Aesthetic measure for user preference elicitation

As a crossing-field of multiple disciplines, aesthetic measure has been applied in many areas like psychology, arts, and neuroscience (Liu, Reference Liu2003; Hartmann et al., Reference Hartmann, Sutcliffe and De Angeli2008; Rigau et al., Reference Rigau, Feixas and Sbert2008; Perlovsky, Reference Perlovsky2010). Generally, a visual scene is seen as stimulus, from which scheme is progressively extracted. It is then perceived as the integration of visual components with key properties. Human eyes are sensitive to some of them, and through further mechanisms, high-level visual perceptions and sense of beauty are generated. Scientists accordingly summarize formula to directly catch the key properties of the target's visual alignment to calculate the aesthetic value. Aesthetic measure was first proposed by Birkhoff (Reference Birkhoff1933). It was further evaluated and tested effectively by Davis (Reference Davis1936). From then on, aesthetic measure has attracted research interests from many areas. Lo et al. (Reference Lo, Ko and Hsiao2015) introduced it into all-in-one stereos form design. Wu and Li (Reference Wu and Li2019) adopted aesthetic measure in interface design inspirations.

As a computational way of aesthetic product form evaluation, aesthetic measure is one of the common approaches addressing the subjectivity and uncertainty of user judgement. Pairwise form design selection is an intuitive way to derive user preference, and prior study further used an interactive genetic algorithm (GA) to learn user preference (Kelly et al., Reference Kelly, Wakefield and Papalambros2011a). Similarly, Ren and Papalambros (Reference Ren and Papalambros2012) viewed the preference elicitation in iterative query schemes as optimization problem and analyzed the query types and algorithms. Ren et al. (Reference Ren, Scott and Papalambros2014) further proposed a query algorithm to select proper query scheme. Another way of using user preference in product form is to analyze relations between coarse- and fine-grained geometric features according to users’ evoked emotions (Orbay et al., Reference Orbay, Fu and Kara2014). As aesthetic measure relies on the prior knowledge of relations between evoked aesthetic feeling and product form, it is very convenient without requiring user experiment in practice and wins in convenience.

Aesthetic dimension formulae are based on mathematic form representation and description. Geometric parameterization by experts is a widely used method (Lo, Reference Lo2018; Zhou et al., Reference Zhou, Ouyang, Su and Zhang2020). Shape grammar is another method which requires higher level of domain expertise (Orsborn et al., Reference Orsborn, Cagan, Pawlicki and Smith2006), given some prior work on automatic extraction of shape grammar (Orsborn et al., Reference Orsborn, Cagan and Boatwright2008; Orsborn and Cagan, Reference Orsborn and Cagan2009). Sketch is intuitive description of product forms, but its graph-based representation for further design exploration is still complex (Fuge et al., Reference Fuge, Yumer, Orbay and Kara2012; Orbay and Kara, Reference Orbay and Kara2012; Orbay et al., Reference Orbay, Yumer and Kara2012). Generally, geometric parameterization is more common and efficient than other methods, and thus, it is applied with aesthetic measure in this paper.

Generally, existing literatures on user preference elicitation or prediction for product form design apply computational methods including aesthetic measure to combat subjectivity and uncertainty. Requiring user experiment and domain expertise in design form representation are major disadvantages of current methods, comparative to aesthetic measure. For current literatures that apply aesthetic measure specifically, how to select proper aesthetic dimensions has not been addressed. This paper proposes to address this with evolutionary algorithm.

Feature selection

Feature selection is an important tool of data preprocessing before building models. More descriptive and valuable features are retained to clean the dataset, which ensure that more concise and comprehensive models are built. Feature selection is fundamental when dealing with high-dimensional data. However, it is also needed when features are redundant or inappropriate. For product form design, major aesthetic dimensions are a small group from all dimensions. In practice, some dimensions may have little contributions to overall aesthetic assessment, or even contradict with each other. Therefore, feature selection methods can be adopted here for aesthetic dimension selection.

Information theory-based, sparse learning-based, and statistical methods are typical methods for feature selection (Li et al., Reference Li, Cheng, Wang, Morstatter, Trevino, Tang and Liu2018). To the best knowledge of authors, existing studies related to feature selection have not been implemented for aesthetic dimension selection yet. This paper proposes to adopt a feature selection method based on the PSO method for this purpose. The PSO-based method transforms feature selection into an optimization problem by using model performances as selection fitness evaluation and solves the problem with evolutionary algorithm. This method is proven effective in many areas, such as gene selection, disease prediction, and so on (Joloudari et al., Reference Joloudari, Saadatfar, Dehzangi and Shamshirband2019; Almomani, Reference Almomani2020; Shukla et al., Reference Shukla, Tripathi, Reddy and Chandramohan2020; Song et al., Reference Song, Zhang, Guo, Sun and Wang2020). As it is easy to implement and fast to converge, it is employed in this paper for aesthetic dimension selection.

PSO algorithm is a heuristic optimization algorithm (Kennedy and Eberhart, Reference Kennedy and Eberhart1995; Shi and Eberhart, Reference Shi and Eberhart1998). Inspired by the birds, the evolutionary algorithm starts from random solutions and goes through iterations to reach final positions. Fitness evaluation is also applied in PSO but different from GAs, PSO does not include cross or mutation operations. Instead, individuals’ experiences and global performances are utilized in determining the direction of update. Typically, an individual's update is determined by inertia, its optimum position, and global optimum position, as shown in Eqs (1) and (2):

(1)$$v_{t + 1} = w\ast v_t + c_1\ast {\rm rand}( ) \ast ( p{\rm best}_t-x_t) + c_2\ast {\rm rand}( ) \ast ( g{\rm best}_t \\-x_t),$$

(2)$$x_{t + 1} = x_t + v_t,$$

where v _t is the individual’ speed at tth iteration step, x _t is its position, pbest_t is its best position up to tth iteration, gbest_t is the best position among individuals, w is the inertia weight, c ₁ is the local optimum weight, c ₂ is the global optimum weight, and rand() gives a random number between 0 and 1.

Genetic algorithm

GA (GA) is a noted evolutionary bionic algorithm for optimization. A population of individuals simulates the nature process of chromosome reproduction by iterative crossing, mutation, and selection. For years, various techniques have enriched GA algorithms and enabled good adaptation to different scenarios. Chromosome initialization can be randomized or in a certain pattern. Cross operation introduces new solutions from two father solutions and also promotes the spread of superior solutions. High crossing probability accelerates convergence. Mutation operation introduces new solutions by random, which ensures the ability to jump out of local optimum for global optimum solutions. Selection operation keeps those more promising chromosomes. The scalar fitness value traditionally used determines the probability of being selected into next iteration.

To solve more complex MOO problems, two types of GA variants are often relied on linear weight-based ones and pareto front-based ones. The first type methods aggregate multiple objects by weights into a single-objective optimization problem, such as random weight genetic algorithm (RWGA) and vector-evaluated genetic algorithm (VEGA). The second type aims to find pareto front solutions, such as multi-objective genetic algorithm (MOGA) and non-dominated sorting genetic algorithm (NSGA). Improved NSGA (NSGA-II) is proposed by Deb to improve the quality of pareto front (Deb et al., Reference Deb, Pratap, Agarwal and Meyarivan2002). After crossing and mutation operations, the fast non-dominance sorting and selection based on crowdedness are employed during each iteration. The selection process embodies elite strategy with a tendency to reach more dispersed pareto solutions. Compared with other MOO algorithms, NSGA-II has the advantage of strong convergence and high speed.

NSGA-II has been successfully applied in various areas. Aydin uses NSGA-II to solve the dynamic demand model for product line design for both remanufactured and new products (Aydin et al., Reference Aydin, Kwong and Ji2014, Reference Aydin, Kwong and Ji2016). Koo introduces NSGA-II into a simulation–optimization model for optimal refinery supply chain design and operation (Koo et al., Reference Koo, Adhitya, Srinivasan and Karimi2008). Kwong uses NSGA-II to solve the comprehensive optimization model to integrate affective design, engineering, and marketing for defining design specifications of new products (Kwong et al., Reference Kwong, Jiang and Luo2016). NSGA-II is also being applied in structural design optimization, thermal performance optimization areas, etc. (Cao et al., Reference Cao, Mihardjo and Parikhani2020; Wu et al., Reference Wu, Huang, Chen, Zhu and Li2022). In the product form design area, evolutionary algorithms are widely integrated in design synthesis (Eckert et al., Reference Eckert, Kelly and Stacey1999; Yang et al., Reference Yang, Zhou, Yu and Yu2019; Zeng et al., Reference Zeng, He, Tang and Wang2020). Hsiao et al. (Reference Hsiao, Chiu and Lu2010) applied GA in coffee machine form design. Kelly et al. (Reference Kelly, Maheut, Petiot and Papalambros2011b) focused on the balance between form and function performances, and employed evolutionary algorithm to generate optimal designs. Such balance was also highlighted and properly handled by evolutionary algorithms in automotive form design (Reid et al., Reference Reid, Gonzalez and Papalambros2010, Reference Reid, Frischknecht and Papalambros2012). Above studies by Hsiao et al. (Reference Hsiao, Chiu and Lu2010), Reid et al. (Reference Reid, Gonzalez and Papalambros2010, Reference Reid, Frischknecht and Papalambros2012), and Kelly et al. (Reference Kelly, Maheut, Petiot and Papalambros2011b) prove the feasibility and value of evolutionary algorithms in form design. However, the design scheme evaluation metrics they use are limited, and correlations between aesthetic dimensions have not been considered. This paper uses a specific dimension selection step to filter out unimportant or correlated aesthetic dimensions and to reduce the number of evaluation metrics. Due to the powerfulness of NSGA-II, it is adopted in this paper for form design parameter optimization.

Parameterization

Product form design follows a principle from whole to part. Overall structure is firstly defined. Then, more detailed parts are being considered. In this paper, we focus on aesthetic form design that is highly valued in overall structure design. Parameterization starts with extracting the overall structure of design target. A complex structure can be viewed as a layout composed of simple objects. Then appropriate parameters are summarized to describe the major geometric characteristics of the layout and objects. These parameters should be sufficient, concise, and easy to use for design modeling. They are supposed to describe major characteristics of product form and as less as possible, for example, lengths, diameters, angles of objects, and ratios of them. Finally, bounds and constraints are defined for them. Currently, only constraints that can be described by parametric inequations are available. The design parameters are denoted as Y = (y ₁, y ₂…y _p), bounds are denoted as D = {d ₁, d ₂…d _p}, and constraints are denoted as F = {f _k}. Equations (3) and (4) need to be satisfied:

(3)$$\forall f_k\in F, \;f_k( Y) < 0,$$

(4)$$y_j\in d_j, \;j = 1, \;2, \;\ldots p.$$

Aesthetic dimension selection based on PSO

Aesthetic dimension selection is based on aesthetic measure and expert decision-making. Several design schemes are prepared for both methods. For each selection sequence, that is, a list of selected aesthetic dimensions, aesthetic measure results of design schemes are compared with expert analysis to evaluate the rationality of that selection. The ones with optimal performance are obtained as final selection solutions. The above process is solved by the PSO algorithm, by setting selection sequences as individuals, and comparison performance as the fitness value.

Aesthetic dimension definition

A total of 11 widely used aesthetic dimensions for 3D objects are considered, that is, balance, equilibrium, symmetry, sequence, cohesion, proportion, simplicity, regularity, economy, homogeneity, and rhythm. These dimensions are proposed in early studies by Ngo and Byrne (Reference Ngo and Byrne1998) and Ngo et al. (Reference Ngo, Lian and Byrne2003) for 2D webpages. Some of them are extended in the later study by Lo et al. (Reference Lo, Ko and Hsiao2015) for 3D objects. In this paper, some dimension formulas for 3D objects are adopted, while other dimension formulas are re-defined according to original 2D definitions of those dimensions. The aesthetic dimensions involved in the proposed approach are illustrated in the following:

1. (1) Balance

Balance represents the even distribution of optical weight. Balance dimension value "BM" is computed by Eqs (5)–(7). Specifically, balance formulae proposed by Lo et al. (Reference Lo, Ko and Hsiao2015) are improved by using the product component with the largest area moment of inertia to define “visual boundary”. Also, the balance level of layout along z-axis BM_z is defined as the ratio of overall area centroid distance to area centroid distance of the product component with the largest area moment of inertia.

(5)$${\rm BM} = 1-\displaystyle{{{\rm B}{\rm M}_x + {\rm B}{\rm M}_y + {\rm B}{\rm M}_z} \over 3},$$

(6)$${\rm B}{\rm M}_z = \displaystyle{{\left\vert {\sum\nolimits_{i = 1}^k {m_{zi}} } \right\vert } \over {\max \vert {m_{zi}} \vert }}\displaystyle{{v_m} \over V},$$

(7)$$m_{zi} = \int {x_i + y_i} dv,$$

where m _zi is the ith object's area moment of inertia along the z-axis, V is the sum of the area of objects, v _m is the area of the product component with the largest area moment of inertia.

(2) Equilibrium

Equilibrium represents the uniform distribution of elements in a target. Equilibrium dimension value "EM" is computed by Eqs (8) and (9) according to Lo et al. (Reference Lo, Ko and Hsiao2015):

(8)$${\rm EM} = 1-\displaystyle{{{\rm E}{\rm M}_x + {\rm E}{\rm M}_y + {\rm E}{\rm M}_z} \over 3},$$

(9)$${\rm E}{\rm M}_x = \displaystyle{{\sum\nolimits_i {x_{ic}-x_c} } \over d},$$

where (x _c, y _c, z _c) is orthogonal coordinates of target centroid, (x _ic, y _ic, z _ic) is orthogonal coordinates of ith object centroid, and d is the span of objects along the x-axis.

(3) Symmetry

Symmetry represents the similarity level of objects on two sides of an axis. Symmetry dimension value "SYM" is computed by Eqs (10)–(17) according to Lo et al. (Reference Lo, Ko and Hsiao2015). Specifically, vertical, horizontal, and radius symmetries along the orthogonal front, right, and top views are averaged. The symmetry level of front view along the vertical axis SYM_FV is calculated by two comparisons: upper right (UR) to upper left (UL) and lower right (LR) to lower left (LL), in terms of orthogonal coordinates, height, width, and polar coordinates.

(10)$$\eqalign{& {\rm SYM} = 1-\left({\displaystyle{{\vert {{\rm SY}{\rm M}_{{\rm FV}}} \vert + \vert {{\rm SY}{\rm M}_{{\rm FH}}} \vert + \vert {{\rm SY}{\rm M}_{{\rm FR}}} \vert + \vert {{\rm SY}{\rm M}_{{\rm RV}}} \vert } \over 9}} \right. \cr & \left. { \displaystyle{{ + \vert {{\rm SY}{\rm M}_{{\rm RD}}} \vert + \vert {{\rm SY}{\rm M}_{{\rm RR}}} \vert + \vert {{\rm SY}{\rm M}_{{\rm TH}}} \vert + \vert {{\rm SY}{\rm M}_{{\rm TD}}} \vert + \vert {{\rm SY}{\rm M}_{{\rm TR}}} \vert } \over 9}} \right),} $$

(11)$$\eqalign{{\rm SY}{\rm M}_{{\rm FV}}& = 1-( {\vert {{{Y}^{\prime}}_{{\rm UR}}-{{Y}^{\prime}}_{{\rm UL}}} \vert + \vert {{{Y}^{\prime}}_{{\rm LR}}-{{Y}^{\prime}}_{{\rm LL}}}} \vert \cr &\quad + \vert {{{Z}^{\prime}}_{{\rm UR}}-{{Z}^{\prime}}_{{\rm UL}}} \vert + \vert {{{Z}^{\prime}}_{{\rm LR}}-{{Z}^{\prime}}_{{\rm LL}}} \vert \cr \,& + \vert {{{B}^{\prime}}_{{\rm UR}}-{{B}^{\prime}}_{{\rm UL}}} \vert + \vert {{{B}^{\prime}}_{{\rm LR}}-{{B}^{\prime}}_{{\rm LL}}} \vert {\rm} + \vert {{{H}^{\prime}}_{{\rm UR}}-{{H}^{\prime}}_{{\rm UL}}} \vert + \vert {{{H}^{\prime}}_{{\rm LR}}-{{H}^{\prime}}_{{\rm LL}}} \vert \cr & { + \vert {{{\Theta }^{\prime}}_{{\rm UR}}-{{\Theta }^{\prime}}_{{\rm UL}}} \vert + \vert {{{\Theta }^{\prime}}_{{\rm LR}}-{{\Theta }^{\prime}}_{{\rm LL}}} \vert }\cr & + \vert {{{R}^{\prime}}_{{\rm UR}}-{{R}^{\prime}}_{{\rm UL}}} \vert + \vert {{{R}^{\prime}}_{{\rm LR}}-{{R}^{\prime}}_{{\rm LL}}} \vert ) {\rm /12},} $$

(12)$$Y_j = \sum\limits_i^{k_j} {\vert {y_{ij}-y_c} \vert } \quad \;j = {\rm UL, \;UR, \;LL, \;LR,}$$

(13)$$Z_j = \sum\limits_i^{k_j} {\vert {z_{ij}-z_c} \vert } \quad \;j = {\rm UL, \;UR, \;LL, \;LR},$$

(14)$$H_j = \sum\limits_i^{k_j} {h_{ij}} \quad \;j = {\rm UL, \;UR, \;LL, \;LR,}$$

(15)$$B_j = \sum\limits_i^{k_j} {b_{ij}} \quad \;j = {\rm UL, \;UR, \;LL, \;LR,}$$

(16)$$\Theta _j = \sum\limits_i^{k_j} {\left\vert {\displaystyle{{z_{ij}-z_c} \over {y_{ij}-y_c}}} \right\vert } \quad \;j = {\rm UL, \;UR, \;LL, \;LR,}$$

(17)$$R_j = \sum\limits_i^{k_j} {\sqrt {{( z_{ij}-z_c) }^2 + {( y_{ij}-y_c) }^2} } \quad \;j = {\rm UL, \;UR, \;LL, \;LR,}$$

where y _ij and z _ij are the orthogonal coordinates of ith product component in jth part, h _ij and b _ij are the height and width of ith product component in jth part, y _c and z _c are the orthogonal coordinates of target centroid, Y _j, Z _j, H _j, B _j, Θ_j, and R _j are processed by max-min normalization into range [0, 1], then they are denoted as ${Y}^{\prime}_j$, ${Z}^{\prime}_j$,${H}^{\prime}_j$, ${B}^{\prime}_j$, ${\Theta }^{\prime}_j$, and ${R}^{\prime}_j$.

(4) Sequence

Sequence represents the alignment that accords with human's habit of eye movement trained by reading. Preferred sequence of human eyes is considered upper left, upper right, lower left, and lower right, which is found and exploited in early studies of screen design (Streveler and Wasserman, Reference Streveler and Wasserman1984; Galitz, Reference Galitz1994). The amount of content is preferred to decrease along the sequence. Sequence dimension value "SQN" is computed by Eqs (18)–(21) extending the formula by Ngo et al. (Reference Ngo, Lian and Byrne2003). Sequence dimension value along x-axis SQN_x measures the similarity of content alignment to the preferred sequence.

(18)$${\rm SQN} = 1-\displaystyle{{{\rm SQ}{\rm N}_x + {\rm SQ}{\rm N}_y + {\rm SQ}{\rm N}_y} \over 3},$$

(19)$${\rm SQ}{\rm N}_x = 1-\displaystyle{{\vert {4-v_{x{\rm UL}}} \vert + \vert {3-v_{x{\rm UR}}} \vert + \vert {2-v_{x{\rm LL}}} \vert + \vert {1-v_{x{\rm LR}}} \vert } \over 8},$$

(20)$$v_{xj} = \left\{{\matrix{ {\matrix{ 4 & {{\rm if}\,w_{xj}\,{\rm is}\,{\rm the}\,{\rm biggest}\,{\rm in}\,W_x} \cr } } \hfill \cr {\matrix{ 3 & {{\rm if}\,w_{xj}\,{\rm is}\,{\rm the}\,{\rm 2rd}\,{\rm biggest}\,{\rm in}\,W_x} \cr } } \hfill \cr {\matrix{ 2 & {{\rm if}\,w_{xj}\,{\rm is}\,{\rm the}\,{\rm 3rd}\,{\rm biggest}\,{\rm in}\,W_x} \cr } } \hfill \cr {\matrix{ 1 & {{\rm if}\,w_{xj}\,{\rm is}\,{\rm the}\,{\rm smallest}\,{\rm in}\,W_x} \cr } } \hfill \cr } } \right.\quad \;j = {\rm UL, \;UR, \;LL, \;LR},$$

(21)$$W_x = { {w_{xj}} } = \left\{{q_j\sum\limits_i^{kj} {a_{xij}} } \right\} \quad \;j = {\rm UL, \;UR, \;LL, \;LR,}$$

where q _j is a weight of quadrant to control the significance of sequence and usually its value is set as 1, and a _xij is the area of ith product component in the jth part along the x-axis.

(5) Cohesion

Cohesion represents the consistency of aspect ratios of the target. The more consistent they are, the more aesthetic contributed (Lo, Reference Lo2018). Cohesion dimension value "COH" is computed by Eqs (22)–(25) extending the formula by Ngo et al. (Reference Ngo, Lian and Byrne2003).

(22)$${\rm COH} = 1-\displaystyle{{{\rm CO}{\rm H}_x + {\rm CO}{\rm H}_y + {\rm CO}{\rm H}_z} \over 3},$$

(23)$${\rm CO}{\rm H}_x = \displaystyle{{\sum\nolimits_{i = 1}^k {t_{xi}} } \over k},$$

(24)$$t_{xi} = \left\{{\matrix{ {\matrix{ {c_{xi}} & {{\rm if\ }c_{xi}} \cr } \le 1} \hfill \cr {\matrix{ {\displaystyle{1 \over {c_{xi}}}} & {{\rm otherwise}} \cr } } \hfill \cr } } \right., $$

(25)$$c_{xi} = \displaystyle{{h_{xi}/b_{xi}} \over {h_{x{\rm layout}}/b_{x{\rm layout}}}},$$

where h _xi is the height of ith object, b _xi is the width of ith object, h _xlayout is the height span of the target, b _xlayout is the width span of the target of the layout along the x-axis, and k is the total number of objects.

(6) Proportion

Proportional aesthetic represents the similarity of target proportions to typical preferred proportions. Some preferred proportions have been discovered by scientists to be aesthetically pleasing and are validated to be universal through a long period (Grigoryan and Agaian, Reference Grigoryan and Agaian2020; De Bartolo et al., Reference De Bartolo, De Luca, Antonucci, Schuster, Morone, Paolucci and Iosa2021). Among those proportions are {1:1, 1:1.414, 1:1.618, 1:1.732, 1:2}. Proportion dimension value "PM" is computed by Eqs (26)–(29), which are simplified expressions according to Ngo et al. (Reference Ngo, Lian and Byrne2003).

(26)$${\rm PM} = \displaystyle{{{\rm P}{\rm M}_{\rm O} + {\rm P}{\rm M}_{\rm E}} \over 2},$$

(27)$${\rm P}{\rm M}_{\rm O} = 1-\displaystyle{{\sum\nolimits_{\,j = 1}^3 {\mathop {\min }\limits_{x\in X} \vert {\,p_{oj}-x} \vert } } \over 3},$$

(28)$${\rm P}{\rm M}_{\rm E} = 1-\displaystyle{{\sum\nolimits_{\,j = 1}^3 {\sum\nolimits_{i = 1}^k {\mathop {\min }\limits_{x\in X} \vert {\,p_{eij}-x} \vert } } } \over {3k}},$$

(29)$$X{\rm} = \{ { \ 1, \;1}{\rm.414, \;1}{\rm.618, \;1}{\rm.732, \;2} \},$$

where p _oj is the overall aspect ratio along the jth orthogonal axis, p _eij is the aspect ratio of ith element along the jth orthogonal axis, and X is a set of preferred proportions.

(7) Simplicity

Simplicity represents the aesthetic feeling of people stimulated by compact and simple scenes (Moshagen and Thielsch, Reference Moshagen and Thielsch2010), which is often referred to as “less the better”. For 3D objects, the number of elements with different geometric poses significantly affects the sense of simplicity, as those elements with the same or similar poses are easily classified as one type, or even grouped together by human eyes. The geometric pose of 3D object is the 3D orientation it exhibits with respect to the ground. Two elements have different geometric poses if the difference between any pair of 3D orientation parameters exceeds the minimum threshold. Minimum thresholds are predefined by experts in the light of geometric features of target. Simplicity dimension value "SMM" is computed by Eq. (30) revising the formulae by Ngo et al. (Reference Ngo, Lian and Byrne2003).

(30)$${\rm SMM} = \displaystyle{1 \over {n_{{\rm pose}}}},$$

where n _pose is the number of elements with different poses.

(8) egularity

Regularity represents the level of conformity under certain rules. For 3D object, geometric rules are considered. Regularity dimension value "RM" is determined by the number of product components being non-parallel or distinct, that is, have different spatial orientations or shapes, as shown in Eqs (31)–(33).

(31)$${\rm RM} = \displaystyle{{{\rm R}{\rm M}_{\rm P} + {\rm R}{\rm M}_{\rm E}} \over 2},$$

(32)$${\rm R}{\rm M}_{\rm P} = 1-\displaystyle{{k_p-1} \over k},$$

(33)$${\rm R}{\rm M}_{\rm E} = 1-\displaystyle{{k_e-1} \over k},$$

where k _p is the number of product components that are non-parallel to each other, k _e is the number of product components that are distinct from each other, and k is the total number of product components.

(9) Economy

Economy represents the amount of resources or cost to realize functions and convey messages to users by the product. As an aesthetic dimension, economy is very customized according to product characteristics and design requirements. For 3D form design solution that consists of solid elements, or referred to as design scheme, this paper calculates the economy dimension value ECO by the volume of design scheme. Economy dimension formula, as shown in Eq. (34), is derived from Ngo et al. (Reference Ngo, Lian and Byrne2003).

(34)$${\rm ECO} = 1-\displaystyle{{V_d} \over {V_{\max }}},$$

where V _d is the volume of geometric parameterized design scheme, and V _max is the largest volume for possible design schemes. V _max is often calculated as the volume of a cube bounded by the value range of design parameters along x axis, y axis, and z axis.

(10) Homogeneity

Homogeneity represents the even distribution of objects in the target. For the 3D target, eight geometric partitions are used, namely, front upper left, front upper right, front lower left, front lower right, back upper left, back upper right, back lower left, and back lower right. The homogeneity dimension value "HM" is computed by the number of objects in eight partitions, as in Eqs (35)–(37), extending the formula proposed by Ngo et al. (Reference Ngo, Lian and Byrne2003).

(35)$${\rm HM} = \displaystyle{W \over {W_{\max }}},$$

(36)$$W = \displaystyle{{k!} \over {\Pi _jk_j!}}, \;\;j = {\rm FUL, \;FUR, \;FLL, \;FLR, \;BUL, \;BUR, \;BLL, \;BLR},$$

(37)$$W_{\max } = \displaystyle{{k!} \over {{\left({\displaystyle{k \over 8}!} \right)}^8}},$$

where k is the total number of objects and k _j is the number of objects with gravity center within jth partition.

(11) Rhythm

Rhythm represents that elements change in a regular pattern, with the order of variation to excite people. Rhythm dimension value "RHM" is computed as the average of rhythm levels of front, right, and top views, as shown in Eq. (38), extending the formula discussed in Ngo et al. (Reference Ngo, Lian and Byrne2003) to 3D objects. The rhythm level of front view RHM_F is calculated by Eqs (39)–(43):

(38)$${\rm RHM} = 1-\displaystyle{{{\rm RH}{\rm M}_{\rm F} + {\rm RH}{\rm M}_{\rm R} + {\rm RH}{\rm M}_{\rm T}} \over 3},$$

(39)$${\rm RH}{\rm M}_{\rm F} = 1-\displaystyle{{\vert {{\rm RH}{\rm M}_y} \vert + \vert {{\rm RH}{\rm M}_z} \vert + \vert {{\rm RH}{\rm M}_{{\rm Farea}}} \vert } \over 3},$$

(40)$$\eqalign{{\rm RH}{\rm M}_y& = ( \vert {{{Y}^{\prime}}_{{\rm UL}}-{{Y}^{\prime}}_{{\rm UR}}} \vert + \vert {{{Y}^{\prime}}_{{\rm UL}}-{{Y}^{\prime}}_{{\rm LR}}} \vert + \vert {{{Y}^{\prime}}_{{\rm UL}}-{{Y}^{\prime}}_{{\rm LL}}} \vert \cr & \quad + \vert {{{Y}^{\prime}}_{{\rm UR}}-{{Y}^{\prime}}_{{\rm LR}}} \vert + \vert {{{Y}^{\prime}}_{{\rm UR}}-{{Y}^{\prime}}_{{\rm LL}}} \vert + \vert {{{Y}^{\prime}}_{{\rm LR}}-{{Y}^{\prime}}_{{\rm LL}}} \vert ) /6}, $$

(41)$$\eqalign{{\rm RH}{\rm M}_z& = ( \vert {{{Z}^{\prime}}_{{\rm UL}}-{{Z}^{\prime}}_{{\rm UR}}} \vert + \vert {{{Z}^{\prime}}_{{\rm UL}}-{{Z}^{\prime}}_{{\rm LR}}} \vert + \vert {{{Z}^{\prime}}_{{\rm UL}}-{{Z}^{\prime}}_{{\rm LL}}} \vert \cr & \quad + \vert {{{Z}^{\prime}}_{{\rm UR}}-{{Z}^{\prime}}_{{\rm LR}}} \vert + \vert {{{Z}^{\prime}}_{{\rm UR}}-{{Z}^{\prime}}_{{\rm LL}}} \vert + \vert {{{Z}^{\prime}}_{{\rm LR}}-{{Z}^{\prime}}_{{\rm LL}}} \vert ) /6}, $$

(42)$$\eqalign{& {\rm RH}{\rm M}_{{\rm Farea}}= ( \vert {{{A}^{\prime}}_{{\rm FUL}}-{{A}^{\prime}}_{{\rm FUR}}} \vert + \vert {{{A}^{\prime}}_{{\rm FUL}}-{{A}^{\prime}}_{{\rm FLR}}} \vert + \vert {{{A}^{\prime}}_{{\rm FUL}}-{{A}^{\prime}}_{{\rm FLL}}} \vert \cr & \quad + \vert {{{A}^{\prime}}_{{\rm FUR}}-{{A}^{\prime}}_{{\rm FLR}}} \vert + \vert {{{A}^{\prime}}_{{\rm FUR}}-{{A}^{\prime}}_{{\rm FLL}}} \vert + \vert {{{A}^{\prime}}_{{\rm FLR}}-{{A}^{\prime}}_{{\rm FLL}}} \vert ) /6}, $$

(43)$$A_{{\rm F}j} = \sum\limits_{i = 1}^{k_{{\rm F}j}} {a_{{\rm F}ij}, \;\;j = {\rm UL, \;UR, \;LL, \;LR}}, $$

where ${Y}^{\prime}_j$ and ${Z}^{\prime}_j$ (j = UL, UR, LL, LR) are calculated by Eqs (12) and (13) then max–min normalized into range [0, 1], a _Fij is the area of ith product component in jth partition of the front view, k _Fj is the number of objects in jth partition of the front view, and A _Fj is normalized into range [0, 1] then it is denoted as ${A}^{\prime}_{{\rm F}j}$.

In practical use, some of above aesthetic dimension formulas need customization. For example, preferred proportions should include those that people are sensitive to according to the type of target. Prior research on screen design (Ngo et al., Reference Ngo, Lian and Byrne2003; Wan et al., Reference Wan, Ji, Wu, Jia, Zhan, Yuan and Wang2021) mostly includes the ones in Eq. (29), while research on aesthetic plastic surgery (Bashour, Reference Bashour2006) highlights golden ratio (1:1.618) and rarely considers the others. Another example is the selection of proper views for 3D object. Aesthetic formulae for balance, equilibrium, symmetry, sequence, cohesion, proportion, and rhythm are based on the orthogonal coordinate system. They compute aesthetic values for orthogonal 2D views by primitive formulae, then take the average as the final aesthetic value for 3D object. Therefore, orientations of orthogonal coordinate systems are essential. They should be decided via expertise to reflect geometric features of 3D forms as well as possible, such as setting z-axis as normal to the profile view for bicycle form design, setting z-axis (vertical axis) along the revolution axis for stereo form design (Lo et al., Reference Lo, Ko and Hsiao2015). For complex cases like surface modeling, more than three views can be aesthetically computed individually, then averaged to be the final aesthetic value.

Above 11 aesthetic dimensions are correlated to some extent. For example, balance and equilibrium both involve locations of elements within an object. Compared with visual balance given the varied weights of elements, the latter only considers the spatial distribution of elements and thus only uses spatial coordinates. Both cohesion and proportion involve aspect ratios of design scheme and elements. The former is to reflect the consistency among elements’ aspect ratios, while the latter is to reflect whether aspect ratios are in conformity with preferred proportions. For further research, such correlations should be considered by experts when specifying the weights of selected dimensions.

Aesthetic measure results of design schemes

Existing form design solutions are defined by the same parameterization as design schemes. The defined design schemes are denoted as {Scheme_j}, j = 1, 2, …n. Selected aesthetic dimensions are arranged in selection sequence and denoted as {s _i}. Aesthetic dimensional values are calculated by aesthetic measure formula and denoted as V = (v _ij)_m×n, where v _ij is the value of ith aesthetic dimension of jth design scheme, m is the number of selected dimensions, and n is the number of design schemes. Aesthetic values of design schemes are the aggregate of considered dimensions, as shown in Eq. (44):

(44)$$A = ( a_j) _n, \;a_j = \displaystyle{{\sum\nolimits_i {v_{ij}} } \over m}.$$

The rank of design schemes by aesthetic measure is determined by Eq. (45):

(45)$$R = ( r_j) _n, \;r_j = \vert \{{ a_t\vert t\ne j, \;a_t > a_j} \} \vert + 1.$$

Expert decision-making

The rank of design schemes is also given by experts to compare the performance of selection sequence. As multiple aesthetic dimension scoring is difficult and subjective, overall aesthetic assessment by experts is used. Involving more experts is beneficial for increasing credibility. Pairwise comparison matrix is adopted here to lower subjectivity and verify their results.

Each expert gives a pairwise comparison matrix $E^t = ( e_{ij}^t )$, where $e_{ij}^t$ is the superiority of the ith design scheme to the jth design scheme given by the t-th expert. Consistency check is conducted for each pairwise comparison matrix, as shown in Eqs (46) and (47). Consistency is approved if CR < 0.1. Aggregated expert decision-making is given by Eq. (46):

(46)$${\rm CI} = \displaystyle{{\lambda _{\max }-n} \over {n-1}},$$

(47)$${\rm CR} = \displaystyle{{{\rm CI}} \over {{\rm RI}}},$$

where λ _max is the maximum eigenvalue of pairwise comparison matrix, n is the matrix dimensionality that is the same as the number of design schemes, and RI is the average random consistency indicator according to the value of n.

(48)$$E = ( e_{ij}) _{n \times n}, \;e_{ij} = \displaystyle{{\sum\nolimits_t {e_{ij}^t } } \over s},$$

where s is the number of experts.

The rank of design schemes by expert decision-making is determined by Eq. (49):

(49)$${R}^{\prime} = ( {r}^{\prime}_j) _n, \;{r}^{\prime}_j = \vert \{{ e_{ij}\vert e_{ij} > 1} \} \vert + 1,$$

Comparison of aesthetic measure and expert decision-making

The principle of aesthetic dimension selection is that the rank of design schemes by aesthetic measure should be the same as rank by expert decision-making. By this way, aesthetic measure techniques are directed by experts to catch the major characteristics of aesthetic design.

Differences in both strategies are calculated in Eqs (50) and (51):

(50)$${\rm Diff} = \displaystyle{{\sum\nolimits_j {\vert {{{r}^{\prime}}_j-r_j} \vert } } \over N},$$

(51)$$N = 2\sum\limits_{i = 1}^{2i \le n} {( n-2i + 1) }, $$

where n is the number of design schemes, and Diff represents the difference level between both strategies between 0 and 1. The closer to 0 means more alike both ranks are and better the selection sequence is.

Aesthetic dimension selection by the PSO algorithm

Selected aesthetic dimensions arranged in the selection sequence is represented by an individual X = (x ₁, x ₂, …, x _m), where x _i is between 0 and 1. The ith aesthetic dimension is included in that sequence if x _i > 0.5, while not included otherwise. Candidate aesthetic dimensions are denoted as {m _i}. The maximum number of iterations are denoted as max_gen. The PSO algorithm to find the best selection sequences is shown in Algorithm 1 (Fig. 1). Fitness calculation of individual X is shown in Algorithm 2 (Fig. 2).

Fig. 1.

Aesthetic dimension selection algorithm based on PSO.

Fig. 2.

Fitness calculation for the individual of aesthetic dimension selection sequence.

Optimization based on NSGA-II

Product form design is conducted by optimizing design parameters with respect to selected aesthetic dimensions. With the design parameters defined in parameterization and aesthetic dimensions selected as S = {s _i}, the optimization problem is formulated as in Eq. (52).

(52)$$\eqalign{& \max s_i( Y) , \;s_i\in S \cr & {\rm s.t.} \left\{{\matrix{ {\forall f_k\in F, \;f_k( Y) < 0} \cr {y_j\in d_j, \;j = 1, \;2, \;\ldots p} \cr } } \right.}. $$

Above is a complex MOO problem with numerous variables and constraints. To solve this, NSGA-II is applied with restrictive cross and mutation operations. The workflow of NSGA-II is shown in Figure 3. Design parameters Y = (y ₁, y ₂, …, y _p) are represented by chromosomes with continuous values. The size of population is defined as Z. Multi-dimensional fitness of chromosome is defined as aesthetic dimension values. During t-th iteration, parent population P _t goes through crossover and mutation operations to generate an offspring population Q _t. Then Q _t and P _t are merged as R _t, which further goes through a fast non-dominated sorting and selection process to produce a new generation P _t+1. After the maximum number of iterations, pareto solutions of final population are obtained as output of the algorithm.

Fig. 3.

NSGA-II algorithm workflow.

Cross operation

Cross operation produces new chromosomes from two existing ones. As shown in Eq. (53), parameters at each position of chromosomes are weighted summed.

(53)$$\left\{{\matrix{ {{\rm chil}{\rm d}_1 = a\ast {\rm paren}{\rm t}_1 + ( 1-a) \ast {\rm paren}{\rm t}_2} \cr {{\rm chil}{\rm d}_2 = a\ast {\rm paren}{\rm t}_2 + ( 1-a) \ast {\rm paren}{\rm t}_1} \cr } } \right.,$$

where a is a random weight, parent₁ and parent₂ are father chromosomes, and child₁ and child₂ are child chromosomes. Constraints in Eq. (52) are checked after crossing. If any constraint is not satisfied for two children, a is again randomly generated until all constraints are satisfied.

Mutation operation

Mutation creates new chromosomes. For a chromosome to be mutated, each parameter y _j has a probability b to be redefined by Eqs (54) and (55).

(54)$$y_j = \bar{d}_j^{\rm L} + {\rm rand}( ) \ast ( \bar{d}_j^{\rm U} -\bar{d}_j^{\rm L} ), $$

(55)$$\bar{d}_j = d_j\cap { y\vert \forall f_k\in F, \;f_k( y_1, \;\ldots,\, y_{\,j-1}, \;y, \;y_{\,j + 1},\ldots, y_p) < 0}, $$

where $\bar{d}_j^{\rm U}$ is the upper bound of $\bar{d}_j$, $\bar{d}_j^{\rm L}$ is the lower bound of $\bar{d}_j$, and rand() gives a random value between 0 and 1. Constraints in Eq. (52) are also checked after mutation, because $\bar{d}_j$ can be non-convex and leads to y _j out of the feasible set. If so, mutation is conducted again with another random value until all constraints are satisfied.

Selection based on dominance and crowdedness

The selection process starts from top dominance levels. Initially, |P _t+1| = 0. If hth level chromosomes can be accommodated, they are selected into P _t+1 as in Eq. (56). Otherwise, R _t,h is sorted by crowdedness in descend. Crowdedness is computed as in Eq. (57). After that, top ones in R _t,h are selected into P _t+1, as in Eq. (58).

(56)$$P_{t + 1} = P_{t + 1}\cup R_{t, h}, \;\quad {\rm if} \vert P_{t + 1}\vert + \vert R_{t, h}\vert < Z,$$

(57)$$\eqalign{& {\rm Cr}( Y_q) = \left\{{\matrix{ {\matrix{ {\sum\limits_i {\displaystyle{{s_i( Y_2) -s_i( Y_1) } \over {s_i( Y_{2Z}) -s_i( Y_1) }}} } \hfill & {{\rm if}\,q = 1} \hfill \cr {\sum\limits_i {\displaystyle{{s_i( Y_{2Z}) -s_i( Y_{2Z-1}) } \over {s_i( Y_{2Z}) -s_i( Y_1) }}} } \hfill & {{\rm if}\,q = 2Z} \hfill \cr {\sum\limits_i {\displaystyle{{s_i( Y_{q + 1}) -s_i( Y_{q-1}) } \over {s_i( Y_{2Z}) -s_i( Y_1) }}} } \hfill & {{\rm otherwise}} \hfill \cr } } \cr } } \right. \cr & s_i( Y_1) < \cdots s_i( Y_{q-1}) < s_i( Y_q) < s_i( Y_{q + 1}) \cdots < s_i( Y_{2Z}) }, $$

(58)$$P_{t + 1} = P_{t + 1}\cup R_{t, h}[ 1, \;2, \;\cdots , \;Z-\vert P_{t + 1}\vert ], $$

where Y _q+1 and Y _q−1 are the neighbors of Y _q by ith aesthetic dimension, R _t,h is the merged population ranked into levels, and R _t,h[1, 2, ⋅ ⋅ ⋅ , Z − |P _t+1|] are the top Z − |P _t+1| chromosomes in R _t,h.

Design validation

Optimized design parameters are validated by designers via modeling. For a certain product form, an exemplar model is built with key parameters accordant with those defined in the section “Parameterization”. The optimized design scheme can be obtained rapidly by loading those parameter values into the exemplar CAD model. The exemplar model focuses on visualizing optimized design to designers. Therefore, unnecessary details are left, and distractive information are avoided. As shown in Figure 5, modeled parts are concise to display major characteristics of form design schemes. Designers visualize and compare the pareto solutions, then select preferable ones for further improvements. Rapid generation of numerous design schemes may also stimulate innovative plans.

Bicycle form design

Bicycle form is a typical case where aesthetics plays an important role. Fashionable styling now has become a key factor to ensure a bicycle prototype's success in the market. Although bicycle forms typically have common structure, consisting of two identical wheels, bars, saddle, chains, and so on, which ensure the basic functions with high dynamic efficiency. However, more details like the arrangement of tubes are not defined. The variance of structures can largely influence users’ sense of beauty and impact users’ satisfaction level. Therefore, bicycle form design is used as case study in this paper. The common structure is decomposed into major components to draw appropriate parameters, as shown in Table 1. Pole type of components that comprise the frame of bicycle is simplified as columns, and the wheel type of components is simplified as rings. Graphical illustration of components is shown in Figure 4. A total of 26 geometric parameters that capture major characteristics of bicycle form are summarized, as listed in Table 2. Bounds for the parameters are defined by their feasible ranges. Constraints include that components should not interfere with each other, the position of saddle must be set between two wheel-centers, paddle trace within the scope of saddle and ground. The last two constraints guarantee mechanical security and ergonomic feasibility.

Fig. 4.

Graphical illustration of components of bicycle form for parameterization.

Table 1.

Decomposition of bicycle form for parameterization

Table 2.

Geometric parameters of bicycle form summarized

There are six bicycle design schemes prepared for expert decision-making in aesthetic dimension selection. Candidate aesthetic dimensions include the 11 aesthetic dimensions introduced in the section “Aesthetic dimension definition”. Evaluated values for aesthetic measure using the formula in the section “Aesthetic dimension definition” are given in Table 3. For expert decision-making, 12 doctorial students who major in industrial engineering are recruited to give pairwise comparison for six bicycles. They have solid knowledge of design theories and experiences of product ideation projects, and are not informed of the research content of this research paper. Their experiences in product design area are ensured. Involving 12 experts are sufficient to ensure credibility while limiting the manpower cost. During the decision-making process, each expert is first given instruction on defining a positive reciprocal matrix as the pairwise comparison matrix. After that, each expert views the bicycle pictures displayed on PC screen in random sequence and finally reports their confirmed pairwise comparison matrixes E ^t in standard form. Rank by expert through Eqs (48) and (49) is obtained as in the final row in Table 3.

Table 3.

Evaluation of design schemes by aesthetic measure and by expert

After expert decision-making, PSO algorithm is applied. A total of 50 trials are run with parameters in Table 4. Experiment is implemented on a PC with Inter Core i7-7700k 4.2 GHz and 16 GB RAM. Among the 50 selection sequences obtained, several selections can offer the same design scheme rank as the rank by expert. These are considered as optimal selections. Among them, those that occur more than four times are listed in Table 5. Frequency denotes the occurrence times of the selection in 50 trials. Listed numbers in a selection denote the indexes of aesthetic dimensions according to Table 3. In Table 5, the selection [1,5,8] contains the least number of aesthetic dimensions. It can be concluded that this selection contains the most representative aesthetic information to accord with the expert decision-making results. At the same time, the least number of dimensions lowers the complexity of the design optimization process later on with NSGA-II. Therefore, this selection is determined as the aesthetic dimensions: balance, cohesion, and regularity.

Table 4.

PSO algorithm parameters

Table 5.

Optimal selections of highest frequencies of 50 trials by the PSO algorithm

Based on selected aesthetic dimensions, NSGA-II is run for bicycle form design optimization. The parameters are shown in Table 6. np denotes the size of population, pc denotes the probability of cross operation, pm denotes the probability of mutation operation at each iteration, and max_iter denotes the maximum number of iterations. These values are defined by finetuning. A total of 20 trials are run, and after each trial, the pareto solutions in the last generation are obtained as the output of the algorithm. On average, 69.25 pareto solutions are obtained.

Table 6.

NSGA-II algorithm parameters

Obtained design parameters are loaded into a bicycle model template created in the Unigraphics NX environment. Among the optimized bicycle design schemes, some examples are shown in Figure 5 with their aesthetic measure results. These schemes clearly display the structural design, despite less important details like chains and brakes. Designers can further verify them aesthetically, choose a preferred one, and adopt the geometry for detailed design.

Fig. 5.

Examples of optimized design schemes.

Method validation

The proposed evolutionary form design method is first validated through subject rating for original and optimized form designs by experts. A total of 10 bicycle designs are used for validation here, including the six bicycle designs for aesthetic measure selection and four randomly selected optimized design. They are rated by those experts included in the pairwise comparison process as discussed in the section “Bicycle form design”. Experts give ratings for their overall aesthetic values in the range of [1–10] after viewing the 10 bicycle pictures in random sequence. Their ratings are aggregated by average and are summarized in Table 7. It is shown that optimized design schemes generally obtain better ratings than the original ones.

Table 7.

Parameter settlement for comparison experiments

Three quantitative criteria are further used to evaluate the effectiveness of the evolutionary product form design method, namely, number of pareto solutions (NOP), variety of pareto crowdedness (VOPC), and computer run time (time). The NOP is defined as the number of pareto solutions. The VOPC is defined as the standard deviation of crowdedness of pareto solutions. Time is the computation run time. Intuitively, the product form solutions are supposed to be numerous, variant, and given by computers as quick as possible. Having more variant solutions means being less clustered and more evenly distributed. Given that the sum of crowdedness of final population solutions is constant, as given by Eq. (59), the standard deviation of crowdedness indicates the variance level of pareto solutions. The lower VOPC means more even distribution and larger variance. Therefore, pareto solutions with a larger NOP, a smaller VOPC, and a shorter time are the better ones.

(59)$$\eqalign{\sum\limits_q {{\rm Cr}( X_q) } & = \sum\limits_{q = 2}^{2Z-1} {\left({\sum\limits_i {\displaystyle{{s_i( X_{q + 1}) -s_i( X_{q-1}) } \over {s_i( X_{2Z}) -s_i( X_1) }}} } \right)}\cr &+ \sum\limits_i {\displaystyle{{s_i( X_2) -s_i( X_1) } \over {s_i( X_{2Z}) -s_i( X_1) }}} + \sum\limits_i {\displaystyle{{s_i( X_{2Z}) -s_i( X_{2Z-1}) } \over {s_i( X_{2Z}) -s_i( X_1) }}} \cr & = \sum\limits_i {\displaystyle{1 \over {s_i( X_{2Z}) -s_i( X_1) }}} \left({\sum\limits_{q = 2}^{2Z-1} {s_i( X_{q + 1}) -s_i( X_{q-1}) }}\right.\cr & \left. \quad + s_i( X_2) -s_i( X_1) + s_i( X_{2Z}) -s_i( X_{2Z-1}) \right)\cr & = \sum\limits_i {\displaystyle{1 \over {s_i( X_{2Z}) -s_i( X_1) }}} ( {2s_i( X_{2Z}) -2s_i( X_1) } ) \cr & = 2\ast {\rm size}( S). } $$

To analyze the effect of np, pc, and pm values on the effectiveness of NSGA-II, three groups of comparison experiments are conducted, respectively, together with the control experiment described above. Parameter settlement for these experiments is summarized in Table 7. The first group consists of three experiments, each with pc set to 0.4, pm set to 0.3, and np chosen from {40, 120, 150}. The second group consists of three experiments, each with np set to 80, pm set to 0.3, and pc chosen from {0.2, 0.6, 0.7}. The third group consists of three experiments, each with np set to 80, pc set to 0.4, and pm chosen from {0.08, 0.2, 0.4}. Each experiment has 20 trials conducted. The average and variance of NOP, VOPC, and time of experiments are shown in Figures 6–8.

Fig. 6.

Comparative experiment results for Group 1 (pc = 0.4, pm = 0.3) with variant np.

Fig. 7.

Comparative experiment results for Group 2 (np = 80, pm = 0.3) with variant pc.

Fig. 8.

Comparative experiment results for Group 3 (np = 80, pc = 0.4) with variant pm.