CN110717264A - Improved strength pareto evolutionary algorithm for multi-objective optimization design of product appearance - Google Patents

Improved strength pareto evolutionary algorithm for multi-objective optimization design of product appearance Download PDF

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CN110717264A
CN110717264A CN201910942687.3A CN201910942687A CN110717264A CN 110717264 A CN110717264 A CN 110717264A CN 201910942687 A CN201910942687 A CN 201910942687A CN 110717264 A CN110717264 A CN 110717264A
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王增
刘卫东
杨明朗
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Abstract

The invention discloses an improved strength pareto evolutionary algorithm for multi-objective optimization design of product appearance, which comprises design analysis and multi-objective optimization design of product appearance, wherein the improved strength pareto evolutionary algorithm comprises product appearance data and product perceptual image data, the main component scoring data of the product appearance outline is obtained by adopting an elliptic Fourier analysis technology, and the perceptual image evaluation mean data of a target adjective is obtained by adopting a perceptual image analysis technology; the multi-objective optimization design of the product shape comprises the following steps of establishing a nonlinear mapping network between a principal component score and an evaluation mean value of the perceptual image of a target adjective by using a genetic algorithm optimization neural network technology, providing a correction operator by using a consistency correlation relation between the principal component score and the evaluation mean value of the perceptual image of the target adjective obtained by using an elliptic Fourier analysis technology, and combining the operator with an improved cross operator and an improved self-adaptive mutation operator to finally form the algorithm of the invention, thereby providing an effective tool for developing the multi-objective optimization design of the product shape.

Description

Improved strength pareto evolutionary algorithm for multi-objective optimization design of product appearance
Technical Field
The invention relates to the field of intelligent design of product appearance, in particular to an improved strength pareto evolutionary algorithm for multi-objective optimization design of product appearance, which is an intelligent design generation algorithm capable of meeting the multi-objective emotional requirement of consumers on the product appearance (if an automobile with luxury, dynamic and elegant appearance is required).
Background
The abundance and homogeneity of the product means that manufacturers face increasingly more severe competition. How to better satisfy the increasingly diverse emotional demands of consumers through product appearance is a key to product design and a necessary requirement for manufacturers to gain competition. Modern perceptual engineering has demonstrated an extremely close relationship between consumers' emotional needs and physical properties of products (such as function and form). As a design method capable of automatically generating the product appearance according to the multi-target emotional requirements of consumers, the multi-target optimization design of the product appearance has more and more important value. The multi-objective optimization algorithm is the key for determining the multi-objective optimization design effect of the product appearance. In order to improve the performance of the multi-objective optimization design of the product appearance, various existing multi-objective optimization algorithms have been tried in the prior art. These studies can be classified into two different types based on the type and application of the multi-objective optimization algorithm. The first category is to aggregate multiple targets and convert the multiple targets into a single target to develop an optimal design. This type of research is simple to implement, but has limited utility to designers and consumers because it limits the search space and eliminates consideration of all possible solutions. The second type of research is the direct use of multi-objective optimization algorithms. Although the application program is more complex, it can better meet the requirements of the multi-objective optimization design of the shape, so that it has become a mainstream trend of research. Such as Yang[1]Multi-objective optimization using a non-dominated sorting genetic algorithm II (NSGA-II) as a multi-objective optimization algorithm and combining a support vector regression machine for a multi-objective fitness function required by NSGA-IIA design method. Su (Chinese character of 'su')[2]And the NSGA-II is also used as a multi-objective optimization algorithm to develop optimization design, and the difference is that the fuzzy neural network is used for providing a fitness function for the NSGA II. Shieh[3]A combination of NSGA-II and quantitative theoretical form I is used. As a widely used second generation classical multi-objective optimization algorithm, NSGA-II has less time complexity and better convergence. However, it does not provide the best diversity compared to other multi-objective optimization algorithms due to its diversity[4]Therefore, the algorithm is still not ideal. In addition to this algorithm, as another well-known second generation classical multi-objective optimization algorithm, an intensity pareto evolutionary algorithm (SPEA2) is also applied to the design of the appearance multi-objective optimization. Li[5]A multi-objective optimization design method for product appearance using SPEA2 as a multi-objective optimization algorithm is proposed. Shieh[6]The comparative research of the multi-objective optimization algorithm which is most suitable for the multi-objective optimization design of the product appearance is completed. This study systematically compared the performance of three classical algorithms, represented by SPEA2, NSGA II, and another multi-objective optimization algorithm, and demonstrated that SPEA2, which provides the most diverse product formats to designers, is the most appropriate multi-objective optimization algorithm for product form design. However, SPEA2 also has insufficient astringency[6]And is easily trapped in local optima due to the inability of SPEA2 to fully utilize the search space due to a fixed evolutionary mechanism[7]And the like. In order to fully utilize the diversity of SPEA2 and avoid the deficiency thereof, the invention provides a new improved intensity pareto evolutionary algorithm ISPEA2, and successfully uses the algorithm in the multi-objective optimization design of product appearance.
Reference documents:
[1]Yang,C.C.Constructing a hybrid Kansei engineering system based onmultiple affective responses:Application to product form design.Computers&Industrial Engineering.2011,60,760–768.
[2] sujianning, Zhannwei, Wujiang Hua, etc. product multiple image modeling evolution design [ J ] computer integrated manufacturing system 2014,20(11): 2675-.
[3]Shieh,M.D.；Li,Y.；Yang,C.C.Product form design model based onmultiobjective optimization and multicriteria decision-making.MathematicalProblems in Engineering.2017,1-15.
[4]Jiang,H.；Kwong,C.K.；Liu,Y.；Ip,W.H.A methodology of integratingaffective design with defining engineering specifications for productdesign.International Journal of Production Research.2014,53,2472–2488.
[5]Li,Y.F.；Shieh,M.D.；Yang,C.C.A posterior preference articulationapproach to Kansei engineering system for product form design.Research inEngineering Design.2019,30,3–19.
[6]Shieh,M.D.；Li,Y.；Yang,C.C.Comparison of multi-objectiveevolutionary algorithms in hybrid Kansei engineering system for product formdesign.Advanced Engineering Informatics.2018,36,31–42.
[7]Zhao,F.Q.；Lei,W.C.；Ma,W.M.；Liu,Y.；Zhang,C.An improved SPEA2algorithm with adaptive selection of evolutionary operators scheme for multi-objective optimization problems.Mathematical Problems in Engineering.2016,1-20.
[8] Wangzhuang, Liuwei Dong, Yangming, Handong, product appearance image design research based on ellipse Fourier [ J/OL ].
http://kns.cnki.net/kcms/detail/11.5946.TP.20190222.0953.008.html.
[9]Wang,G.L.；Wu,J.H.；Wu,J.H.；Wang,X.H.A comparison between the linearneural network method and the multiple linear regression method in themodeling of continuous data.Journal of Computers.2011,6,2143–2148.
[10]Ding,S.F.；Su,C.Y.；Yu,J.Z.An optimizing BP neural networkalgorithm based on genetic algorithm.Artificial Intelligence Review.2011,36,153–162.
[11]Mukaka,M.Statistics corner:A guide to appropriate use ofcorrelationcoefficient in medical research.Malawi medical journal:the journalof Medical Association of Malawi.2012,24,69–71.
Disclosure of Invention
The method comprises two parts of design analysis and product appearance multi-objective optimization design, wherein the design analysis part is a preliminary stage basis and mainly provides accurate and objective quantitative data for the core algorithm of the method. Because the data comprises two kinds of product appearance data and product perceptual image data, the invention adopts the ellipse Fourier analysis technology to obtain the main component score data of the product appearance outline, and uses the perceptual image analysis technology to obtain the perceptual image evaluation mean data of the target adjective. The product appearance multi-objective optimization design part is the core algorithm part of the invention. A genetic algorithm optimized neural network (GABP) technique is used to establish a non-linear mapping network between principal component scores and perceptual image evaluation means of target adjectives, which will be further used as a fitness function for a core multi-objective optimization algorithm. Then, the invention utilizes the consistency correlation relationship between the principal component score obtained by the elliptic Fourier analysis technology and the evaluation mean value of the perceptual image of the target adjective to creatively provide a new correction operator. The operator is combined with the improved crossover operator and the improved adaptive mutation operator to finally form the improved strength pareto evolutionary algorithm ISPEA2 for the multi-objective optimization design of the product shape, the algorithm framework of which is shown in FIG. 1, and the specific technical process and steps are as follows:
1 analysis of design
1.1 elliptic Fourier analysis
The elliptic Fourier analysis of the product appearance comprises four steps of image preprocessing, elliptic Fourier descriptor calculation, elliptic Fourier descriptor normalization and principal component analysis, and the specific process is shown in FIG. 2.
1.1.1 image preprocessing
After image preprocessing, a two-dimensional contour point coordinate sequence z (i) composed of x and y coordinates of a discrete sampling points is obtained, and is expressed as z (i) ═ x (i), y (i) ═ 0,1, …, a-1 (1)
1.1.2 elliptic Fourier descriptor computation
For the coordinate sequence z (i) of the two-dimensional contour points, the two-dimensional contour points are expanded into
Figure BDA0002223343670000031
In the formula:
Figure BDA0002223343670000032
is the abscissa of the center point of the profile,is the ordinate of the profile center point, N is the harmonic frequency, N is the maximum harmonic frequency, and t is the arc length parameter. Obtaining four approximate ellipse Fourier coefficients a of the triangle function of the new interpolation point coordinate function by a Riemann summation methodxn、bxn、ayn、bynIs composed of
Figure BDA0002223343670000041
In the formula: a is the number of new interpolation points, and four coefficients correspond to the same harmonic frequency n. Thus, an arbitrarily closed contour can be described as a set of elliptic Fourier coefficients containing n harmonic frequencies
F is an elliptic Fourier description sub-coefficient matrix.
1.1.3 elliptical Fourier descriptor normalization
The primary elliptic Fourier descriptors of the appearances of the products are inconsistent and need to be further processed to obtain an elliptic Fourier descriptor matrix F with normalized position, direction and size*. Let the normalized coefficients of the nth harmonic be
Figure BDA0002223343670000043
and
Figure BDA0002223343670000044
Comprises the following steps:
(1) setting harmonic DC component
Figure BDA0002223343670000045
Namely, the center point of the contour is moved to the origin of coordinates, and the position normalization can be realized.
(2) The elliptic fourier coefficients are transformed according to the following equation, achieving a directional normalization:
Figure BDA0002223343670000046
in the formula: a is1,b1,c1And d1Are the four harmonic coefficients of the first ellipse,
Figure BDA0002223343670000047
Figure BDA0002223343670000048
(3) calculate the first ellipse E*Then dividing each descriptor matrix by E*Normalizing the size:
1.1.4 principal Components analysis
Principal component analysis is a technique for obtaining a low-dimensional representation of multivariate data, and is used in the present invention to obtain principal component score data for all product profiles. By using the principal component score as the quantitative feature of the product profile, quantitative analysis of the product profile can be well realized. Using principal component analysis to obtain a matrix F with a sample number q*Reducing vitamin to m dimension
Figure BDA0002223343670000051
Z is the principal component score matrix for refining the outline information of the product, and at the moment, the cumulative variance contribution rate of each principal component reaches a certain set value.
1.2 perceptual image analysis
As a quantitative analysis technique for the emotional needs of consumers, perceptual vocabulary analysis is often used to identify key imagery words that describe the consumer's perception of the appearance of a product. The invention adopts the technology to carry out the perceptual image analysis, and the steps are as follows:
step 1: ideograms, i.e., adjectives, are collected and the subject is organized for informal investigations to pick out the appropriate adjectives.
Step 2: the expert was invited to set up a focal group and the most representative adjectives were selected using delphi.
And step 3: representative adjectives are combined with the product samples to form a questionnaire. And (4) performing questionnaire survey to evaluate the perception of the expert on the product image by using the semantic difference table and the 7-point Likter table, and calculating the average value of all expert evaluation scores of each product sample.
And 4, step 4: and combining the perceptual image evaluation mean matrix and the principal component score matrix of the target adjective, and performing correlation analysis on the combined matrix to obtain a correlation coefficient. Then introducing the relation number into the image target comprehensive scoring model[8]To calculate a final target image composite score.
And 5: and determining the target adjectives and the sequence thereof according to the final target image comprehensive score obtained in the step 4.
2 product appearance multi-objective optimization design
2.1 genetic Algorithm optimized neural network construction
The principal component score and the perceptual image evaluation mean (input and output variables, respectively) of the target adjective are continuous variable data. Wang (Wang)[9]Proved that the neural network model has excellent prediction and fitting capability on continuous data, and the genetic algorithm optimization neural network (GABP) technology can better predict the mapping relation between nonlinear input and output variables[10]. Therefore, the present invention adopts the GABP technique as an improvementAnd constructing a high-precision neural network by using a network construction technology of a target optimization algorithm. The detailed implementation procedure of this technique can be found in the literature [10 ]]。
2.2 improved intensity pareto evolutionary Algorithm (ISPEA2)
The optimization of the traditional SPEA2 algorithm is realized mainly by improving the crossover operator, the adaptive mutation operator and the correction operator, and the improved strength pareto evolutionary algorithm with better algorithm convergence and diversity is obtained. The implementation flow is shown in fig. 3, which includes seven main steps:
step 1: initializing a population and an external population, and setting generation numbers.
Step 2: the neural network trained by the GABP is used to calculate fitness values for the multi-objective images corresponding thereto. Then, the fitness value is assigned to the sum PtAnd AtEach of the individuals in (a). Specifically, P is first given according to a dominating relationshiptAnd AtOf each individual, wherein st (i) may be defined as:
St(i)＝|{j|j∈Pt+At∧i＞j}| (8)
in the formula: the symbol > represents the pareto dominant relationship. Then, a coarse fitness value is calculated from the intensity values of each individual as:
Figure BDA0002223343670000061
then, the individual density is calculated to distinguish individuals having the same rough fitness value:
Figure BDA0002223343670000062
in the formula:
Figure BDA0002223343670000063
the distance between the ith individual and the gamma-th adjacent individual after the distances between the individual i and all other individuals are sorted in ascending order is shown. Finally, the fitness value of an individual may be expressed as:
F(i)＝R(i)+D(i) (11)
and step 3: identifying all non-disadvantaged individuals and using context selection techniques from PtAnd AtIn which they are copied to At+1. If A ist+1Is larger than its maximum allowable size, and a is filled with the best non-inferior solution by the truncation operationt+1. Otherwise, with PtAnd AtDominant individual population of (1) to At+1In (1).
And 4, step 4: if the stopping criterion is met, use At+1The non-dominant individuals in (a) act as the final pareto solution set and terminate.
And 5: from A by binary tournament selectiont+1Selecting individuals and putting the individuals into a mating pool.
Step 6: calculating by respectively applying improved crossover operator, improved adaptive mutation operator and correction operator to obtain result population Pt+1。
The calculation process for each operator is as follows:
(1) let principal component score vector
Figure BDA0002223343670000064
Andthe two descendant individuals after the cross operation is carried out by the improved cross operator are expressed as
Figure BDA0002223343670000066
In the formula: α is a random number uniformly distributed over the interval [ -0.5,1.5], and α + β ═ 1; ξ is the number of principal components.
(2) Let principal component score vector
Figure BDA0002223343670000068
For the parent individuals of the t generation to be genetically manipulated, the offspring individuals after mutation manipulation through the improved adaptive mutation operator are represented as
Figure BDA0002223343670000071
In the formula: ms is the coefficient of variation scale; md is the decay rate of the variation; cn is the current number of iterations; mn is the maximum number of iterations; rn1Is a standard normally distributed random number; ub and lb are the upper and lower limits of the gene boundary, respectively.
(3) The modifier modifies the results of the crossover and mutation operations according to the set modification probabilities according to the following equation:
Figure BDA0002223343670000072
Figure BDA0002223343670000073
in the formula: whereinRepresenting offspring individuals obtained after crossover and mutation operations, wherein
Figure BDA0002223343670000076
Is thatIs that
Figure BDA0002223343670000078
Is that
Figure BDA0002223343670000079
i is a number containing all principal components; i.e. i*Corresponding to the principal componentThe main component number which has consistent correlation coefficient with the evaluation mean value of the perceptual image of the target adjective; mc ofi*Is the ith*Correction coefficients of the individual principal components; rn2Is the interval [4, 5]]A random number of (c);is the ith*The correlation coefficient of the z-th target adjective of the principal component, and
Figure BDA00022233436700000711
when a correlation is considered to exist between two variables[11](ii) a Z is the number of target adjectives. By using
Figure BDA00022233436700000712
The replacement corresponds to
Figure BDA00022233436700000713
I of (a)*And the operation of modifying the operator can be completed by the position gene.
The operators proposed in the present invention work as follows for the overall algorithm. Firstly, the value range of alpha and beta in the improved crossover operator is not limited to [0,1 ] in the traditional algorithm]Can make its search space cover
Figure BDA00022233436700000714
Andall neighborhoods of (b) improve the probability of searching for more different descendants, thereby ensuring that the ISPEA2 algorithm has better diversity. Second, the adaptive mutation operator is improved so that the mutation rate can be dynamically changed as the number of iterations increases. In the initial stage of iteration, the global optimal solution is searched faster with a larger variation rate, and when the iteration is finished, the local optimal solution is searched more finely with a smaller variation rate. This helps to improve the convergence of the ISPEA2 algorithm. Finally, the correction operator helps achieve the goal of continuously improving the score of the target adjective in the iteration. Due to the fact that
Figure BDA00022233436700000716
And rn2Value range of (2) determines
Figure BDA00022233436700000717
The value of the correction coefficient is in a reasonable range, so that the ith relation is included no matter whether positive or negative is relevant*The individual genes of the numbered principle component scores may evolve with a preset probability of modification in each iteration in the direction of increasing the target adjective score. Therefore, the introduction of the correction operator can obviously improve the convergence of the algorithm. With reasonably preset modification probabilities, the ISPEA2 algorithm can achieve a good balance between convergence and diversity.
In conclusion, the ISPEA2 algorithm for the product appearance multi-objective optimization design is formed by improving the crossover operator, the adaptive mutation operator and the correction operator. The algorithm successfully makes up the defects of the original SPEA2 algorithm and retains the advantage of diversity.
The invention has the advantages of
The invention overcomes the problem of insufficient performance of the traditional multi-target algorithm in the multi-target optimization design of the product appearance, and has the following advantages:
(1) in order to eliminate main algorithm obstacles restricting the multi-objective optimization design of the product appearance, the invention provides a novel ISPEA2 multi-objective optimization algorithm. By using a novel correction operator introduced by the technical advantages of the elliptic Fourier analysis, an improved crossover operator and an improved adaptive mutation operator, the ISPEA2 has better convergence and diversity than the original algorithm, which is the key for ensuring the innovation and application value of the method.
(2) The invention forms a product appearance multi-objective optimization design integration method taking an ISPEA2 algorithm as a core and combining an elliptic Fourier technology. The method fully utilizes the advantages of the two technologies, so that the method has excellent comprehensive performance on the multi-objective optimization design of the product appearance.
(3) The invention has the advantages that the invention can provide effective method and tool for designers to develop the multi-objective optimization design of the product appearance.
Drawings
Fig. 1 is a schematic diagram of a framework of an improved intensity pareto evolutionary algorithm proposed by the present invention.
FIG. 2 is a flow chart of an elliptical Fourier analysis of a product profile.
Fig. 3 is a flowchart of an implementation of the improved intensity pareto evolutionary algorithm (ISPEA2) provided by the present invention.
Fig. 4 is a graph of 45 representative samples.
Fig. 5 is a diagram of an image preprocessing process.
Fig. 6 is a representative sample silhouette image.
Fig. 7 is a representative sample profile.
Fig. 8 is a graph of the effect of profile reconstruction of sample No. 1 at different harmonic numbers.
FIG. 9 is a graph of neural network prediction error for three target adjectives.
Fig. 10 is a graph showing the variation trend of the C and S index values of the two algorithms under 30 independent runs.
FIG. 11 is a diagram of an automatic simulation system for the proposed algorithm.
Detailed Description
The invention discloses an improved strength pareto evolutionary algorithm for multi-objective optimization design of product appearance, which comprises two parts of design analysis and multi-objective optimization design of product appearance, wherein the data of the two parts comprise product appearance data and product perceptual image data;
the algorithm step of the product appearance multi-objective optimization design part is that a genetic algorithm optimization neural network GABP technology is used for establishing a nonlinear mapping network between principal component scores and a perceptual image evaluation mean value of a target adjective, and the network is further used as a fitness function of a core multi-objective optimization algorithm; then, a correction operator is proposed by utilizing the consistency correlation relationship between the principal component score obtained by the elliptic Fourier analysis technology and the perceptual image evaluation mean value of the target adjective, and the operator is combined with the improved crossover operator and the improved adaptive mutation operator, so that the improved strength pareto evolutionary algorithm ISPEA2 for the multi-objective optimization design of the product appearance is finally formed.
In the design analysis part, the specific steps are S1.1 elliptic Fourier analysis
The elliptic Fourier analysis of the product appearance comprises four steps of image preprocessing, elliptic Fourier descriptor calculation, elliptic Fourier descriptor normalization and principal component analysis:
s1.1.1 image preprocessing
After image preprocessing, a two-dimensional contour point coordinate sequence z (i) composed of x and y coordinates of a discrete sampling points is obtained, and is expressed as z (i) ═ x (i), y (i) ═ 0,1, …, a-1 (1)
S1.1.2 elliptic Fourier descriptor computation
For the coordinate sequence z (i) of the two-dimensional contour points, the two-dimensional contour points are expanded into
Figure BDA0002223343670000091
In the formula:is the abscissa of the center point of the profile,
Figure BDA0002223343670000093
is the ordinate of the profile center point, N is the harmonic frequency, N is the maximum harmonic frequency, and t is the arc length parameter. Obtaining four approximate ellipse Fourier coefficients a of the triangle function of the new interpolation point coordinate function by a Riemann summation methodxn、bxn、ayn、bynIs composed of
Figure BDA0002223343670000094
In the formula: a is the number of new interpolation points, four coefficients correspond to the same harmonic frequency n, and any closed contour is described by a group of elliptic Fourier coefficients containing n harmonic frequencies
Figure BDA0002223343670000101
F is an elliptic Fourier description sub-system number matrix;
s1.1.3 elliptic Fourier descriptor normalization
Let the normalized coefficients of the nth harmonic be
Figure BDA0002223343670000102
and
Figure BDA0002223343670000103
Comprises the following steps:
setting harmonic DC component
Figure BDA0002223343670000104
Namely, the center point of the outline is moved to the origin of coordinates, and the position normalization can be realized;
the elliptic fourier coefficients are transformed according to the following equation, achieving a directional normalization:
Figure BDA0002223343670000105
in the formula: a is1,b1,c1And d1Are the four harmonic coefficients of the first ellipse,
Figure BDA0002223343670000106
Figure BDA0002223343670000107
Figure BDA0002223343670000108
calculate the first ellipse E*Then dividing each descriptor matrix by E*Will be largeSmall normalization:
Figure BDA0002223343670000109
s1.1.4 principal component analysis
Using principal component analysis to obtain a matrix F with a sample number q*Reducing vitamin to m dimension
Z is a principal component score matrix which refines the outline information of the product;
s1.2, analyzing the perceptual image, which comprises the following specific steps:
step 1: collecting ideographs, i.e. adjectives, and organizing the subject to perform informal investigation to pick out appropriate adjectives;
step 2: inviting experts to establish a focus group, and selecting the most representative adjectives by using a Delphi method;
and step 3: combining the representative adjectives with the product samples to form a questionnaire; using the semantic difference table and the 7-point Likter table to perform perception of questionnaire survey evaluation experts on the product images, and calculating the mean value of all expert evaluation scores of each product sample;
and 4, step 4: combining the perceptual image evaluation mean matrix and the principal component score matrix of the target adjective, and performing correlation analysis on the combined matrix to obtain a correlation coefficient; then introducing the relation number into the image target comprehensive scoring model[8]To calculate a final target image composite score;
and 5: and determining the target adjectives and the sequence thereof according to the final target image comprehensive score obtained in the step 4.
In the product shape multi-objective optimization design part, the specific steps are
S2.1, optimizing the neural network construction by the genetic algorithm, and constructing a high-precision neural network by adopting a GABP (gain-based Back propagation) technology as a network construction technology of the multi-target optimization algorithm;
s2.2 improved intensity pareto evolutionary algorithm ISPEA2
The optimization of the traditional SPEA2 algorithm is realized by improving a crossover operator, an adaptive mutation operator and a correction operator, and the improved strength pareto evolution algorithm is obtained, and comprises seven steps:
step 1: initializing a population and an external population, and setting generation numbers;
step 2: the neural network trained by the GABP is used for calculating the fitness value of the multi-target image corresponding to the neural network, and then the fitness value is distributed to the sum PtAnd AtEach of the individuals in (a); specifically, P is first given according to a dominating relationshiptAnd AtOf each individual, wherein st (i) may be defined as:
St(i)＝|{j|j∈Pt+At∧i＞j}| (8)
in the formula: the symbol > represents the pareto dominant relationship, and then a coarse fitness value is calculated from the intensity values of each individual as:
Figure BDA0002223343670000111
then, the individual density is calculated to distinguish individuals having the same rough fitness value:
in the formula:the distance between the ith individual and the gamma adjacent individuals after the distance between the individual i and all other individuals is sorted in ascending order is shown, and finally, the fitness value of the individual can be expressed as:
F(i)＝R(i)+D(i) (11)；
and step 3: identifying all non-disadvantaged individuals and using context selection techniques from PtAnd AtIn which they are copied to At+1If A ist+1Is larger than the maximum size allowed by the operation, and is used for the truncation operationOptimal non-degraded padding At+1Otherwise, with PtAnd AtDominant individual population of (1) to At+1Performing the following steps;
and 4, step 4: if the stopping criterion is met, use At+1The non-dominant individual in (a) as the final pareto solution set and terminates;
and 5: from A by binary tournament selectiont+1Selecting individuals and putting the individuals into a mating pool;
step 6: calculating by respectively applying improved crossover operator, improved adaptive mutation operator and correction operator to obtain result population Pt+1(ii) a The calculation process for each operator is as follows:
1) let principal component score vector
Figure BDA0002223343670000121
And
Figure BDA0002223343670000122
the two descendant individuals after the cross operation is carried out by the improved cross operator are expressed as
Figure BDA0002223343670000123
Figure BDA0002223343670000124
In the formula: α is a random number uniformly distributed over the interval [ -0.5,1.5], and α + β ═ 1; ξ is the number of principal components;
2) let principal component score vector
Figure BDA0002223343670000125
For the parent individuals of the t generation to be genetically manipulated, the offspring individuals after mutation manipulation through the improved adaptive mutation operator are represented as
Figure BDA0002223343670000126
In the formula: ms is the coefficient of variation scale; md is the decay rate of the variation; cn is the current number of iterations; mn is the maximum number of iterations; rn1Is a standard normally distributed random number; ub and lb are the upper and lower limits of the gene boundary, respectively;
3) the modifier modifies the results of the crossover and mutation operations according to the set modification probabilities according to the following equation:
Figure BDA0002223343670000127
Figure BDA0002223343670000128
Figure BDA0002223343670000129
in the formula: wherein
Figure BDA00022233436700001210
Representing offspring individuals obtained after crossover and mutation operations, wherein
Figure BDA00022233436700001211
Is that
Figure BDA00022233436700001212
Figure BDA00022233436700001213
Is that
Figure BDA00022233436700001214
Figure BDA00022233436700001215
Is that
Figure BDA00022233436700001216
i is a number containing all principal components; i.e. i*The main component score and the evaluation mean value of the perceptual image of the target adjective have consistencyA principal component number of the correlation coefficient;
Figure BDA00022233436700001217
is the ith*Correction coefficients of the individual principal components; rn2Is the interval [4, 5]]A random number of (c);
Figure BDA00022233436700001218
is the ith*The correlation coefficient of the z-th target adjective of the principal component, and
Figure BDA00022233436700001219
when a correlation is considered to exist between two variables[11](ii) a Z is the number of target adjectives, by
Figure BDA00022233436700001220
The replacement corresponds to
Figure BDA00022233436700001221
I of (a)*And the operation of modifying the operator can be completed by the position gene.
The following further illustrates the embodiments of the method of the present invention by means of the drawings and practical examples. Because the automobile appearance is the most representative design object in the appearance design of modern products, the method is applied to the multi-objective optimization design of the automobile appearance so as to verify the effectiveness of the method on the multi-objective optimization design of the modern products.
Firstly, a front side view of a sedan is widely collected, 45 samples are reserved on the basis of ensuring the shape discrimination and the brand difference of an automobile sample to form a total sample of a post vehicle as a research object (fig. 4), and specific brands and models are shown in table 1.
TABLE 1 make and model of representative samples
Figure BDA0002223343670000131
(1) Elliptic Fourier analysis
Image preprocessing as shown in FIG. 5After processing, we obtain a sequence of contour point coordinates for a representative sample. As can be seen in FIG. 6, the tire sizes of the different representative samples did not vary significantly, and therefore their effect on the product image was negligible. Therefore, the extracted car profile does not contain the tire information (fig. 7). We perform an elliptical Fourier analysis on the sequence of contour point coordinates according to equations (1-6) and obtain a normalized elliptical Fourier descriptor matrix F for each sample*. Table 2 shows the normalized elliptical fourier descriptor matrix for sample No. 1.
Table 2.1 normalized elliptic fourier descriptor matrix of sample No. (n ═ 64)
The effect was observed by reconstructing the sample profile by inverse fourier transform, as shown in fig. 8. By comparison with the original profile, the harmonic number n is determined to be 64 because at this harmonic number, the reconstructed profile has most of the characteristics of the original profile and is very noisy. Then, F of each sample*The matrix of dimension 64 × 4 is transformed into row vectors, and then the row vectors of all samples are combined into a matrix of dimension 45 × 256. And performing principal component analysis on the matrix to obtain a principal component score matrix of all samples. To ensure high-precision reconstruction of the sample contour and reduce computational complexity, the number of principal components is set to 7 based on the cumulative variance contribution rate. At this time, the cumulative variance contribution rate exceeded 99.95% (see table 3), and the principal component score matrix was a matrix of dimension 45 × 7 (table 4).
TABLE 3 variance contribution ratio of each principal component
Figure BDA0002223343670000141
TABLE 4 principal component score of representative samples
Figure BDA0002223343670000151
(2) Image analysis
Through informal investigations of 16 industrial design professional students (8 males and 8 females), 36 related adjectives were preliminarily selected (table 5). Then, the 11 most representative adjectives were determined by the focus group. The mean matrix of perceptual image evaluations for representative adjectives was calculated after completion of the questionnaire at the focal group (table 6). Finally, a final target image composite score and a ranking of 11 representative adjectives were obtained by using the image target composite score model (table 7). The top three adjectives were set as the target adjectives in this case study, namely "luxury", "stable" and "modern".
TABLE 5.36 related adjectives
Figure BDA0002223343670000152
TABLE 6 perceptual image evaluation mean matrix for representative adjectives
Figure BDA0002223343670000153
Figure BDA0002223343670000161
TABLE 7 final objective image composite score and 11 representative adjective rankings
Figure BDA0002223343670000171
(3) Construction of genetic algorithm optimization neural network
And constructing three genetic algorithm optimization neural networks to respectively predict the calculation scores of the three target adjectives. The number of neurons of the input layer, the hidden layer and the output layer is respectively 7,10 and 1. And generating optimal initial weights and threshold values of the three neural networks by adopting a genetic algorithm. After training is completed, these neural networks are further used as fitness functions to predict the calculated scores of the three target adjectives in ISPEA 2. Table 8 reflects the specific parameter values for the genetic algorithm to optimize the neural network, and fig. 9 shows the prediction error for three neural networks when the number of iterations is 20,000.
TABLE 8 genetic Algorithm and neural network Algorithm parameters
Figure BDA0002223343670000172
As can be seen from fig. 9, for the three target adjectives, there is a small error between the predicted and true values for all samples. Furthermore, we further used Root Mean Square Error (RMSE) to evaluate the performance of these neural networks, defined as
Figure BDA0002223343670000173
In the formula: ns is the number of samples;
Figure BDA0002223343670000174
and psiiThe predicted and true values of the neural network for the ith sample, respectively.
Table 9 reflects that the RMSE values for the three neural networks dropped rapidly as the number of iterations increased. After 20,000 periods, the final rms error values were 0.0455,0.0534, and 0.0526, respectively, all within an acceptable tolerance of less than 0.1, indicating satisfactory convergence performance for each network.
TABLE 9 root mean square error of three neural networks at different iterations
Figure BDA0002223343670000181
(4) Multi-objective optimization design of product appearance
Table 10 shows the correlation coefficient between the principal component score and the mean of the evaluation of perceptual images of the three target adjectives, and it can be seen that the principal component 1 and the principal component 2 are positively correlated and negatively correlated with the three target adjectives, respectively. Through a large number of simulation tests, determining the modification probability to be 0.3 can achieve a good balance between algorithm convergence and diversity. A total of 30 pareto solutions (i.e., topographical design solutions) were obtained by operating ISPEA2 according to the key parameters in table 11, with principal component scores and target adjective calculation scores as shown in table 12.
TABLE 10 correlation coefficient of principal component score and mean evaluation of perceptual image of three target adjectives
Indicates correlation, indicates significant correlation.
TABLE 11 Algorithm parameters of ISPEA2 and SPEA2
Figure BDA0002223343670000183
TABLE 12.30 principal component scores and target adjective calculation scores for pareto solutions
Figure BDA0002223343670000184
Figure BDA0002223343670000191
(5) Performance validation of ISPEA2
To verify the performance of ISPEA2, ISPEA2 and SPEA2 were run independently 30 times (table 11) under the same parameter settings, and the performance of the two algorithms were compared by using the usual convergence index, interval index and coverage index. Let A ═ A1,A2,…,AA) Is the solution set obtained from ISPEA2 or SPEA2,is a true pareto frontier solution set, and the three indices can be calculated as follows:
the convergence index C represents the average distance from the solution set obtained by the algorithm to the true pareto front and can be calculated as
Figure BDA0002223343670000193
Figure BDA0002223343670000194
In the formula: wherein d isiIs from A to P*The minimum normalized euclidean distance of each solution of (a); f. ofz maxAnd fz minRespectively true pareto frontier solution set P*The maximum and minimum function values of the z-th objective function in (1). The smaller the value of C, the better the convergence. When C ═ 0, this means that all solutions in a are on the true pareto front.
The distance index S represents the distance between two continuous solutions and can measure the diversity performance of the algorithm, and the calculation method is as follows
Figure BDA0002223343670000201
In the formula:
Figure BDA0002223343670000202
is diIs measured. A smaller S-value indicates that the obtained solution is more evenly distributed throughout the target space.
Coverage index C (A)1,A2) The resulting solution representing one algorithm dominates the resulting solution of the other algorithm, which is calculated as follows
Figure BDA0002223343670000203
In the formula: c (A)1,A2) Is represented by A2In the formula A1The number of solutions dominated with A2All the ratios of the solution numbers in (A), C2,A1) The opposite is indicated. If C (A)1,A2)≥C(A2,A1) Then A1Ratio A2Good results are obtained. After calculating the above-mentioned indexes, we calculated the mean and standard deviation of the C and S indexes and the metric index value C (A) of the two algorithms1,A2) (Table 13), and plots the change trend of the index values of C and S under 30 independent runs (FIG. 10).
TABLE 13 mean and standard deviation of C and S indices in ISPEA2 and SPEA2 algorithms, and C (A)1,A2) Value of
As can be seen from table 13 and fig. 10, the C index value of ISPEA2 is significantly smaller than SPEA2, and the S index value is also lower than SPEA2, indicating that the convergence of ISPEA2 is significantly improved and the diversity is also enhanced to some extent. Furthermore, C (A)2,A1) 0 means that there is no pareto solution in SPEA2 that can dominate the ISPEA2 solution. Higher C (A)1,A2) The indicators indicate that the pareto solution of ISPEA2 significantly dominates the pareto solution of SPEA2, and that the pareto solution of ISPEA2 is closer to the true pareto front and more widely distributed. Thus, the comparison demonstrates that ISPEA2 has better convergence and diversity. It is noted that the standard deviations of the two algorithms are very close, indicating that the stability of the two algorithms is relatively consistent.
An automatic simulation system of the algorithm provided by the invention is developed based on MATLAB software. The system provides a visual interface that enables designers to design and view results intuitively and efficiently (fig. 11). The realization and the operation of the simulation system further prove that the method can be used as a general design tool with better innovation and application values.

Claims (3)

1. An improved strength pareto evolutionary algorithm for multi-objective optimization design of product appearance is characterized in that: the method comprises two parts of design analysis and product appearance multi-objective optimization design, wherein the data of the two parts comprise product appearance data and product perceptual image data, the main component score data of the product appearance outline is obtained by adopting an elliptical Fourier analysis technology, and the perceptual image evaluation mean data of a target adjective is obtained by adopting a perceptual image analysis technology;
the algorithm step of the product appearance multi-objective optimization design part is that a genetic algorithm optimization neural network GABP technology is used for establishing a nonlinear mapping network between principal component scores and a perceptual image evaluation mean value of a target adjective, and the network is further used as a fitness function of a core multi-objective optimization algorithm; then, a correction operator is proposed by utilizing the consistency correlation relationship between the principal component score obtained by the elliptic Fourier analysis technology and the perceptual image evaluation mean value of the target adjective, and the operator is combined with the improved crossover operator and the improved adaptive mutation operator, so that the improved strength pareto evolutionary algorithm ISPEA2 for the multi-objective optimization design of the product appearance is finally formed.
2. The improved strength pareto evolutionary algorithm for multi-objective design optimization of product form factors of claim 1, characterized by: in the design analysis part, the specific steps are
S1.1 elliptic Fourier analysis
The elliptic Fourier analysis of the product appearance comprises four steps of image preprocessing, elliptic Fourier descriptor calculation, elliptic Fourier descriptor normalization and principal component analysis:
s1.1.1 image preprocessing
After image preprocessing, a two-dimensional contour point coordinate sequence z (i) consisting of x and y coordinates of a discrete sampling points can be obtained and expressed as
z(i)＝(x(i),y(i))(i＝0,1,…,a-1) (1)
S1.1.2 elliptic Fourier descriptor computation
For the coordinate sequence z (i) of the two-dimensional contour points, the two-dimensional contour points are expanded into
Figure FDA0002223343660000011
In the formula:
Figure FDA0002223343660000012
is the abscissa of the center point of the profile,is the ordinate of the profile center point, N is the harmonic frequency, N is the maximum harmonic frequency, and t is the arc length parameter. Obtaining four approximate ellipse Fourier coefficients a of the triangle function of the new interpolation point coordinate function by a Riemann summation methodxn、bxn、ayn、bynIs composed of
Figure FDA0002223343660000021
In the formula: a is the number of new interpolation points, four coefficients correspond to the same harmonic frequency n, and any closed contour is described by a group of elliptic Fourier coefficients containing n harmonic frequencies
Figure FDA0002223343660000022
F is an elliptic Fourier description sub-system number matrix;
s1.1.3 elliptic Fourier descriptor normalization
Let the normalized coefficients of the nth harmonic be
Figure FDA0002223343660000023
Comprises the following steps:
setting harmonic DC component
Figure FDA0002223343660000024
Namely, the center point of the outline is moved to the origin of coordinates, and the position normalization can be realized;
the elliptic fourier coefficients are transformed according to the following equation, achieving a directional normalization:
Figure FDA0002223343660000025
in the formula: a is1,b1,c1And d1Are the four harmonic coefficients of the first ellipse,
Figure FDA0002223343660000026
Figure FDA0002223343660000027
Figure FDA0002223343660000028
calculate the first ellipse E*Then dividing each descriptor matrix by E*Normalizing the size:
Figure FDA0002223343660000029
s1.1.4 principal component analysis
Using principal component analysis to obtain a matrix F with a sample number q*Reducing vitamin to m dimension
Figure FDA0002223343660000031
Z is a principal component score matrix which refines the outline information of the product;
s1.2, analyzing the perceptual image, which comprises the following specific steps:
step 1: collecting ideographs, i.e. adjectives, and organizing the subject to perform informal investigation to pick out appropriate adjectives;
step 2: inviting experts to establish a focus group, and selecting the most representative adjectives by using a Delphi method;
and step 3: combining the representative adjectives with the product samples to form a questionnaire; using the semantic difference table and the 7-point Likter table to perform perception of questionnaire survey evaluation experts on the product images, and calculating the mean value of all expert evaluation scores of each product sample;
and 4, step 4: combining the perceptual image evaluation mean matrix and the principal component score matrix of the target adjective, and performing correlation analysis on the combined matrix to obtain a correlation coefficient; then introducing the relation number into the image target comprehensive scoring model[8]To calculate a final target image composite score;
and 5: and determining the target adjectives and the sequence thereof according to the final target image comprehensive score obtained in the step 4.
3. The improved strength pareto evolutionary algorithm for multi-objective design optimization of product form factors of claim 1, characterized by: in the product shape multi-objective optimization design part, the specific steps are
S2.1, optimizing the neural network construction by the genetic algorithm, and constructing a high-precision neural network by adopting a GABP (gain-based Back propagation) technology as a network construction technology of the multi-target optimization algorithm;
s2.2 improved intensity pareto evolutionary algorithm ISPEA2
The optimization of the traditional SPEA2 algorithm is realized by improving a crossover operator, an adaptive mutation operator and a correction operator, and the improved strength pareto evolution algorithm is obtained, and comprises seven steps:
step 1: initializing a population and an external population, and setting generation numbers;
step 2: the neural network trained by the GABP is used for calculating the fitness value of the multi-target image corresponding to the neural network, and then the fitness value is distributed to the sum PtAnd AtEach of the individuals in (a); specifically, P is first given according to a dominating relationshiptAnd AtOf each individual, wherein st (i) may be defined as:
St(i)＝|{j|j∈Pt+At∧i＞j}| (8)
in the formula: the symbol > represents the pareto dominant relationship, and then a coarse fitness value is calculated from the intensity values of each individual as:
Figure FDA0002223343660000032
then, the individual density is calculated to distinguish individuals having the same rough fitness value:
Figure FDA0002223343660000033
in the formula:
Figure FDA0002223343660000041
the distance between the ith individual and the gamma adjacent individuals after the distance between the individual i and all other individuals is sorted in ascending order is shown, and finally, the fitness value of the individual can be expressed as:
F(i)＝R(i)+D(i) (11)；
and step 3: identifying all non-disadvantaged individuals and using context selection techniques from PtAnd AtIn which they are copied to At+1If A ist+1Is larger than its maximum allowable size, and a is filled with the best non-inferior solution by the truncation operationt+1Otherwise, with PtAnd AtDominant individual population of (1) to At+1Performing the following steps;
and 4, step 4: if the stopping criterion is met, use At+1The non-dominant individual in (a) as the final pareto solution set and terminates;
and 5: from A by binary tournament selectiont+1Selecting individuals and putting the individuals into a mating pool;
step 6: calculating by respectively applying improved crossover operator, improved adaptive mutation operator and correction operator to obtain result population Pt+1(ii) a The calculation process for each operator is as follows:
1) let principal component score vectorAnd
Figure FDA0002223343660000043
if the two parents are the tth generation individuals to be genetically manipulated, the crossover operation is carried out by improving the crossover operatorThe latter two progeny individuals are represented as
Figure FDA0002223343660000044
Figure FDA0002223343660000045
In the formula: α is a random number uniformly distributed over the interval [ -0.5,1.5], and α + β ═ 1; ξ is the number of principal components;
2) let principal component score vector
Figure FDA0002223343660000046
For the parent individuals of the t generation to be genetically manipulated, the offspring individuals after mutation manipulation through the improved adaptive mutation operator are represented as
Figure FDA0002223343660000047
In the formula: ms is the coefficient of variation scale; md is the decay rate of the variation; cn is the current number of iterations; mn is the maximum number of iterations; rn1Is a standard normally distributed random number; ub and lb are the upper and lower limits of the gene boundary, respectively;
3) the modifier modifies the results of the crossover and mutation operations according to the set modification probabilities according to the following equation:
Figure FDA0002223343660000049
Figure FDA0002223343660000051
in the formula: wherein
Figure FDA0002223343660000052
Representing offspring individuals obtained after crossover and mutation operations, wherein
Figure FDA0002223343660000053
Is that
Figure FDA0002223343660000055
Is that
Figure FDA0002223343660000057
Is that
Figure FDA0002223343660000058
i is a number containing all principal components; i.e. i*The main component number which corresponds to the main component score and has a consistent correlation coefficient with the evaluation mean value of the perceptual image of the target adjective;
Figure FDA0002223343660000059
is the ith*Correction coefficients of the individual principal components; rn2Is the interval [4, 5]]A random number of (c);
Figure FDA00022233436600000510
is the ith*The correlation coefficient of the z-th target adjective of the principal component, and
Figure FDA00022233436600000511
when a correlation is considered to exist between two variables[11](ii) a Z is the number of target adjectives, by
Figure FDA00022233436600000512
The replacement corresponds to
Figure FDA00022233436600000513
I of (a)*And the operation of modifying the operator can be completed by the position gene.
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