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This study aims to research the influence of non-probabilistic design variables on interval robust optimization of electric multiple units (EMU) brake module, therefore obtain the reasonable of design variables of the EMU brake module.

A robust optimization model of the EMU brake module based on interval analysis is established. This model also considers the dimension tolerance of design variables, and it uses symmetric tolerance to describe the uncertainty of design variables. The interval order relation and possibility degree of interval number are employed to deal with the uncertainty of objective function and constraint condition, respectively. On this basis, a multiobjective robust optimization model in view of interval analysis is established and applied to the robust optimization of the EMU brake module.

Compared with the traditional method and the method proposed in the reference, the maximum stress fluctuation of the EMU brake module structure is smaller after using the method proposed in this paper, which indicates that the robustness of the maximum stress of the structure has been improved. In addition, the weight and strength of the structure meet the design requirements. It shows that this method and model introduced in this research have certain feasibility.

This study is the first attempt to apply the robust optimization model based on interval analysis to the optimization of EMU structure and obtain the optimal solution set that meets the design requirements. Therefore, this study provides an idea for nonprobabilistic robust optimization of the EMU structure.

The purpose of this paper is to propose a robust optimization strategy to deal with the aerodynamic optimization issue, which does not need a large sum of information on the uncertainty of input parameters.

Interval numbers were adopted to describe the uncertain input, which only requires bounds and does not necessarily need probability distributions. Based on the method, model outputs were also regarded as intervals. To identify a better solution, an order relation was used to rank interval numbers.

Based on intervals analysis method, the uncertain optimization problem was transformed into nested optimization. The outer optimization was used to optimize the design vector, and inner optimization was used to compute the interval of model outputs. A flying wing aircraft was used as a basis for uncertainty optimization through the suggested optimization strategy, and optimization results demonstrated the validity of the method.

In aircraft conceptual design, the uncertain information of design parameters are often insufficient. Interval number programming method used for uncertainty analysis is effective for aerodynamic robust optimization for aircraft conceptual design.

The purpose of this paper is to propose a unified optimization design method and apply it to handle the brake squeal instability involving various uncertainties in a unified framework.

Fuzzy random variables are taken as equivalent variables of conventional uncertain variables, and a unified response analysis method is first derived based on level-cut technique, Taylor expansion and central difference scheme. Next, a unified reliability analysis method is developed by integrating the unified response analysis and fuzzy possibility theory. Finally, based on the unified reliability analysis method, a unified reliability-based optimization model is established, which is capable of optimizing uncertain responses in a unified way for different uncertainty cases.

The proposed method is extended to perform squeal instability analysis and optimization involving various uncertainties. Numerical examples under eight uncertainty cases are provided and the results demonstrate the effectiveness of the proposed method.

Most of the existing methods of uncertainty analysis and optimization are merely effective in tackling one uncertainty case. The proposed method is able to handle the uncertain problems involving various types of uncertainties in a unified way.

The interval multi-objective optimization problems (IMOPs) are universal and vital uncertain optimization problems. In this study, an interval multi-objective grey wolf optimization algorithm (GWO) based on fuzzy system is proposed to solve IMOPs effectively.

First, the classical genetic operators are embedded into the interval multi-objective GWO as local search strategies, which effectively balanced the global search ability and local development ability. Second, by constructing a fuzzy system, an effective local search activation mechanism is proposed to save computing resources as much as possible while ensuring the performance of the algorithm. The fuzzy system takes hypervolume, imprecision and number of iterations as inputs and outputs the activation index, local population size and maximum number of iterations. Then, the fuzzy inference rules are defined. It uses the activation index to determine whether to activate the local search process and sets the population size and the maximum number of iterations in the process.

The experimental results show that the proposed algorithm achieves optimal hypervolume results on 9 of the 10 benchmark test problems. The imprecision achieved on 8 test problems is significantly better than other algorithms. This means that the proposed algorithm has better performance than the commonly used interval multi-objective evolutionary algorithms. Moreover, through experiments show that the local search activation mechanism based on fuzzy system proposed in this study can effectively ensure that the local search is activated reasonably in the whole algorithm process, and reasonably allocate computing resources by adaptively setting the population size and maximum number of iterations in the local search process.

This study proposes an Interval multi-objective GWO, which could effectively balance the global search ability and local development ability. Then an effective local search activation mechanism is developed by using fuzzy inference system. It closely combines global optimization with local search, which improves the performance of the algorithm and saves computing resources.

The purpose of this paper is to formulate the reliability optimization problem in stochastic and interval domain and also to solve the same under different stochastic set up.

Stochastic programming technique has been used to convert the chance constraints into deterministic form and the corresponding problem is transformed to mixed-integer constrained optimization problem with interval objective. Then the reduced problem has been converted to unconstrained optimization problem with interval objective by Big-M penalty technique. The resulting problem has been solved by advanced real coded genetic algorithm with interval fitness, tournament selection, intermediate crossover and one-neighbourhood mutation.

A new optimization technique has been developed in stochastic domain and the concept of interval valued parameters has been integrated with the stochastic setup so as to increase the applicability of the resultant solution to the interval valued nonlinear optimization problems.

The concept of probability distribution with interval valued parameters has been introduced. This concept will motivate the researchers to carry out the research in this new direction.

The application of genetic algorithm is extended to solve the reliability optimization problem in stochastic and interval domain.

Engineering system design and optimization problems are usually multi-objective and constrained and have uncertainties in the inputs. These uncertainties might significantly degrade the overall performance of engineering systems and change the feasibility of the obtained solutions. This paper aims to propose a multi-objective robust optimization approach based on Kriging metamodel (K-MORO) to obtain the robust Pareto set under the interval uncertainty.

In K-MORO, the nested optimization structure is reduced into a single loop optimization structure to ease the computational burden. Considering the interpolation uncertainty from the Kriging metamodel may affect the robustness of the Pareto optima, an objective switching and sequential updating strategy is introduced in K-MORO to determine (1) whether the robust analysis or the Kriging metamodel should be used to evaluate the robustness of design alternatives, and (2) which design alternatives are selected to improve the prediction accuracy of the Kriging metamodel during the robust optimization process.

Five numerical and engineering cases are used to demonstrate the applicability of the proposed approach. The results illustrate that K-MORO is able to obtain robust Pareto frontier, while significantly reducing computational cost.

The proposed approach exhibits great capability for practical engineering design optimization problems that are multi-objective and constrained and have uncertainties.

A K-MORO approach is proposed, which can obtain the robust Pareto set under the interval uncertainty and ease the computational burden of the robust optimization process.

Uncertainty is inevitable in real-world engineering optimization. With an outer-inner optimization structure, most previous robust optimization (RO) approaches under interval uncertainty can become computationally intractable because the inner level must perform robust evaluation for each design alternative delivered from the outer level. This paper aims to propose an on-line Kriging metamodel-assisted variable adjustment robust optimization (OLK-VARO) to ease the computational burden of previous VARO approach.

In OLK-VARO, Kriging metamodels are constructed for replacing robust evaluations of the design alternative delivered from the outer level, reducing the nested optimization structure of previous VARO approach into a single loop optimization structure. An on-line updating mechanism is introduced in OLK-VARO to exploit the obtained data from previous iterations.

One nonlinear numerical example and two engineering cases have been used to demonstrate the applicability and efficiency of the proposed OLK-VARO approach. Results illustrate that OLK-VARO is able to obtain comparable robust optimums as to that obtained by previous VARO, while at the same time significantly reducing computational cost.

The proposed approach exhibits great capability for practical engineering design optimization problems under interval uncertainty.

The main contribution of this paper lies in the following: an OLK-VARO approach under interval uncertainty is proposed, which can significantly ease the computational burden of previous VARO approach.

This paper aims to seek the optimal proportion of female executives in corporate management teams, and to analyze the threshold effect of the proportion of female executives on the enterprise market value and enterprise management performance by using a panel threshold regression model. The purpose of this paper is to obtain the optimal interval, during which female executives exert positive effects on enterprise market value and enterprise management performance.

Based on the data of listed companies in SSE from 2003 to 2012, this paper conducts theoretical and empirical analysis by using a panel threshold regression model.

This paper proves that the proportion of female executives has a threshold effect on the enterprise market value and enterprise management performance. The results show that the proportion of female executives has an optimal interval. In other words, during the 53.8-68.4 percent interval, the proportion of female executives exerts the least negative effect on the enterprise market value and the most positive effect on the enterprise management performance.

In this paper, the non-linear relationship between female executives, enterprise market value and enterprise management performance has been verified, and the optimization interval of the female executives’ proportion has been figured out as well.

This study aims to propose an improved affine interval truncation algorithm to restrain interval extension for interval function.

To reduce the occurrence times of related variables in interval function, the processing method of interval operation sequence is proposed.

The interval variable is evenly divided into several subintervals based on correlation analysis of interval variables. The interval function value is modified by the interval truncation method to restrain larger estimation of interval operation results.

Through several uncertain displacement response engineering examples, the effectiveness and applicability of the proposed algorithm are verified by comparing with interval method and optimization algorithm.

We present a detailed explanation of a mathematical method and numerical technique applied to solve an irregular non‐linear fitting problem that results from attempts to model the calorimetric profiles generated by the binding of phenolic ligands to the insulin hexamer. The method employed uses a non‐traditional approach to modeling data. Rather than start with a simplified model, we use a hierarchical tree of physical models with different degrees of sophistication. Starting with the model of highest dimension, we work our way to an optimum model which is of a lower dimension and is less complex. The algorithm uses two complementary techniques. First, a sensitivity analysis in the vicinity of the optimal point for each model is used to estimate errors in the parameters; that, in turn, provides the user with insight for model simplification. Second, we utilize the optimized model in the prediction of new experimental curves. The core of the method combines a strategy based on the proper split of the initial global numerical task into three locally independent subtasks, and induces a specific split in the search space. The application of three different optimization techniques (two parametric and one variational) with an alternating objective function defined in corresponding subspaces, in combination with the search along the hierarchical tree of mathematical models, enables us to overcome difficult computational problems, including over‐parametrization. We have obtained very accurate fits to a number of calorimetric curves, resulting in a quantitative description of intrinsic functional (free ligand concentration) and vector (equilibrium coefficients and enthalpies of binding) parameters. These quantitative results can now be used to improve the stability of insulin formulations. We believe that, with small modifications to the model, the method and algorithms presented in this article can be applied to other protein‐ligand systems.