Taguchi's method for multi‐objective optimization problems

Taguchi's method is a quality design technique whose applications in numerical single‐objective optimization have been recently exploited. In this article, a novel multi‐objective (MO) algorithm based on Taguchi's technique is illustrated and its performances assessed. Validation is performed through a comparison between the presented algorithm and a MO genetic algorithm (GA) based optimization, first on different sets of test functions and then on a practical antenna array synthesis problem. Results indicate a generally better behavior of the proposed algorithm in terms of convergence and spreading over the Pareto front with respect to the GA benchmark. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE , 2013.     

Multiobjective optimization on adhesive bonding of aluminum‐carbon fiber laminate

This work presents a multi‐objective optimization methodology to find compromise adhesive bonding schemes that possess a great shear load and a low percentage of remaining fiber in the bonding. The joining overlap, adhesive type, and prior surface finishing are considered. The Pareto front of the multi‐objective response surface model is found with an Nondominated Sorting Genetic algorithm. The adhesive bonding factors are the adhesive (MP55420, Betamate 120, and DC‐80), the surface finishing (acetone cleaned and atmospheric plasma), and the overlapping distance of the test coupons.     

Multi‐objective design optimization of antennas for reflection,size, and gain variability using kriging surrogates and generalized domain segmentation

Cost‐efficient multi‐objective design optimization of antennas is presented. The framework exploits auxiliary data‐driven surrogates, a multi‐objective evolutionary algorithm for initial Pareto front identification, response correction techniques for design refinement, as well as generalized domain segmentation. The purpose of this last mechanism is to reduce the volume of the design space region that needs to be sampled in order to construct the surrogate model, and, consequently, limit the number of training data points required. The recently introduced segmentation concept is generalized here to allow for handling an arbitrary number of design objectives. Its operation is illustrated using an ultra‐wideband monopole optimized for best in‐band reflection, minimum gain variability, and minimum size. When compared with conventional surrogate‐based approach, segmentation leads to reduction of the initial Pareto identification cost by over 20%. Numerical results are supported by experimental validation of the selected Pareto‐optimal antenna designs.     

Multi‐objective memetic algorithm: comparing artificial neural networks and pattern search filter method approaches

In this work, two methodologies to reduce the computation time of expensive multi‐objective optimization problems are compared. These methodologies consist of the hybridization of a multi‐objective evolutionary algorithm (MOEA) with local search procedures. First, an inverse artificial neural network proposed previously, consisting of mapping the decision variables into the multiple objectives to be optimized in order to generate improved solutions on certain generations of the MOEA, is presented. Second, a new approach based on a pattern search filter method is proposed in order to perform a local search around certain solutions selected previously from the Pareto frontier. The results obtained, by the application of both methodologies to difficult test problems, indicate a good performance of the approaches proposed.     

Engineering an efficient two‐phase local search algorithm for the co‐rotating twin‐screw extruder configuration problem

The twin‐screw configuration problem arises during polymer extrusion and compounding. It consists in defining the location of a set of pre‐defined screw elements along the screw axis in order to optimize different, typically conflicting objectives. In this paper, we present a simple yet effective stochastic local search (SLS) algorithm for this problem. Our algorithm is based on efficient single‐objective iterative improvement algorithms, which have been developed by studying different neighborhood structures, neighborhood search strategies, and neighborhood restrictions. These algorithms are embedded into a variation of the two‐phase local search framework to tackle various bi‐objective versions of this problem. An experimental comparison with a previously proposed multi‐objective evolutionary algorithm shows that a main advantage of our SLS algorithm is that it converges faster to a high‐quality approximation to the Pareto front.     

基于动态调整的多目标粒子群优化算法

为了改善多目标粒子群优化算法生成的最终Pareto前端的多样性和收敛性,提出了一种针对多目标粒子群算法进化状态的检测机制.通过对外部Pareto解集的更新情况进行检测,进而评估算法的进化状态,获取反馈信息来动态调整进化策略,使得算法在进化过程中兼顾近似Pareto前端的多样性和收敛性.最后,在ZDT系列测试函数中,将本文算法与其他4种对等算法比较,证明了本文算法生成的最终Pareto前端在多样性和收敛性上均有显著的优势.     

Recent advances in rapid multiobjective optimization of expensive simulation models in microwave and antenna engineering by Pareto front exploration

Practical engineering design problems are inherently multiobjective, that is, require simultaneous control of several (and often conflicting) criteria. In many situations, genuine multiobjective optimization is required to acquire comprehensive information about the system of interest. The most popular solution techniques are population‐based metaheuristics, however, they are not practical for handling expensive electromagnetic (EM)‐simulation models in microwave and antenna engineering. A workaround is to use auxiliary response surface approximation surrogates but it is challenging for higher‐dimensional problems. Recently, a deterministic approach has been proposed for expedited multiobjective design optimization of expensive models in computational EMs. The method relies on variable‐fidelity EM simulations, tracking the Pareto front geometry, as well as response correction. The algorithm sequentially generates Pareto‐optimal designs using a series of constrained single‐objective optimizations. The previously obtained design is used as a starting point for the next iteration. In this work, we review this technique and its modification based on space mapping surrogates. We also propose new variations exploiting adjoint sensitivities, as well as response features, which can be attractive depending on availability of derivatives or the characteristics of the system responses that need to be handled. We also discuss several case studies involving various antenna and microwave components.     

Meta‐heuristic multi‐ and many‐objective optimization techniques for solution of machine learning problems

Recently, multi‐ and many‐objective meta‐heuristic algorithms have received considerable attention due to their capability to solve optimization problems that require more than one fitness function. This paper presents a comprehensive study of these techniques applied in the context of machine learning problems. Three different topics are reviewed in this work: (a) feature extraction and selection, (b) hyper‐parameter optimization and model selection in the context of supervised learning, and (c) clustering or unsupervised learning. The survey also highlights future research towards related areas.     

Pareto-optimal solutions for multicriteria optimization of a chemical engineering process using a diploid genetic algorithm

In many, if not most, optimization problems, industrialists are often confronted with multi‐objective decision problems. For example, in manufacturing processes, it may be necessary to optimize several criteria to take into account all the market constraints. Hence, the purpose is to choose the best trade‐offs among all the defined and conflicting objectives. This paper presents a multi‐objective optimization procedure based on a diploid genetic algorithm, which yields an optimal zone containing the solution under the concept of Pareto dominance. Pair‐wise points are compared, and non‐dominated points are collected in the Pareto region. Then a ranking is established, and the decision maker selects the first‐best solution. Finally, the procedure is applied to the chemical engineering process of cattle feed manufacture.     

Boundary‐based optimization strategies for design of fragment‐type isolation structure in compact MIMO systems

Fragment‐type structures have been used to acquire high isolation in compact multiple‐input and multiple‐output (MIMO) systems. In this paper, two novel optimization strategies, boundary‐based two‐dimensional (2D) median filtering operator and boundary‐based 2D weighted sum filtering operator, are proposed to design fragment‐type isolation structures first when specific boundary conditions are considered in engineering designs. Second, two computer aided optimization techniques are proposed through combining these two operators with MOEA/D‐GO (multi‐objective evolutionary algorithm based on decomposition combined with enhanced genetic operators), respectively. Finally, fragment‐type isolation structures of a compact MIMO PIFAs (planar inverted‐F antennas) system operating at 2.345‐2.36 GHz are designed. Comparison results show that more alternative designs could be found at the expense of searching speed, and both better front‐back‐ratio and wider impedance bandwidth are observed.     

多目标优化Knee前沿搜索方法研究进展

多目标优化算法是近年来进化计算研究领域的一个热点,大多数的多目标优化算法试图找到问题的完整的Pareto前沿.然而,随着待优化问题目标个数的增加,算法需要更大的种群规模才能合理地描绘出完整的Pareto前沿.显然这样不仅增加了算法的运行时间,更增加了(决策者)最终解的选择难度.因此,聚焦于搜索Pareto前沿上的特定区...     

求解VLSI布图规划问题的多目标粒子群优化算法

布图规划在超大规模集成电路(VLSI)物理设计过程中具有重要作用,它是一个多目标组合优化问题且被证明是一个NP问题。为了有效解决布图规划问题,本文提出一个多目标粒子群优化(PSO)算法。该算法采用序列对表示法对粒子进行编码,根据遗传算法交叉算子的思想对粒子更新公式进行了修改;引入Pareto最优解的概念和精英保留策略,并设计了一个基于表现型共享的适应值函数以维护种群的多样性。仿真实验通过对MCNC标准问题的测试表明了本文算法是可行且有效的。     

A multiobjective approach for maximizing the reach or GRP of different brands in TV advertising

This paper focuses on a multiobjective optimization problem in TV advertising from an advertising agency's perspective, which involves deciding on which commercial breaks to air the ads of various brands to jointly maximize reach or gross rating point (GRP) for the different brands subject to budget constraints, brand competition constraints, and other scheduling constraints. We present a multiobjective integer programming formulation of this problem and develop and implement algorithms for generating provably Pareto‐optimal solutions. We also develop reduction and visualization procedures to aid a decision maker in choosing suitable subsets of the Pareto‐optimal solutions obtained. Numerical experiments on five TV advertising problems involving 20–40 objective functions and thousands of decision variables and constraints demonstrate the effectiveness of the proposed formulation and solution methods in generating Pareto‐optimal objective vectors that reflect brand priorities and that are well distributed along the Pareto front.     

Application of bacteria foraging algorithm for the design optimization of multi‐objective Yagi‐Uda array

The design of antenna array with desirable multiple performance parameters such as directivity, input impedance, beam width, and side‐lobe level using any optimization algorithm is a highly challenging task. Bacteria Foraging Algorithm (BFA), as reported by electrical engineers, is the most robust and efficient algorithm in comparison with other presently available algorithms for global optimization of multi‐objective, multi‐parameter design problems. The objective of this article is to apply this new optimization technique, BFA, in the design of Yagi‐Uda array for multi‐objective design parameters. We optimize length and spacing for 6 and 15 elements array to achieve higher directivity, pertinent input impedance, minimum 3‐dB beam width, and maximum front to back ratio both in the E and H planes of the array. At first, we develop a Method of Moments code in MATLAB environment for the Yagi‐Uda array structure for obtaining the above design parameters and then coupled with the BFA for the evaluation of the optimized design parameters. Detail simulation results are included to confirm the design criteria. © 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE , 2010.     

Multi‐objective optimal motion control of a laboratory helicopter based on parallel simple cell mapping method

This paper presents a study of multi‐objective optimal design of nonlinear control systems and has validated the control design with a twin rotor model helicopter. The gains of the porportional integral differential (PID) control are designed in the framework of multi‐objective opitmization. Eight design paramaters are optimized to minimize six time‐domain objective objective functions. The study of multi‐objective optimal design of feedback control with such a number of design paramaters and objective functions is rare in the literature. The Pareto optimal solutions are obtained by the proposed parallel simple cell mapping method consisting of a robust Pareto set recovery algorithm and a rolling subdivision technique. The proposed parallel simple cell mapping algorithm has two features: the number of cells in the invariant set grows linearly with the rolling subdivisions, and the Pareto set is insensitive to the inital set of seed cells. The current control design is compared with the classical LQE control for linear systems, and is also experimentally validated. The current design provides improved control performance as compares with the LQR control, and is applicable to complex nonlinear systems.     

ADAPTIVE RANKS CLONE AND k‐NEAREST NEIGHBOR LIST–BASED IMMUNE MULTI‐OBJECTIVE OPTIMIZATION

Artificial immune systems (AIS) are computational systems inspired by the principles and processes of the vertebrate immune system. The AIS‐based algorithms typically exploit the immune system's characteristics of learning and adaptability to solve some complicated problems. Although, several AIS‐based algorithms have proposed to solve multi‐objective optimization problems (MOPs), little focus have been placed on the issues that adaptively use the online discovered solutions. Here, we proposed an adaptive selection scheme and an adaptive ranks clone scheme by the online discovered solutions in different ranks. Accordingly, the dynamic information of the online antibody population is efficiently exploited, which is beneficial to the search process. Furthermore, it has been widely approved that one‐off deletion could not obtain excellent diversity in the final population; therefore, a k‐nearest neighbor list (where k is the number of objectives) is established and maintained to eliminate the solutions in the archive population. The k‐nearest neighbors of each antibody are founded and stored in a list memory. Once an antibody with minimal product of k‐nearest neighbors is deleted, the neighborhood relations of the remaining antibodies in the list memory are updated. Finally, the proposed algorithm is tested on 10 well‐known and frequently used multi‐objective problems and two many‐objective problems with 4, 6, and 8 objectives. Compared with five other state‐of‐the‐art multi‐objective algorithms, namely NSGA‐II, SPEA2, IBEA, HYPE, and NNIA, our method achieves comparable results in terms of convergence, diversity metrics, and computational time.     

Generalized Pareto ranking bisection for computationally feasible multiobjective antenna optimization

Multiobjective optimization (MO) allows for obtaining comprehensive information about possible design trade‐offs of a given antenna structure. Yet, executing MO using the most popular class of techniques, population‐based metaheuristics, may be computationally prohibitive when full‐wave EM analysis is utilized for antenna evaluation. In this work, a low‐cost and fully deterministic MO methodology is introduced. The proposed generalized Pareto ranking bisection algorithm permits identifying a set of Pareto optimal sets of parameters representing the best trade‐offs between considered objectives. The subsequent designs are found by iterative partitioning of the intervals connecting previously obtained designs and executing Pareto‐ranking‐based poll search. The initial approximation of the Pareto front found using the bisection procedure is subsequently refined to the level of the high‐fidelity EM model of the antenna at hand using local optimization. The proposed framework overcomes a serious limitation of the original, recently reported, bisection algorithm, which was only capable of considering two objectives. The generalized version proposed here allows for handling any number of design goals. An improved poll search procedure has also been developed and incorporated. Our algorithm has been demonstrated using two examples of UWB monopole antennas with four figures of interest taken into account: structure size, reflection response, total efficiency, and gain variability.     

Analysing the factors that influence the Pareto frontier of a bi‐objective supply chain design problem

In this paper a bi‐objective multi‐product model for the design of a production/distribution supply chain logistic network with four echelons is considered. The proposed optimization model minimizes the total cost of the network (including the fixed cost to open facilities and the transportation costs between them) and the total CO₂ emissions. Five factors (network size, product complexity, cost variability, CO₂ emissions generation and over‐capacity) are considered for the experimental framework. The problem is solved using the ε‐constraint method and the resulting Pareto frontiers (PF) are characterized using five new metrics specifically developed for analysing how those factors affect the resulting optimal configurations. The results show that over‐capacity and product complexity are the two most influential factors regarding the characteristics of the PF, and that their effects are in the same direction: more complexity and capacity mean a wider set of optima alternatives, some close to the ideal point, and in general with a smaller number of links used.     

Evolutionary algorithms for the multi‐objective test data generation problem

Automatic test data generation is a very popular domain in the field of search‐based software engineering. Traditionally, the main goal has been to maximize coverage. However, other objectives can be defined, such as the oracle cost, which is the cost of executing the entire test suite and the cost of checking the system behavior. Indeed, in very large software systems, the cost spent to test the system can be an issue, and then it makes sense by considering two conflicting objectives: maximizing the coverage and minimizing the oracle cost. This is what we did in this paper. We mainly compared two approaches to deal with the multi‐objective test data generation problem: a direct multi‐objective approach and a combination of a mono‐objective algorithm together with multi‐objective test case selection optimization. Concretely, in this work, we used four state‐of‐the‐art multi‐objective algorithms and two mono‐objective evolutionary algorithms followed by a multi‐objective test case selection based on Pareto efficiency. The experimental analysis compares these techniques on two different benchmarks. The first one is composed of 800 Java programs created through a program generator. The second benchmark is composed of 13 real programs extracted from the literature. In the direct multi‐objective approach, the results indicate that the oracle cost can be properly optimized; however, the full branch coverage of the system poses a great challenge. Regarding the mono‐objective algorithms, although they need a second phase of test case selection for reducing the oracle cost, they are very effective in maximizing the branch coverage. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation

This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective AWS method to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the -dimensional objective space. It is demonstrated that the method produces a well-distributed Pareto front mesh for effective visualization, and that it finds solutions in nonconvex regions. Two numerical examples and a simple structural optimization problem are solved as case studies. Presented as paper AIAA-2004-4322 at the 10th AIAA-ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, August 30–September 1, 2004