COMBINING EVOLUTIONARY ALGORITHMS AND ...

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JS Gero (ed): 2006, Design Computing and Cognition’06 © Springer, Dordrecht, pp. 521540.

COMBINING EVOLUTIONARY ALGORITHMS AND SHAPE GRAMMARS TO GENERATE BRANDED PRODUCT DESIGN

MEI CHOO ANG, HAU HING CHAU, ALISON MCKAY AND ALAN DE PENNINGTON University of Leeds, United Kingdom

Abstract. Shape grammars have been used to generate new branded product design shapes in accordance with designer preferences in a number of product domains. In parallel, evolutionary algorithms have been established as random search techniques to evolve and optimize designs to meet specific requirements. The research reported in this paper investigated the use of a combined approach, bringing together the shape synthesis capability from shape grammars and the evolution and optimization capability from evolutionary algorithms, to support the generation and evaluation of new product shapes. A system architecture for the integration of shape grammars with evolutionary algorithms is presented. Prototype software based on this architecture is described and demonstrated using a Coca-Cola bottle grammar as a case study.

1. Introduction A product is an artifact that is manufactured and sold by an enterprise to its customers (Ulrich and Eppinger 2000; Pahl and Beitz 2001). The success of an enterprise depends on its ability to identify customers’ needs and create products that meet these needs quickly and at low cost. However, the consumer market is filled with mass-produced products that are virtually indistinguishable from one another. As technologies become mature, customers take the basic performance of products for granted and begin to look for other properties such as price, value, prestige, appearance, brand reputation and convenience. They tend to purchase by brand name rather than technical distinctions (Norman 1998). As a consequence, enterprises have to brand their products distinctively and promote their brands to gain market share. Brand identity becomes an essential strategy to increase competitiveness. Branded products are delivered to customers through product development processes. A number of different product development processes are proposed in the literature. For example, Ulrich & Eppinger

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(2000) divide product development processes into six phases: planning, concept development, system-level design, detail design, testing and refinement, and production ramp-up. Typically enterprises strive to improve their product development processes by producing more designs, more quickly, at lower cost and higher quality. The achievement of these goals enables enterprises to respond better to customer demand. A key to achieve these goals lies in the synthesis of new product shapes that both conform to brand identity and meet specific functional requirements, for example, a given volume for a bottle. Shape grammar research for product design has focused on the development of product shape or external concept designs and has not stressed the evaluation of the generated designs with respect to functional requirements. Parametric shape grammars have been used to generate branded product design concepts conforming to brand identity but, again, without an explicit relationship to functional requirements. In addition, the sequences of shape grammar rules needed to generate new design concepts have been selected manually. Evolutionary algorithm research for product design has focused on the evaluation of the generated designs with respect to functional requirements but not on the maintenance of the style or external appearance of products. The computational approaches of evolutionary algorithms that automatically search and evaluate designs are capable of replacing the manual effort of rule selection and design evaluation needed in the shape grammar design process. This paper presents the results of research that explored the incorporation of evolutionary algorithms into a shape grammar-based design system. The evolutionary algorithms were used to evaluate generated shapes with respect to a functional requirement. The results of these evaluations were then used to inform the identification of shape grammar rule sequences that were used to create the next generation of shapes. A system architecture for the integration of shape grammars with evolutionary algorithms is presented. Prototype software based on this architecture was built and is demonstrated in this paper using a Coca-Cola bottle grammar as a case study. 2. Shape Grammars 2.1. BACKGROUND

Shape grammars were first introduced by Stiny and Gips in 1972. Shape grammars consist of a set of finite number of shapes S, a set of labels L, an initial shape I and a set of shape rules R that define spatial relationships between shapes (Stiny 1980). It is a formal method to generate shapes through a sequence of rule applications beginning from an starting initial shape, I. Rules take the forms A → B, where A and B are both shapes. As a

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demonstration, a simple shape grammar given by Stiny (1976) is used and illustrated in Figure 1. By applying rule 1 twice and rule 2 once on the initial shape, the resulting shape is shown step by step in the Figure 2.

Figure 1. Simple shape grammar

Figure 2. An example pattern generated from the simple shape grammar 2.2. IMPLEMENTATIONS OF SHAPE GRAMMAR IN GENERATING BRANDED PRODUCT DESIGNS

The visual elements of brand identity can be regarded as an integrated system that includes shapes, colors, and typography/contents (Perry and Wisnom 2002; Wheeler 2003). In the sequence of cognition, the human brain acknowledges and remembers shapes first (Wheeler 2003). Thus, the product shape portrays product identities and gives significant impact to its market share (Perry and Wisnom 2002; Wheeler 2003). Shape grammars have been used to design the shape of consumer products; the first example in the literature was a coffeemaker grammar (Agarwal and Cagan 1998). The coffeemaker grammar was able to generate four existing branded models of coffeemaker but it did not address the issue of style conformance to one particular brand. It gave similar features among them but not distinct features to allow brand differentiation. The first attempt to capture a brand style using shape grammar was the Dove soap bar grammar (Chau 2002). Other examples are the Harley-Davidson Motorcycle (Pugliese and Cagan, 2002 ) and Buick automobile grammars (McCormack, Cagan and Vogel 2004), a Cola-Cola bottle grammar (Chau et al. 2004), and a Personal Care Products grammar (Chen et al. 2004). Table 1 gives a comparison of the shape grammars introduced in this paragraph. In most cases, the shape grammars have focused on the development of product shapes and external forms rather than the satisfaction of functional requirements.

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TABLE 1. Summary and comparisons of research work in shape grammar for product designs Personal Care Products (2004) Personal Care Container Dove, Elvive, Sasson, Gliss Kur, Trevor Sorbie, H & S1

Grammar Name

Coffeemaker (1998)

Dove (2002)

HarleyDavidson (2002)

Buick (2004)

Coca-Cola (2004)

Product Scope

Coffee Maker

Soap

Motorcycle

Car

Beverage Bottle

Brand Link

Krups, Black & Decker, Proctol Silex, Braun

Dove

HarleyDavidson

Buick

Coca-Cola

2D2

3D

2D

2D

2D

3D

100

12

45

63

12

14

Transformation Rule Representation

Components of product

Outline contour of product

Components of product

Components of product

Partitioning of product

Crosssection of product

Essence of brand characteristics

1. Heater Unit 2. Filter 3. Base Unit 4. Water Storage Unit 5. Burner Unit

Entire Product

1. 45degree Vtwin engine 2. Teardropshaped fuel tank.

1. Grill 2. Hood flow lines 3. Outer hood 4. Fenders 5. Middle hood 6. Emblem

Entire product

Entire product

Generation of New Product

Yes

No3

Yes

Yes

Yes

Yes

Shape Generation Criteria

Functional Requirements and Manufacturing Cost

Nil

User Preference (aesthetic)

Designer Preference (aesthetic)

Nil

Nil

Manual

Manual

Manual

Manual

Manual

Manual

No

No

Yes

Yes

No

No

Shape/ Geometric Representation in Rules Number of Rules

Rule Utilisation Method Distinct Identity of Brand Shape 4

1

H&S – Head & Shoulder. Although the rules were in 2D, the authors of this grammar showed that it is possible to interpret the resulting shapes in a 3D form. 3 Dove grammar was not used to generate new Dove shape but was used to generate existing shape of other branded soap. 4 Distinct identity of Brand Shape is defined as clearly recognisable shape to a particular brand among the user of the product scope. 2

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Shape grammars can be used to generate shapes that conform to brand identity. The generation of such shapes entails a sequence of rule applications. Each rule application involves the selection of rules, identification of sub shapes, implementation of the rule and generation of new shapes. Currently these steps are done manually. This research investigated the use of evolutionary algorithms to perform the rule selection step and determination of parameters automatically while satisfying functional requirement and parameter constraints. 3. Evolutionary Algorithm 3.1. BACKGROUND

There are three main biological evolutionary systems that constitute evolutionary algorithms. These three main biological evolutionary algorithms are: evolution strategies, evolutionary programming and genetic algorithms (Whitley 2001). These biological evolutionary systems were introduced independently by several computer scientists in the 1950s and 1960. Evolution strategies were introduced by Ingo Rechenberg in Germany in 1960s and were further developed by Jams-Paul Schwefel. Evolutionary programming was introduced by Fogel, Owens, and Walsh in 1966. Genetic algorithms were introduced by John Holland in the 1960s and developed further by Holland himself together with his students and colleagues at the University of Michigan (Mitchell 1997). These pioneers shared the same idea that the evolution process could be simulated and used as an optimization tool for engineering problems. The general approach in all these systems was to evolve a population of candidate solutions to a given problem using operators inspired by natural genetic variation and natural selection. Since these inventions, there has been widespread of interactions among researchers and evolutionary algorithms have been extensively applied to solve many engineering problems. Terminologies described in evolutionary algorithms are normally analogous to their genetic counterparts in biology. An individual is an encoded solution to some problem. Typically, an individual solution is represented as a string (or string of strings) corresponding to a biological genotype. This genotype defines an individual organism when it is expressed (decoded) into a phenotype. The genotype is composed of one or more chromosomes, where each chromosome is composed of separate genes which take on certain values (alleles). A locus identifies a gene’s position within the chromosome. This evolutionary algorithms terminology is summarised in Table 2. A set of genotypes is termed as a population. Three major evolutionary operators operate on an evolutionary algorithm’s population. These major evolutionary operators are recombination, mutation, and selection. In general terms, recombination exchanges genetic material between a pair of parents’ chromosomes. Mutation flips (replaces) a symbol

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at a randomly chosen locus with a randomly chosen new symbol. Mutation does not happen on every individual; it is executed whenever the mutation probability of an individual is higher than mutation rate. The selection gives individuals with higher fitness a higher probability of contributing one or more offspring in the succeeding generation. The processes of recombination, mutation and selection for reproduction continue until some conditions are met (for example, it reaches the maximum generation). An evolutionary algorithm requires both an objective and a fitness function. The objective function defines the evolutionary algorithm’s optimal condition in the problem domain. On the other hand, the fitness function (in the algorithm domain) measures how ‘well’ a particular solution satisfies that condition and assigns a real-value to that solution. TABLE 2. Explanation of evolutionary algorithms terms Evolutionary Algorithms Explanation Chromosome (string, individual) Solution (coding), part of a complete genotype Genes (bits) Part of solution Locus Position of gene Alleles Values of gene Phenotype Decoded solution Genotype Encoded solution

Historically, evolutionary algorithms have been used for functional optimization, control and machine learning (Goldberg 1989). As such, initial applications of evolutionary algorithms in design were largely focused on the optimization of design parameters. However, more recent research in evolutionary algorithms has been related to the generation of forms or creative designs (Rosenman 1997; Bentley et al. 2001; Renner and Ekárt 2003). Integration of evolutionary algorithm approaches and shape grammars has been attempted in architectural design (Chouchoulas 2003), structural design (Gero et al. 1994) and product design (Lee and Tang 2004). Their work used genetic algorithms to explore design possibilities and shape grammar to provide a syntactic generation method. Chouchoulas used genetic algorithms and shape grammars to evolve architectural layouts. In his work, he generated room layout designs for high rise building that were evaluated against a number of functional requirements (Chouchoulas 2003). He applied simple rectangular oblongs to represent abstract internal room organizations on different floors which required further refinements to complete a building layout. His work was not linked to any existing architectural style. Gero et al. (1994) has used genetic algorithms to produce new generations (by evolution) of a beam structure shape grammar starting from an initial shape grammar. They showed that the evolved shape grammar was able to produce better beam structures than the initial shape grammar. The performance of the evolved shape grammar in each generation was ranked computationally by comparing two conflicting

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physical properties (maximise moment of inertia, minimise beam section perimeter) in the shapes that were generated from the grammar. Lee has also used genetic algorithms to evolve shape grammars for particular types of consumer product; shapes generated from the evolved shape grammars were evaluated manually based on human preference by looking at the designs generated in each generation. Evolutionary algorithms have been successfully applied to many real world problems. However, existing applications have shown that standard evolutionary algorithm approaches alone are not able to achieve the desired results; customization of evolutionary algorithms, by the incorporation of domain specific measures, is needed (Nagendra et al. 1996). The evolutionary algorithm used in the research presented in this paper was customised to enable the evaluation of alternative bottle shapes with respect to their volume. The brief review above discussed about existing evolutionary algorithms that have been integrated with shape grammar. Another important integration system that combines optimisation techniques and shape grammar is shape annealing introduced by Cagan and Mitchell in 1993. Shape annealing is integration between simulated annealing and shape grammar. The search process in simulated annealing is different from evolutionary algorithms in that it borrows ideas from physical processes rather than biology. The use of the shape annealing approach was shown in geodesic dome style designs (Shea and Cagan 1997) and truss designs (Shea and Cagan 1999). In the application of truss design, Shea and Cagan (1999), have used specific grammar rules to generate golden triangles; this has allowed truss structures reflecting style of golden proportions to be built. 4. Integrating Evolutionary Algorithms and Shape Grammars to Generate Branded Product Design to meet Functional Requirement Two main integration interfaces are used to evolve branded product designs: the encoding and decoding interfaces. The encoding interface uses shape grammar rules, initial shapes, parameters and constraints to provide the blueprint for the genotype to be used by the evolutionary algorithm. Each shape rule has its associated shapes, parameters and constraints. The decoding interface allows the evolutionary algorithm to decode the genotype into a phenotype. The phenotype is the actual representation of shape rules and parameters needed to generate the actual design shapes. This encoding process is needed during the early planning stage of evolutionary algorithms. Genotype is coded into a 2-dimensional array of data structure and in the context of shape grammar rules and parameters. As an example, a genetic representation is explained using a case study in Section 5. The decoding interface has two purposes: 1) it allows the evolutionary algorithm to evaluate and rank the phenotype performance with respect to functional requirements during the fitness assignment process; 2) information from it allows final shape to be produced using shape grammar implementation software. A system architecture for the integration of shape grammars with

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evolutionary algorithms is given in Figure 3. 5. A Case Study On A 2-D Coca-Cola Bottle Grammar This case study demonstrated the application of the integrated architecture shown in Figure 3 to produce some viable product shapes that conformed to the Coca-Cola brand identity. The evolutionary algorithm was specifically developed by combining customised evolutionary algorithm sub-functions: recombination, mutation and selection procedures. The representation was a combination of rule numbers (reference number in integers) and their associated parameters (floating point numbers). The case study was based on Coca-cola bottle shape grammar as shown in Figure 4 (Chau et al. 2004).

Figure 3. An evolutionary algorithm and shape grammar integration architecture

The Coca-Cola bottle shape grammar provides information on specific bottle sections, characteristics of the shape and contour in each section, and the relationships between sections. There are no specific rules to generate values for different diameters and heights. There are also no specific constraints on diameters and heights in order to maintain brand image, it is

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still an open issues as to how to maintain brand image. In this application, each bottle section is described using diameters and heights. Diameters and heights provide the start point and end point of each connecting curve. Curves in each bottle section are formed by three points. The curves are determined manually after completing the evolutionary process by approximating the curve shape used in the shape grammar rules. Build the main body

Construct the upper part

Rule 1 $ ! ! ! ! # ! ! ! ! "

Rule 21 Rule 22 Rule 3

Modify the main body

Construct the bottom

Construct the lower part

Rule 41 $ ! ! ! # ! ! ! "

Rule 51 Rule 52

$ ! ! ! # ! ! ! "

Rule 61 Rule 62

Construct the label region

Construct the cap

Rule 71 $ ! ! ! # ! ! ! "

Rule 81 Rule 82

Figure 4. Shape rules for the Coca-Cola bottle grammar, reproduced from

Chau et

al. (2004)

In the implementation, the diameters and heights of each section are set within finite ranges ([minwidth, maxwidth] and [minheight, maxheight]), and currently their values are minwidth = minheight = 0 cm and maxwidth =

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maxheight = 10cm. These parameters were incorporated to facilitate the calculation of the volume (functional requirement) for the bottle shapes produced by the prototype system. The use of volume was a demonstration of one possible application of evolutionary algorithms to generate product designs that both conform to a style captured in shape grammar rules and meet a given functional requirement. 5.1. GENETIC REPRESENTATION

Five sections or parts are used to define the Coca-Cola bottle: cap, upper part, label region, lower part and bottom, Figure 5. There are a total of seven rule groups in the Coca-Cola bottle grammar, Figure 4. Starting from the rules for building the main body, there are other rules for construction of the upper part, modification of the main body, construction of the bottom, construction of the lower part, construction of the label region and construction of the cap. A rule group may contain more than one rule, for example, the construction of the upper part contains three separate rules that produce different shapes on top of the main body.

Figure 5. Graphical illustration of Coca-Cola bottle reproduced from (Chau et al. 2004)

Additional parameters are used to describe the height and diameters of each bottle section (refer Figure 6). For example, three parameters are used to describe the upper part section: bottom diameter (Dia Φ1,2), top diameter (Dia Φ2,2) and height (Height3,2).With these additional parameters for diameters and heights in each bottle part, a genetic representation was built to represent the genotype to encode these seven rule groups and their parameters. The genotype (genetic representation) resembled a (m x n) matrix. In this case study, a 5x7 matrix is used and illustrated in Figure 7. The construction sections and associated rule numbers are given in Table 3. Based on the shape rules of the Coca-Cola bottle grammar, there is more than one rule in each rule group but only one rule can be selected to be executed in a given computation step. In the construction of a bottle, the use of a rule from groups RG1, RG2, RG3 and RG4 is compulsory to produce a valid bottle design because every bottle must have a body, an upper part, a bottom and a cap. The rules in groups RG5, RG6 and RG7 are executed to produce variation in the bottle designs and their use is optional.

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Figure 6. Diameters and heights of the bottle parts

The structures of bottle sections in each rule group vary, Table 3, and they were generalised to facilitate the calculation of the total volume in that the curvilinear parts were simplified into linear lines. Thus, the volume calculated was an approximation to the actual volume. The shape of the bottom and cap were not included in the volume calculations.

Figure 7. Genotype of the Coca-Cola bottles TABLE 3. Properties in each rule group Rule Group (RG)

Construction part

Rules

Structure

RG1

Main body

1

Cylinder

RG2

Upper bottle part

21, 22, 3

Frustum

RG3

Bottle bottom

51, 52

RG4

Bottle cap

81, 82

RG5

41

Cylinder

RG6

Modification on the main body Label region

71

Cylinder

RG7

Lower part

61, 62

Two frustums

The corresponding parameters in each rule group depend on the resulting shape when one of the rules in a rule group is applied. In this particular case study, an evolutionary algorithm was used to generate rule sequences and associated parameters to achieve the total volume of 500ml. Each body part

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had its own parameters: diameters and heights. Both the body part diameters and the body part heights were set in the range [minwidth, maxwidth] and [minheight, maxheight]. The initial population was generated randomly. The probability of selecting a given rule in a given rule group was equal; there was no bias to any particular rule. For example, in rule group two (RG2), there were three rules that could have been selected: 21, 22, and 3, the probability of each being selected was equal, and the probability for each rule was 1/3. The parameters generated were also random and the diameters of the connecting bottle parts were made equal to ensure that the bottle designs were valid. 5.3. EVALUATION AND FITNESS ASSIGNMENT

The objective function for this particular case study was to minimize the difference between the bottle volume () and a desired target volume (). Mathematically, it can be written as equation (1) which is equivalent to equation (2). (1)

(2)

A constant C was added to g(v) to ensure that the objective function took only positive values in its domain (Michalewicz 1996). The volume of a bottle, v, refers to the total volume that the bottle can contain. The volume calculation does not consider the cap and bottom as these do not normally contain the content of the bottle. Each bottle had four possible body parts to be included in the volume calculation (refer section 5.1). The volume of each body part was summed to obtain the total volume of a bottle. The structures of the body parts were varied and could be either a cylinder or a frustum. Individuals in every generation were evaluated based on the volume of the bottle in their phenotype. Fitness assignment is the process of assigning selection probability to each of the individuals in the population. The selection probability is calculated using equation (3). (3)

Where p

i

is the selection probability for individual i; v

i

is the total

volume for individual i; and F is the total fitness of the population, given by equation (4).

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This method is also known as the proportional fitness assignment (4) approach (Goldberg 1989; Michalewicz 1996). 5.4. SELECTION AND PAIRING

The selection procedure was based on stochastic universal sampling (Baker 1987). It provided zero bias and minimum spread. In the procedure (Pohlheim 2005), the individuals were mapped to contiguous segments of a line where each individual's segment was proportional in size to its selection probability. Some equally spaced pointers are placed over the line as many as there are individuals to be selected. If the number of individuals to be selected is N, then the spacing between the pointers is 1/N. The position of the first pointer is given by a randomly generated number in the range [0, 1/N]. Table 4 shows an example of 10 individuals and their corresponding selection probability. If six individuals are to be selected from a population of ten individuals, then N = 6 and pointer spacing, 1/N = 0.167. A random number are generated from the range of [0, 0.167]. As shown in Figure 8, using the index (from 1 to 10), the individuals to be selected in the example are 1, 2, 3, 4, 6, and 8. These individual will later be paired up to undergo the reproduction process in recombination and mutation operations. TABLE 4. Selection probability Individual index Selection probability

1 0.18

2 0.16

3 0.15

4 0.13

5 0.11

6 0.09

7 0.07

8 0.06

9 0.03

10 0.02

Figure 8: Stochastic universal sampling, reproduced from Pohlheim (2005) 5.5. GENETIC OPERATIONS

There were one genetic recombination and one mutation operation in this case study. The recombination operation was a modification of the single point crossover (explained in section 5.5.1) whereas the mutation operation could operate on both rules and parameters associated with the rules (explained in section 5.5.2).

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5.5.1. Recombination The recombination operation began with the random selection of a pair of parents who were chosen for reproduction. Then, a crossover point, represented by an integer, was randomly generated in the range [1, m-1]. The crossover point was the starting location where genetic material between parents was swapped. Figure 9 shows a single-point crossover operating on Parent1 and Parent2; each parent is cut and recombined with a piece of the other. The crossover operation involved in this case study was a modification on this single-point crossover and is illustrated in Figure 10. Two parents P1 and P2 were selected and a crossover point was located in a position equal to 2. This implied that rule numbers in RG1 and RG2 were maintained in the same positions of P1 and P2, but rule numbers RG3 to RG7 were swapped. The resulting chromosomes of the offspring: C1 and C2 are also shown in Figure 10. After the crossover operation, the diameters of the adjacent body parts were usually different. In order to produce a smooth transition between body parts, their diameters were averaged to obtain a new diameter value. Crossover point Before crossover Parent1 (P1) Parent2 (P2) After crossover Child1 (C1) Child2 (C2)

Figure 9. Single-point crossover

5.5.2. Mutation The mutation operator could change any chosen rule into a different rule in the same group. For example, rule 21 could become rule 22 or 3. The associated rule parameters could be altered into different values. The rule parameters were real numbers, and they were randomly mutated in a predefined range as described in section 5.1. The mutation location was an integer, chosen randomly in the range [1, m-1]. The mutation operation was executed with a probability equal to the mutation rate. 6. Experimental Results The case study was coded and implemented following the evolutionary algorithm and shape grammar integration architecture given in Figure 3. A display of the program output is shown in Figure 11. The results shown in the program output can be viewed by the user by scrolling to view the other

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near best solutions in the list of top ten solutions. The last results displayed by the output were the best solutions as shown in Figure 11.

Figure 10. Recombination operations

The best solution found after 1000 generations for a population size of 50 individuals, crossover rate 0.5 and mutation rate 0.5, is highlighted in the Figure 11. The best solution shows that the evolutionary algorithm found a bottle volume of 499.92 ml. Based on the shape rules and associated parameters; the bottle shape was generated using the Coca-Cola bottle grammar implementation software developed by Chau (Chau et al. 2004). The resulting shape is shown in Figure 12.

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Best solution

Figure 11: The output interface in the implementation and the best solution (in cm)

Figure 12. The best solution modelled using Coca-Cola shape grammar implementation

The resulting shape has a similar style to the Coca-Cola contour bottle in that it has an upper part which follows an earlier style of Coca-Cola bottle, as well as a standard label region and a lower part that imitates the wellknown Coca-Cola contour section. It is possible to generate different solutions with volumes within a small tolerance that have different forms. Figure 13 shows the results of best solutions as the generation number increases. These results are obtained by setting the population size to be 100, recombination rate 0.5 and mutation rate 0.7. The details of rule sequences and associated parameters of selected results are given in Table 5. These details show that the different volumes have different rule sequences and associated parameters and would therefore give different forms if modelled graphically using Coca-Cola shape grammar implementation.

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TABLE 5. Detailed results of bottle volumes, rule sequences and associated parameters Population size = 100 Crossover rate = 0.5 Mutation rate = 0.7 Generation Volume Rule sequence, Rule and parameters Number 50 507.5007 1 21 52 82 1 07.91666 07.72321 21 07.91666 03.25223 04.91313 52 07.91666 82 03.25223 100 503.1757 1 3 52 82 41 61 1 06.38506 09.14664 3 06.38506 01.66892 05.50197 52 06.38506 82 01.66892 41 06.38506 07.89625 01.25040 61 06.38506 04.07032 02.90657 150 497.7858 1 21 3 52 82 41 71 61 1 06.41059 09.18485 3 06.41059 02.16005 05.65698 52 06.41059 82 02.16005 41 06.41059 07.55557 01.62928 71 06.29811 01.62928 61 06.41059 04.06910 03.19670 500 500.8721 1 22 52 82 41 71 61 1 06.20483 09.65275 22 06.20483 02.18049 06.35109 52 06.20483 82 02.18049 41 06.20483 07.81973 01.83302 71 06.17222 01.83302 61 06.20483 04.25747 04.33448 1000 500.0054 1 21 52 82 41 71 61 1 06.00181 07.83138 21 06.00181 02.69377 05.80848 52 06.00181 82 02.69377 41 06.00181 07.98335 01.58121 71 05.76209 01.58121 61 06.00181 04.80297 03.53589

04.98967

04.35887

03.48525

04.44746

7. Discussions and Conclusions The research reported in this paper has demonstrated that evolutionary algorithms can be used to generate a number of shape grammar rule sequences and associated parameters automatically and the designs can be evaluated with respect to a single functional requirement (volume of the bottle). The case study showed that it is possible to integrate evolutionary algorithms and shape grammars to deliver product shapes with a particular style and that meet functional requirements. This work can be further expanded to investigate and compare results with other solution from samebrand and other competing brands. This case study is only a starting point to investigate issues in the design of branded product. Considerations of other quantifiable evaluation criteria are currently underway.

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Figure 13. The best solutions converging to 500ml when generation number increases

Acknowledgements The authors would like to thank Ms XiaoJuan Chen for her insightful comments and assistance on the reported research. The authors would also like to express their appreciation to Ministry of Science, Technology and Innovation of Malaysia and Universiti Kebangsaan Malaysia for their scholarship and financial support.

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MC ANG, HH CHAU, A MCKAY AND A DE PENNINGTON