, mathematically expressed as follows: ; in Indicates subtasks The initial multi-task sequence reliability calculation function, For subtasks Initial multi-task sequence reliability; Indicates subtasks The initial multi-task sequence The highest subtask number; operation Indicates to exist arrive Perform cumulative multiplication calculation; Subtasks The subtask reliability calculation function.

[0047] The above formula for calculating the reliability of a multi-task sequence reflects the "one-vote veto" characteristic of a multi-task sequence: if any sub-task fails, the entire sequence fails. Therefore, the decision must take into account the reliability requirements of all sub-tasks.

[0048] S6. Define the component replacement strategy and the initial spare parts carrying quantity as decision variables, define spare parts resource constraints and construct the feasible replacement set for each maintenance stage, and calculate the remaining spare parts quantity after each maintenance stage.

[0049] As a preferred embodiment of step S6, it specifically includes: Each replacement operation during a maintenance phase must ensure that the consumption of spare parts does not exceed the current remaining quantity of spare parts. Therefore, the spare parts resource constraint is defined as follows: ; in, Indicates that the repair phase has been completed. The remaining amount of spare parts after the event. For dimension A vector of all 1s Used to calculate maintenance phase Spare parts consumption.

[0050] Based on spare parts resource constraints, the feasible replacement set is defined as follows: ;

[0051] in Indicates based on maintenance phase Remaining spare parts The obtained maintenance stage The set of feasible replacements, which includes all component replacement strategies that satisfy spare parts resource constraints.

[0052] The remaining quantity of spare parts after repair is updated as follows: ; in, After the maintenance phase The remaining amount of spare parts.

[0053] S7. Define an objective function that aims to maximize the reliability of the multi-task sequence while optimizing the component replacement strategy. ( (Maximum number for maintenance phase) and initial spare parts carrying quantity This ensures the highest possible success rate for the entire multi-task sequence under the constraint of spare parts resources.

[0054] As a preferred embodiment of step S7, it specifically includes:

[0055] The objective function is defined as follows: ; in, This indicates the reliability of the multi-task sequence of the system (starting from the first subtask). This represents the combined performance levels of all system components at the start of the first subtask. This represents all subtasks in the system (subtask 1-subtask 2). A multi-task sequence composed of ) Indicates the need to make Maximum initial spare parts carrying capacity Component replacement strategies for each maintenance stage This enables the coordinated optimization of "spare parts supply quantity" and "component replacement timing," ensuring that the dynamic programming solution process always revolves around the core objective of "maximizing the reliability of multi-task sequences," and avoiding the decision-making process from deviating from the optimal direction.

[0056] The objective function defined in this step is the optimization criterion and core guideline for collaborative decision-making in this invention. Its core function is to set a clear optimization objective for the construction and solution of the subsequent dynamic programming basic equations. The subsequent dynamic programming solution process is essentially based on this objective function, and under the constraint of spare parts resources in each maintenance stage, it solves the problem in reverse recursion. The combination of decision variables that is maximized.

[0057] Steps S5 and S6 define the decision variables (including component replacement strategy and initial spare parts carrying quantity) and constraints (including spare parts resource constraints and post-replacement component status update constraints). Combined with the objective function defined in step S7, they jointly complete the construction of the joint decision model.

[0058] S8. Combining the joint decision-making model, design and solve the basic equations of dynamic programming based on the defined objective function to obtain the optimal component replacement strategy and the optimal multi-task sequence reliability of each sub-task in each maintenance stage, and construct the optimal strategy table.

[0059] As a preferred embodiment of step S8, the specific process includes: S81. Construct the basic equations of dynamic programming, defined as follows: when When the fundamental equation of dynamic programming is the terminal stage equation, its mathematical representation is as follows: ; when At this point, the fundamental equations of dynamic programming are intermediate-stage equations, mathematically represented as follows: ; in, Subtasks The reliability of the optimal multi-task sequence at the end. This indicates the system's optimal multitasking sequence reliability (after all subtasks have been completed). Indicates in subtask At the end, based on the combined performance levels of all components of the current system. After the maintenance phase Remaining spare parts and subsequent subtask sequences For the reliability of subsequent optimal multi-task sequences Taking conditional expectations, its core function is to incorporate the optimal reliability expectation of the future maintenance phase into the decision evaluation of the current maintenance phase, thereby realizing the reverse recursive logic of dynamic programming. This indicates that all components of the system are in the subtask. Performance level combination at the time of technology After the maintenance phase Remaining spare parts Maintenance phase Component replacement strategy The obtained system components in the subtask The initial performance level combination, Indicates components In subtask The state transition probability during the period For subtasks The duration; Indicates components In subtask From the initial performance level during the period Transfer to the performance level at the end The probability of the component Transition rate matrix The solution is obtained using the inverse Laplace transform, and the mathematical representation of the solution process is as follows: ; in, Indicates components The state transition matrix, whose elements correspond to the transition probabilities between different states (i.e., between different performance levels); Indicates components Transition rate matrix Matrix exponent; The inverse Laplace transform is the core mathematical operation for converting the solution from the complex frequency domain back to the state transition matrix in the time domain. The variables in the complex frequency domain of the Laplace transform have no actual physical meaning and are merely formal parameters in the transformation process. Representation and transition rate matrix An identity matrix of the same dimension.

[0060] Since the last subtask has no subsequent tasks, the optimal value depends only on the last subtask. Given the reliability of the component, the component replacement strategy that maximizes this reliability is selected as the optimal component replacement strategy. .

[0061] S82, Execute the dynamic programming optimization algorithm.

[0062] S821. Input basic parameters: Transition rate matrix of each component Multi-task sequence (including each sub-task) , Initial spare parts carrying capacity upper limit (This upper bound is used to limit the range of initial spare parts carrying capacity values ​​for subsequent traversal optimization, that is, in step S823, it will be...) Iterate through all possible initial spare parts carrying quantities within the interval. and for each We perform dynamic programming to solve the problem and ultimately select the sequence of tasks that maximizes reliability. As the optimal initial spare parts carrying quantity, avoid meaningless infinite traversal; this upper bound is taken as the combination of task number × component number and initial system performance level when there is no prior information;

[0063] S822, Preprocessing Calculations: Based on the Fokker-Planck equations, the system's general generating function, the subtask reliability calculation function, and the state transition matrix in step S13. Solve for the probability distribution of each component at different performance levels. System performance level set and probability distribution at different performance levels Subtask reliability Feasible replacement sets for each maintenance stage .

[0064] S823, Terminal Initialization Phase: set up Iterate through all state combinations Subtask Before starting, the system was in the maintenance phase. After completion, the remaining quantity of all spare parts that meet the spare parts resource constraints. Combination of all possible performance levels of components (Even at the terminal stage) Components may still be at various discrete performance levels, and the remaining spare parts also have multiple feasible values. Therefore, it is necessary to traverse all feasible state combinations under this stage. The maintenance stage is solved based on the terminal stage equations. Optimal component replacement strategy and its corresponding subtasks Optimal multi-task sequence reliability at the end .

[0065] S824, Perform recursive calculations: set up Iterate through all state combinations Solving the maintenance stage equations based on intermediate stage equations Optimal component replacement strategy and its corresponding subtasks Optimal multi-task sequence reliability at the end .

[0066] S825. Construct the optimal strategy table:

[0067] After the iteration is completed, the solution results of all iteration stages are sorted out, and an optimal strategy table is constructed. The optimal strategy table is a combination of subtask number and the performance level of the component at the end of the previous subtask. Spare parts remaining quantity Given the input, output the optimal component replacement strategy for the current maintenance phase. And the reliability of the optimal multi-task sequence for the current subtask .

[0068] This invention aims to maximize the reliability of multi-task sequences and uses a dynamic programming optimization algorithm to achieve global coordination of maintenance resources, thereby improving reliability and ensuring the stable completion of high-demand sub-tasks.

[0069] S9. Set different initial spare parts carrying quantities, select the optimal initial spare parts carrying quantity based on the optimal multi-task sequence reliability output by the optimal strategy table, and the optimal initial spare parts carrying quantity and the optimal component replacement strategy constitute the optimal decision to realize spare parts supply and replacement.

[0070] As a preferred embodiment of step S9, the specific process includes: S91, Set different initial spare parts carrying quantities Input the optimal strategy table, combined with the initial performance levels of the components. Generate different initial spare parts carrying capacity The optimal multi-task sequence reliability of the corresponding subtask 1 .

[0071] S92. Select the option that maximizes the reliability of the multi-task sequence. Maximum initial spare parts carrying capacity Optimal initial spare parts carrying quantity .

[0072] More specifically, if there are multiple optimal decisions (different component replacement strategies or initial spare parts carrying capacity correspond to the same maximum multi-task sequence reliability), then the decision with less spare parts consumption is selected first, reserving more spare parts for subsequent tasks.

[0073] Steps S8 and S9 are based on the dynamic programming optimization algorithm to construct and solve the basic equations of dynamic programming, and finally obtain the optimal component replacement strategy and the optimal initial spare parts carrying quantity, which constitute the optimal decision.

[0074] This invention achieves an optimal balance between reliability and cost by clearly defining the initial spare parts carrying capacity, avoiding insufficient or redundant spare parts waste. Furthermore, the optimal strategy table constructed by this invention can be quickly queried directly through "state input → decision output" without complex calculations. It is adaptable to engineering scenarios in multiple fields such as power, aviation, and shipbuilding, offering convenient deployment and strong practicality, providing a reliable and efficient technical solution for multi-modal system multi-task operation.

[0075] To fully verify the effectiveness and reliability of the collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios proposed in this invention, this embodiment takes a commonly used fluid transport system in engineering as the simulation object. Through a combination of theoretical calculation and Monte Carlo simulation, a full-process, multi-dimensional simulation analysis is carried out from system modeling, parameter setting, strategy solving to result verification.

[0076] The specific information about the simulation software is as follows: Software Name: MATLAB; Version information: 9.8.0.1380330 (R2020a) Update 2; Core toolboxes: Statistics and Machine Learning Toolbox, Control System Toolbox.

[0077] (1) Simulation object and system structure

[0078] The simulation object is a fluid transport system consisting of three pipe components, and the system performance is characterized by mass flow rate (unit: ton / min). The system structure is "component 1 and component 2 connected in parallel, and then connected in series with component 3". The parallel branch ensures the basic transport capacity, and the series component 3 is the core pressurization / flow limiting unit. Its performance limit directly determines whether the system can meet the high-demand sub-tasks. Figure 2 This intuitively illustrates the "parallel + series" structural relationship of the fluid transport system in this embodiment.

[0079] To visually represent the system's operation flow and maintenance timing in multi-tasking mode, the simulation references the actual system working logic and plots a diagram illustrating the relationship between time and performance level in multi-tasking mode, as shown below. Figure 3 As shown. Figure 3 The alternating cyclical relationship of "maintenance phase - subtask" is clearly demonstrated: the system starts from the initial state, first going through the first ( During the maintenance phase, a component replacement decision is made, and then the first ( The subtask runs; after the subtask finishes, the second subtask begins. During the maintenance phase, this process continues until all maintenance is completed. Each maintenance stage and Each task has a sub-task, and the failure of any one sub-task will cause the entire task sequence to fail.

[0080] (2) Simulation parameter setting

[0081] By combining the actual working parameters and degradation patterns of the fluid transport system, the core parameters of each component, system, and multi-task sequence are identified to ensure the realism and representativeness of the simulation.

[0082] a. Component parameter settings: Performance levels of each component (visible) Figure 2 The transition rate and initial state probability are determined based on actual experimental data.

[0083] Component 1: 2 performance levels ( ), transfer rate Probability distribution at the initial performance level Subsequent moments Probability distribution of component 1 at two performance levels , The calculation formula is as follows: ; in, This represents the base of the natural logarithm, used here in the exponential function. In this context, the probability evolution law that characterizes the exponential decay of component performance over time is a typical expression of the state transition probability in a continuous-time Markov process.

[0084] Component 2: 2 performance levels ( ), transfer rate Probability distribution at the initial performance level Subsequent moments Probability distribution of component 2 at two performance levels , The calculation formula is as follows: ;

[0085] Component 3: 3 performance levels ( ), transfer rate , Probability distribution at the initial performance level Subsequent moments Probability distribution of component 3 at 3 performance levels , , The calculation formula is as follows:

[0086] b. System performance and UGF modeling

[0087] The system performance level is calculated using the Universal Generating Function (UGF). The expressions for the UGF of each component and the system UGF are as follows: Component 1UGF: ; Component 2UGF: ; Component 3UGF: ; System UGF: ; in, The system structure function represents the system structure when the system is a series system. The corresponding general generating operator; The system structure function represents the system structure when the system is a parallel system. The corresponding general generating operator.

[0088] System performance level set Then the system at each time Probability distribution of these 5 performance levels , , , , The calculation is as follows: .

[0089] c. Multi-task sequence parameter settings

[0090] Setting up a multi-task sequence The subtasks are defined as follows: Subtask 1: ; Subtask 2: ; Subtask 3: ; in , , The system performance level requirements for subtasks 1-3. , , The duration of subtasks 1-3.

[0091] d. Component state transition matrix

[0092] State transition matrix of each component Solving using the inverse Laplace transform: Component 1 state transition matrix: ; Component 2 state transition matrix: ; Component 3 state transition matrix:

[0093] ;

[0094] e. Subtask reliability expression

[0095] Subtask reliability of subtasks 1-3 , , Calculate the probability of "performance ≥ requirements": .

[0096] (3) Simulation steps

[0097] a. Initialize parameters

[0098] Input the transfer rate matrix of each component Multi-task sequence System initial performance level combination (Corresponding to component 1 being intact, component 2 being faulty, and component 3 being degraded), and setting an upper limit for the initial spare parts carrying capacity. .

[0099] b. Calculate core basic parameters

[0100] The probability distribution of the components at each performance level, the system UGF and the probability distribution of the system at each performance level, the reliability of the subtasks, the state transition matrix, and the feasible replacement set for each maintenance stage are calculated.

[0101] c. Solving for the optimal strategy table

[0102] A dynamic programming optimization algorithm is used to calculate the optimal decision for each maintenance stage by working backward from the last subtask (k=3), thus forming an optimal strategy table.

[0103] d. Monte Carlo simulation. For Each simulation was performed 1000 times, and the number of successful tasks and simulation reliability were statistically analyzed.

[0104] e. Comparison and Analysis

[0105] Compare the errors between theoretical and simulated values; compare the reliability differences between multi-task mode and single-task mode; identify the "growth-saturation" law of reliability with the change of spare parts quantity, and thus update and optimize the upper bound of the initial spare parts carrying quantity (this is an engineering optimization and verification correction of the conservative upper bound used as the basic parameter input in the previous text, i.e., taking the number of tasks × the number of parts; the initial spare parts carrying quantity upper bound input in the previous text was a conservative value set to define the initial traversal range of dynamic programming and ensure that the optimal solution is not missed, which has the problem of taking too large a value and increasing computational redundancy; this step finds the critical spare parts quantity where the reliability no longer increases significantly with the increase of spare parts quantity, and uses this critical value as the optimized initial spare parts carrying quantity upper bound, which not only retains the core requirement of "covering the optimal solution", but also greatly reduces the traversal range of subsequent dynamic programming, reduces computational complexity, and realizes the closed loop of "conservative input - simulation optimization - precise definition").

[0106] (4) Simulation results and analysis

[0107] a. Optimal Strategy Table

[0108] The optimal policy table obtained by solving the dynamic programming optimization algorithm is shown in Table 1 below, which contains the optimal decision under all possible state combinations. This table is categorized by "subtask number". Spare parts remaining quantity The combination of performance levels at the end of a subtask on a component The input is "the optimal component replacement strategy for the current maintenance phase". Optimal multitasking sequence reliability of the current subtask " is the output.

[0109] Table 1. Example of the optimal strategy in this embodiment

[0110] b. Relationship between initial spare parts carrying capacity and multi-task sequence reliability

[0111] Maintain the initial performance level combination of the system Unchanged, initial spare parts carrying capacity Increase gradually from 0 to 10. Figure 4 The graph visually presents the relationship between theoretical calculations and Monte Carlo simulations regarding the reliability of multi-task sequences. The horizontal axis represents the initial spare parts carrying capacity. (Value range 0-10), the vertical axis represents the reliability of the multi-task sequence (value range 0-1). The solid line in the graph represents the theoretical value obtained from theoretical calculations, and the dashed line represents the Monte Carlo value obtained from Monte Carlo simulations. Both curves show a consistent trend of "0 → rapid growth → steady state": when... or When the multi-task sequence reliability is 0, the reliability of the multi-task sequence is 0. The reliability of multi-task sequences increases rapidly; when At this point, the reliability of the multi-task sequence tends to a steady-state value of 0.9115. Figure 4 The impact of the initial spare parts carrying capacity on the reliability of multi-task sequences is clearly demonstrated.

[0112] c. Consistency verification between theoretical and simulation results

[0113] To quantitatively verify the accuracy of the multimodal system theoretical model constructed in this invention, Table 2 presents a comparison of the relative errors between the theoretical values ​​and the Monte Carlo simulation values ​​under different initial spare parts carrying capacities. This table includes the initial spare parts carrying capacities. The table presents a complete picture of the error situation under different initial spare parts carrying quantities, including (0-10), theoretical calculation values, simulation calculation values, and error (%). As can be seen from Table 2, all relative errors are controlled within a small range (maximum not exceeding 1.5%), for example... The time error is 1.22%. The time error is only 0.55%, indicating that the multimodal system theoretical model established in this invention is highly consistent with the actual system operation law, and the accuracy and reliability of the model have been fully verified.

[0114] Table 2. Comparison of relative errors between theoretical values ​​and Monte Carlo simulation values ​​under different initial spare parts carrying capacities. Initial spare parts carrying capacity 0 1 2 3 4 5 6 7 8 9 10 Theoretical value 0 0 0.8149 0.9063 0.9114 0.9115 0.9115 0.9115 0.9115 0.9115 0.9115 Monte Carlo simulation values 0 0 0.841 0.901 0.901 0.903 0.9 0.916 0.911 0.914 0.918 Relative error (%) 0 0 -3.09 0.55 1.11 0.89 1.22 -0.55 0.05 -0.33 -0.76

[0115] d. Comparison of multitasking and single-tasking modes

[0116] To verify the superiority of the decision-making method of this invention in a multi-task mode, a comparative simulation was designed, under the same parameter conditions (initial spare parts carrying capacity). The optimal decisions for single-task and multi-task modes were run in the same Monte Carlo simulation experiment (1000 simulations), and the results were compared. Figure 5 Presented.

[0117] Figure 5The horizontal axis represents the initial spare parts carrying capacity. The vertical axis represents system task reliability (multi-task mode: multi-task sequence reliability; single-task mode: single-task reliability). The solid line represents the simulation results in the multi-task mode (the method of this invention), and the dashed line represents the simulation results in the single-task mode. From Figure 5 It is clear that, with all initial spare parts carried, the system task reliability in the multi-task mode is higher than that in the single-task mode. This is because the single-task mode only optimizes locally for the "next task", which can easily lead to premature consumption of spare parts. In contrast, the multi-task mode makes overall decisions from the perspective of the entire task sequence, which can allocate resources more reasonably and ensure the completion of subsequent high-demand sub-tasks, fully demonstrating the core advantages of the method of this invention.

[0118] (5) Simulation conclusions

[0119] Through full-process simulation analysis of the fluid transport system, the following conclusions can be drawn: The multimodal system modeling method proposed in this invention (based on Markov process and UGF) can accurately describe the performance degradation law of components and systems. The error between theoretical calculation results and Monte Carlo simulation results is less than 1.5%, and the modeling accuracy is reliable. The dynamic programming optimization algorithm can effectively solve the optimal initial spare parts carrying quantity and optimal component replacement strategy in multi-task mode. The generated optimal strategy table can be directly used for engineering decision-making and has strong practicality. The determination of the upper bound of the initial spare parts quantity provides a quantitative basis for engineering spare parts configuration and avoids the problem of insufficient or redundant spare parts. The decision-making in multi-task mode is better than that in single-task mode and can significantly improve the reliability of system task sequence. It is applicable to multi-task scenarios such as power, aviation, and shipbuilding.

[0120] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "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 this application. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of those different embodiments or examples.

[0121] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus or device (such as a computer-based system, a processor-included system or other system that can fetch and execute instructions from, an instruction execution system, apparatus or device).

[0122] The above embodiments provide a detailed description of the present invention. Specific examples have been used to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of the present invention. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A collaborative decision-making method for spare parts supply and replacement in a multi-modal system under a multi-task scenario, characterized in that, include: Construct a set of component performance levels, and obtain the transition rate matrix of the components and the probability distribution of the components at each performance level; Define the system structure function, construct the system performance level set, construct the general generating function of each component and the general generating function of the system, and map the probability distribution of each component at each performance level to the probability distribution of the system at each performance level; Define a multitasking mode; the multitasking mode includes a multitasking sequence consisting of several subtasks, and the parameters of each subtask in the multitasking sequence include system performance level requirements and duration; Define the component replacement state equation, and calculate the performance level of each component at the beginning of the current subtask based on the performance level of each component at the end of the previous subtask and the component replacement strategy. Calculate task reliability; the task reliability includes the sub-task reliability of each sub-task and the multi-task sequence reliability of the system, calculated based on the performance level of each component at the beginning of each sub-task and the probability distribution of the system at each performance level; Define the component replacement strategy and the initial spare parts carrying quantity as decision variables, define spare parts resource constraints and construct the feasible replacement set for each maintenance stage, and calculate the remaining spare parts quantity after each maintenance stage; Define an objective function that aims to maximize the reliability of the multi-task sequence while optimizing the component replacement strategy and the initial spare parts carrying capacity. Based on the defined objective function, design and solve the basic equations of dynamic programming to obtain the optimal component replacement strategy and the optimal multi-task sequence reliability of each sub-task in each maintenance stage, and construct the optimal strategy table. Different initial spare parts carrying quantities are set, and the optimal initial spare parts carrying quantity is selected based on the optimal strategy table. The optimal initial spare parts carrying quantity and the optimal component replacement strategy constitute the optimal decision, thereby realizing spare parts supply and replacement.

2. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 1, characterized in that: The construction of the component performance level set, obtaining the component's transfer rate matrix and the probability distribution of the component at each performance level, includes: The components are sorted by performance level from smallest to largest to construct a set of component performance levels, which is mathematically represented as follows: ； in Indicates components A set of component performance levels; Indicates components The Performance levels, sorted from smallest to largest. For components Total number of performance levels; Define a transition rate matrix to describe the state transition characteristics of each component. The transition rate matrix is ​​defined as follows: ； in, Indicates components The transition rate matrix; Indicates components By the performance level Degenerate to the first performance level The transfer rate, and ; The probability distribution of each component at different performance levels is defined mathematically as follows: ； in Indicates components At any moment The probability distribution, Each represents a component At any moment In the first performance level The probability of and satisfying ; This represents the transpose operation; the probability distribution follows the Fokker-Planck equation, mathematically represented as follows: ； in, For components At any moment probability distribution The first derivative with respect to time represents the component. The rate at which the probability of being at each performance level changes over time.

3. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 1, characterized in that: The definition of the system structure function, the construction of the system performance level set, and the construction of the general generation functions for each component and the system general generation function include: The system structure function maps the performance levels of each component to the overall system performance level. When the system is a series system, the system structure function is defined as follows: The system structure function, when the system is a parallel system, is defined as follows: ,in Represents the system structure function. Indicates at time From components , The system performance level is jointly determined by the performance levels of the components. , Representing time respectively part , Performance level, This indicates taking the minimum value of each performance level within the parentheses; The system performance levels are sorted from smallest to largest to construct a set of system performance levels, which is mathematically represented as follows: ； in, Represents the set of system performance levels; The system's first Performance levels, sorted from smallest to largest. The total number of system performance levels; The general generating functions for each component are constructed, and their mathematical representation is as follows: ； in, Indicates components General generating functions, , This represents the total number of components. Indicates components At any moment In the first performance level The probability, For components Total number of performance levels; Formal variables of general generating functions, Indicates components The Performance level; Indicates in Take 1 to Within range Perform cumulative calculations; Constructing general generation functions for the system The mathematical representation is as follows: ； in, Representation and system structure function The corresponding tensor product operation; Indicates components In the first performance level The probability, For components Total number of performance levels; Indicates in Take 1 to Within range Perform cumulative multiplication calculation; This indicates the use of system structure functions. Performance levels of each component After mapping to system performance level, with The power form represents the system's performance level; Indicates the system at time... In the first performance level The probability distribution; The system's first Performance level.

4. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 2, characterized in that: The component replacement strategy is mathematically represented as follows: ； in, Indicates the maintenance phase Component replacement strategy; Indicates the maintenance phase Components Part replacement instructions The total number of components. , Indicates that during the maintenance phase Components Replace with brand new parts. This indicates that no replacement will be performed; The component replacement state equation is defined as follows: ； in, Indicates components In subtasks Initial performance level; Indicates components In subtasks Performance level at the end Indicates components The Performance level; subtasks The maintenance phase begins at the beginning. At the end, subtask The end is the maintenance phase. At the beginning.

5. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 4, characterized in that: The reliability of a subtask is the sum of conditional probabilities that the system's performance level meets the system's performance level requirements at the end of the subtask, mathematically represented as follows: ； in, This indicates that all components of the system are in the subtask. The initial combination of performance levels is defined as follows: , Indicates components In subtasks Initial performance level; Subtasks ; Subtasks The subtask reliability calculation function, For subtasks Subtask reliability; Subtasks When the system ends, it is in the first position. performance level The conditional probability, based on the system being in the th position. performance level probability distribution and subtasks Duration calculation, The total number of system performance levels; For indicator functions, This indicates that the condition is met. The value is 1 if the value is 1, otherwise the value is 0. For subtasks The system performance level requirements; The reliability of the multi-task sequence is the product of the reliability of each sub-task, mathematically represented as follows: ； in, Indicates subtasks The initial multi-task sequence reliability calculation function, For subtasks Initial multi-task sequence reliability; Indicates subtasks The initial multi-task sequence The highest subtask number; operation Indicates to exist arrive Perform cumulative multiplication within the range; Subtasks The subtask reliability calculation function.

6. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 4, characterized in that: The spare parts resource constraint is defined as the spare parts consumption not exceeding the current remaining spare parts quantity, and is mathematically expressed as follows: ； in, Indicates that the repair phase has been completed. The remaining amount of spare parts after the event. For dimension A vector of all 1s Used to calculate maintenance phase Spare parts consumption; The set of feasible replacements is defined as follows: ； in Indicates based on maintenance phase Remaining spare parts The obtained maintenance stage The set of feasible replacements; The remaining quantity of spare parts is calculated using the following equation: ； in, After the maintenance phase The remaining amount of spare parts.

7. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 5, characterized in that: The objective function is defined as follows: ； in, Indicates the reliability of the system's multi-task sequence. This represents the combined performance levels of all system components at the start of the first subtask. This represents a multitasking sequence consisting of all subtasks in the system. Indicates the need to make Maximum initial spare parts carrying capacity Component replacement strategies for each maintenance stage , This is the maximum number for the maintenance phase.

8. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 7, characterized in that: The process involves designing and solving the basic equations of dynamic programming based on the defined objective function to obtain the optimal component replacement strategy for each maintenance stage and the optimal multi-task sequence reliability for each corresponding sub-task, including: The basic equations of dynamic programming are constructed and defined as follows: when When the fundamental equation of dynamic programming is the terminal stage equation, its mathematical representation is as follows: ； when When the fundamental equations of dynamic programming are intermediate stage equations, their mathematical representation is as follows: ； in, Subtasks The reliability of the optimal multi-task sequence at the end. This indicates the reliability of the optimal multi-task sequence in the system. This indicates that all components of the system are in the subtask. The combination of performance levels at the end is defined as , Indicates components In subtasks Performance level at the end Indicates that the repair phase has been completed. The remaining amount of spare parts after the event. Indicates the maintenance phase Component replacement strategy Indicates based on maintenance phase Remaining spare parts The obtained maintenance stage The set of feasible replacements; Indicates in subtask At the end, based on the combined performance levels of all components of the current system. After the maintenance phase Remaining spare parts and subsequent subtask sequences For the reliability of subsequent optimal multi-task sequences Take the conditional expectation; This indicates that all components of the system are in the subtask. Performance level combination at the time of technology After the maintenance phase Remaining spare parts Maintenance phase Component replacement strategy The obtained system components in the subtask The initial performance level combination, Indicates components In subtasks The state transition probability during the period For subtasks The duration; Indicates components In subtasks From the initial performance level during the period Transfer to the performance level at the end The probability of the component Transition rate matrix The solution is obtained using the inverse Laplace transform, and the mathematical representation of the solution process is as follows: ； in, Indicates components The state transition matrix, whose elements correspond to components The transition probability between different performance levels; Indicates components Transition rate matrix Matrix exponent; Indicates the inverse Laplace transform; Represents the complex frequency domain variables of the Laplace transform; Representation and transition rate matrix Identity matrices of the same dimension; set up traverse all state combinations Solving the maintenance phase equations based on the terminal phase equations Optimal component replacement strategy and its corresponding subtasks Optimal multi-task sequence reliability at the end ; set up traverse all state combinations Solving the maintenance stage equations based on intermediate stage equations Optimal component replacement strategy and its corresponding subtasks Optimal multi-task sequence reliability at the end ; After the iteration is completed, the solution results of all iteration stages are sorted out and the optimal strategy table is constructed. The optimal strategy table takes the subtask number, the performance level combination at the end of the previous subtask of the component, and the remaining spare parts as input, and outputs the optimal component replacement strategy and the optimal multi-task sequence reliability of the current maintenance stage.

9. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 8, characterized in that: The setting of different initial spare parts carrying quantities and the selection of the optimal initial spare parts carrying quantity based on the optimal strategy table include: Set different initial spare parts carrying quantities Input the optimal strategy table, combined with the initial performance levels of the components. Generate different initial spare parts carrying capacity The optimal multi-task sequence reliability of the corresponding subtask 1 ; Choose to optimize the reliability of the multi-task sequence. Maximum initial spare parts carrying capacity Optimal initial spare parts carrying quantity .

10. The collaborative decision-making method for spare parts supply and replacement of multimodal systems in multi-task scenarios according to claim 9, characterized in that: If multiple optimal decisions exist, the decision with the lowest spare parts consumption should be selected first.

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