Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.     

An entropy-based three-stage approach for multi-objective system reliability optimization considering uncertainty

This article presents a three-stage approach for solving multi-objective system reliability optimization problems considering uncertainty. The reliability of each component is considered in the formulation as a component reliability estimate in the form of an interval value and discrete values. Component reliability may vary owing to variations in the usage scenarios. Uncertainty is described by defining a set of usage scenarios. To address this problem, an entropy-based approach to the redundancy allocation problem is proposed in this study to identify the deterministic reliability of each component. In the second stage, a multi-objective evolutionary algorithm (MOEA) is applied to produce a Pareto-optimal solution set. A hybrid algorithm based on k-means and silhouettes is performed to select representative solutions in the third stage. Finally, a numerical example is presented to illustrate the performance of the proposed approach.     

Stability analysis and regularization of uncertain linear multi-objective integer optimization problems

This article presents a systematic approach to analysing linear integer multi-objective optimization problems with uncertainty in the input data. The goal of this approach is to provide decision makers with meaningful information to facilitate the selection of a solution that meets performance expectations and is robust to input parameter uncertainty. Standard regularization techniques often deal with global stability concepts. The concept presented here is based on local quasi-stability and includes a local regularization technique that may be used to increase the robustness of any given efficient solution or to transform efficient solutions that are not robust (i.e. unstable), into robust solutions. An application to a multi-objective problem drawn from the mining industry is also presented.     

Sequential interactive evolution for finding high-quality topologies

Finding a diverse set of high-quality (HQ) topologies for a single-objective optimization problem using an evolutionary computation algorithm can be difficult without a reliable measure that adequately describes the dissimilarity between competing topologies. In this article, a new approach for enhancing diversity among HQ topologies for engineering design applications is proposed. The technique initially selects one HQ solution and then searches for alternative HQ solutions by performing an optimization of the original objective and its dissimilarity with respect to the previously found solution. The proposed multi-objective optimization approach interactively amalgamates user articulated preferences with an evolutionary search so as sequentially to produce a set of diverse HQ solutions to a single-objective problem. For enhancing diversity, a new measure is suggested and an approach to reducing its computational time is studied and implemented. To illustrate the technique, a series of studies involving different topologies represented as bitmaps is presented.     

Highly reliable optimal solutions to multi-objective problems and their evolution by means of worst-case analysis

In the current article, high reliability in the presence of uncertainty is of interest. Therefore, no violation of constraints by any solution, although uncertainty exists, is mandatory. The article studies uncertainties in which the boundaries of uncertainty are known. To allow a high reliability, the notion of worst-violation set is introduced. Moreover, two possible measures to assess the extent of the violation of the constraints by a solution, which is subjected to uncertainty, are suggested. One of these measures is then introduced into a multi-objective evolutionary algorithm (MOEA) in order to search for optimal reliable solutions. It is shown that the approach applies a search towards solutions with optimal performances while taking into account high reliability. The suggested approach is the only one available so far (to the authors’ best knowledge), which treats reliability through evolutionary multi-objective search, while not assuming any probability distribution of the uncertainty.     

Parallel simple cell mapping for multi-objective optimization

In this article the Parallel Simple Cell Mapping (pSCM) is presented, a novel method for the numerical treatment of multi-objective optimization problems. The method is a parallel version of the simple cell mapping (SCM) method which also integrates elements from subdivision techniques. The classical SCM method exhibits nice properties for parallelization, which is used to speed up computations significantly. These statements are underlined on some classical benchmark problems with up to 10 decision variables and up to 5 objectives and provide comparisons to sequential SCM. Further, the method is applied on illustrative examples for which the method is also able to find the set of local optimal solutions efficiently, which is interesting in multi-objective multi-modal optimization, as well as the set of approximate solutions. The latter is of potential interest for the decision maker since it comprises an extended set of possible realizations of the given problem.     

Shape design of internal cooling passages within a turbine blade

The article concerns the optimization of the shape and location of non-circular passages cooling the blade of a gas turbine. To model the shape, four Bezier curves which form a closed profile of the passage were used. In order to match the shape of the passage to the blade profile, a technique was put forward to copy and scale the profile fragments into the component, and build the outline of the passage on the basis of them. For so-defined cooling passages, optimization calculations were carried out with a view to finding their optimal shape and location in terms of the assumed objectives. The task was solved as a multi-objective problem with the use of the Pareto method, for a cooling system composed of four and five passages. The tool employed for the optimization was the evolutionary algorithm. The article presents the impact of the population on the task convergence, and discusses the impact of different optimization objectives on the Pareto optimal solutions obtained. Due to the problem of different impacts of individual objectives on the position of the solution front which was noticed during the calculations, a two-step optimization procedure was introduced. Also, comparative optimization calculations for the scalar objective function were carried out and set up against the non-dominated solutions obtained in the Pareto approach. The optimization process resulted in a configuration of the cooling system that allows a significant reduction in the temperature of the blade and its thermal stress.     

Time-response-based evolutionary optimization

Solutions to engineering problems are often evaluated by considering their time responses; thus, each solution is associated with a function. To avoid optimizing the functions, such optimization is usually carried out by setting auxiliary objectives (e.g. minimal overshoot). Therefore, in order to find different optimal solutions, alternative auxiliary optimization objectives may have to be defined prior to optimization. In the current study, a new approach is suggested that avoids the need to define auxiliary objectives. An algorithm is suggested that enables the optimization of solutions according to their transient behaviours. For this optimization, the functions are sampled and the problem is posed as a multi-objective problem. The recently introduced algorithm NSGA-II-PSA is adopted and tailored to solve it. Mathematical as well as engineering problems are utilized to explain and demonstrate the approach and its applicability to real life problems. The results highlight the advantages of avoiding the definition of artificial objectives.     

An efficient local improvement operator for the multi-objective wireless sensor network deployment problem

Wireless sensor network layout, also known as sensor node deployment, is a complex NP-complete optimization task that determines most of the functioning features of a wireless sensor network. Coverage, connectivity and lifetime (handled through its opposing parameter, power consumption), are three of the most important characteristics of the service, and are taken into consideration in this article within a multi-objective approach of the problem. Leveraging on the specific properties of the wireless sensor nodes and networks, the Proximity Avoidance Coverage-preserving Operator (PACO) for local improvement is presented, described and tested. The testbed consists of a set of state-of-the-art multi-objective optimization algorithms with different configurations, and problem instances of varying size. In all the scenarios, and more specially in the algorithmic settings that already produce high performance solutions, PACO has proven to be a robust enhancement to the raw optimization technique, without requiring additional computation, that easily scales through problem complexity.     

产品操作界面元素布局多目标优化设计

目的针对产品设计中操作界面布局设计时存在的随意性、不确定性大等问题,使用多目标优化设计的方法,寻找界面元素的最优布置,以提升界面的使用舒适性和人机交互效率。方法在分析了工效学准则和界面布局美度评价准则的基础上,确立了层次性、相关性、简洁性和舒适性四个界面布局基本原则,并依据原则构建了界面元素布局多目标优化数学模型,在此模型的基础上采用改进的遗传算法,建立基于遗传算法的界面元素布局多目标优化方法。结果给出产品操作界面布局设计的基本原则,提出一种基于遗传算法的产品操作界面元素布局的多目标优化方法及流程。结论提出的布局原则和优化方法能较好地协助设计师获得满足设计需求的布局方案,实例结果表明了理论模型的合理性与遗传算法对于界面元素布局多目标优化问题求解的有效性。     

A particle swarm optimizer for constrained multi-objective engineering design problems

This article presents a particle swarm optimizer (PSO) capable of handling constrained multi-objective optimization problems. The latter occur frequently in engineering design, especially when cost and performance are simultaneously optimized. The proposed algorithm combines the swarm intelligence fundamentals with elements from bio-inspired algorithms. A distinctive feature of the algorithm is the utilization of an arithmetic recombination operator, which allows interaction between non-dominated particles. Furthermore, there is no utilization of an external archive to store optimal solutions. The PSO algorithm is applied to multi-objective optimization benchmark problems and also to constrained multi-objective engineering design problems. The algorithmic effectiveness is demonstrated through comparisons of the PSO results with those obtained from other evolutionary optimization algorithms. The proposed particle swarm optimizer was able to perform in a very satisfactory manner in problems with multiple constraints and/or high dimensionality. Promising results were also obtained for a multi-objective engineering design problem with mixed variables.     

A bispace parameterization method for shape optimization of thin‐walled curved shell structures with openings

Simultaneous shape optimization of thin‐walled curved shell structures and involved hole boundaries is studied in this paper. A novel bispace parameterization method is proposed for the first time to define global and local shape design variables both in the Cartesian coordinate system and the intrinsic coordinate system. This method has the advantage of achieving a simultaneous optimization of the global shape of the shell surface and the local shape of the openings attached automatically on the former. Inherent problems, for example, the effective parameterization of shape design variables, mapping operation between two spaces, and sensitivity analysis with respect to both kinds of design variables are highlighted. A design procedure is given to show how both kinds of design variables are managed together and how the whole design flowchart is carried out with relevant formulations. Numerical examples are presented and the effects of both kinds of design variables upon the optimal solutions are discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

一种基于近似模型管理的多目标优化方法

针对实际工程优化问题,为了提高效率和减少近似模型带来的误差,提出一种基于模型管理的多目标优化方法。利用加强径向基插值函数在整个寻优区域内构造目标和约束的近似模型,结合微型多目标遗传算法寻找当前非支配解。通过模型管理方法更新近似模型,并控制由于近似模型带来的误差和更新次数,最后将误差控制在一定范围内的多个非支配解当作实际问题的解。在测试函数中验证了此方法的效率及非支配解的精度和分布的均匀性。最后成功应用于车身薄壁构件的耐撞性优化中,表明了可用于求解复杂的工程优化问题。     

Effective multi-objective optimization with the coral reefs optimization algorithm

In this article a new algorithm for multi-objective optimization is presented, the Multi-Objective Coral Reefs Optimization (MO-CRO) algorithm. The algorithm is based on the simulation of processes in coral reefs, such as corals' reproduction and fight for space in the reef. The adaptation to multi-objective problems is a process based on domination or non-domination during the process of fight for space in the reef. The final MO-CRO is an easily-implemented and fast algorithm, simple and robust, since it is able to keep diversity in the population of corals (solutions) in a natural way. The experimental evaluation of this new approach for multi-objective optimization problems is carried out on different multi-objective benchmark problems, where the MO-CRO has shown excellent performance in cases with limited computational resources, and in a real-world problem of wind speed prediction, where the MO-CRO algorithm is used to find the best set of features to predict the wind speed, taking into account two objective functions related to the performance of the prediction and the computation time of the regressor.     

A new multi-objective particle swarm optimizer using empirical movement and diversified search strategies

Most real-world optimization problems involve the optimization task of more than a single objective function and, therefore, require a great amount of computational effort as the solution procedure is designed to anchor multiple compromised optimal solutions. Abundant multi-objective evolutionary algorithms (MOEAs) for multi-objective optimization have appeared in the literature over the past two decades. In this article, a new proposal by means of particle swarm optimization is addressed for solving multi-objective optimization problems. The proposed algorithm is constructed based on the concept of Pareto dominance, taking both the diversified search and empirical movement strategies into account. The proposed particle swarm MOEA with these two strategies is thus dubbed the empirical-movement diversified-search multi-objective particle swarm optimizer (EMDS-MOPSO). Its performance is assessed in terms of a suite of standard benchmark functions taken from the literature and compared to other four state-of-the-art MOEAs. The computational results demonstrate that the proposed algorithm shows great promise in solving multi-objective optimization problems.     

Multi-objective robust design optimization of fatigue life for a welded box girder

To reduce the scatter of fatigue life for welded structures, a robust optimization method is presented in this study based on a dual surrogate modelling and multi-objective particle swam optimization algorithm. Considering the perturbations of material parameters and environment variables, the mean and standard deviation of fatigue life are fitted using dual surrogate modelling and selected as the objective function to be minimized. As an example, a welded box girder is presented to reduce the standard deviation of fatigue life. A set of non-dominated solutions is produced through a multi-objective particle swam optimization algorithm. A cognitive approach is used to select the optimum solution from the Pareto sets. As a comparative study, traditional single objective optimizations are also presented in this study. The results reduced the standard deviation of the fatigue life by about 16.5%, which indicated that the procedure improved the robustness of the fatigue life.     

An improved version of the multiple trajectory search for real value multi-objective optimization problems

Multi-objective optimization is widely used in science, engineering and business. In this article, an improved version of the multiple trajectory search (MTS) called MTS2 is presented and successfully applied to real-value multi-objective optimization problems. In the first step, MTS2 generates M initial solutions distributed over the solution space. These solutions are called seeds. Some seeds with good objective values are selected as foreground seeds. Then, MTS2 chooses a suitable region search method for each foreground seed according to the landscape of the neighbourhood of the seed. During the search, MTS2 focuses its search on some promising areas specified by the foreground seeds. The performance of MTS2 was examined by applying it to solve the benchmark problems provided by the Competition of Performance Assessment of Constrained/Bound Constrained Multi-Objective Optimization Algorithms held at the 2009 IEEE Congress on Evolutionary Computation.     

Hybridizing evolutionary strategies with continuation methods for solving multi-objective problems

Two techniques for the numerical treatment of multi-objective optimization problems—a continuation method and a particle swarm optimizer—are combined in order to unite their particular advantages. Continuation methods can be applied very efficiently to perform the search along the Pareto set, even for high-dimensional models, but are of local nature. In contrast, many multi-objective particle swarm optimizers tend to have slow convergence, but instead accomplish the ‘global task’ well. An algorithm which combines these two techniques is proposed, some convergence results for continuous models are provided, possible realizations are discussed, and finally some numerical results are presented indicating the strength of this novel approach.     

Formulation and application of weight-function-based physical programming

Physical programming is effective in multi-objective optimization since it assists the designer to find the most preferred solution. Preference-function-based physical programming (PFPP) abandons the weighted-sum approach and its performance in generating Pareto solutions is susceptible to the transformation of pseudo-preferences. With the aim of integrating a weighted-sum approach into physical programming and generating well-distributed Pareto solutions, a weight-function-based physical programming (WFPP) method has been proposed. The approach forms a weight function for each normalized criterion and uses the variable weighted sum of all criteria as the aggregate objective function. Implementation for numerical and engineering design problems indicates that WFPP works as well as PFPP. The design process of generating Pareto solutions by WFPP is further presented, where the pseudo-preferences are allowed to transform in different ranges. Examples and results demonstrate that solutions generated by WFPP have better diversity performance than those of PFPP, especially when the pseudo-preferences are far from the true Pareto front.     

Multi-objective optimization of engineering systems using game theory and particle swarm optimization

This study proposes particle swarm optimization (PSO) based algorithms to solve multi-objective engineering optimization problems involving continuous, discrete and/or mixed design variables. The original PSO algorithm is modified to include dynamic maximum velocity function and bounce method to enhance the computational efficiency and solution accuracy. The algorithm uses a closest discrete approach (CDA) to solve optimization problems with discrete design variables. A modified game theory (MGT) approach, coupled with the modified PSO, is used to solve multi-objective optimization problems. A dynamic penalty function is used to handle constraints in the optimization problem. The methodologies proposed are illustrated by several engineering applications and the results obtained are compared with those reported in the literature.