Decomposition of fuzzy exponential mathematical quantitative process in industrial manufacturing design

Abstract

Small and medium-sized manufacturing enterprises have the characteristics of large numbers and small scales. Problems such as backward manufacturing technology, lack of talents, small amount of information resources, and insufficient product research and development capabilities have severely restricted the development of enterprises. The backward manufacturing design model cannot adapt to the development trend of modern manufacturing informatization. This paper proposes and designs a fuzzy inference model and fuzzy inference engine algorithm with threshold. In order to describe the numerical multiple input and multiple output variables in the industrial manufacturing design industry, the relevant experience is used to make numerical reasoning decisions. Applying fuzzy sets and fuzzy theory to the expert system, a fuzzy rule model containing the membership function information and thresholds of the corresponding fuzzy sets is proposed and established, and a fuzzy reasoning system suitable for numerical and uncertain reasoning decisions is constructed. The improved grey relational analysis method is used to decompose and evaluate the exponential mathematical quantitative process of manufacturing enterprises. Based on the fuzzy Decision Analytic Network Process (DANP) method to calculate the relative weight of the influencing factors in the evaluation system, the evaluation index of the enterprise is obtained. Starting from the industrial manufacturing design process, this article constructs a relatively comprehensive and reasonable enterprise exponential mathematical quantitative process decomposition evaluation system. Considering that there are complex interactions between the various influencing factors in the system, the fuzzy Decision Making Trial and Evaluation Laboratory (DEMATEL) method is selected to process the direct impact matrix of the evaluation system, and the causal relationship between the indicators is obtained. The fuzzy exponential gray correlation method is used to evaluate the quantitative process of industrial manufacturing design, avoiding the shortcomings of traditional methods that only consider ideal values.

Keywords

Industrial manufacturing design quantitative process decomposition DANP fuzzy reasoning

1 Introduction

The manufacturing industry is an important pillar of the development of human society. The development of the manufacturing industry reflects the economic and comprehensive strength of a country or region [1]. Since entering the information age, the manufacturing sector has undergone earth-shaking changes due to the influence of knowledge economy, global thinking, networked economy and innovative technology [2]. The manufacturing concept has developed from large-scale production and low-cost production to the current stage of knowledge innovation. With the development of economic globalization, global informatization and network service, the use of information technology and computer network technology, especially the rapid development of Internet technology, has become the main direction of manufacturing research in various countries [3, 4]. In this era, in order to adapt to the ever-changing market and meet the increasingly personalized and diversified needs of users, manufacturing modes such as virtual manufacturing, agile manufacturing, industrial manufacturing design, green manufacturing and networked manufacturing have gradually developed [5, 6].

As the trend of “dataization” and “cloudification” of manufacturing industry is sweeping the world, a large amount of manufacturing resources will be stored in the cloud resource pool (manufacturing resource database) [7]. Therefore, data mining research and research on the manufacturing resources in the resource pool will be carried out. Resources that meet specific needs can be quickly found, which has become an urgent problem that needs to be resolved [8]. At the same time, due to the heterogeneity and independence of manufacturing resources, the description of many resources is not clear, and it is difficult to analyze the relevance of resources with some single attributes, which limits the optimization of related specific resources in the later period [9]. Therefore, there are generally ambiguities among manufacturing resources in the cloud mode. How to perform a scientific and reasonable cluster analysis of fuzzy manufacturing resources to enhance the relevance and integration of data resources, so as to optimize the allocation of resources or services in the future, has become the only way for the current cloud manufacturing to move towards digitalization and intelligence [10]. Due to the fuzziness of data information in some research fields, current scholars at home and abroad mostly use fuzzy clustering methods to cluster data [11, 12]. Relevant scholars introduced fuzzy clustering into load balancing technology, and made corresponding improvements to the clustering objective function based on the system characteristics of virtual clusters [13]. The researchers introduced the fuzzy C-means clustering algorithm to the analysis and optimization of directional data, and verified the advantages of this method in data analysis by comparing the clustering results [14]. Related scholars transformed the problem of community structure division into a sample classification problem in fuzzy clustering algorithm, and verified its feasibility with examples [15]. However, there is still a lack of in-depth research on the clustering and classification of manufacturing resources. Facing the networked manufacturing system, related scholars proposed to use the related theory of fuzzy mathematics to dynamically cluster and divide a specific manufacturing resource, which strengthened the requirement of layered packaging and retrieval of manufacturing resources in practical applications, but did not consider the dynamics [16]. The computational complexity of the clustering algorithm itself affects cloud services. Researchers have proposed a variety of resource classification methods based on the characteristics of resources in the networked manufacturing environment, but they have not given the mathematical methods used in the specific classification process [17, 18]. It has been reported that although the genetic algorithm is combined with the fuzzy clustering algorithm, the manufacturing resources are grouped based on the processing technology, which improves the search efficiency of processing equipment, but there is a lack of research and discussion on other fuzzy clustering methods [19, 20].

In this paper, the specific input value in the application field of fuzzy inference system is fuzzified to obtain the corresponding membership degree, combined with fuzzy rules for fuzzy inference, and the obtained fuzzy set inference result is defuzzified into clear specific output value, which directly drives the application field implementing agencies. The constructed rule-based fuzzy inference system is very suitable for processing multiple-input multiple-output (MIMO) numerical reasoning in the field, and is good at handling and describing numerical, uncertain, inaccurate, and empirical fuzzy language concepts. This paper uses fuzzy Decision Making Trial and Evaluation Laboratory (DEMATEL) method, fuzzy DANP method and improved grey relational analysis method to evaluate and analyze the exponential mathematical quantitative process decomposition of manufacturing enterprises. The fuzzy DEMATEL method is used to construct the pairwise influence relationship matrix of each factor in the evaluation system, calculate the influence degree and the degree of influence of each index, and then determine its centrality and cause degree, and construct a causal relationship diagram of influencing factors. In the identification of key factors, traditional methods are not used to directly characterize all affected factors as non-key factors. Instead, the cause degree, centrality and influence degree of each index are comprehensively analyzed, and the cause degree is slightly less than zero but the factors with a higher degree of ranking were also identified as key factors, and finally a total of seven key factors were identified, making the analysis results more comprehensive and scientific.

The rest of this article is organized as follows. Section 2 discusses the related work of the quantitative process decomposition of industrial manufacturing design. Section 3 constructs the fuzzy reasoning model in industrial manufacturing design and designs the reasoning quantification algorithm. Section 4 conducts an empirical analysis. Section 5 summarizes the full text.

2 Related work

2.1 Uncertain knowledge representation based on weighted fuzzy petri net

In the production and operation process of the industrial manufacturing design system, there is a large amount of dynamic uncertainty information, such as the random occurrence of machine failure, the dynamic change of the buffer in-process level, etc., which are difficult to accurately describe the production knowledge. Therefore, the use of fuzzy knowledge to describe and analyze the actual operation of the production system can be more in line with the thinking mode of the production system manager. However, the traditional fuzzy Petri net does not consider the degree of influence of the preconditions on the conclusion. In the actual knowledge representation of the production system, a conclusion may be caused by multiple preconditions with different contributions, and one precondition may also produce multiple conclusions with different credibility.

Therefore, in order to more accurately express and describe the energy-saving knowledge of the industrial manufacturing design system in the operation process, this paper makes certain improvements to the relevant parameters in the fuzzy Petri on the basis of the existing research. An 11-tuple Weighted fuzzy Petri nets are used to realize the energy-saving knowledge representation and reasoning of the industrial manufacturing design system.

When describing the energy-saving operation knowledge of the industrial manufacturing design system, this research mainly uses four types of fuzzy production rules, which are now summarized as: single condition single conclusion, multiple condition single conclusion, multiple rule single conclusion, single rules and conclusions. For these four types of energy-saving rules, the corresponding fuzzy Petri net mapping model is introduced.

2.2 Manufacturing resource allocation framework

Manufacturing resources are composed of soft manufacturing resources and hard manufacturing resources. Among them, soft manufacturing resources generally refer to non-hardware resources in manufacturing activities, and their function is to ensure the normal operation and operation of the manufacturing process during the entire manufacturing life cycle, such as software resources, service resources, human resources, knowledge resources, financial resources, etc.; manufacturing resources generally refer to all physical hardware resources directly related to the manufacturing process involved in manufacturing activities.

From the perspective of the complete theory of cloud manufacturing and the technical development framework of the cloud manufacturing service platform, the optimization of manufacturing resources involves multiple technologies and functional modules, as shown in Fig. 1.

Fig.1

Framework for optimal allocation of resources in industrial manufacturing design.

2.3 Key technologies for optimal allocation of manufacturing resources

The optimal configuration of manufacturing resources in the cloud mode requires the support of multiple key functional modules, and the theoretical and technical system design of related functional modules is shown in Fig. 2.

Fig.2

The hierarchical system of optimal allocation of industrial manufacturing design resources.

The manufacturing resources, providers, and access content are described in the form of manufacturing services, and the manufacturing services are virtualized and encapsulated. In this process, how to verify the structure and semantics of the described manufacturing services is an important part of encapsulation.

Clustering analysis of manufacturing resources stored in the database is a data mining technology. Although this part of the module is not a necessary research content for the key technologies of cloud manufacturing, a preliminary cluster analysis is performed on cloud manufacturing resources to enhance the data. The correlation between them can pave the way for the later resource allocation, improve the efficiency of resource retrieval and optimize the configuration effect, thereby enhancing the intelligent level of cloud manufacturing.

The matching of resource supply and demand is the core of the framework of resource allocation. How to quickly, accurately, and efficiently match the most suitable resources or tasks for the supply and demand sides of manufacturing resources in the massive cloud resource pool, and perform intelligent combination of tasks, is the key to achieving “manufacturing as a service” in cloud manufacturing.

2.4 Product family collaborative optimization design system structure

The life cycle of a modern product refers to the entire process from the formation of the product to the demise of the product, and then to the regeneration of the product. The life cycle of a product is an open dynamic process system, which generally includes the acquisition of raw materials, product planning and manufacturing, product sales and distribution, product use and maintenance, recycling, reuse and treatment of waste products. It is in the process system that the product has a meaningful connection with people and the environment. For example, marketers convert products into commodities in a market environment, users use products to create a reasonable lifestyle, while recyclers dismantle and recycle waste products to convert products into usable renewable resources. The function of the product system is realized in the process of interaction and coordination of this kind of person-product-environment.

The systematic thinking mode and system behavior of product design have important practical significance in today’s increasingly complicated “human-society-nature” system relationship. First of all, system design is a powerful guarantee for maintaining ecological balance and seeking the sustainable development of man-society-nature. Secondly, system design is an effective method to ensure the realization of product functional meaning. Only by starting from the product life cycle and digging out the meaning of the product and the external environment in the ever-evolving lifestyle, can we carry out a reasonable product positioning and maximize the value of the product. Finally, system design is an effective way to form products. Product positioning has made a limited definition of the final form of the product, but through system analysis, system elements and structure coordination can create diversified design solutions. It is an effective way to form a new product to find the best solution through system integration and optimization among multiple solutions. The structure of the product design system is shown in Fig. 3.

Fig.3

Product design system structure.

3 Establishment of fuzzy inference model and inference quantitative algorithm in industrial manufacturing design

3.1 Fuzzy set and membership function

The fuzzy system is based on fuzzy logic. Compared with traditional binary logic, fuzzy logic can more accurately reflect people’s thinking and better describe natural language. Fuzzy logic can more effectively describe and deal with uncertainties such as approximations and imprecision in reality. When we are unable to accurately and quantitatively describe the known information sources, or when the traditional quantitative methods are too complicated, the methods related to the fuzzy system show strong effectiveness and usability.

In traditional set theory, an element either absolutely belongs to the specified set, or absolutely does not belong to the specified set, either. The fuzzy set theory, as a generalization of the classic set theory, can have an inaccurate relationship between elements and a specified set, allowing elements to belong to the specified set to a certain extent; the degree of membership is defined as the degree of membership, and the value range of the degree of membership is [0, 1]. With the help of fuzzy sets and membership functions, mathematical models can be used to quantitatively describe uncertainty. Triangular and Gaussian membership functions are the most widely used.

The triangle membership function is: $u (x) = {\begin{matrix} 1 - | m - x | / σ & | m - x | < σ \\ 0 & | m - x | ⩾ σ \end{matrix}$ (1)

Among them, m represents the center of the fuzzy set, and σ represents the width of the fuzzy set.

The Gaussian membership function is: $u (x) = e^{- \frac{(x - c)^{2}}{σ^{2}}}$ (2)

Among them, c represents the center of the fuzzy set, and σ represents the width of the fuzzy set.

The supporting set Supp(A) of fuzzy set A on the universe U is a clear set containing all the elements in U whose membership value is greater than 0 in A, namely: $S (A) = {x \in U | u_{A} (x) ⩾ 0}$ (3)

If the supporting set Supp(A) of fuzzy set A contains only one single point in U, the fuzzy set A is called a single point fuzzy set or fuzzy single value.

3.2 Fuzzy inference system

Fuzzy inference system is a system based on fuzzy rules, also called fuzzy quantizer (when fuzzy technology is used for quantization). Figure 4 shows the general structure of fuzzy inference system. It can be seen from Fig. 4 that a fuzzy inference system is mainly composed of 4 basic functional modules as shown below.

Fig.4

General structure of fuzzy inference system.

Fuzzy system is essentially an expert system that contains a number of “IF-THEN” formal language rules. The most used fuzzy rules are fuzzy “IF-THEN” rules, also known as Mamdani fuzzy rules.

In the general form of the Mamdani fuzzy rule model, the language containing expert knowledge is quantified, and the corresponding fuzzy inference engine can use language information to obtain the output of the system. In the general structure of the above-mentioned Mamdani rule model, the input (condition) and output (conclusion) of the rule are both fuzzy sets. The general engineering experience and the input and output of the actual system are all clear quantities. Therefore, from the perspective of simplification and convenient application, a single-point fuzzy set is used in the subsequent part of the fuzzy rule.

Fuzzy implication expresses the relationship between the antecedent (condition) and the conclusion (postpart) in the rule. The fuzzy implication operation executes the implication operator according to the membership value of the antecedent of the rule, and finally calculates the membership value of the inference conclusion. Based on the existing fuzzy implication relations and fuzzy rules, the system can execute the fuzzy inference process. Since fuzzy implication relations can have different definitions, the choice of fuzzy implication operators that need to be adapted will also be very different.

The expression of the fuzzy minimum implication operation (Mamdani) method is: $R = A \cdot B = \int_{X \cdot Y} \frac{u_{A} (x) \land u_{B} (x)}{(x, y)}$ (4)

3.3 Design of fuzzy inference model and inference quantitative algorithm

When constructing the Mamdani fuzzy inference engine, when choosing different fuzzy relationship meanings and fuzzy implication methods, synthesis operators, fuzzification and defuzzification methods, the fuzzy inference algorithm used by the fuzzy inference engine is different, which brings about the quantitative effect of inference. In the most important and common Mamdani fuzzy inference engine in practical applications, the fuzzy relationship uses Rc (smaller) operation, the synthesis operator uses “∨ – ∧” (maximum-minimum), single-point fuzzification, and weighted average method to resolve. Under this method configuration, all fuzzy rules will produce a total fuzzy relationship as follows: $R = \cup_{l = 1}^{r} R_{l}$ (5)

The resulting inference result is a fuzzy set, and the accurate value output can be calculated after the weighted average method is used to resolve the fuzzification. The proposed process and algorithm of fuzzy inference based on fuzzy IF-THEN rules and input are summarized as follows:

According to the specific value of the user input variable and the membership function of the input language variable to each input fuzzy subset (input variable language value), you obtain each input language variable to each input fuzzy sub set.

For each fuzzy rule, it is necessary to check whether the membership degree of each input linguistic variable in its predecessor to the corresponding input fuzzy subset is greater than or equal to the corresponding sub-condition threshold. After passing the threshold test, the AND operation can be continued to obtain the synthetic credibility or activation degree of the antecedents of each rule.

According to the calculated synthetic credibility or activation degree of each rule’s antecedent, combined with the rule credibility, the membership degree of the conclusion variable to the corresponding output fuzzy subset is obtained. After the conclusion threshold is tested, the conclusion of the activated rule is obtained.

Finally, the output is obtained by defuzzification, and the fuzzy set of the activated rules is combined and superimposed, and then the clear and specific output value is obtained by the weighted average method.

3.4 Fuzzy division of input space

All the input language variables in the antecedent (condition) of the fuzzy inference rule and all the output language variables in the latter (conclusion) respectively constitute the fuzzy input space and the fuzzy output space. Before designing fuzzy rules, it is necessary to plan and give a set of several basic fuzzy subsets based on the system requirements and experience in the domain of each input language variable. This process is called the fuzzy division of the input space of the fuzzy system. For each input language variable, the support set Supp(A) of each basic fuzzy subset defined covers a section of the input variable domain, so the input variable domain is divided into domains. The basic fuzzy subsets defined above have one-to-one correspondence between intervals, and these intervals often overlap each other. The number of fuzzy divisions directly determines the precision and detail of fuzzy inference. The selected fuzzy language value (language name) often has a specific meaning, such as NB (negative large), NM (negative medium), NS (negative small), ZO (zero), PS (positive small), PM (positive middle), PB (Zhengda), or other similar language values. The number of fuzzy divisions also directly determines the maximum number of fuzzy inference rules. As the number of fuzzy divisions increases, the maximum number of fuzzy inference rules increases exponentially. Fuzzy division should not only meet the requirements of inference quantification accuracy, but should not be too fine. The selection of the membership function of each fuzzy subset of the input linguistic variables in the antecedents (conditions) of the fuzzy inference rule determines to a large extent the accuracy of the fuzzy system’s approximation of arbitrary functions; an empirical conclusion is that when the membership function curves of two adjacent fuzzy subsets of the domain to which a language variable belongs intersect near the midpoint of the overlapping area of the two, the approximation accuracy of the fuzzy inference system is often very high, otherwise, the accuracy is low.

4 Empirical analysis

4.1 Causal correlation analysis of green evaluation factors of manufacturing enterprises

We construct the direct influence matrix of the factors affecting the exponential mathematical quantitative process decomposition of manufacturing enterprises. According to fuzzy semantic conversion, the obtained expert score is converted into the corresponding triangular fuzzy number, and then the obtained triangular fuzzy number is calculated according to the center of gravity method to resolve the fuzzy formula, and the corresponding clear number is obtained. In the case of effectively reducing the subjective deviation of the expert scoring, the specific value that can be calculated by the DEMATEL matrix. The relationship between the influence value of fuzzy semantics and the clarity number is shown in Fig. 5.

Fig.5

The relationship between the influence value of fuzzy semantics and the clarity number.

We calculate the influence degree D and the influence degree R of the influencing factors of the manufacturing enterprise, and calculate the centrality D + R and the cause degree D-R on this basis. Since the centrality represents the sum of the influence and the affected degree, the importance of the factor is positively related to the value of the centrality, that is, the greater the centrality, the greater the importance of the corresponding factor. The degree of cause is the difference between the degree of influence and the degree of influence, which represents the net influence value of the corresponding factor on the influencing factors of the entire manufacturing enterprise. If the degree of cause is positive, the corresponding factor is the cause factor. If the degree of cause is negative, the corresponding factor is the result factor. The calculation result of fuzzy DEMATEL is shown in Fig. 6.

Fig.6

Calculation results of fuzzy DEMATEL.

According to the centrality and cause degree, we draw a causal relationship diagram of the influencing factors of the exponential mathematical quantitative process decomposition of manufacturing enterprises, as shown in Fig. 7. Through the causality diagram, it is possible to intuitively identify the influence and importance of each factor on the evaluation system of the exponential mathematical quantitative process decomposition of the entire manufacturing enterprise.

Fig.7

Causality diagram.

Combining the causality diagram drawn by centrality and cause degree, we can intuitively observe whether each factor belongs to the cause factor or the result factor and the importance of the factor to the entire evaluation system. The analysis of the degree of influence and the degree of influence can effectively identify the key influencing factors of the exponential mathematical quantitative process decomposition of manufacturing enterprises. This article will give a detailed description of important cause factors, result factors, and key influencing factors.

Determination of the relative weights of factors affecting the exponential mathematical quantitative process decomposition of manufacturing companies

Although the fuzzy DEMATEL method can effectively deal with the inter-influence relationship between the influencing factors of the exponential mathematical quantitative process decomposition of various manufacturing enterprises, and can identify the contribution of each indicator to the entire evaluation system, it ignores the problem of different weights of indicators in different dimensions. In order to comprehensively consider the problem of different index weights, in this section, the ANP method is combined with the fuzzy DEMATEL method, namely Decision Analytic Network Process (DANP). We calculate the relative weight value of each factor, and more accurately identify the key factors in the evaluation system of the factors affecting the exponential mathematical quantitative process decomposition of manufacturing enterprises.

This paper constructs an evaluation index system from five dimensions: manufacturing capacity, recycling capacity, economic benefit, ecological benefit, and green management system. Now, according to these five dimensions, the comprehensive influence matrix obtained in step 3 of the previous fuzzy DEMATEL method is divided into 25 sub-matrices. Each sub-matrix is standardized, and then transposed and placed back to the corresponding position in the original matrix. Then, the unweighted super matrix of the factors affecting the exponential mathematical quantification process of the manufacturing enterprise is obtained.

Similar to the process of calculating the comprehensive influence matrix of the secondary indicators of the exponential mathematical quantitative process decomposition of the manufacturing enterprise, firstly, experts are collected to summarize the scoring opinions of the degree of mutual influence between the two indicators, and then the fuzzy semantic conversion table is used to convert them into fuzzy evaluation, as shown in Fig. 8.

Fig.8

Triangular fuzzy evaluation of first-level indicators.

We unfuzzy the triangular fuzzy number, obtain the clear number of the influence degree between each first-level index, and construct the direct influence matrix of the first-level index. And we standardize and limit the direct influence matrix respectively, and obtain the comprehensive influence matrix of the first-level index of the enterprise exponential mathematical quantitative process decomposition influencing factor evaluation.

We standardize the comprehensive influence matrix of the first-level index factors, and obtain the weights of the first-level index of the influencing factors in the exponential mathematical quantitative process of manufacturing enterprises, as shown in Fig. 9.

Fig.9

Weights of first-level indicators.

We construct a weighted super matrix, calculate the unweighted matrix and the first-level index weights of the influencing factors of the exponential mathematical quantification process of the manufacturing enterprise, and multiply the value of the obtained unweighted matrix with the first-level index weight value of the corresponding block to obtain the index-type mathematical quantification of the manufacturing enterprise decomposition influencing factors weighted super matrix.

According to the calculation result of the limit matrix, the relative weight of each indicator in the entire evaluation system is obtained, as shown in Fig. 10. Manufacturing companies should start from the perspective of how to improve economic and ecological benefits, make decision-making adjustments and industrial upgrades.

Fig.10

The relative weights of the influencing factors of the exponential mathematical quantitative process decomposition of manufacturing enterprises.

5 Conclusion

In order to describe the numerical multiple-input multiple-output (MIMO) variables and their empirical knowledge in the industrial manufacturing design industry and make numerical reasoning decisions, this paper applies fuzzy sets and fuzzy theory to the expert system, and proposes and establishes the corresponding fuzzy rule model. Using the fuzzy relationship of Rc operation, the “∨ – ∧” (maximum-minimum) synthesis operator, single-point fuzzification and weighted average method to defuzzify the method, we design the fuzzy inference engine inference algorithm, and finally construct the fuzzy reasoning system. On the basis of the fuzzy DANP method to calculate the relative weight value of each indicator, the improved grey relational analysis is used to evaluate the exponential mathematical quantitative process decomposition of manufacturing enterprises. Through the expert scoring method, the company’s performance in various indicators is scored, the ideal value grayscale and the tolerance value grayscale are calculated, and then the evaluation index of each company can be obtained, which can analyze the horizontal competitiveness of the company and make the best select. Key factor identification and index weight calculation results show that companies should focus on improving the maturity of the recycling system during the recycling process, improving the quality of recycled old parts, vigorously researching and developing core manufacturing technologies during the manufacturing process, and paying attention to whether the relevant machines are advanced in the industry. In the internal management, we should pay attention to the construction of energy saving and emission reduction management system. By improving the gray correlation method to calculate the evaluation index of each enterprise, and comprehensively analyzing the two angles, the selection of the enterprise with the highest evaluation index is obtained, and a more scientific and reasonable plan selection model is constructed.

Footnotes

Acknowledgments

Western Project of China National Social Science Fund in 2019: Research on Yunnan Minority Costume Symbols Based on Lotman Semiosphere Theory (19XMZ066).

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