Improved fmea method based on possibility chance constrained data envelopment analysis framework

Abstract

The application discloses an improved FMEA method based on a possibility chance-constrained data envelopment analysis framework, first, the risk-induced factors are introduced into FMEA, second, a general DEA framework including series, parallel and network structure forms is constructed to build the structure correlation between the risk and the induced factors, finally, the PCCP model corresponding to the Me measure is introduced into FMEA, with the assistance of the PCCP model, the decision maker shows a flexible attitude to the inaccurate parameters and / or the chance constraints, so that the decision maker can obtain stable failure mode priority ranking. The method of the application can support the decision maker to obtain stable priority ranking of the failure mode, under the DEA framework of various structure forms, the different optimism pessimism coefficients and confidence levels of the decision maker are considered, for the priority ranking of the failure mode, the framework of the method of the application has stable performance, which is not only limited to the field of production system, but also can be applied to risk analysis and environment evaluation of a supply chain network and the like.

Description

Technical Field

[0001] This invention belongs to the field of reliability technology, specifically relating to an improved FMEA method based on a probability-chance constraint data envelopment analysis framework. Background Technology

[0002] Manufacturing is a pillar and strategic industry of a nation. The production system is the most typical form of production and processing organization in manufacturing. Generally, production systems can be categorized into series, parallel, and network structures based on product manufacturing processes or workstations. In a production system, the workstation, as the basic unit of production operation, typically includes product processing equipment, product testing equipment, and the product itself. The performance of the production system plays a crucial role in ensuring product quality. For example, in semiconductor manufacturing, improper operation often leads to numerous potential failure modes (such as selecting inappropriate processing materials, abnormal temperature and humidity settings in the processing workshop, and incorrect production rate settings for processing equipment). These inevitably induce product quality abnormalities (such as abnormal copper foil thickness, short circuits in product circuits, misaligned aperture positions, and weak noise immunity). Clearly, eliminating potential failure modes in the production system is one of the effective ways to ensure product quality. However, due to the numerous failure modes and the complexity of the production process, decision-makers often cannot accurately identify all potential failure modes and take corrective measures to eliminate them. In this situation, prioritizing potential failure modes in the production system becomes one of the urgent challenges to be addressed.

[0003] Failure Mode and Effects Analysis (FMEA), as a promising risk analysis tool, has been widely used to prioritize potential failure modes in production systems. Traditionally, FMEA prioritizes potential failure modes based on the arithmetic product of three risk factors: severity, detectability, and frequency of occurrence. Each risk factor is estimated using a precise value from 1 to 10. However, due to the complexity of production systems and the lack of sufficient knowledge and experience among decision-makers, accurately assessing risk factors is very difficult. Therefore, much research has focused on using uncertainty theories (such as fuzzy set theory) to overcome these challenges. On the other hand, the precipitating factors caused by risks (such as the loss costs and recovery time suffered by the production system after a failure mode occurs) have received little attention in FMEA implementations over the past few years. Furthermore, the structural correlation between risks and precipitating factors has been rarely investigated. This correlation can help decision-makers accurately prioritize potential failure modes. Typically, the aforementioned losses are considered as the loss costs and recovery time for the system and production. Loss costs usually represent the product scrap losses that these failure modes may cause during the production process. Recovery time is considered as the time required to handle these failure modes and ensure that the production system returns to normal operation. Therefore, a promising approach to overcoming the aforementioned challenges is to compare the magnitude of the loss costs and recovery times induced by risks (or failure modes). In most production scenarios, the greater the loss costs and recovery times caused by a particular failure mode, the higher its priority. In this way, prioritizing potential failure modes in a production system transforms into a risk analysis problem for a production system with multiple inputs (i.e., multiple failure modes) and multiple outputs (i.e., inducing factors such as loss costs and recovery times caused by risks).

[0004] Data Envelopment Analysis (DEA) is a nonparametric method that encompasses multiple inputs and outputs. Generally, DEA uses the relative efficiency of multiple decision units (e.g., failure modes) to rank them. This efficiency is considered the ability to induce triggering factors. Traditionally, DEA treats the evaluation object (e.g., a production system) as a black box, where input items (e.g., risks or failure modes) are provided to output items (e.g., triggering factors). Numerous studies have reportedly extended traditional DEA to relational DEA (e.g., serial, parallel, and network structure DEA). In engineering scenarios, due to a lack of sufficient knowledge and experience, and the complexity of production systems, decision-makers' observations of risk factors often exhibit potential randomness and imprecision. An effective method to address the randomness and imprecision of parameters is the Possibilistic Chance-Constrained Programming (PCCP) model. When historical data is available, PCCP models can employ fuzzy random distributions (e.g., trapezoidal and triangular random distributions) to compensate for decision-makers' lack of knowledge and experience regarding risk factors. In PCCP models, the Me measure has been a widely used measurement model in the field. Essentially, by introducing optimistic and pessimistic parameters, the Me measure enables decision-makers to exhibit a flexible attitude towards imprecise parameters and / or opportunity constraints.

[0005] To date, although DEA ​​has been widely applied in the field of FMEA, the following techniques have been rarely studied in FMEA. First, existing DEA-FMEA methods are mainly limited to serial DEA structures, especially single-stage DEA, thus failing to characterize the structural correlations between risks and triggering factors in parallel and network DEA structures. Second, opportunity-constrained DEA, particularly PCCP-based DEA, has been largely absent from FMEA research in the past decade, preventing decision-makers from expressing flexible attitudes towards imprecise parameters and / or opportunity constraints. In this context, the prioritization of potential failure modes may not yield stable results. Summary of the Invention

[0006] To address the aforementioned problems in existing technologies, this invention proposes an improved FMEA method based on a probability-chance constraint data envelopment analysis framework.

[0007] The technical solution adopted in this invention is: an improved FMEA method based on a probability-chance constraint data envelopment analysis framework, the specific steps of which are as follows:

[0008] S1. Identify the potential failure modes, possible risk factors, and triggering factors in the production system, construct the structural form of the production system, and form a correspondence model between the production system and relational data envelopment analysis;

[0009] S2. Based on the knowledge and experience of experts, and combined with relevant historical data from the production process, assess potential failure mode risk factors and triggering factors to obtain a risk input and output matrix;

[0010] S3. Based on the correspondence model in step S1 and combined with the risk input and output matrices obtained in step S2, construct an optimization model with the goal of maximizing the expected efficiency or expected virtual output of the failure mode (i.e., maximizing the potential consequences caused by the failure mode), and obtain the priority ranking of potential failure modes in the production system.

[0011] Furthermore, step S1 specifically includes the following:

[0012] S101. Clarify the causes and potential consequences of potential failure modes in the production system, and construct an FMEA assessment table;

[0013] S102. Identify potential failure modes FM = {FM j |FM1,FM2,...,FM J}(j=1,2,…,J), with I (i=1,2,…,I) potential risk factors and R (r=1,2,…,R) potential triggering factors;

[0014] Among them, FM1, FM2, ..., FM J This represents J fault modes.

[0015] S103. Construct a correspondence model between the production system and the DEA, and match the workstations in the production system with the stages in the DEA.

[0016] Furthermore, in step S1, the inducing factors are specifically: the production system suffers losses in cost and recovery time due to the occurrence of risk or failure modes.

[0017] Furthermore, step S2 specifically includes the following:

[0018] S201. Define the decision-maker's assessment values ​​for risk factors and triggering factors, and use a trapezoidal random distribution to assess the risk factors and triggering factors for each failure mode;

[0019] Where, ξ=(ξ (1) ,ξ (2) ,ξ (3) ,ξ (4) ξ represents a trapezoidal random distribution. (·)(·=1,2,3,4) follows a uniform distribution U(a (·) ,b (·) U represents a uniform distribution, derived from a (·) (i.e., minimum value) and b (.) The two parameters (i.e., maximum value) are defined.

[0020] S202. Based on the definition of the expected value of the trapezoidal random distribution in the Me measure of the PCCP model, the expected value of ξ (i.e., E) can be calculated. Me [ξ]);

[0021] Using the Me measure to clearly equivalence the opportunity constraints:

[0022]

[0023]

[0024] Where Me{…} denotes the chance constraint operator under the Me measure, χ denotes the pre-defined unknown variable, λ denotes the optimism-pessimism coefficient which follows a uniform distribution U(0,1), and α denotes the decision-maker's confidence level in the imprecise parameters and / or chance constraints which follows a uniform distribution U(0.5,1).

[0025] S203. The risk factors and triggering factors for each failure mode are evaluated using a trapezoidal random distribution, resulting in the risk input matrix X = [X...]. ij ] I×J and risk output matrix Y = [Y rj ] R×J ;

[0026] Where i = 1, 2, ..., I, j = 1, 2, ..., J, r = 1, 2, ..., R, X ij Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the j-th failure mode. rj It is also a set of trapezoidal random distributions, representing the observed value of the r-th inducing factor for the j-th failure mode.

[0027] Furthermore, step S3 specifically includes the following:

[0028] S301. Divide the structural forms of the production system;

[0029] The production system with multiple workstations in series, parallel and network structures forms a multi-stage series, parallel and network structure DEA based on the correspondence model in step S103.

[0030] S302. Establish a multi-stage serial, parallel and network structure DEA framework based on the PCCP model, with the optimization objective of maximizing the expected efficiency or expected virtual output of the failure mode, and the constraint of the decision-maker’s attitude toward risk input and risk propagation.

[0031] The FMEA method within the DEA framework is proposed, specifically including the following three categories:

[0032] (1) FMEA method based on multi-stage (single-stage and two-stage) cascaded DEA:

[0033] a. FMEA method based on single-stage DEA;

[0034] Risk input matrix X = [X ij ] I×J and risk output matrix Y = [Y rj ] R×J These correspond to the input and output terms in the DEA, respectively. The structural correlation between X and Y is induced by the stages in the DEA. The optimization model below aims to maximize the expected efficiency (i.e., ...) of the k-th (k∈J) failure mode under the Me measure. ):

[0035]

[0036] Among them, X ik Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. rk It is also a set of trapezoidal random distributions, representing the observed values ​​of the r-th triggering factor for the k-th failure mode; u r and v i Let represent the multipliers of the r-th output term and the i-th input term, respectively, and ε represent a very small, pre-defined constant; constraints Ensure that the expected efficiency of the k-th failure mode is [0,1], with chance constraints. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. The expected level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates policymakers' pessimistic view of risk propagation during the DEA phase. α represents the level of conservatism, which follows a uniform distribution U(0,1); α represents the decision-maker's confidence level in imprecise parameters and / or chance constraints, which follows a uniform distribution U(0.5,1).

[0037] b. FMEA based on two-stage cascaded DEA;

[0038] Risk input matrix X = [X ij ]I×J and risk output matrix Y = [Y rj ] R×J These correspond to the input and output terms of DEA, respectively. Matrix Z = [Z hj ] H×J (h = 1, 2, ..., H) represents the intermediate term in DEA, Z hj Let X be a set of trapezoidal random distributions, representing the observations of the h-th intermediate factor for the j-th failure mode, where H represents the number of intermediate factors; Z represents the observations of each failure mode by the decision-maker after X has undergone the first stage of the DEA. The relationship between X, Z, and Y is considered a structural correlation caused by the two stages in the DEA. The optimization model below aims to maximize the expected efficiency (i.e., ..., the efficiency of the k-th (k∈J) failure mode under the Me measure) in the DEA measure. ):

[0039]

[0040] Among them, X ik Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. rk It is also a set of trapezoidal random distributions, representing the observed values ​​of the r-th triggering factor for the k-th failure mode; u r v i and w h Let r represent the multipliers of the r-th output term, the i-th input term, and the h-th intermediate term, respectively; ε represents a pre-defined, very small constant; constraints Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. The expected level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates policymakers' pessimistic view of risk propagation during the two phases of DEA. The conservative level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates policymakers' pessimistic view of risk propagation during the first phase of DEA. The conservative level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates policymakers' pessimistic view of risk propagation during the second phase of DEA. α represents the level of conservatism, which follows a uniform distribution U(0,1); α represents the decision-maker's confidence level in imprecise parameters and / or chance constraints, which follows a uniform distribution U(0.5,1).

[0041] (2) FMEA method based on multi-stage parallel DEA:

[0042] Risk input matrix X = [X ij ] I×J and risk output matrix Y = [Y rj ] R×J These correspond to the input and output terms of DEA, respectively. Branch input term X (n) (n = 1, 2, ..., N), satisfying The total number of branches N can take any integer value not less than 2. Let represent the weight of the input item in the nth branch, which follows a uniform distribution U(0,1) and satisfies ... θ n Let represent the weight of the output item of the nth branch, which follows a uniform distribution U(0,1) and satisfies . The relationship between X and Y is considered a structural correlation, caused by two stages in DEA. The optimization model below aims to maximize the expected efficiency (i.e., ...) of the k-th (k∈J) failure mode under the Me measure. ):

[0043]

[0044] Among them, X ik Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. rk It is also a set of trapezoidal random distributions, representing the observed values ​​of the r-th triggering factor for the k-th failure mode; u r and v i Let r represent the multipliers of the r-th output term and the i-th input term, respectively; ε represents a very small, pre-defined constant; constraints. Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. The desired level is represented by a uniform distribution U(0,1); constraints This represents the relaxation constraints corresponding to all stages. (Constraint set) s represents the set of relaxed constraints corresponding to the sub-stage. k This represents the slack variables for all stages when evaluating failure mode k. Represents the slack variables corresponding to stage n; chance constraint set This indicates that policymakers are pessimistic about the spread of risk at all stages. The conservative level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates that policymakers are pessimistic about the risk propagation in the nth sub-phase. α represents the level of conservatism, which follows a uniform distribution U(0,1); α represents the decision-maker's confidence level in imprecise parameters and / or chance constraints, which follows a uniform distribution U(0.5,1).

[0045] (3) FMEA method based on network structure DEA;

[0046] Risk input matrix X = [X ij ] I×J The input item corresponding to DEA. Branch input item X (l) (l=1,2,...,L), satisfying The total number of inputs L can take any integer value not less than 3. Let represent the weight of the input item in the l-th branch, which follows a uniform distribution U(0,1) and satisfies . Branch output item Y (t) (t=1,2,...,T), satisfying The total number of branch outputs T can take any integer value not less than 5, ζ t (ζ t ~U(0,1)) represents the weight of the output item of the t-th branch, and satisfies Risk output matrix Y (3) Y (4) and Y (5) These are the output items in DEA. X, Y (3) Y (4) and Y (5) The relationship between these factors is considered a structural correlation, which is caused by the three stages in DEA. The optimization model below aims to maximize the expected efficiency (i.e., ...) of the k-th (k∈J) failure mode under the Me measure. ):

[0047]

[0048] Among them, X ik Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. rk It is also a set of trapezoidal random distributions, representing the observed value of the r-th triggering factor for the k-th failure mode; and These are the multipliers of the five branch output terms of the r-th input term, v. i It is the multiplier of the i-th input item. and These are the multipliers of the three branch inputs of the i-th input term, respectively, where ε represents a very small pre-defined constant; constraints Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. The expected level is represented by a uniform distribution U(0,1); the chance constraint set... and This indicates policymakers' pessimistic view of the risk transmission in the sub-phases, among which and Representing the level of conservatism, the set of opportunity constraints This indicates that policymakers are pessimistic about the spread of risk at all stages. The conservative level is represented by α, which represents the decision-maker's confidence level in imprecise parameters and / or chance constraints, and follows a uniform distribution U(0.5,1).

[0049] S303. Based on the definition of expected value and the handling of opportunity constraints in step S202, the three types of FMEA methods based on the DEA framework in step S302 are converted into clear equivalent forms.

[0050] The beneficial effects of this invention are as follows: First, the method of this invention introduces risk-induced factors into FMEA. Second, it constructs a general DEA framework including series, parallel, and network structures to establish the structural correlation between risk and induced factors. Through this correlation, decision-makers can obtain two benefits: first, accurately determining the priority ranking of failure modes; and second, thoroughly understanding the structural form of risk propagation in the production system. Finally, it introduces a PCCP model corresponding to the Me metric into FMEA. With the assistance of the PCCP model, decision-makers exhibit a flexible attitude towards imprecise parameters and / or opportunity constraints, thereby helping them obtain a stable priority ranking of failure modes. The method of this invention can support decision-makers in obtaining a stable priority ranking of failure modes. Under various DEA framework structures, considering different optimism / pessimism coefficients and confidence levels of decision-makers, the framework of this invention has stable performance in prioritizing failure modes. It is not limited to the field of production systems but can also be applied to risk analysis and environmental assessment of supply chain networks. Attached Figure Description

[0051] Picture 1 This is a diagram illustrating the correspondence between the production system and the DEA in an embodiment of the present invention.

[0052] Picture 2 This is a schematic diagram of a multi-stage DEA and multi-station production system in an embodiment of the present invention.

[0053] Picture 3 This is a schematic diagram of the FMEA method based on single-stage DEA in an embodiment of the present invention.

[0054] Picture 4This is a schematic diagram of the FMEA method based on two-stage DEA in an embodiment of the present invention.

[0055] Picture 5 This is a schematic diagram of the FMEA method based on parallel structure DEA in an embodiment of the present invention.

[0056] Picture 6 This is a schematic diagram of the FMEA method based on network structure DEA in an embodiment of the present invention.

[0057] Picture 7 This is a schematic diagram of a semiconductor manufacturing system with multiple structural forms in an embodiment of the present invention.

[0058] Picture 8 This is a graph showing the calculation results of the FMEA method based on single-stage DEA in an embodiment of the present invention.

[0059] Picture 9 The graph shows the calculation results of the FMEA method based on two-stage DEA in this embodiment of the invention.

[0060] Picture 10 The graph shows the calculation results of the FMEA method based on parallel structure DEA in this embodiment of the invention.

[0061] Picture 11 This is a graph showing the calculation results of the FMEA method based on network structure DEA in this embodiment of the invention. Detailed Implementation

[0062] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0063] In Embodiment 1 of the present invention, the specific steps are as follows:

[0064] S1. Identify the potential failure modes, possible risk factors, and triggering factors in the production system, construct the structural form of the production system, and form a correspondence model between the production system and relational data envelopment analysis;

[0065] S2. Based on the knowledge and experience of experts, and combined with relevant historical data from the production process, assess potential failure mode risk factors and triggering factors to obtain a risk input and output matrix;

[0066] S3. Based on the correspondence model in step S1 and combined with the risk input and output matrices obtained in step S2, construct an optimization model with the goal of maximizing the expected efficiency or expected virtual output of the failure mode (i.e., maximizing the potential consequences caused by the failure mode), and obtain the priority ranking of potential failure modes in the production system.

[0067] In this embodiment, step S1 specifically includes the following:

[0068] S101. Clarify the causes and potential consequences of potential failure modes in the production system, and construct an FMEA assessment table;

[0069] As shown in Table 1, this embodiment presents the general form of the FMEA evaluation table, along with an example:

[0070] Table 1

[0071]

[0072] S102. Identify potential failure modes FM = {FM j |FM1,FM2,...,FM J}(j=1,2,...,J), with I (i=1,2,...,I) potential risk factors and R (r=1,2,...,R) potential triggering factors;

[0073] Among them, FM1, FM2, ..., FM J This represents J fault modes.

[0074] S103. Construct a correspondence model between the production system and the DEA, and match the workstations in the production system with the stages in the DEA.

[0075] Taking a multi-station serial production system as an example, such as Picture 1 As shown, risks will propagate along the structural form of the production system, thus forming a risk input matrix X and a risk output matrix Y.

[0076] In this embodiment, step S2 specifically includes the following:

[0077] S201. Define the decision-maker's assessment values ​​for risk factors and triggering factors, and use a trapezoidal random distribution to assess the risk factors and triggering factors for each failure mode;

[0078] Where, ξ=(ξ (1) ,ξ (2) ,ξ (3) ,ξ (4) ξ represents a trapezoidal random distribution. (·) (·=1,2,3,4) follows a uniform distribution U(a (·) ,b (·) U represents a uniform distribution, derived from a (·) (i.e., minimum value) and b (·) The two parameters (i.e., maximum value) are defined.

[0079] S202. Based on the definition of the expected value of the trapezoidal random distribution in the Me measure of the PCCP model, the expected value of ξ (i.e., E) can be calculated.Me [ξ]);

[0080] Using the Me measure to clearly equivalence the opportunity constraints:

[0081]

[0082]

[0083] Where Me{...} denotes the chance constraint operator under the Me measure, χ denotes the pre-defined unknown variable, λ denotes the optimism-pessimism coefficient, which follows a uniform distribution U(0,1), and α denotes the decision-maker's confidence level in the imprecise parameters and / or chance constraints, which follows a uniform distribution U(0.5,1).

[0084] S203. The risk factors and triggering factors for each failure mode are evaluated using a trapezoidal random distribution, resulting in the risk input matrix X = [X...]. ij ] I×J and risk output matrix Y = [Y rj ] R×J ;

[0085] Where i = 1, 2, ..., I, j = 1, 2, ..., J, r = 1, 2, ..., R, X ij Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the j-th failure mode. rj It is also a set of trapezoidal random distributions, representing the observed value of the r-th inducing factor for the j-th failure mode.

[0086] In this embodiment, step S3 specifically includes the following:

[0087] S301. Divide the structural forms of the production system;

[0088] The production system with multiple workstations in series, parallel and network structures forms a multi-stage series, parallel and network structure DEA based on the correspondence model in step S103.

[0089] like Picture 2 As shown, Picture 2 (a) shows a schematic diagram of a multi-stage serial DEA and a multi-station serial production system; Picture 2 (b) A schematic diagram of a multi-stage parallel DEA and a multi-station parallel production system is shown; Picture 2 (c) A schematic diagram of the network structure DEA and the network structure production system is shown. As described in step S103, for any structured production system, its workstations correspond to the stages in the DEA.

[0090] S302. Establish a multi-stage serial, parallel and network structure DEA framework based on the PCCP model, with the optimization objective of maximizing the expected efficiency or expected virtual output of the failure mode, and the constraint of the decision-maker’s attitude toward risk input and risk propagation.

[0091] The FMEA method based on the DEA framework proposed in this embodiment specifically includes the following three categories:

[0092] (1) FMEA method based on multi-stage (single-stage and two-stage) cascaded DEA:

[0093] a. FMEA method based on single-stage DEA, as shown in the diagram. Picture 3 As shown;

[0094] Risk input matrix X = [X ij ] I×J and risk output matrix Y = [Y rj ] R×J These correspond to the input and output terms in the DEA, respectively. The structural correlation between X and Y is induced by the stages in the DEA. The optimization model below aims to maximize the expected efficiency (i.e., ...) of the k-th (k∈J) failure mode under the Me measure. ):

[0095]

[0096] Among them, X ik Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. rk It is also a set of trapezoidal random distributions, representing the observed values ​​of the r-th triggering factor for the k-th failure mode; u r and v i Let represent the multipliers of the r-th output term and the i-th input term, respectively, and ε represent a very small, pre-defined constant; constraints Ensure that the expected efficiency of the k-th failure mode is [0,1], with chance constraints. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. The expected level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates policymakers' pessimistic view of risk propagation during the DEA phase. α represents the level of conservatism, which follows a uniform distribution U(0,1); α represents the decision-maker's confidence level in imprecise parameters and / or chance constraints, which follows a uniform distribution U(0.5,1).

[0097] b. FMEA method based on two-stage cascaded DEA, as shown in the diagram. Picture 4 As shown;

[0098] Risk input matrix X = [X ij ] I×J and risk output matrix Y = [Y rj ] R×J These correspond to the input and output terms of DEA, respectively. Matrix Z = [Z hj ] H×J (h = 1, 2, ..., H) represents the intermediate term in DEA, Z hj Let X be a set of trapezoidal random distributions, representing the observations of the h-th intermediate factor for the j-th failure mode, where H represents the number of intermediate factors; Z represents the observations of each failure mode by the decision-maker after X has undergone the first stage of the DEA. The relationship between X, Z, and Y is considered a structural correlation caused by the two stages in the DEA. The optimization model below aims to maximize the expected efficiency (i.e., ..., the efficiency of the k-th (k∈J) failure mode under the Me measure) in the DEA measure. ):

[0099]

[0100] Among them, X ik Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. rk It is also a set of trapezoidal random distributions, representing the observed values ​​of the r-th triggering factor for the k-th failure mode; u r v i and w h Let r represent the multipliers of the r-th output term, the i-th input term, and the h-th intermediate term, respectively; ε represents a pre-defined, very small constant; constraints Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. The expected level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates policymakers' pessimistic view of risk propagation during the two phases of DEA. The conservative level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates policymakers' pessimistic view of risk propagation during the first phase of DEA. The conservative level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates policymakers' pessimistic view of risk propagation during the second phase of DEA. α represents the level of conservatism, which follows a uniform distribution U(0,1); α represents the decision-maker's confidence level in imprecise parameters and / or chance constraints, which follows a uniform distribution U(0.5,1).

[0101] (2) The FMEA method based on multi-stage parallel DEA is illustrated in the diagram below. Picture 5 As shown:

[0102] Risk input matrix X = [X ij ] I×J and risk output matrix Y = [Y rj ] R×J These correspond to the input and output terms of DEA, respectively. Branch input term X (n) (n = 1, 2, ..., N), satisfying The total number of branches N can take any integer value not less than 2. Let represent the weight of the input item in the nth branch, which follows a uniform distribution U(0,1) and satisfies ... θ n Let represent the weight of the output item of the nth branch, which follows a uniform distribution U(0,1) and satisfies . The relationship between X and Y is considered a structural correlation, caused by two stages in DEA. The optimization model below aims to maximize the expected efficiency (i.e., ...) of the k-th (k∈J) failure mode under the Me measure. ):

[0103]

[0104] Among them, X ik Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. rk It is also a set of trapezoidal random distributions, representing the observed values ​​of the r-th triggering factor for the k-th failure mode; u r and v i Let r represent the multipliers of the r-th output term and the i-th input term, respectively; ε represents a very small, pre-defined constant; constraints. Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. The desired level is represented by a uniform distribution U(0,1); constraints This represents the relaxation constraints corresponding to all stages. (Constraint set) s represents the set of relaxed constraints corresponding to the sub-stage. k This represents the slack variables for all stages when evaluating failure mode k. Represents the slack variables corresponding to stage n; chance constraint set This indicates that policymakers are pessimistic about the spread of risk at all stages. The conservative level is represented by a uniform distribution U(0,1); the chance constraint set... This indicates that policymakers are pessimistic about the risk propagation in the nth sub-phase. α represents the level of conservatism, which follows a uniform distribution U(0,1); α represents the decision-maker's confidence level in imprecise parameters and / or chance constraints, which follows a uniform distribution U(0.5,1).

[0105] (3) FMEA method based on network structure DEA, as shown in the diagram. Picture 6 As shown:

[0106] Risk input matrix X = [X ij ] I×J The input item corresponding to DEA. Branch input item X (l) (l=1,2,...,L), satisfying The total number of inputs L can take any integer value not less than 3. Let represent the weight of the input item in the l-th branch, which follows a uniform distribution U(0,1) and satisfies . Branch output item Y (t) (t=1,2,...,T), satisfying The total number of branch outputs T can take any integer value not less than 5, ζ t (ζ t ~U(0,1)) represents the weight of the output item of the t-th branch, and satisfies Risk output matrix Y (3) Y (4) and Y (5) These are the output items in DEA. X, Y (3) Y (4) and Y (5) The relationship between these factors is considered a structural correlation, which is caused by the three stages in DEA. The optimization model below aims to maximize the expected efficiency (i.e., ...) of the k-th (k∈J) failure mode under the Me measure. ):

[0107]

[0108] Among them, X ik Y is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. rk It is also a set of trapezoidal random distributions, representing the observed value of the r-th triggering factor for the k-th failure mode; and These are the multipliers of the five branch output terms of the r-th input term, v. i It is the multiplier of the i-th input item. and These are the multipliers of the three branch inputs of the i-th input term, respectively, where ε represents a very small pre-defined constant; constraints Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. The expected level is represented by a uniform distribution U(0,1); the chance constraint set... and This indicates policymakers' pessimistic view of the risk transmission in the sub-phases, among which and Representing the level of conservatism, the set of opportunity constraints This indicates that policymakers are pessimistic about the spread of risk at all stages. The conservative level is represented by α, which represents the decision-maker's confidence level in imprecise parameters and / or chance constraints, and follows a uniform distribution U(0.5,1).

[0109] S303. Based on the definition of expected value and the handling of opportunity constraints in step S202, the FMEA method based on the DEA framework in step S302 is transformed into a clear equivalent form.

[0110] In Embodiment 2 of the present invention, a multi-station semiconductor manufacturing system is used as Embodiment 2, and the specific steps are as follows:

[0111] S1. Identify the potential failure modes, possible risk factors, and triggering factors in a multi-station semiconductor manufacturing system; construct a manufacturing system that covers series, parallel, and network structures; and form a corresponding DEA model for series, parallel, and network structures.

[0112] S2. Using trapezoidal random distribution, assess potential failure mode risk factors and triggering factors to obtain risk input and risk output matrices. At the same time, based on expert knowledge and experience, determine the parameters required in the FMEA method based on the DEA framework, such as expected level and conservative level.

[0113] S3. Based on the obtained risk input and risk output matrices and related parameters, the FMEA method based on the DEA framework is used to prioritize potential failure modes in semiconductor manufacturing systems with various structural forms.

[0114] Semiconductor manufacturing systems, as typical multi-station production systems, can be customized into series, parallel, and network structures due to variations in their production processes. For example... Picture 7 As shown, the specific production process is given, and four structural forms are considered in this example.

[0115] Drilling and copper plating stations are the primary production processes, while deburring and plasma cleaning stations are auxiliary processes. In some production practices, because the risks and hazards of auxiliary processes are smaller than those of primary processes, the risks posed by auxiliary processes can often be ignored. Therefore, semiconductor manufacturing systems can evolve into various structural forms. Furthermore, other types of manufacturing systems (such as LCD panel manufacturing systems and wafer manufacturing systems), as long as they can be classified into multiple structural forms and the corresponding risk input matrix, risk output matrix, and related parameters are obtained, are all applicable to this invention.

[0116] The steps of the embodiments of the present invention are described in detail below:

[0117] (1) Identify potential failure modes in the semiconductor manufacturing system and create an FMEA assessment table. Determine the risk factors and triggering factors of each failure mode, and construct a model relating the semiconductor manufacturing system to the DEA (Definition of Analysis).

[0118] a. Based on expert knowledge and experience, as well as relevant historical data from the production process, potential failure modes are identified, and these failure modes are tracked and located to determine their potential consequences, ultimately forming an FMEA assessment table, as shown in Table 2. This embodiment lists five common potential failure modes and describes the potential consequences that each failure mode may cause.

[0119] Table 2

[0120]

[0121] b. Label the fault modes in Table 2 as FM1, FM2, FM3, FM4 and FM5 in sequence.

[0122] c. The risk factors for failure modes are identified as severity, detectability, and frequency of occurrence; the triggering factors are identified as the losses and recovery time that the semiconductor manufacturing system will face if the failure mode occurs.

[0123] d. Classify the semiconductor manufacturing system into structural forms, namely: series, parallel, and network structures, and form a corresponding DEA model. It is worth noting that in actual operating conditions, the potential failure modes differ due to variations in the structural form of the manufacturing system and the specific production process. In the examples of this embodiment, to facilitate the illustration of the applicability and advantages of the invention, the failure modes in Table 2 were adopted for different structural forms of the manufacturing system, and risk analysis and priority ranking of failure modes were then carried out.

[0124] (2) Assess the risk factors and triggering factors of potential failure modes, and obtain the parameters required in the FMEA method based on the DEA framework (e.g., expected level and conservative level):

[0125] a. Risk factors and triggering factors of potential failure modes are evaluated using a trapezoidal random distribution, and the observed values ​​of risk input matrix X and risk output Y are obtained, as listed in Tables 3 and 4, respectively.

[0126] Table 3

[0127]

[0128] Table 4

[0129] (U(1,2),U(2,3),U(3,4),U(4,5)) (U(1.5,2.5),U(2.5,3),U(3,4.5),U(4.5,5.5)) (U(2,3),U(3,4),U(4,5),U(5,6)) (U(2.5,3.5),U(3.5,4),U(4.5,5.5),U(5.5,6)) (U(0.9,1.3),U(1.5,2),U(2,3),U(3,4)) (U(1.5,2.3),U(2.3,3),U(3,3.5),U(3.5,4.5)) (U(2.2,3.2),U(3.2,4.2),U(4.2,5.2),U(5.2,6.2)) (U(1.4,2.4),U(2.4,3.4),U(3.4,4.5),U(5.5,6)) (U(1.2,1.3),U(1.4,1.6),U(1.7,1.9),U(1.9,2.1)) (U(1.5,2.5),U(2.5,3),U(3,3.5),U(3.5,4))

[0130] b. For the FMEA method based on two-stage cascaded DEA, the observed values ​​of matrix Z are obtained using a trapezoidal random distribution, as shown in Table 5.

[0131] Table 5

[0132]

[0133] c. Based on expert knowledge and experience, determine the expected level and conservative level in the FMEA method based on the DEA framework, as shown in Table 6.

[0134] Table 6

[0135]

[0136] (3) Based on the data obtained in step (2), the FMEA method based on the DEA framework of the invention is used to conduct a risk assessment of the semiconductor manufacturing system and determine the priority ranking of potential failure modes. It is worth noting that in order to obtain the changing trend of the priority ranking of each failure mode under random conditions, the method of the invention was executed 100 times in this embodiment. Of course, the number of executions can be appropriately adjusted according to the specific research object and case of the decision-maker.

[0137] a. Using the FMEA method based on single-stage DEA, the potential failure modes in Table 2 were prioritized, and the results are as follows: Picture 8 As shown.

[0138] b. Using the FMEA method based on two-stage cascaded DEA, the potential failure modes in Table 2 were prioritized, and the results are as follows: Picture 9 As shown.

[0139] c. Using the FMEA method based on parallel structure DEA, the potential failure modes in Table 2 are prioritized, and the results are as follows: Picture 10 As shown.

[0140] d. Using the FMEA method based on network structure DEA, the potential failure modes in Table 2 were prioritized, and the results are as follows: Picture 11 As shown.

[0141] Table 7 summarizes the failure mode prioritization results under different DEA framework structures. From Table 7, the following conclusions can be drawn: First, regardless of the method used (i.e., the DEA framework structure), FM2 consistently exhibits the highest expected efficiency and the smallest expected efficiency standard deviation. Second, for parallel DEA structures, FM2 possesses non-DEA efficiency. This phenomenon indicates that parallel DEA structures may lead to a loss of expected efficiency. As a result, among the four DEA frameworks, the average expected efficiency of parallel DEA structures is the lowest. However, the loss of expected efficiency does not affect the overall failure mode prioritization.

[0142] Table 7

[0143]

[0144] (Δ represents the mean of the expected efficiency or standard deviation of all valid solutions across 100 attempts)

[0145] Picture 8 (a)-11(a) depicts the changing trends in failure mode prioritization under different DEA framework structures. Picture 8 (a)-11(a) shows that, except for the parallel DEA structure, FM2 exhibits the most stable expected efficiency variation trend. Within the parallel DEA framework, FM2 and FM4 both have the highest priority. For any DEA structure, the expected efficiency variation trends of FM3 and FM5 are always similar, and these two failure modes have the lowest priority. Furthermore, in most cases, FM4 and FM1 consistently rank second and fourth in priority, respectively. In addition, from... Picture 8 As can be seen from (b)-11(b), for any DEA structure, the value of λ is always greater than α, and the value of λ is always greater than 0.5. This result implies that the FMEA method based on the DEA framework proposed in this invention is an optimistic method. Of course, this result will also encourage decision-makers to accept the prioritization results of the failure modes proposed in this invention.

[0146] In summary, the method of this invention corresponds the workstations in a multi-station series, parallel, and network structured production system with the stages in a multi-stage series, parallel, and network structured data envelopment analysis (DEA), forming a correspondence model between the production system and DEA. Simultaneously, the method considers the structural correlation between risks (e.g., failure modes) and triggering factors (e.g., the loss costs and recovery time suffered by the production system after a failure mode occurs), introducing a probability-chance constraint model into DEA to effectively address the potential randomness and imprecision of decision-makers' observations of risk factors. The method of this invention enables decision-makers to express flexible attitudes towards imprecise parameters and / or chance constraints, thereby assisting them in obtaining a stable failure mode prioritization.

Claims

1. An improved FMEA method based on a probability and chance constraint data envelopment analysis framework, the specific steps of which are as follows: S1. Identify the potential failure modes, possible risk factors, and triggering factors in the production system, construct the structural form of the production system, and form a correspondence model between the production system and relational data envelopment analysis; S101. Clarify the causes and potential consequences of potential failure modes in the production system, and construct an FMEA assessment table; S102. Identify potential failure modes. Possible One risk factor and One triggering factor; in, , , , express One failure mode; S103. Construct a correspondence model between the production system and the DEA, and match the workstations in the production system with the stages in the DEA. S2. Based on the knowledge and experience of experts, and combined with relevant historical data from the production process, assess potential failure mode risk factors and triggering factors to obtain a risk input and output matrix; S201. Define the decision-maker's assessment values ​​for risk factors and triggering factors, and use a trapezoidal random distribution to assess the risk factors and triggering factors for each failure mode; in, Represents a set of trapezoidal random distributions. , Following a uniform distribution , Represents a uniform distribution, with a minimum value. and maximum value These two parameters are defined; S202. Based on the definition of the expected value of the trapezoidal random distribution in the Me measure of the PCCP model, the expected value is calculated. The expected value, i.e. ; Using the Me measure to clearly equivalence the opportunity constraints: (1)； (2)； in, This represents the opportunity constraint operator under the Me measure. This represents a pre-defined unknown variable. This represents the optimism-pessimism coefficient, which follows a uniform distribution. , This represents the decision-maker's confidence level regarding imprecise parameters and / or opportunity constraints, and it follows a uniform distribution. ; S203. The risk factors and triggering factors for each failure mode are evaluated using a trapezoidal random distribution to obtain the risk input matrix. and risk output matrix ; in, , , , It is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the j-th failure mode. It is also a set of trapezoidal random distributions, representing the observed values ​​of the r-th inducing factor for the j-th failure mode; S3. Based on the correspondence model in step S1, and combined with the risk input and output matrices obtained in step S2, construct an optimization model with the goal of maximizing the expected efficiency or expected virtual output of the failure modes, and obtain the priority ranking of potential failure modes in the production system. S301. Divide the structural forms of the production system; The production system with multiple workstations in series, parallel and network structures forms a multi-stage series, parallel and network structure DEA based on the correspondence model in step S103; S302. Establish a multi-stage serial, parallel and network structure DEA framework based on the PCCP model, with the optimization objective of maximizing the expected efficiency or expected virtual output of the failure mode, and the constraint of the decision-maker’s attitude toward risk input and risk propagation. S303. Based on the definition of expected value and the handling of opportunity constraints in step S202, the three types of FMEA methods based on the DEA framework in step S302 are converted into clear equivalent forms.

2. The improved FMEA method based on a probability-chance constraint data envelopment analysis framework according to claim 1, characterized in that, In step S1, the triggering factors are specifically: the production system suffers losses in cost and recovery time due to the occurrence of risk or failure modes.

3. The improved FMEA method based on a probability-chance constraint data envelopment analysis framework according to claim 1, characterized in that, In step S302, the specific details are as follows: This paper proposes an FMEA method based on the DEA framework, which includes the following three categories: (1) FMEA method based on multi-stage cascaded DEA: a. FMEA method based on single-stage DEA; Risk Input Matrix and risk output matrix These correspond to the input and output terms in DEA, respectively. and The structural correlation between them is induced by the stages in the DEA; the following optimization model aims to maximize the first stage in the Me measure. The expected efficiency of each failure mode, i.e. : (3)； in, It is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. It is also a set of trapezoidal random distributions, representing the observed value of the r-th triggering factor for the k-th failure mode; and Let r represent the multipliers of the r-th output term and the i-th input term, respectively. Represents a very small, pre-defined constant; constraint Ensure that the expected efficiency of the k-th failure mode is [0,1], with chance constraints. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. This represents the expected level, which follows a uniform distribution. Opportunity constraint set This indicates policymakers' pessimistic view of risk propagation during the DEA phase. This represents the level of conservatism, which follows a uniform distribution. ; This represents the decision-maker's confidence level regarding imprecise parameters and / or opportunity constraints, and it follows a uniform distribution. ; b. FMEA method based on two-stage cascaded DEA; Risk Input Matrix and risk output matrix The matrix corresponds to the input and output terms of DEA, respectively. , This represents the intermediate term in DEA. It is a set of trapezoidal random distributions, representing the observation value of the h-th intermediate factor for the j-th failure mode, where H represents the number of intermediate factors; This indicates that after X has gone through the first stage of DEA, the decision-maker re-evaluates the observations of each failure mode; , and The relationship between them is considered a structural correlation, which is caused by two stages in DEA; the following optimization model aims to maximize the first stage under the Me measure. The expected efficiency of each failure mode, i.e. : (4)； in, It is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. It is also a set of trapezoidal random distributions, representing the observed value of the r-th triggering factor for the k-th failure mode; , and Let r represent the multipliers of the r-th output term, the i-th input term, and the h-th intermediate term, respectively. Represents a very small, pre-defined constant; constraint Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. This represents the expected level, which follows a uniform distribution. Opportunity constraint set This indicates policymakers' pessimistic view of risk propagation during the two phases of DEA. This represents the level of conservatism, which follows a uniform distribution. Opportunity constraint set This indicates policymakers' pessimistic view of risk propagation during the first phase of DEA. This represents the level of conservatism, which follows a uniform distribution. Opportunity constraint set This indicates policymakers' pessimistic view of risk propagation during the second phase of DEA. This represents the level of conservatism, which follows a uniform distribution. ; This represents the decision-maker's confidence level regarding imprecise parameters and / or opportunity constraints, and it follows a uniform distribution. ; (2) FMEA method based on multi-stage parallel DEA: Risk Input Matrix and risk output matrix These correspond to the input and output terms of DEA, respectively, and the branch input terms. , ,satisfy The total number of branches N takes any integer value not less than 2. This represents the weight of the input item in the nth branch, which follows a uniform distribution. and satisfy ; This represents the weight of the output item of the nth branch, which follows a uniform distribution. and satisfy ; and The relationship between them is considered a structural correlation, which is caused by two stages in DEA; the following optimization model aims to maximize the first stage under the Me measure. The expected efficiency of each failure mode, i.e. : (5)； in, It is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. It is also a set of trapezoidal random distributions, representing the observed value of the r-th triggering factor for the k-th failure mode; and Let r represent the multipliers of the r-th output term and the i-th input term, respectively. Represents a very small, pre-defined constant; constraint Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. This represents the expected level, which follows a uniform distribution. ;constraint This represents the relaxation constraints corresponding to all stages; constraint set This represents the set of relaxed constraints corresponding to the sub-stage. This represents the slack variables for all stages when evaluating failure mode k. Represents the slack variables corresponding to stage n; chance constraint set This indicates that policymakers are pessimistic about the spread of risk at all stages. This represents the level of conservatism, which follows a uniform distribution. Opportunity constraint set This indicates that policymakers are pessimistic about the risk propagation in the nth sub-phase. This represents the level of conservatism, which follows a uniform distribution. ; This represents the decision-maker's confidence level regarding imprecise parameters and / or opportunity constraints, and it follows a uniform distribution. ; (3) FMEA method based on network structure DEA; Risk Input Matrix Corresponding to the input items of DEA, branch input items , ,satisfy The total number of inputs L in the main branch can be any integer value not less than 3. This represents the weight of the input item in the l-th branch, which follows a uniform distribution. and satisfy Branch output items , ,satisfy The total number of outputs T can be any integer value not less than 5. Let represent the weight of the output item of the t-th branch, and satisfy . Risk output matrix , and For output items in DEA; , , and The relationship between these is considered a structural correlation, which is caused by the three stages in DEA; the following optimization model aims to maximize the first stage under the Me measure. The expected efficiency of each failure mode, i.e. : (6)； in, It is a set of trapezoidal random distributions, representing the observed values ​​of the i-th risk factor for the k-th failure mode. It is also a set of trapezoidal random distributions, representing the observed value of the r-th triggering factor for the k-th failure mode; and These are the multipliers of the five branch output terms of the r-th input term, in sequence. It is the multiplier of the i-th input item. and These are the multipliers of the three branch input terms of the i-th input term, in sequence. Represents a very small, pre-defined constant; constraint Ensure the expected efficiency of the k-th failure mode is [0,1]; chance constraint. and This indicates that policymakers are optimistic about the risk inputs at each stage of the DEA process. This represents the expected level, which follows a uniform distribution. Opportunity constraint set , and This indicates policymakers' pessimistic view of the risk transmission in the sub-phases, among which , and Representing the level of conservatism, the set of opportunity constraints This indicates that policymakers are pessimistic about the spread of risk at all stages. Indicates a conservative level. This represents the decision-maker's confidence level regarding imprecise parameters and / or opportunity constraints, and it follows a uniform distribution. .

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