DEMATEL-BWM fusion assembly sequence evaluation method based on Bayesian probability model

Abstract

The invention relates to a DEMATEL-BWM fusion assembly sequence evaluation method based on a Bayesian probability model. Assembly sequence evaluation is carried out on a complex mechanical object. Collecting data, establishing a direct influence matrix, normalizing the direct influence matrix into a standard influence matrix, performing Bayesian updating, and performing exponential transformation processing on an output posteriori mean value to obtain a preference matrix; determining optimal and worst elements, performing pairwise comparison to obtain optimal and worst vectors, modeling the optimal and worst vectors as polynomial distribution, and modeling the output weight vectors as Dirichlet distribution; processing input of multiple decision groups through a Bayesian hierarchical model, establishing a joint probability model, and performing decomposition through conditional independence and probability chain rules; the prior distribution and observation data are used, posterior distribution is obtained through Bayesian updating, and a confidence level is used to generate a weighted directed graph; and outputting the aggregation weight, and evaluating the assembly sequence. The problem that an assembly sequence evaluation method is low in efficiency is solved, the inconsistency of subjective judgment is reduced, and the method is more fault-tolerant, concise and efficient for deviation.

Description

Technical Field

[0001] The present invention relates to an assembly sequence technology, and particularly to a DEMATEL-BWM fusion assembly sequence evaluation method based on a Bayesian probability model. Background Art

[0002] As the last link in the machining and manufacturing process of a machine, assembly directly affects the final performance and lifespan of a product. An assembly sequence is the assembly order planned for an assembly body according to certain constraint rules, and is an important content in the field of assembly sequence planning. For a product, there are many different feasible assembly sequences, and different assembly sequences determine the assembly cost, quality, efficiency, and safety. As the complexity of the assembly body increases and the number of parts included increases, the number of feasible assembly sequences also increases exponentially. How to select a most reasonable solution that can meet the requirements in terms of efficiency, cost, accuracy, etc. among numerous sequences depends crucially on whether all the obtained solutions can be comprehensively and effectively evaluated.

[0003] After establishing an evaluation system, current assembly sequence evaluation methods mainly use fuzzy processing methods to calculate index weights, including the analytic hierarchy process, principal component analysis, entropy weight method, weight distribution method, etc. However, when the number of selected indicators is large, such as in the analytic hierarchy process, a very large and complex judgment matrix needs to be constructed, and pairwise comparisons of many indicators need to be carried out, which will lead to difficult determination of weights and weight deviations due to the mutual dependence relationship between indicators. The problems are as follows:

[0004] 1. Directly using survey data for analysis and calculation is not accurate enough;

[0005] 2. When the number of people from whom data is sourced is too large, the calculation is difficult;

[0006] Current assembly sequence evaluation methods generally use the subjective experience of experts to compare indicators and allocate weights, and the indicators are compared multiple times, with a complicated process. Therefore, there are various problems, including a large and complex judgment matrix construction, strong subjectivity, obvious expert judgment deviations, insufficient handling of the mutual dependence relationship between indicators, low calculation efficiency, and difficulty in dealing with large-scale sequences, etc., and it is difficult to meet the requirements of modern manufacturing for rapid decision-making and real-time optimization. Summary of the Invention

[0007] Aiming at the problems of complex process, large subjective deviation, and low efficiency in current assembly sequence evaluation methods, a DEMATEL-BWM fusion assembly sequence evaluation method based on a Bayesian probability model is proposed.

[0008] The technical solution of the present invention is as follows:

[0009] A DEMATEL-BWM fusion assembly sequence evaluation method based on Bayesian probability model is used to evaluate the assembly sequence for complex mechanical objects; the decision-making trial and evaluation laboratory (DEMATEL) method is used to collect data according to the established assembly sequence evaluation system, establish a direct influence matrix, normalize it into a standardized influence matrix to eliminate the influence of dimension, use the centrality as the observed data for Bayesian updating, perform exponential transformation on the output posterior mean to obtain a preference matrix; determine the best and worst elements according to the preference matrix, conduct pairwise comparisons, obtain the best-worst vector and model it as a multinomial distribution, and then model the output weight vector as a Dirichlet distribution; process the inputs of multiple decision-making groups through a Bayesian hierarchical model, establish a joint probability model, and decompose it through conditional independence and probability chain rules; use the prior distribution and observed data to obtain the posterior distribution through Bayesian updating, generate a weighted directed graph using the confidence level to show the relationship between criteria; output the aggregated weight to evaluate the assembly sequence.

[0010] Further, it includes the following steps:

[0011] Step 1: For the machinery with complex assembly process, select constraints according to actual requirements, construct a constraint system for evaluating the assembly sequence; according to the constructed constraint system, score each element in the system using a database or technician opinions to obtain a direct influence matrix.

[0012] Step 2: Normalize the direct influence matrix to obtain a standardized relationship influence matrix, then multiply it by the inverse matrix of (the identity matrix - the standardized relationship influence matrix) to obtain a comprehensive influence matrix, and calculate the centrality, that is, the total correlation strength of the factor in the system; conduct Bayesian updating, first obtain the prior mean and prior covariance through the direct influence matrix, then observe the centrality and calculate the posterior distribution parameters to obtain the posterior mean.

[0013] Step 3: Use the exponential approximation method to process all the posterior means in the decision-making group to obtain a data display of the importance degree of indicators, and further process it to obtain a preference ranking applicable to the best-worst method; regard the input items of the best-worst method: the preference degree of the best to others and the preference degree of others to the worst as probability distributions, model them using a multinomial distribution, and represent the output weight vector of the best-worst method with a Dirichlet distribution.

[0014] Step 4: Establish a Bayesian hierarchical model, including an input layer, an individual weight layer, an aggregated weight layer, and a concentration parameter part, so as to model the prior distribution of the aggregated weight, set the concentration parameter using a gamma distribution, indicating the degree to which the individual weight is close to the aggregated weight.

[0015] Step 5: Establish a joint probability model, combine the prior of the aggregated weight, the conditional distribution of the weight of each decision maker, and the likelihood of the observed data to calculate the joint probability distribution.

[0016] Step 6: When calculating the posterior distribution, use the MCMC (Markov Chain Monte Carlo) method for posterior distribution sampling. The tool used is the JAGS sampler. Introduce a credible ranking, and evaluate the advantages and disadvantages of pairwise criteria by calculating the confidence level. Specifically, for each pair of criteria, calculate through the posterior distribution, and then estimate the confidence level through the mean value of MCMC sampling. Finally, output the weight distribution of each decision group, the aggregated weight distribution, the credible ranking and confidence level between criteria, and display them as a weighted directed graph;

[0017] Step 7: Establish an assembly sequence evaluation function according to the requirements of the actual assembly, calculate using the output aggregated weight, and finally obtain the score of the assembly sequence, so as to select the optimal assembly sequence.

[0018] Furthermore, it specifically includes the following steps:

[0019] Step 1: For machinery with complex assembly processes, construct a constraint system for evaluating assembly sequences according to actual production requirements. Based on the constructed constraint system, analyze using the opinions of multiple decision-makers to obtain multiple direct influence matrices, and conduct random grouping;

[0020] Step 2: Normalize the direct influence matrix to obtain a canonical relationship influence matrix, then multiply it by the inverse matrix of (the identity matrix - the canonical relationship influence matrix) to obtain a comprehensive influence matrix, and calculate the centrality. Use the direct influence matrix for Bayesian update to obtain the prior mean and prior covariance, and then observe the centrality and calculate the posterior distribution parameters to obtain the posterior mean;

[0021] Step 3: Use the exponential approximation method to process the posterior mean to obtain the ranking of index importance degrees. The formula is as follows:

[0022]

[0023] where max_val is the maximum value in the posterior mean, min_val is the minimum value in the posterior mean, a is the base of the exponent, and x is the baseline offset;

[0024] Each decision group obtains a set of preference rankings, and aggregates them to obtain a preference matrix applicable to the best-worst method. Regard the preference degree of the best to others (A B ) and the preference degree of others to the worst (A W ) in the input items of the best-worst method as probability distributions, and model them using the multinomial distribution. The probability mass function of A W is as follows:

[0025]

[0026] Because A BCompared with A W Therefore, inverse modeling of weights is used, that is:

[0027]

[0028] For the output weight vector w of the best-worst method, since the weight vector must satisfy the properties of non-negativity and sum to one, a Dirichlet distribution is used for modeling; given the parameter ∝∈R n , the Dirichlet distribution of the weight w is defined as:

[0029]

[0030] Step 4: Model the prior distribution of the aggregated weights through a Bayesian hierarchical model; the Bayesian hierarchical model includes an input layer, an individual weight layer, an aggregated weight layer, and a concentration parameter; the input layer is: assume there are K decision groups, and within each set of preference rankings obtained for each decision group, compare to obtain the best criterion to other criteria and other criteria to the worst criterion These two pairwise comparison vectors follow a multinomial distribution, that is:

[0031]

[0032] Each w in the individual weight layer k is modeled using a reparameterized Dirichlet distribution, that is:

[0033] w k ∣w agg ~Dirichlet(γ·w agg )

[0034] In the group weight layer, w agg is the mean of the individual weights and is modeled using an uninformative Dirichlet distribution, that is:

[0035] w agg ~Dirichlet(α), α = 1

[0036] The concentration parameter γ is modeled using a gamma distribution, that is:

[0037] γ~Gamma(a,b)

[0038] In the hierarchical model, γ represents the degree to which the individual weights are close to the aggregated weights, controlling the closeness of w k and w agg ;

[0039] Step 5: Calculate the joint probability distribution by establishing a joint probability model, combining the prior P(w agg ) of the aggregated weights, the conditional distribution P(w k ∣wagg ) Likelihood of Observation Data Combined, in line with Bayes' rule and the conditional independence assumption, the formula is as follows:

[0040]

[0041] Considering the independence of variables, applying Bayes' rule to it and decomposing it through conditional independence and the probability chain rule, the calculation formula is as follows:

[0042]

[0043] Step 6: Calculate the posterior distributions of the aggregation weight, individual weight, and concentration parameter, perform posterior distribution sampling using the Markov chain Monte Carlo method, with the tool JAGS, and introduce a credible ranking; first, define the credibility O:

[0044] O = (c i , c j , R, d)

[0045] where R is the relationship between the standard c i and c j , that is, >, < or =; d ∈ [0, 1] represents the confidence level of the relationship;

[0046] Evaluate the advantages and disadvantages of pairwise criteria by calculating the confidence level of the criterion weights. The method is: by designing a new Bayesian test, based on this test, the confidence level of each credible ranking can be found; this test is based on the posterior distribution of w agg ; calculate the confidence level that c i is superior to c j as:

[0047]

[0048] where P(w agg ) is the posterior distribution of w agg , if holds, then I is 1, otherwise it is zero; this kind of integral can be approximated by the samples obtained by the MCMC method; assume Q samples are obtained from the posterior distribution, and the confidence level can be calculated as:

[0049]

[0050] where is the q-th sample of w agg from the MCMC samples; for each pair of criteria, a confidence level that one is superior to the other can be calculated; obviously, P(c i > c j ) + P(c j > c i) = 1; Therefore, if and only if P(c i > c j ) > 0.5, c u is more important than c j ; Therefore, a threshold of 0.5 is applied to the confidence ranking;

[0051] Specifically, for each pair of criteria, the posterior distribution is calculated, and then the confidence level is estimated by the mean of MCMC sampling. The final output is the weight distribution, aggregated weight distribution, credible ranking and confidence level between criteria for each decision group, and is presented as a weighted directed graph;

[0052] Step 7: Establish an evaluation function using the constraint system obtained by aggregating the weights of the output, calculate each assembly sequence, and finally obtain a score to select the optimal assembly sequence.

[0053] Furthermore, in Step 1, the constraint system includes multiple modules such as the final assembly accuracy, efficiency, cost, and safety of the assembly process, which are modified and refined according to the actual assembly.

[0054] Furthermore, in Step 2, the Bayesian update formula is as follows:

[0055]

[0056] where μ post is the posterior mean, Σ0 is the prior covariance matrix, Σ noise is the noise covariance matrix, μ0 is the prior mean, and M is the centrality.

[0057] Furthermore, in Step 2, the prior distribution of the Bayesian update definition factor is a multivariate Gaussian distribution. The prior mean is set as a zero vector to represent an unbiased hypothesis; the prior covariance is set as the identity matrix to represent that the factors are independent and have the same variance; the noise covariance is set as 0.1 times the prior covariance.

[0058] Furthermore, in Step 3, the purpose of the exponential approximation method is to convert the difference relationship data into discrete priorities for easy comparison; the process is as follows: First, determine the maximum and minimum values, then calculate the scaling base a and the logarithmic scale starting point x, compress the ratio into 8 equally spaced logarithmic intervals, and finally output the priority in integer form.

[0059] Furthermore, in Step 5, the nodes of the weighted directed graph represent decision criteria, and the size of the nodes represents the uncertainty of the ranking; the edges represent the ranking relationship between the criteria, and the labels of the edges show the probability of the ranking.

[0060] Further, for the posterior distribution sampling in step 6, these samples represent the posterior distributions of the weight vector w and the aggregated weight vector; the collected samples are reshaped and their average value is calculated to obtain the aggregated weight vector, which represents the relative importance of the criteria.

[0061] Further, it is characterized in that for the assembly sequence evaluation function in step 7, according to the actual number of indicators, the corresponding calculated weights are allocated, and the formula is as follows:

[0062] F = ω1·n1 + ω2·n2 + … + ω n ·n n

[0063] where F is the final score of the sequence, n represents the measurement data corresponding to each indicator. For example, concentricity and parallelism can be selected for the assembly accuracy aspect; the number of tool changes and the number of direction changes can be selected for the assembly efficiency aspect, etc.; ω represents the weights corresponding to each indicator.

[0064] The beneficial effects of the present invention are as follows:

[0065] (1) For the traditional assembly sequence evaluation method, when multiple decision-makers make pairwise comparisons of multiple criteria, inconsistent judgments are likely to occur. This method ensures the optimal consistency of the comparison matrix through a mathematical optimization model, reduces the inconsistency in subjective judgments, is more tolerant of deviations, and is simple and efficient.

[0066] (2) Through the decision-making trial and evaluation laboratory method improved by Bayesian, the numerous direct influence matrices are grouped and calculated, and then the results are summarized to obtain the preference matrix required by the best-worst method. And through the design of the algorithm, there is no need to re-model when changing the number of evaluation indicators, which simplifies the steps.

[0067] (3) The traditional method for calculating the weights of the assembly sequence system requires pairwise comparisons of each indicator, with a large amount of calculation. This method only needs to make two groups of judgments, greatly reducing the calculation workload and being applicable to multi-criteria decision-making problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 is the implementation flowchart of the assembly sequence evaluation method of the present invention;

[0069] Figure 2 is the example diagram of the fusion algorithm of the present invention;

[0070] Figure 3 is the example diagram of the credible ranking weighted directed graph of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0071] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is implemented on the premise of the technical solution of the present invention, and gives the detailed implementation manner and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0072] Bayesian-BWM (Bayesian Best-Worst Method) is a decision-making method based on Bayesian theory, mainly used to solve multi-attribute decision-making problems. Especially when dealing with multiple alternatives and multiple evaluation criteria, it is used to evaluate and select the optimal alternative.

[0073] 1. What is BWM (Best-Worst Method)?

[0074] BWM is a method widely used in multi-criteria decision-making. Its core idea is to rank the relative importance of each evaluation criterion by comparing the "best" and "worst" criteria. The specific steps are as follows: Select the most important criterion (Best): Select the one you think is the most important from all the evaluation criteria. Select the least important criterion (Worst): Select the one you think is the least important from all the evaluation criteria. Compare the relative importance of other criteria with Best and Worst: For each evaluation criterion, make the following two comparisons: The relative importance with Best: That is, the priority of other criteria compared with the most important criterion. The relative importance with Worst: That is, the priority of other criteria compared with the least important criterion. In this way, BWM can estimate the importance ranking of each criterion with relatively few judgments. The advantage of this method is that it is relatively intuitive and avoids complex many-to-many comparisons.

[0075] 2. What is Bayesian-BWM?

[0076] Bayesian-BWM is based on the traditional BWM method and adds the Bayesian method to handle the uncertainty problem in decision-making. The core of Bayesian theory is to use prior information and newly obtained data to update the probability estimate of the occurrence of an event. Simply put, Bayesian-BWM is to add the idea of probability theory to the framework of BWM, considering uncertainty and ambiguity.

[0077] Features of Bayesian-BWM:

[0078] Introduces uncertainty: In BWM, quantitative judgments are usually required, such as determining how much more important one criterion is compared to another. In reality, people's judgments are often fuzzy and may not be able to give precise numerical values. Bayesian-BWM can model these fuzzy judgments by introducing probability distributions. Use of prior information: The Bayesian approach emphasizes updating the decision-making process through existing information. Bayesian-BWM not only relies on the decision-maker's subjective judgment but also takes into account existing relevant data or historical information, making the decision-making process more scientific. Uncertainty of decisions: In reality, the results of many decisions are uncertain. Bayesian-BWM quantifies this uncertainty, making the decision results more robust (able to make relatively reasonable decisions even in situations with high uncertainty).

[0079] Specific operations: 1. Determine the priority of evaluation criteria: Similar to BWM, first select the most important criterion (Best) and the least important criterion (Worst). 2. Bayesian inference: Consider the judgment of the relative importance of each criterion as an event with uncertainty and use the Bayesian method to update it. That is, based on existing knowledge and data, conduct a probability assessment of the importance of different criteria. If there is prior knowledge (such as previous decision-making experience or research data in related fields), these information can be used as prior probabilities and input into the Bayesian model. By observing new data (such as the decision-maker's judgment or actual situation), update the posterior probability. 3. Calculate the importance ranking of criteria: Through the results updated by Bayesian inference, the importance of each criterion can be more accurately evaluated, thus obtaining a more reasonable ranking.

[0080] Source: Title: Bayesian best-worst method: A probabilistic group decision making model.

[0081] (https: / / www.sciencedirect.com / science / article / pii / S0305048318313963)

[0082] A method for evaluating assembly sequences by integrating the decision-making trial and evaluation laboratory method and the best-worst method based on a Bayesian probability model, including the following steps:

[0083] Step 1: For machinery with complex assembly processes, select constraints according to actual requirements and construct a constraint system for evaluating assembly sequences. According to the constructed constraint system, use a database or technician opinions to score each element in the system to obtain a direct influence matrix;

[0084] Step 2: Normalize the direct influence matrix to obtain the standardized relationship influence matrix, then multiply it by the inverse matrix of (the identity matrix - the standardized relationship influence matrix) to obtain the comprehensive influence matrix, and calculate the centrality (the total correlation strength of factors in the system). Conduct Bayesian update. First, obtain the prior mean and prior covariance through the direct influence matrix, then observe the centrality and calculate the posterior distribution parameters to obtain the posterior mean;

[0085] Step 3: Use the exponential approximation method to process all posterior means in the decision-making group to obtain the data display of the importance degree of indicators, and further process it to obtain the preference ranking applicable to the best-worst method. Regard the input items of the best-worst method: the preference degree of the best to others and the preference degree of others to the worst as probability distributions, model them using the multinomial distribution, and represent the output weight vector of the best-worst method using the Dirichlet distribution;

[0086] Step 4: Establish a Bayesian hierarchical model, including four parts: the input layer, the individual weight layer, the aggregated weight layer, and the concentration parameter, so as to model the prior distribution of the aggregated weight, and set the concentration parameter using the gamma distribution to represent the degree to which the individual weight is close to the aggregated weight;

[0087] Step 5: Establish a joint probability model, combine the prior of the aggregated weight, the conditional distribution of the weight of each decision-maker, and the likelihood of the observed data to calculate the joint probability distribution;

[0088] Step 6: When calculating the posterior distribution, use MCMC (Markov Chain Monte Carlo) for posterior distribution sampling, and the tool used is JAGS (Just Another Gibbs Sampler). Introduce the credible ranking, evaluate the advantages and disadvantages of pairwise criteria by calculating the confidence level, specifically by calculating through the posterior distribution for each pair of criteria, and then estimating this confidence level through the mean of the MCMC sampling. Finally, output the weight distribution of each decision-making group, the aggregated weight distribution, and the credible ranking and confidence level between the criteria, and display them as a weighted directed graph;

[0089] Step 7: Establish an assembly sequence evaluation function according to the requirements of the actual assembly, calculate using the output aggregated weight, and finally obtain the score of the assembly sequence, so as to select the optimal assembly sequence.

[0090] The present invention mainly includes: using the improved Decision Making Trial and Evaluation Laboratory (DEMATEL) method and the Best-Worst Method (BWM) to evaluate the assembly sequences for complex mechanical objects. In the part of the DEMATEL method, data is collected according to the established assembly sequence evaluation system, a direct influence matrix is established, and it is normalized into a standardized influence matrix to eliminate the influence of dimensions. The centrality is used as the observed data for Bayesian updating, and the exponential transformation is performed on the output posterior mean to obtain the preference matrix; according to the preference matrix, the best and worst elements are determined, pairwise comparisons are made, the best-worst vector is obtained and modeled as a multinomial distribution, and then the output weight vector is modeled as a Dirichlet distribution; the input of multiple decision-making groups is processed through a Bayesian hierarchical model, a joint probability model is established, and it is decomposed through conditional independence and the probability chain rule. Using the prior distribution (usually a non-informative Dirichlet distribution) and the observed data, the posterior distribution is obtained through Bayesian updating, and a weighted directed graph is generated using the confidence level to show the relationship between the criteria; the aggregated weights are output to evaluate the assembly sequences.

[0091] This embodiment describes a DEMATEL-BWM fusion assembly sequence evaluation method based on a Bayesian framework, including the following steps:

[0092] Step 1: For a machine with a complex assembly process, a constraint system for evaluating the assembly sequence is constructed according to the actual production requirements. Based on the constructed constraint system, the opinions of multiple decision-makers are analyzed to obtain multiple direct influence matrices, and random grouping is performed.

[0093] In Step 1, the constraint system includes multiple modules such as the final assembly accuracy, the efficiency, cost, and safety of the assembly process, which are modified and refined according to the actual assembly.

[0094] Step 2: The direct influence matrix is normalized to obtain a standardized relationship influence matrix, and then multiplied by the inverse matrix of (the identity matrix - the standardized relationship influence matrix) to obtain a comprehensive influence matrix, and the centrality (the total correlation strength of a single criterion in the system) is calculated. Bayesian updating is performed using the direct influence matrix (defining the prior distribution of the factors as a multivariate Gaussian distribution, setting the prior mean as a zero vector to represent an unbiased hypothesis; setting the prior covariance as the identity matrix to represent that the factors are independent and have the same variance; setting the noise covariance as 0.1 times the prior covariance) to obtain the prior mean and prior covariance, and then the centrality is observed and the posterior distribution parameters are calculated to obtain the posterior mean.

[0095] The Bayesian updating formula in Step 2 is as follows:

[0096]

[0097] where μ post is the posterior mean, Σ0 is the prior covariance matrix, Σ noiseis the noise covariance matrix, μ0 is the prior mean, and M is the centrality.

[0098] Step 3: Use the exponential approximation method to process the posterior mean to obtain the ranking of index importance. The formula is as follows:

[0099]

[0100] where max_val is the maximum value in the posterior mean, min_val is the minimum value in the posterior mean, a is the base of the exponent, and x is the reference offset.

[0101] Each decision group obtains a set of preference rankings, which are aggregated to obtain a preference matrix applicable to the best-worst method. The preference degree of the best in the input items of the best-worst method to the others (A B ) and the preference degree of the others to the worst (A W ) are regarded as probability distributions, and a multinomial distribution is used to model them. The probability mass function of A W is as follows:

[0102]

[0103] Because A B is relative to A W , the inverse of the weight is used for modeling, that is:

[0104]

[0105] For the output weight vector w of the best-worst method, since the weight vector must satisfy the non-negativity and the property of summing to one, a Dirichlet distribution is used for modeling. Given the parameter ∝∈R n , the Dirichlet distribution of the weight w is defined as:

[0106]

[0107] The purpose of the exponential approximation method in Step 3 is to convert the difference relationship data into discrete priorities for easy comparison. The process is as follows: First, determine the maximum and minimum values, then calculate the scaling base a and the logarithmic scale starting point x, compress the ratio into 8 equally spaced logarithmic intervals, and finally output the integer form priority.

[0108] Step 4: Use a Bayesian hierarchical model to model the prior distribution of the aggregated weights. The Bayesian hierarchical model includes an input layer, an individual weight layer, an aggregated weight layer, and a concentration parameter. The input layer is as follows: Assume there are K decision groups. Compare the internal of a set of preference rankings obtained for each decision group to obtain (best criterion to other criteria) and (other criteria to worst criterion) these two pairwise comparison vectors, which follow a multinomial distribution, that is:

[0109]

[0110] Each \(w\) in the individual weight layer k is modeled using a reparameterized Dirichlet distribution, i.e.:

[0111] \(w\) k \(\mid w\) agg \(\sim Dirichlet(\gamma\cdot w\) agg )

[0112] \(w\) in the group weight layer agg is the mean of the individual weights and is modeled using an uninformative Dirichlet distribution, i.e.:

[0113] \(w\) agg \(\sim Dirichlet(\alpha), \alpha = 1\)

[0114] The concentration parameter \(\gamma\) is modeled using a gamma distribution, i.e.:

[0115] \(\gamma\sim Gamma(a, b)\)

[0116] In the hierarchical model, \(\gamma\) represents the degree to which the individual weights are close to the aggregated weights, controlling the proximity between \(w\) k and \(w\) agg . Specifically, the larger \(\gamma\) is, the stronger the dispersion of the individual weights.

[0117] Step 5: Calculate the joint probability distribution by establishing a joint probability model, and combine the prior \(P(w\) agg ), the conditional distribution of the weights of each decision - maker \(P(w\) k \(\mid w\) agg ), and the likelihood of the observed data . This conforms to Bayes' rule and the conditional independence assumption, and the formula is as follows:

[0118]

[0119] Considering the independence of the variables, apply Bayes' rule to it and decompose it using the conditional independence and probability chain rules. The calculation formula is as follows:

[0120]

[0121]

[0122] Step 6: Calculate the posterior distributions of the aggregation weights, individual weights, and concentration parameters. Use the Markov Chain Monte Carlo method for posterior distribution sampling, with the tool JAGS, and introduce credible ordering. Since the working mode of MCDM (Multi-Criteria Decision Making) is that if the weight of one criterion, or the mean weight of group decision-making, is higher than another criterion, then this criterion is more important than the other criterion. Therefore, the concept of credible ordering is introduced, which can calibrate the degree to which one criterion is superior to another. Having the posterior distribution of weights will help measure the confidence in the relationships between various criteria. The difference between credible ordering and other ordering methods is that: Credibility ordering is based on a distribution, namely the Dirichlet distribution of w agg to calculate the confidence; while other ordering methods usually take two numbers / intervals and try to find out which one is more advantageous. First, define the credibility O:

[0123] O = (c i , c j , R, d)

[0124] where R is the relationship between the criteria c u and c j , that is, >, < or =; d ∈ [0, 1] represents the confidence in the relationship.

[0125] Evaluate the superiority of pairwise criteria by calculating the confidence level of the criterion weights. The method is: By designing a new Bayesian test, based on this test, the confidence in each credible ordering can be found. This test is based on the posterior distribution of w agg . Calculate the confidence that c i is superior to c j as:

[0126]

[0127] where P(w agg ) is the posterior distribution of w agg . If holds, then I is 1, otherwise it is zero. This integral can be approximated by the samples obtained by the MCMC method. Let Q samples be obtained from the posterior distribution, and the confidence can be calculated as:

[0128]

[0129] where is the q-th sample of w agg from the MCMC samples. For each pair of criteria, a confidence that one is superior to the other can be calculated. Obviously, P(c i > c j ) + P(c j > c i) = 1. Therefore, if and only if P(c i > c j ) > 0.5, c i is more important than c j . Therefore, a threshold of 0.5 is applied to the confidence ranking.

[0130] Specifically, for each pair of criteria, the posterior distribution is calculated, and the confidence level is estimated by the mean of MCMC sampling. The final output is the weight distribution of each decision group, the aggregated weight distribution, the credible ranking and confidence level between criteria, and is presented as a weighted directed graph, where the nodes represent decision criteria and the size of the nodes represents the uncertainty of the ranking; the edges represent the ranking relationship between criteria, and the labels of the edges show the probability of the ranking.

[0131] The posterior distribution sampling in Step 6, these samples represent the posterior distribution of the weight vector w and the aggregated weight vector. The collected samples are reshaped and their mean is calculated to obtain the aggregated weight vector, which represents the relative importance of the criteria.

[0132] Step 7: Establish an evaluation function using the constraint system obtained by aggregating the weights of the output, calculate each assembly sequence, and finally obtain a score to select the optimal assembly sequence.

[0133] The assembly sequence evaluation function in Step 7, according to the actual number of indicators, assigns the corresponding weights calculated, and the formula is as follows:

[0134] F = ω1·n1 + ω2·n2 + … + ω n ·n n

[0135] where F is the final score of the sequence, n represents the measured data corresponding to each indicator. For example, coaxiality and parallelism can be selected for assembly accuracy; the number of tool changes and the number of direction changes can be selected for assembly efficiency, etc. ω represents the weights corresponding to each indicator.

[0136] Implementation example:

[0137] A complex mechanical deviation analysis method based on a variable structure dynamic Bayesian network, including the following steps:

[0138] Step 1: Construct a constraint system for evaluating assembly sequences.

[0139] According to the rules in the actual assembly process, select the evaluation function indicators according to the following structure:

[0140] Human factors: Consider the skill level and experience of the operators, as well as their training and safety measures.

[0141] Machine factors: Evaluate the accuracy, reliability and maintenance status of the mechanical equipment used.

[0142] Part factors: Analyze the attributes of the assembled parts that may affect the assembly process.

[0143] Environmental factors: Consider the working environment, such as temperature, humidity, and cleanliness.

[0144] Result judgment: Select indicators for the measurement error accuracy required for the final assembled product.

[0145] Step 2: Construction and normalization of the direct influence matrix

[0146] Based on expert opinions and combined with the database, data collection is carried out for each selected indicator, and a direct influence matrix is constructed. The direct influence matrix is normalized to obtain a standardized relationship influence matrix. Using the Bayesian probability model, the prior mean and prior covariance are obtained from the direct influence matrix. Then, the centrality is observed and the posterior distribution parameters are calculated to obtain the posterior mean.

[0147] Step 3: Exponential approximation method and preference matrix

[0148] The exponential approximation method is used to process the posterior mean to obtain the ranking of indicator importance, which is converted into a preference matrix applicable to the best-worst method.

[0149] Step 4: Bayesian hierarchical model

[0150] A Bayesian hierarchical model is established, including an input layer, an individual weight layer, a group weight layer, and a concentration parameter. The prior distribution of the aggregated weights is modeled.

[0151] Step 5: Establishment of a joint probability model

[0152] Using Bayes' rule, each variable is decomposed through conditional independence and the probability chain rule.

[0153] Step 6: Posterior distribution sampling and credible ranking

[0154] The Markov chain Monte Carlo method is used for posterior distribution sampling. Credible ranking is introduced, and the superiority and inferiority of pairwise criteria are evaluated by calculating the confidence level of the criterion weights, and a weighted directed graph is output for easy observation. The arrows between indicators indicate the comparison relationship, with the importance decreasing from top to bottom, and the numbers on the connection lines being the corresponding confidence levels, as Figure 3 shown.

[0155] Step 7: Calculation of the assembly sequence evaluation function

[0156] Using the output weights and combined with the indicators obtained according to the requirements of the actual assembly, an assembly sequence evaluation function is established, and the score of the assembly sequence is calculated. Based on the score results, the optimal assembly sequence is selected.

[0157] The above-described embodiments merely represent one implementation mode of the present invention. The description is relatively specific and detailed, but it should not be construed as a limitation on the scope of the invention patent. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several modifications and improvements can still be made, and these all fall within the protection scope of the present invention. Therefore, the protection scope of the present invention patent shall be subject to the appended claims.

Claims

1. A DEMATEL-BWM fusion assembly sequence evaluation method based on Bayesian probability model, characterized in that, Evaluate the assembly sequence for complex mechanical objects; conduct decision-making experiments and evaluation experiments, collect data according to the established assembly sequence evaluation system, establish a direct influence matrix, normalize it into a standardized influence matrix, use centrality as the observed data for Bayesian updating, perform exponential transformation on the output posterior mean to obtain a preference matrix; determine the best and worst elements according to the preference matrix, conduct pairwise comparisons, obtain the best-worst vector and model it as a multinomial distribution, and then model the output weight vector as a Dirichlet distribution; process the input of multiple decision-making groups through a Bayesian hierarchical model, establish a joint probability model, and decompose it through conditional independence and probability chain rules; use the prior distribution and observed data to obtain the posterior distribution through Bayesian updating, and generate a weighted directed graph using the confidence level to show the relationship between criteria; Output the aggregated weights and evaluate the assembly sequence.

2. The DEMATEL - BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 1, wherein The steps are as follows: Step 1: For machinery with complex assembly processes, select constraints according to actual requirements to construct a constraint system for evaluating the assembly sequence; according to the constructed constraint system, score each element in the system using a database or technician opinions to obtain a direct influence matrix; Step 2: Normalize the direct influence matrix to obtain a standardized relationship influence matrix, and then multiply it by the inverse matrix of (the identity matrix - the standardized relationship influence matrix) to obtain a comprehensive influence matrix, and calculate the centrality, that is, the total association strength of the factor in the system; conduct Bayesian updating, first obtain the prior mean and prior covariance through the direct influence matrix, then observe the centrality and calculate the posterior distribution parameters to obtain the posterior mean; Step 3: Use the exponential approximation method to process all posterior means in the decision-making group to obtain a data display of the importance degree of indicators, and further process it to obtain a preference ranking applicable to the best-worst method; regard the input items of the best-worst method: the preference degree of the best to others and the preference degree of others to the worst as probability distributions, model them using a multinomial distribution, and represent the output weight vector of the best-worst method with a Dirichlet distribution; Step 4: Establish a Bayesian hierarchical model, including an input layer, an individual weight layer, an aggregated weight layer, and a concentration parameter part, so as to model the prior distribution of the aggregated weights, set the concentration parameter using a gamma distribution, indicating the degree to which the individual weights are close to the aggregated weights; Step 5: Establish a joint probability model, combine the prior of the aggregated weights, the conditional distribution of the weights of each decision maker, and the likelihood of the observed data to calculate the joint probability distribution; Step 6: When calculating the posterior distribution, use the MCMC (Markov Chain Monte Carlo) method to sample the posterior distribution, and the tool used is the JAGS sampler. Introduce a credible ranking, and evaluate the advantages and disadvantages of pairwise criteria by calculating the confidence level. Specifically, for each pair of criteria, calculate through the posterior distribution, and then estimate this confidence level through the mean value of the MCMC sampling. Finally, output the weight distribution of each decision-making group, the aggregated weight distribution, and the credible ranking and confidence level between criteria, and display them as a weighted directed graph; Step 7: Establish an assembly sequence evaluation function according to the requirements of the actual assembly, calculate using the output aggregated weights, and finally obtain the score of the assembly sequence, so as to select the optimal assembly sequence.

3. The DEMATEL-BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 2, characterized in that, Specifically, it includes the following steps: Step 1: For machinery with complex assembly processes, construct a constraint system for evaluating assembly sequences according to actual production requirements. Based on the constructed constraint system, analyze using the opinions of multiple decision-makers to obtain multiple direct influence matrices and perform random grouping. Step 2: Normalize the direct influence matrix to obtain a normalized relationship influence matrix, then multiply it by the inverse matrix of (the identity matrix - the normalized relationship influence matrix) to obtain a comprehensive influence matrix, and calculate the centrality. Use the direct influence matrix for Bayesian update to obtain the prior mean and prior covariance, then observe the centrality and calculate the posterior distribution parameters to obtain the posterior mean. Step 3: Use the exponential approximation method to process the posterior mean to obtain the ranking of index importance. The formula is as follows: Where max_val is the maximum value in the posterior mean, min_val is the minimum value in the posterior mean, a is the base of the exponent, and x is the reference offset. Each decision group obtains a set of preference rankings, which are aggregated to obtain a preference matrix applicable to the best-worst method. The preference degree of the best in the input items of the best-worst method over the others (A B ) and the preference degree of the others over the worst (A W ) are regarded as probability distributions and modeled using the multinomial distribution. The probability mass function of A W is as follows: Because of A B versus A W relatively, so inverse modeling with weights is used, that is: For the output weight vector w of the best-worst method, since the weight vector must satisfy the non-negativity and the property of summing to one, the Dirichlet distribution is used for modeling; given the parameter ∝ ∈ R n , the Dirichlet distribution of the weight w is defined as: Step 4: Model the prior distribution of the aggregation weights through a Bayesian hierarchical model; the Bayesian hierarchical model includes an input layer, an individual weight layer, an aggregation weight layer, and a concentration parameter; the input layer is: assuming there are K decision-making groups, compare within a set of preference rankings obtained for each decision-making group to obtain the best criterion to other criteria and other criteria to the worst criterion These two pairwise comparison vectors follow a multinomial distribution, that is: Each \(w\) in the individual weight layer k is modeled using a reparameterized Dirichlet distribution, i.e.: w k |w agg ~Dirichlet(γ·w agg ) w in the group weight layer agg is the mean of the individual weights and is modeled using an uninformative Dirichlet distribution, i.e.: w agg ~ Dirichlet(α), α = 1 The concentration parameter γ is modeled using the gamma distribution, that is: γ~Gamma(a,b) In the hierarchical model, γ represents the degree to which the individual weights are close to the aggregated weight, controlling the proximity between w k and w agg ; Step 5: Calculate the joint probability distribution by establishing a joint probability model, and combine the prior of the aggregation weight P(w agg ), the conditional distribution of the weight of each decision maker P(w k ∣w agg ), and the likelihood of the observed data jointly, which conforms to the Bayes' rule and the conditional independence assumption. The formula is as follows: Considering the independence of variables, use Bayes' rule for it, and decompose it through conditional independence and the probability chain rule. The calculation formula is as follows: Step 6: Calculate the posterior distributions of the aggregated weights, individual weights, and concentration parameters, use the Markov chain Monte Carlo method for posterior distribution sampling, the tool used is JAGS, and introduce credible sorting; first define the credibility O: O = (c i , c j , R, d) where R is the relationship between standard c i and c j i.e., >, < or =; d ∈ [0, 1] represents the confidence level of the relationship; Evaluate the advantages and disadvantages of pairwise criteria by calculating the confidence level of criterion weights. The method is as follows: Design a new Bayesian test, based on which the confidence of each credible ranking can be found; this test is based on the posterior distribution of w agg ; calculate the confidence that c i is better than c j as: where P(w agg ) is the posterior distribution of w agg . If holds, then I is 1, otherwise it is zero; this integral can be approximated by samples obtained using the MCMC method; let Q samples be obtained from the posterior distribution, and the confidence level can be calculated as: where is the q-th sample from the MCMC samples; for each pair of criteria, a confidence level that one is better than the other can be calculated; obviously, P(c agg >c i ) + P(c j >c j ) = 1; i ) = 1; Therefore, if and only if P(c i >c j )>0.5, c i is more important than c j ; therefore, a threshold of 0.5 is applied to the confidence ranking; Specifically, it is manifested as calculating for each pair of criteria through the posterior distribution, then estimating the confidence level through the mean of MCMC sampling, and finally outputting the weight distribution, aggregated weight distribution of each decision group, as well as the credible sorting and confidence level between criteria, and presenting it as a weighted directed graph. Step 7: Establish an evaluation function using the constraint system with aggregated weights obtained from the output, calculate for each assembly sequence, and finally obtain the score, so as to select the optimal assembly sequence.

4. The DEMATEL - BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 3, characterized in that In Step 1, the constraint system includes multiple modules such as the final assembly accuracy, efficiency, cost, and safety of the assembly process, and is modified and refined according to the actual assembly.

5. The DEMATEL-BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 3, wherein The Bayesian update formula in Step 2 is as follows: where μ post is the posterior mean, Σ0 is the prior covariance matrix, Σ noise is the noise covariance matrix, μ0 is the prior mean, and M is the centrality.

6. The DEMATEL - BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 3, characterized in that, In Step 2, the prior distribution of the Bayesian update defines the factors as a multivariate Gaussian distribution, sets the prior mean as a vector of all zeros, indicating an unbiased hypothesis; sets the prior covariance as the identity matrix, indicating that the factors are independent and have the same variance; sets the noise covariance as 0.1 times the prior covariance.

7. The DEMATEL - BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 3, wherein The purpose of the exponential approximation method in Step 3 is to convert the difference relationship data into discrete priorities for easy comparison; the process is as follows: first determine the maximum and minimum values, then calculate the scaling base a and the logarithmic scale starting point x, compress the ratio into 8 equally spaced logarithmic intervals, and finally output the integer form priority.

8. The DEMATEL-BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 3, characterized in that, In Step 5, the nodes of the weighted directed graph represent decision criteria, and the size of the nodes represents the uncertainty of the ranking; the edges represent the ranking relationship between the criteria, and the labels of the edges show the probability of the ranking.

9. The DEMATEL-BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 3, wherein Posterior distribution sampling in step 6, where these samples represent the posterior distributions of the weight vector w and the aggregated weight vector; reshape the collected samples and calculate their average value to obtain the aggregated weight vector, which represents the relative importance of the criteria.

10. The DEMATEL-BWM fusion assembly sequence evaluation method based on the Bayesian probability model according to claim 3, wherein The assembly sequence evaluation function in step 7, according to the actual number of indicators, assigns the corresponding calculated weights, and the formula is as follows: F = ω1·n1 + ω2·n2 + … + ω n ·n n Where F is the final score of the sequence, n represents the measured data corresponding to each indicator. For example, coaxiality and parallelism can be selected for assembly accuracy; the number of tool changes and the number of direction changes can be selected for assembly efficiency, etc.; ω represents the weights corresponding to each indicator.

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