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The weighted sum of objective functions is one of the simplest fitness functions widely applied in evolutionary algorithms (EAs) for multiobjective programming. However, EAs with this fitness function cannot find uniformly distributed solutions on the entire Pareto front for nonconvex and complex multiobjective programming. In this paper, a novel EA based on adaptive multiple fitness functions and adaptive objective space division is proposed to overcome this shortcoming. The objective space is divided into multiple regions of about the same size by uniform design, and one fitness function is defined on each region by the weighted sum of objective functions to search for the nondominated solutions in this region. Once a region contains fewer nondominated solutions, it is divided into several sub-regions and one additional fitness function is defined on each sub-region. The search will be carried out simultaneously in these sub-regions, and it is hopeful to find more nondominated solutions in such a region. As a result, the nondominated solutions in each region are changed adaptively, and eventually are uniformly distributed on the entire Pareto front. Moreover, the complexity of the proposed algorithm is analyzed. The proposed algorithm is applied to solve 13 test problems and its performance is compared with that of 10 widely used algorithms. The results show that the proposed algorithm can effectively handle nonconvex and complex problems, generate widely spread and uniformly distributed solutions on the entire Pareto front, and outperform those compared algorithms.

In modern techniques and science one meets design problems for multi-factor experiments in a large experimental region where the underlying model is unknown. These problems needs space filling designs. The uniform experimental design seeks its design points to be uniformly scattered on the experimental domain and is one kind of space filling designs that can be used for computer experiments and also for industrial experiments when the underlying model is unknown. In this paper we shall introduce the theory and method of the uniform design and related data analysis and modelling methods. Applications of the uniform design to industry and other areas are discussed.

In this paper, a new variable selection procedure is introduced for the analysis of uniform design and computer experiment. The new procedure is distinguished from the traditional ones in such a way that it deletes insignificant variables and estimates the coefficients of significant variables simultaneously. The new procedure has an oracle property (Fan and Li⁸). It is better than the best subset variable selection in terms of computational cost and model stability. It is superior to the stepwise regression because it does not ignore stochastic errors during the course of selecting variables. The proposed procedure is illustrated by two examples, one is a typical example of uniform design, and the other one is a classical example for computer experiment.

The relationship between the reliability probabilities of the component and the system is hard to get. If this relationship can be obtained easily, the reliability of the system can be calculated by using the reliability structures of the components. The common method to express this relationship is using the linear correlation index, which only shows the linear correlation between the components failure rather than the relationship between high nonlinear functions. In order to describe this relationship accurately and calculate the system reliability using the component reliability structures, a Uniform Design (UD)-Saddlepoint Approximation (SA)-based system reliability analysis method is proposed. The system reliability analysis method is decomposed to three simple steps: (1) calculating the weight coefficient which represents the contribution rate of each component to system reliability, (2) approximating Cumulant Generation Function (CGF) of each component, (3) calculating CGF of the system and approximating the system reliability with SA method. The weight coefficient of each component is derived from UD method, and a variable interval selection method is developed to decrease the required number of samples and increase the accuracy of the weight coefficients. First-Order Saddlepoint Approximation (FOSA) method or Mean-Value First-Order Saddlepoint Approximation (MVFOSA) method is used to analyze the CGF of a component performance function. Then the CGF of the system can be obtained by the weighted addition law by combining the CGFs of components performance functions with the weight coefficients. Finally, the system reliability can be approximated by SA method. Four examples are employed to demonstrate that the new method outperforms other methods for system reliability analysis in terms of efficiency and accuracy.

This study aimed to improve the micromotion of a Semados dental implant system subjected to dynamic chewing loads. Micromotion of the Semados dental implant system with basic dimensions was obtained using dynamic finite element analysis, and five parameters of the implant were selected as the control factors to be improved. A uniform design method was employed to construct a set of experimental simulations. Next, for each experimental simulation, the dynamic finite analysis package ANSYS/LS-DYNA was employed to simulate the behavior of the Semados dental implant model subjected to dynamic chewing loads and then determine the maximum micromotion of the cortical and cancellous bones. Finally, the best design of the experimental simulations that caused the smallest amount of micromotion was selected as the improved design version. Compared to the original design, which experienced micromotion of 74.53 μm, the improved version experienced micromotion of 45.41 μm. The rate of improvement was 39.1%.

How to detect the topological degree (TD) of a function is of vital importance in investigating the existence and the number of zero values in the function, which is a topic of major significance in the theory of nonlinear scientific fields. Usually a sufficient refinement of the boundary of the polyhedron decided by Boult and Sikorski algorithm (BS) is needed as prerequisite when the well known method of Stenger and Kearfott is chosen for computing TD. However two linchpins are indispensable to BS, the parameter δ on the boundary of the polyhedron and an estimation of the Lipschitz constant K of the function, whose computations are analytically difficult. In this paper, through an appropriate scheme that transforms the problems of computing δ and K into searching optimums of two non-differentiable functions, a novel differential evolution algorithm (DE) combined with established techniques is proposed as an alternative method to computing δ and K. Firstly it uses uniform design method to generate the initial population in feasible field so as to have the property of large scale convergence, without better approximation of the unknown parameter as iterative initial point. Secondly, it restrains the normal DE's local convergence limitation virtually through deflection and stretching of objective function. The main advantages of the put algorithm are its simplicity and its ability to work by using function values solely. Finally, details of applying the proposed method into computing δ and K are given, and experimental results on two benchmark problems in contrast to the results reported have demonstrated the promising performance of the proposed algorithm in different scenarios.

Uniform design has become a standard tool in experimental design over the last decade. Uniform design is particularly powerful when the specific form of the response to differing factor levels is unknown. Different criteria such as L₂-discrepancy and modifications thereof like the centered and wrap-around L₂-discrepancies are used to assess the uniformity of U-type designs. Recent advantages in the analysis of these criteria allow for efficient calculation and the derivation of lower bounds. In this contribution, these results are used to construct two-step sequential U-type designs. The construction is based on the optimization heuristic threshold accepting. The results are compared with theoretical lower bounds and ad hoc heuristics for the generation of sequential designs.

Discrepancy is a kind of important measure used in experimental design. Recently, a so-called discrete discrepancy has been applied to evaluate the uniformity of factorial designs. In this paper, we review some recent advances on application of the discrete discrepancy to several common experimental designs and summarize some important results.

Sensitivity analysis plays an important role to help engineers gain knowledge of complex model behaviors and make informed decisions regarding where to spend engineering effort. In design under uncertainty, probabilistic sensitivity analysis (PSA) is performed to quantify the impact of uncertainties in random variables on the uncertainty in model outputs. One of the most challenging issues for PSA is the intensive computational demand for assessing the impact of probabilistic variations. An efficient approach to PSA is presented in this article. Our approach employs the Kolmogorov-Smirnov (KS) distance to quantify the importance of input variables. The saddlepoint approximation approach is introduced to improve the efficiency of generating cumulative distribution functions (CDFs) required for the evaluation of the KS distance. To further improve efficiency, optimized uniform samples are used to replace the direct Monte Carlo simulations for determining the cumulant generating function (CGF) in saddlepoint approximation. Efficient construction of a uniform design necessary to generate the “best” samples in a multidimensional space is presented. Our approach is illustrated with a structural design problem. It has the potential to be the most beneficial for high dimensional engineering design problems that involve expensive computer simulations.

This paper elaborates the kernel selection problem in the majorization framework by Zhang, Fang, Li & Sudjianto (2004) for experimental designs. For designs with qualitative factors, the row-wise coincidence distribution and its raw, central and factorial moments are studied. Under the effects hierarchy principle, two protocols are recommended to employ power and exponential kernels, which are shown equivalent to some classical criteria for fractional factorial designs and uniform designs, respectively. In addition, an extension of majorization framework is given to uniform designs with quantitative factors under wraparound discrepancy criterion.

Consider a design of experiments with mixtures: 0 ⩽ a_i < x_i < b_i ⩽ 1, 1 ⩽ i ⩽ s, x₁ + ⋯ + x_s = 1, where a_i, b_i, 1 ⩽ i ⩽ s are given constants. A method is proposed to treat this model by the theory of uniform distribution in number theory.

Fast ion conductors of Li₂O~SiO₂~V₂O₅ system have been prepared by using Li₂O, SiO₂ and V₂O₅ as starting materials. The optimal ratio of starting materials was designed by Uniform Design. The conductivities of both electric and ionic were investigated. The highest ion conductivity is 1.5 × 10^-4S/cm at ambient temperature for the above lithium fast ion conductor system, the electronic conductivity is 4 orders of magnitude lower than the ionic conductivity.

Compound recipe was the important form in traditional Chinese medicine. A formula is mainly composed of more than two herbs along with application of the single herb. At present, statistics tools were used to optimize the dose or proportion of formula. Here we showed a method based on uniform design and quadratic polynomial regression for the dose and proportion analysis in GeGenQingLian Decoction(GGQLD).The result obtained the different indicators reflected the different principal drug, the different indicators appeared the distinct mechanism of GGQLD in treating the ulcerative colitis(UC).