Shape optimization of preform tools in forging of aerofoil using a metamodel-assisted multi-island genetic algorithm

Preform design plays an important role in improving the material flow, mechanical properties and reducing defects for forgings with complex shapes. In this paper, a study on shape optimization of preform tools in forging of an airfoil is carried out based on a multi-island genetic algorithm combined with a metamodel technique. An optimal Latin hypercube sampling technique is employed for sampling with the expected coverage of parameter space. Finite element (FE) simulations of multistep forging processes are implemented to obtain the objective function values for evaluating the forging qualities. For facilitating the optimization process, a radial basis function surrogate model is established to predict the responses of the hot forging process to the variation of the preform tool shape. In consideration of the compromise between different optimal objectives, a set of Pareto-optimal solutions are identified by the suggested genetic algorithm to provide more selections. Finally, according to the proposed fitness function, the best solution of multi-objective optimization on the Pareto front is confirmed and the corresponding preform tool shape proves optimal performances with substantially improved forging qualities via FE validation.     

Multi-objective scheduling problems: Determination of pruned Pareto sets

There are often multiple competing objectives for industrial scheduling and production planning problems. Two practical methods are presented to efficiently identify promising solutions from among a Pareto optimal set for multi-objective scheduling problems. Generally, multi-objective optimization problems can be solved by combining the objectives into a single objective using equivalent cost conversions, utility theory, etc., or by determination of a Pareto optimal set. Pareto optimal sets or representative subsets can be found by using a multi-objective genetic algorithm or by other means. Then, in practice, the decision maker ultimately has to select one solution from this set for system implementation. However, the Pareto optimal set is often large and cumbersome, making the post-Pareto analysis phase potentially difficult, especially as the number of objectives increase. Our research involves the post Pareto analysis phase, and two methods are presented to filter the Pareto optimal set to determine a subset of promising or desirable solutions. The first method is pruning using non-numerical objective function ranking preferences. The second approach involves pruning by using data clustering. The k-means algorithm is used to find clusters of similar solutions in the Pareto optimal set. The clustered data allows the decision maker to have just k general solutions from which to choose. These methods are general, and they are demonstrated using two multi-objective problems involving the scheduling of the bottleneck operation of a printed wiring board manufacturing line and a more general scheduling problem.     

水平轴风力机叶片最优体型设计

从降低风力机成本及提高风力机运行可靠性的角度,建立以最小的叶片质量获得最大的年发电量两个目标函数,以叶片体型参数及运行参数等关键参数为设计变量,以叶片的强度为主要约束条件的多目标优化设计数学模型,采用Pareto遗传算法对某1.5MW风力机叶片进行最优体型设计研究。与原设计叶片相比,最优体型设计方案提高了风力机的年发电量、减轻了叶片质量,说明优化设计模型合理有效。     

Decomposition-based multi-objective differential evolution particle swarm optimization for the design of a tubular permanent magnet linear synchronous motor

This article proposes a decomposition-based multi-objective differential evolution particle swarm optimization (DMDEPSO) algorithm for the design of a tubular permanent magnet linear synchronous motor (TPMLSM) which takes into account multiple conflicting objectives. In the optimization process, the objectives are evaluated by an artificial neural network response surface (ANNRS), which is trained by the samples of the TPMSLM whose performances are calculated by finite element analysis (FEA). DMDEPSO which hybridizes differential evolution (DE) and particle swarm optimization (PSO) together, first decomposes the multi-objective optimization problem into a number of single-objective optimization subproblems, each of which is associated with a Pareto optimal solution, and then optimizes these subproblems simultaneously. PSO updates the position of each particle (solution) according to the best information about itself and its neighbourhood. If any particle stagnates continuously, DE relocates its position by using two different particles randomly selected from the whole swarm. Finally, based on the DMDEPSO, optimization is gradually carried out to maximize the thrust of TPMLSM and minimize the ripple, permanent magnet volume, and winding volume simultaneously. The result shows that the optimized TPMLSM meets or exceeds the performance requirements. In addition, comparisons with chosen algorithms illustrate the effectiveness of DMDEPSO to find the Pareto optimal solutions for the TPMLSM optimization problem.     

Aerothermal shape optimization for a double row of discrete film cooling holes on the suction surface of a turbine vane

A multiple-objective optimization is implemented for a double row of staggered film holes on the suction surface of a turbine vane. The optimization aims to maximize the film cooling performance, which is assessed using the cooling effectiveness, while minimizing the corresponding aerodynamic loss, which is measured with a mass-averaged total pressure coefficient. Three geometric variables defining the hole shape are optimized: the conical expansion angle, compound angle and length to diameter ratio of the non-diffused portion of the hole. The optimization employs a non-dominated sorting genetic algorithm coupled with an artificial neural network to generate the Pareto front. Reynolds-averaged Navier–Stokes simulations are employed to construct the neural network and investigate the aerodynamic and thermal optimum solutions. The optimum designs exhibit improved performance in comparison to the reference design. The optimization methodology allowed investigation into the impact of varying the geometric variables on the cooling effectiveness and the aerodynamic loss.     

Trade-off analysis approach for interactive nonlinear multiobjective optimization

When solving multiobjective optimization problems, there is typically a decision maker (DM) who is responsible for determining the most preferred Pareto optimal solution based on his preferences. To gain confidence that the decisions to be made are the right ones for the DM, it is important to understand the trade-offs related to different Pareto optimal solutions. We first propose a trade-off analysis approach that can be connected to various multiobjective optimization methods utilizing a certain type of scalarization to produce Pareto optimal solutions. With this approach, the DM can conveniently learn about local trade-offs between the conflicting objectives and judge whether they are acceptable. The approach is based on an idea where the DM is able to make small changes in the components of a selected Pareto optimal objective vector. The resulting vector is treated as a reference point which is then projected to the tangent hyperplane of the Pareto optimal set located at the Pareto optimal solution selected. The obtained approximate Pareto optimal solutions can be used to study trade-off information. The approach is especially useful when trade-off analysis must be carried out without increasing computation workload. We demonstrate the usage of the approach through an academic example problem.     

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.     

Multi-objective optimization of a stirling-type pulse tube refrigerator

A genetic algorithm (GA) optimization method which is coupled to a one-dimensional finite volume method is proposed and implemented as a computer program for the modeling and optimization of a stirling-type pulse tube refrigerator (PTR). The multi-objective optimization procedure is applied to provide the optimization design parameters which are charge pressure, operating frequency, and temperature of after-cooler as well as swept volume of compressor. The procedure is selected to obtain the maximum coefficient of performance (COP) and the minimum cooling temperature (T_cold) as two objective functions. In order to validate the simulation code, the results were compared with the results of other models and experiments. The results showed a reasonably well agreement between simulation output and experimental data. The results of optimal designs are a set of multiple optimum solutions, called Pareto optimal solutions. Moreover, the closed form relations between two objectives are derived for Pareto optimal solutions of pulse tube refrigerator. Finally, a sensitivity analysis of the variation of each design parameter on both objective functions was carried out as well and the results are presented. As a result, the COP is more sensitive than T_cold in the optimum design points. The frequency of refrigerator is the most sensitive factor which affects the COP even with little changes.     

Level Diagrams analysis of Pareto Front for multiobjective system redundancy allocation

Reliability-based and risk-informed design, operation, maintenance and regulation lead to multiobjective (multicriteria) optimization problems. In this context, the Pareto Front and Set found in a multiobjective optimality search provide a family of solutions among which the decision maker has to look for the best choice according to his or her preferences. Efficient visualization techniques for Pareto Front and Set analyses are needed for helping decision makers in the selection task.In this paper, we consider the multiobjective optimization of system redundancy allocation and use the recently introduced Level Diagrams technique for graphically representing the resulting Pareto Front and Set. Each objective and decision variable is represented on separate diagrams where the points of the Pareto Front and Set are positioned according to their proximity to ideally optimal points, as measured by a metric of normalized objective values. All diagrams are synchronized across all objectives and decision variables. On the basis of the analysis of the Level Diagrams, we introduce a procedure for reducing the number of solutions in the Pareto Front; from the reduced set of solutions, the decision maker can more easily identify his or her preferred solution.     

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.     

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.     

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.     

基于差异演化算法和多属性决策的机电系统可靠性多目标优化设计

针对机电系统可靠性设计问题,以可靠性和费用(或体积等)最优为目标建立可靠性设计的多目标优化模型．提出了自适应多目标差异演化算法,该算法提出了自适应缩放因子和混沌交叉率,采用改进的快速排序方法构造Pareto最优解,采用NSGA-II的拥挤操作对档案文件进行消减．采用自适应多目标差异演化算法获得多目标问题的Pareto最优解,利用TOPSIS方法对Pareto最优解进行多属性决策．实际工程结果表明：自适应多目标差异演化算法调节参数更少,且求得的Pareto最优解分布均匀;采用基于TOPSIS的多属性决策方法得到的结果合理可行．     

MONSS: A multi-objective nonlinear simplex search approach

This article presents a novel methodology for dealing with continuous box-constrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple elements of the Pareto optimal set. The search is directed by a well-distributed set of weight vectors, each of which defines a scalarization problem that is solved by deforming a simplex according to the movements described by Nelder and Mead's method. Considering an MOP with n decision variables, the simplex is constructed using n+1 solutions which minimize different scalarization problems defined by n+1 neighbor weight vectors. All solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a proper representation of the Pareto optimal front. In this article, it is shown that a well-designed strategy using just mathematical programming techniques can be competitive with respect to the state-of-the-art multi-objective evolutionary algorithms against which it was compared.     

基于流场分析的某割草车节能优化设计

为了提高某割草车的工作效率,降低其长时间工作时的能源损耗,对其割台中的刀盘和刀片进行建模与仿真分析,通过对刀盘角区和刀片形状进行优化,达到节能的效果。首先,对割草过程中刀盘内的流场情况进行数值模拟,分析了刀片和刀盘所受的压力、刀片的速度以及刀片所受扭矩;其次,基于流场分析规律,对刀盘角区和刀片的形状进行优化,提出3个优化方案;然后,以扭矩的大小作为方案是否节能的主要评价指标,并与原方案进行对比分析,得到节能效果最优的设计方案;最后,将优化后的刀片加工成形,装在实车上进行试验,试验结果与仿真结果的误差小于5%,说明仿真结果正确,同时试验结果验证了该优化方案具有良好的节能效果。仿真与试验结果表明：刀片形状对割草机工作过程中的流场及扭矩等有重要影响,异形刀片有助于减小流场涡流与风阻,从而减小刀片上的扭矩,达到节能提效的目的。     

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.     

Uncertainty analysis of trade‐offs between multiple responses using hypervolume

When multiple responses are considered in process optimization, the degree to which they can be simultaneously optimized depends on the optimization objectives and the amount of trade‐offs between the responses. The normalized hypervolume of the Pareto front is a useful summary to quantify the amount of trade‐offs required to balance performance across the multiple responses. To quantify the impact of uncertainty of the estimated response surfaces and add realism to what future data to expect, 2 versions of the scaled normalized hypervolume of the Pareto front are presented. To demonstrate the variation of the hypervolume distributions, we explore a case study for a chemical process involving 3 responses, each with a different type of optimization goal. Results show that the global normalized hypervolume characterizes the proximity to the ideal results possible, while the instance‐specific summary considers the richness of the front and the severity of trade‐offs between alternatives. The 2 scaling schemes complement each other and highlight different features of the Pareto front and hence are useful to quantify what solutions are possible for simultaneous optimization of multiple responses.