# Research on the Optimization Technology of Mechanical Product Design Plan Based on Fuzzy Comprehensive Evaluation Analysis Method

Authors: Zhijie Wang; Yan Cao; Hu Qiao; Yujia Wu; Miao Liu
Journal: Scientific Programming
 (2022)

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## Abstract

In the scheme design stage of mechanical products, several design solutions need to be evaluated so that the best design solution can be selected from them. Because the evaluation of mechanical product scheme design is a multilevel, multiattribute and contains many fuzzy models, fuzzy comprehensive evaluation becomes an advantageous model for the evaluation of mechanical product scheme design. In this article, a fuzzy comprehensive evaluation method is constructed, and a combination of the subjectivity of determining weights by hierarchical analysis method and the objectivity of determining weights by entropy value method is used to calculate the weights of evaluation indexes, and a combined assignment method is established to improve the reliability of the evaluation method. Then, the best solution is selected by calculating the comprehensive evaluation value of each product design solution. Finally, the validity, reasonableness, and feasibility of the evaluation model are verified through the evaluation and selection of three design solutions for a gearbox reducer, which also provides a new method for the evaluation and selection of other product design solutions.


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## Body

## 1. Introduction

In the conceptual design phase of mechanical products, schematic design is a key aspect of product design, and the quality of the product design scheme directly affects the final product. The product solution design stage is a complex, fully defined, and innovative design reasoning process [1–4]. According to the product demand information, there may be a variety of product solutions designed, and the evaluation of the scheme often involves a variety of technical indicators. How to consider the impact of various uncertainty factors on the weight of the indicators, it is necessary to establish a perfect product design program evaluation model, and comprehensively evaluate many technical indicators. Thus, it can obtain the optimal mechanical product design scheme, which is also the key to ensuring product performance and further design [5–8]. A comprehensive evaluation of mechanical product design solutions can effectively ensure the quality of the design and also enable the designer to preferably select the best solution among many design solutions that meet the target requirements in all aspects of performance [9–11].There are various evaluation methods commonly used in product solution design, such as data envelopment analysis (DEA) [12], robust design [13], analytical hierarchy process (AHP) [14], grey correlation analysis [15], artificial neural network [16], analytic network process [17], etc. Chen et al. [18] developed a decision model that combines the design criteria of IF World Design Guidelines and multiattribute decision methods, which can help decision-makers to systematically evaluate and improve the performance of product designs. Yuan et al. [19] proposed a new fuzzy integrated evaluation method for configuration solutions integrating customer requirements based on fuzzy set theory. Li and Zhang [20] proposed a method combining the Kano model (KM), hierarchical analysis method (AHP), and quality function unfolding (QFD) method with intuitionistic fuzzy sets (IFS) to solve the design decision problem of new product development, and finally the reliability and scientificity of the method were verified by examples. Xu et al. [21] described the evaluation problem of the overall mechanical product design solution as an incomplete multiparameter decision problem and established a decision method for the evaluation of mechanical product solution design by information entropy and ordered weighted average operator. Yumoto [22] established an AHP-based product selection based on the decision rules of a rough set of qualitative evaluation decision support system. Du et al. [23] proposed an improved approach to the traditional quantitative Kano model that facilitates the personalization of products for heterogeneous customer classification groups and can realize accurate marketing. At present, most of the research on the evaluation methods of mechanical product design solutions is done by applying fuzzy theory, fuzzy processing of qualitative indicators, and using methods such as expert scoring to rank the overall evaluation value of design solutions. In the evaluation process, the setting of indicator weights mostly uses empirical methods, which has a strong subjectivity. In this article, the characteristics of technical indicators of mechanical product design solutions are analyzed, hierarchical analysis and information entropy are introduced to describe the uncertainty of indicator weights, the comprehensive evaluation value of design solutions is applied, a preferential model and solution method are constructed for the evaluation of product design solutions, and finally the effectiveness of the method is verified through the evaluation of a gearbox reducer design solution as an example. This evaluation method can not only improve the efficiency by fully scientific design evaluation but also reduce the design cost.


## 2. Construction of Fuzzy Comprehensive Evaluation Analysis Method

### 2.1. Fuzzy Comprehensive Evaluation System

Fuzzy hierarchical analysis is an evaluation method formed by integrating fuzzy mathematics and hierarchical analysis. The complexity of the objective world and the ever-changing nature of eternal motion give the world a random uncertainty and a more general uncertainty, that is fuzziness. Therefore, based on the affiliation theory of fuzzy mathematics, qualitative evaluation is transformed into quantitative evaluation, that is, fuzzy mathematics is used to make an overall evaluation of things or objects that are subject to multiple factors [24]. This method can effectively improve the objectivity and scientific nature of the design evaluation, and the specific evaluation process is:(1)
Determining the set of evaluation indicatorsUIn order to facilitate weight allocation and evaluation, the evaluation indicators can be divided intom subsets according to the attributes of the evaluation indicators, and each category is regarded as a single evaluation indicator and called the first-level evaluation indicator. The first-level evaluation index can set the subordinate second-level evaluation index, and the second-level evaluation index can set the subordinate third-level evaluation index, and so on. Denoted as U1,U2,…,Us, it should satisfy:(1)U=U1∪U2∪…∪UsUi∩Uj=∅i≠j,i,j=1,2,…,s.(2)
Determining the rubric setV of evaluation indicatorsThe set of rubricsV=v1,v2,…,vn is the evaluation grade standard and the set of rubric grades is composed of the total evaluation results that the evaluator may make to the evaluated object. In this article, the rubric set V is divided into five levels: good, better, average, qualified, and unqualified, and its level score corresponds to 90, 70, 50, 30, and 10.(3)
Determining the weight vector of evaluation indicatorsThe weight is a quantitative value that compares and weighs the relative importance of the factors in the evaluated thing as a whole. In the fuzzy comprehensive evaluation, the weights will have a great influence on the final evaluation results, and different weights will sometimes get completely different conclusions.(4)
Establishing the fuzzy comprehensive evaluation matrixRThe fuzzy comprehensive evaluation matrix composed of the affiliation degree of each evaluation index relative to the evaluation set is:(2)R=r11r12⋯r1nr21r22⋯r2n⋮⋮⋱⋮rm1rm2⋯rmn0≤rij≤1.(5)
Calculating the evaluation vectorB of the product design solutionAfter determining the weights of each evaluation index, the evaluation vectorB of the product design solution is obtained by synthesizing the resulting weight vector W of each evaluation index with the evaluation matrix R of the corresponding evaluation indicators.(3)B=W•R=ω1,ω2,⋯,ωnr11r12⋯r1nr21r22⋯r2n⋮⋮⋱⋮rm1rm2⋯rmn=b1,b2,⋯,bn.(6)
Calculating the comprehensive evaluation valueK of product design solutionsThe fuzzy comprehensive evaluation matrixB is transformed into a comprehensive evaluation value K, and different solutions are selected according to their K values.(4)K=B·VT.After getting the evaluation value, a comparison of multiple solutions will give an intuitive and quantitative understanding.


### 2.2. Calculation of Integrated Weights

The methods for calculating the weights of product design evaluation indicators are mainly divided into subjective and objective methods. The final results obtained when applying the subjective method are highly subjective, while the objective method is a method that uses attribute indicators to determine the weights. In order to get more objective and reliable weights of product design evaluation indexes, this article combines subjective and objective methods and proposes a combined weighting method. The subjective method applies the hierarchical analysis method and the objective method applies the entropy value method, then the weights obtained from both are combined, and then the comprehensive weights are derived to provide reliable calculated weights for the subsequent evaluation of product design solutions.


#### 2.2.1. Analytical Hierarchy Analysis to Determine the Weights

In the early 1970s, American operations researcher Professor Saaty proposed the analytical hierarchy process (AHP) [25]. After years of development, this method is now a more mature combination of qualitative and quantitative analysis of multi-criteria decision-making methods. AHP is easy to use, reliable, and practical. It has been widely used at home and abroad after years of development. The principle of AHP is similar to the process of thinking and judging when deciding; when analyzing a problem, the decision-maker has to establish a hierarchical recursive system structure for the influencing factors of the constraint object, so that the decision-maker can rationalize the complex problem.(1) Construct Decision-Making Indicator System. The establishment of decision indicators is generally determined according to the professional knowledge or engineering practice experience of the evaluator, but the indicators selected in this way are somewhat arbitrary, and the number of decision indicators usually selected is too large, with a serious overlap of information between them and poor representativeness. Therefore, it is necessary to select decision indicators in accordance with the principles of completeness and coordination, scientificity and mutual exclusivity, feasibility and sensitivity, the combination of dynamic and static, the combination of qualitative and quantitative principles, and the decision indicator system of the final constructed product design scheme.(2) Constructing Judgment Matrix. After the recursive hierarchy has been established, the affiliation between the upper and lower elements is determined. The next step is to determine the weights of the elements at each level. The relative importance of the different indicators is determined by a two-by-two comparison of the indicators at the next level that affects the indicators at the upper level. In this article, the 1 to 9 scale method of hierarchical analysis [26] (Table 1) is used to construct a judgment matrix by comparing each evaluation index in the evaluation system between two so as to calculate the weights corresponding to the evaluation indexes at the criterion level and each subcriterion level.Table 1
Scale value table.
Scale valueThe value ofaji1Evaluation indicatorA is the same importance as B2Evaluation indicatorA is slightly important than B3Evaluation indicatorA is slightly important than B4Evaluation indicatorA is more important than B5Evaluation indicatorA is obviously important than B6Evaluation indicatorA is very important than B7Evaluation indicatorA is fiercely important than B8Evaluation indicatorA is more fiercely important than B9Evaluation indicatorA is fiercely important than B(3) Consistency Test of Judgment Matrix. The judgment matrix constructed by a two-by-two comparison does not necessarily have consistency, and a consistency test is required to control the deviation generated by the judgment matrix within a certain range before proceeding to calculate the weights. At present, the maximum characteristic root λmax of the judgment matrix is generally applied to test the consistency of the judgment. The consistency index is calculated according to the following expression:(5)CI=λmax−mm−1,where CI is the consistency index of the judgment matrix, λmax is the maximum eigenvalue of the judgment matrix A, and m is the order of the judgment matrix A.The average random consistency indexRI is introduced to measure the allowable range of inconsistency of judgment matrices of different orders. The RI values of judgment matrices of order 1 to 9 are shown in Table 2.Table 2
RI values of high-order average random consistency index.
m12345678910RI0.000.000.580.901.121.241.321.411.451.48Determining the allowable range of inconsistency and calculating the inconsistency ratioCR test formula for the judgment matrix A is performed according to the following expression:(6)CR=CIRI,where CR is the consistency ratio and RI is the random consistency index.When the calculatedCR ≤ 0.1, the judgment matrix can be considered consistent, indicating a reasonable weight assignment.(4) Calculation of Weights. There are many methods to calculate the final weight vector for hierarchical analysis, among which the easiest and most commonly used method is the eigenvector method; so in this article, we choose the eigenvector as the method to calculate the weight vector and has Perron’s theorem [27] as its theoretical basis.Perron’s theorem: IfA=aijn×n is a positive matrix and ρA is its spectral radius, then the following conditions are satisfied.(1)
The largest eigenvalueλmax of A exists, is unique, and λmax=ρA;(2)
The normalized eigenvectorλmax corresponding to ω=W1,W2,…,WnT is a positive vector, that is every element of ω is greater than zero.The relative importance weight vectorω is first obtained by solving the eigenequation of matrix A in terms of the largest eigenvalue λmax and its corresponding eigenvector and normalizing it.The eigenvectors are obtained by solving the eigenvalues of matrixA and their corresponding eigenvectors λmax and then normalizing them to obtain the relative importance weight vector ω.(7)A−λmaxIω=0.The eigenvector corresponding toλ is(8)ω=ω1,ω2,…,ωnT.The feature vectorω corresponding to the maximum eigenvalue λmax is the weight vector of the scheme set.(5) Multilevel Indicator Weights. The calculation method of the weight vector in the comprehensive evaluation can obtain the weight of each decision indicator to the upper decision factor layer, the weight of each indicator to the total decision target is needed in the decision, and the weight of the decision factor to the total decision target can be calculated by AHP. Let there be a total of s decision indicators in the decision factor layer and the weight D of the indicators in the factor layer obtained by AHP, then the weight of the final decision indicator on the total decision objective is equal to the product of the weight of the corresponding decision indicator and the weight of the upper decision factor, which is denoted as:(9)ωj=ωje•ωjz,j=1,2,…,s,where ωje is the weight of the decision factor C corresponding to the indicator Cj, ωjz is the weight of decision factor C to the overall objective of the decision, ωj is the weight of indicator Cj on the total objective of the decision.


#### 2.2.2. Entropy Evaluation Method for Determining Weights

In information theory, entropy is a measure of uncertainty. The greater the amount of information, the smaller the uncertainty and the smaller the entropy; conversely, the greater the entropy [28]. According to the properties of entropy, the randomness and the degree of disorder of a scheme can be judged by calculating the entropy value. The entropy evaluation method (EEM) is an objective assignment method, which determines the index weights according to the size of the information provided by the observations of each index and can reduce the influence of the subjectivity that exists in the hierarchical analysis method on the analysis results [29].(1)
Constructing the indicator matrixAssuming that there aren evaluation levels and m evaluation indicators, the evaluation indicators are scored according to the expert opinions, which constitute an n × m indicator matrix A.(10)A=a11a12⋯a1ma21a22⋯⋯⋯⋯⋯⋯an1an2⋯anmwhereaij is the value of the jth indicator of the ith program.(2)
Normalizing the indicator matrixSince the units of measurement of each indicator are not uniform, it is necessary to standardize them before using them to calculate the composite indicators so as to solve the problem of homogenization of the different qualitative indicator values. The indicator matrixA is normalized to obtain the normalization matrix P.(11)P=p11p12⋯p1mp21p22⋯p2m⋮⋮⋱⋮pn1pn2⋯pnmpij=qij∑i=1nqijpositive indicatorsqij=aij−aminamax−aminpositive indicatorsqij=amax−aijamax−amini=1,…,n,j=1,…m(3)
Calculating the information entropy valueEj of the jth indicator(12)Ej=−1lnn∑i=1npijlnpiji=1,…,n,j=1,…,m,where: 0 ≤Ej ≤ 1.(4)
Calculating the information entropy redundancyDj(13)Dj=1−Ej.(5)
Calculating the entropy weight of each indicator(14)ωj=Dj∑j=1mDj.


#### 2.2.3. Determining the Comprehensive Weights of Indicators

The hierarchical analysis method and entropy value method have obtained subjective weights and objective weights, respectively, and it is necessary to recombine the subjective and objective weights to get more accurate comprehensive weights. Then the comprehensive weight of each evaluation index is calculated by the following formula [30].(15)Wj=ωja·ωje∑j=1mωja·ωjeAHP:ωa=ω1a,ω1a,…,ωmaEEM:ωe=ω1e,ω1e,…,ωme.


## 2.1. Fuzzy Comprehensive Evaluation System

Fuzzy hierarchical analysis is an evaluation method formed by integrating fuzzy mathematics and hierarchical analysis. The complexity of the objective world and the ever-changing nature of eternal motion give the world a random uncertainty and a more general uncertainty, that is fuzziness. Therefore, based on the affiliation theory of fuzzy mathematics, qualitative evaluation is transformed into quantitative evaluation, that is, fuzzy mathematics is used to make an overall evaluation of things or objects that are subject to multiple factors [24]. This method can effectively improve the objectivity and scientific nature of the design evaluation, and the specific evaluation process is:(1)
Determining the set of evaluation indicatorsUIn order to facilitate weight allocation and evaluation, the evaluation indicators can be divided intom subsets according to the attributes of the evaluation indicators, and each category is regarded as a single evaluation indicator and called the first-level evaluation indicator. The first-level evaluation index can set the subordinate second-level evaluation index, and the second-level evaluation index can set the subordinate third-level evaluation index, and so on. Denoted as U1,U2,…,Us, it should satisfy:(1)U=U1∪U2∪…∪UsUi∩Uj=∅i≠j,i,j=1,2,…,s.(2)
Determining the rubric setV of evaluation indicatorsThe set of rubricsV=v1,v2,…,vn is the evaluation grade standard and the set of rubric grades is composed of the total evaluation results that the evaluator may make to the evaluated object. In this article, the rubric set V is divided into five levels: good, better, average, qualified, and unqualified, and its level score corresponds to 90, 70, 50, 30, and 10.(3)
Determining the weight vector of evaluation indicatorsThe weight is a quantitative value that compares and weighs the relative importance of the factors in the evaluated thing as a whole. In the fuzzy comprehensive evaluation, the weights will have a great influence on the final evaluation results, and different weights will sometimes get completely different conclusions.(4)
Establishing the fuzzy comprehensive evaluation matrixRThe fuzzy comprehensive evaluation matrix composed of the affiliation degree of each evaluation index relative to the evaluation set is:(2)R=r11r12⋯r1nr21r22⋯r2n⋮⋮⋱⋮rm1rm2⋯rmn0≤rij≤1.(5)
Calculating the evaluation vectorB of the product design solutionAfter determining the weights of each evaluation index, the evaluation vectorB of the product design solution is obtained by synthesizing the resulting weight vector W of each evaluation index with the evaluation matrix R of the corresponding evaluation indicators.(3)B=W•R=ω1,ω2,⋯,ωnr11r12⋯r1nr21r22⋯r2n⋮⋮⋱⋮rm1rm2⋯rmn=b1,b2,⋯,bn.(6)
Calculating the comprehensive evaluation valueK of product design solutionsThe fuzzy comprehensive evaluation matrixB is transformed into a comprehensive evaluation value K, and different solutions are selected according to their K values.(4)K=B·VT.After getting the evaluation value, a comparison of multiple solutions will give an intuitive and quantitative understanding.


## 2.2. Calculation of Integrated Weights

The methods for calculating the weights of product design evaluation indicators are mainly divided into subjective and objective methods. The final results obtained when applying the subjective method are highly subjective, while the objective method is a method that uses attribute indicators to determine the weights. In order to get more objective and reliable weights of product design evaluation indexes, this article combines subjective and objective methods and proposes a combined weighting method. The subjective method applies the hierarchical analysis method and the objective method applies the entropy value method, then the weights obtained from both are combined, and then the comprehensive weights are derived to provide reliable calculated weights for the subsequent evaluation of product design solutions.


### 2.2.1. Analytical Hierarchy Analysis to Determine the Weights

In the early 1970s, American operations researcher Professor Saaty proposed the analytical hierarchy process (AHP) [25]. After years of development, this method is now a more mature combination of qualitative and quantitative analysis of multi-criteria decision-making methods. AHP is easy to use, reliable, and practical. It has been widely used at home and abroad after years of development. The principle of AHP is similar to the process of thinking and judging when deciding; when analyzing a problem, the decision-maker has to establish a hierarchical recursive system structure for the influencing factors of the constraint object, so that the decision-maker can rationalize the complex problem.(1) Construct Decision-Making Indicator System. The establishment of decision indicators is generally determined according to the professional knowledge or engineering practice experience of the evaluator, but the indicators selected in this way are somewhat arbitrary, and the number of decision indicators usually selected is too large, with a serious overlap of information between them and poor representativeness. Therefore, it is necessary to select decision indicators in accordance with the principles of completeness and coordination, scientificity and mutual exclusivity, feasibility and sensitivity, the combination of dynamic and static, the combination of qualitative and quantitative principles, and the decision indicator system of the final constructed product design scheme.(2) Constructing Judgment Matrix. After the recursive hierarchy has been established, the affiliation between the upper and lower elements is determined. The next step is to determine the weights of the elements at each level. The relative importance of the different indicators is determined by a two-by-two comparison of the indicators at the next level that affects the indicators at the upper level. In this article, the 1 to 9 scale method of hierarchical analysis [26] (Table 1) is used to construct a judgment matrix by comparing each evaluation index in the evaluation system between two so as to calculate the weights corresponding to the evaluation indexes at the criterion level and each subcriterion level.Table 1
Scale value table.
Scale valueThe value ofaji1Evaluation indicatorA is the same importance as B2Evaluation indicatorA is slightly important than B3Evaluation indicatorA is slightly important than B4Evaluation indicatorA is more important than B5Evaluation indicatorA is obviously important than B6Evaluation indicatorA is very important than B7Evaluation indicatorA is fiercely important than B8Evaluation indicatorA is more fiercely important than B9Evaluation indicatorA is fiercely important than B(3) Consistency Test of Judgment Matrix. The judgment matrix constructed by a two-by-two comparison does not necessarily have consistency, and a consistency test is required to control the deviation generated by the judgment matrix within a certain range before proceeding to calculate the weights. At present, the maximum characteristic root λmax of the judgment matrix is generally applied to test the consistency of the judgment. The consistency index is calculated according to the following expression:(5)CI=λmax−mm−1,where CI is the consistency index of the judgment matrix, λmax is the maximum eigenvalue of the judgment matrix A, and m is the order of the judgment matrix A.The average random consistency indexRI is introduced to measure the allowable range of inconsistency of judgment matrices of different orders. The RI values of judgment matrices of order 1 to 9 are shown in Table 2.Table 2
RI values of high-order average random consistency index.
m12345678910RI0.000.000.580.901.121.241.321.411.451.48Determining the allowable range of inconsistency and calculating the inconsistency ratioCR test formula for the judgment matrix A is performed according to the following expression:(6)CR=CIRI,where CR is the consistency ratio and RI is the random consistency index.When the calculatedCR ≤ 0.1, the judgment matrix can be considered consistent, indicating a reasonable weight assignment.(4) Calculation of Weights. There are many methods to calculate the final weight vector for hierarchical analysis, among which the easiest and most commonly used method is the eigenvector method; so in this article, we choose the eigenvector as the method to calculate the weight vector and has Perron’s theorem [27] as its theoretical basis.Perron’s theorem: IfA=aijn×n is a positive matrix and ρA is its spectral radius, then the following conditions are satisfied.(1)
The largest eigenvalueλmax of A exists, is unique, and λmax=ρA;(2)
The normalized eigenvectorλmax corresponding to ω=W1,W2,…,WnT is a positive vector, that is every element of ω is greater than zero.The relative importance weight vectorω is first obtained by solving the eigenequation of matrix A in terms of the largest eigenvalue λmax and its corresponding eigenvector and normalizing it.The eigenvectors are obtained by solving the eigenvalues of matrixA and their corresponding eigenvectors λmax and then normalizing them to obtain the relative importance weight vector ω.(7)A−λmaxIω=0.The eigenvector corresponding toλ is(8)ω=ω1,ω2,…,ωnT.The feature vectorω corresponding to the maximum eigenvalue λmax is the weight vector of the scheme set.(5) Multilevel Indicator Weights. The calculation method of the weight vector in the comprehensive evaluation can obtain the weight of each decision indicator to the upper decision factor layer, the weight of each indicator to the total decision target is needed in the decision, and the weight of the decision factor to the total decision target can be calculated by AHP. Let there be a total of s decision indicators in the decision factor layer and the weight D of the indicators in the factor layer obtained by AHP, then the weight of the final decision indicator on the total decision objective is equal to the product of the weight of the corresponding decision indicator and the weight of the upper decision factor, which is denoted as:(9)ωj=ωje•ωjz,j=1,2,…,s,where ωje is the weight of the decision factor C corresponding to the indicator Cj, ωjz is the weight of decision factor C to the overall objective of the decision, ωj is the weight of indicator Cj on the total objective of the decision.


### 2.2.2. Entropy Evaluation Method for Determining Weights

In information theory, entropy is a measure of uncertainty. The greater the amount of information, the smaller the uncertainty and the smaller the entropy; conversely, the greater the entropy [28]. According to the properties of entropy, the randomness and the degree of disorder of a scheme can be judged by calculating the entropy value. The entropy evaluation method (EEM) is an objective assignment method, which determines the index weights according to the size of the information provided by the observations of each index and can reduce the influence of the subjectivity that exists in the hierarchical analysis method on the analysis results [29].(1)
Constructing the indicator matrixAssuming that there aren evaluation levels and m evaluation indicators, the evaluation indicators are scored according to the expert opinions, which constitute an n × m indicator matrix A.(10)A=a11a12⋯a1ma21a22⋯⋯⋯⋯⋯⋯an1an2⋯anmwhereaij is the value of the jth indicator of the ith program.(2)
Normalizing the indicator matrixSince the units of measurement of each indicator are not uniform, it is necessary to standardize them before using them to calculate the composite indicators so as to solve the problem of homogenization of the different qualitative indicator values. The indicator matrixA is normalized to obtain the normalization matrix P.(11)P=p11p12⋯p1mp21p22⋯p2m⋮⋮⋱⋮pn1pn2⋯pnmpij=qij∑i=1nqijpositive indicatorsqij=aij−aminamax−aminpositive indicatorsqij=amax−aijamax−amini=1,…,n,j=1,…m(3)
Calculating the information entropy valueEj of the jth indicator(12)Ej=−1lnn∑i=1npijlnpiji=1,…,n,j=1,…,m,where: 0 ≤Ej ≤ 1.(4)
Calculating the information entropy redundancyDj(13)Dj=1−Ej.(5)
Calculating the entropy weight of each indicator(14)ωj=Dj∑j=1mDj.


### 2.2.3. Determining the Comprehensive Weights of Indicators

The hierarchical analysis method and entropy value method have obtained subjective weights and objective weights, respectively, and it is necessary to recombine the subjective and objective weights to get more accurate comprehensive weights. Then the comprehensive weight of each evaluation index is calculated by the following formula [30].(15)Wj=ωja·ωje∑j=1mωja·ωjeAHP:ωa=ω1a,ω1a,…,ωmaEEM:ωe=ω1e,ω1e,…,ωme.


## 2.2.1. Analytical Hierarchy Analysis to Determine the Weights

In the early 1970s, American operations researcher Professor Saaty proposed the analytical hierarchy process (AHP) [25]. After years of development, this method is now a more mature combination of qualitative and quantitative analysis of multi-criteria decision-making methods. AHP is easy to use, reliable, and practical. It has been widely used at home and abroad after years of development. The principle of AHP is similar to the process of thinking and judging when deciding; when analyzing a problem, the decision-maker has to establish a hierarchical recursive system structure for the influencing factors of the constraint object, so that the decision-maker can rationalize the complex problem.(1) Construct Decision-Making Indicator System. The establishment of decision indicators is generally determined according to the professional knowledge or engineering practice experience of the evaluator, but the indicators selected in this way are somewhat arbitrary, and the number of decision indicators usually selected is too large, with a serious overlap of information between them and poor representativeness. Therefore, it is necessary to select decision indicators in accordance with the principles of completeness and coordination, scientificity and mutual exclusivity, feasibility and sensitivity, the combination of dynamic and static, the combination of qualitative and quantitative principles, and the decision indicator system of the final constructed product design scheme.(2) Constructing Judgment Matrix. After the recursive hierarchy has been established, the affiliation between the upper and lower elements is determined. The next step is to determine the weights of the elements at each level. The relative importance of the different indicators is determined by a two-by-two comparison of the indicators at the next level that affects the indicators at the upper level. In this article, the 1 to 9 scale method of hierarchical analysis [26] (Table 1) is used to construct a judgment matrix by comparing each evaluation index in the evaluation system between two so as to calculate the weights corresponding to the evaluation indexes at the criterion level and each subcriterion level.Table 1
Scale value table.
Scale valueThe value ofaji1Evaluation indicatorA is the same importance as B2Evaluation indicatorA is slightly important than B3Evaluation indicatorA is slightly important than B4Evaluation indicatorA is more important than B5Evaluation indicatorA is obviously important than B6Evaluation indicatorA is very important than B7Evaluation indicatorA is fiercely important than B8Evaluation indicatorA is more fiercely important than B9Evaluation indicatorA is fiercely important than B(3) Consistency Test of Judgment Matrix. The judgment matrix constructed by a two-by-two comparison does not necessarily have consistency, and a consistency test is required to control the deviation generated by the judgment matrix within a certain range before proceeding to calculate the weights. At present, the maximum characteristic root λmax of the judgment matrix is generally applied to test the consistency of the judgment. The consistency index is calculated according to the following expression:(5)CI=λmax−mm−1,where CI is the consistency index of the judgment matrix, λmax is the maximum eigenvalue of the judgment matrix A, and m is the order of the judgment matrix A.The average random consistency indexRI is introduced to measure the allowable range of inconsistency of judgment matrices of different orders. The RI values of judgment matrices of order 1 to 9 are shown in Table 2.Table 2
RI values of high-order average random consistency index.
m12345678910RI0.000.000.580.901.121.241.321.411.451.48Determining the allowable range of inconsistency and calculating the inconsistency ratioCR test formula for the judgment matrix A is performed according to the following expression:(6)CR=CIRI,where CR is the consistency ratio and RI is the random consistency index.When the calculatedCR ≤ 0.1, the judgment matrix can be considered consistent, indicating a reasonable weight assignment.(4) Calculation of Weights. There are many methods to calculate the final weight vector for hierarchical analysis, among which the easiest and most commonly used method is the eigenvector method; so in this article, we choose the eigenvector as the method to calculate the weight vector and has Perron’s theorem [27] as its theoretical basis.Perron’s theorem: IfA=aijn×n is a positive matrix and ρA is its spectral radius, then the following conditions are satisfied.(1)
The largest eigenvalueλmax of A exists, is unique, and λmax=ρA;(2)
The normalized eigenvectorλmax corresponding to ω=W1,W2,…,WnT is a positive vector, that is every element of ω is greater than zero.The relative importance weight vectorω is first obtained by solving the eigenequation of matrix A in terms of the largest eigenvalue λmax and its corresponding eigenvector and normalizing it.The eigenvectors are obtained by solving the eigenvalues of matrixA and their corresponding eigenvectors λmax and then normalizing them to obtain the relative importance weight vector ω.(7)A−λmaxIω=0.The eigenvector corresponding toλ is(8)ω=ω1,ω2,…,ωnT.The feature vectorω corresponding to the maximum eigenvalue λmax is the weight vector of the scheme set.(5) Multilevel Indicator Weights. The calculation method of the weight vector in the comprehensive evaluation can obtain the weight of each decision indicator to the upper decision factor layer, the weight of each indicator to the total decision target is needed in the decision, and the weight of the decision factor to the total decision target can be calculated by AHP. Let there be a total of s decision indicators in the decision factor layer and the weight D of the indicators in the factor layer obtained by AHP, then the weight of the final decision indicator on the total decision objective is equal to the product of the weight of the corresponding decision indicator and the weight of the upper decision factor, which is denoted as:(9)ωj=ωje•ωjz,j=1,2,…,s,where ωje is the weight of the decision factor C corresponding to the indicator Cj, ωjz is the weight of decision factor C to the overall objective of the decision, ωj is the weight of indicator Cj on the total objective of the decision.


## 2.2.2. Entropy Evaluation Method for Determining Weights

In information theory, entropy is a measure of uncertainty. The greater the amount of information, the smaller the uncertainty and the smaller the entropy; conversely, the greater the entropy [28]. According to the properties of entropy, the randomness and the degree of disorder of a scheme can be judged by calculating the entropy value. The entropy evaluation method (EEM) is an objective assignment method, which determines the index weights according to the size of the information provided by the observations of each index and can reduce the influence of the subjectivity that exists in the hierarchical analysis method on the analysis results [29].(1)
Constructing the indicator matrixAssuming that there aren evaluation levels and m evaluation indicators, the evaluation indicators are scored according to the expert opinions, which constitute an n × m indicator matrix A.(10)A=a11a12⋯a1ma21a22⋯⋯⋯⋯⋯⋯an1an2⋯anmwhereaij is the value of the jth indicator of the ith program.(2)
Normalizing the indicator matrixSince the units of measurement of each indicator are not uniform, it is necessary to standardize them before using them to calculate the composite indicators so as to solve the problem of homogenization of the different qualitative indicator values. The indicator matrixA is normalized to obtain the normalization matrix P.(11)P=p11p12⋯p1mp21p22⋯p2m⋮⋮⋱⋮pn1pn2⋯pnmpij=qij∑i=1nqijpositive indicatorsqij=aij−aminamax−aminpositive indicatorsqij=amax−aijamax−amini=1,…,n,j=1,…m(3)
Calculating the information entropy valueEj of the jth indicator(12)Ej=−1lnn∑i=1npijlnpiji=1,…,n,j=1,…,m,where: 0 ≤Ej ≤ 1.(4)
Calculating the information entropy redundancyDj(13)Dj=1−Ej.(5)
Calculating the entropy weight of each indicator(14)ωj=Dj∑j=1mDj.


## 2.2.3. Determining the Comprehensive Weights of Indicators

The hierarchical analysis method and entropy value method have obtained subjective weights and objective weights, respectively, and it is necessary to recombine the subjective and objective weights to get more accurate comprehensive weights. Then the comprehensive weight of each evaluation index is calculated by the following formula [30].(15)Wj=ωja·ωje∑j=1mωja·ωjeAHP:ωa=ω1a,ω1a,…,ωmaEEM:ωe=ω1e,ω1e,…,ωme.


## 3. Fuzzy Comprehensive Evaluation Based on Mechanical Product Design Scheme Optimization

### 3.1. Establishment of Fuzzy Comprehensive Evaluation System

The article takes the evaluation of a gearbox reducer design scheme as an example, and three preliminary design options are available [31].(1)
A two-stage reduction: this scheme can make full use of space to reduce the center distance, but it is bound to increase the supporting devices such as shafts, gears, and bearings for the first-stage drive.(2)
Single-stage transmission, with the clutch arranged in the middle end and the box components divided into two large parts, namely the box and the front cover; the pinion is hollow-set on the long shaft, and the input power is transmitted to the pinion through the long shaft and the clutch closure on it.(3)
Single-stage transmission, the structure of scheme (2) is partially adjusted: the input shaft adopts torsion shaft transmission, the pinion gear is empty set on the torsion shaft, which makes the reducer have the characteristics of small size; the clutch is arranged at the rear end, the box combination surface is changed from longitudinal section to transverse section, which improves the assembly and disassembly performance of the clutch parts and output shaft parts; however, the process is more difficult.By reviewing the information, a total of 2 principles of technical indicatorsT and economic indicators E were identified as the first-level indicators of the evaluation index system. Performance indicator T1, processability indicator T2, and service-oriented indicator T3 constitute the secondary indicators of technical indicator T, and labor cost E1 and time cost E2 constitute the secondary indicators of economic indicator E. And center distance T11 and weight T12 constitute the tertiary indicators of performance indicator T1, ease of installation T21 and ease of processing T22 constitute the tertiary indicators of processability indicator T2, and ease of maintenance T31 constitutes the tertiary indicators of serviceability indicator T3. The processing cost E11 and material cost E12 are the tertiary indicators of labor cost, and the time of trial production and production start-up E21 and design progress E11 are the tertiary indicators of time cost [31] (Figure 1).Figure 1
Gearbox reducer design scheme evaluation index system.


### 3.2. Method of Calculating the Comprehensive Weights of the Evaluation Indexes of the Design Scheme

(1)
AHP method to determine the evaluation index weights of gearbox reducer design schemeFor gearbox product development, according to historical data and the experience of relevant engineering designers, two comparisons are made using the 1–9 scale method to construct the judgment matrix of evaluation indicators at all levels, calculate the weights under a single criterion based on the work formulas (5)–(8), and conduct consistency tests on the judgment matrix of order n > 2 (Table 3).(2)
EEM to determine the weight of gearbox reducer design evaluation indexBy inviting experts to make a two-by-two comparison of the evaluation indexes at each level of the gearbox reducer design evaluation system shown in Figure1 using the 1 to 9 scale method, the judgment matrix A of the evaluation indexes at each level is constructed.(16)A=11/721/2111/2337184774991/21/811/31/21/21/51121/4312215511/721/2111/23311/721/2111/23321/451221771/31/911/51/31/31/7111/31/911/51/31/31/711The entropy and entropy weights of each three-level index under the gearbox reducer design scheme can be obtained by equations (10)–(13) (Table 4).(3)
Comprehensive weight determination of evaluation indicatorsThe comprehensive weight vector of security principle evaluation indexes can be derived from equation (14) as(17)W1=0.090,0.320,0.069,0.115,0.086,0.082,0.136,0.051,0.051.Table 3
Indicator weights and consistency test.
Tertiary level indicator weight and consistency testEvaluation indicatorsWeight vectorCRConsistency checkT1(0.125, 0.875)——T2(0.25, 0.75)——T3———E1(0.25, 0.75)——E2(0.5, 0.5)——Secondary level indicator weight and consistency testEvaluation indicatorsWeight vectorCRConsistency checkT(0.73, 0.19, 0.08)0.058PassE(0.83, 0.17)——Primary level indicator weight and consistency testEvaluation indicatorsWeight vectorCRConsistency checkOverall solution(0.75, 0.25)——Final weight vectorW=0.068,0.480,0.036,0.107,0.060,0.052,0.155,0.021,0.021Table 4
Entropy and entropy weight of the third-level indicator.
Entropy(0.642, 0.482, 0.677, 0.609, 0.642, 0.642, 0.616, 0.743, 0.743)Entropy weights(0.112, 0.162, 0.101, 0.122, 0.112, 0.112, 0.119, 0.080, 0.080)


### 3.3. Determining the Affiliation MatrixR of the Product Design Solution

In this article, the affiliation matrix of the literature [32] on the three design options of the gearbox reducer is directly chosen as shown in Table 5.Table 5
The affiliation matrix of each design solution.
SchemeAffiliation matrixRScheme 1R1=100000000.20.8000.20.60.20.40.60000000.40.60000.80.20000.80.20.20.40.4000.80.2000Scheme 2R2=0.20.60.200.2000.40.40.200.20.60.200.40.6000000.20.60.20.20.40.40000.20.60.200.80.20000.80.2000Scheme 3R3=0.40.6000000.40.6000.20.60.200.20.40.4000.20.60.20000.40.40.2000.40.40.200.20.40.4000.20.40.400


### 3.4. Determining the Evaluation Matrix B of the Product Design Solution

The evaluation matrixB for each gearbox reducer design option can be calculated according to (3).Scheme 1:(18)B1=W•R1=0.1870,0.0996,0.0342,0.3142,0.3650.Scheme 2:(19)B2=W•R2=0.1620,0.2172,0.3190,0.2206,0.0812.Scheme 3:(20)B3=W•R3=0.0966,0.2934,0.3606,0.2494,0.


### 3.5. Determining the Comprehensive Evaluation Value of the Product Design Solution K

According to (5), the evaluation score K can be calculated for each gearbox reducer design solution.Scheme 1:(21)K1=B1•VT=0.1870,0.0996,0.0342,0.3142,0.365090,70,50,30,10T=38.588.Scheme 2:(22)K2=B2•VT=0.1620,0.2172,0.3190,0.2206,0.081290,70,50,30,10T=53.146.Scheme 3:(23)K3=B3•VT=0.0966,0.2934,0.3606,0.2494,090,70,50,30,10T=54.744.The final rating results of the gearbox reducer design scheme calculated by the above method areK3 > K2 > K1, with scheme 3 being the best, scheme 2 the second best, and scheme 1 the worst. This is consistent with the evaluation results of literature [32, 33], and from the actual analysis, scheme 1 uses two-stage reduction, which requires an additional level of transmission system supporting device, while scheme 2 uses a single-stage transmission and scheme 3 is an improved design based on scheme 2, so the evaluation results are also consistent with the actual situation.


## 3.1. Establishment of Fuzzy Comprehensive Evaluation System

The article takes the evaluation of a gearbox reducer design scheme as an example, and three preliminary design options are available [31].(1)
A two-stage reduction: this scheme can make full use of space to reduce the center distance, but it is bound to increase the supporting devices such as shafts, gears, and bearings for the first-stage drive.(2)
Single-stage transmission, with the clutch arranged in the middle end and the box components divided into two large parts, namely the box and the front cover; the pinion is hollow-set on the long shaft, and the input power is transmitted to the pinion through the long shaft and the clutch closure on it.(3)
Single-stage transmission, the structure of scheme (2) is partially adjusted: the input shaft adopts torsion shaft transmission, the pinion gear is empty set on the torsion shaft, which makes the reducer have the characteristics of small size; the clutch is arranged at the rear end, the box combination surface is changed from longitudinal section to transverse section, which improves the assembly and disassembly performance of the clutch parts and output shaft parts; however, the process is more difficult.By reviewing the information, a total of 2 principles of technical indicatorsT and economic indicators E were identified as the first-level indicators of the evaluation index system. Performance indicator T1, processability indicator T2, and service-oriented indicator T3 constitute the secondary indicators of technical indicator T, and labor cost E1 and time cost E2 constitute the secondary indicators of economic indicator E. And center distance T11 and weight T12 constitute the tertiary indicators of performance indicator T1, ease of installation T21 and ease of processing T22 constitute the tertiary indicators of processability indicator T2, and ease of maintenance T31 constitutes the tertiary indicators of serviceability indicator T3. The processing cost E11 and material cost E12 are the tertiary indicators of labor cost, and the time of trial production and production start-up E21 and design progress E11 are the tertiary indicators of time cost [31] (Figure 1).Figure 1
Gearbox reducer design scheme evaluation index system.


## 3.2. Method of Calculating the Comprehensive Weights of the Evaluation Indexes of the Design Scheme

(1)
AHP method to determine the evaluation index weights of gearbox reducer design schemeFor gearbox product development, according to historical data and the experience of relevant engineering designers, two comparisons are made using the 1–9 scale method to construct the judgment matrix of evaluation indicators at all levels, calculate the weights under a single criterion based on the work formulas (5)–(8), and conduct consistency tests on the judgment matrix of order n > 2 (Table 3).(2)
EEM to determine the weight of gearbox reducer design evaluation indexBy inviting experts to make a two-by-two comparison of the evaluation indexes at each level of the gearbox reducer design evaluation system shown in Figure1 using the 1 to 9 scale method, the judgment matrix A of the evaluation indexes at each level is constructed.(16)A=11/721/2111/2337184774991/21/811/31/21/21/51121/4312215511/721/2111/23311/721/2111/23321/451221771/31/911/51/31/31/7111/31/911/51/31/31/711The entropy and entropy weights of each three-level index under the gearbox reducer design scheme can be obtained by equations (10)–(13) (Table 4).(3)
Comprehensive weight determination of evaluation indicatorsThe comprehensive weight vector of security principle evaluation indexes can be derived from equation (14) as(17)W1=0.090,0.320,0.069,0.115,0.086,0.082,0.136,0.051,0.051.Table 3
Indicator weights and consistency test.
Tertiary level indicator weight and consistency testEvaluation indicatorsWeight vectorCRConsistency checkT1(0.125, 0.875)——T2(0.25, 0.75)——T3———E1(0.25, 0.75)——E2(0.5, 0.5)——Secondary level indicator weight and consistency testEvaluation indicatorsWeight vectorCRConsistency checkT(0.73, 0.19, 0.08)0.058PassE(0.83, 0.17)——Primary level indicator weight and consistency testEvaluation indicatorsWeight vectorCRConsistency checkOverall solution(0.75, 0.25)——Final weight vectorW=0.068,0.480,0.036,0.107,0.060,0.052,0.155,0.021,0.021Table 4
Entropy and entropy weight of the third-level indicator.
Entropy(0.642, 0.482, 0.677, 0.609, 0.642, 0.642, 0.616, 0.743, 0.743)Entropy weights(0.112, 0.162, 0.101, 0.122, 0.112, 0.112, 0.119, 0.080, 0.080)


## 3.3. Determining the Affiliation MatrixR of the Product Design Solution

In this article, the affiliation matrix of the literature [32] on the three design options of the gearbox reducer is directly chosen as shown in Table 5.Table 5
The affiliation matrix of each design solution.
SchemeAffiliation matrixRScheme 1R1=100000000.20.8000.20.60.20.40.60000000.40.60000.80.20000.80.20.20.40.4000.80.2000Scheme 2R2=0.20.60.200.2000.40.40.200.20.60.200.40.6000000.20.60.20.20.40.40000.20.60.200.80.20000.80.2000Scheme 3R3=0.40.6000000.40.6000.20.60.200.20.40.4000.20.60.20000.40.40.2000.40.40.200.20.40.4000.20.40.400


## 3.4. Determining the Evaluation Matrix B of the Product Design Solution

The evaluation matrixB for each gearbox reducer design option can be calculated according to (3).Scheme 1:(18)B1=W•R1=0.1870,0.0996,0.0342,0.3142,0.3650.Scheme 2:(19)B2=W•R2=0.1620,0.2172,0.3190,0.2206,0.0812.Scheme 3:(20)B3=W•R3=0.0966,0.2934,0.3606,0.2494,0.


## 3.5. Determining the Comprehensive Evaluation Value of the Product Design Solution K

According to (5), the evaluation score K can be calculated for each gearbox reducer design solution.Scheme 1:(21)K1=B1•VT=0.1870,0.0996,0.0342,0.3142,0.365090,70,50,30,10T=38.588.Scheme 2:(22)K2=B2•VT=0.1620,0.2172,0.3190,0.2206,0.081290,70,50,30,10T=53.146.Scheme 3:(23)K3=B3•VT=0.0966,0.2934,0.3606,0.2494,090,70,50,30,10T=54.744.The final rating results of the gearbox reducer design scheme calculated by the above method areK3 > K2 > K1, with scheme 3 being the best, scheme 2 the second best, and scheme 1 the worst. This is consistent with the evaluation results of literature [32, 33], and from the actual analysis, scheme 1 uses two-stage reduction, which requires an additional level of transmission system supporting device, while scheme 2 uses a single-stage transmission and scheme 3 is an improved design based on scheme 2, so the evaluation results are also consistent with the actual situation.


## 4. Conclusions

Evaluation decision plays a crucial role in the design process, and its effectiveness directly affects the direction and results of the design progress. In the decision-making process, when determining the weights of each evaluation index, combining the subjectivity of the hierarchical analysis method to determine the weights and the objectivity of the entropy weight method to determine the weights, the decision-makers can evaluate the design solutions more scientifically and accurately and improve the reliability of the design solution evaluation. The final design scheme, Scheme 3, was preferentially selected after evaluating three designs for a gearbox reducer, which is not only consistent with the literature evaluation results [28, 29] but is also in line with the actual analysis results. This method confirms its scientificity, validity, and reliability, thereby reducing the product development cycle and improving its quality.



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*Source: 1458003-2022-08-29.xml*