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A hierarchical co-evolutionary approach to conceptual design

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A hierarchical co-evolutionary approach to conceptual design

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Content A HIERARCHICAL CO-EVOLUTIONARY APPROACH TO CONCEPTUAL DESIGN by Wei Li A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) August 2006 Copyright 2006 Wei Li Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3236523 Copyright 2006 by Li, Wei All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3236523 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication To My Parents and My Wife Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I would like to express my gratitude to those people. Without their support and encouragement, I would not have been able to achieve this goal. My first and foremost thankfulness goes to my thesis advisor, Dr. Yan Jin. It is he who teaches me to be a scholar with his knowledge and patient. His enthusiasm and insight inspire me to make progress every day. It would be the most valuable experience in my life under his supervision. 1 would like to thank the other members on my dissertation committee, Dr. Stephen Lu, Dr. Geoffrey Shiflett, and Dr. Behrokh Khoshnevis for their critical comments and insightful suggestions. 1 am also thankful to Dr. David E. Goldberg for his valuable advice when serving as a member of my Ph.D. qualifying exam committee. I would like to thank my colleagues in IMPACT laboratory; in particular, Dr. Mohammad Reza Danesh Dezfuli, Dr. Oren Benami, Dr. Kai-Lu Wang, Dr. Li Zhao, Dr. Pawat Chusilp, Yoishiro Suzuki, Nan Jing, Jing Zhang, Jieying Zhang, Mathieu Gestlin, Daehwan Kim, Qingfeng Li, George Zouein, Hui Wang, and Hung-Fu Chang for their helps with my study in USC and their comments on my research. Thanks also to Silvana Martinez-Vergas for her timely administrative support. The deepest thanks go to my wonderful parents, who cultivate and educate me with their endless love. My final, and most heartfelt, acknowledgment must go to my lovely wife Yuan. Her support, encouragement, and companionship have turned my study and research journey into a pleasure. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Dedication............................................................................................................................ii Acknowledgements........................................................................................................... iii List of Tables.................................................................................................................... vii List of Figures..................................................................................................................viii Abstract................................................................................................................................x 1 Introduction.................................................................................................................1 1.1 The Problems and Motivation............................................................................ 1 1.2 Research Issues................................................................................................... 5 1.3 Thesis Overview.................................................................................................7 1.4 Thesis Organization............................................................................................8 2 Related W o rk .............................................................................................................9 2.1 Design methods..................................................................................................10 2.1.1 Axiomatic design..................................................................................... 1 1 2.1.2 Systematic approach................................................................................ 13 2.1.3 Functional representation.........................................................................14 2.1.4 Grammar-based Approach.......................................................................15 2.1.5 Co-evolutionary model............................................................................ 17 2.2 Evolutionary Computations.................... ........................................................19 2.2.1 Genetic Algorithms..................................................................................20 2.2.2 Genetic Programming..............................................................................23 2.2.3 Applications of Evolutionary Computation...........................................27 2.3 Fuzzy Set Theory..............................................................................................28 2.3.1 Definition..................................................................................................29 2.3.2 Operations on fuzzy set........................................................................... 29 2.3.3 Fuzzy wei ghted average......................................................................... 30 2.3.4 Fuzzy preference relations..................................................................... 31 2.3.5 Applications of fuzzy set theory.............................................................33 2.4 Conclusion....................................................................................................... 35 3 Conceptual D esign.................................................................................................. 36 3.1 Conceptual design process..............................................................................36 3.2 Evaluation in conceptual design......................................................................39 3.3 Points of departure............................................................................................41 3.4 Conclusion.......................................................................................................42 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 A Hierarchical Co-Evolutionary Approach to Conceptual Design............... 44 4.1 The HiCED Framework...................................................................................44 4.2 Design Concept.................................................................................................46 4.3 The Co-Evolutionary Design Process.............................................................47 4.4 Hierarchical Exploration.................................................................................. 50 4.5 Function and Means Library............................................................................51 4.5.1 Functions...................................................................................................52 4.5.2 Means........................................................................................................53 4.5.3 Constraints............................................................................................... 54 4.6 Evaluation of design concepts.........................................................................55 4.7 Conclusion........................................................................................................ 58 5 Algorithms of H iC ED .............................................................................................59 5.1 Function Elaboration.........................................................................................60 5.1.1 General function elaboration rule set (G FE)........................................ 61 5.1.2 Action-related function elaboration rule set..........................................64 5.2 Co-Evolutionary design................................................................................... 66 5.2.1 Genetic modeling of function structures................................................67 5.2.2 Constrained genetic programming operations.......................................70 5.2.3 Fitness functions for function structuring..............................................75 5.2.4 Genetic modeling of m eans.................................................................... 81 5.2.5 Fitness functions for means selection.................................................... 83 5.3 Fuzzy evaluation on means connectivity........................................................85 5.3.1 Weight of means connectivity................................................................87 5.3.2 Preference model for design evaluation.................................................89 5.4 Interaction with designers................................................................................ 92 5.5 Conclusion........................................................................................................ 95 6 Test and Evaluation............................................................................................... 96 6.1 System Implementation and Setup................................................................. 96 6.2 Experiment Design...........................................................................................99 6.3 Hypotheses.......................................................................................................101 6.4 Case Study 1: Personal Transporter Design.................................................102 6.4.1 Function Library and Means Library....................................................102 6.4.2 Analysis of Results................................................................................ 103 6.4.3 Fuzzy Evaluation..................................................................................117 6.5 Case Study 2: Testing Machine Design........................................................ 124 6.5.1 Function Library.....................................................................................125 6.5.2 Function Elaboration.............................................................................. 125 6.5.3 Function Structuring.............................................................................. 126 6.5.4 Discussion............................................................................................... 128 6.6 Conclusion.......................................................................................................128 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 Contributions........................................................................................................129 7.1 Summary of Research Results...................................................................... 129 7.2 Contributions..................................................................................................131 7.3 Recommendation for the Future W ork.........................................................133 Bibliography...................................................................................................................135 Appendix A: Greedy Search Algorithm for Function Expansion Rule (FER) .144 Appendix B: Action-Based Function Elaboration Rules....................................... 146 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table 4-1: Flow set in functional basis.......................................................................... 52 Table 4-2: Actions in functional basis............................................................................53 Table 6-1: Parameter setting for co-evolutionary design..............................................97 Table 6-2: Available means for <guide><ME> and <stop><ME>........................... 105 Table 6-3: Solutions with Requirements of Low Cost and Light W eight.................107 Table 6-4: Means Selection with Long Travel Range Requirement......................... 107 Table 6-5: Solutions after library size changes............................................................ 108 Table 6-6: Effect of Control Variables on Function Structuring................................110 Table 6-7: Optional solutions for simple mechanical transporter design..................120 Table 6-8: Fuzzy Connection Relations among Means.............................................. 121 Table 6-9: Function library for testing machine design.............................................. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure 2-1: A model of Co-evolutionary Design...........................................................17 Figure 2-2: Relationship between design and evolution...............................................20 Figure 2-3: Common operators for GA.......................................................................... 23 Figure 2-4: An example of AD F..................................................................................... 24 Figure 2-5: Genetic programming operations................................................................25 Figure 2-6: General structure for genetic algorithms.................................................... 26 Figure 2-7: Fuzzy set....................................................................................................... 29 Figure 2-8: Example of Fuzzy Preference Relation...................................................... 32 Figure 4-1: Component of HiCED..................................................................................45 Figure 4-2: Power train design........................................................................................ 47 Figure 4-3: Design Process o f HiCED........................................................................... 48 Figure 5-1: Topological relationships in function structure.........................................69 Figure 5-2: A chromosome model of function structure..............................................69 Figure 5-3: An invalid genetic operation....................................................................... 70 Figure 5-4: Illustrating constrained crossover operation..............................................73 Figure 5-5: Illustrating constrained mutation operation...............................................74 Figure 5-6: A chromosome model of m eans.................................................................82 Figure 5-7: Genetic operators for means evolution...................................................... 82 Figure 5-8: An example of function structure................................................................ 87 Figure 5-9: Connection relation derived from function decomposition..................... 88 Figure 5-10: Topological relationships in function structure.......................................89 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5-11: Means connectivity based on function structure..................................... 90 Figure 6-1: System Architecture of HiCED...................................................................97 Figure 6-2: An example of function structure generated by the System.................... 98 Figure 6-3: An Example of Convergence Curves for Co-Evolutionary Design 98 Figure 6-4: Experiment Design....................................................................................... 99 Figure 6-5: Function and means library........................................................................102 Figure 6-6: An Example of Grammar-based Function Elaboration........................... 103 Figure 6-7: Two Alternative Function Structures at Level 3 ......................................105 Figure 6-8: Function Structure Regarding Weight and Cost Requirements 106 Figure 6-9: Function Structure for Long Travel Range Requirement.......................107 Figure 6-10: Effective Distribution of Fitness Function Weight..............................110 Figure 6-11: Weight Distribution for Constrained Genetic operators..................... 115 Figure 6-12: Convergence Curve for Function Structuring...................................... 115 Figure 6-13: Function means connectivity...................................................................118 Figure 6-14: Linguistic scale of weight value............................................................119 Figure 6-15: Linguistic scale of fuzzy connection.......................................................120 Figure 6-16: Overall connectivity level........................................................................122 Figure 6-17: Function elaboration for testing machine............................................... 126 Figure 6-18: Distribution of Fitness Function Weight................................................ 126 Figure 6-19: Function structure for a testing machine................................................ 127 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Conceptual design is a key activity in product development. However, the limited understanding of the conceptual design process and the lack of quantitative information at this stage of design pose major difficulties for effective design concept generation and evaluation. A conceptual design support tool is needed. The objective of this research is to take a computational approach to supporting effective design concept generation and evaluation at the conceptual design stage. With this objective in mind, a hierarchical co-evolutionary conceptual design model, called HiCED, is proposed. In the model, the conceptual design process is viewed as a process in which the required functions and solution means evolve in parallel across different levels of elaboration hierarchy. A zigzag design process is adopted as the overall co-evolutionary design process. To automate the process of the conceptual design, two reasoning mechanisms are developed, i.e., a grammar-based mechanism for function elaboration and a genetic algorithm (GA) and genetic programming (GP) based co-evolutionary algorithm for function structuring and means selection. The hierarchical co-evolutionary development allows designers to make the maximum use of the available information for effective evaluation. A set of fitness functions have been investigated to gain the insights of what is important to a design. In order to deal with the highly qualitative information at the early stage of design, a fuzzy preference model is developed to rank design alternatives based on the function-means connectivity. The developed framework has x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. been tested with two conceptual design problems. The results demonstrate the effectiveness of the HiCED model in supporting conceptual design. The research made the following contributions to the field of engineering design research. First, it advances the understanding of the conceptual design process and provides a computational support for design concept generation and evaluation. Second, the proposed model provides insights into what are important to a design. Third, this research provides a fuzzy evaluation approach to utilizing the subjective and incomplete information at the conceptual design stage. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Introduction 1.1 The Problems and Motivation Conceptual design is a process to determine fundamental concepts and constraints from which the subsequent designs, i.e., embodiment design and detail design (Pahl & Beitz, 1996) will follow. The results of the conceptual design will directly affect design cycle time, cost and quality. It has been shown that conceptual design costs only 3% of product development resource, but it determines almost 60% of product’s features such as performance, manufacturability, and cost (Vollbracht, 1988). In the area of mechanical engineering, conceptual design can be very complex. A general approach to designing complex mechanical systems is to break the design problems into more manageable sub-problems, solve the less complex sub-problems, and then develop the overall design by synthesizing the sub-solutions (Pahl & Beitz, 1996; Suh, 2001; Maimon et al., 1996). This method of divide-and- conquer (Coremen et al., 2001) usually involves multi-layered problem and solution spaces in which the required functions are generated and elaborated and then the means are identified to fulfill the functions. There are two general ways to handle the mapping between the required functions and means. One is “function first”, that is, elaborating functions and generating a function elaboration hierarchy first and then identifying desired means (or working principles in Pahl & Beitz’s term) for only the leaf-level functions. The final design concept is developed by selecting the most desirable combinations of the leaf-level means. Systematic design (Pahl & Beitz, 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1996) is an example of this approach in which leaf-level functions and the corresponding means form a morphology chart for determining final design concepts. The other general approach to developing design concepts is “zigzag”, as indicated by Suh (Suh, 2001) in axiomatic design. In the zigzag approach, identification and mapping of means for fulfilling the required functions are carried out at every level of function decomposition. Before a given function is decomposed into lower level ones, its corresponding means (or design parameters in Suh’s term) are explored and determined. The selected means provides useful information to guide the process of decomposing the given function. In this way, both functions and means are developed concurrently. Unlike the electronic circuit design and some architecture design problems, both required functions and candidate means in the mechanical engineering domain are numerous and their relationships are much more complex. Required functions can be decomposed in many different ways. At each level of function decomposition hierarchy, there exist multiple mappings from functions to their corresponding means. Although expert designers can skillfully perform these design tasks relying on their experiences, there are two fundamental problems. First, it is possible that an expert designer may not be able to go through all reachable solutions due to time constraints and mental limitations. Second, in most industrial sectors, access to expert designers is limited. Many design tasks are carried out by young and less experienced designers whose limited knowledge and skill may lead to degraded 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. designs. What is needed by the industry is a conceptual design support tool that can make experienced designers more efficient and novice designers more effective. In the past two decades, engineering community has benefited considerably from the advance of computer and information technologies. However, most computer based design tools developed to date are for later design stages rather than the conceptual design. Developing tools to support the conceptual design remains a major challenge for design researchers. Efforts have been made for conceptual design support. Two general approaches have been taken. One is information management support, which enhances designers’ capabilities of documenting and managing their design information (Berkelman et al, 1995; Shilit et al, 1998; Klein, 1997). Although information management support can increase the efficiency of the design process and allow designers more time to think about their potential alternatives, it cannot enlarge the solution space that the designers can potentially reach. The other approach is more aggressive and attempts to generate design concepts automatically for designers. Grammar based approach (Schmidt and Cagan, 1997) and other computational search based approaches (Maher, 2001; Zhou et al, 2002) have been applied to generate design alternatives. While the results of the research to date have shown the potential of using computers to generate alternatives for relatively simple design problems, dealing with complex design problems that involve remains to be a major challenge. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Automating design concept generation is difficult for two major reasons. First, the current understanding of design concept generation process is limited, and this lack of understanding makes it difficult to apply any existing design process model directly for computer based concept generation. Although research in design theory and methodology has developed methods for designers to follow (Pahl & Beitz, 1996; Suh, 2001; Kitamura, 2002; Akao, 1995), these methods specify only the steps that designers should follow and provide little guidance for designing within each step and for moving between the steps. Cognitive models of conceptual design process have been explored recently (Benami et al., 2000; Cross et al., 1997; Chuslip et al., 2004), but it will still take some time for these models to be useful for building computational and automated design tools. Another important issue related to providing conceptual design support is the lack of quantitative information for design concept evaluation. The concerns at the early stage of design are mainly focused on how to identify the most appropriate design principles rather than technical details (Pahl & Beitz, 1996). The general aspects and essential problems are identified and the properties are only known qualitatively. Compared with the later stages of design, objectives at the conceptual design stage are more abstract, search space is more open, but the available information for design concept evaluation is limited. The information is inherently incomplete and subjective at this stage. This situation makes it difficult to establish effective evaluation criteria and poses a major difficulty for developing automated tools for conceptual design. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To date, there is not a good model of conceptual design to support design automation because of the limited knowledge of the conceptual design process and the lack of quantitative information for effective evaluation. The overall goal of this research is to develop a new computational approach and tool that support conceptual design by providing automated exploration of design space and formation of design concepts. The core task is to develop a model of the conceptual design process that allows design concepts to be generated from initial design requirements, an algorithm for efficient design concept generation, and effective evaluation criteria for choosing best design concepts. 1.2 Research Issues The importance of conceptual design is drawing designers’ attention to the need of a framework which can not only provide insights into conceptual design, but also helps develop a computational tool to augment traditional conceptual design activities. The challenges are: how can we develop an effective model o f the conceptual design process to support design automation? and what criteria should be adopted and how do these criteria impact design concept generation? To meet these challenges, following research issues must be addressed: l) Modeling conceptual design process. A model that can support conceptual design automation needs to provide detailed guidance for designing within each step and for moving between the steps, which can not be done by the existing design methods. Upon this model, effective 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. algorithms can be expected to develop for design concept generation and evaluation. 2) Developing algorithms for design concept generation. Conceptual design involves the evolution of function structures and means across multiple levels of decomposition hierarchy. In the mechanical engineering design domain, the potential solutions spaces are huge but available information in limited. An effective algorithm for conceptual design automation should make the maximum use of available information to design concept generation and evaluation. The algorithms must take care of two aspects: first, function decomposition and design concept evolution across different levels of decomposition hierarchy; second, design effectiveness and efficiency. 3) Devising evaluation methods for assessing the generated design concepts. A major reason of lacking conceptual design support is that conceptual design is characterized by the highly qualitative information. The incomplete and subjective information makes it difficult to establish effective evaluation criteria for design concept generation. To develop automated tools of conceptual design, evaluation methods must be investigated and the insight of what is important to the design should be gained. These research issues raise following initial research questions: 1) How can we model conceptual design as an evolutionary process? 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2) What is the role of decomposition in evolving design concepts? 3) How can we mingle decomposition with evolution? 4) What criteria or fitness function should be adopted for evaluating design concepts? 5) How will the fitness functions impact the design results? 6) How can we deal with the incomplete and subjective information at the early stage of design? 1.3 Thesis Overview This thesis presents a framework for automating the process of the conceptual design. To address the research questions mentioned above, a hierarchical co-evolutionary approach is proposed to support the exploration of design space, formation of design concepts and effective evaluation at the early stage. The approach adopts a zigzag process in which functions and means co-evolve in parallel across different levels of decomposition hierarchy. A set of fitness functions have been identified. And a case study is conducted to validate the model and investigate the impact of evolutionary mechanisms on design concept generation. Overall, the main academic objectives are: 1) To develop an evolutionary mechanism for concept generation during the conceptual design process; 2) To investigate the role of decomposition in evolving design concepts; 3) To develop a method that can make the maximum use of the available design information for design concept evaluation; 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4) To study what criteria are important to design concept generation; 5) To explore fuzzy logic based methods for design concept evaluation The main technology objective of this research is: 6) To develop a computational approach that can support design concept generation and evaluation at the conceptual design stage 1.4 Thesis Organization The thesis consists of seven chapters. Following is a summary of the rest of the chapters: Chapter 2 presents an extensive review of the literature that is related to this research work. The review covers research from the following research fields: design methods, fuzzy set theory and evolutionary computation. Chapter 3 particularly describes current state of conceptual design from both research point of view and industry point of view. Then we point out the limitation of current research and address the need for a new approach. Chapter 4 presents the overall framework o f our proposed approach. In our research, conceptual design process is modeled as a co-evolutionary process of functions and means across different levels of elaboration hierarchy. The model makes the maximum use of the available information for design concept generation and evaluation. Chapter 5 describes the co-evolutionary algorithms based on the model. Grammar based rules guide functional elaboration, and GA&GP based algorithms 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. develop functions and means at each level of elaboration hierarchy. A set of fitness functions is identified for design concept evaluation. A fuzzy evaluation approach is proposed to rank the alternatives based on the function-means connectivity in terms of the subjective and incomplete information. Chapter 6 uses an example to validate and demonstrate the proposed model. A series of cases are designed to investigate the impact of user’s requirements, size of the libraries, fitness functions, genetic parameter settings and means connectivity on design concept generation. The results of the case study are discussed in detail. Chapter 7 concludes this thesis by summarizing the findings and contributions, and recommending the future work. 2 Related Work My research suggests a hierarchical co-evolutionary approach {HiCED) to automating the process of design concept generation and evaluation during the early phase of design. The development of the theoretical framework for HiCED is drawn up on three fields of research, namely design methods, evolutionary computation and fuzzy set theory. In the field of design methods, researchers try to prescribe how design should be represented and how it should be done. The research provides a basis upon which our proposed approach establishes. In the field of evolutionary computing, researchers use the mechanisms that are inspired by biological evolution in engineering design and have demonstrated 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. their success in design exploration, design decision-making, design optimization and so on. Conceptual design can be characterized by its level of uncertainty in terms of the lack of quantitative information and poorly defined specifications. Fuzzy logic allows the designers to use a more natural way to represent their qualitative knowledge of the design and make their decisions based on their confidence on the knowledge that they utilized. The approach proposed in my research will utilize the methods of evolutionary computation and fuzzy set theory to explore design space and provide guidance between design steps for the conceptual design. This chapter presents an extensive review of the related literature in these research fields. 2.1 Design methods Research on design methods explores what are the appropriate ways to engineering design. Axiomatic design is developed in an effort to make engineering design process as logical as possible. Systematic approach describes the design process from an engineering practice point of view. Functional representation provides a method to describe the contents of design and achieve the repeated and meaningful results from the perspective of functional modeling. Grammar-based approach can support functional decomposition and design synthesis by combining grammar rules and search strategy, and co-evolutionary design can explore design by switching design focus between problem and solution design spaces. In this section, we will 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. review these methods and discuss their relations with design concept generation at the stage of the conceptual design. 2.1.1 Axiomatic design Axiomatic Design (Suh, 1990, 1998, 2001) is a general methodology that helps designers structure and understand design problems, thereby facilitates the synthesis and analysis o f suitable design requirements, solutions and processes (Yang et al., 2000). In Axiomatic Design, four domains, namely customer domain (CNs), functional domain (FRs), physical domain (DPs), and process domain (PVs) are proposed. Design process is organized as a “zig-zag” process in which the mapping is carried out from left to right between the domains whereas decompositions are conducted top-down within the confines of each domain. The activity of the conceptual design takes place between the functional domain and the physical domain. The objective of the conceptual design is to search for a set of design parameters (DPs) in the physical domain that satisfy their corresponding function requirements (FRs) in the functional domain. Axiomatic design also provides a consistent framework from which the metrics of design alternatives can be quantified (Yang et al., 2000). To assist designers to make proper decisions when facing multiple alternatives available during the decompositions and mappings, two design axioms are defined as a rational basis for evaluating the proposed solution candidates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Axiom 1 (The independent Axiom): Maintain the independence of the functional requirements. Axiom 2 (The Information Axiom): Minimizing the information content of the design. The first axiom focuses on the nature of the mapping between “what is required” and “how to achieve it”. It states that the independence of functional requirements must be always maintained. When there are two or more functional requirements, the optimal solutions should be the ones that each functional requirement can be satisfied without affecting other functional requirements. Decoupling must be executed if FRs are coupled or become interdependent in the proposed design. The second axiom establishes information content as a relative measure for evaluating and comparing solutions that satisfy the first axiom. It states that among those designs that satisfy the independent axiom, the solution which has the highest probability of success is the best design. Axiomatic design is a general guideline for conceptual design and a powerful tool in dealing with complex design problems. A lot of efforts have been made to apply Axiomatic Design to engineering design problems, for example, machine control systems (Lee, 2001) and suspension system (Deo, 2003). In these research studies, computational techniques have been devised to facilitate the determination of the independence of FRs when selecting DPs, but it still heavily depends on the design experience to implement the mapping process. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1.2 Systematic approach The systematic design approach (Pahl & Beitz, 1996) proposes design process based on the observations of solution processes from real design situations through many years of engineering practice. The activities of design can be roughly classified into four stages: 1) planning and clarifying the task; 2) conceptual design; 3) embodiment design; and 4) detail design. Conceptual design is a stage of design process to determine a solution principle by the identification of essential problems through establishing function structure, and searching for appropriate working principles and their combinations. The first step in the conceptual design is to abstract the requirement list into an essential problem statement. Personal preferences are eliminated; requirements that have no direct links with functions and essential constraints are omitted; quantitative data are transformed into qualitative mode and at last the problem is formulated into solution neutral terms. In function structure design, functions are represented as blocks in a function diagram based on the flows of energy, material and signal. The function diagram represents the relationships between inputs and outputs of the required function independently of the solution. A complex overall function is usually decomposed into sub-functions with less complexity. Depending on the complexity of the problem, the sub-functions are further decomposed until the details of specifications can be acquired. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The third step of conceptual design is to develop working structure. Desired working principles are selected to implement the corresponding functions in function structure and these principles are eventually combined into a working structure. The concretization of working structure will lead to the solution principles. Some general applicable evaluation methods have been proposed for the determination of solutions. Different from Axiomatic Design, systematic design tries to avoid involving physical implementations when developing function structures at the conceptual design stage. It argues that the functional design should be solution-neutral so that designers can search for optimal solutions without any constraints. Although systematic design prescribes the process how conceptual design should be done, it does not provide specific guidelines or criteria for concept evaluation. The design concept generation and evaluation are still human experience based. 2.1.3 Functional representation A formal functional representation provides a method to describe the contents of design and achieves the repeated and meaningful results from functional modeling, which can help designers to understand and represent function structures without relying on physical implementation. The researchers considered that the misrepresentation in function structure design is a key reason to cause the practice ineffectively and inefficiently. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A standardized set function-related terminology, or a functional basis (Stone et al., 1999, 2000; Hirtz et al., 2001a, 2001b), has been developed for design capture and reuse. Similar to systematic design, functions in the functional basis are categorized by its action, and the relations between functions by the contents of the flow at different levels. Based on these research results, designers can enhance and expand their practice in design repositories, design synthesis and general product modeling. Sridharan (Sridharan, 2004) introduced an approach to function structuring by combining function representation and graph grammars. And Bryant (Bryant et al., 2005a) utilized the repository of the existing design knowledge to automate the process of design concept generation and evaluation based on the generated function structure. However, without taking into consideration of the implementations of the functions, the functional representation makes the evaluation of function structures and final design concepts difficult. 2.1.4 Grammar-based Approach Grammar is a way of combining symbols into a meaningful expression in human language system. Similar phenomena can also be found in design where various shapes and components can be arranged to fulfill the desired design by following certain grammar rules. Grammar based design started around two decades ago. Since then, it has attracted design researchers when people started realizing its ability in 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. design exploration through a flexible expression of grammar rules (Schmidt et al., 1995, 1997). In mechanical design synthesis, grammar has been adopted as a formal definition of syntactic structure of mechanical systems (Schmidt et al., 1995). From the technical point of view, design grammar is a production system that generates designs according to a set of user-defined rules (Stiny, 1980; Gips et al., 1980; Chase et al., 2002). In engineering design, grammar has successfully been applied to generate various kinds o f mechanical artifacts such as coffeemakers (Agarwal et al., 2000), epicyclic gear train (Li et al., 2001), clocks (Starling et al., 2002), sheet metal (Soman et al., 2002), etc. Partially due to its origin from shape grammar, most of these applications of grammatical approach are based on shape or form grammars. Some researchers have introduced grammar of function and explored their application together with form grammar in relatively specific engineering domains (Schmidt et al., 1995, 1997; Starling et al., 2002). It has been demonstrated that function grammar provides a computational support for conceptual design but may lead to a large space of design alternatives (Schmidt et al., 1995, 1997). If an efficient search algorithm can be devised, the approach can facilitate to extend designers’ reach of design space and find optimal solutions. Grammar based approach has been proved a useful tool in supporting the automation of the conceptual design (Schmidt et al., 1995, 1997). However, its performance is much degraded because of the large number of randomly generated 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. data. It makes the approach difficult to be applied in the mechanical design domain, where candidate functions and means are usually huge. 2.1.5 Co-evolutionary model Co-evolution is a term to describe the natural process in which two or more species affect each other intimately in evolution. The biological phenomenon in nature inspires the emergence of co-evolutionary computing (Richard, 1995; Paresis, 1998). Co-evolutionary design, proposed by Maher (Maher, 1994), provides a creative approach to solve design problems in which requirements and solutions evolve in parallel. To Maher’s point of view, design can not only be considered as a search process for well-defined problems in which design focus is not changed with the development of the design, but also as an exploration process where design focus switches between different design spaces. A computational model of co-evolutionary approach was developed for design exploration (Maher et al., 1996,2001), as illustrated in Figure 2-1. Requirements Interaction Solutions Figure 2-1: A model of Co-evolutionary Design 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Figure 2-1, Co-evolutionary design occurs through the change of search focus between problem space and solution space. When design problems rise, solution space is searched based on the evaluation using problem space as the source of focus. The activity is corresponding to the downward arrow in Figure 2.1. If no satisfactory solutions are found given the design requirements, solution space will serve as the basis of evaluation for searching in the problem space. It is corresponding to upward arrow in Figure 2.1. After the search for relevant requirements in problem space, desired solutions have to be searched in solution space again with respect to the change of problem space. The process will continue until satisfactory solutions are found. The co-evolutionary model has attracted researchers’ attentions in different domains where traditional evolutionary algorithms are limited because of considerable computational cost associated with complex design problems (Bentley, 1999). Potter (2000) described a framework for automatically evolving sub components as a collection of cooperating species. And Rubenstein-Montano (2000) employed the co-evolutionary technique to enhance decision-making in complex negotiation situations. The research results have shown that co-evolutionary strategy can be of significant benefit within the adaptive search domain when complex engineering design problems arise. The co-evolutionary method provides a computational model to explore design by dynamically making revisions to fitness function. However, it is only applied into some “simple” system designs, where building blocks are simple and 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solutions are generated in bottom-up hierarchical approach. But conceptual design in the mechanical engineering domain is complex. It usually takes a top-down hierarchical approach and involves multiple-layered decompositions and mappings. 2.2 Evolutionary Computations That evolution and design process share many similar characteristics has been recognized for a long time (Holland, 1975; Goldberg, 1989, 1991; Koza, 1992, 1994; Fogel et al., 1996; Parmee, 1997; Bentley, 1999). Evolutionary design exploits evolutionary searching for a space of solutions analogous to biological evolutionary searching for a space of gene sequences (Figure 2-2) (Lee et al., 2001). Since the 1960’s, there has been an increasing interests in evolutionary design when people realized its ability in objective optimization and adaptive search. New researches (Bentley, 1999; Parmee, 1997; Maher, 2001) have shown that evolutionary computation not only can work well in search and optimization, but also have the ability in design exploration and creative design. The applications (Bentley, 1999; Koza, 1999) have demonstrated that certain novel designs can be automatically synthesized. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Solution space Encoding Genotype space Candidate solutions Chromosomes Decoding Evaluation & selection Genetic operation Figure 2-2: Relationship between design and evolution Two main avenues of evolutionary algorithms are widely known and applied into science domains and engineering design domains currently. They are genetic algorithm (GA) (Holland, 1975; Goldberg, 1989) and genetic programming (GP) (Koza, 1992, 1994, 1999). Same underlying evolutionary mechanisms are shared but different encoding schemas are applied. Genetic algorithms uses binary (or real - Michalewicz, 1994) encoding schema but genetic programming uses hierarchical computer programs with dynamically varying size and shape. 2.2.1 Genetic Algorithms Genetic algorithms (GA) are developed for global stochastic optimization and search based on the evolutionary mechanism of natural selection and genetics (Lee et al., 2001). They were introduced by Holland in 1975, and then advanced extensively (Goldberg, 1989; Michalewicz, 1994; Maher, 1996; Parmee, 1997; Bentley, 1998) because their great potential in design optimization and exploration can often outperform the conventional techniques. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To make GA applicable, three elements must be taken into account. They are encoding scheme, fitness function and genetic operators. Encoding scheme: encoding schema is a definition of the solution representation in the genotype space. In GA, the solutions are usually represented as binary strings (Goldberg, 1989) or real-number strings (Michalewicz, 1994). A coding scheme is needed to map the candidate solutions into the coded strings. Based on the coded strings, genetic operators can be applied to maintain and manipulate the population of candidate solutions. A typical chromosome in GA is represented as: 1 0 0 1 1 Fitness function: a fitness function is defined to quantify a chromosome to see how fit it is as a solution to a problem. By fitness evaluation, a particular individual can be ranked against other chromosomes. A chromosome with a higher fitness value has a higher chance to breed new generations with other chromosomes. Genetic operators: A genetic operator is a process used in genetic algorithms to maintain genetic diversity (www.worldiq.com). The most common used genetic operators are selection, crossover (recombination) and mutation. Selection schema is a bias to select individuals from the population for the reproduction of fitter individuals. It is necessary to adopt a selection scheme with adequate selection pressure so that a quick convergence is achieved. There are two most popular selection schemes, namely, fitness proportion selection and tournament selection. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fitness proportionate selection, also known as roulette-wheel selection, uses a relative fitness to determine the selection probability of an individual. Through the evaluation of the fitness functions, each chromosome is assigned a fitness value. In fitness proportionate selection, this fitness level is associated with a probability of selection. Candidate solutions with a higher fitness will be less likely to be eliminated (www.worldiq.com). Tournament selection runs a tournament among a few individuals and selects the best for the next generation reproduction. In the tournament selection, selection pressure can be easily adjusted by modifying the tournament size. The higher the tournament size is, the smaller a weak individual has chances to be selected (www.worldiq.com). Crossover is a genetic operator to generate new offsprings from two parent chromosomes by copying the selected coding from each parent (Mitchell, 1997). The chromosomes are selected based on the selection scheme. Crossover is the most important genetic operator for a GA. It is a driving force to explore the search space (Holland, 1975; Lee et al., 2001). Mutation is a genetic operator to introduce random changes into the population with a low possibility. By random changes to the population, mutation can serve a crucial role in generating the new context which is lost during the initialization and overcoming the local optimization. By mutation, more search regions can be reached and premature convergence can be avoided (Lee et al., 2001). 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Some common genetic operators in a GA are shown in Figure 2-3 (Mitchell, 1997). Single-point crossover: Two-point crossover: Initial strings Crossover mask 11101001000 — 11111000000 00001010101 - ^ ^ 11101001000 - " " ^ 00111110000 00001010101- ^ ^ Uniform crossover: Point mutation 111010Q10QQ- 00Q01010101- 11101Q01000 10011010011 Offspring 11101010101 00001001000 11001011000 00101000101 10001000100 01101011001 -+11101101000 Figure 2-3: Common operators for GA 2.2.2 Genetic Programming Genetic programming (GP) is a subfield of evolutionary computing pioneered by Cramer in 1985 and then explored in depth by Koza (Koza, 1992, 1994). GP is an extension of the conventional GA and uses similar methods as GA. A main difference between GA and GP is that GA uses list structures, often of fixed size, to represent solutions, while GP uses tree structures which can vary in size and shape (www.worldiq.com). 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The flexibility of GP can make it fit to a wide variety of problems, but it also lead to a closure problem. That is how to generate and preserve valid programs given the design knowledge. Whigham (Whigham, 1995), O’Nei (O’Nei, 2000) and Rodrigues (Rodrigues, 2002) extended GP by using grammar-guided genetic operations to ensure that only valid or bias-based solutions are generated. The terminal and function sets are two important components of genetic programming. The terminals are a set to form all end (leaf) nodes in a genetic programming tree. A terminal might be a variable, a constant or a function with no arguments. The function set consists of a set of operators used in a genetic program, e.g. add, subtract, multiply, and divide. They act as the branch points in the GP tree, linking other functions or terminals. progn defun progn 'A RG0.ARG1 ARG2 ADF0 values ADF0 ARG1 ARG2 Figure 2-4: An example of ADF Another important concept in GP is Automatically Defined Function (ADF) (Koza, 1994). ADF is a function (i.e. subroutine, procedure or module) that is dynamically evolved during a run of GP and which may be called by a calling 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. program that is simultaneously being evolved. GP with ADF can solve regularity- rich problems in a hierarchical way where computer programming involves a similar process of reuse when a subroutine is repeatedly called from a calling program (Koza, 1994). An illustration of ADF is shown in Figure 2-4, and common operators in GP are shown in Figure 2-5(A-C). sin (A) Program tree representation in GP sin sin sin (B) Crossover operation in GP Figure 2-5: Genetic programming operations 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C) Mutation operation in GP Figure 2-5: Continued Stopping criteria met? Evaluation the population The best chrom osom e is taken as solution Evaluate the offspring Create initial population Populate chrom osom es according to selection scheme Apply genetic operation to chrom osom es to generate offspring Figure 2-6: General structure for genetic algorithms GP shares the same process to evolve solutions as GA. The evolutionary algorithms maintain a population of solution candidates which are represented by suitable data structure - bit string or symbols. At the beginning, some initial 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. population is created randomly. By genetic operation, new population is bred. The population evolves through successive generations. During each generation, the chromosomes are evaluated by fitness function and selected based on the selection mechanism. With the running of the algorithms, the population’s average fitness value will improve under selection pressure and the generated best individual will approximate to the optimal or near-optimal solution (Kee et al., 2001). The general structure of evolutionary algorithms is shown in Figure 2-6. 2.2.3 Applications of Evolutionary Computation Evolutionary computational techniques have been investigated in the conceptual design (Goldberg, 1991; Parmee, 1997; Bentley, 1998; Lee et al., 2001). Cvetkovic (Cvetkovic, 2001) studied the problems in the conceptual engineering design and the possible use of adaptive search techniques. Bonnie (Bonnie, 2000) presented a genetic technique that assists decision maker to develop strategies for decision making in complex, multi-objective and multi-person negotiation situations. Topology structural design (Chapman et al., 1994; Nassef, 2002), design parameter optimization (Saxena, 2002; Lyu, 2003; Deb et al., 2004) and robust design (Hacker, 2002; Kazancioglu, 2003) are well-known applications for GAs at the conceptual design stage. Genetic algorithms are also demonstrated as a powerful tool to support design automation. Autogenetic Design Theory (ADT) (Vajna, 2002) facilities the integration of intuition, creativity and artificial intelligence into conventional design 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. process. Zhou (Zhou, 2002) proposed a general architecture for synthesis Micro Electronic Mechanical System (MEMS). Another application of design synthesis is dynamic system (Goodman et al., 2002; Fan et al., 2003) which were implemented by taking advantage of genetic programming. Bennett (Bennett, 1999) and Koza (Koza, 1999) have made a considerable progress in automating the synthesis of certain categories of purely digital circuits. These researches justify that evolutionary methods can support a range of activities from conceptual exploration, decision making to final product definition (Parmee, 1997). They exhibit a great potential in automating the conceptual design process. However, the extant successful applications of evolutionary methods are mostly in the area of parametric design and deal with the issue of optimization. The design problems in these applications are usually well defined, performance measurements easy to identify and function elaboration not needed. 2.3 Fuzzy Set Theory Fuzzy set theory is an extension o f the mathematical concept of a set. Zadeh (Zadeh, 1965) proposed a grade of membership, which allows for the gradual assessment of the membership of elements in relation to a set. The grade of membership of all its elements is defined as a fuzzy set. A grade of membership is normally a real number between 0 and 1. In contrast, an element has a deterministic condition in relation to a classic set — it either belongs or does not belong to a set. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.1 Definition A fuzzy set is usually defined as follows: A = {(x,^(x))| i e X ) (2-1) The membership function t]A (jc)quantifies the grade of membership of the elements x in relation to the set X. The grade of membership is normally restricted between 0 and 1. An element mapping to the value 0 means that the member is not included in the given set, and value 1 indicates a fully included member. Values between 0 and 1 characterize the fuzzy members (Figure 2-7) 1.2 1 0.8 u(x) 0.6 0.4 0.2 0 4 5 6 Fuzzy s e t Figure 2-7: Fuzzy set 2.3.2 Operations on fuzzy set Given fuzzy set A and B, the fuzzy set operations (Jantzan, 1998) are defined as: a) The intersection of A and B A n B = A min B (2-2) where operation min is a item-to-item minimum comparison between corresponding elements between set A and B. b) The union of A and B 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A u B = A max B (2-3) where operation max is a item-to-item maximum comparison between corresponding elements between set A and B c) The complement of A A = 1 - A (2-4) The sum of each membership value of A and its corresponding value in A’s complement set is 1. 2.3.3 Fuzzy weighted average In conceptual design evaluation, the rating criteria and their corresponding weighting values are vague or can not be precisely determined. Fuzzy weighted average (FWA) is proposed to compute the weighted sum of the criteria. The fuzzy weighted average is frequently expressed as follows: i> ,* , y = f { x l,x 2,...,xn,w „w 2,...,wn) = —n (2-5) < = i Where x l,x 2,..,xnare fuzzy numbers in fuzzy sets Al,A 2,...,An; wt,w 2,...,wnare fuzzy weights in fuzzy set Wx,W2,...,Wn. Dong and Wong (Dong, 1987) described a computational algorithm based on the a-cut representation of fuzzy sets and interval analysis. However, this algorithm is very cumbersome with the increase of information. Liou and Wang (Liou, 1992) improved the algorithms by introducing the following notations: 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and proving the following theorems: min : /( x ,, x2 ,...,x„, w,, w2 wn) = min : f L (w,, w2 wn) (2-8) m ax: /(x , ,x 2,...,xn,wt,w2,...,wn) = m ax: f v (w,, w2 wn) (2-9) where [a,, 6,] is the end point of the intervals of x ,. Based on their work, Lee and Park (Lee, 1997) proposed a more efficient fuzzy weighted average (EFWA) by reducing the number of comparisons. 2.3.4 Fuzzy preference relations A fuzzy preference relation R on a set A is a fuzzy set on the product AxA, that is characterized by a membership function (Wang, 1997; Zimmerman, 1987): tjR :A xA -+ [0,\] (2-10) Let P{a,b)e R be fuzzy preference relation between a and b, where a,b e A , then P(a,b) and Pip,a) are reciprocal: P(a,b)+P(b,a)= 1 (2-11) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fuzzy Preference Relation 1.2 1 0.8 u(x) 0.6 0.4 8 0.2 0 0 0.2 0.6 0.8 0.4 1 Figure 2-8: Example of Fuzzy Preference Relation The fuzzy preference relations P(a,b) and P(b,a) are calculated as follows (Wang, 1997; Tseng and Klein, 1989): H a,b)= V ’ D(a,0)+D(b,0) P(b,a) = E M ± £ k ! S M (2 -, 3) V ’ D(o,0)+D(6,0) where D(a,b) is the area where a dominates b (area 5 and 3 in Figure 2-8), D(b,a) is the area where b dominates a (area 1 and 2 in Figure 2-8), D(a,0) is the area of a (area 1, 4 and 5 in Figure 2-8), D(b,0) is the area of b (area 2, 3, and 4 in Figure 2-8), D(a n b,0) is the area where a and b are indifferent (area 4 in Figure 2-8). 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.5 Applications of fuzzy set theory Fuzzy set (Zadeh, 1965; Zimmerman, 1987; Klir and Yuan, 1995) is an extension of classic set theory and uses the grade of membership for all its members. It has been widely applied in the fields of design evaluation (Wang, 1997; Thurston, 1992; Vanegas, 2001) and business decision making (Tseng and Klein, 1989; Lin et al., 2005), where solutions must be derived from a substantial amount of imprecise or vague information. As development progresses, engineering design becomes more and more complex. A design task usually involves multiple objectives to be optimized, whereas these objectives are often in conflict with each other. Fuzzy logic provides a more natural way to represent the various multi-objective optimization problems. That is, when we cannot select each criterion maximally due to the conflicts, we can optimize each of them to a certain extent. Thurston and Carnahan (Thurston and Carnahan, 1992) proposed the usage of fuzzy set theory in multiple criteria engineering design evaluation. Fuzzy weighted average (FWA) was applied to find the overall desirability of the alternatives. Other similar applications can be found in bearing selection (Chen, 1996), bumper beam material selection (Vanegas, 2001), valve selection problem (Wang, 1997) and handle for closing a window (Vanegas, 2005). Fuzzy weighted average (FWA) is commonly used to compute the overall desirability of the alternatives in a design evaluation in terms of fuzzy rating criteria and the weights of their corresponding importance. Dong and Wong (Dong and 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Wang, 1987) first proposed an algorithm to compute FWA based on the a-cut representation of fuzzy sets and interval analysis. Later, Liou and Wang (Liou and Wang, 1992) suggested an improved fuzzy weighted average algorithm to simplify the computational process. Lee and Park (Lee and Park, 1997) improved the calculation process by reducing the number of comparisons and arithmetic operations to 0(n log n) . Nevertheless, operations on FWA tend to increase unnecessarily the imprecision. A new FWA (NFWA) (Vanegas and Labib, 2001, 2005) is proposed to obtain overall desirability levels less imprecise and more realistic than the conventional FWA. One of the major challenges of conceptual design is determining how to select the “best” design concepts against others for further development in the latter design stages. Researchers have proposed different ranking methods for the selection of alternatives based on the design desirability, which is represented as a fuzzy number. Wang (Wang, 1997) suggested three preference modes based on the outranking approach to discriminate the alternatives. Dong and Wong (Dong and Wong, 1987) defined a fuzzy preference relation as a degree of outranking associated with each pair of alternatives A and B. While in the research (Vanegas and Labib, 2001, 2005), an equivalent crisp number for each fuzzy number is determined so that the fuzzy ranking problem becomes a simple ordering of real numbers. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.4 Conclusion We have reviewed related research in the fields of design method, evolutionary computations and fuzzy set theory that provide a theoretical foundation and a starting point of the development of our research. Design methods are a basis upon which our research is established, and evolutionary computations provide a powerful tool by which design can be automatically explored. Fuzzy set theory provides a natural way to represent the fuzzy information at the early stage of design. In the next chapter, we will particularly investigate the current status of the conceptual design and point out the need of a new approach to extending the designer’s reach of design space. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 Conceptual Design The design problems are usually complex and ill-defined at the conceptual design stage in the mechanical design domain. The activity of the conceptual design involves design concept exploration across multiple levels of decomposition hierarchy. At each level of decomposition, designers search through possible solution spaces, determine potential solutions or partial solutions and assess their desirability with regard to the constraints and requirements. The design concepts evolve as designers gain better understanding of functional requirements and the corresponding means to achieve these functions. However, conceptual design is characterized by the uncertainty in terms of the lack of quantitative information and poorly defined specification. Therefore, conceptual design is highly “risky” because decisions concerning design directions must be made at this very early stage of design in the face of the uncertainty. A poor judgment could cause more redesign cycles to be undertaken when inconsistencies are identified at later design stages, such as embodiment design and detail design (Pahl & Beitz, 1996). A design support tool is needed for effective design concept generation and evaluation at the conceptual design stage. This chapter analyzes the state of our knowledge on conceptual design and proposes a new approach to supporting conceptual design. 3.1 Conceptual design process Conceptual design is carried out after tasks are clarified and specification established. Initial alternative solutions are generated based on designers’ experience or 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. blueprints of similar products. Then the candidate solutions are assessed against evaluation criteria which are derived from the requirements, constraints, expert knowledge or creativity of the designers. If the solutions cannot match the expected requirements, new solutions will have to be created. The activities of modifications may include introducing new ideas into previous solutions, making changes based on previous solutions or exchanging ideas among solutions. The new solutions will be usually better than the preceding ones. The iteration process of trial-and-error continues until satisfactory solutions are found at last. At the conceptual design stage, essential problems are abstracted and the task-specific constraints and details are removed. The high-level function requirements are abstract and usually cannot be fulfilled by existing technological solutions directly. Decomposition becomes a general approach to complex system designs (Pahl & Beitz, 1996; Suh, 2001; Maimon et al., 1996). Through hierarchical development, functions are elaborated and more realizable means are found. The conceptual design process shares a lot of similarities with the evolutionary process. In nature, three mechanisms are generally considered responsible for the process of evaluation, namely selection, mutation and crossover (Holland, 1975; Goldberg, 1989; Lee, 2001). Crossover produces offsprings by copying selected coding from two parents. It is similar to the designer’s work when he creates new solutions by combining various properties of products based on his knowledge, experience or the use of brain storming. Mutation causes random modifications of an individual with a low possibility. A similar phenomenon can be 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. found in the conceptual design where a designer generates “small” adjustment or brings new ideas to the original solutions by his intuition and knowledge. In the conceptual design, evaluation criteria play the same role as the selection pressure in nature to determine the design directions. The similarities have motivated a lot of research studies (Koza, 1992; Chapman et al., 1994; Maher, 1996; Parmee, 1997; Bennett, 1999; Zhou, 2002) to explore the conceptual design by evolutionary techniques. However, traditional evolutionary techniques require fixed measurement metric and the design usually evolves within a solution space. This property limits the applications of evolutionary computations at the conceptual design stage, where design problems are usually ill-defined, and exploration involves multiple levels of decompositions and mappings. Human design process distinguishes from nature evolution in two aspects. First, decomposition is an artificial technique for continuous development of the understanding of the design problems and the specifications of the functional requirements. It is a top-down hierarchal approach to design problems. But in nature, species evolve from single forms to complex organism. We can view it as a bottom- up development process. Second, design is apparently both directional and able to take enormous leaps. It is far from the random generation and slow phenotypic change in natural evolution (Bentley, 1999). The potential of evolutionary techniques for the conceptual design and the differences between natural evolution and human design raise the following research questions: 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1) Can we model the process of automated conceptual design as an evolutionary process? Should the automated conceptual design process follow human design process, or natural evolutionary process, or both? 2) How can we reconcile the “directional” human design process and the “bottom-up” natural evolution process? 3) How can we make design solutions quickly converge if evolutionary model is adopted? 3.2 Evaluation in conceptual design Conceptual design is complex particularly in the field of mechanical design. It is hard or even impossible to directly find solutions or means to realize the abstract top level functions. A practical approach to complex engineering design is to decompose higher level functions into lower level ones and then identify implementable means to fulfill the lowest level functions. The decomposition may go through several levels and design be explored across multiple solution spaces. Designers must make an appropriate decision before they proceed into next level of decomposition hierarchy. For example, in the “function-first” method (Pahl and Beitz, 1996), as mentioned in Chapter 1, a designer needs the knowledge on what function structure is appropriate at each level of decomposition and what means are most desirable combinations for the leaf-level functions. Whilst in “zigzag” design process (Suh, 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2001), a designer must gain the knowledge of how to identify and map means to the required functions at every level of function decomposition. No matter “function first” or “zigzag” approach, making decisions on “which way to decompose functions” and “what combination of means should be adopted” requires effective evaluation mechanisms. However, the available information of design at the conceptual stage is inevitably ambiguous and even inconsistent due to the qualitative nature at the idea firming-up stage. In the current practice, designers make their decisions heavily depending on their experience. No effective ways are available for them to assess design concepts until the design concepts are transformed into a more concrete form, such as embodiment design or detail design (Pahl & Beitz, 1996). The poor evaluation could lead to inconsistencies that will be identified in the later design stages and new design cycles have to be conducted to deal with them. It has long been recognized that means connectivity plays an important role in design performance (Ullman, 1997). Four structural interactions (Rajagopalan et al., 2005) have been identified in an effort to outline an assembly model method that is repeatable. Bryant (Bryant, 2005a, 2005b) presented an automated design tool that makes use of the repository (Bohm and Stone, 2004) of existing design knowledge for concept evaluation based on a function-component connectivity matrix. However, in reality, the resources may not be accessed by designers or designers may have a different vision of the design because the information is usually subjective or incomplete. Therefore, function-means connectivity is better treated as fuzzy 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. variables rather than as crisp ones. A major challenge the designers face at the conceptual design is how to effectively evaluate the generated design concepts in face of the subjective or fuzzy information. Four fundamental research questions in developing effective methods for design concept evaluation at the conceptual design stage should be addressed: 1) How can we capture design information and make the maximum use of it? 2) What evaluation criteria should be adopted to evolve design concepts? 3) How will these criteria impact the design solutions? 4) How the subjective and incomplete information affect design evaluation? 3.3 Points of departure To date, there are no effective support tools for design automation at conceptual design stage. Traditional design methods prescribe the process of conceptual design that designers should follow (Pahl & Beitz, 1996; Suh, 2001), but they do not offer a practical model for automating conceptual design. All the work on decomposition and mapping is presumed human experience based. Functional Representation (Stone et al., 1999, 2000; Hirtz et al., 1998, 2001) has been applied to function structuring by combining function representation and graph grammars. However, without considering the implementation of the functions, function structure remains difficult to evaluate in terms of correctness and effectiveness. Grammar-based approach (Schmidt et al., 1995, 1997) is optimally directed based on large number of randomly generated data. But in mechanical design domain, the candidates of 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. function and means are usually huge. The large number candidates make the approach difficult to be applied. Besides, grammar-based approach does not provide a mechanism for design evaluation and for evaluation criteria investigation. Co- Evolutionary Design (Maher, 1994; 1996) is only limited to simple system design problems, where solutions evolve with the flat and incremental approach. However, Conceptual design in mechanical engineering domain usually involves multiple- layered solution spaces across different levels of decomposition hierarchy. Without an improvement on the current co-evolutionary model, it is hard for the model to be applied directly to complex mechanical designs. In this research, we propose a framework of conceptual design modeling to support design concept generation and effective evaluation. The framework focuses on two aspects: co-evolutionary mechanism and evaluation methods. The Co- evolutionary mechanism addresses how conceptual design is modeled as a co- evolutionary process and how design concepts are generated automatically under this model. And evaluation methods focus on how design solutions are effectively assessed. In the following chapter, we will present the details of the framework of our hierarchical co-evolutionary approach to conceptual design. 3.4 Conclusion Because o f the limited understanding o f conceptual design and the lack o f quantitative information, no mature models are available for supporting conceptual design automation to date. In this chapter, we looked into the process o f conceptual 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. design and investigated two aspects which hinder computational approach to design at this stage. Then a set of research questions are raised based on the current limitation. In next chapter, we will focus on the conceptual design modeling and propose a hierarchical evolutionary framework for design concept generation and evaluation. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 A Hierarchical Co-Evolutionary Approach to Conceptual Design Our main research objective is to develop a new approach to deepen the understanding of the conceptual design process and provide a computational method for design concept generation and evaluation. With this objective in mind, a hierarchical co-evolutionary approach to conceptual design (HiCED) was developed. The basic idea underlying this approach is that the conceptual design process can be seen as a process o f co-evolution o f both function structures and means across different levels o f elaboration hierarchy. At each elaboration level, functions and their structures serve as a basis for identifying desired means, and in turn, the means can help function structuring and elaboration until satisfactory design concepts are generated. In this chapter, we present the framework of HiCED and describe how design concepts are generated and design knowledge utilized for evaluating the generated design concept. 4.1 The HiCED Framework The proposed HiCED is composed of three main components, namely function and means library, co-evolutionary mechanism, and fitness functions, as shown in Figure 4-1. The Function and Means Library serves as a knowledge base of what can be specified as required functions, and what are the possible technologies or solutions to 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. achieve these functions. It provides a basis for function elaboration and co-evolution among functions and means. The richness of the libraries determines the extent to which the design space can reach. In our research, we expect that an agent-based system can be developed to fuel the library by collecting information from designers’ design-log, design results, and relevant websites. Since the focus of our current research is on the other two components, we use a given library for the purpose of testing. Top level function Requirem ents & Constraints Collected functions and m eans Design ConceDt(s) Function & Means Library Fitness Functions • For function structuring, • For m eans selection Co-Evolution Mechanism • Zigzag design process • G ram m ar based function elaboration, • G P -G A based function structuring and means mapping) Figure 4-1: Component of HiCED The co-evolutionary mechanism provides procedures for generating the design concepts. A zigzag design process is adopted as the overall co-evolutionary design process and a grammar-based approach is applied for function elaboration. A co-evolutionary design algorithm is developed to explore the function and means space at each level of elaboration hierarchy for developing function structures and searching for the best matched means. The fitness functions are derived from general principles and the acquired information during the design process. They serve as evaluation criteria for design 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. concept generation. There are two categories of fitness functions: those for evaluating means by using functions as the evaluation focus; and those for evaluating function structures by using selected means as the evaluation focus. The fitness functions determine the direction of desired solutions. 4.2 Design Concept Conceptual design determines the fundamental concepts and constraints for later design stages (Paul and Beitz, 1996). In HiCED framework, design concepts are specified as function structure together with a set of combined means to fulfill these functions. Function structure is an expression of the designer’s intent of how the functions are arranged to satisfy the user’s requirements, while the solutions or means are physical implementation of the required functions. Neither functions nor means can provide a deterministic solution. Both function structure and means need to be identified at the conceptual design stage. A typical example to illustrate how a function structure affects design solutions can be found in a power train design example for hybrid vehicle (Jin, 2005). Figure 4-2(a) shows a function structure of series power train design, and Figure 4-2(b) is a function structure for parallel power train design. Although there exist same functions in the functional domain, different function structures lead to different products. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fuel C o n v e r t SLiilCEtoME \ ME i E x h a u s t 5 M E j C o r v e r t | E E I f lM E t o E E —* Convert Distribute EE Store I ME ME to EE EE 4 EE f Distribute EE E E j Store T E E 1 — S fL . C o n v e r t [ ME to EE f JEE EE EE to ME M E ME j F tie i l Convert r EE to ME I ME p jj m e j M E Atr Convert JCEjoM E ME (a) Series Hybrid (b) Parallel Hybrid Figure 4-2: Power train design 4.3 The Co-Evolutionary Design Process Conceptual design involves two important activities, namely, developing function structures and selecting desired means to fulfill the function structures (Pahl and Beitz, 1996; Suh, 2001). The exploration of functions and means follows a zigzag design process (Suh, 2001). The parallel exploration of functions and means can be viewed as a hierarchical co-evolutionary process. We can not make appropriate decisions if only one aspect is taken into account. The relationship between functions and means are following: 1. Functions identify what means are needed. Each means can fulfill certain functions. Only those means that satisfy the required functions are considered as candidates. 2. Means can provide certain parametric information for function structuring. In systematic design (Pahl & Beitz, 1996), function structure is developed without considering how it can be embodied. The only information available is function flows. It poses a difficulty for designers to make an 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. effective evaluation because of the lack of quantitative information. Taking into consideration of certain parametric information from means fertilizes designers with more criteria for function structuring. 3. Function structure provides certain logic information and physical relationships for means evaluation, whilst the information can not be obtained directly from the means. For example, designers can infer means connectivity relationships based on the connectivity of functions they fulfill in the corresponding function structure. If some means can not satisfy the connectivity relationships, then the means can not emerge in the solutions at the same time. ML FL PF-MS Co-Evolution PF-FsS Q-[FsM]S PF-FS Select Means (GA) Map Functions To Means (Search) Compose Function Structure (GP) Check & Select L eg en d : PF-: Partially F e a sib le ML: M e an s Library FS: Function S e t F-: F e a sb ile FL: F unction Library F sS : Function S tru ctu re S e t 0 - : O ptim al Q: R e q u ire m e n ts M S: M e an s S e t f0: O verall Function [FsM]: Function S tru c tu re a n d M e a n s P air Figure 4-3: Design Process of HiCED The design process of HiCED is shown in Figure 4-3. The exploration process of the conceptual design begins with a given overall function to be achieved and its associated requirements and constraints. Following are the steps of the process: 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. Based on the grammar rules, the overall function is elaborated into lower level sub-functions that are more specific and form a partially feasible function set {PF-FS). 2. At every level of the elaboration hierarchy, for every function in PF-FS, all its feasible means in the library are identified and form a partially feasible means set (PF-MS). Since the number of feasible means for each function can be so numerous for mechanical design problems, the PF-MS set can be very large. 3. A co-evolutionary algorithm is devised here to search for the optimal solutions from the means space (i.e., PF-MS) and the function space (i.e., PF-FS). o First, genetic programming (GP) is employed to develop a set of all partially feasible function structures (PF-FsS). o After the PF-FsS are developed, their information is used to identify feasible combinations o f the partially feasible means by genetic algorithm (GA). o In light of the information of the identified feasible combinations o f the partially feasible means, the system goes back to further evolve better PF-FsS. o The search focus switches between function space and means space until satisfactory feasible function structure and means pair sets (F- FsMS) are found. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. Select optimal function structure and means pairs whose fitness values are higher than an allowable threshold. If the selected pairs contain means that are not implemented, then go to Step 1 . 4.4 Hierarchical Exploration Hierarchical exploration is a key in our co-evolutionary approach to conceptual design support. For most “creative design” problems that require creation of new design concepts, the high-level function requirements are usually abstract and cannot be directly fulfilled by existing technological solutions. Function elaboration is needed to make the high level design problem (i.e., function) into more tractable sub problems (i.e., sub-functions). It is worth mentioning that we use the term function “elaboration” instead of function “decomposition” because when designers generate lower level functions to fulfill the higher level ones, it is not really “decomposing” in the sense of “dividing”. It is a process that involves creating a set of functions and their relations that on the one hand fulfill the overall functional requirements and on the other hand are more realizable by existing technologies or means. The function elaboration process can be viewed as a process of continuous development of the understanding of the design problem and specification of details of the requirements. The function structure hierarchy is both a tool and a result of such function elaboration process. The ultimate goal of design is not to find what functions are needed but to find solutions or means to realize the functions. It has been proposed that identifying 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. means, implementable or not, at each level of function elaboration hierarchy provides important information and knowledge to facilitate the elaboration of the function structures to the next level (Suh, 1990). In our proposed HiCED framework, the best means for each function at a given level of function structure hierarchy evolve based on the information of both function structure itself and the available means. Once selected, the means at a given level of function structure hierarchy have two roles: 1) they facilitate the evolution process of best function structures, and 2) they help further elaborating the evolved best function structures into lower level ones. Hierarchical approach is not a part of evolutionary process. However, integrating hierarchical approach into evolutionary process is of mutual benefits to both of them. The means evolution helps the determination of function elaboration, whilst the hierarchical exploration facilitates the capture of intermediate design information, which can be used for design concept evaluation at each level of elaboration hierarchy. 4.5 Function and Means Library The function and means library provides a knowledge basis for functional elaboration and design concept exploration at the conceptual design stage. Although it is not a focus in our current research, the design of the libraries still needs attention because the richness of information in the library determines the quality of the conceptual design. Besides, it must be general enough so that the libraries can be 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. easily extended to any specific mechanical design problems. In this section, we will present what necessary information is required in our library. 4.5.1 Functions In HiCED, a function is represent as a set / = { (action)(object), {input _ flow}, {output _ flow}} (4-1) where, <action> denotes the operation to be performed, <objecf> is the object to be acted upon, and the attributes <input Jlow> and <output Jlow> are flows of energy, material and signal (Pahl and Beitz, 1996). In order to make the function library generic enough, the types of action comply with the function set defined in the function basis, and types of object and flow comply with the flow set defined in the function basis (Stone et al., 1999, 2000; Hirtz et al., 2001a, 2001b). To simplify the expression in this paper, we only use action-object pairs to represent the functions. Table 4-1 shows the common basis of flows and Table 4-2 is the common actions used in functional basis (Stone et al., 1999, 2000; Hirtz et al., 2001a, 2001b). Class (Primary) Secondary Material Human, Gas, Liquid, Solid, Plasma, Mixture Signal Status, Control Energy Human, Acoustic, Biological, Chemical, Electrical, Electromagnetic, Hydraulic, Magnetic, Mechanical, Pneumatic, Radioactive/Nuclear, Thermal Table 4-1: Flow set in functional basis 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Class (Primary) Secondary Branch Separate, Distribute, Divide, Extract, Remove Channel Import, Export, Transfer, Transport, Transmit, Guide Connect Couple, Mix Control Magnitude Actuate, Regulate, Change, Stop Convert Convert Provision Store, Supply Signal Sense, Indicate, Process Support Stabilize, Secure, Position Table 4-2: Actions in functional basis 4.5.2 Means Means are available technologies that satisfy the required functions. In HiCED, a means is defined as follows: means { description {applicable function} (5-2) {high - level consistent means} {(attribute)(value)}} where, application functions are a set of functions for which the means can be a carrier, high-level consistent means set contains those means of which this means is a subtype; and attributes are a set of properties the means has. The applicable functions set can help map functions to desired means. And the high-level consistent means set can be used to determine whether the corresponding function that this means is supposed to satisfy is still decomposable. The information of means can be acquired from product descriptions published on the webs, catalogs or designer’s logs. At present, we assume an 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. experienced designer has collected the necessary knowledge about the studied case and filled it into the library. 4.5.3 Constraints A constraint indicates certain relationships between a specific means and other objects, such as another means. There are two categories of the constraints. One is a problem-specific constraint, which comes up with a specific design task after the requirements are determined. For example, the weight of the transporter can be not larger than 501b. Another type of the constraint is restrictions among means which is independent of specific designs. For example, “steering wheel” can not connect to “pedal brake” directly. Because the second constraint is closely related to the usage of means, we include the type of constraints into the means library. In HiCED, a problem-independent constraint is defined as: constr = {ml, / l , m2, / 2 , relation, ContrValue} (4-3) The expression indicates a constraint that means ml which satisfies function f \ , and means m2 which satisfies function / 2 must have a relation relation which is restricted by the value ConstrValue. The acquirement of the constraints is an activity heavily depending on designers’ experience and expertise. Besides, the constraints continuously change with the advancement of the technologies. So our function and means library is designed as an interactive one so that designers can modify its contents when conflicts are identified. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.6 Evaluation of design concepts Solutions at each level of elaboration hierarchy consist of function structures and a set of combined means to fulfill these functions. In the mechanical engineering design domain, the number of the candidate means is numerous and their relationships are much more complex. There usually exist various mappings from the functions to their corresponding means. Different mapping methods could lead to different solutions. So the performance measurement should be a function of the function structures, the means and the mapping methods. A general expression of performance evaluation at the conceptual design stage is defined as follows: where Ffxf indicates a function structure, M a set of means, and M ,,x M a mapping function from functional domain to means domain. The lack of quantitative information at the conceptual design stage poses a major difficulty in design concept evaluation. How to effectively assess the design performance based on the highly qualitative information is a challenge in our research. There are several evaluation methods in use, including experience-based evaluation, experiment-based evaluation and mathematical evaluation (Tan, 2000). Experience evaluation is a human-centered activity. Much effort on information management support (Berkelman et al, 1995; Shilit et al, 1998; Klein, 1997) and knowledge representation (Davis et al, 1993; Esterline et al, 1995) has been done for transforming the personal information into formal representation for design (4-4) 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. automation. Experiment evaluation may lead to high cost on time and resource. Mathematical evaluation employs mathematical model and analysis for design concept evaluation, such as function-component matrix (Bohm, 2003, 2004; Bryant et al., 2005) and fuzzy logic (Wang, 1987; Thurston, 1992; Vanegas, 2001). The work has proved that design tools like function-component matrix can be generated to enhance the evaluation of the design. In this section, we propose some generic expressions of performance measurement in mathematical model. The measurements include evaluation of function structure, means and their mappings. And a fuzzy evaluation model will be proposed in next chapter. 1. Evaluation of function structure PF =W,.Eh ,Ffxf (4-5) where Ffyf is a function structure to be evaluated; WF = [w, w2 A w j is a weighting value matrix for each evaluation function for the function structure; and = [e, e2 A en\ i s an evaluation function matrix. Each element in this matrix represents an evaluation function for function structuring, for example, function flow consistency, which is used to check whether each flow of a function in a function structure can be satisfied by other functions or the overall function. 2. Evaluation of means P m =WmEmM (4-6) where M is a set of means to be evaluated; W M =[w, w2 A w„] is a weighting value matrix for each evaluation function for selecting means; and 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Em = [e, e2 A enJ is an evaluation function matrix. Each element in this matrix represents an evaluation function for selecting desired means to match the required functions, for example, the independent axiom in Axiomatic Design (Suh, 2001). 3. Evaluation of the mappings P ^ = ^ MEF xM M rxM (4-7) where M F xM is a mapping function from the function structure Ff,f to the means set M; WrxM = [w, w2 A w„] is a weighting value matrix for each evaluation method to assess the mapping; and E,,yM = [e, e2 A e j 7 is an evaluation function matrix. Each element in this matrix represents an evaluation function for the mapping from the function structure to the means, for example, the evaluation based on the function-component matrix (Bryant et al., 2005). 4. Evaluation of system performance The system performance is a weighted sum of the evaluations of function structure, means and their mappings (Equation 4-8). P.sys ~ [WF WM w FxM \ p f Pm P fxM Y (4-8) In this section, some general expressions about system performance are defined. They provide a basis of our computational approach to design concept evaluation. In next chapter, I will present the details of how the evaluation methods are implemented and how they are integrated into our co-evolutionary algorithms for design concept evaluation. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.7 Conclusion In this chapter, we presented a hierarchical co-evolutionary approach to conceptual design. In this framework, the conceptual design process is viewed as a process of the co-evolution of function structures and means across different levels of elaboration hierarchy. Three basic components: function and means library, co- evolutionary mechanism and evaluation methods, are discussed. The benefits of hierarchical development are studied and some general expressions for design concept evaluation are defined. In the next chapter, we will present how the co- evolutionary algorithms are devised and how the evaluation methods are implemented for design concept generation and evaluation based on the model. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Algorithms of HiCED In the previous chapter, a hierarchical co-evolutionary framework is proposed for design concept generation and evaluation at the conceptual design stage. In our framework, means evolve in parallel with functions across multiple levels of functional elaboration hierarchy. To realize the co-evolutionary design process, we developed a grammar-based mechanism for function elaboration and a GA&GP based co-evolutionary algorithm for function structuring and means selection. Then a fuzzy preference method is described to enhance the assessment in terms of the incomplete and subjective information. Before we advance to the details of the algorithms, two issues must be addressed. One important issue for design automation is inclusive. Because of the constraints of time resource, the extent that designers can potentially reach is limited. Successful algorithms must take into account all possible alternatives before appropriate decisions are made later. The second issue of the design automation is efficiency. As mentioned before, an effective algorithm must search all potential design spaces for desired solutions. In the mechanical design domain, both required functions and candidate means are numerous and their relationships are much more complex. The complexity leads to a huge number o f candidate solutions. The time spent on searching for solutions will rocket exponentially with the increase of functions and means. Although the advancement of computer and information techniques provides much benefit, it is 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. still time consuming for design exploration at the conceptual design stage. A trade off between effectiveness and efficiency must be taken into consideration in designing the co-evolutionary algorithms. To gain design inclusiveness whilst keeping design efficiency, action-related function elaboration rules are designed with general function elaboration rules in grammar-based functional elaboration. And genetic algorithms (GA&GP) are employed to search through large design spaces for appropriate solutions. We will describe these algorithms at length in the following two subsections respectively. 5.1 Function Elaboration In nature, function is an artificial term to describe physiological activities of an organ or body part (www.answer.com). Whilst in engineering design, it is defined as the transformation of flows (Wang, 2002). Functional elaboration is a process of continuously improving the understanding of design problems. The process follows a top-down hierarchical approach rather than bottom-up development process, which is a typical process o f natural evolution. The difference leads to that if a hierarchical strategy is employed, the evolutionary computation can not be applied directly in design concept exploration where design concepts are formed across different levels of elaboration hierarchy. There is a need of a functional elaboration method to drive design evolving from more abstract high-level problem definitions to more detailed low-level design specifications. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The grammar-based approach has been proved useful in functional decomposition and design synthesis (Schmidt et al., 1995, 1997; Li et al., 2001; Sridharan, 2004). However, current research can not satisfy the requirements both in effectiveness and efficiency for an automated design tool at the conceptual design stage. The shape grammars (Schmidt et al., 1995, 1997) lead to huge number of randomly generated data, which make design evaluation more difficult or even impossible for complex design problems. And the application of the formalism of a graph grammar into function structuring (Sridharan, 2004) relies on designers’ experience to create the grammar rules. It also could result in a large number of candidate solutions without providing effective measurement methods. In our model, grammar based approach is employed to determine how functions are elaborated into sub-functions. O f these grammar rules, we distinguish between general elaboration rules that are generally applicable and similar to shape grammar rules (Schmidt et al., 1995, 1997; Li et al., 2001), and action-based elaboration rules that can be applied only in relatively narrow situations and more like the rules found in production/expert systems (Jin et al., 1992). In this subsection, we present the definitions of grammar rules and discuss how these rules are applied to functional elaboration. 5.1.1 General function elaboration rule set (GFE) The general function elaboration rule set is used under such a situation that no design experience can be referred. Unlike the shape grammars (Schmidt et al., 1995, 1997) 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which randomly generate candidate solutions, our general rules only search for the possible sub function set on the basis of function flows to satisfy the higher level functions. The constrained search space will contribute to design efficiency. The general function elaboration rule set is composed of four sub rules for function elaboration: Rule 1: Function expansion rule (FER) Fm j is an unsatisfactory function elaboration set for function j\ at level i + 1 . j is the number o f sub functions in function set Fi + X ■ . A major issue in applying this rule to function elaboration is how to choose elaboration set F,+ij+i better satisfy the top level function /,. In HiCED, a greedy search algorithm is introduced to select the most compatible function f+ij+ 1 . First, we examine how many unmapped input flows— i.e., the input flows that cannot find any feasible providing function—and unmapped output flows— i.e., the output flows that cannot find any feasible receiving function—are there in the function set Fi+ ij, and then choose a function f+ij+i that can most effectively reduce the number of unmapped input and output flows. The details of the greedy search algorithm can be found in Appendix A. the next function f+ij+i from the function library and make the new function 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rule 2: Function reduction rule (FRR) O h ■ F i + i j O O / i + l j + l O F i + l j This rule removes function f+ij+i from the original elaboration set. One issue with the greedy algorithm mentioned above is that after a function f+ij+i is added into the elaboration set, the function may introduce new unmapped input flows and/or output flows that cannot be satisfied by any other functions in the function library. In this case, we need to remove the function from the function elaboration set and look for other functions for expansion. Rule 3: Refinement-based expansion rule (RER) Following Rule 1, only those functions that can reduce the number of unmapped input flows and/or output flows of the existing function structure have the chance to be selected. Any other functions, such as channel function (Stone et al., 2000) which has same input and output flows, are excluded even though they may contribute to the improvement of the function structures in other ways. Rule 3 is employed to enhance function elaboration by adding or inserting a “refinement” function into the elaboration set generated by Rule 1. O — O 0 - 0 - 0 f+1 /i+l,k /t+lj /i+l.l /i+l,k 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rule 4: Elaboration termination rule (ETR) F i+ l.m * 0 ^ + U For any function /, at i,h layer, if each input flow of a function f Mk in the functional elaboration set Fj+| m comes from either an output of another function in 1 m or f i j , and each output flow of f j+ lk goes to either an input of another function in Fl+ ] mor f Lj , then the function elaboration for terminates. 5.1.2 Action-related function elaboration rule set In a certain design circumstance, function elaboration can be determined upon designer’s experience or design knowledge. It is trivial to apply the general rules and makes design inefficient. Action related function elaboration set contains a set of grammar rules which are applicable in the most of design cases. They usually are developed from design experience or design knowledge and could change with the advancement of techniques or design pattern. The specific rules facilitate the elaboration and make it more efficient. Three action related function elaboration rule sets are summarized here: Action-based function elaboration rule set (AFE) In the engineering domain, it is often the case that a higher-level action A C T h can be divided into a set of low-level actions [ACTn ,ACTn ,...,ACTyn] in the 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sense that the completion of the low-level action set implies the completion of the high level action (Jin et al., 1992). Based on the semantic expansion, an action-based elaboration is defined as: if f = (ACTh)(object) and ACTh ={ACTn ,ACTi2,...,ACTln} - elaboration (r r ^ i (5-1) => / ------------------------- H / 1. / 2.- f n \ where f = {ACTp )(object), (l < / < n) Action based requirement elaboration rule set (ARE) In some cases, the execution of a certain action either depends on or will lead to other actions. The original function set can be elaborated by “including” new actions while keeping the original action. The rule set contains a set of sub rules related to some specific actions. A specific ARE rule is defined as follows: if / , = (ACT\ ){object) and ACTi => {.ACTX ; ACT2 ,...ACT„ } (5-2) =* / , e X t" > { / „ / 2 where f ] = {ACT'^(object) For example, action import will lead to two possible actions: change or transmit. So the action inclusion results of function (import)(energy) are two possible function sets {(import^energy^change^energy}} or \fjmport}(energyj, (transmiTjlenergy^. Predefined function elaboration rule set (PFE) Sometimes it is heuristically known that certain function can be elaborated into a number of other functions. The result of PFE is that the original function is replaced by the new function set. Its definition is highly dependent on designer’s 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. experience. For example the function (move)(object) can be replaced with (,generate)(ME),(guide)(ME), and (stop)(ME). Here ME refers to mechanical energy. The PFE rule is defined as follows: if f \ ={ACT\)(object) and =>{/2, / 3,...} (5-3) . j. elaboration < r r ) = > f \ ----------------------> \ h . h > • •• ) O f these grammar rules, general function elaboration rules lead to the variety and novelty by combining greedy search and refinement-based expansion rule, but it requires much computation resource and makes search slow. On the other hand, action-related function elaboration rules make the elaboration more efficient with the help of designer’s experience. The drawback is that the search is conducted within a limited space and innovative opportunities may be lost. By integrating action-related function elaboration rules into the general grammar rules, our grammar based approach can better achieve the effectiveness and efficiency. On the other hand, it raises another issue: how to balance the uses of general rules and action-related rules for both novelty and efficiency. It will be our future research topic. 5.2 Co-Evolutionary design From a computational perspective, function elaboration and function-means mapping may lead to a large number of possible solutions. While human designers can find their satisfactory function structures and design concepts based on their experience and expertise, automating this process requires a powerful search mechanism that can make the maximum use of the available design information. In FtiCED, we 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. developed a co-evolutionary algorithm to explore both the function space and the means space. The practicability of evolutionary techniques (Goldberg, 1989; Koza, 1992; Maher, 1996; Bentley, 1999) have proved that they are particularly good at exploring optimal solutions in a time that is orders of magnitude less than a full solutions taken by using other algorithms such as exhaust search (Lee et al., 2001a). The advantage of the evolutionary computation in design exploration and the similarity between design and natural evolution motivate us to develop a co- evolutionary algorithm to support the automation of the conceptual design. In HiCED, desired means are selected by using genetic algorithm (GA) (Goldberg, 1989) in parallel with function structuring by using genetic programming (GP) (Koza, 1992). The details of co-evolutionary algorithms are presented in the following two sections. 5.2.1 Genetic modeling of function structures As shown in Figure 4-3, after a partially feasible function set is generated through function elaboration, composing function structures by GP is the next challenging task. There has been research that uses GP to automate configurations of electrical circuits (Koza, 1999; Bennett, 1999) or micro electronic mechanical system (Fan et al., 2003; Goodman et al., 2002) that are similar to function structure (all of them can be seen as a cyclic graph). However, circuit configuration problems are relatively simple in the sense that they have only a few building blocks, such as resistors, capacitors and inductors, and one type of (wire) connection to deal with. Function 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. structures of mechanical systems involve a lot more function blocks and they can be connected by different functional flows, namely energy, material and signal (Paul & Beitz, 1996). In addition, after an electrical circuit is generated by GP, all its parameters are determined, so designers can easily measure its performance. But in functional design, the information is still highly qualitative. Fitness functions are much difficult to identify. The difficulties in function structuring exist in two fields: modeling and evaluation. We will introduce how a set of fitness functions are defined for function structure evaluation in the next sub-section. And the focus in this part is on how function set and terminal set are designed to form a valid function structure. One of main challenges in constructing a valid function structure by GP is how to create a transformation mechanism that can map a genetic programming tree into a graph-like function structure. As mentioned above, function structures can be complex. A same function can be employed multiple times in a function structure, and each function can connect to other functions by any number of three types of flows. In order to convert a tree-like GP structure to a graph-like function structure, we introduced a concept called proxy node in our GP model. For example, function f A can have one or more proxies, denoted as f A\ in a GP tree. The proxy nodes will be considered as the same of their original nodes when they are converted back to functional structures. To model the topology of function structures, we introduce two relationships, namely Connect, or Parallel. Connect indicates that a function is linked to another 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by a function flow t; while Parallel stands for a situation that two functions have no direct connections. The topological relationships in a function structure are shown in Figure 5-1. /. /. fi h (A) Connect (B) Parallel Figure 5-1: Topological relationships in function structure In the HiCED GP model, four genetic programming functions are defined and they are three Connect functions and one Parallel function. Connect-E(/A , / B , e): connect / A to /B by energy flows e. Connect-M(/A,/B, m)\ connect/A to /B by material flows m. C o n n ect-I^ ,^ , s): connect f A to/b by information flows s. Parallel (/A , / B ) :/A and/ b are not connected directly. CfParaUdf^) f i JB fc / d i M J E (fC o n n ect-E f) ( f P a r a l l e f f ) (fconnectff) Connect- (A) Function structure (B) Genetic Programming Tree Form P arallel(Parallel(C onnect-E(fA ,fc ), Connect-M (fA Connect-E(fH , f D')) (C) Genetic Programming Function Form Figure 5-2: A chromosome model of function structure 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5-2 illustrates an example of a function structure (Figure 5-2(a)) and its representations in the forms of GP tree (Figure 5-2(b)) and the GP function (Figure 5-2(c)). 5.2.2 Constrained genetic programming operations A major issue about genetic programming is closure problem (Rodrigues, 2002), that is, the generation and preservation of valid programs. A functional flow which connects two functions together must be compatible with one of the outputs of the previous function and one of the inputs of the next one. It is a very high possibility that impractical function structures are produced if the genetic operations are randomly selected. For example, the crossover operation in Figure 5-3 will lead to two infeasible function structures because of the incompatible functional flows. As a result of randomly selected genetic operations, much computation cost is spent on searching for feasible solutions rather than the optimal ones. C Connect-M ) ------ Connect-E crossover point crossover point Figure 5-3: An invalid genetic operation A lot of attempts (Whigham, 1995; O’Nei, 2000; Rodrigues, 2002) have been made to enhance GP by applying type restriction languages, for example, context free grammar (Rodrigues, 2002) in genetic operators and ensure that only valid programs are generated. With the use of constrained syntactic structure (Koza, 1994, 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1999), automatically defined functions (ADF) can be implemented in automating the synthesis of certain categories of digital circuits. To maintain the consistency of the genetic tree and reduce the cost due to the possible generation of impractical function structures by genetic programming, the constrained genetic operations “initialization”, “crossover” and “mutation” are introduced in the HiCED GP model. Constrained Initialization: program =>< exp > < exp >=> parallel(< exp >, < exp >) | connect(< var >, < var >, < flow >) < flow >=> material \ engery \ signal < var >=> function | proxy(function) | empty (5-4) Given the top level function TopFunction connect{< var 1 >, < var 2 >, < flow >) => varl = fu nction an d var 2 = function an d < fla w > e output(\& r\) an d < flo w > e in pu t(yar2) • varl = fu nction an d var 2 = em pty an d < fla w > e output(v arl) and < flo w > e outputifopF unction) v arl = em pty an d v a r2 = function an d < flo w > e in pu t(\ax2) an d < flo w > e input(TopFunction) In GP trees, internal nodes are genetic programming functions {Parallel and Connect), and terminals (var in Equation 5-4) are the functions of a given function elaboration set, their proxies or an empty node. If a function is connected to an empty node by the flow t, it means that the function has an interaction with the outside environment by the flow t rather than with the internal functions in the function structure. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Different from the random generation, the constrained initialization process builds up a genetic tree by applying a set of rules to the leaf nodes until no more rules is possible to be applied. Following the rules, function Parallel only takes genetic functions {Parallel or Connect) as its argument, whilst function Connect only allows two terminals as its arguments. Considering the factors about function flows which are used to connect different functions, four constraints are imposed on function Connect: 1) Both of its two arguments can not be empty at the same time; 2) If both its arguments are functions from the given function elaboration set, then the one of the output flows of the first function and the one of the input flows of the second function must satisfy the functional flow constrained by the Connect function. 3) If its first argument is a function, whilst the second is an empty node, which means the function interacts with the outside environment by a functional flow defined by the Connect function. Then one of the output flows of the first function must be compatible with one of the output flows of the top-level function. 4) If the first argument is an empty node, whilst the second is a function, then one of the input flow of the second function must be consistent with not only one of the input flow of its top level function, but also the one constrained by the Connect function. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Constrained Crossover. crossover = > crossover(< exp >, < exp >) | crossover(< var >, < var >) If the crossover point selected in the first genetic programming tree is on a function, then the crossover point in the second tree must be on a function. The same rule is applied to terminals, but the situations are more complex. crossover point 1 ^C onnect-E ^ crossover point 4 / a / b fc / d Figure 5-4: Illustrating constrained crossover operation 1) The parent function of the crossover point in the first genetic programming tree and the one in the second tree must have connection type (energy, material or signal). 2) If the crossover point in the first genetic programming tree is the point 1 (Figure 5-4) and the point 3 in the second tree; or the point 2 in the first tree and the point 4 in the second tree, then the crossover can be operated safely. 3) But if the crossover point is the point 2 in the first tree, whist the point 3 in the second tree, the operation can be performed only if one of the output flow of the function is consistent with the functional flows defined by its parent function (Connect-E in Figure 5-4) and one of the input flows of the function fc is consistent with the flow defined by its parent function. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4) If the crossover point is the point 1 in the first tree, whist the point 4 in the second tree, the operation can be performed only if one of the output functional flow of the function / D is consistent with the functional flows defined by its parent function and one of the input functional flows of the function / a is consistent with the flow defined by its parent function. 5) After the crossover, the newly generated trees can not contain such a Connect function which has both empty nodes as its arguments. Constrained Mutation: mutation => mutate{< exp >, < exp >) | crossover(< var >, < var >) If the mutation point selected in the genetic programming tree is on a function, then the generated mutant must be a function. The same rule is applied to terminals, but more constraints are imposed: Connect-E mutation poinl mutant Figure 5-5: Illustrating constrained mutation operation 1) If the mutation point in the genetic programming tree is the point 1 (Figure 5-5), then the one of the output functional flows of the mutant must be compatible with the flow defined by the parent genetic function of the mutation point (Connect-E in Figure 5-5). 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2) If the mutation point in the genetic programming tree is the point 2 (Figure 5-5), then one of the input flows of the mutant must be consistent the functional flow constrained by its parent Connect function. 3) After the mutation operation, the newly generated trees can not contain such a Connect function which has both empty nodes as its arguments. 5.2.3 Fitness functions for function structuring How can we evaluate the fitness of a given function structure is a major challenge in our research because there is little quantitative information available. Our long term goal is to develop analytical methods to simulate the functional performance of function structures (Wang and Jin, 2002; Wang 2002). In the current research, we introduce a set of measurements that can be used to assess the validity and overall desirability of a given function structure based on the topology of the structure and the information of the means selected to fulfill the functions. Using means information to evaluate the function structure is a part of the co-evolution in HiCED. Following (Pahl & Beitz, 1996), a valid function structure must maintain three basic properties: 1) each input (or output) flow of a function in a function structure should come from (or goes to) one or more other functions, or outside of the system; 2) the function structure should have same input and output flows as its top function; and 3) a lower level function structure should be consistency with its higher level ones. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Mapping means to the functions in a function structure provides a source of information to assess the desirability of a given function structure. As will be described in the next two subsections, designers can attach general desirability information the means that can be used to help evaluate functions structures. GP can cause function structure dimension to expand randomly because of randomly selected genetic operations. Designers, based on their experience, can determine the desirable range of the dimensions in terms of both number of functions and numbers and types of function flows. Let f o p be the top-level function, Fs = <F, R> the function structure to be evaluated, where F = {/}, f 2 , ...} is a function set which forms the function structure and R = {r/t r2 , ...} is a relation set that includes all relations among functions. To evaluate Fs, We have identified seven main fitness function components, namely, function flows consistency (FC), global consistency (GC), means-based consistency (A/C), structural dimension (SD), hierarchical consistency (HC), function variety (FV) and grammar rule based consistency (RC) for function structure evaluation. Function flow consistency (FC) For a function in Fs, all its input flows V /7 ,.„ e input(f) should come either from one or more other functions in the structure or from the outside of the system being designed. If the latter is the case, then the input flows of the function should be consistent with those of the top-level function. That is, V/7„, e input(f) and / e F , 3 f'e F or f u tp 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. so that f l m e o utput(f) or f l m e input{flo p ) The similar relations apply to output flows y/7„„, e o u p u t( f ), too. That is, V A u , G ouput(f) and / e F , 3 /'e F or f M p so that f l o u , e in p u t(f) or Jlo u l e ouput(fM p) The fitness function for function flow consistency (FC) is defined as //,( = unmatch _ flo w (f ), (5-5) / where unmatch _ flow{ f i) is the number of unmatched flows of j ] . Global consistency (GC) GC is defined to check whether the generated function structure Fs has the same input and output function flows as the top-level function. If the consistency is maintained, then we have Vfl,l > p -I n ^ input(flo r ), e input (Fs), so that f l lo p _ m = ; and -o u t e input ( f lo p ), 3 / 7 £ output (Fs), so that f l lo p -o u l — f l F s — o u t * Therefore, we introduce the following GC fitness function: .fc,, = -unmatch _ flow{f„,n ,F r) (5-6) where unmatch _ flow [flop, Fs) returns the number of unmatched flows. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Means-based consistency (MC) After a combination of matching means Me is selected, the selected means can be used to evaluate the function structure Fs. This fitness function reflects how function structures co-evolve as the matching means evolve. For any two functions f j and / . in Fs, their corresponding means m, and mj in Me must satisfy the relation defined by f and f j , such as Connect-E or Connect-M. The fitness function for MC is defined as: M c . f M C = -^ u n m a tc h _means[mj ,m t ,r0) (5-7) Structure dimension (SD) One feature of GP is that the dimension of the GP tree expands randomly if no dimensional control mechanism is applied. A large dimension of the GP tree usually involves a large number of duplications of functions and makes the function structure ineffective. In HiCED, we introduce the following structure dimension control fitness function. where rjj = relation[fj, / . ) e /? and f , f j e F , and unmatch means\ 1 if mf and mf are incompatible with rt j 0 otherwise f N lolul> K d x N t otherwise distinct distinct (5-8) 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Nto ta i is the number of functions in Fs\ and N distinct is the number of distinct functions in the partially feasible function set PF-FS (see Figure 4.3), and K j is a ratio, by which some level of duplications is allowed. In our current case study, Kd = 1. The usage of the exponential function in SD can impose a stricter restriction on structural dimension so that function structures with reasonable dimension can converge quickly. Function variety (FV) Although each function has an equal chance to be selected into a function structure by GP, the application of SD will exclude functions from the structure as many as possible. FV is introduced to encourage each distinct function in PF-FS to be used in the function structure. Let N distm ct-F s be the number of distinct functions in the function structure Fs and N distinct the number of distinct functions in the partially feasible function set PF- FS, then the fitness function of FV is defined as: fI V = ~(N d is tin c t ~ ^J,tincl-fs ) (5-9) Hierarchical consistency (HC) The evolutionary mechanism in GP breeds the new individual randomly. There is a high possibility that lower level function structure is not consistent with their higher level ones. H C is used to maintain the consistency between function structures at different levels of elaboration hierarchy. For function structure Fs at a given of elaboration hierarchy, we have Vf,fj and 3 relation^, f ) = rtJ . If f and j) are 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. elaborated into lower level function sets F, and Fs in a lower-level function structure Fs', f ' —d a h o r a le >F and f J —e la h H r a le ^ . p . ; respectively, then in F s\ the function subsets Fj and F} must also maintain the relation(F„ Fj) = rtJ . We define the HC fitness function as follows: Fs fHC = unmatch{relation{Fx , F2)? relation(f{ , f 2)) (5-10) cj where /, e ,a h o m ,e > Ft and f 2 e la h o r a ,e >F2 and ^ matches F [0 otherwise Grammar rule based consistency (RC) The applications of certain function elaboration grammar rules, such as ARE, may introduce certain relationships between the functions at the lower level of elaboration. RC is defined to maintain such relationships of the relevant functions in a generated function structure. For in Fs, let rl } = relation(f„,fj) denotes the flow relations between f and fj, and rrl } = rule relationif,, ,fi) denotes the relations specified by the grammar rule that led to the inclusion off and f . rrt J = 0 if no such was applied. We define the RC fitness function as follows: Fs fnc = unmatch(relation{fx, f 2), rule _ relation(f, f 2)) (5-11) •J / \ H if rx^ 0 and r2^ 0 and rx ^ r2 where unmatcmr., r ,) = - i [0 otherwise 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Summarizing the above, we introduce the following overall fitness function for function structure evaluation as the weighted sum of the above fitness function components: fis = wr ' ‘fix "* ■ wac 'fee ' fM( ' + wsn 'fsn ^ j ^ + wfv ' f i v + wne ' fm ■ + wr c ' I r c We will discuss at length how these fitness functions impact design concept generation, and how these weighting values should be distributed in a case study later. 5.2.4 Genetic modeling of means At a given level of function elaboration hierarchy, function structures and their corresponding combinations of means co-evolve over time. As described in Section 4.3, choosing a set of means for a given functional structure in HiCED is carried out in two steps. One is to search for all feasible means for each of the functions and the other is to select the best combinations of the means from the sets of the feasible means. Identification of feasible means can be done based on pre-coded knowledge in the function and means library. To select the best combinations, a GA based algorithm is employed. In this algorithm, the means are encoded into strings o f binary bits, as shown in Figure 5-6(A). The length of bits for each means is dynamically determined by the total number of the means available for a specific function. For a given function elaboration set {/}, with the corresponding feasible means sets {M/, ..., M„} 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. illustrated in Figure 5-6(A), the means combinations are modeled as chromosomes (Goldberg, 1989) shown in Figure 5-6(B). M eans se t M i fo r function f Means M t Coding m l 001 ' m5 010 GA chrom osom e m odel M eans set M„ f o r function f. Means M„ Coding m2 0001 - m4 0010 Figure 5-6: A chromosome model of means (A) crossover operator crossover point crossovenpoint ijp i M eans se t M / f o r function f Means Mi Coding m l 000 m5 001 m6 010 mlO 011 0 0 0 ^ 0 0 0 0 | nn\ | m: | ... | m i. [ I 0 I 0 I 0 I' o i 111 mutation nnm t (B) mutation operator 0 "^io i o i 0 0 0 l.o I 1 I 0 0 0 Figure 5-7: Genetic operators for m eans evolution The evolution of means in GA is determined by a set of genetic operations, namely crossover and mutation (Goldberg, 1989). In HiCED, we used a single-point 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. crossover operator (Michell, 1997) to recombine the selected individuals of current population. The genetic operators are illustrated in Figure 5-7. 5.2.5 Fitness functions for means selection Developing effective fitness functions at the conceptual design stage is difficult due to the limited availability of quantitative information. Our co-evolutionary approach is designed to utilize the information not only from the means space but also from the function space. Function structures provide the information of required logical and physical relationships, represented as flows of energy, material and information, between the corresponding function-carrying means. Besides, although much parametric or geometric information is not obtainable, certain simple performance measures, such as weight, cost, and desirability can be associated to various means. Given a function structure = {F,R}, where F - is a function set and R - {r, } a relation set, we have a feasible combination ofpartially feasible means Me = To evaluate the fitness of Me, we introduce the following fitness functions. Means connection rule (MC) For any two means m, and m; (m, e Me, mj e Me) and their corresponding functions f and f ( f e F s ,f e Fs), and flow relation rt J = relation {f,,f}, we have the following fitness function: M e . . I m c = unmatch _means[ml , m j (5-13) i.j 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where unmatch means 0 if m, and m) are compatible with rt J 1 if mi and mj are not compatible with rt J This means connection rule (MC) fitness function measures the compatibility of means connection based on the function structure. This is how the function structure information is utilized for means evaluation during the co-evolution process. Means performance (MP) The fitness function is used to measure the difference between the performance of a designed system and the stated objectives. It can be applied only when such simple performance measures are available. Let the design requirements b e g = { < 7, q2,— \, and the simple performance measuresP = {pt,p 2 . We define the means performance (MP) fitness function as: Means preference (MPR) One can attach general preferences to certain means. The preference can be related to designers’ confidence (Ullman, 2000) or frequency of the use. When preference information is available, we have the following means preference (MPR) fitness function: (5-14) q, and p, are the ith requirement and the ith measured performance respectively, and w is the weighting value for the ith requirement of the deigned system. fm > R = - £ 0 - a ) (5-15) 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where p : indicates the preference of the designer to the use of the means mj in the design. The overall fitness function for evaluating a means combination Me is a weighted sum of individual fitness functions: / = W MC ■ / MC + W M ,> • f u p + W M,‘ R • fu P R ( 5 ‘ 1 6 ) 5.3 Fuzzy evaluation on means connectivity In the genetic model for means selection, means performance (MC) is defined to measure how the selected means to affect the system performance. However, the fitness function only takes effect under specific design conditions where the design performance can be determined by certain individual design properties, such as weight, cost, etc. Most evaluations about means connections are only experience- based or experiment-based. Besides, as we emphasize before, conceptual design is characterized by the lack of quantitative information. The vague or subjective information poses more difficulties for designers to make appropriate decisions. The situation raises two research questions: 1) How can we evaluate the overall system performance on the basis of available means? 2) How can we deal with the subjective or incomplete knowledge at the conceptual design stage? In this section, we present a fuzzy preference model for means evaluation in terms of the subjective or imprecise information available to the designers. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The connection relations among means have an important effect on the quality of the design. Research studies (Ullman, 1997; Rajagopalan et al, 2005) have addressed the importance of means connection. Four structural interactions (Rajagopalan et al., 2005) have been identified in an effort to outline an assembly model method that is repeatable. Bryant (Bryant, 2005a, 2005b) proposed an automated design tool that makes use of the repository of existing design knowledge for concept evaluation based on a function-component connectivity matrix. In his research, means connectivity is explicitly determined by the repository (Bohm and Stone, 2004). However, in reality, the resources may not be accessed by designers or designers may have a different understanding because the information is usually subjective or incomplete. Therefore, function-means connectivity is better treated as fuzzy variables rather than as crisp ones. The HiCED model adopts that means evolve in parallel with functions at the early stage of design (Li and Jin, 2005). The results of function evolution are function structures. The relationships among functions can provide logical and/or physical information for means connections. The connection relations of means are usually vague at the conceptual design stage. For example in the function structure shown in Figure 5-8, we have the means “hand brake” to implement the function <stop> <ME>. If a designer selects “human” for the function <generate><ME>, then the connection between the two means is acceptable. But if he/she decides to use a motor as the energy source, the connection becomes unclear because the designer cannot determine how powerful the motor is at this stage. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E <generate> m F.^ <stop> TE <ME> <ME> Figure 5-8: An example of function structure The vagueness in means connectivity is usually described with some linguistic terms, such as “impossible”, “possible”, “very possible,” etc. Those linguistic terms can be represented and manipulated with fuzzy set theory. In this section, we propose an approach to evaluating designs based on the fuzzy connectivity of means at the conceptual stage of design. 5.3.1 Weight of means connectivity In HiCED, a functional connection between two functions is introduced in one of three ways: inherited from the higher-level function structures, generated by executing functional grammar rules, or generated randomly through genetic operations. In a valid function structure, the lower level function structures must be consistent with the higher level ones. For example in the following function structure (Figure 5-9), the connection from /, to f 2 in the lower-level function structure (b) is directly inherited from its higher-level function structure (a), meaning that the connection has already been established at the higher level (i.e., level (a)) and thus needs to be maintained at the current level (i.e., level (b)). Therefore, it is more important for the means that implement function /j and f 2 to maintain the connectivity defined by /, and f 2. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) Higher-level function (b) Lower-level function Figure 5-9: Connection relation derived from function decomposition Second, when the grammar rules are applied in elaborating higher level functions, specific functional flow relations may be introduced. As we introduced before, three sets of function grammar rules, namely action-based function elaboration rules, action specific expansion rules and requirement-based function elaboration rules (Li and Yan, 2005) have been identified to facilitate function elaboration in HiCED. Since these rules encode both formal and experiential knowledge, it is important for the implementing means to respect the corresponding function relations. For example, an action specific expansion rule is presented below. The means to be selected should satisfy the connection prescribed by the rule consequently. (generate)(ME}—A S I ' >((supply)(E),(generate)(ME)j . Compared to the connections produced by random generation by genetic operations, the connections generated from the higher-level function structures and the function grammar rules have more “knowledge” embedded. It is more important to make sure that the corresponding means connections respect these functional connections. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3.2 Preference model for design evaluation In terms of the fuzzy means connectivity and fuzzy weight of connections at the conceptual design stage, we propose a fuzzy preference model for design evaluation based on the means connectivity, which is inferred from the generated function structure from GP model. From the topological point of view, there are two basic topological relationships in a function structure: series and parallel. Series is a configuration where two or more functions are connected one by one via functional flows, whereas parallel indicates that two or more functions (or sub function structures) have no direct interactions via functional flows (Figure 5-10). (a) Serial (b) Parallel Figure 5-10: Topological relationships in function structure The relationships reflected in the physical domain are the means connections (Figure 5-11). In Figure 5-11, jc is a fuzzy number for means connectivity and w is the corresponding importance of the connection. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X|/W,| X2/W2I O T, 1 « 2 | W3 X4/w . (a) Serial (b) Parallel Figure 5-11: Means connectivity based on function structure A design evaluation based on the function-means connectivity is conducted as follows. Step 1: Evaluate overall connectivity Designers want to select solutions with the most compatible connection. With this objective in mind, we apply fuzzy weighted average (FWA) to calculate the connectivity level for series connection and fuzzy intersection operation for parallel relationship of means. For example, the total connectivity for Figure 5-11(a) is c - x,w, +x2w2 wx + w2 (5-17) and the connectivity level for Figure 5-11(b) is: c = x3 o jc4 (5-18) The underlying idea of this approach is to evaluate how successful the design could be based on the means connectivity. For the series connection of means, the FWA represents the average degree of success based on the designers’ knowledge of the design. But for the parallel structure, design success is restricted by the poorest branches. 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i Step 2: Determine fuzzy preference relation o f alternatives In the area of mechanical engineering, the desired functions can be fulfilled by numerous means. Thus, the number of alternative solutions generated from HiCED is large. After the first step, each alternative has a fuzzy number to represent its degree of success based on the connectivity, for example s,, s2,...,sn. Then we use the Equation 2-12 to determine their preference p {s,,Sj) Step 3: Select “best” alternatives One disadvantage of the fuzzy weighted average is that it unnecessarily increases the level of imprecision (Vanegas and Labib, 2001). So it is not appropriate to rate the final solutions directly based on fuzzy preference relations calculated from step 2. Two fuzzy preference models are used to discriminate the alternatives into preference and indifference sets. Assume a and b are connectivity levels, and the two models are: 1) Strict preference S \/a,beA, and a> 0 a S b o p{a,b)> a (5-19) 2) Indifference preference I \Ja,b&A, and a> 0 a I b < = > p(a,b)< a (5-20) The threshold a is used to discriminate two alternatives between strict preference and indifference. If the difference between a and b is greater than a , then it is convincing that candidate a is better than b. That is, designers have sufficient 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. evidence to believe a, and b can be removed. But if the difference between a and b does not exceed a, then it is considered that solution a and b are of the same level of importance in terms of the connectivity. If there are no other evaluation criteria available for further ranking, both solution a and b should be kept until enough information is obtained. The selection of threshold a expresses the degree of imprecision that a designer believes that solution a is preferable to b. If he (or she) is confident in the connectivity among the means he (or she) has selected, a lesser value is set. In our model, only the condition that alternative a is better than b is taken into account, that is, p(a,b)> 0.5. 5.4 Interaction with designers Design is a human-centered activity. However, because of resource constraints and mental limitations, it is impractical or impossible for designers to go through all reachable solutions. In order to improve the quality of a design, a design support tool is needed to facilitate designers in design concept generation and evaluation. In our research, we proposed a computational approach to conceptual design, in which grammar rules guide function elaboration, and a GA&GP co-evolutionary algorithm evolves design solutions at each level of elaboration hierarchy. Although this approach implements the automation of conceptual design to some extent, human designers are indispensable in determining design directions and 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. design quality. The influences of human designers on a design manifest in the following aspects: 1) Function and means library: the design quality is dependent on the richness of information in the library. What knowledge is necessary for design concept generation and evaluation is solely decided by human designers’ experience and expertise. The pre-defmed knowledge is needed to fuel the library before a design can be conducted by HiCED. 2) Action-related grammar rules: action-related rules are a set of elaboration patterns which designers often adopt for certain functions. The definitions of the rules heavily depend on designers’ experience or expertise. Besides, the rules change with the development of technologies. Although general function elaboration rules can generate sub functions to satisfy the required functions, the action-related functions provided by designers can facilitate the design. 3) Design requirements and constraints: the requirements and constraints are identified by designers after planning and clarifying the task (Pahl & Beitz, 1996). Then they are transformed into such a format that our HiCED can recognize. The activities of identification and transformation are human-based work, and an interface is required to accept the transformed data from designers. 4) Fuzzy connectivity: In HiCED, a fuzzy preference model based on function-means connectivity for means evaluation is proposed. The fuzzy 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. relationships between means represent the grade of a designer’s uncertainty on his decision. Different designers, because of his knowledge and the degree of the understanding of the design, have different confidence in his determination. So this information is contingent with respect to different designers or scenarios, and cannot be pre-coded into our knowledge library. The HiCED fuzzy preference evaluation algorithm depends on designers’ input for decision makings. 5) The final decisions on solution selection: The HiCED framework is proposed to support conceptual design by generating multiple design concepts with respect to fitness functions. The final decision on which concept to choose is still made by designers. There can be situations where none of the HiCED generated design concepts is acceptable. One possible reason for these situations can be due to the designers’ limited understanding of the design problem, which led to the ill-defined requirements, constraints, and the uncertainty of means connectivity. In these cases, using or running HiCED model can be a learning process for the designers to further understand the design problems. When the designers have better knowledge of the design, new updated information can be input into the model to improve the design. The interaction between designers and the system benefit each other mutually. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.5 Conclusion This chapter describes the algorithms to realize the HiCED model. With the action- related rules, our general functional grammar rules can more effectively elaborate functions from more abstract solution spaces to more specific solutions spaces. GA&GP based co-evolutionary algorithms can search efficiently through the large solution spaces and produce alternative solutions. Fitness functions are identified to evaluate the candidates based on design knowledge and experience. In order to deal with the incomplete and subjective information at the conceptual design stage, a fuzzy preference model is complemented to rank the alternatives based on function- means connectivity. At last, the interaction between our algorithms and designers are discussed. In the following section, two case studies, designing a simple mechanical transporter and a testing machine, are presented to illustrate how these grammar rules, fitness functions, co-evolutionary algorithms and fuzzy preference model effectively work together for design concept generation and evaluation. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 Test and Evaluation In this chapter, two design cases are presented to demonstrate the capability of our proposed approach in design concept generation and evaluation at the conceptual design stage. First, we chose to expand Ullman’s bicycle design problem (Ullman, 2003) to a problem of designing a self-powered personal transporter. This expansion creates a bigger design space for HiCED to generate different kinds of personal transporters. The goal of the case study is to investigate (1) how design results correspond to different design scenarios or requirements, (2) How design solutions respond to the change of the size of the library, (3) how design solutions are sensitive to the fitness functions, (4) how design solutions respond to genetic parameter settings, and (5) how the subjective and incomplete information is utilized to rank the alternatives. Second, we adopted the example of designing a testing machine (Pahl & Beitz, 1996) to verify the findings of how design solutions are sensitive to the fitness functions in function structuring because constructing a function structure by GP is a big challenge in our research. 6.1 System Implementation and Setup Figure 6-1 illustrates the system architecture of HiCED. In the case of designing a self-powered personal transporter, the function library was created based on the function structures developed by the mechanical engineering students in their design projects for a senior level design methodology course. The means in the means library were collected from commonly used mechanical vehicles, such as bicycle, 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. scooter, and skating board. As an input to HiCED, we set the top-level function to be <transport><X>. Function decorngqsjti.o_n_, R e q u ire m e n ts O v e ra ll fu n c tio n s F u n c tio n d e c o m p o s itio n f u n c t i o n s m a p to m e a n s lib rary D e v e lo p fu n ctio n s tr u c tu r e b y G P Designer Computer F itn e s s S e le c t m e a n s b y G A G e n e tic m o d u le .e le c t checl Satisfactory Figure 6-1: System Architecture of HiCED The HiCED framework has been implemented in Java language. The major parameters for co-evolutionary design are shown in Table 6-1: GP for function structuring GA for means selection Number of generation: 100 Population size: 2000 Initial population: grow Initial depth: 5 Max depth: 10 Selection: tournament Crossover: 60% Mutation: 5% - 50% Reproduction: 10% Number of generation: 50 Population size: 30 Crossover: 90% Mutation: 5% Reproduction: 5% Table 6-1: Param eter setting for co-evolutionary design 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 •'Tf e i atcw ste ME Input from £ b y £ ficm S ty- X O trtpn? t o n o tlf 4 try M E to ac<U 5 fcy M E « 4 • <,»op - M E Input ln )ta itod# 3 by M E OlltptU a * - ■ irtud* - M E 1 In(xit frc ta no d e 3 by M E O utput *■6 X 1 Input f)trtpru to iK’< S e " b y M E ♦ r trsuitJtst M S - Lllpitt fic ia c o d f £ by* M E O ut put: * 5 v n p p ty E Utpw Oi.tjxit to tteti* ? trv E to ucd« 3 &v X ME E - * <*«*>p}y* Figure 6-2: An example of function structure generated by the System The system generates function structures in a text format that can be represented as a schematic function structure. Figure 6-2 illustrates the mapping between the text and schematic representations of a function structure. C onvergence Curve B est Value v s. G eneration 0 itir p 25 r T ~ .................. -...................... -45 - * ................... ................................................. 11 21 31 41 51 61 71 81 91 Generation (a) Function structuring by GP (b) Means selection by GA Figure 6-3: An Example of Convergence Curves for Co-Evolutionary Design Our system runs on a Pentium-4 2.2GHz PC with 512M memory. Initially there are 14 functions in function library and 41 means in means library. It took about an hour for HiCED to generate optimal solutions. Figure 6-3 shows an 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. example of convergence curves for function structuring and means selection for designing a self-powered personal transporter. 6.2 Experiment Design As shown in Figure 6-4, our experiment design has two independent variables, two dependent variables and three control variables. For example, in the case of designing a self-powered personal transporter, the independent variable design problem has only one value, i.e., the top-level function <transport><X>. For user requirements, we have three design cases for testing how HiCED can generate different design solutions in response to different user requirements. The three design cases are 1) design for low cost; 2) design for light weight; and 3) design for long travel range, which is translated to choosing the designs that require less power from riders. U ser’ s R equirem ents 1 D esign Problem Independent Variables HiCED Design Process Size of Library G enetic P a ra m e te r setting F itness Functions Function & Co-Evolutionary Fitness Means Library Mechanism Evaluation Control Variables F itness of design ► Running Tim e ► Dependent Variables Figure 6-4: Experiment Design Two parameters were chosen as dependent variables, i.e., the fitness o f design and the program running time. The fitness of design is a relative value which 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. indicates how a candidate solution is similar to the “desired” ones which are constrained by the fitness functions. If two alternatives have the same fitness values, an experience designer will be consulted to judge the solutions. It is conceivable that the design solutions of HiCED depend on the size and variety of the function and means library. In our experiment, we tested the response of HiCED solutions to the size of the means library in designing a personal transporter. With everything else fixed, the variation of the size of means library from having 41 means to 32 led to very different design results. The second control variable in Figure 6-4 is genetic parameter settings. In our experiment study, we are interested in understanding how the mutation rate can help or hinder the results of design. As will be discussed below, we changed the rate of mutation from 5% to 50% and gained interesting insights. The main focus of our case study was to investigate how different distributions of the weights of the fitness functions impact the design results. Through this investigation, we hope to gain insights on what are important for generating design concepts. We pay specific attentions to function structure formation and evaluation. Compared with means selection by GA, the exploration of functional space by GP is more difficult because of the unavailability of quantitative information and structural complexity. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3 Hypotheses To investigate design concept generation with regard to the control variables, i.e., fitness functions and genetic parameters, we developed following hypotheses. 1) Function-based consistency is a determinant factor for function structuring because it guarantees that each flow of a function in the function structure is satisfied by other functions in function structure or the top-level function. 2) Global consistency is a major factor for function structuring. One most important evaluation criterion for function structuring is that the derived function structure must have the same inputs and outputs as its top-level function. So we consider global consistency a key to form a valid function structure. 3) Limitation on the dimension of function structures is important for GP to develop feasible function structures. The limitation can avoid meaningless repetition of functions in a function structure and facilitate the process to find optimal solutions. 4) Hierarchical development is helpful for design concept generation. We believe that the intermediate information from elaboration levels provides instructive guide for further design concept generation and evaluation. 5) Increase of mutation fraction helps find optimal solutions during the conceptual design because the mutation can bring about new ideas and extend the reach in design space. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 Case Study 1: Personal Transporter Design In the following section, we use the scenarios of designing a self-powered personal transporter to verify our hypothesis. Based on the results, findings of what is important to a design are concluded. 6.4.1 Function Library and Means Library ID Function /, <transportxx> A <move><X> A <support><X> A <generate><ME> A <guide><ME> A <stop><ME> A <secure><X> A <transmit><ME> A <supply><E> f m <control><E> / „ <input><E> /,2 <change><E> A, <transmit><x> /.« <convert><E to ME> ID Means Applicable function High level Consistent means Weight value Cost value pedal drive A 0 n/a n/a W I4 pedal gear f\u W 2 3 3 W li, pedal A y 1 1 . 1 1 M 26 chain A n as 2 2 (A) Function library (B) Mean library Figure 6-5: Function and means library In HiCED, function and means library serves as a basis of solution spaces for design exploration. In our current research, we assume that the function and means library has been established and it contains rich enough information for design concept generation and evaluation. Partial of function library and means library used in this example is shown in Figure 6-5. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Currently we assume the knowledge is readily available. The next step of our research will investigate how to acquire the knowledge for design concept generation. 6.4.2 Analysis of Results In this section, we discuss the results of our case study and address the questions of how co-evolutionary method helps designers find feasible solutions; how the model appropriately responds to different user’s requirements; and how different fitness functions and the settings of genetic parameters influence design concept generation. 6.4.2.1 Results of function elaboration Grammar-based approach is adopted in HiCED for function elaboration. Considering that one major factor of the automation is its efficiency throughout large solution spaces, action-related elaboration rules are assigned a higher priority than general function elaboration rules. <transport><X> Rule A I <support><X> I GFE: General Function Elaboration Rule I <generate><ME>| | <guide><ME>| I <stop><ME>l I <secure><xTll<transmit><ME>l I <supply><E> I I <aenerate><ME> I I <control><£> | I <input><E> I I <chanqe><e> I l<convert><E to ME>| I <transmit><ME> I RnlfiD IFE & Rule E Figure 6-6: An Example of Grammar-based Function Elaboration 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6-6 presents an example of how function grammar rules are applied to functional elaboration. The action based grammar rules applied during the process are shown in detail in Appendix B, and the general function elaboration rules (GFEs) are those discussed in section 5.1.1. The elaboration proceeds in a width-first sequence. After elaborating functions at one level, design focus will shift to explore functions and means by co-evolutionary techniques. The leaf functions derived from function elaboration will become a basis for constructing a function structure and searching for desired means to satisfy these functions. At present, whether a function can be elaborated further is determined by its physical implementation. If its corresponding means consists of some sub type means, then we view this function is not atomic and has to be broken for more concrete implementation. The zigzag process of function elaboration and co-evolution will continue until no functions are decomposable. 6.4.2.2 Co-evolution of functions and means The key idea of HiCED is that the conceptual design process can be viewed as a co- evolutionary process of functions and means at each level of elaboration hierarchy. Functions and means can help each other find feasible solutions by providing more information for their evaluation. The following example shows the effect of co evolution. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figures 6-7(A) and 6-7(B) are two alternative function structures generated by the genetic programming algorithm at the level 3 of the elaboration hierarchy. Both of them are “equally good” if only functional flow compatibility is considered for evaluation. M E ME M E ME ME TE <generate> <ME> <guide> <ME> <generate> <M E> <secure> <X > <transmit> <ME> <guide> <ME> <stop> <ME> ME ME M E (A) Alternative 1 (B) Alternative 2 Figure 6-7: Two Alternative Function Structures at Level 3 Function Means <guide><ME> steering wheel guide wheel <stop><ME> cramp brake pedal brake friction with ground Table 6-2: Available means for <guidexM E> and <stop><ME> When the candidate means that implement the functions <stop><ME> and <guide><ME>, shown in Table 6-2, are taken into consideration, however, the distinction becomes obvious. Based on design experience, we know that it is not feasible to connect the means for <guide><ME> to the means for <stop><ME> directly. Once means are considered, function structures are re-evaluated based on the new means information. The final result shows that alternative 1 is better than 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. alternative 2 because it does not require a direct connection of means between the two functions. This example demonstrates that our co-evolutionary model makes the maximum use of available information for effective design concept evaluation during the process of design concepts generation. 6.4.2.3 User’s requirements Three scenarios are designed to test how our system responds to different user’s requirements, namely, light weight requirement, low cost requirement, and long travel range requirement. With respect to the requirements of light weight and low cost, the system came up with the same function structure (Figure 6-8), but different means combinations (Table 6-3). While for the requirement of long travel range, the system generated a different function structure (Figure 6-9) and different means combination (Table 6-4), which is similar to a bicycle design. x * ■ ME ► ME — ► TE — ► X — ► Figure 6-8: Function Structure Regarding Weight and Cost Requirements 106 <secure> <X> ME. <control> <E> <input> <E> <convert> <E to ME> ME <transmit> ME <ME> <transmit> <ME> <guide> <ME> <stop> <ME> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Function Solution w ith M inim um cost Solutions w ith m inim um w eight Solution 1 Solution 2 solution <secure><X> board shoes shoes <transmit><ME> wheels wheels wheels <control><E> human human human <input><E> feet feet feet <convert><E to ME> shaft shaft shaft <transmit><ME> wheels wheels wheels <guide><ME> guide wheels guide wheels guide wheels <stop><ME> friction friction friction Table 6-3: Solutions with Requirements of Low Cost and Light Weight ME ME TE <input> <E> <control> <E> <secure> < x> <convert> <E to ME> <transmit> <ME> <transmit> <ME> <guide> <ME> <stop> <ME> ME ME ME. Figure 6-9: Function Structure for Long Travel Range Requirement Function M eans <secure><X> saddle <transm it><M E> frame and wheel <control><E> human <input><E> feet <change><E> pedal gear <convert><E to ME> gear <transm it><M E> chain <guide><ME> handler bar <stop><ME> Cramp brake Table 6-4: Means Selection with Long Travel Range Requirement 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A careful examination of the concepts reveals that the three requirement scenarios led to three different designs. The tests showed that by changing the user requirements, we can obtain different design concepts from HiCED. 6.4.2.4 Size of library It is conceivable that the design concepts depend on the richness of the function and means library, or the size of the library. In this case, we check how our HiCED model can react to the change of the size of the library. We still use same requirements as described in Section 6.4.2.3 but different size of means library. Function Solutions w ith m inim um cost Solutions w ith m inim um w eight Solution solution <secure><X> board board <transmit><ME> wheels wheels <control><E> human human <input><E> feet feet <convert><E to ME> shaft shaft <transmit><ME> wheels wheels <guide><M E> guide wheels guide wheels <stop><ME> friction friction Table 6-5: Solutions after library size changes In the previous case, there are total 41 means in means library and the function <generate><ME> can be implemented by three means, namely, “pedal drive”, “slide drive” and “sketch drive”. In this test, the means “slide drive” and its related sub-means are removed from the means library. The size of means library reduces to 32. Compared with the previous results, only one solution is generated 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with respect to the minimum cost requirement, and different outcome is produced with respect to the requirement of light weight (Table 6-5). It can be observed that design concept generation is sensitive to the change of the size of means library. A minor change would lead to very different outcomes. But because of our small size of means library, the difference of time cost for the search for solutions is imperceptible. 6.4.2.5 Control variables Function structure generation and evaluation is a most difficult task in our co- evolutionary mechanism because of the lack of quantitative information. Currently seven fitness function components have been identified for function structure evaluation. In this subsection we discuss how these control variables determine the formation of function structures. Our experiments were conducted in the following procedure. First, we assign each fitness function component an equal weight (see Equation 5-12). Then we test each fitness function component in two situations while keeping all other component weights the same: one with a smaller weight and another with a bigger weight. By comparing the solutions in different weighting cases, we can gain insights on how sensitive the function structuring is to each of the components. From the sensitivity study, the “best” weight distribution for the fitness function, as shown in Figure 6-10, can be obtained. After that, we check how mutation fraction affects the generation of function structure. In our test, mutation 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fraction was changed from 5% to 60% with an increase step of 5%. In the following, we first presents how these control variables influenced the function structuring results in Table 6-6 and then discuss the reasons behind the influence. Weight Distribution FC GC MC SD HC FV Fitness Function Figure 6-10: Effective Distribution of Fitness Function Weight Test conditions Observations Equal weight distribution No feasible solutions were generated. Function connection compatibility was always violated. Weight of Function-based Consistency (FC) A smaller weight of FC led to function connection incompatibility; and the increased weight of FC improved function connection compatibility. Weight of Global Consistency (G O The change of weight of GC did not influence I/O relationship between function structure and its top function. Weight of Structural Dimension (SD) A smaller weight of SD caused the dimension to expand randomly, while a bigger weight of SD could maintain a reasonable size of dimension for function structures. Table 6-6: Effect of Control Variables on Function Structuring 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Test conditions Observations Weight of Function variety (FV) A smaller weight of FV caused function structure design to exclude some functions and made the function structure more probably to be infeasible. A bigger weight of FV helped find more feasible solutions. Weight of Hierarchical Consistency (HC) A bigger weight of HC helped make lower-level function structure be consistent with the higher level ones; but a very big weight of HC caused some functions be excluded from the function structures. Weight of Decomposition Consistency (DC) A bigger weight of DC could maintain the function relationships coming from function decomposition in its corresponding function structure design. Mutation fraction Less feasible solutions are generated when mutation fraction is less than 5%. From 5% to 50%, with the increase of mutation fraction, more feasible solutions are produced. But solutions became worse (compared with the “best one” generated) when mutation fraction is greater than 60%. Table 6-6: Continued Function flow consistency is a basic criterion to form a valid function structure. Any input, or output, flow of a function must be an output, or input, flow of another function, or it must be inherited from the top-level function. A smaller weight of Function-based Consistency (FC) cannot guarantee a valid function structure due to the possible function flow inconsistencies. FC is a very important factor in function structuring. G lobal consistency (GC) is a criterion to determine whether the generated function structure has the same input and output flows with the top-level function. The result shown in Table 6-6 contradicts our hypothesis that GC is an important 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. factor. Our further cross-examination revealed that the effect of GC was weakened by Function Variety (FV). From function elaboration point of view, if more types of functions are included into a function structure, then it is likely that the Global consistency is satisfied. Although means provide some complementary information for function structuring, the key factors in determining a feasible function structure are still function flow related. Means-based Consistency (MC) has a weaker effect on function structure development compared with FC. Structural dimension (SD) limitation prevents function structure from expanding randomly. The experiment result is consistent with our hypotheses that feasible solutions can be generated only when an effective dimension control mechanism is applied. Intermediate information provides much information for further developing solutions. Without hierarchical approach, the information will be lost and evaluation will be more difficult. The test result that bigger weight of Hierarchical Consistency (HC) and Decomposition Consistency (DC) are required proved our hypotheses. Function variety (FV) aims to counteract the side-effect of SD which tends to exclude functions from the function structure. The test result showed that the more distinct functions are included into a function structure design, the higher possibility feasible solutions are found because it extends the reach of design space. However the side effect of FV is that the running time increases significantly. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Mutation can play a crucial role in generating new concepts. It is analogous to human design where a designer applies intuitive ideas or generates a modification of his/her solution. The experiment proved our guess that keeping pouring new design context into a design will be helpful to design concept generation. However, feasible solutions cannot be yielded if the mutation fraction is set too high, probably due to the lack o f focus. This result indicates that in the conceptual design, more new ideas are encouraged in order to make an effective search for optimal solutions and, at the same time, the focus of search still needs to be maintained. To summarize, our experiments using HiCED in designing a self-powered personal transporter has led to the following findings: 1) Connection consistency is more important than global consistency; 2) The information of elaboration hierarchy is important for design concept generation; 3) More functions tend to help generate more feasible solutions, but increase the calculation cost significantly. 4) More mutation is helpful for finding feasible solutions, but too much may lead to no-convergence; 5) Stability is very sensitive to dimension constraint. 6.4.2.6 C onstrained G enetic O perations In the above case study, we tested the impact of control variables on design concept generation under the design environment where genetic operations were selected 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. randomly. But this brings about another research issue, that it, impractical function structures are produced with a very high possibility and much computation resource is cost on what alternatives are feasible rather than what are optimal. In this sub section, we will check how the constrained genetic operations affect design concept generation. To investigate design concept generation with respect to the constrained genetic operations, we developed following hypothesis. 1) It will take a longer time to select suitable arguments for the constrained genetic operations, so the running time for design concept generation will be longer than the normal genetic model. 2) The constrained genetic operations avoid impractical solutions, so it will have a quicker convergence than the normal model. 3) The mutation fraction plays a weaker role in design concept generation than the normal model because of more knowledge brought in by the constrained genetic operations. Our experiments follow the same procedure as described as Section 6.4.2.5 to compare the effects between the constrained genetic operations and normal genetic operations. The “best” weight distribution for the fitness function with respect to the constrained genetic operations is shown in Figure 6-11. And the comparison of convergence curve between them is shown in Figure 6-12. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Weight Distribution i f FC GC MC SD FV DC HC Finess Function Figure 6-11: Weight Distribution for Constrained Genetic operators Convergence -104 -106 © -108 | -110 f i -1 1 2 (fi -114 -116 -118 G enerations Convergence Curve ___________ > -60 o .90 1 11 21 31 41 51 61 71 81 91 Gtntrations (A) Constrained Genetic Operations (B) Normal Genetic Operations Figure 6-12: Convergence Curve for Function Structuring By comparing the design concepts generated by the constrained genetic operations with the ones produced by the normal genetic operations, we can gain almost same conclusions as described in Section 6.4.2.5, but two distinct differences are found. 1) Convergence and running time In the normal model, the optimal solutions were generated after the 15th generation (Figure 6-12(B)), and the model took around 1 hour to search for function 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. structures from 215 function elaboration sets. While with the constrained genetic operations, the optimal solutions were generated from the 12th generation (Figure 6- 12(A)), but the model took more than two hours to search for solutions within the same solution space. The results are consistent with our hypothesis. The slower design concept generation by the constrained genetic model is because more computational resource is cost on selecting suitable arguments for the constrained genetic operations. Due to the limit of the size of function and means library, we did not see the advantages from the constrained genetic operations. On the contrary, the normal genetic model seems more efficient. In our future research, we will study whether the search will benefit from the constrained genetic model with larger libraries. 2) Mutation Fraction In the constrained genetic model, a smaller mutation fraction (5%~20%) is enough to generate feasible solutions, while a too large (>35%) mutation fraction will slow down the calculation and make solution non-converged. The observation is different from the one from the normal genetic model, where a larger mutation fraction is desired. The mutation operation can help introduce new design context into a design. However, under the constrained genetic model, only practical function connections are allowed. Every mutant is a feasible sub function structure. Given a limited functional space, a higher mutation fraction can not bring more ideas than a lower one. By comparing the difference between the two models, we can gain the insight 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. into the conceptual design. Mutation (or new ideas) is more important in design concept generation where design information is lack or ad-hoc than where more knowledge is available. 6.4.3 Fuzzy Evaluation In the previous case studies for means selection, we assume that all information available for performance evaluation is deterministic. However, it is impractical in reality because conceptual design is characterized by the incomplete or subjective information. It is natural to represent the information by fuzzy logic. In this section, we present an example to illustrate how our fuzzy evaluation model utilizes the imprecise information for design concept evaluation. In the model of HiCED, means evolve in parallel with functions at each level of the elaboration hierarchy. Function structures provide connection information for means evaluation. At the conceptual design stage, the means connectivity is fuzzy. The case study shows how means are assessed by a fuzzy preference model based on their connectivity with respect to their corresponding functional connectivity introduced in the function structure. 6.4.3.1 Function structure Function structures are generated by GP after function elaborations. In this case, let us take a look at the function structures at elaboration level 5 (Figure 6-13(B)). 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X E X ME ME ME ME TE <secure> <X> <generate> <ME> < s u p p l y > <E> <transmit> <ME> < s t O p > <ME> cguide> <ME> (A) Function stracture at elaboration level 4 (B) <secure> <X> Wl ME w2 w l w l w l <con»rol> <E> <input> E <change> E <convert> ME <transmit> <E> <E> <E to ME> <ME> w2 ME w2 <tnansmit> <ME> < g u i d e > <ME> <stop> <ME> ME ► ME — ► TE — ► X -► (B) Function structure in elaboration level 5 Figure 6-13: Function means connectivity The connection relation between (control)(E) and (input)(£) is prescribed by the action-based function elaboration rule (Li and Jin, 2005). The connection relations between function (transmitj^ME'j and (guide'j^ME'j , and between ('transmit)(ME) and (stop)(ME) are determined by the higher level function structure (Figure 6-13(A)). In the lower level of elaboration, these relations must be maintained when appropriate means are selected. They are more important than other connections produced by genetic programming. The weight (importance) 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of connections is expressed in the linguistic terms of “important” and “normal” (Figure 6-14). Fuzzy w eight num ber Normal Important Figure 6-14: Linguistic scale of weight value 6.4.3.2 Alternatives In the mechanical engineering design domain, the number of applicable means for each function is numerous. Due to the mental limitation or incomplete information, their connection relations can be vague. A finite set of fuzzy numbers is used to express these imprecision connection levels among means, namely, “very impossible”, “impossible”, “fairly impossible”, “neutral”, “fairly possible”, “possible” and “very possible”. The linguistic scale is used to transform a linguistic term into a fuzzy number (Figure 6-15). 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — Very impossible — Impossibler Fairly impossible Neutral — Fairly possible — • — Possible —i — Very possible 0 0.2 0.4 0.6 0.8 1 Figure 6-15: Linguistic scale of fuzzy connection In HiCED, the means library provides a knowledge base of what can be specified as required function (Li and Jin, 2005). In this example, four candidate solutions are generated (Table 6-7) and the connection relations (7 means very possible, and 1 means very impossible) among means requires designers to input for fuzzy analysis later (Table 6-8). Function Solution (a) Solution (b) Solution (c) Solution (d) <secure><X> Saddle Board Board Seat <transmit><M E> Frame Wheel Frame Wheel <control><E> Human Human Human Hand <input><E> Pedal Spring Lever Row <change><E> Pedal Arm Lever Gear Level <convert><E to ME> Gear Spring Gear Friction <transmit><ME> Chain Spring Gear Body <guide><M E> Steering Wheel Guide Wheel Steering Wheel Body <stop><M E> Pedal Brake Cramp Brake Friction Friction Table 6-7: Optional solutions for simple mechanical transporter design 1 2 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Saddle 7 Board 6 Board 5 Seat 6 Frame Wheel Frame Wheel Human 7 Human 7 Human 7 Hand 7 Pedal Spring Lever Row Pedal 7 Spring 6 Lever 4 Row 6 Arm Lever Gear Level Arm 6 Lever 6 Gear 7 Lever 4 Gear Spring Gear Friction Gear 6 Spring 7 Gear 7 Friction 4 Chain Spring Steering Body Chain 7 Spring 2 Gear 4 Body 7 Steering wheel Guide Wheel Friction Guide Wheel Chain 4 Spring 2 Body 5 Pedal Brake Cramp Brake Friction Table 6-8: Fuzzy Connection Relations among Means (7: Very possible; 6: Possible; 5: Fairly possible, 4: Neutral; 3: Fairly Impossible; 2: Impossible; 1: Very Impossible) 6.4.3.3 Evaluation result Given the function structure and means connectivity, the overall connection levels for solution a, b, c and d is shown in Figure 6-16. According to equation (2-12), we can obtain that 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p(ct,b) = 0.91 p(a,c) = 0.55 p(a,d) = 0.7 p(c,b) = 0.92 p (c,d ) = 0.59 p(d,b) = 0.85 ( 6 - 1 ) C onnectivity Figure 6-16: Overall connectivity level The overall fuzzy preference relations can be determined a s a fy c fy d fy b (expression a < t > b indicates that a is preferable to b). When the preference threshold a = 0.6, it is observed that the preference for alternative a and c are indifferent, and both of them are preferable to alternative b. Although alternative c and d are indifferent, alternative a is preferable to d. Thus, candidate b and d can be removed, and a and c are kept for further refinement. But if the designer is more confident in his (or her) knowledge on the means connectivity shown in Table 6-7, then he (or she) may choose a lower preference threshold, for example, a = 0.5. Under such an assumption, alternative a is obviously 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. preferable to all other alternatives b, c, and d. Therefore, only a is kept for further design consideration, and b, c and d are dropped. 6.4.3.4 Discussion Unlike the traditional evaluation methods in means connection, which usually consider the connection relations as crisp variables, our fuzzy preference model uses fuzzy numbers to express the connectivity level among means to accommodate the incomplete and subjective information at the early stage of design. Based on the fuzzy values of the connectivity between the means, the designer can compare two alternatives, based on the connectivity, by calculating the preference relation between the two alternative using Equation (2-12). The preference relations, e.g., those shown in Equation (6-1), provide a first-hand comparison information based on the given fuzzy connectivity knowledge. In addition to the fuzziness of connectivity knowledge, there is another layer of subjectivity that makes the information at early stage of design imprecise, i.e., designers’ confidence on their knowledge about the means connectivity. Making decisions based on Equation (6-1), for example, depends on how confident the decision-maker is about the connection knowledge of Table 6-7. In our approach the threshold a is introduced to handle the issue of the subjective confidence. If a designer is not confident in his knowledge on means connectivity, a larger threshold can be set. In the above example, with a higher threshold a = 0.6, solution b and d are discarded and both a and c are kept for further consideration. But if the designer 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is comfortable with means connectivity he or she assigned in Table 6-7, a lower threshold ( a closer to 0.5) can be selected so that more alternatives are removed from further consideration. For example, given the threshold a = 0.5 , only alternative a is selected for further as the final design concept. If two or more alternatives are kept for further consideration, i.e., they are indifferent given the current knowledge and the confidence on the knowledge, then further information is needed and the evaluation should go beyond the means connectivity. For example, given the threshold a = 0.6, alternative a and c are indifferent if only the means connectivity is taken into account. But if the weight of the transporter is also considered a major factor for design evaluation, the solution a may be a better choice because gears are usually heavier. 6.5 Case Study 2: Testing Machine Design Function structuring is a major challenge in our HiCED model because of the highly qualitative information and complex functional relationships. Although some findings have been identified in the previous case study, are they specific or general to any design problems? In order to verify our conclusions of what is important to a design, we employed a same example of designing a testing machine (Paul & Beitz, 1996) to verify our findings. To simplify the testing, we only focus on investigating the design concept generation in functional domain because means selection is relatively simpler compared to function structuring. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.5.1 Function Library The functions in the testing machine design have been provided by Paul and Beitz (machine (Paul & Beitz, 1996). In order to make the functions general enough for any specific mechanical design cases, we rewrote the functions based on function basis (Stone et al., 1999, 2000; Hirtz et al., 2001a, 2001b). For example, the function “Test specimen” is transformed into a more general expression as “<sense><X>”, where X refers to any physical objects. The function library for testing machine used in this example is shown in Table 6-9. Functions provided by Paul and Beitz Functions in HiCED Test specimen <sense><X> Load specimen <load><X> <measure><X> Change energy into force and movement <convert><E to ME> Measure force <measure><force> Measure deformation <measure><displacement> Adjust energy flow <change><ME> Hold specimen <hold><X> Amplify measurements <increase><Signal> Compare target with actual values <compare><Status> Table 6-9: Function library for testing machine design 6.5.2 Function Elaboration Figure 6-17 presents the result of how function grammar rules are applied to functional elaboration in designing a testing machine. The action based grammar rules applied during the process are shown in detail in Appendix B, and the general function elaboration rules (GFEs) are those discussed in section 5.1.1. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <load> <measure> <convert> <increase> <change> <compare> <X> <X> <E to ME> <Signal> <ME> <Status> ule G <hold> <load> <measure> <measure> <X> <X> <force> <displace> Figure 6-17: Function elaboration for testing machine From the elaboration, we found that although the order of function elaboration between the HiCED and Paul and Beitz’s, same results were reached at last. 6.5.3 Function Structuring The objective of the case study 2 is to verify the conclusions of what is important to conceptual design that we have found in designing a personal transporter. We adopted a similar distribution of fitness function weight (Figure 6-18), and the running results are shown in Figure 6-19(A) and Figure 6-19(B). W eig h t D istrib u te Figure 6-18: Distribution of Fitness Function Weight 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 'target 'eform <change> <E> <convert> <E to ME> <measure> " <X> <load> <X> <increase> <signal> <compare> <status (C) Function structure at elaboration level 2 'target 'deform <load> <X> <hold> <X> <change> <E> •* <increase> * <signal> <compare> <status <convert> <E to ME> <measure> <displacement> <measure> <force> (D) Function Structure at elaboration level 3 Figure 6-19: Function structure for a testing machine The function structure generated by HiCED model is similar to the solutions proposed by Paul and Beitz in Systematic Approach (Paul & Beitz, 1996). 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.5.4 Discussion In this section, we redesigned the function structure for a testing machine by our proposed HiCED. The similar design solution while different approach proves the capability of the HiCED model in effective design concept generation and evaluation at the conceptual design stage. Besides, the case study confirms the following conclusions which were found from the previous case study: 1) Connection consistency is more important than global consistency; 2) The information of elaboration hierarchy is important for design concept generation; 3) More mutation is helpful for finding feasible solutions, but too much may lead to no-convergence; 4) Stability is very sensitive to dimension constraint. 6.6 Conclusion In this chapter, we used two examples to test our HiCED model in design concept generation and evaluation. In our experiments, we investigated how design concepts are affected by user’s requirements, the richness of function library, fitness functions, genetic parameters and fuzzy means connectivity. The results reveal the insight of what are important to a design at the conceptual design stage. In the last chapter, we will summarize the findings and contributions and then recommend for the future research work. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 Contributions This chapter summarizes the research results, lists the contributions and suggests the potential future work. The primary work about this research includes developing a co-evolutionary model for design concept generation and evaluation, identifying the impact of various fitness functions on design concepts, and proposing a fuzzy evaluation method to deal with the incomplete and subjective information at conceptual design stage. The research contributes to the research fields of design methodology and genetic programming. Opportunities for future research include design knowledge acquisition, the further exploration of evaluation methods, and the improvement of the algorithms to deal with the scaling problem. 7.1 Summary of Research Results As design problems become more complex and design lead time more pressing, designers need supporting tools to expand their reach in the design space and increase the number and quality of their design concepts. Our research takes a hierarchical co-evolutionary approach to help designers explore design space and develop design concepts by automatically generating desirable function structures and their mappings to the embodiment means. The approach adopts a zigzag design process in which grammar rules are applied to elaborate higher level functions and GA and GP based algorithms are employed to let function structures and means co- evolve into design concepts. The concepts of co-evolution and fitness of this approach worked well to deal with the two major issues of the conceptual design 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mentioned in the introduction section, i.e., unclear process for design concept generation and the lack of quantitative information for conceptual design evaluation. The HiCED prototype system was developed and the case study results demonstrated the effectiveness of the proposed approach and how the HiCED system responded to the variations of users’ requirements, fitness function weights, and mutation rates. Modeling the conceptual design process as a co-evolutionary process and composing effective fitness functions were two major challenges in our research. Integrated use of general grammar rules and action-based heuristic grammar rules can balance design novelty and computational efficiency for function elaboration and structuring. Developing design concepts across different levels of elaboration hierarchy can help make the best use of available information for effective design concept generation and evaluation. The co-evolutionary algorithm together with the multi-component fitness function allows best function structures and their mapping means combination co-evolve at each level of elaboration hierarchy. The testing results showed that our approach provides 1) a process model for automated design concept generation; 2) an effective fitness function scheme for effective concept evaluation; and 3) the insights of what is important, through fitness function weight distribution, for concept generation. Conceptual design is characterized by the lack of quantitative information. In our research, we treat means connectivity as an important source of information for evaluating design concepts. To enhance HiCED capability in dealing with the subjective and incomplete information, a fuzzy preference model is devised for 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. design evaluation that allows designers to code their qualitative knowledge about means connectivity into fuzzy relations among the means and make their selection decisions based on their confidence on the knowledge that they utilized. The case study has shown the mechanism and the effectiveness of our proposed approach. 7.2 Contributions The primary contributions of this research to the area of design methods are: 1) The research advances the understanding of the conceptual design process and provides a computational support for design concept generation and evaluation at the early stage of design. Comparing to other automated design methods (Maher, 1994; Schmidt, 1995, 1997; Parmee, 1997; Koza, 1999; Goodman et al., 2002; Sridharan, 2004), our HiCED model is more effective for complex mechanical design at conceptual design stage because it can make the maximum use of available information by co-evolving design concepts across different levels of elaboration hierarchy. 2) The research develops a set of fitness function for design concept evaluation. By investigating the impact effect of fitness functions on design concept generation, our model provides insights into what matters most to a design. The understanding can help designers more focus on the most critical parts of a design and improve the design quality. 3) Another major contribution of this research is to provide a fuzzy model for design evaluation based on the means connectivity at the conceptual 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. design stage. Means connectivity plays an important role in design success. However, due to the incomplete and subjective information, few research studies have been conducted to study how means connectivity affects the quality of a design. Our model transforms the subjective means connection information into fuzzy preference relations among alternatives and helps designers to select “best” ones. 4) The research establishes a foundation and offers a starting point for further design study. The HiCED model of conceptual design provides a flexible computational environment for us to conduct more extensive research on design concept generation and evaluation. Such research may include investigating how different fitness functions may impact the novelty, variety, quantity and quality (Shah et al., 2003; Chusilp and Jin, 2005) of design concept generation. The contributions of this research to the area of genetic programming include: 5) Our research opens a new avenue for developing function structure by genetic programming. One of the major challenges of developing function structures by GP is how to transform a genetic programming tree into a graph-like function structure. Our HiCED algorithm defines proxy nodes to implement the conversion. The proxy nodes are merged into their original nodes when they are converted back into function structure. 6) Another important issue about the genetic programming is closure problem (Rodrigues, 2002). Our algorithm introduces a set of constrained 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. genetic operations, by which only practical alternatives can be produced. The constrained genetic operations can allow the evaluation to more focus on what is optimal rather than what is feasible. 7.3 Recommendation for the Future Work The proposed HiCED model raises several important research issues and reveals many research opportunities on the computational support for conceptual design. In this section, I address various research issues in future work: 1) Our current HiCED system is limited by the small size of function and means library and the limited number of genetic functions. It is conceivable that as the size of the libraries increase and more genetic functions are introduced, the computation time will increase dramatically. Our future research aims to improve the co-evolutionary algorithm to deal with the scale-up issue. 2) The HiCED model depends on the richness of function and means library. How to efficiently acquire the necessary knowledge for design concept generation and evaluation remains a research topic. The future work on knowledge acquisition could be in the direction of exploring an intelligent agent-based approach in which agents attempt to pick up useful functions and means from the databases and various documents through data-mining and text-mining techniques. 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3) The HiCED model provides a flexible environment to investigate the impact of fitness functions on design concept generation. However, limited to case studies, the relation between novelty, variety, quantity and quality of design solutions and fitness functions has not been be well explored. More experiments need to be conducted to identify correlations between the fitness functions, contents of the function and means library, and the above mentioned design metrics. 4) The HiCED model develops a set of fitness functions and fuzzy evaluation methods for design concept evaluation. However, most of them focus on what can be feasible rather than what are the best. The limitation of current assessment raised an important issue for the future work: how can we devise more effective evaluation methods to rank alternatives in the face of the lack of the quantitative information at conceptual design stage? 5) Functional elaboration is implemented by two types of grammar rules: the general functional elaboration rules and the action-related functional elaboration rules. The general rules can lead to the variety and novelty, but it costs much computational resource and makes the search slow; whilst action- related rules can facilitate the elaboration, but limit innovative opportunities. How to balance the uses of the general rules and action-related rules for both novelty and efficiency is a future research topic. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography Akao, Y. 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Kluwer Academic Publishers, Boston Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A: Greedy Search Algorithm for Function Expansion Rule (FER) To facilitate the greedy search algorithm, two accessorial functions are defined. Function unm atch_output(f) returns those outgoing flows of function/that are not compatible with any other functions’ input flows in an elaboration set, and function unmatch _in p u t{f) returns those input flows of function / that do not match any other function’s output flows in an elaboration set. Assume fin { f) and fo u t(f) are input flow set and output flow set of a function / respectively. The search algorithm can be described as the following: 1) Calculate total unmatched output flows number(Ful ( ) UFoul - unmatch_output{fk ) k=Kft eFitlJ) 2) Calculate total unmatched input flows number(FjtX p UFin = ^ unmatch _ input(fk ) k=Kf„^FMi) 3) Calculate compatibility of function/from the function library with input flows fi„ and output flows f o ut\ C> = number(UFout n /•„)+ number(UFin n f out) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4) Select a candidate function fj+\j+\ with the maximum Cf,+ij+i and fi+ \j+ 1 £ Fm . Add this function into the elaboration function set Fj+ ]j . 5) Update unm atch_output(f) and unm atch_input(f) in F;+ly+1: for Vt e fin , do unmatch _ output{f) = unmatch _ output{f ) - t for Vt e /out, do unmatch _ input(f) = unmatch _ input(f ) - l In the above equations number is a function returning the size of a set. For any function f - t in FM j+ x , if unmatch _ output( / ) = 0 and unmatch _ input{f) = 0 Then the function elaboration process terminates. Otherwise, repeat step from (1) to (5) until all input and output flows of any function in the elaboration set are satisfied by other functions. 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B: Action-Based Function Elaboration Rules Rule A (transport){X) — A H ' >{(movg), {support^jx) => |{move )(x),(support)(X)} Rule B {move )(x) — — — >{(generate)(ME), (guide)(ME), (stop)(ME)} Rule C (generate}(ME}—^E^}(supply}^E} ^generate} (M E }\^ Rule D (supply}(E) —/l/’ / > {(controfy,{input)\E} => \lcontrol)(E},(input)(E)} Rule E (import} (E} —A R E --> | (import}(E}, (change} {E} J (import}(E} —A R E ->j(import}(E},(transmit}(E} j /?»/e F (sense}(X) —'4 A /' > ^load), ( measure )tx) =• \(Ioad )(X ),( measure > W I Rule G (measure}(^X} ——— >{(measure}(force}, (measure)(displacement)} Rule H (load)(x) A R ,i > {(hold){X), (load)(X)} 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Li, Wei (author)

Core Title A hierarchical co-evolutionary approach to conceptual design

School Graduate School

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, mechanical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Jin, Yan (committee chair), Khoshnevis, Behrokh (committee member), Lu, Stephen C-Y. (committee member), Shiflett, Geoffrey (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-584578

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