An Improved FMEA Method Based on Fuzzy Hierarchical Analysis, Fuzzy VIKOR Method and PAM Clustering

Abstract

This invention discloses an improved FMEA method based on fuzzy hierarchical analysis, fuzzy VIKOR method, and PAM clustering. The method addresses shortcomings in traditional FMEA in three aspects: failure mode rating, risk factor weight determination, and failure mode importance ranking. First, triangular fuzzy numbers are used to make the failure mode evaluation more realistic. Second, comprehensive weights are applied to improve the weights of risk factors. Then, the fuzzy VIKOR method is used to improve the FMEA, and a comprehensive evaluation value combining group benefit value and individual regret value is used as the ranking basis. Finally, the PAM clustering algorithm is used to perform cluster analysis on failure modes based on failure mode RPN values ​​and comprehensive evaluation values. According to this invention, this method is applied to risk assessment of intelligent manufacturing systems, determining the importance of each failure mode in the intelligent manufacturing system and identifying key risk sources, thus verifying the effectiveness of the improved method.

Description

Technical Field

[0001] This invention relates to the technical field of reliability analysis methods, and in particular to an improved FMEA method based on fuzzy hierarchical analysis, fuzzy VIKOR method and PAM clustering. Background Technology

[0002] Failure Mode and Effects Analysis (FMEA), as a systematic, group-oriented, structured, proactive, and effective reliability analysis and risk assessment method, can be used to identify potential failure modes in systems, products, processes, and services, evaluate the impact of different failure modes, and allocate limited resources to improve or eliminate failure modes. It has been widely applied in enterprise quality management and risk analysis, proving effective in improving the reliability of enterprise products, services, and processes. In traditional FMEA, an expert panel scores the severity (S), occurrence (O), and detection (D) of failure modes based on actual conditions, and then multiplies these scores to calculate a risk priority number (RPN), which is used to rank the failure modes.

[0003] The VIKOR (Vlsekriterijumska Optimizacija I Kompromisno Resenje) method, also known as the multi-criteria compromise solution ranking method, is a compromise ranking method. It can overcome the shortcomings of the ranking method that approximates the ideal solution, while taking into account the maximization of group benefits and the minimization of individual regrets of opposing opinions. It can also fully consider the subjective preferences of decision-makers. The fuzzy VIKOR method is an extension of the VIKOR method in fuzzy environments and is mainly used to solve discrete fuzzy multi-criteria problems with incommensurability and conflicting criteria.

[0004] Clustering refers to dividing a set of data objects into multiple clusters composed of objects with high similarity according to certain rules. After decades of development, cluster analysis is now widely used in image processing, speech recognition, data analysis, and other fields. The PAM (Partition Around Medoids) clustering algorithm, also known as the K-center algorithm, uses a centroid to represent a cluster and can effectively handle outlier data points. The PAM clustering algorithm uses the distance from non-centroids to the cluster center to measure the clustering effect. While it is highly complex, its logic is relatively simple, making it suitable for clustering small databases.

[0005] Traditional FMEA has several shortcomings. First, it assumes that the three risk factors have equal weights and multiplies them, without considering the relative weights of each factor or the differences between them in different application contexts. Second, it uses precise numbers to assess the risk factors of failure modes, relying primarily on expert opinions, which are difficult for experts to provide in uncertain environments. Third, traditional FMEA may yield the same calculated RPN values, but the actual risk implications of the failure modes could be completely different, resulting in low accuracy in the ranking results. Summary of the Invention

[0006] To address the shortcomings of existing technologies, the present invention aims to provide an improved FMEA method based on fuzzy hierarchical analysis, fuzzy VIKOR method, and PAM clustering. This method, applied to risk assessment of intelligent manufacturing systems, determines the importance of each failure mode in the intelligent manufacturing system, identifies key risk sources, and verifies the effectiveness of the improved method. To achieve the above-mentioned objectives and other advantages of the present invention, an improved FMEA method based on fuzzy hierarchical analysis, fuzzy VIKOR method, and PAM clustering is provided, comprising the following steps:

[0007] S1. Assemble an FMEA team based on the requirements to analyze the operation and structure of the research object;

[0008] S2. The FMEA team, through brainstorming and literature review, identified m potential failure modes for the analysis object: FM1, FM2, ..., FM m ;

[0009] S3. Each expert in the FMEA team evaluates the three risk factors of each failure mode: severity (S), occurrence (O), and detectability (D). The risk factors of the failure mode are evaluated using five linguistic variables: very high (VH), high (H), moderate (M), very low (VL), and low (L). The evaluation is carried out according to the evaluation criteria corresponding to the linguistic variables (as shown in Table 1), and the expert scoring table is obtained.

[0010] S4. Based on the traditional FMEA scoring values ​​corresponding to the fuzzy language variables (as shown in Table 2), the expert scoring table is converted into the scoring values ​​of each expert for each risk factor in the traditional FMEA. The arithmetic mean of all expert scores for each risk factor is calculated as the value of each risk factor in the traditional FMEA and multiplied to obtain the risk priority number (RPN) of failure modes in the traditional FMEA.

[0011] S5. Based on Table 1, convert the linguistic variables in the expert scoring table into corresponding triangular fuzzy numbers, obtaining the triangular fuzzy numbers as follows: The FMEA team has a total of k members, and the weight of the p-th member is α. p The j-th risk factor corresponding to the i-th failure mode is The team's score for this item. in Thus, the comprehensive fuzzy evaluation matrix is ​​obtained.

[0012] S6. Determine the subjective weights of risk factors using fuzzy analytic hierarchy process (FAHP): The FMEA team conducted pairwise comparative evaluations of the three risk factors. The triangular fuzzy numbers corresponding to the linguistic variables of the risk factor evaluations are shown in Table 3, resulting in the pairwise comparison matrix. Each of them All are a set of triangular fuzzy numbers (a ij ',a ij ”,a ij ”'), Calculate the fuzzy comprehensive degree of the i-th object in Then calculate the two fuzzy synthesis degrees S1 and S2, where The degree of probability between the two when S2≥S1 It can be equivalently expressed as:

[0013]

[0014] Then calculate a triangular ambiguity number S that is greater than k triangular ambiguity numbers S. i The probability degree V(S≥S1,S2,...,Sk) of (i=1,...,k) k )=V[(S≥S1)&(S≥S2)&...&(S≥S k )]=minV(S≥S i ) can then be used to obtain d'(A i )=minV(S i ≥S j For each risk factor i = 1, 2, 3, j = 1, 2, 3, i ≠ j, after normalization, the subjective weights of each risk factor are obtained:

[0015] Table 1. Evaluation criteria and corresponding triangular fuzzy numbers for linguistic variables.

[0016]

[0017] Table 2. Fuzzy linguistic variables corresponding to traditional FMEA scores.

[0018]

[0019] Table 3. Triangular Fuzzy Numbers Corresponding to Linguistic Variables in Risk Factor Evaluation

[0020]

[0021] Secondly, the objective weights of the risk factors are determined according to the entropy weight method: the comprehensive fuzzy evaluation matrix is ​​then divided according to... Defuzzification yields a definite confidence matrix X = (x ij ) m×3 ; Calculate the information entropy of the j-th risk factor. in: k = 1 / lnm; therefore, its objective weight is

[0022] Finally, the comprehensive weight of the risk factors is determined using the multiplicative synthesis method: The comprehensive weight of the j-th risk factor is obtained by using the multiplicative synthesis method.

[0023] S7. The optimal and worst values ​​of each evaluation index, Calculate the maximum group effect value for each failure mode. and minimum personal regret value Then obtain the comprehensive evaluation value. in v is the weight for maximizing group utility;

[0024] S8, Failure Mode After defuzzing the values, the S, R, and Q values ​​are obtained and sorted in ascending order to generate three sequences. After obtaining the sorting results, the following two conditions must be met: (1) Q A (2)-Q A (1)≥1 / (m-1), where Q A (1) is the Q value of the optimal solution in the sorting results, Q A (2) is the Q value of the second-best solution in the ranking result, and m is the total number of failure modes; (2) the S value or R value of the best solution must also be optimal, so as to ensure that the solution with the smallest Q value is the best solution; if the above conditions cannot be satisfied at the same time, a compromise solution can be obtained, and there are the following two cases: (3) if condition (1) is satisfied, a set of compromise solutions is obtained: A(1), A(2), which are all the most important failure modes; (4) if condition (1) is not satisfied but (2) is satisfied, a set of compromise solutions is obtained: A(1), ..., A(M), which are all the most important failure modes, where M is the value of the solution according to the formula Q. A (M)-Q A (1) The maximum value of M is determined by <1 / (m-1);

[0025] S9. Using the RPN value and Q value as two feature values ​​describing the i-th failure mode, we obtain the description matrix F.

[0026] Input: Expected number of clusters k', a dataset containing m data objects;

[0027] Output: k' clusters that minimize the sum of dissimilarity between all points and their nearest centroid;

[0028] S91. Randomly select k' points from m data objects as the initial center set;

[0029] S92. Calculate the distance from each non-representative object to each center point and assign it to the cluster closest to it. This invention uses Euclidean distance d. ij =[(x i -x j ) 2 +(y i -y j ) 2 ] 1 / 2 Perform distance calculations;

[0030] S93, For each non-central object O h The following process is executed sequentially: using the current point O h Replace one of the center points O i And calculate the total cost function generated by the replacement. If it is less than 0, then replace; otherwise, do not replace the center point.

[0031] The cost function is based on each non-selected object O j Calculations are performed using different formulas for the following four scenarios:

[0032] First scenario: O j Currently belongs to O i In the cluster represented by points, O j2 For O j The second closest point to the center, and O j Away from O j2 O h Near, at this time C jih =d(O j O j2 )-d(O j O i );

[0033] Second scenario: O j Currently belongs to O i In the cluster represented by points, O j2 For O j The second closest point to the center, and O j Away from O h O j2 Near, at this time C jih =d(O j O h )-d(Oj O i );

[0034] The third scenario: O j Currently belongs to another non-O i Instead, O j2 In the cluster representing points, and O j Away from O j2 O h Near, at this time C jih =0;

[0035] Fourth case: O j Currently belongs to another non-O i Instead, O j2 In the cluster representing points, and O j Away from O h O j2 Near, at this time C jih =d(O j O h )-d(O j O j2 );

[0036] S94. Obtain a set of k' center points, and re-divide all objects into the clusters closest to them according to the minimum distance principle;

[0037] S10. Analyze the failure modes of the research objects based on the ranking and clustering results, and allocate resources and manage risks for the prevention and monitoring of different failure modes according to the categories and rankings.

[0038] Compared with existing technologies, the advantages of this invention are as follows: First, it uses triangular fuzzy numbers to make the evaluation of failure modes more realistic. Second, it improves the weighting of risk factors using comprehensive weighting. Third, it improves FMEA using fuzzy VIKOR and combines the comprehensive evaluation value of group benefit value and individual regret value as the ranking basis. Finally, it uses the PAM clustering algorithm to perform cluster analysis on failure modes based on the failure mode RPN value and the comprehensive evaluation value. The reliability and risk assessment analysis of an intelligent manufacturing system using this method verifies its effectiveness and applicability. Attached Figure Description

[0039] Figure 1 This is a schematic diagram of the method for improving FMEA based on fuzzy hierarchical analysis, fuzzy VIKOR method and PAM clustering according to the present invention.

[0040] Figure 2 This is a PAM clustering result diagram of failure modes in intelligent manufacturing systems based on the improved FMEA method of fuzzy hierarchical analysis, fuzzy VIKOR method and PAM clustering according to the present invention. Detailed Implementation

[0041] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0042] Reference Figure 1-2 A method for improving FMEA based on fuzzy hierarchical analysis, fuzzy VIKOR method and PAM clustering, including:

[0043] This embodiment analyzes and evaluates Intelligent Manufacturing System (IMS).

[0044] In this embodiment, the FMEA team consisted of four experts: Expert 1, IMS implementation consultant K1; Expert 2, IT consulting expert K2; Expert 3, IT project senior manager K3; and Expert 4, university scholar K4 in the field of intelligent manufacturing. First, the Work Breakdown Structure (WBS) method was used to analyze the intelligent manufacturing system from top to bottom. Combined with brainstorming, expert interviews, and a review of relevant literature, risks were identified in the intelligent manufacturing system. It was initially divided into a three-layer structure: Enterprise Resource Planning (ERP), Manufacturing Execution System (MES), and Factory Control Layer. Each layer was then further analyzed. Finally, the common risks of the three layers were analyzed and summarized separately to determine the failure modes of the intelligent manufacturing system and analyze their impact. For ease of description, the failure modes are denoted by numbers FMi (i = 1, 2, ..., 30). The obtained failure modes and their consequences and causes are shown in Tables 4, 5, 6, and 7.

[0045] Each expert on the FMEA team evaluated the risk factors of the failure mode using five linguistic variables based on the evaluation criteria and corresponding triangular fuzzy numbers in Table 1. The resulting expert scoring table is shown in Table 8. Based on the traditional FMEA scoring values ​​corresponding to the fuzzy linguistic variables (as shown in Table 2), the expert scoring table was transformed into scores for each risk factor by each expert in traditional FMEA. The arithmetic mean of all expert scores for each risk factor was calculated as the numerical value for each risk factor in traditional FMEA, and these values ​​were multiplied to obtain the RPN value of the failure mode in traditional FMEA, as shown in Table 9.

[0046] Table 4 ERP Failure Modes, Consequences, and Causes

[0047]

[0048] Table 5 MES Failure Modes, Consequences, and Causes

[0049]

[0050] Continued from Table 5

[0051]

[0052] Continued from Table 5

[0053]

[0054] Table 6 Failure Modes of the Factory Control Layer and Their Corresponding Consequences and Causes

[0055]

[0056] Table 7 Common Failure Modes and Their Corresponding Consequences and Causes

[0057]

[0058] Table 8 Expert Scoring Sheet

[0059]

[0060] Continued from Table 8

[0061]

[0062] Table 9 Traditional FMEA Scoring Results

[0063]

[0064] Continued from Table 9

[0065]

[0066] The scores from various experts were collected and summarized. The linguistic variables in the expert scoring table were then converted into triangular fuzzy numbers according to the transformation relationships in Table 1. Since each expert scored from different perspectives, this embodiment sets each expert to have the same weight: α1 = 0.25, α2 = 0.25, α3 = 0.25, α4 = 0.25, to obtain more accurate evaluation results. The scores of each risk factor for each failure mode were summarized and processed. The weighted average of the scores from the four experts yielded the comprehensive fuzzy evaluation matrix, as shown in Table 10.

[0067] Table 10 Comprehensive Fuzzy Evaluation Matrix

[0068]

[0069] When determining the weights of risk factors, the subjective weights are first determined using the fuzzy analytic hierarchy process (AHP). After four experts score the failure modes, the risk factors are compared pairwise using the scoring criteria mentioned in Table 3. The scores are then aggregated and weighted averaged for each group. Similarly, each expert is assigned the same weight, resulting in the final subjective evaluation matrix of risk factors, as shown in Table 11. The calculated values ​​are: V(S1≥S2)=1, V(S1≥S3)=1, V(S2≥S1)=0.355, V(S2≥S3)=0.975, V(S3≥S1)=0.334, V(S3≥S2)=1, d′(A1)=1, d′(A2)=0.355, d′(A3)=0.334. After normalization, the subjective weights are: Secondly, the objective weights are calculated using the entropy weight method. The composite weight of risk factors can be calculated using the multiplicative synthesis method.

[0070] Table 11 Weighted average of risk factors

[0071]

[0072] Based on the comprehensive fuzzy evaluation table, the failure modes are ranked using the fuzzy VIKOR method. First, the fuzzy optimal and worst values ​​of each risk factor are determined, and then their fuzzy maximum group benefit values ​​are calculated. and minimum individual regret value according to Value calculation for each failure mode The values ​​are then defuzzified to obtain the explicit values ​​of S, R, and Q, which are then sorted in ascending order, resulting in Table 12. Among these, Q... A (2)-Q A (1) = 0.186 ≥ 1 / 29, indicating that FM20 satisfies condition (1), while the S value of FM20 does not satisfy condition (2). Therefore, A(1) and A(2) are the compromise solutions to this problem, i.e., FM20 and FM26 are the compromise solutions to this problem. Since the failure modes need to be ranked, FM20 is in the first or second position in the ranking of S, R, and Q values, while FM26 ranks second in Q value but ranks lower in S and R values. Therefore, in this case, the failure modes are ranked as follows: FM20 first and FM26 second.

[0073] Table 12 specifies the Q, S, and R values ​​for each failure mode.

[0074]

[0075] The description matrix F of the failure mode is obtained based on the RPN value of the failure mode and the comprehensive evaluation value Q.

[0076]

[0077] Based on the actual situation, failure modes are divided into three categories: relatively important, moderately important, and relatively unimportant. Therefore, let k' = 3. The PAM clustering method is used to cluster the failure modes according to the description matrix F, such as... Figure 2 As shown.

[0078] Depend on Figure 2 It can be seen that "Cluster1" includes FM5, FM6, FM7, FM8, FM9, FM10, FM16, FM19, FM20, FM22, FM23, FM26, FM27, and FM28. This type has a high RPN value and a low Q value, indicating that it has a strong impact on the operation of the intelligent manufacturing system, whether under the traditional FMEA method or the improved FMEA method. The greatest resources should be invested in prevention and monitoring. After tracing the causes of failure modes, the main causes are hardware failures and network transmission errors. In the future, appropriate improvements should be made by enhancing employees' backup awareness and regularly testing and optimizing hardware. Cluster 2, comprising FM1, FM2, FM3, FM4, FM13, FM15, FM17, FM21, FM24, and FM25, exhibits moderate RPN and Q values ​​for failure modes, indicating a less significant impact on the intelligent manufacturing system compared to the previous cluster. Therefore, excessive resources are not required for prevention and monitoring of this cluster. The primary causes of failures in Cluster 2 are internal software errors, human error, and management-level risks. To prevent internal software errors, regular software upgrades and timely backups are necessary. Human error can be mitigated by enhancing employee awareness of fault prevention. Management-level risks can be addressed by requiring more cautious decision-making from management. Cluster 3, comprising FM11, FM12, FM14, FM18, FM29, and FM30, has very low RPN and high Q values, indicating a relatively weak impact on the intelligent manufacturing system. Appropriate resources are sufficient for prevention and monitoring of this cluster. The main reasons for cluster 3 include management-level risks and natural disasters that are difficult to classify. Natural disasters are rare, and the management process can minimize the investment of resources in them, but it is still necessary to have contingency plans in place so that timely responses can be made to reduce losses when they occur.

[0079] In summary, "Cluster1" is a relatively important failure mode cluster, "Cluster2" is a failure mode cluster with moderate impact, and "Cluster3" is a relatively unimportant failure mode cluster. This clustering result is quite reasonable.

[0080] Through case studies, this invention is applicable and effective in reliability analysis.

[0081] The number of devices and processing scale described herein are for the purpose of simplifying the description of the invention, and applications, modifications and variations thereof will be apparent to those skilled in the art.

[0082] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. They can be applied to various fields suitable for the present invention. For those skilled in the art, other modifications can be easily made. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and illustrations shown and described herein.

Claims

1. A method for improving FMEA based on fuzzy AHP, fuzzy VIKOR and PAM clustering, characterized in that, It comprises the following steps: S1, according to the demand, the FMEA team is organized, and the operation and structure of the research object are analyzed; S2, the FMEA team determines a plurality of potential failure modes of the analysis object through brainstorming and consulting materials; S3, each expert in the FMEA team evaluates the severity, occurrence and detectability of each failure mode, and evaluates the risk factors of the failure mode through five language variables of very high, high, medium, very low and low, and obtains an expert scoring table according to the evaluation standard corresponding to the language variable; S4, according to the traditional FMEA scoring value corresponding to the fuzzy language variable, the expert scoring table is converted into the scoring value of each risk factor in the traditional FMEA, and the arithmetic mean of the scoring values of all experts for each risk factor is calculated as the value of each risk factor in the traditional FMEA and multiplied to obtain the risk priority number RPN of the failure mode in the traditional FMEA; S5, according to the triangular fuzzy number corresponding to the language variable, the language variable in the expert scoring table is converted into the corresponding triangular fuzzy number; S6, the comprehensive weight of the risk factor is determined according to the fuzzy analytic hierarchy process and the entropy weight method; S7, the maximum group effect value and the minimum individual regret value of each failure mode are calculated from the optimal value and the worst value of each evaluation index, and then the comprehensive evaluation value is obtained; S8. After defuzzifying the maximum group effect value, minimum individual regret value, and comprehensive evaluation value of the failure mode, obtain the S, R, and Q values ​​and sort them in ascending order to generate three sequences. After obtaining the sorting results, the following two conditions must be met: (1) Q A (2)-Q A (1)≥1 / (m-1), where Q A (1) is the Q value of the optimal solution in the sorting results, Q A (2) is the Q value of the second-best solution in the sorting result, and m is the total number of failure modes; (2) the S value or R value of the best solution must also be optimal, so as to ensure that the solution with the smallest Q value is the best solution; if the above conditions cannot be satisfied at the same time, a compromise solution can be obtained. S9, the RPN value and the Q value are taken as two characteristic values for describing the i-th failure mode, a description matrix F is obtained, and the PAM clustering method is used to cluster into k' classes; S10, according to the sorting result and the clustering result, the failure modes of the research object are analyzed, and the resource allocation for prevention and monitoring and risk management of different failure modes are allocated according to the class and the order.

2. The method for improving FMEA based on fuzzy AHP, fuzzy VIKOR and PAM clustering of claim 1, wherein, The triangular fuzzy number in step S5 is There are k members in the FMEA team, and the weight of the pth member is α p The jth risk factor corresponding to the ith failure mode is The score of the team for the item is Wherein Thus, the comprehensive fuzzy evaluation matrix is obtained 3. The method for improving FMEA based on fuzzy AHP, fuzzy VIKOR and PAM clustering of claim 1, wherein, The method for determining the comprehensive weight of the risk factor according to the fuzzy analytic hierarchy process and the entropy weight method in step S6 comprises the following steps: S61、FMEA team pairwise comparison evaluation of three risk factors, get pairwise comparison matrix wherein each is a set of triangular fuzzy numbers, calculate the fuzzy comprehensive degree of the ith object, then calculate the possibility degree between two fuzzy comprehensive degrees S1, S2, S2≥S1, then calculate the possibility degree V of one triangular fuzzy number S greater than k triangular fuzzy numbers S i , i=1,...,k, after normalization, get the subjective weight of each risk factor; S62, determining the objective weight of the risk factor according to the entropy weight method: the comprehensive fuzzy evaluation matrix is de-fuzzified to obtain a clear confidence matrix, the information entropy of the j-th risk factor is calculated, and thus the objective weight thereof is obtained; S63, determining the comprehensive weight of the risk factor according to the multiplication synthesis method: the multiplication synthesis method is used to determine the comprehensive weight, and the comprehensive weight of the j-th risk factor is obtained.

4. The method for improving FMEA based on fuzzy AHP, fuzzy VIKOR and PAM clustering of claim 1, wherein, In step S8, there are two cases: (3) if condition (1) is satisfied, a set of compromise solutions is obtained: A(l), A(2), i.e. both are the most important failure modes; (4) if condition (1) is not satisfied but condition (2) is satisfied, a set of compromise solutions is obtained: A(l),..., A(M), i.e. all are the most important failure modes, where M is the maximum value of M determined according to the formula Q A (M) - Q A (1) <1 / (m-1) determined maximum M value.

5. The method for improving FMEA based on fuzzy AHP, fuzzy VIKOR and PAM clustering of claim 1, wherein, The PAM clustering method in step S9 comprises the following steps: S91, randomly selecting k' points in m data objects as initial center sets; S92, calculating the Euclidean distance of each non-representative object to each center point, and distributing it to the nearest cluster; S93. For each non-central object O h , the following procedure is executed in turn: replace one of the central points O h with the current point O i , and calculate the total cost function resulting from the replacement, and if it is less than 0, replace the central point, otherwise do not replace the central point; S94, obtaining a set of k' center points, and reclassifying all objects into the nearest cluster according to the minimum distance principle.

6. The method for improving FMEA based on fuzzy AHP, fuzzy VIKOR and PAM clustering of claim 5, wherein, The cost function depends on each non-selected object O j The calculation is performed by different formulas for the following four cases: The first case: O j Currently belongs to O i For the cluster of representative points, O j2 For O j The second center point, and O j Away from O j2 O h Close, at this time C jih =d(O j ,O j2 )-d(O j ,O i ); Second case: O j Current belongs to O i For the cluster of representative points, O j2 For O j The second center point of proximity, and O j Away from O h O j2 Close, at this time C jih =d(O j ,O h )-d(O j ,O i ); Third case: O j Currently belongs to another non-O i But O j2 In the cluster of representative points, and O j From O j2 O h Close, at this time C jih = 0; Fourth case: O j Currently belongs to another non-O i But O j2 is a representative point in the cluster, and O j is the center of the cluster h is closer to O j2 than O jih = d(O j , O h ) - d(O j , O j2 ).

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