Aggregate Objective Functions and Pareto Frontiers: Required Relationships and Practical Implications

This paper addresses the problem of capturing Pareto optimal points on non-convex Pareto frontiers, which are encountered in nonlinear multiobjective optimization problems in computational engineering design optimization. The emphasis is on the choice of the aggregate objective function (AOF) of the objectives that is employed to capture Pareto optimal points. A fundamental property of the aggregate objective function, the admissibility property, is developed and its equivalence to the coordinatewise increasing property is established. Necessary and sufficient conditions for such an admissible aggregate objective function to capture Pareto optimal points are derived. Numerical examples illustrate these conditions in the biobjective case. This paper demonstrates in general terms the limitation of the popular weighted-sum AOF approach, which captures only convex Pareto frontiers, and helps us understand why some commonly used AOFs cannot capture desirable Pareto optimal points, and how to avoid this situation in practice. Since nearly all applications of optimization in engineering design involve the formation of AOFs, this paper is of direct theoretical and practical usefulness.     

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 design optimization of antenna structures using sequential domain patching with automated patch size determination

In this article, a simple yet efficient and reliable technique for fully automated multi-objective design optimization of antenna structures using sequential domain patching (SDP) is discussed. The optimization procedure according to SDP is a two-step process: (i) obtaining the initial set of Pareto-optimal designs representing the best possible trade-offs between considered conflicting objectives, and (ii) Pareto set refinement for yielding the optimal designs at the high-fidelity electromagnetic (EM) simulation model level. For the sake of computational efficiency, the first step is realized at the level of a low-fidelity (coarse-discretization) EM model by sequential construction and relocation of small design space segments (patches) in order to create a path connecting the extreme Pareto front designs obtained beforehand. The second stage involves response correction techniques and local response surface approximation models constructed by reusing EM simulation data acquired in the first step. A major contribution of this work is an automated procedure for determining the patch dimensions. It allows for appropriate selection of the number of patches for each geometry variable so as to ensure reliability of the optimization process while maintaining its low cost. The importance of this procedure is demonstrated by comparing it with uniform patch dimensions.     

A MULTIOBJECTIVE INTERACTIVE SEQUENTIAL HYBRID OPTIMIZATION TECHNIQUE FOR DESIGN DECISION MAKING

The paper proposes an interactive method for obtaining a solution to a multiple objective design decision making problem. The focus is on generating Pareto solutions including those that are in the non-convex region, and are desirable to obtain in an engineering design context. After the generation of a small subset of the Pareto solutions, the designer's feedback is elicited in order to eliminate part of the subset. The process is repeated until il iteratively narrows down the Pareto solution set to a size small enough so that the designer is able to easily select a final solution. The advantage of this approach is that the designer can view a few sample points from the Pareto set before zooming into the region preferred and without expending computation time in generating a complete Pareto set. The process has been demonstrated with the help of an example, the design of a fleet of ships, that has mixed-discrete variables and hence a genetic algorithm is used as the optimizer.     

Adapting the Hypervolume Quality Indicator to Quantify Trade‐offs and Search Efficiency for Multiple Criteria Decision Making Using Pareto Fronts

When choosing a best solution based on simultaneously balancing multiple objectives, the Pareto front approach allows promising solutions across the spectrum of user preferences for the weightings of the objectives to be identified and compared quantitatively. The shape of the complete Pareto front provides useful information about the amount of trade‐off between the different criteria and how much compromise is needed from some criterion to improve the others. Visualizing the Pareto front in higher (3 or more) dimensions becomes difficult, so a numerical measure of this relationship helps capture the degree of trade‐off. The traditional hypervolume quality indicator based on subjective scaling for multiple criteria optimization method comparison provides an arbitrary value that lacks direct interpretability. This paper proposes an interpretable summary for quantifying the nature of the relationship between criteria with a standardized hypervolume under the Pareto front (HVUPF) for a flexible number of optimization criteria, and demonstrates how this single number summary can be used to evaluate and compare the efficiency of different search methods as well as tracking the search progress in populating the complete Pareto front. A new HVUPF growth plot is developed for quantifying the performance of a search method on completeness, efficiency, as well as variability associated with the use of random starts, and offers an effective approach for method assessment and comparison. Two new enhancements for the algorithm to populate the Pareto front are described and compared with the HVUPF growth plot. The methodology is illustrated with an optimal screening design example, where new Pareto search methods are proposed to improve computational efficiency, but is broadly applicable to other multiple criteria optimization problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Computation of pareto optima in structural design

The structural design problem is acknowledge to be commonly multi-criteria in nature. The various bases for multi-criteria optimization methodologies are outlined and a computationally viable method for generating Pareto optimal solutions is adopted for the structural design problem where the criteria may be non-commensurable. A numerical example on optimal truss design illustrating non-commensurable criteria is given.     

Using Genetic Algorithms to Design Experiments: A Review

Genetic algorithms (GAs) have been used in many disciplines to optimize solutions for a broad range of problems. In the last 20 years, the statistical literature has seen an increase in the use and study of this optimization algorithm for generating optimal designs in a diverse set of experimental settings. These efforts are due in part to an interest in implementing a novel methodology as well as the hope that careful application of elements of the GA framework to the unique aspects of a designed experiment problem might lead to an efficient means of finding improved or optimal designs. In this paper, we explore the merits of using this approach, some of the aspects of design that make it a unique application relative to other optimization scenarios, and discuss elements which should be considered for an effective implementation. We conclude that the current GA implementations can, but do not always, provide a competitive methodology to produce substantial gains over standard optimal design strategies. We consider both the probability of finding a globally optimal design as well as the computational efficiency of this approach. Copyright © 2014 John Wiley & Sons, Ltd.     

A unified parametric design approach to structural shape optimization

This paper presents a general parametric design approach for 2-D shape optimization problems. This approach has been achieved by integrating practical design methodologies into numerical procedures. It is characterized by three features: (i) automatic selection of a minimum number of shape design variables based on the CAD geometric model; (ii) integration of sequential convex programming algorithms to solve equality constrained optimization problems; (iii) efficient sensitivity analysis by means of the improved semi-analytical method. It is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities. Numerical solutions are performed to demonstrate the applicability of the proposed approach. A discussion of the results is also given:     

A probabilistic computational framework for bridge network optimal maintenance scheduling

This paper presents a probabilistic computational framework for the Pareto optimization of the preventive maintenance applications to bridges of a highway transportation network. The bridge characteristics are represented by their uncertain reliability index profiles. The in/out of service states of the bridges are simulated taking into account their correlation structure. Multi-objective Genetic Algorithms have been chosen as numerical tool for the solution of the optimization problem. The design variables of the optimization are the preventive maintenance schedules of all the bridges of the network. The two conflicting objectives are the minimization of the total present maintenance cost and the maximization of the network performance indicator. The final result is the Pareto front of optimal solutions among which the managers should chose, depending on engineering and economical factors. A numerical example illustrates the application of the proposed approach.     

A new efficient convergence criterion for reducing computational expense in topology optimization: reducible design variable method

A new efficient convergence criterion, named the reducible design variable method (RDVM), is proposed to save computational expense in topology optimization. There are two types of computational costs: one is to calculate the governing equations, and the other is to update the design variables. In conventional topology optimization, the number of design variables is usually fixed during the optimization procedure. Thus, the computational expense linearly increases with respect to the iteration number. Some design variables, however, quickly converge and some other design variables slowly converge. The idea of the proposed method is to adaptively reduce the number of design variables on the basis of the history of each design variable during optimization. Using the RDVM, those design variables that quickly converge are not considered as design variables for the next iterations. This means that the number of design variables can be reduced to save the computational costs of updating design variables. Then, the iteration will repeat until the number of design variables becomes 0. In addition, the proposed method can lead to faster convergence of the optimization procedure, which indeed is a more significant time saving. It is also revealed that the RDVM gives identical optimal solutions as those by conventional methods. We confirmed the numerical efficiency and solution effectiveness of the RDVM with respect to two types of optimization: static linear elastic minimization, and linear vibration problems with the first eigenvalue as the objective function for maximization. Copyright © 2012 John Wiley & Sons, Ltd.     

A global single-loop deterministic approach for reliability-based design optimization of truss structures with continuous and discrete design variables

A single-loop deterministic method (SLDM) has previously been proposed for solving reliability-based design optimization (RBDO) problems. In SLDM, probabilistic constraints are converted to approximate deterministic constraints. Consequently, RBDO problems can be transformed into approximate deterministic optimization problems, and hence the computational cost of solving such problems is reduced significantly. However, SLDM is limited to continuous design variables, and the obtained solutions are often trapped into local extrema. To overcome these two disadvantages, a global single-loop deterministic approach is developed in this article, and then it is applied to solve the RBDO problems of truss structures with both continuous and discrete design variables. The proposed approach is a combination of SLDM and improved differential evolution (IDE). The IDE algorithm is an improved version of the original differential evolution (DE) algorithm with two improvements: a roulette wheel selection with stochastic acceptance and an elitist selection technique. These improvements are applied to the mutation and selection phases of DE to enhance its convergence rate and accuracy. To demonstrate the reliability, efficiency and applicability of the proposed method, three numerical examples are executed, and the obtained results are compared with those available in the literature.     

考虑双目标的JIT生产系统看板数量设定

针对现有JIT系统看板数量决策问题研究多以单目标为主的不足,提出了一种基于实验设计的双目标JIT生产系统看板数量设定方法。该方法同时考虑了高订单满足率和低系统平均在制品水平的双目标优化,以B公司CR油嘴JIT生产系统为例,建立了该JIT生产线的Witness仿真模型以实现数据的收集,以各看板循环回路的看板数量和看板容量进行水平设定,并进行正交实验设计及数据的直观分析处理,然后采用全因子实验的方法,基于帕累托最优的思想获得生产系统看板数量帕累托最优解,形成近似最优看板数量组合的帕累托最优前沿。生产管理人员可根据不同的生产计划和绩效目标从组合中选择合适的看板数量。最后的研究结果验证了该方法的可行性和有效性。     

Simulated annealing-based multiobjective algorithms and their application for system reliability

The paper presents five simulated annealing based multiobjective algorithms - SMOSA, UMOSA, PSA, PDMOSA and WMOSA. All of these algorithms aim to find a Pareto set of solutions of a system reliability multiobjective optimization problem in a short time. In each algorithm the solution vector explores its neighborhood in a way similar to that of Classical Simulated Annealing. All the algorithms are problem-specific and if the true Pareto-optimal set has few solutions, UMOSA, SMOSA, PSA and WMOSA work better than PDMOSA. In some cases, PSA works the best. The computational cost is least in the case of the WMOSA algorithm since it does not need to use the penalty function approach to handle the constraints, and is the maximum with PDMOSA since it requires two sets of Pareto solutions.     

Surrogate-based multi-objective optimization of a composite laminate with curvilinear fibers

A variable stiffness design can increase the structural performance of a composite plate and provides flexibility for trade-offs between structural properties. In this paper, we examine the simultaneous optimization of stiffness and buckling load of a composite laminate plate with curvilinear fiber paths. The problem, which falls in the area of multi-objective optimization, is formulated and solved through a surrogate-based optimization algorithm capable of finding the set of optimum Pareto solutions. We integrate surrogate modeling into an evolutionary algorithm to reduce the high computational cost required to solve the optimization process. The results show that a curvilinear fiber path can increase both buckling load and stiffness simultaneously over the quasi-isotropic laminate. Furthermore, the optimum direction for varying the fiber angle is dependent on the loading direction and boundary conditions. The results for a plate under uniform compression with free transverse edges shows that varying the fiber orientation perpendicular to the loading direction can increase the buckling load by 116% with respect to that of a quasi-isotropic laminate.     

A new optimization algorithm based on a combination of particle swarm optimization,convergence and divergence operators for single-objective and multi-objective problems

Particle swarm optimization (PSO) is a randomized and population-based optimization method that was inspired by the flocking behaviour of birds and human social interactions. In this work, multi-objective PSO is modified in two stages. In the first stage, PSO is combined with convergence and divergence operators. Here, this method is named CDPSO. In the second stage, to produce a set of Pareto optimal solutions which has good convergence, diversity and distribution, two mechanisms are used. In the first mechanism, a new leader selection method is defined, which uses the periodic iteration and the concept of the particle's neighbour number. This method is named periodic multi-objective algorithm. In the second mechanism, an adaptive elimination method is employed to limit the number of non-dominated solutions in the archive, which has influences on computational time, convergence and diversity of solution. Single-objective results show that CDPSO performs very well on the complex test functions in terms of solution accuracy and convergence speed. Furthermore, some benchmark functions are used to evaluate the performance of periodic multi-objective CDPSO. This analysis demonstrates that the proposed algorithm operates better in three metrics through comparison with three well-known elitist multi-objective evolutionary algorithms. Finally, the algorithm is used for Pareto optimal design of a two-degree of freedom vehicle vibration model. The conflicting objective functions are sprung mass acceleration and relative displacement between sprung mass and tyre. The feasibility and efficiency of periodic multi-objective CDPSO are assessed in comparison with multi-objective modified NSGAII.     

平流层飞艇的多目标优化与决策

该文建立了以平流层飞艇阻力最小、自重最轻、极限承载力最大及刚度最大为优化目标的多目标优化模型;采用强度Pareto进化算法(SPEA)进行了多目标优化设计;基于优化所得的Pareto解集,采用基于信噪比的决策方法选择满足实际需要的最终方案。结果表明:采用的SPEA算法是合理有效的,可以得到非劣解分布较均匀的Pareto曲面;通过基于信噪比的决策方法,可从非劣解集中获得满足实际要求的最稳健设计方案。     

ROBUSTNESS OPTIMIZATION FOR CONSTRAINED NONLINEAR PROGRAMMING PROBLEMS∗

In realistic situations engineering designs should take into consideration random aberrations from the stipulated design variables arising from manufacturing variability. Moreover, many environmental parameters are often stochastic in nature. Traditional nonlinear optimization attempts to find a deterministic optimum of a cost function and does not take into account the effect of these random variations on the objective. This paper attempts to devise a technique for finding optima of constrained nonlinear functions that are robust with respect to such variations. The expectation of the function over a domain of aberrations in the parameters is taken as a measure of ‘robustness’ of the function value at a point. It is pointed out that robustness optimization is ideally an attempt to trade off between ‘optimality’ and ‘robustness’. A newly-developed multi-criteria optimization technique known as Normal-Boundary Intersection is used to find evenly-spaced points on the Pareto curve for the ‘optimality’ and ‘robustness’ criteria. This Pareto curve enables the user to make the trade-off decision explicitly, free of arbitrary ‘weighting’ parameters.

This paper also formulates a derivative-based approximation for evaluating the expected value of the objective function on the nonlinear manifold defined by the state equations for the system. Existing procedures for evaluating the expectation usually involve numerical integration techniques requiring many solutions of the state equations for one evaluation of the expectation. The procedure presented here bypasses the need for multiple solutions of the state equations and hence provides a cheaper and more easily optimizable approximation to the expectation. Finally, this paper discusses how nonlinear inequality constraints should be treated in the presence of random parameters in the design. Computational results are presented for finding a robust optimum of a nonlinear structural optimization problem.     

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.     

Design optimization using multiple dominance relations

A challenge in engineering design is to choose suitable objectives and constraints from many quantities of interest, while ensuring an optimization is both meaningful and computationally tractable. We propose an optimization formulation that can take account of more quantities of interest than existing formulations, without reducing the tractability of the problem. This formulation searches for designs that are optimal with respect to a binary relation within the set of designs that are optimal with respect to another binary relation. We then propose a method of finding such designs in a single optimization by defining an overall ranking function to use in optimizers, reducing the cost required to solve this formulation. In a design under uncertainty problem, our method obtains the most robust design that is not stochastically dominated faster than a multiobjective optimization. In a car suspension design problem, our method obtains superior designs according to a k-optimality condition than previously suggested multiobjective approaches to this problem. In an airfoil design problem, our method obtains designs closer to the true lift/drag Pareto front using the same computational budget as a multiobjective optimization.