Understanding knee points in bicriteria problems and their implications as preferred solution principles
TL;DR: In this article, an attempt is made to improve understanding of a knee point and investigate the properties of a bicriteria problem that may exhibit a knee on its Pareto-optimal front.
Abstract: A knee point is almost always a preferred trade-off solution, if it exists in a bicriteria optimization problem. In this article, an attempt is made to improve understanding of a knee point and investigate the properties of a bicriteria problem that may exhibit a knee on its Pareto-optimal front. Past studies are reviewed and a couple of new definitions are suggested. Additionally, a knee region is defined for problems in which, instead of one, a set of knee-like solutions exists. Edge-knee solutions, which behave like knee solutions but lie near one of the extremes on the Pareto-optimal front, are also introduced. It is interesting that in many problem-solving tasks, despite the existence of a number of solution methodologies, only one or a few of them are commonly used. Here, it is argued that often such common solution principles are knee solutions to a bicriteria problem formed with two conflicting goals of the underlying problem-solving task. The argument is illustrated on a number of tasks, such as ...
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TL;DR: A knee point-driven EA to solve MaOPs by showing that knee points are naturally most preferred among nondominated solutions if no explicit user preferences are given and enhancing the convergence performance in many-objective optimization.
Abstract: Evolutionary algorithms (EAs) have shown to be promising in solving many-objective optimization problems (MaOPs), where the performance of these algorithms heavily depends on whether solutions that can accelerate convergence toward the Pareto front and maintaining a high degree of diversity will be selected from a set of nondominated solutions. In this paper, we propose a knee point-driven EA to solve MaOPs. Our basic idea is that knee points are naturally most preferred among nondominated solutions if no explicit user preferences are given. A bias toward the knee points in the nondominated solutions in the current population is shown to be an approximation of a bias toward a large hypervolume, thereby enhancing the convergence performance in many-objective optimization. In addition, as at most one solution will be identified as a knee point inside the neighborhood of each solution in the nondominated front, no additional diversity maintenance mechanisms need to be introduced in the proposed algorithm, considerably reducing the computational complexity compared to many existing multiobjective EAs for many-objective optimization. Experimental results on 16 test problems demonstrate the competitiveness of the proposed algorithm in terms of both solution quality and computational efficiency.
624 citations
Additional excerpts
...NSGA-II [6] and SPEA2 [7] are two representative Pareto-based MOEAs, which have been shown to be very effective in solving MOPs having two or three objectives....
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01 Jul 2013
TL;DR: The evaluation of the proposed procedure against extreme conditions having a dense (as high as 91 %) placement of obstacles indicates its robustness and efficiency in solving complex path planning problems.
Abstract: A multi-objective vehicle path planning method has been proposed to optimize path length, path safety, and path smoothness using the elitist non-dominated sorting genetic algorithm--a well-known soft computing approach. Four different path representation schemes that begin their coding from the start point and move one grid at a time towards the destination point are proposed. Minimization of traveled distance and maximization of path safety are considered as objectives of this study while path smoothness is considered as a secondary objective. This study makes an extensive analysis of a number of issues related to the optimization of path planning task-handling of constraints associated with the problem, identifying an efficient path representation scheme, handling single versus multiple objectives, and evaluating the proposed algorithm on large-sized grids and having a dense set of obstacles. The study also compares the performance of the proposed algorithm with an existing GA-based approach. The evaluation of the proposed procedure against extreme conditions having a dense (as high as 91 %) placement of obstacles indicates its robustness and efficiency in solving complex path planning problems. The paper demonstrates the flexibility of evolutionary computing approaches in dealing with large-scale and multi-objective optimization problems.
161 citations
TL;DR: This work combines a recently developed method for evolving virtual organisms with an information-theoretic metric of morphological complexity in order to investigate how the complexity of morphologies, which are evolved for locomotion, varies across different environments.
Abstract: Whether, when, how, and why increased complexity evolves in biological populations is a longstanding open question. In this work we combine a recently developed method for evolving virtual organisms with an information-theoretic metric of morphological complexity in order to investigate how the complexity of morphologies, which are evolved for locomotion, varies across different environments. We first demonstrate that selection for locomotion results in the evolution of organisms with morphologies that increase in complexity over evolutionary time beyond what would be expected due to random chance. This provides evidence that the increase in complexity observed is a result of a driven rather than a passive trend. In subsequent experiments we demonstrate that morphologies having greater complexity evolve in complex environments, when compared to a simple environment when a cost of complexity is imposed. This suggests that in some niches, evolution may act to complexify the body plans of organisms while in other niches selection favors simpler body plans.
131 citations
Cites background from "Understanding knee points in bicrit..."
...Another possibility would be to find the knee point [71] on each Pareto front....
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TL;DR: A new soft-thresholding evolutionary multiobjective algorithm (StEMO) is presented, which uses a soft-Thresholding technique to incorporate two additional heuristics: one with greater chance to increase speed of convergence toward the PF, and another with higher probability to improve the spread of solutions along thePF, enabling an optimal solution to be found in the knee region.
Abstract: This paper addresses the problem of finding sparse solutions to linear systems. Although this problem involves two competing cost function terms (measurement error and a sparsity-inducing term), previous approaches combine these into a single cost term and solve the problem using conventional numerical optimization methods. In contrast, the main contribution of this paper is to use a multiobjective approach. The paper begins by investigating the sparse reconstruction problem, and presents data to show that knee regions do exist on the Pareto front (PF) for this problem and that optimal solutions can be found in these knee regions. Another contribution of the paper, a new soft-thresholding evolutionary multiobjective algorithm (StEMO), is then presented, which uses a soft-thresholding technique to incorporate two additional heuristics: one with greater chance to increase speed of convergence toward the PF, and another with higher probability to improve the spread of solutions along the PF, enabling an optimal solution to be found in the knee region. Experiments are presented, which show that StEMO significantly outperforms five other well known techniques that are commonly used for sparse reconstruction. Practical applications are also demonstrated to fundamental problems of recovering signals and images from noisy data.
125 citations
TL;DR: In this article, the authors used the simplified super folding element theory to estimate the energy dissipation of angle elements and developed theoretical expressions of the mean crushing force for three types of tubes under dynamic loading.
Abstract: The triangular tubes with multi-cell were first studied on the aspects of theoretical prediction and crashworthiness optimization design under the impact loading. The tubes׳ profiles were divided into 2-, 3-, T-shapes, 4-, and 6-panel angle elements. The Simplified Super Folding Element theory was utilized to estimate the energy dissipation of angle elements. Based on the estimation, theoretical expressions of the mean crushing force were developed for three types of tubes under dynamic loading. When taking the inertia effects into account, the dynamic enhancement coefficient was also considered. In the process of multiobjective crashworthiness optimization, Deb and Gupta method was utilized to find out the knee points from the Pareto solutions space. Finally, the theoretical prediction showed an excellent coincidence with the numerical optimal results, and also validated the efficiency of the crashworthiness optimization design method based on surrogate models.
118 citations
References
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Book•
01 Jan 1990TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.
Abstract: From the Publisher:
The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Like the first edition,this text can also be used for self-study by technical professionals since it discusses engineering issues in algorithm design as well as the mathematical aspects.
In its new edition,Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity,and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage.
As in the classic first edition,this new edition of Introduction to Algorithms presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers. Further,the algorithms are presented in pseudocode to make the book easily accessible to students from all programming language backgrounds.
Each chapter presents an algorithm,a design technique,an application area,or a related topic. The chapters are not dependent on one another,so the instructor can organize his or her use of the book in the way that best suits the course's needs. Additionally,the new edition offers a 25% increase over the first edition in the number of problems,giving the book 155 problems and over 900 exercises thatreinforcethe concepts the students are learning.
21,651 citations
Book•
01 Jan 2001
TL;DR: This text provides an excellent introduction to the use of evolutionary algorithms in multi-objective optimization, allowing use as a graduate course text or for self-study.
Abstract: From the Publisher:
Evolutionary algorithms are relatively new, but very powerful techniques used to find solutions to many real-world search and optimization problems. Many of these problems have multiple objectives, which leads to the need to obtain a set of optimal solutions, known as effective solutions. It has been found that using evolutionary algorithms is a highly effective way of finding multiple effective solutions in a single simulation run. · Comprehensive coverage of this growing area of research · Carefully introduces each algorithm with examples and in-depth discussion · Includes many applications to real-world problems, including engineering design and scheduling · Includes discussion of advanced topics and future research · Features exercises and solutions, enabling use as a course text or for self-study · Accessible to those with limited knowledge of classical multi-objective optimization and evolutionary algorithms The integrated presentation of theory, algorithms and examples will benefit those working and researching in the areas of optimization, optimal design and evolutionary computing. This text provides an excellent introduction to the use of evolutionary algorithms in multi-objective optimization, allowing use as a graduate course text or for self-study.
12,134 citations
Book•
01 Mar 1981
TL;DR: In this paper, the authors present a classification of MADM methods by data type and propose a ranking method based on the degree of similarity of the MADM method to the original MADM algorithm.
Abstract: I. Introduction.- II. Multiple Attribute Decision Making - An Overview.- 2.1 Basics and Concepts.- 2.2 Classifications of MADM Methods.- 2.2.1 Classification by Information.- 2.2.2 Classification by Solution Aimed At.- 2.2.3 Classification by Data Type.- 2.3 Description of MADM Methods.- Method (1): DOMINANCE.- Method (2): MAXIMIN.- Method (3): MAXIMAX.- Method (4): CONJUNCTIVE METHOD.- Method (5): DISJUNCTIVE METHOD.- Method (6): LEXICOGRAPHIC METHOD.- Method (7): LEXICOGRAPHIC SEMIORDER METHOD.- Method (8): ELIMINATION BY ASPECTS (EBA).- Method (9): LINEAR ASSIGNMENT METHOD (LAM).- Method (10): SIMPLE ADDITIVE WEIGHTING METHOD (SAW).- Method (11): ELECTRE (Elimination et Choice Translating Reality).- Method (12): TOPSIS (Technique for Order Preference by Similarity to Ideal Solution).- Method (13): WEIGHTED PRODUCT METHOD.- Method (14): DISTANCE FROM TARGET METHOD.- III. Fuzzy Sets and their Operations.- 3.1 Introduction.- 3.2 Basics of Fuzzy Sets.- 3.2.1 Definition of a Fuzzy Set.- 3.2.2 Basic Concepts of Fuzzy Sets.- 3.2.2.1 Complement of a Fuzzy Set.- 3.2.2.2 Support of a Fuzzy Set.- 3.2.2.3 ?-cut of a Fuzzy Set.- 3.2.2.4 Convexity of a Fuzzy Set.- 3.2.2.5 Normality of a Fuzzy Set.- 3.2.2.6 Cardinality of a Fuzzy Set.- 3.2.2.7 The mth Power of a Fuzzy Set.- 3.3 Set-Theoretic Operations with Fuzzy Sets.- 3.3.1 No Compensation Operators.- 3.3.1.1 The Min Operator.- 3.3.2 Compensation-Min Operators.- 3.3.2.1 Algebraic Product.- 3.3.2.2 Bounded Product.- 3.3.2.3 Hamacher's Min Operator.- 3.3.2.4 Yager's Min Operator.- 3.3.2.5 Dubois and Prade's Min Operator.- 3.3.3 Full Compensation Operators.- 3.3.3.1 The Max Operator.- 3.3.4 Compensation-Max Operators.- 3.3.4.1 Algebraic Sum.- 3.3.4.2 Bounded Sum.- 3.3.4.3 Hamacher's Max Operator.- 3.3.4.4 Yager's Max Operator.- 3.3.4.5 Dubois and Prade's Max Operator.- 3.3.5 General Compensation Operators.- 3.3.5.1 Zimmermann and Zysno's ? Operator.- 3.3.6 Selecting Appropriate Operators.- 3.4 The Extension Principle and Fuzzy Arithmetics.- 3.4.1 The Extension Principle.- 3.4.2 Fuzzy Arithmetics.- 3.4.2.1 Fuzzy Number.- 3.4.2.2 Addition of Fuzzy Numbers.- 3.4.2.3 Subtraction of Fuzzy Numbers.- 3.4.2.4 Multiplication of Fuzzy Numbers.- 3.4.2.5 Division of Fuzzy Numbers.- 3.4.2.6 Fuzzy Max and Fuzzy Min.- 3.4.3 Special Fuzzy Numbers.- 3.4.3.1 L-R Fuzzy Number.- 3.4.3.2 Triangular (or Trapezoidal) Fuzzy Number.- 3.4.3.3 Proof of Formulas.- 3.4.3.3.1 The Image of Fuzzy Number N.- 3.4.3.3.2 The Inverse of Fuzzy Number N.- 3.4.3.3.3 Addition and Subtraction.- 3.4.3.3.4 Multiplication and Division.- 3.5 Conclusions.- IV. Fuzzy Ranking Methods.- 4.1 Introduction.- 4.2 Ranking Using Degree of Optimality.- 4.2.1 Baas and Kwakernaak's Approach.- 4.2.2 Watson et al.'s Approach.- 4.2.3 Baldwin and Guild's Approach.- 4.3 Ranking Using Hamming Distance.- 4.3.1 Yager's Approach.- 4.3.2 Kerre's Approach.- 4.3.3 Nakamura's Approach.- 4.3.4 Kolodziejczyk's Approach.- 4.4 Ranking Using ?-Cuts.- 4.4.1 Adamo's Approach.- 4.4.2 Buckley and Chanas' Approach.- 4.4.3 Mabuchi's Approach.- 4.5 Ranking Using Comparison Function.- 4.5.1 Dubois and Prade's Approach.- 4.5.2 Tsukamoto et al.'s Approach.- 4.5.3 Delgado et al.'s Approach.- 4.6 Ranking Using Fuzzy Mean and Spread.- 4.6.1 Lee and Li's Approach.- 4.7 Ranking Using Proportion to The Ideal.- 4.7.1 McCahone's Approach.- 4.8 Ranking Using Left and Right Scores.- 4.8.1 Jain's Approach.- 4.8.2 Chen's Approach.- 4.8.3 Chen and Hwang's Approach.- 4.9 Ranking with Centroid Index.- 4.9.1 Yager's Centroid Index.- 4.9.2 Murakami et al.'s Approach.- 4.10 Ranking Using Area Measurement.- 4.10.1 Yager's Approach.- 4.11 Linguistic Ranking Methods.- 4.11.1 Efstathiou and Tong's Approach.- 4.11.2 Tong and Bonissone's Approach.- V. Fuzzy Multiple Attribute Decision Making Methods.- 5.1 Introduction.- 5.2 Fuzzy Simple Additive Weighting Methods.- 5.2.1 Baas and Kwakernaak's Approach.- 5.2.2 Kwakernaak's Approach.- 5.2.3 Dubois and Prade's Approach.- 5.2.4 Cheng and McInnis's Approach.- 5.2.5 Bonissone's Approach.- 5.3 Analytic Hierarchical Process (AHP) Methods.- 5.3.1 Saaty's AHP Approach.- 5.3.2 Laarhoven and Pedrycz's Approach.- 5.3.3 Buckley's Approach.- 5.4 Fuzzy Conjunctive/Disjunctive Method.- 5.4.1 Dubois, Prade, and Testemale's Approach.- 5.5 Heuristic MAUF Approach.- 5.6 Negi's Approach.- 5.7 Fuzzy Outranking Methods.- 5.7.1 Roy's Approach.- 5.7.2 Siskos et al.'s Approach.- 5.7.3 Brans et al.'s Approach.- 5.7.4 Takeda's Approach.- 5.8 Maximin Methods.- 5.8.1 Gellman and Zadeh's Approach.- 5.8.2 Yager's Approach.- 5.9 A New Approach to Fuzzy MADM Problems.- 5.9.1 Converting Linguistic Terms to Fuzzy Numbers.- 5.9.2 Converting Fuzzy Numbers to Crisp Scores.- 5.9.3 The Algorithm.- VI. Concluding Remarks.- 6.1 MADM Problems and Fuzzy Sets.- 6.2 On Existing MADM Solution Methods.- 6.2.1 Classical Methods for MADM Problems.- 6.2.2 Fuzzy Methods for MADM Problems.- 6.2.2.1 Fuzzy Ranking Methods.- 6.2.2.2 Fuzzy MADM Methods.- 6.3 Critiques of the Existing Fuzzy Methods.- 6.3.1 Size of Problem.- 6.3.2 Fuzzy vs. Crisp Data.- 6.4 A New Approach to Fuzzy MADM Problem Solving.- 6.4.1 Semantic Modeling of Linguistic Terms.- 6.4.2 Fuzzy Scoring System.- 6.4.3 The Solution.- 6.4.4 The Advantages of the New Approach.- 6.5 Other Multiple Criteria Decision Making Methods.- 6.5.1 Multiple Objective Decision Making Methods.- 6.5.2 Methods of Group Decision Making under Multiple Criteria.- 6.5.2.1 Social Choice Theory.- 6.5.2.2 Experts Judgement/Group Participation.- 6.5.2.3 Game Theory.- 6.6 On Future Studies.- 6.6.1 Semantics of Linguistic Terms.- 6.6.2 Fuzzy Ranking Methods.- 6.6.3 Fuzzy MADM Methods.- 6.6.4 MADM Expert Decision Support Systems.- VII. Bibliography.
8,629 citations
Book•
26 Sep 2011
TL;DR: This paper is concerned with the development of methods for dealing with the role of symbols in the interpretation of semantics.
Abstract: 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.
4,976 citations
"Understanding knee points in bicrit..." refers background or methods in this paper
...Knee points are well-recognized by multi-objective optimization researchers (Das 1999, Miettinen 1999, Branke et al. 2004, Mattson et al. 2004, Deb and Sundar 2006, Rachmawati and *Corresponding author....
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...Theoretically, such problems give rise to a set of optimal solutions, known as Pareto-optimal solutions, which together constitute a trade-off front (Miettinen 1999, Deb 2001)....
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...The domination between two solutions is defined as follows (Miettinen 1999, Deb 2001)....
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...For a two-objective problem, this refers to identifying the points that are dominated (by Definition 2.1) with respect to the transformed objectives (Miettinen 1999, Deb 2001) given as follows: F1 = f1 + 1 α f2, F2 = 1 β f1 + f2....
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Book•
01 Jan 1979
TL;DR: On MADM Methods Classification.
Abstract: I. Introduction.- II. Basic Concepts and Foundations.- 1. Definitions.- 1.1 Terms for MCDM Environment.- 1.2 MCDM Solutions.- 2. Models for MADM.- 2.1 Noncompensatory Model.- 2.2 Compensatory Model.- 3. Transformation of Attributes.- 3.1 Quantification of Fuzzy Attributes.- 3.2 Normalization.- 4. Fuzzy Decision Rules.- 4.1 Definition of Fuzzy Set.- 4.2 Some Basic Operations of Fuzzy Sets.- 5. Methods for Assessing Weight.- 5.1 Eigenvector Method.- 5.2 Weighted Least Square Method.- 5.3 Entropy Method.- 5.4 Linmap.- III. Methods for Multiple Attribute Decision Making.- 1. Methods for No Preference Information Given.- 1.1.1 Dominance.- 1.1.2 Maximin.- 1.1.3 Maximax.- 2. Methods for Information on Attribute Given.- 2.1 Methods for Standard Level of Attribute Given.- 2.1.1 Conjunctive Method (Satisficing Method).- 2.1.2 Disjunctive Method.- 2.2 Methods for Ordinal Preference of Attribute Given.- 2.2.1 Lexicographic Method.- 2.2.2 Elimination By Aspects.- 2.2.3 Permutation Method.- 2.3 Methods for Cardinal Preference of Attribute Given.- 2.3.1 Linear Assignment Method.- 2.3.2 Simple Additive Weighting Method.- 2.3.3 Hierarchical Additive Weighting Method.- 2.3.4 ELECTRE Method.- 2.3.5 TOPSIS.- 2.4 Methods for Marginal Rate of Substitution of Attributes Given.- 2.4.1 Hierarchical Tradeoffs.- 3. Methods for Information on Alternative Given.- 3.1 Methods for Pairwise Preference Given.- 3.1.1 LINMAP.- 3.1.2 Interactive Simple Additive Weighting Method.- 3.2 Method for Pairwise Proximity Given.- 3.2.1 Multidimensional Scaling with Ideal Point.- IV. Applications.- 1. Commodity Selection.- 2. Facility Location (Siting) Selection.- 3. Personnel Selection.- 4. Project Selection.- 4.1 Environmental Planning.- 4.2 Land Use Planning.- 4.3 R & D Project.- 4.4 Water Resources Planning.- 4.5 Miscellaneous.- 5. Public Facility Selection.- V. Concluding Remarks.- On MADM Methods Classification.- On Applications of MADM.- On Multiple Objective Decision Making (MODM) Methods.- On Multiattribute Utility Theory (MAUT).- A Choice Rule for MADM Methods.- A Unified Approach to MADM.- On Future Study.- VI. Bibliography.- Books, Monographs, and Conference Proceedings.- Journal Articles, Technical Reports, and Theses.
2,380 citations
"Understanding knee points in bicrit..." refers background in this paper
...…cases, the front obtained by the Pareto-optimal solutions is such that the choice of a single preferred solution is not straightforward and a multi-criteria decision-making methodology is needed to choose a single preferred solution systematically (Hwang and Masud 1979, Chankong and Haimes 1983)....
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