Design for Six Sigma through collaborative multiobjective optimization

This paper presents a global methodology for designing product for Six Sigma. First, we combine a feasibility-modeling technique with an interactive multiobjective algorithm taking into account the decision maker’s preferences (IMOP) to generate several Pareto-optimal solutions that maintain a probability of constraint satisfaction. These solutions are called reliable Pareto-optimal solutions.The solutions found by the algorithm fulfill as much as possible the decision makers’ requirements. Second, we develop a procedure for choosing a solution for implementation from among the reliable Pareto-optimal solutions generated by the algorithm. This procedure is based on the robust design and philosophy of Six Sigma. Finally, the critical characteristics are identified to help the managers develop the manufacturing system and its related control plans in order to achieve quality products. The proposed methodology is applied to vehicle crash-worthiness design optimization for side impact with structural weight and front door velocity under side impact as objectives.     

Crashworthiness design of vehicle by using multiobjective robust optimization

Although deterministic optimization has to a considerable extent been successfully applied in various crashworthiness designs to improve passenger safety and reduce vehicle cost, the design could become less meaningful or even unacceptable when considering the perturbations of design variables and noises of system parameters. To overcome this drawback, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on multiple crashworthiness criteria, where several different sigma criteria are adopted to measure the variations. As an example, a full front impact of vehicle is considered with increase in energy absorption and reduction of structural weight as the design objectives, and peak deceleration as the constraint. A multiobjective particle swarm optimization is applied to generate robust Pareto solution, which no longer requires formulating a single cost function by using weighting factors or other means. From the example, a clear compromise between the Pareto deterministic and robust designs can be observed. The results demonstrate the advantages of using multiobjective robust optimization, with not only the increase in the energy absorption and decrease in structural weight from a baseline design, but also a significant improvement in the robustness of optimum.     

Crashworthiness design of multi-component tailor-welded blank (TWB) structures

Crashworthiness of tailor-welded blank (TWB) structures signifies an increasing concern in lightweight design of vehicle. Although multiobjective optimization (MOO) has to a considerable extent been successfully applied to enhance crashworthiness of vehicular structures, majority of existing designs were restricted to single or uniform thin-walled components. Limited attention has been paid to such non-uniform components as TWB structures. In this paper, MOO of a multi-component TWB structure that involves both the B-pillar and inner door system subjected to a side impact, is proposed by considering the structural weight, intrusive displacements and velocity of the B-pillar component as objectives, and the thickness in different positions and the height of welding line of B-pillar as the design variables. The MOO problem is formulated by using a range of different metamodeling techniques, including response surface methodology (RSM), artificial neural network (ANN), radial basis functions (RBF), and Kriging (KRG), to approximate the sophisticated nonlinear responses. By comparison, it is found that the constructed metamodels based upon the radial basis function (RBF, especially multi-quadric model, namely RBF-MQ) fit to the design of experiment (DoE) checking points well and are employed to carry out the design optimization. The performance of the TWB B-pillar and indoor panel system can be improved by optimizing the thickness of the different parts and height of the welding line. This study demonstrated that the multi-component TWB structure can be optimized to further enhance the crashworthiness and reduce the weight, offering a new class of structural/material configuration for lightweight design.     

Multiobjective optimization for crash safety design of vehicles using stepwise regression model

In automotive industry, structural optimization for crashworthiness criteria is of special importance. Due to the high nonlinearities, however, there exists substantial difficulty to obtain accurate continuum or discrete sensitivities. For this reason, metamodel or surrogate model methods have been extensively employed in vehicle design with industry interest. This paper presents a multiobjective optimization procedure for the vehicle design, where the weight, acceleration characteristics and toe-board intrusion are considered as the design objectives. The response surface method with linear and quadratic basis functions is employed to formulate these objectives, in which optimal Latin hypercube sampling and stepwise regression techniques are implemented. In this study, a nondominated sorting genetic algorithm is employed to search for Pareto solution to a full-scale vehicle design problem that undergoes both the full frontal and 40% offset-frontal crashes. The results demonstrate the capability and potential of this procedure in solving the crashworthiness design of vehicles.     

一种新的基于进化策略的多目标优化算法

用进化策略求解多目标优化问题时，为了提高解在决策变量空间中的搜索能力和保证Pareto前沿的多样性，提出了一种新的基于进化策略的多目标优化算法。运用自适应变异步长的进化策略，使解在决策变量空间中进行全局和局部搜索；并引入非劣解按一定比例进入下一代的方法，使完全被占优的个体有机会参与到下一代的繁殖，保持了解在Pareto前沿的多样性。该算法在保证解在决策空间多样性的同时，也保持了Pareto前沿的多样性。仿真实验表明，该算法具有良好的搜索性能。     

Factor screening and multivariable crashworthiness optimization for vehicle side impact by factorial design

This paper demonstrates the application of factor screening to multivariable crashworthiness design of the vehicle body subjected to the side impact loading. Crashworthiness, influenced unequally by disparate factors such as the structural dimensions and material parameters, represents a natural benchmark criterion to judge the passive safety quality of the automobile design. In order to single out the active factors which pose a profound influence on the crashworthiness of vehicle bodies subjected to the side impact loading, the unreplicated saturated factorial design is adopted to tackle the obstacle from the factor screening due to its huge benefits in the efficiency and accuracy. In this paper, two different kinds of vehicles are analyzed by the unreplicated saturated factorial design for multivariable crashworthiness and the optimization results enhance the crashworthiness of vehicle. This method overcomes the limitations of design variables selection which depends on experience, and solves the in-efficiency problems caused by the direct optimization design without the selection of variables. It will shorten the design cycles, decrease the development costs and will have a certain reference value for the improvement of the vehicle’s crashworthiness performance.     

Reliability-Based Optimization Using Evolutionary Algorithms

Uncertainties in design variables and problem parameters are often inevitable and must be considered in an optimization task if reliable optimal solutions are sought. Besides a number of sampling techniques, there exist several mathematical approximations of a solution's reliability. These techniques are coupled in various ways with optimization in the classical reliability-based optimization field. This paper demonstrates how classical reliability-based concepts can be borrowed and modified and, with integrated single and multiobjective evolutionary algorithms, used to enhance their scope in handling uncertainties involved among decision variables and problem parameters. Three different optimization tasks are discussed in which classical reliability-based optimization procedures usually have difficulties, namely (1) reliability-based optimization problems having multiple local optima, (2) finding and revealing reliable solutions for different reliability indices simultaneously by means of a bi-criterion optimization approach, and (3) multiobjective optimization with uncertainty and specified system or component reliability values. Each of these optimization tasks is illustrated by solving a number of test problems and a well-studied automobile design problem. Results are also compared with a classical reliability-based methodology.     

Structural optimization with crashworthiness constraints

An automated structural design methodology has been devised which simultaneously considers design criteria associated with both linear elastic and crashworthiness loading conditions. This method is developed within the context of a nonlinear mathematical programming based structural optimization capability using an efficient two-phased crashworthiness analysis technique. Specially constructed nonlinear approximations for the crashworthiness constraints are employed to further reduce the computational burden during the optimization process. This methodology is demonstrated on an automobile structural design problem. It is shown that more mass efficient designs can be obtained by simultaneously considering elastic and crashworthiness design criteria as compared to a sequential approach in which the structure is first designed for the elastic loads and then modified to satisfy the crashworthiness criteria.     

Adaptive weighted-sum method for bi-objective optimization: Pareto front generation

This paper presents a new method that effectively determines a Pareto front for bi-objective optimization with potential application to multiple objectives. A traditional method for multiobjective optimization is the weighted-sum method, which seeks Pareto optimal solutions one by one by systematically changing the weights among the objective functions. Previous research has shown that this method often produces poorly distributed solutions along a Pareto front, and that it does not find Pareto optimal solutions in non-convex regions. The proposed adaptive weighted sum method focuses on unexplored regions by changing the weights adaptively rather than by using a priori weight selections and by specifying additional inequality constraints. It is demonstrated that the adaptive weighted sum method produces well-distributed solutions, finds Pareto optimal solutions in non-convex regions, and neglects non-Pareto optimal solutions. This last point can be a potential liability of Normal Boundary Intersection, an otherwise successful multiobjective method, which is mainly caused by its reliance on equality constraints. The promise of this robust algorithm is demonstrated with two numerical examples and a simple structural optimization problem.     

Multivariable crashworthiness optimization of vehicle body by unreplicated saturated factorial design

Crashworthiness, influenced unequally by disparate factors such as the structural dimensions and the material parameters, represents a natural benchmark criterion to judge the passive safety quality of the automobile design. The unreplicated saturated factorial design has enjoyed a remarkable success in the factor screening of different industrial regions due to its huge benefits in the efficiency and accuracy. In order to single out the active factors which pose a profound impact on the crashworthiness, this paper introduces an unreplicated saturated factorial design to tackle the obstacle from the factor screening during the multivariable crashworthiness optimization design of the whole vehicle body. Three unreplicated saturated factorial design methods, including the normal or half-normal probability plot method, Dong93 method, and PSZ method, are employed to capture the active factors while D-optimal design is presented to obtain the design sampling points and to construct the response surface model for the crashworthiness optimization problem. Finally, multi-island genetic algorithm (MIGA) and sequential quadratic programming (SQP)-NLPQL are utilized to obtain the Pareto set of the optimal solution for the multivariable crashworthiness optimization design of the vehicle body under the full-scale frontal impact loading.