Multi-variety processing group line productivity reliability prediction method based on Markov reward process
By using a Markov reward process model and a multidimensional probability generating function, the problem of evaluating equipment failures and process interactions in multi-product mixed production lines was solved, and efficient and reliable prediction of multi-product processing lines was achieved.
Patent Information
- Application Number
- CN202511104806.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-07
- Publication Date
- 2025-11-18
AI Technical Summary
Existing technologies struggle to accurately describe the interaction between random equipment failures and parallel processes in multi-product mixed-line production, and lack methods for evaluating the probability of joint completion by analytically calculating within an acceptable computational limit.
A Markov reward process model is adopted. By constructing a model that includes the failure state of processing equipment and output reward, the reliability of the production line in completing multi-variety orders within the specified delivery cycle is calculated. The joint completion probability of multi-variety output is analyzed by using a multidimensional probability generating function and multiple partial derivatives.
It enables accurate calculation of the probability of a production line completing its tasks on time in multi-variety processing lines, improving the real-time nature of capacity forecasting and the accuracy of delivery reliability assessment.
Smart Images
Figure CN120975489A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a technology in the field of flexible manufacturing, specifically a method for predicting the reliability of production capacity of multi-product processing lines based on Markov reward processes. Background Technology
[0002] As the manufacturing industry moves towards multi-variety, small-batch, and flexible production, mixed-line production modes have emerged in fields such as aerospace. Mixed-line production lines are typically characterized by complex processes and varied order types, resulting in intricate relationships between equipment and parts within the workshop. The industry generally uses the probability that the cumulative output of each product type within a given delivery cycle simultaneously meets order requirements as a crucial reliability indicator for measuring the delivery capability of mixed-line production lines. However, existing methods either rely on large-scale simulations or are only applicable to simplified scenarios. There is a lack of an evaluation method that can accurately describe the random failures of equipment and the parallel interaction of multiple product processes, while also analytically calculating the probability of joint completion within an acceptable computational limit. Summary of the Invention
[0003] This invention addresses the shortcomings of existing technologies in assessing the probability of timely completion of multi-variety orders by proposing a method for predicting the reliability of multi-variety processing line capacity based on a Markov reward process. By constructing a Markov reward process model that includes processing equipment failure states and output rewards, the reliability of the production line in completing multi-variety orders within a specified delivery cycle can be calculated, enabling accurate calculation of the probability that the production line will complete the production tasks of each variety on time within a given cycle.
[0004] This invention is achieved through the following technical solution:
[0005] This invention relates to a method for predicting the reliability of production capacity in a multi-product processing line based on a Markov reward process. In the offline stage, based on order requirements and the processing equipment in the multi-product processing line, the method identifies the process path and corresponding processing equipment for each product to be processed. It estimates the failure rate and repair rate of each piece of equipment based on historical failure data, constructs a Markov transition rate matrix, and generates a multidimensional reward rate vector by calculating the production rate of each product under different equipment combinations. After further constructing a multidimensional Markov reward process, the probability generating function (PGF) is obtained. In the online stage, based on the user-input order target output, the method performs multiple partial derivatives of the PGF at zero and accumulates them to obtain the joint completion probability of all products simultaneously meeting the target within the production window, thus achieving production capacity prediction.
[0006] The aforementioned multi-product processing line refers to a production line that processes two or more product varieties simultaneously on the same production line, with each product having different processing technology and processing path.
[0007] The aforementioned processing equipment refers to a key manufacturing unit that undertakes one or more processing steps in the production process and whose operating status directly affects output capacity. Specifically, it consists of M processing equipment units constituting an equipment set E = {E1, ..., E...} M}
[0008] The process path refers to the sequence of equipment required to complete all processing steps of a certain product. The product has a positive output rate only when all equipment in the path is in normal working order.
[0009] The aforementioned order task requirements refer to: the set of quantity requirements (D1, D2, ..., D) submitted by customers for several different types of products within the predetermined delivery period. N ), where: N represents the number of product types to be processed, D i Let be the target output that needs to be achieved for the i-th product.
[0010] The aforementioned combination of devices in different states refers to: an M-dimensional binary vector ω = (x1, ..., x) consisting of all device states. M ), totaling 2 M Species, of which: x k This indicates the equipment status, i.e., the operating condition of a single piece of equipment. When x k =1 when device E k Normal; when x k =0 when device E k It is currently in a fault-locked and out-of-service state.
[0011] The Markov transition rate matrix Each device is an independent, continuous-time Markov chain, and only one device will undergo a state transition at any given time. Among them: Q k Let be the transfer rate matrix of device k. λ k Let μ be the failure rate of device k. k The repair rate of device k; I (·) It is an identity matrix of size (·) × (·). For Kronecker and, For Kronecker product.
[0012] The reward rate vector is the set of instantaneous production capacities for all product types when the system state is ω. It is assumed that the arrival event of product i follows a parameter r. i For a Poisson process with (ω), the reward rate vector is specifically: r(ω) = (r 1 (ω),r 2 (ω),…,r N (ω)), where: ρ i Let be the nominal output rate of the i-th product under the condition that all relevant equipment is functioning normally and there are no bottleneck constraints.
[0013] The Markov reward process refers to a stochastic process formed by coupling the Markov transition rate matrix with the reward rate vector, which can describe the cumulative change of yield of multiple varieties over time.
[0014] The aforementioned probability generating function is the multi-variety cumulative output vector (X) generated by the constructed Markov reward process at any time T. 1 (T),…,X N The probability generating function of (T) is as follows: Where: X i (T) represents the cumulative qualified output of the i-th product within the time interval [0,T]. Y(t) is the Markov state at time t; θ i Let be the independent variable of the generating function corresponding to the i-th product; t is the continuous time parameter; T is the length of the evaluation period determined by the contract delivery terms; Let p(0) be the mathematical expectation operator, 0 ≤ t ≤ T, and p(0) be the initial system state vector. 1 is a column vector with all elements equal to 1, and a diagonal matrix. Its diagonal element Right now
[0015] The aforementioned joint completion probability refers to the probability that, at the end of the evaluation period length T, the cumulative yield of each variety simultaneously equals or exceeds its target yield requirement. in:
[0016] This invention relates to a multi-product processing line capacity prediction system for implementing the above-mentioned method, comprising: an equipment state transition rate generation module, a process path-capacity parameter configuration module, a Markov reward process modeling and generating function solving module, and a multi-product joint reliability assessment module. The equipment state transition rate generation module receives historical fault and maintenance records from the production site and generates a rate matrix describing the relationships between all possible states of the entire production line. The process path-capacity parameter configuration module reads the process list and order information from the enterprise resource planning system, extracts the sequence of equipment required for each product, and, combined with standard cycle time or simulation calibration results, provides the ideal reliability of each product. The system calculates the output rate per unit time under operating conditions and generates a capacity table corresponding to the system state. The Markov reward process modeling and generating function solving module integrates data from the equipment state transition rate generation module and the process path-capacity parameter configuration module to automatically construct a Markov reward process model that can simultaneously characterize random equipment failures and multi-product output. Then, it calculates a probability generating function that describes the global output statistics. The multi-product joint reliability assessment module performs partial derivative and accumulation operations on the generating function based on the user-set delivery cycle and order output target, and quickly outputs the joint completion probability that the output of each product variety simultaneously reaches or exceeds the demand within a specified time, thus obtaining the reliability assessment results of the multi-product processing line. Technical effect
[0017] This invention abstracts a multi-product processing line into a Markov reward process model. By combining the Markov reward process with a multi-dimensional probability generating function, it maps the process path, equipment availability, and product capacity into the reward function. Furthermore, it extracts the joint distribution of output by analytically examining the multiple partial derivatives of the probability generating function at zero. Compared to existing technologies, this invention can analytically calculate the probability of simultaneous achievement of output targets for multiple products within an order window. Compared to existing technologies, this invention adapts to the production characteristics of multi-product processing lines, significantly improving the real-time performance of capacity forecasting and the accuracy of delivery reliability assessment. Attached Figure Description
[0018] Figure 1 This is a flowchart of the present invention;
[0019] Figure 2 A schematic diagram of the product to be processed and the processing technology;
[0020] Figure 3 This is a schematic diagram of the state transition of a multi-product processing line consisting of four processing machines;
[0021] Figure 4 and Figure 5 This is a schematic diagram illustrating the effect of an example. Detailed Implementation
[0022] like Figure 1As shown in the figure, this embodiment relates to a method for predicting the reliability of multi-product processing line capacity based on a Markov reward process, including:
[0023] Step 1, as follows Figure 2 As shown, the order requirements and key processing equipment are determined, and the state space of the processing equipment is constructed, specifically including:
[0024] 1.1 Based on the mid-year production schedule of a certain aviation parts factory, it is determined that two types of annular thin-walled parts need to be completed simultaneously within the delivery cycle: output shaft valve assembly D1 and bearing valve assembly D2.
[0025] 1.2 Based on the comparison of on-site processes and equipment lists, four pieces of equipment with the greatest impact on delivery capability were selected as evaluation objects: precision coordinate boring and milling machining center E1, CNC vertical grinder E2, CNC turning and milling composite machine tool E3, and EDM forming machine E4.
[0026] Step 2: Collect and organize historical fault records, maintenance records and start-up logs of all key processing equipment on the multi-product mixed production line, and estimate the failure rate and repair rate of the processing equipment, as shown in Table 1.
[0027] Table 1. Failure Rate and Repair Rate of Processing Equipment
[0028] Step 3: Perform process path analysis and capacity configuration for each product to be processed, forming a mapping table from system status to capacity. Specifically: Product D1 needs to pass through E1→E2→E3→E4 in sequence; Product D2 needs to pass through E1→E3→ E4.
[0029] Step 4, construct the system state transition rate matrix Q, specifically as follows: in:
[0030] When k = 1, 2, 3, 4, the system state transition rate matrix
[0031] Step 5, generate the reward rate vector r(ω) = (r 1 (ω),r 2 (ω)) and the working status of the four devices E1, E2, E3, and E4 in the system: ω=(x1,x2,x3,x4)∈{0,1} 4 ,in: ρ1 = 0.15 and ρ2 = 0.1 represent the production capacity of product D1 and product D2 under ideal operating conditions, respectively. {·}The indicator function takes the value 1 when the condition is met and 0 otherwise. The resulting reward rate vector is shown in Table 2.
[0032] Table 2. Reward Rate Vector Equipment status <![CDATA[r 1 (oh)]]> <![CDATA[r 2 (oh)]]> 0000 0 0 0001 0 0 0010 0 0 0011 0 0 0100 0 0 0101 0 0 0110 0 0 0111 0 0 1000 0 0 1001 0 0 1010 0 0 1011 0 0 1100 0 0 1101 0 0.1 1110 0 0 1111 0.15 0.1
[0033] Step 6: Establish a multidimensional Markov reward process model, specifically: obtain the diagonal matrix from step 5. The system state transition speed in step 4 Summing the rate matrix Q and the Λ matrix, when all key processing equipment is initially functioning normally... Where: e 15 Let G(θ1,θ2;t)=p(0)exp[(Q+Λ(θ1,θ2))t]1 represent a 16-dimensional standard basis vector with the 15th bit (counting from 0) being 1 and the rest being 0. Construct a multidimensional Markov reward process model based on G(θ1,θ2;t)=p(0)exp[(Q+Λ(θ1,θ2))t]1.
[0034] Step 7: Perform matrix exponential operation on the multidimensional Markov reward process model constructed in Step 6 to obtain the analytical expression of the probability generating function containing formal variables for the cumulative output of multiple varieties.
[0035] Step 8: Given the user-inputted target output for orders as D1 = 10 and D2 = 7, take multiple partial derivatives of the probability generation function at zero and sum them up to obtain the joint completion probability of multiple product outputs simultaneously meeting the target in different production windows, as shown in Table 3. Figure 4 As shown, the relationship between demand threshold and reliability under a given production window is as follows: Figure 5 As shown.
[0036] In practice, the processing capacity can be used as a reference. Figure 5 Select an appropriate combination of quantities of products to be processed.
[0037] Table 3. Probability of Simultaneous Completion of Production Targets for Multiple Varieties
[0038] Compared with existing technologies, this invention combines three techniques: Markov reward process, multidimensional probability generation function, and joint partial derivative extraction probability. This overcomes the limitations of traditional single-variety approximation in evaluating mixed-line interactions, significantly improving the accuracy of reliability assessment results for multi-variety processing lines and shortening the assessment time.
[0039] The above-described specific implementations can be partially adjusted by those skilled in the art in different ways without departing from the principles and purpose of the present invention. The scope of protection of the present invention is defined by the claims and is not limited to the above-described specific implementations. All implementation schemes within the scope of the claims are bound by the present invention.
Claims
1. A method for predicting the reliability of multi-product processing line capacity based on Markov reward processes, characterized in that, In the offline phase, based on order task requirements and processing equipment in multi-variety processing lines, the process path and corresponding processing equipment for each product to be processed are identified. Based on historical fault data, the failure rate and repair rate of each piece of equipment are estimated to construct a Markov transition rate matrix. By generating a multidimensional reward rate vector for each product under different equipment combinations, and further constructing a multidimensional Markov reward process, the probability generation function is obtained by solving the problem. In the online phase, based on the target output of the order input by the user, multiple partial derivatives of the probability generation function are taken at zero and accumulated to obtain the joint completion probability of all varieties simultaneously meeting the target within the production window, thus realizing capacity prediction.
2. The method for predicting the reliability of multi-product processing line capacity based on Markov reward processes according to claim 1, characterized in that, The aforementioned multi-product processing line refers to a production line that processes two or more product varieties simultaneously on the same production line, with each product having a different processing technology and processing path; The aforementioned processing equipment refers to a key manufacturing unit that undertakes one or more processing steps in the production process and whose operating status directly affects output capacity. Specifically, it consists of M processing equipment units constituting an equipment set E = {E1, ..., E...} M }; The process path refers to the sequence of equipment that must be passed through to complete all processing steps of a certain product. The product has a positive output rate if and only if all equipment in the path is in normal working order. The aforementioned order task requirements refer to: the set of quantity requirements (D1, D2, ..., D) submitted by customers for several different types of products within the predetermined delivery period. N ), where: N represents the number of product types to be processed, D i Let be the target output that the i-th product needs to achieve; The aforementioned combination of devices in different states refers to: an M-dimensional binary vector ω = (x1, ..., x) consisting of all device states. M ), totaling 2 M Species, of which: x k This indicates the equipment status, i.e., the operating condition of a single piece of equipment. When x k =1 when device E k Normal; when x k =0 when device E k It is currently in a fault-locked and out-of-service state.
3. The method for predicting the reliability of multi-product processing line capacity based on Markov reward processes according to claim 1, characterized in that, The Markov transition rate matrix Each device is an independent, continuous-time Markov chain, and only one device will undergo a state transition at any given time. Among them: Q k Let be the transfer rate matrix of device k. λ k Let μ be the failure rate of device k. k The repair rate of device k; I (·) It is an identity matrix of size (·) × (·). For Kronecker and, For Kronecker product.
4. The method for predicting the reliability of multi-product processing line capacity based on Markov reward processes according to claim 1 or 3, characterized in that, The aforementioned reward rate vector is the set of instantaneous production capacities for all product types when the system state is ω, specifically: r(ω) = (r 1 (ω),r 2 (ω),…,r N (ω)), where: ρ i Let be the nominal output rate of the i-th product under the condition that all relevant equipment is functioning normally and there are no bottleneck constraints.
5. The method for predicting the reliability of multi-product processing line capacity based on Markov reward processes according to claim 1 or 3, characterized in that, The Markov reward process refers to a stochastic process formed by coupling the Markov transition rate matrix with the reward rate vector, which can describe the cumulative change of yield of multiple varieties over time.
6. The method for predicting the reliability of multi-product processing line capacity based on Markov reward processes according to claim 1 or 3, characterized in that, The aforementioned probability generating function is the multi-variety cumulative output vector (X) generated by the constructed Markov reward process at any time T. 1 (T),…,X N The probability generating function of (T) is as follows: Where: X i (T) represents the cumulative qualified output of the i-th product within the time interval [0,T]. Y(t) is the Markov state at time t; θ i Let be the independent variable of the generating function corresponding to the t-th product; t is the continuous time parameter; T is the length of the evaluation period determined by the contract delivery terms; Let p(0) be the mathematical expectation operator, 0 ≤ t ≤ T, and p(0) be the initial system state vector. 1 is a column vector with all elements equal to 1, and a diagonal matrix. Its diagonal element Right now 7. The method for predicting the reliability of multi-product processing line capacity based on Markov reward processes according to claim 6, characterized in that, The aforementioned joint completion probability refers to the probability that, at the end of the evaluation period T, the cumulative yield of each variety simultaneously equals or exceeds its target yield requirement. in:
8. A multi-product processing line capacity prediction system for implementing the method of any one of claims 1-7, characterized in that, include: The system comprises several modules: an equipment state transition rate generation module, a process path-capacity parameter configuration module, a Markov reward process modeling and generating function solution module, and a multi-product joint reliability assessment module. Specifically: the equipment state transition rate generation module receives historical fault and maintenance records from the production site and generates a rate matrix describing the transition relationships between all possible states of the entire production line; the process path-capacity parameter configuration module reads the process list and order information from the enterprise resource planning system, extracts the equipment sequence required for each product, and, combined with standard cycle time or simulation calibration results, provides the unit-time output rate of each product under ideal operating conditions, and based on this... The system generates a capacity table corresponding to system states. The Markov reward process modeling and generating function solving module integrates data from the equipment state transition rate generation module and the process path-capacity parameter configuration module to automatically construct a Markov reward process model that can simultaneously characterize random equipment failures and multi-product output. Subsequently, it calculates a probability generating function that describes the global output statistics. The multi-product joint reliability assessment module performs partial derivative and accumulation operations on the generating function based on the user-set delivery cycle and order output target, and quickly outputs the joint completion probability that the output of each product simultaneously reaches or exceeds the demand within a specified time, thus obtaining the reliability assessment results of the multi-product processing line.