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On Maximizing Solution Diversity in a Multiobjective Multidisciplinary Genetic Algorithm for Design Optimization
Abstract
This article presents a new method for multiobjective multidisciplinary design optimization. The method obtains an estimate of the Pareto frontier with maximum solution diversity using a quality index, referred to as entropy index. Unlike previous methods that maintain diversity in the solution set heuristically, our method improves overall quality of solutions by explicitly optimizing the entropy index at every system-level iteration, and then using this information to bias the search process toward obtaining a solution set with maximum diversity. Our method utilizes a multiobjective genetic algorithm as an optimizer in each subproblem of a multidisciplinary optimization problem. To demonstrate the proposed approach, we applied our method to a mechanical design problem of a speed reducer and the results are compared with those obtained by a few other multiobjective optimization methods. A minimal set of quality indexes is used to compare the diversity and optimality of the obtained solution sets from the different methods on a quantitative basis.
... As a result, the multiplicative speckle noises are not totally vanished around the boundary of oil spill footprints. In this prospective, multi-objective optimization algorithm can involve in entropy metric (Gunawan et al., 2004) to preserve the diversity among different solution to minimize the influence of the look-alikes and multiplicative speckle noise (Lathi, 1968;10 Marghany, 2001;Zhang et al., 2013). ...
... Take the advantage of E-MMGA of preserving the diversity of solution set (Gunawan et al., 2004) and solving the multidisciplinary of uncertainty of random oil spill footprint discrimination in SAR data. The uniqueness of this study is to deal with entropy of 15 oil spill detection as multi-objective Genetic Algorithm (GA). ...
... Comprehending Coello et al. (2002), the multi-objective optimization (MOP) has already been successfully adopted to solve uncertainty of object detection in SAR images (Marghany, 2014a). In general, MOP consists of n decision variable parameters, k objective functions and m constraints (Gunawan et al., 2004). Multi-objective Optimization (Marghany, 2014b;20 Gunawan et al., 2004) aims at conducting optimization for a range of functions as follows ...
Oil spill pollution has a substantial role in damaging the marine ecosystem. Oil spill that floats on top of water, as well as decreasing the fauna populations, affects the food chain in the ecosystem. In fact, oil spill is reducing the sunlight penetrates the water, limiting the photosynthesis of marine plants and phytoplankton. Moreover, marine mammals for instance, disclosed to oil spills their insulating capacities are reduced, and so making them more vulnerable to temperature variations and much less buoyant in the seawater. This study has demonstrated a design tool for oil spill detection in SAR satellite data using optimization of Entropy based Multi-Objective Evolutionary Algorithm (E-MMGA) which based on Pareto optimal solutions. The study also shows that optimization entropy based Multi-Objective Evolutionary Algorithm provides an accurate pattern of oil slick in SAR data. This shown by 85 % for oil spill, 10 % look-alike and 5 % for sea roughness using the receiver-operational characteristics (ROC) curve. The E-MMGA also shows excellent performance in SAR data. In conclusion, E-MMGA can be used as optimization for entropy to perform an automatic detection of oil spill in SAR satellite data.
... Cette méthode s'appuie sur une décomposition de l'optimisation au niveau disciplinaire. La première version proposée (Gunawan et al. 2003) ne prenait pas en compte les couplages et était limitée aux systèmes hiérarchiques (Gunawan et al. 2004). ...
... Analyse du problème Gunawan, Farhang-Mehr, et Azarm (2004) comparent les résultats obtenus sur ce problème avec EM-MOGA et MOGA. Ils agrègent les fonctions f 1,1 et f 2,1 en une unique fonction F 1 représentant le volume total du réducteur de vitesse (figure 4.21). ...
... Les solutions de compromis C 1 (figure 4.23) devraient, en principe, ressembler aux solutions présentées figure 4.21. Gunawan, Farhang-Mehr, et Azarm (2004) trouvent cependant une répartition des solutions moins dispersée dans la première discipline (il y a moins de solutions dominées pour cette discipline). Ceci peut s'expliquer par le fait que les méthodes basées sur les algorithmes évolutionnaires multiobjectifs telles qu'EM-MGA ou COSMOS (dont les optimisations disciplinaires sont réalisées indépendamment par des algorithmes évolutionnaires multiobjectifs distincts) ne permettent pas de converger vers l'ensemble des solutions efficaces du problème global. ...
Multidisciplinary Design OptimiZation refers to the design and optimization of complex engineering systems, requiring simultaneous work of at least two disciplines. Each of them could have more than one objective to achieve. The traditional methods do not consider the case where each discipline has a multiobjective optimization problem to solve. Recently, methods which transform the multidisciplinary design optimization problem into a multiobjective optimization problem have been proposed. These methods are based on evolutionary multiobjective optimization algorithms. Unfortunately, the set of solutions found does not reflect the preferences of the disciplines: solutions can be globally efficient whereas they are locally dominated. Using the properties of order relations, we propose four definitions of the compromise between disciplines which take into account the grouping of the objectives. The theoretical properties of these compromises are studied, and especially their capacity to converge to the wanted solution set when they are used within an evolutionary algorithm. The compromises are integrated within an evolutionary multiobjective optimization algorithm. Experimental analysis of this algorithm with the four compromises are conducted. They confirm the theoretical predictions and show the relevance of the obtained solutions.
... Classical convex optimization techniques such as the steepest descent, conjugate gradient, and Newton-based optimization methods (Boyd & Vandenberghe, 2004) are some of the most well-known methods in this category, while many other gradient-based approaches have been developed in the past decades too (Swisher et al., 1984;Coello, 1999). Among various multi-variable optimization techniques one can name the Pareto optimality method initially designed for economic efficiency analysis which found its application to many engineering optimization problems too (Censor, 1977;Gunawan et al., 2004). ...
Optimal design of complex multibody mechanical systems involving numerous design parameters and constraints is a challenging problem. System parameters are typically determined through a recursive process which leads to a set of nominal values. While these nominal values satisfy design constraints they generally result in suboptimal performance. In practice, nominal design values can undergo a final adjustment step to improve performance while ensuring that design constraints are not violated. In this paper, we present a framework for design parameter variation to optimize the performance of a multibody system. As an example, we apply the proposed method to an exploration rover and show that the proposed approach provides superior results compared to the standard sensitivity derivative approach.
... For example, set-based approaches have been advocated in the automotive industry [9]. Multiobjective genetic algorithms have been applied to multidisciplinary problems as a means of generating a variety of solutions for each subproblem, overcoming the convergence difficulties of gradient-based approaches, and incorporating discrete variables; however, extensive iterations between subproblems are still required [10][11][12][13][14]. In a different set-based approach, robust design techniques have been utilized to generate ranged sets or intervals of design specifications that can be shared with collaborating designers [15][16][17]. ...
Multiscale design problems are typically decomposed into a hierarchy of distributed and strongly coupled sub-problems, each solved by design teams with specialized knowledge and tools. There are two contrasting approaches to formulating and solving such problems: (1) highly iterative exchanges of point solutions, as in most optimization-based approaches, and (2) minimally iterative exchanges of richer solutions, as in set-based approaches. The flexibility-based method is a representative set-based approach, developed by the authors, in which designers exchange targets and sets of Pareto solutions. In this paper, we explore the effect of these alternative approaches on overall lead time and lead time variability in a noisy, non-ideal design environment. Discrete event simulations are utilized to evaluate multiscale design lead time and its sensitivity to the level of noise in the design environment, including distracting interruptions and secondary design jobs. The results of the simulations indicate that the lead times of minimally iterative multiscale design processes are more robust to noisy design environments than highly iterative alternatives. Accordingly, the results suggest that it may be advantageous to favor richer, but less frequent, exchanges of information in a multiscale design process, even if more processing time is required within each sub-problem to generate those sets of information.
... This method is not concurrent and does not accommodate arbitrarily shaped design regions. Genetic algorithms have been applied to decompose and/or solve multidisciplinary problems22232421,25]. When they are used to solve multidisciplinary problems, they are setbased in terms of maintaining a diverse population of potential solutions, but they are resource intensive and do not provide a means of explicitly representing conditional relationships between variables. ...
A set-based approach to collaborative design is presented, in which Bayesian networks are used to represent promising regions of the design space. In collaborative design exploration, complex multilevel design problems are often decomposed into distributed subproblems that are linked by shared or coupled parameters. Collaborating designers often prefer conflicting values for these coupled parameters, resulting in incompatibilities that require substantial iteration to resolve, extending the design process lead time without guarantee of achieving a good design. In the proposed approach to collaborative design, each designer builds a locally developed Bayesian network that represents regions of interest in his design space. Then, these local networks are shared and combined with those of collaborating designers to promote more efficient local design space search that takes into account the interests of one's collaborators. The proposed method has the potential to capture a designer's preferences for arbitrarily shaped and potentially disconnected regions of the design space in order to identify compatible or conflicting preferences between collaborators and to facilitate a compromise if necessary. It also sets the stage for a flexible and concurrent design process with varying degrees of designer involvement that can support different designer strategies such as hill-climbing or region identification. The potential benefits are the capture of expert knowledge for future use as well as conflict identification and resolution. This paper presents an overview of the proposed method as well as an example implementation for the design of an unmanned aerial vehicle.
... Set-based coordination strategies have taken several forms, including: robust design techniques for generating ranged sets or intervals of design specifications that can be shared with collaborating designers101112131415; fuzzy set theory [16,17] for modeling uncertain or imprecise parameters (such as preferences for performance variables) during negotiation; metamodeling approaches for zooming into regions of interest [18]; and game theoretic approaches for coordinating the competitive reactions of designers to one another's decisions19202122232425262728. Other methods such as multiobjective genetic algorithms provide a set of distinct possibilities as opposed to a range2930313233, but they are resource intensive. In the set-based method (SBM) used in this paper, designers collaborate by exchanging targets for shared parameters, followed by Pareto sets of solutions that represent achievable tradeoffs between coupled parameters343536. ...
A set-based multiscale and multidisciplinary design method has been proposed in which distributed designers manage interdependencies by exchanging targets and Pareto sets of solutions. Prior research has shown that the set-based method (SBM) has the potential to reduce the number of costly iterations between design teams, relative to centralized optimization approaches, while expanding the variety of high-quality, system-wide solutions. These results have been obtained with representative examples in a laboratory setting. The goal of this research is to investigate whether similar results are obtained from an industrial trial, implemented in an industry design environment. The SBM is applied to the design of a downhole module for our industrial partners at Schlumberger, a developer of oilfield tools and services. The design was conducted on location at Schlumberger by an intern who converted the existing Schlumberger design process into a set-based design process. Results indicate that the SBM delivers the benefits predicted in the laboratory, along with a host of advantageous side effects, such as a library of back-up design options for future design projects.
... Thereafter, these designs are filtered based on pairwise comparisons of their associated performances so that only nondominated designs remain subject to further consideration (Mattson et al., 2004). In enhancement of the first, a second approach makes additional use of genetic or evolutionary algorithms to further improve the initial designs compared to a mere sampling (Narayanan and Azarm, 1999; Gunawan et al., 2004). Both approaches are similar in that they provide the designer with a set of nondominated solutions, but in general do not guarantee that any of these solution is also an optimal design. ...
The high dimensionality encountered in engineering design optimization due to large numbers of performance criteria and specifications
leads to cumbersome and sometimes unachievable trade-off analyses. To facilitate those analyses and enhance decision-making
and design selection, we propose to decompose the original problem by considering only pairs of criteria at a time, thereby
making trade-off evaluation the simplest possible. For the final design integration, we develop a novel coordination mechanism
that guarantees that the selected design is also preferred for the original problem. The solution of an overall large-scale
problem is therefore reduced to solving a family of bicriteria subproblems and allows designers to effectively use decision-making
in merely two dimensions for multicriteria design optimization.
... A set-based design philosophy has been advocated in the automotive industry[9], and set-based approaches have assumed several forms. For example, multiobjective genetic algorithms have been applied to multidisciplinary problems as a means of generating a variety of solutions for each sub problem, overcoming the convergence difficulties of gradient-based approaches, and incorporating discrete variables; however, extensive iterations between sub-problems are still required[10][11][12][13][14]. In a different set-based approach, robust design techniques have been utilized to generate ranged sets or intervals of design specifications that can be shared with collaborating designers[15][16][17]. ...
Multilevel design problems are typically decomposed into a hierarchy of distributed and strongly coupled sub-problems, each solved by design teams with specialized knowledge and tools. There are two contrasting approaches to formulating and solving such collaborative design problems: (1) highly iterative exchanges of single design solutions, as in point-based optimization approaches, and (2) minimally iterative exchanges of multiple solutions, as in set-based approaches. In this article, the effects of these alternative approaches on the overall lead time of a design process are explored. A discrete event simulation is developed to evaluate the lead times of highly iterative and minimally iterative multilevel design strategies and the sensitivity of those lead times to the level of noise in the design environment for a range of designer work loads. Designer work loads include not only the multilevel design task of interest, but also secondary design jobs that consume designer time. Noise is represented as variability in task processing times, arrival rates, and iteration levels. An example design process for an unmanned aerial vehicle is used to compare set-based and point-based design strategies. The results of the simulations indicate that the lead times of minimally iterative, set-based design processes are more robust to busy design environments than highly iterative, point-based alternatives. Accordingly, it may be advantageous to favor richer, but less frequent, exchanges of information in a multilevel design process, even if more effort is required to generate those sets of information.
Multidisciplinary design optimization (MDO) involves solving problems that feature multiple subsystems or disciplines, which is an important characteristic of many complex real-world problems. Whilst a range of single-objective benchmark problems have been proposed for MDO, there exists only a limited selection of multi-objective benchmarks, with only one of these problems being scalable in the number of disciplines. In this paper, we propose a new multi-objective MDO test suite, based on the popular ZDT bi-objective benchmark problems, which is scalable in the number of disciplines and design variables. Dependencies between disciplines can be defined directly in the problem formulation, enabling a diverse set of multidisciplinary topologies to be constructed that can resemble more realistic MDO problems. The new problems are solved using a multidisciplinary feasible architecture which combines a conventional multi-objective optimizer (NSGA-II) with a Newton-based multidisciplinary analysis solver. Empirical findings show that it is possible to solve the proposed ZDT-MDO problems but that multimodal problem landscapes can pose a significant challenge to the optimizer. The proposed test suite can help stimulate more research into the neglected but important topic of multi-objective multidisciplinary optimization.
Many-objective programs have often been reported in the literature as mathematical optimization tools to model and solve decision-making problems in many areas of human activity. However, in the presence of many objectives, the solution of the overall problem in its entirety may be challenging or even impossible, and decomposition of the problem into subproblems with a smaller number of criteria becomes appealing provided solving the subproblems can be coordinated and related to solving the initial problem. Based on an overview of selected decomposition and coordination methods, we put forward a list of steps that may be followed with certain desirable properties for a decomposition–coordination technique to effectively support the decision-maker and allow for an interactive tradeoff analysis. We support this process with theoretical results pertaining to a quasi-separable many-objective program and illustrate this process on an engineering design example problem.
Secondary power system based on gas turbine engine is usually used for starting the aircraft main engine at ground and finds similar application in air during the emergency. The secondary power system comprises of an auxiliary power unit (APU) and an air starter. APU provides stand-alone power to the main engine subsystems and air to the air starter for starting main engine. The main rotor system of APU is to be designed for safe operation during the ground critical speed crossing and maneuver loading. The weight of such a rotor system is an important design criterion. In the current work, multi-objective optimization using hybrid genetic algorithm (HGA) is employed for simultaneously minimizing the response at critical speed, the response during maneuvering, and shaft weight of the APU rotor system used in secondary power system of an aircraft. The shaft considered is a stepped one and is modeled using Rayleigh beam elements with three disks. Effects due to gyroscopic effect are also included in the analysis. A Pareto front is generated, and subsequently, an optimal solution is selected for the problem. The comparison with typical two-objective optimization result, having response at critical speed and shaft weight as objectives and response during maneuvering as constraint, has shown that the novel three-objective optimization has provided the designer with improved design choices for the selection of final compromise solution.
The novelty of this study is the use of a multi-objective evolutionary algorithm for the automatic detection of hydrodynamic turbulent boundaries overlying coral reefs. The procedure is implemented using sequences of Flock-1 satellite data acquired in the Red Sea. The study demonstrates that implementing Pareto-optimal solutions allows for the generation of accurate coral reef-water interface patterns. This is validated by a Pareto-optimal front and the receiver-operating characteristic (ROC) curve. The Pareto-optimal front indicates a significant relationship between hydrodynamic turbulent boundaries, macroalgae, and coral reefs. The ROC curves confirm the finding of the Pareto-optimal front that hydrodynamic turbulent boundary layers and macroalgae are caused by coral reefs. The performance accuracy is identified with an under-curve area of 90%. In conclusion, the multi-objective evolutionary algorithm has the applicability for the automatic detection of hydrodynamic turbulent boundary layers to coral reef studies.
Microelectromechanical systems (MEMS) are a highly multidisciplinary field and this has large implications on their applications and design. Designers are often faced with the task of balancing the modelling, simulation and optimisation that each discipline brings in order to bring about a complete whole system. In order to aid designers, strategies for navigating this multidisciplinary environment are essential, particularly when it comes to automating design synthesis and optimisation. This paper outlines a new multi-objective and multidisciplinary strategy for the application of engineering design problems. It employs a population-based evolutionary approach that looks to overcome the limitations of past work by using a non-hierarchical architecture that allows for interaction across all disciplines during optimisation. Two case studies are presented, the first focusing on a common speed reducer design problem found throughout the literature used to validate the methodology and a more complex example of design optimisation, that of a MEMS bandpass filter. Results show good agreement in terms of performance with past multi-objective multidisciplinary design optimisation methods with respect to the first speed reducer case study, and improved performance for the design of the MEMS bandpass filter case study.
The successful application of multiobjective optimization to engineering problems has motivated studies of more complex systems involving multiple subsystems and design disciplines, each with multiple design criteria. Complex system design requires participation of different teams that are highly specialized within each discipline and subsystem. Such a high differentiation results in limited sharing of information among the design teams. The mathematical modeling and the solution algorithm proposed in this paper address the issue of coordinating multiple design problems that negotiate according to conflicting criteria. The design of the layout of hybrid vehicles is formulated as a bilevel decomposed problem including a vehicle level and a battery level in concert with the specialization of the respective design teams required at each level. An iterative algorithm, the Multiobjective Decomposition Algorithm (MODA) is proposed, whose generated sequences are shown to converge to efficient designs for the overall design problem under certain conditions examined in the context of the block coordinate descent method and the method of multipliers. MODA applied to the hybrid electric design problem captures the bilevel tradeoffs originating by the conflicting objectives at the vehicle and battery levels.
The design of a complex engineering system typically involves tradeoffs among multiple design criteria or disciplinary performance to achieve the optimal design. The design process is usually an iterative procedure with individual discipline sub-systems designed concurrently to meet target values assigned from the system level. One of the most challenging issues is the large number of iterations in this design process, especially when uncertainty is taken into account. To improve the design concurrency while maintaining preferred tradeoffs at the system level, a new method is developed that identifies proper targets based on disciplinary design capability information while optimizing the design goal at the system level. The design capability of a discipline or criterion is represented by the achievable area bounded by its Pareto frontier. Using target values obtained from this method using Pareto information, the number of design iterations can be reduced in both deterministic and probabilistic design scenarios compared to existing approaches, such as Analytical Target Cascading (ATC). To demonstrate applications and benefits of the developed method this approach is applied to the design of a two-bar truss structure.
NASA’s future space exploration systems will include a highly complex Integrated Systems Health Management (ISHM) capability, which can detect, predict, isolate and respond to system and component failures in order to improve safety and maintainability. An ISHM system, as a whole, consists of several subsystems that monitor different components of a space mission. Due to the complex and multidisciplinary nature of designing ISHM, there seems to be a lack of formal methodologies to design an optimal (or near-optimal) ISHM for a given system of systems. In this research, we propose a new methodology to design and optimize ISHM as a distributed system with multiple interacting disciplines as well as multiple conflicting design objectives (i.e. Figures Of Merit or FOMs). This specialized multidisciplinary design approach can be used to optimize the effectiveness of ISHM systems for future NASA missions. We assume a hierarchical design protocol, where each subsystem communicates with other subsystems only in a top-down tree structure. At the top level, the overall performance of the mission consists of system-level variables, parameters, objectives, and constraints that are shared throughout the system and by all subsystems. Each subsystem will then comprise of these shared values in addition to those values that are specific to subsystems. As a specific case study, we take the example of designing an ISHM capability for X-34 reusable launch vehicle in two levels. The proposed approach, referred to as ISHM Multidisciplinary and Multiobjective System Analysis & Optimization (or ISHM MMSA&O), has a hierarchical structure to pass up or down shared values between the two levels with system-level and subsystem-level optimization routines.
This paper examines the use of graph based evolutionary algorithms (GBEAs) to find multiple acceptable solutions for heat transfer in engineering systems during the optimization process. GBEAs are a type of evolutionary algorithm (EA) in which a topology, or geography, is imposed on an evolving population of solutions. The rates at which solutions can spread within the population are controlled by the choice of topology. As in nature geography can be used to develop and sustain diversity within the solution population. Altering the choice of graph can create a more or less diverse population of potential solutions. The choice of graph can also affect the convergence rate for the EA and the number of mating events required for convergence. The engineering system examined in this paper is a biomass fueled cookstove used in developing nations for household cooking. In this cookstove wood is combusted in a small combustion chamber and the resulting hot gases are utilized to heat the stove’s cooking surface. The spatial temperature profile of the cooking surface is determined by a series of baffles that direct the flow of hot gases. The optimization goal is to find baffle configurations that provide an even temperature distribution on the cooking surface. Often in engineering, the goal of optimization is not to find the single optimum solution but rather to identify a number of good solutions that can be used as a starting point for detailed engineering design. Because of this a key aspect of evolutionary optimization is the diversity of the solutions found. The key conclusion in this paper is that GBEA’s can be used to create multiple good solutions needed to support engineering design.
Sensitivity analysis has received significant attention in engineering design. While sensitivity analysis methods can be global, taking into account all variations, or local, taking into account small variations, they generally identify which uncertain parameters are most important and to what extent their effect might be on design performance. The extant methods do not, in general, tackle the question of which ranges of parameter uncertainty are most important or how to best allocate investments to partial uncertainty reduction in parameters under a limited budget. More specifically, no previous approach has been reported that can handle single-disciplinary multi-output global sensitivity analysis for both a single design and multiple designs under interval uncertainty. Two new global uncertainty metrics, i.e., radius of output sensitivity region and multi-output entropy performance, are presented. With these metrics, a multi-objective optimization model is developed and solved to obtain fractional levels of parameter uncertainty reduction that provide the greatest payoff in system performance for the least amount of “investment”. Two case studies of varying difficulty are presented to demonstrate the applicability of the proposed approach.
Multidisciplinary optimization (MDO) has proved to be a useful tool for engineering design problems. Multiobjective optimization
has been introduced to strengthen MDO techniques and deal with non-comparable and conflicting design objectives. A large majority
of papers on multiobjective MDO have been applied in nature. This paper develops theory of multiobjective MDO and examines
relationships between efficient solutions of a quasi-separable multiobjective multidisciplinary optimization problem and efficient
solutions of its separable counterpart. Equivalence of the original and separable problems in the context of the Kuhn-Tucker
constraint qualification and efficiency conditions are proved. Two decomposition approaches are proposed and offer a possibility
of finding efficient solutions of the original problem by only finding efficient solutions of the subproblems. The presented
results are related to algorithms published in the engineering literature on multiobjective MDO.
For many products, the design process is a complex system involving the interaction of many distributed design activities that need to be carefully coordinated. This research develops a new tool, called a Bayesian network classifier, to improve one specific aspect of this challenge: quantitatively capturing a consensus of which designs are feasible options for meeting system-wide engineering requirements. Classifiers enable designers to independently develop and share maps of the feasible regions of their design space, enabling set-based collaborative design. The method is set-based in that resources are used to thoroughly understand design tradeoffs before commitment is made to a final design. The method is collaborative because the maps are coordinated between design teams to represent the mutually feasible design space of all stake-holders. The benefits are a more thorough understanding of the system-wide design problem across team boundaries as well as knowledge capture for future re-use, potentially leading to faster product development and higher quality products.
The increasing complexity of engineering systems has sparked rising interest in multidisciplinary optimization (MDO). This paper surveys recent publications in the field of aerospace, in which the interest in MDO has been particularly intense. The primary c hallenges in MDO are computational expense and organizational complexity. Accordingly, this survey focuses on various methods used by different researchers to address these challenges. The survey is organized by a breakdown of MDO into its conceptual components, reflected in sections on mathematical modelling, approximation concepts, optimization procedures, system sensitivity, and human interface. Because the authors' primary area of expertise is in the structures discipline, the majority of the references focus on the interaction of this discipline with others. In particular, two sections at the end of this review focus on two interactions that have recently been pursued with vigour: the simultaneous optimization of structures and aerodynamics and the simultaneous optimization of structures with active control.
A standard technique for generating the Pareto set in multicriteria optimization problems is to minimize (convex) weighted sums of the different objectives for various different settings of the weights. However, it is well-known that this method succeeds in getting points from all parts of the Pareto set only when the Pareto curve is convex. This article provides a geometrical argument as to why this is the case.Secondly, it is a frequent observation that even for convex Pareto curves, an evenly distributed set of weights fails to produce an even distribution of points from all parts of the Pareto set. This article aims to identify the mechanism behind this observation. Roughly, the weight is related to the slope of the Pareto curve in the objective space in a way such that an even spread of Pareto points actually corresponds to often very uneven distributions of weights. Several examples are provided showing assumed shapes of Pareto curves and the distribution of weights corresponding to an even spread of points on those Pareto curves.
This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem, aimed at capturing the tradeoff among the various conflicting objectives. It can be rigorously proved that this method is completely independent of the relative scales of the functions and is quite successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of 'weights', a property which the popular method of linear combinations lacks. Further, this method can be easily extended in the case of more than two objectives while retaining the computational efficiency of continuation-type algorithms, which is an improvement over homotopy techniques for tracing the tradeoff curve.
Many, if not most, optimization problems have multiple objectives.
Historically, multiple objectives have been combined ad hoc to form a
scalar objective function, usually through a linear combination
(weighted sum) of the multiple attributes, or by turning objectives into
constraints. The genetic algorithm (GA), however, is readily modified to
deal with multiple objectives by incorporating the concept of Pareto
domination in its selection operator, and applying a niching pressure to
spread its population out along the Pareto optimal tradeoff surface. We
introduce the Niched Pareto GA as an algorithm for finding the Pareto
optimal set. We demonstrate its ability to find and maintain a diverse
“Pareto optimal population” on two artificial problems and
an open problem in hydrosystems
This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem. Such points collectively capture the trade-off among the various conflicting objectives. It is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of parameters, a property which the popular method of minimizing weighted combinations of objective functions lacks. Further, this method can handle more than two objectives while retaining the computational efficiency of continuation-type algorithms. This is an improvement over continuation techniques for tracing the trade-off curve since continuation strategies cannot easily be extended to handle more than two objectives. This research was partially supported by the Dept. of Energy, DOE Grant DE-FG03-95ER25257, the Air Force, Air Force Grant F49620-95-1-021...
List of Figures. List of Tables. Preface. Foreword. 1. Basic Concepts. 2. Evolutionary Algorithm MOP Approaches. 3. MOEA Test Suites. 4. MOEA Testing and Analysis. 5. MOEA Theory and Issues. 3. MOEA Theoretical Issues. 6. Applications. 7. MOEA Parallelization. 8. Multi-Criteria Decision Making. 9. Special Topics. 10. Epilog. Appendix A: MOEA Classification and Technique Analysis. Appendix B: MOPs in the Literature. Appendix C: Ptrue & PFtrue for Selected Numeric MOPs. Appendix D: Ptrue & PFtrue for Side-Constrained MOPs. Appendix E: MOEA Software Availability. Appendix F: MOEA-Related Information. Index. References.
Limitations of the crude Monte-Carlo method for synthesis problems are described. A new method for random checking of the admissible solutions is proposed. Advantages and peculiarities of the last method are described. Also application of dynamic programming for optimal synthesis explained. The methods are illustrated by examples. The results obtained by different methods are confronted.ZusammenfassungProbleme der optimalen Synthese, gelöst mittels nichtlinearer Programmierung und statistischer Methoden: J. Golinski Die Grenzen der Monte-Carlo-Methode für optimale Getribesynthese werden auseinandergesetzt. Es wird eine neue Untersuchungsmethode vorgeschlagen, deren Eigenschaften und Vorzüge beschrieben werden. Auch die Anwendung der dynamischen Programmierung zur Getriebesynthese wird behandelt. Alle Methoden werden durch Beispiele illustriert, und die mittels verschiedener Methoden gewonnenen Ergebnisse werden verglichen.РефератПpoблeмы oптимaлянoгo cинтeэa, peшeнныe пpи пoмoши нeлинeйнoгo пpoгpaммиpoвaнния и выbopoчныч мeтoдoв: Д-p Я. Гoлинcкиь. в paбoтe пoкaэaнo oгoaничeeниe иpцбшoкeннoгo мeтoдa мoнтe Кapлo для пpoблeм cинтeea. Пpeдлaгaeтcя нoбый мeтoд выбopoчнoгo кoнтpoля дoпycтимыч peшeний. Oпиcaн aлгopитм и пpивeдeны peэyлятaты иccлeдoвaний влияния paэличныч пapaмeтpoв пpocecca. Oбьяcняeтcя пpoг¶ммиpoвaниe для oптимaльнoгo cинтeэa. Meтoды иллюcтpиpyютcя пимepaми. Coпocтaвляютcя тaкзe peэyльтaты эaдaч paэличными мeтoдaми.
In this paper, several new set quality metrics are introduced that can be used to evaluate dir ''goodness'' of all observed Pareto solution? set. These metrics, which are formulated in closed-form and geometrically illustrated, include hyperarea difference, Pareto spread accuracy of an observed Pareto frontier; number of distinct choices and cluster : The metrics should enable a designer to either monitor the quality of an observed Pareto solution set as obtained by a multiobjective optimization method, or compare the quality of observed Pareto solution sets as reported by. different multiobjective optimization methods. A A vibrating platform example is used to demonstrate the calculation of these metrics for an observed Pareto solution set.
A new method is presented to solve multi-objective multidisciplinary optimization (M-MDO) problems. This M-MDO method is applicable to multi-objective optimization problems that can be decomposed hierarchically into multi-objective subproblems and whose objective functions are either separable or additively separable. In the decomposition, the subproblems may have both common and unique objectives. The method uses a multiobjective genetic algorithm (MOGA) to optimize the multi-objective subproblems; hence, it is referred to as a multi-objective multidisciplinary genetic algorithm (M-MGA). It is shown that for any Pareto point of the original (single-level) problem, M-MGA generates at least one point that is noninferior with respect to that Pareto point. Also a comparison is shown between the computational complexity of M-MGA and a single-level MOGA in terms of number of functions calls. The M-MGA is demonstrated by two engineering examples: the design of a speed reducer and the design of a payload for an undersea autonomous vehicle. In both examples, the generated solutions are similar to solutions generated by solving the examples as single-level problems. M-MGA produces relatively the same solutions from one M-MGA run to another.
This investigation focuses on the development of modifications to the Collaborative Optimization (CO) approach to multidisciplinary systems design, that will provide solution capabilities for multiobjective problems. The primary goal of this paper is to provide a comprehensive overview and development of mathematically rigorous optimization strategies for MultiObjective Collaborative Optimization (MOCO). Collaborative Optimization strategies provide design optimization capabilities to discipline designers within a multidisciplinary design environment. To date these CO strategies have primarily been applied to system design problems which have a single objective function. Recent investigations involving multidisciplinary design simulators have reported success in applying CO to multiobjective system design problems. In this research three MultiObjective Collaborative Optimization (MOCO) strategies are developed, reviewed and implemented in a comparative study. The goal of this effort is to provide an in depth comparison of different MOCO strategies available to system designers. Each of the three strategies makes use of parameter sensitivities within multilevel solution strategies. in implementation studies, each of the three MOCO strategies is effective in solving a multiobjective multidisciplinary systems design problem. Results indicate that these MOCO strategies require an accurate estimation of parameter sensitivities for successful implementation. In each of the three MOCO strategies these parameter sensitivities are obtained using post-optimality analysis techniques.
In trying to solve multiobjective optimization problems, many traditional methods scalarize the objective vector into a single objective. In those cases, the obtained solution is highly sensitive to the weight vector used in the scalarization process and demands that the user have knowledge about the underlying problem. Moreover, in solving multiobjective problems, designers may be interested in a set of Pareto-optimal points, instead of a single point. Since genetic algorithms (GAs) work with a population of points, it seems natural to use GAs in multiobjective optimization problems to capture a number of solutions simultaneously. Although a vector evaluated GA (VEGA) has been implemented by Schaffer and has been tried to solve a number of multiobjective problems, the algorithm seems to have bias toward some regions. In this paper, we investigate Goldberg's notion of nondominated sorting in GAs along with a niche and speciation method to find multiple Pareto-optimal points simultaneously. The proof-of-principle results obtained on three problems used by Schaffer and others suggest that the proposed method can be extended to higher dimensional and more difficult multiobjective problems. A number of suggestions for extension and application of the algorithm are also discussed.
Preface. Acknowledgements. Notation and Symbols. Part I: Terminology and Theory. 1. Introduction. 2. Concepts. 3. Theoretical Background. Part II: Methods. 1. Introduction. 2. No-Preference Methods. 3. A Posteriori Methods. 4. A Priori Methods. 5. Interactive Methods. Part III: Related Issues. 1. Comparing Methods. 2. Software. 3. Graphical Illustration. 4. Future Directions. 5. Epilogue. References. Index.
This work proposes a quantitative, non-parametric interpre- tation of statistical performance of stochastic multiobjective optimizers, including, but not limited to, genetic algorithms. It is shown that, accord- ing to this interpretation, typical performance can be defined in terms analogous to the notion of median for ordinal data, as can other measures analogous to other quantiles. Non-parametric statistical test procedures are then shown to be useful in deciding the relative performance of different multiobjective optimizers on a given problem. Illustrative experimental results are provided to support the discussion.
Bell System Technical Journal, also pp. 623-656 (October)
Four constraint handling improvements for Multi-Objective Genetic Algorithms (MOGA) are proposed. These improvements are made
in the fitness assignment stage of a MOGA and are all based upon a “Constraint-First-Objective-Next" model. Two multi-objective
design optimization examples, i.e. a speed reducer design and the design of a fleet of ships, are used to demonstrate the
improvements. For both examples, it is shown that the proposed constraint handling techniques significantly improve the performance
of a baseline MOGA.
This paper presents some improvements to Multi-Objective Genetic Algorithms (MOGAs). MOGA modifies certain operators within the GA itself to produce a multiobjective optimization technique. The improvements are made to overcome some of the shortcomings in niche formation, stopping criteria and interaction with a design decision-maker. The technique involves filtering, mating restrictions, the idea of objective constraints, and detecting Pareto solutions in the non-convex region of the Pareto set. A step-by-step procedure for an improved MOGA has been developed and demonstrated via two multiobjective engineering design examples: (i) two-bar truss design, and (ii) vibrating platform design. The two-bar truss example has continuous variables while the vibrating platform example has mixed-discrete (combinatorial) variables. Both examples are solved by MOGA with and without the improvements. It is shown that MOGA with the improvements performs better for both examples in terms of the number of function evaluations.
A recently developed active set algorithm for tracing parametrized optima is adapted to multi-objective optimization. The algorithm traces a path of Kuhn-Tucker points using homotopy curve tracking techniques, and is based on identifying and maintaining the set of active constraints. Second order necessary optimality conditions are used to determine nonoptimal stationary points on the path. In the bi-objective optimization case the algorithm is used to trace the curve of efficient solution (Pareto optima). As an example, the algorithm is applied to the simultaneous minimization of the weight and control force of a ten-bar truss with two collocated sensors and actuators, with some interesting results.
We present a new method for solving a multi-level multi-objective optimization problem that is hierarchically decomposed into several sub-problems. The method preserves diversity of Pareto solutions by maximizing an entropy metric, a quantitative measure of distribution quality of a set of solutions. The main idea behind the method is to optimize the sub-problems independently using a Multi-Objective Genetic Algorithm (MOGA) while systematically using the entropy values of intermediate solutions to guide the optimization of sub-problems to the overall Pareto solutions. As a demonstration, we applied the multi-level MOGA to a mechanical design example: the design of a speed reducer. We also solved the example in its equivalent single-level form by a MOGA. The results show that our proposed multi-level multi-objective optimization method obtains more Pareto solutions with a better diversity compared to those obtained by the single-level MOGA.
A unified overview is given of problem formulation approaches for the optimization of multidisciplinary coupled systems. The overview includes six fundamental approaches upon which a large number of variations may be made. Consistent approach names and a compact approach notation are given. The approaches are formulated to apply to general nonhierarchic systems. The approaches are compared both from a computational viewpoint and a managerial viewpoint. Opportunities for parallelism of both computation and manpower resources are discussed. Recommendations regarding the need for future research are advanced.
In this paper, we study the problem features that may cause a multi-objective genetic algorithm (GA) difficulty in converging to the true Pareto-optimal front. Identification of such features helps us develop difficult test problems for multi-objective optimization. Multi-objective test problems are constructed from single-objective optimization problems, thereby allowing known difficult features of single-objective problems (such as multi-modality, isolation, or deception) to be directly transferred to the corresponding multi-objective problem. In addition, test problems having features specific to multi-objective optimization are also constructed. More importantly, these difficult test problems will enable researchers to test their algorithms for specific aspects of multi-objective optimization.
Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mid-eighties in an attempt to stochastically solve problems of this generic class. During the past decade, a variety, of multiobjective EA (MOEA) techniques have been proposed and applied to many scientific and engineering applications. Our discussion's intent is to rigorously define multiobjective optimization problems and certain related concepts, present an MOEA classification scheme, and evaluate the variety of contemporary MOEAs. Current MOEA theoretical developments are evaluated; specific topics addressed include fitness functions, Pareto ranking, niching, fitness sharing, mating restriction, and secondary populations. Since the development and application of MOEAs is a dynamic and rapidly growing activity, we focus on key analytical insights based upon critical MOEA evaluation of current research and applications. Recommended MOEA designs are presented, along with conclusions and recommendations for future work.
An entropy-based metric is presented to assess the diversity of
solutions in a multi-objective optimization technique. This metric
quantifies the 'goodness' of a solution set in terms of its distribution
quality over the Pareto-optimal frontier. As a demonstration via a
three-objective test example, the entropy metric is used as a means of
comparing two multi-objective genetic algorithms
The paper reviews several genetic algorithm (GA) approaches to
multi objective optimization problems (MOPs). The keynote point of GAs
to MOPs is designing efficient selection/reproduction operators so that
a variety of Pareto optimal solutions are generated. From this
viewpoint, the paper reviews several devices proposed for multi
objective optimization by GAs such as the parallel selection method, the
Pareto based ranking, and the fitness sharing. Characteristics of these
approaches have been confirmed through computational experiments with a
simple example. Moreover, two practical applications of the GA
approaches to MOPs are introduced briefly
This paper describes a non-generational genetic algorithm for multiobjective optimization. The fitness of each individual in the population is calculated incrementally based on the degree in which it is dominated in the Pareto sense, or close to other individuals. The closeness of individuals is measured using a sharing function. The performance of the algorithm presented is compared to previous efforts on three multiobjective optimization problems of growing difficulty. The behavior of each algorithm is analyzed with regard to the visited search space, the quality of the final population attained, and the percentage of non-dominated individuals in the population through time. According to all these performance measures, the algorithm presented clearly outperforms previous efforts based on generational genetic algorithms. 1 INTRODUCTION It can be said that true optimization must be multiobjective; real problems usually involve more than one objective function to b...
The application of evolutionary algorithms (EAs) in multiobjective optimization is currently receiving growing interest from researchers with various backgrounds. Most research in this area has understandably concentrated on the selection stage of EAs, due to the need to integrate vectorial performance measures with the inherently scalar way in which EAs reward individual performance, that is, number of offspring.
In this review, current multiobjective evolutionary approaches are discussed, ranging from the conventional analytical aggregation of the different objectives into a single function to a number of population-based approaches and the more recent ranking schemes based on the definition of Pareto optimality. The sensitivity of different methods to objective scaling and/or possible concavities in the trade-off surface is considered, and related to the (static) fitness landscapes such methods induce on the search space. From the discussion, directions for future research in multiobjective fitness assignment and search strategies are identified, including the incorporation of decision making in the selection procedure, fitness sharing, and adaptive representations.
Multi-Objective Optimization Using Evolutionary Algorithms Diversity assessment of pareto optimal solutions: an entropy approach
- K Deb
- A Farhang-Mehr
- S Azarm
Deb, K. (2001). Multi-Objective Optimization Using Evolutionary Algorithms. Chichester, UK: John Wiley and Sons. Farhang-Mehr, A., Azarm, S. (2002). Diversity assessment of pareto optimal solutions: an entropy approach. In: CD-ROM Proc of IEEE World Congress on Computational Intelligence. Honolulu, Hawaii, May 12–17.
A niched pareto genetic algorithm for multi-objective optimization
- J Horn
- N Nafploitis
- D E Goldberg