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.

https://bi.snu.ac.kr/Info/EC/Multiobjective%20Optimization.pdf

Muiltiobjective optimization using nondominated sorting in genetic algorithms

A survey of current continuous nonlinear multi-objective optimization (MOO) concepts and methods is presented. It consolidates and relates seemingly different terminology and methods. The methods are divided into three major categories: methods with a priori articulation of preferences, methods with a posteriori articulation of preferences, and methods with no articulation of preferences. Genetic algorithms are surveyed as well. Commentary is provided on three fronts, concerning the advantages and pitfalls of individual methods, the different classes of methods, and the field of MOO as a whole. The Characteristics of the most significant methods are summarized. Conclusions are drawn that reflect often-neglected ideas and applicability to engineering problems. It is found that no single approach is superior. Rather, the selection of a specific method depends on the type of information that is provided in the problem, the user’s preferences, the solution requirements, and the availability of software.

Survey of multi-objective optimization methods for engineering

Having developed multiobjective optimization algorithms using evolutionary optimization methods and demonstrated their niche on various practical problems involving mostly two and three objectives, there is now a growing need for developing evolutionary multiobjective optimization (EMO) algorithms for handling many-objective (having four or more objectives) optimization problems. In this paper, we recognize a few recent efforts and discuss a number of viable directions for developing a potential EMO algorithm for solving many-objective optimization problems. Thereafter, we suggest a reference-point-based many-objective evolutionary algorithm following NSGA-II framework (we call it NSGA-III) that emphasizes population members that are nondominated, yet close to a set of supplied reference points. The proposed NSGA-III is applied to a number of many-objective test problems with three to 15 objectives and compared with two versions of a recently suggested EMO algorithm (MOEA/D). While each of the two MOEA/D methods works well on different classes of problems, the proposed NSGA-III is found to produce satisfactory results on all problems considered in this paper. This paper presents results on unconstrained problems, and the sequel paper considers constrained and other specialties in handling many-objective optimization problems.

/pdf/an-evolutionary-many-objective-optimization-algorithm-using-2bj8j4euce.pdf

An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints

A unified view of metaheuristics This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling. It presents the main design questions for all families of metaheuristics and clearly illustrates how to implement the algorithms under a software framework to reuse both the design and code. Throughout the book, the key search components of metaheuristics are considered as a toolbox for: Designing efficient metaheuristics (e.g. local search, tabu search, simulated annealing, evolutionary algorithms, particle swarm optimization, scatter search, ant colonies, bee colonies, artificial immune systems) for optimization problems Designing efficient metaheuristics for multi-objective optimization problems Designing hybrid, parallel, and distributed metaheuristics Implementing metaheuristics on sequential and parallel machines Using many case studies and treating design and implementation independently, this book gives readers the skills necessary to solve large-scale optimization problems quickly and efficiently. It is a valuable reference for practicing engineers and researchers from diverse areas dealing with optimization or machine learning; and graduate students in computer science, operations research, control, engineering, business and management, and applied mathematics.

/pdf/metaheuristics-from-design-to-implementation-1vqdfv8nk9.pdf

Metaheuristics: From Design to Implementation

Introduction to linear and nonlinear programming

Multiobjective Decision Making: Theory and Methodology

Load forecast errors can yield suboptimal unit commitment decisions. The economic cost of inaccurate forecasts is assessed by a combination of forecast simulation, unit commitment optimization, and economic dispatch modeling for several different generation/load systems. The forecast simulation preserves the error distributions and correlations actually experienced by users of a neural net-based forecasting system. Underforecasts result in purchases of expensive peaking or spot market power; overforecasts inflate start-up and fixed costs because too much capacity is committed. The value of improved accuracy is found to depend on load and generator characteristics; for the systems considered here, a reduction of 1% in mean absolute percentage error (MAPE) decreases variable generation costs by approximately 0.1%-0.3% when MAPE is in the range of 3%-5%. These values are broadly consistent with the results of a survey of 19 utilities, using estimates obtained by simpler methods. A conservative estimate is that a 1% reduction in forecasting error for a 10,000 MW utility can save up to $1.6 million annually.

Analysis of the value for unit commitment of improved load forecasts

For intermediate and high risk prostate cancer, both the prostate gland and seminal vesicles are included in the clinical target volume. Internal motion patterns of these two organs vary, presenting a challenge for adaptive treatment. Adaptive techniques such as isocenter repositioning and soft tissue alignment are effective when tumor volumes only exhibit translational shift, while direct re-optimization of the intensity-modulated radiation therapy (IMRT) plan maybe more desirable when extreme deformation or differential positioning changes of the organs occur. Currently, direct re-optimization of the IMRT plan using beamlet (or fluence map) has not been reported. In this study, we report a novel on-line re-optimization technique that can accomplish plan adjustment on-line. Deformable image registration is used to provide position variation information on each voxel along the three dimensions. The original planned dose distribution is used as the 'goal' dose distribution for adaptation and to ensure planning quality. Fluence maps are re-optimized via linear programming, and a plan solution can be achieved within 2 min. The feasibility of this technique is demonstrated with a clinical case with large deformation. Such on-line ART process can be highly valuable with hypo-fractionated prostate IMRT treatment.

http://iopscience.iop.org/article/10.1088/0031-9155/53/3/011/pdf

On-line re-optimization of prostate IMRT plans for adaptive radiation therapy

Many multiple criteria decision making methods have been proposed and applied to water planning. Their purpose is to provide information on tradeoffs among objectives and to help users articulate value judgments in a systematic, coherent, and documentable manner. The wide variety of available techniques confuses potential users, causing inappropriate matching of methods with problems. Experiments in which water planners apply more than one multicriteria procedure to realistic problems can help dispel this confusion by testing method appropriateness, ease of use, and validity. We summarize one such experiment where U.S. Army Corps of Engineers personnel used several methods to screen urban water supply plans. The methods evaluated include goal programming, ELECTRE I, additive value functions, multiplicative utility functions, and three techniques for choosing weights (direct rating, indifference tradeoff, and the analytical hierarchy process). Among the conclusions we reach are the following. First, experienced planners generally prefer simpler, more transparent methods. Additive value functions are favored. Yet none of the methods are endorsed by a majority of the participants; many preferred to use no formal method at all. Second, there is strong evidence that rating, the most commonly applied weight selection method, is likely to lead to weights that fail to represent the trade-offs that users are willing to make among criteria. Finally, we show that decisions can be as or more sensitive to the method used as to which person applies it. Therefore, if who chooses is important, then so too is how a choice is made.

Vira Chankong

Papers

Multiobjective Decision Making: Theory and Methodology

Analysis of the value for unit commitment of improved load forecasts

On-line re-optimization of prostate IMRT plans for adaptive radiation therapy

Does choice of multicriteria method matter? An experiment in water resources planning

Decision Making with Multiple Objectives