THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

In evolutionary multiobjective optimization, maintaining a good balance between convergence and diversity is particularly crucial to the performance of the evolutionary algorithms (EAs). In addition, it becomes increasingly important to incorporate user preferences because it will be less likely to achieve a representative subset of the Pareto-optimal solutions using a limited population size as the number of objectives increases. This paper proposes a reference vector-guided EA for many-objective optimization. The reference vectors can be used not only to decompose the original multiobjective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In the proposed algorithm, a scalarization approach, termed angle-penalized distance, is adopted to balance convergence and diversity of the solutions in the high-dimensional objective space. An adaptation strategy is proposed to dynamically adjust the distribution of the reference vectors according to the scales of the objective functions. Our experimental results on a variety of benchmark test problems show that the proposed algorithm is highly competitive in comparison with five state-of-the-art EAs for many-objective optimization. In addition, we show that reference vectors are effective and cost-efficient for preference articulation, which is particularly desirable for many-objective optimization. Furthermore, a reference vector regeneration strategy is proposed for handling irregular PFs. Finally, the proposed algorithm is extended for solving constrained many-objective optimization problems.

A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization

Over the last three decades, a large number of evolutionary algorithms have been developed for solving multi-objective optimization problems. However, there lacks an upto-date and comprehensive software platform for researchers to properly benchmark existing algorithms and for practitioners to apply selected algorithms to solve their real-world problems. The demand of such a common tool becomes even more urgent, when the source code of many proposed algorithms has not been made publicly available. To address these issues, we have developed a MATLAB platform for evolutionary multi-objective optimization in this paper, called PlatEMO, which includes more than 50 multiobjective evolutionary algorithms and more than 100 multi-objective test problems, along with several widely used performance indicators. With a user-friendly graphical user interface, PlatEMO enables users to easily compare several evolutionary algorithms at one time and collect statistical results in Excel or LaTeX files. More importantly, PlatEMO is completely open source, such that users are able to develop new algorithms on the basis of it. This paper introduces the main features of PlatEMO and illustrates how to use it for performing comparative experiments, embedding new algorithms, creating new test problems, and developing performance indicators. Source code of PlatEMO is now available at: http://bimk.ahu.edu.cn/index.php?s=/Index/Software/index.html.

PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization [Educational Forum]

Achieving balance between convergence and diversity is a key issue in evolutionary multiobjective optimization. Most existing methodologies, which have demonstrated their niche on various practical problems involving two and three objectives, face significant challenges in many-objective optimization. This paper suggests a unified paradigm, which combines dominance- and decomposition-based approaches, for many-objective optimization. Our major purpose is to exploit the merits of both dominance- and decomposition-based approaches to balance the convergence and diversity of the evolutionary process. The performance of our proposed method is validated and compared with four state-of-the-art algorithms on a number of unconstrained benchmark problems with up to 15 objectives. Empirical results fully demonstrate the superiority of our proposed method on all considered test instances. In addition, we extend this method to solve constrained problems having a large number of objectives. Compared to two other recently proposed constrained optimizers, our proposed method shows highly competitive performance on all the constrained optimization problems.

An Evolutionary Many-Objective Optimization Algorithm Based on Dominance and Decomposition

Over the last three decades, a large number of evolutionary algorithms have been developed for solving multiobjective
optimization problems. However, there lacks an up-to-date and comprehensive software platform for
researchers to properly benchmark existing algorithms and for practitioners to apply selected algorithms to solve
their real-world problems. The demand of such a common tool becomes even more urgent, when the source code
of many proposed algorithms has not been made publicly available. To address these issues, we have developed
a MATLAB platform for evolutionary multi-objective optimization in this paper, called PlatEMO, which includes
more than 50 multi-objective evolutionary algorithms and more than 100 multi-objective test problems, along with
several widely used performance indicators. With a user-friendly graphical user interface, PlatEMO enables users
to easily compare several evolutionary algorithms at one time and collect statistical results in Excel or LaTeX files.
More importantly, PlatEMO is completely open source, such that users are able to develop new algorithms on the
basis of it. This paper introduces the main features of PlatEMO and illustrates how to use it for performing comparative
experiments, embedding new algorithms, creating new test problems, and developing performance indicators.
Source code of PlatEMO is now available at: http://bimk.ahu.edu.cn/index.php?s=/Index/Software/index.html.

PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization

This letter suggests an approach for decomposing a multiobjective optimization problem (MOP) into a set of simple multiobjective optimization subproblems. Using this approach, it proposes MOEA/D-M2M, a new version of multiobjective optimization evolutionary algorithm-based decomposition. This proposed algorithm solves these subproblems in a collaborative way. Each subproblem has its own population and receives computational effort at each generation. In such a way, population diversity can be maintained, which is critical for solving some MOPs. Experimental studies have been conducted to compare MOEA/D-M2M with classic MOEA/D and NSGA-II. This letter argues that population diversity is more important than convergence in multiobjective evolutionary algorithms for dealing with some MOPs. It also explains why MOEA/D-M2M performs better.

/pdf/decomposition-of-a-multiobjective-optimization-problem-into-nd2ag4ghk7.pdf

Decomposition of a Multiobjective Optimization Problem Into a Number of Simple Multiobjective Subproblems

Brain storm optimization (BSO) is a new kind of swarm intelligence algorithm inspired by human creative problem solving process. Human being is the most intelligent organism in the world and the brainstorming process popularly used by them has been demonstrated to be a significant and promising way to create great ideas for problem solving. BSO transplants the brainstorming process in human being into optimization algorithm design and gains successes. BSO generally uses the grouping, replacing, and creating operators to produce ideas as many as possible to approach the problem global optimum generation by generation. In this paper, we propose two novel designs to enhance the conventional BSO performance. The first design of the modified BSO (MBSO) is that it uses a simple grouping method (SGM) in the grouping operator instead of the clustering method to reduce the algorithm computational burden. The second design is that MBSO uses a novel idea difference strategy (IDS) in the creating operator instead of the Gaussian random strategy. The IDS not only contains open minded element to avoid the ideas being trapped by local optima, but also can match the search environment to create better new ideas for problem solving. Experiments have been conducted to illustrate the effectiveness and efficiency of the MBSO algorithm. Moreover, the contributions of SGM and IDS are investigated to show how and why MBSO can perform better than BSO.

A modified brain storm optimization

An effective allocation of search effort is important in multiobjective optimization, particularly in many-objective optimization problems (MaOPs). This paper presents a new adaptive search effort allocation strategy for multiobjective evolutionary algorithm based on decomposition MOEA/D-M2M, a recent MOEA/D algorithm for challenging MaOPs. This proposed method adaptively adjusts the subregions of its subproblems by detecting the importance of different objectives in an adaptive manner. More specifically, it periodically resets the subregion setting based on the distribution of the current solutions in the objective space such that the search effort is not wasted on unpromising regions. The basic idea is that the current population can be regarded as an approximation to the Pareto front (PF) and thus one can implicitly estimate the shape of the PF and such estimation can be used for adjusting the search focus. The performance of proposed algorithm has been verified by comparing it with eight representative and competitive algorithms on a set of degenerated MaOPs with disconnected and connected PFs. Performances of the proposed algorithm on a number of nondegenerated test instances with connected and disconnected PFs are also studied.

Adaptively Allocating Search Effort in Challenging Many-Objective Optimization Problems

In most multiobjective evolutionary algorithms (MOEA) based on aggregating objectives, the weight vectors are user-supplied or generated randomly, and they are static in the algorithms. If the Pareto front (PF) shape is not complex, the algorithms can find a set of uniformly distributed Pareto optimal solutions along the PF; otherwise, they might fail. A dynamic weight design method based on the projection of the current nondominant solutions and equidistant interpolation is proposed in this paper. Even if the PF is complex, we can find evenly distributed Pareto optimal solutions by this method. Some test instances are constructed to compare the performance of the MOEA/D using dynamic weight design method with that of MOEA/D. The results indicate that the dynamic weight design method can dramatically improve the performance of the algorithms.

A multiobjective evolutionary algorithm using dynamic weight design method

By dividing the multiobjective optimization of the decision space into several small regions, this paper proposes multi-objective optimization algorithm based on sub-regional search, which makes individuals in same region operate each other by evolutionary operator and the information between the individuals of different regions exchange through their offsprings re-divided into regions again. Since the proposed algorithm utilizes the sub-regional search, the computational complexity at each generation is lower than the NSGA-II and MSEA. The proposed algorithm makes use of the max-min strategy with determined weight as fitness functions, which make it approach evenly distributed solution in Pareto front. This paper presents a kind of easy technology dealing with the constraint, which makes the proposed algorithm solved unconstrained multiobjective problems can also be used to solve constrained multiobjective problems. The numerical results, with 13 unconstrained multiobjective optimization testing instances and 10 constrained multiobjective optimization testing instances, are shown in this paper.

Hai-Lin Liu

Papers

Decomposition of a Multiobjective Optimization Problem Into a Number of Simple Multiobjective Subproblems

A modified brain storm optimization

Adaptively Allocating Search Effort in Challenging Many-Objective Optimization Problems

A multiobjective evolutionary algorithm using dynamic weight design method

The multiobjective evolutionary algorithm based on determined weight and sub-regional search