Dependent Interval Variables Research Articles

Non-probabilistic optimization model of engineering structures with dependent interval variables

To solve the design problems of engineering structures with multiple conflicting performance indices under uncertainties, a non-probabilistic optimization model which takes into consideration the correlation of uncertainties was proposed in this paper. The multidimensional parallelepiped model was utilized to describe the uncertainties and deal with their correlation. Further, a novel concept of feasible discriminant of interval constraints was proposed to identify the feasible solutions, and a competitive strategy for design vectors was proposed to achieve equilibrium design of structures. To improve the efficiency of optimization and promote its applications in engineering, a direct solution framework was constructed in this paper. This framework was based on response surface model and integrated the interval analysis with the genetic algorithm. The effectiveness and superiority of the proposed approach were illustrated by two numerical examples. Finally, the proposed method was applied to structural design of the upper and lower beams of forming machine, and the design of lightweight and equilibrium were achieved. This structural design example demonstrated the applicability and effectiveness of the proposed method in engineering.

Dimension Reduction Method-Based RBDO for Dependent Interval Variables

Reliability-based design optimization (RBDO) under epistemic uncertainty (i.e., imprecise probabilistic information), especially in the presence of dependency of input variables, is a challenging problem. In this paper, we propose a dimension reduction-based RBDO framework considering dependent interval variables, which is pursued in a purely probabilistic manner. Most probable point (MPP) based dimension reduction method (DRM) is used for reliability evaluation due to its ability to circumvent the shortcomings of poor approximation by first order reliability method (FORM) and pronounced computational complexity by second order reliability method (SORM). For modeling correlation of input variables, copula is used instead of true joint cumulative density function (CDF). A flexible Johnson family of distributions is used to handle the stochastic but poorly known epistemic uncertainty. In order to handle the uncertainty in correlation measures, arisen due to interval data, expert suggested bounds of correlation measures have been recommended. For the overall RBDO problem, a decoupled approach to optimization is explored. Two numerical examples — one mathematical problem and one engineering problem — have been solved to properly explicate the proposed RBDO process. It is demonstrated that correlations in input variables have significant impact on the optimal design solutions.

A New Interval Multi-Objective Optimization Method for Uncertain Problems with Dependent Interval Variables

In this paper, a new interval multi-objective optimization (MOO) method integrating with the multidimensional parallelepiped (MP) interval model has been proposed to handle the uncertain problems with dependent interval variables. The MP interval model is integrated to depict the uncertain domain of the problem, where the uncertainties are described by marginal intervals and the degree of the dependencies among the interval variables is described by correlation coefficients. Then an efficient multi-objective iterative algorithm combining the micro multi-objective genetic algorithm (MOGA) with an approximate optimization method is formulated. Three numerical examples are presented to demonstrate the efficiency of the proposed approach.

An inverse reliability analysis method for reliability-based design optimization with random and dependent interval variables constrained within ellipsoids

ABSTRACTIn practical design problems, interval variables exist. Many existing methods can handle only independent interval variables. Some interval variables, however, are dependent. In this work, dependent interval variables constrained within a multi-ellipsoid convex set are considered and incorporated into reliability-based design optimization (RBDO). An efficient RBDO method is proposed by employing the sequential single-loop procedure, which separates the coupled reliability analysis procedure from the deterministic optimization procedure. In the reliability analysis procedure, a single-loop optimization for the inverse reliability analysis is performed, and an efficient inverse reliability analysis method for searching for the worst-case most probable point (WMPP) is developed. The search method contains two stages. The first stage deals the situation where the WMPP is on the boundary of the feasible region, while the second stage accommodates the situation where the WMPP is inside the feasible region by interpolation. Three examples are used for a demonstration.

High dimensional model representation for hybrid reliability analysis with dependent interval variables constrained within ellipsoids

Random variables could be dependent, and so could interval variables. To accommodate dependent interval variables, this work develops an efficient hybrid reliability analysis method to handle both random and dependent interval input variables. Due to the dependent interval variables, the reliability analysis needs to perform the probability analysis for every combination of dependent interval variables. This involves a nested double-loop procedure and dramatically decreases the efficiency. The proposed method decomposes the nested double loops into sequential probability analysis (PA) loop and interval analysis (IA) loop. An efficient IA algorithm based on the cut-HDMR (High Dimensional Model Representation) expansion is developed and a sampling strategy with the leave-one-out technique without extra calls of the limit-state function is proposed. The proposed method has good accuracy and efficiency as demonstrated by two engineering examples.

Author's reply to X. Yang's discussion of ‘reliability‐based design optimization with dependent interval variables’

Author's reply to X. Yang's discussion of ‘reliability‐based design optimization with dependent interval variables’

Discussion of “Reliability‐based design optimization with dependent interval variables” by Xiaoping Du, International Journal for Numerical Methods in Engineering 2012; 91:218–228

SUMMARYIn the subject paper, a reliability‐based design optimization (RBDO) model with both random and dependent interval uncertainties was proposed based on the First Order Reliability Method. The lower bound of reliability defined in Equation (9) of the subject paper was utilized as the constraint in this RBDO model. The author claimed that it is the minimum reliability with both random and interval variables. However, we prove that it is not the minimum value. It is therefore suggested that the minimum reliability should be used in the RBDO model. Copyright © 2014 John Wiley & Sons, Ltd.

A new nonlinear interval programming method for uncertain problems with dependent interval variables

This paper proposes a new nonlinear interval programming method that can be used to handle uncertain optimization problems when there are dependencies among the interval variables. The uncertain domain is modeled using a multidimensional parallelepiped interval model. The model depicts single-variable uncertainty using a marginal interval and depicts the degree of dependencies among the interval variables using correlation angles and correlation coefficients. Based on the order relation of interval and the possibility degree of interval, the uncertain optimization problem is converted to a deterministic two-layer nesting optimization problem. The affine coordinate is then introduced to convert the uncertain domain of a multidimensional parallelepiped interval model to a standard interval uncertain domain. A highly efficient iterative algorithm is formulated to generate an efficient solution for the multi-layer nesting optimization problem after the conversion. Three computational examples are given to verify the effectiveness of the proposed method.

Reliability‐based design optimization with dependent interval variables

SUMMARYReliability‐based design optimization (RBDO) ensures design requirements are satisfied at required reliability levels in the presence of uncertainty. The recent RBDO methodologies can accommodate two types of uncertainty – aleatory type, represented by random variables, and epistemic type, represented by interval variables. However, only independent interval variables can be handled. The objective of this research is to introduce dependent interval variables into RBDO. The dependency of interval variables comes from equality and inequality constraints. With the intervals, the reliability is also an interval, and its lower bound is considered to preserve conservativeness. An efficient optimization method is developed to solve RBDO problems. The method is an extension of the existing framework of sequential optimization and reliability analysis with the inclusion of inequality constraints of interval variables. The new method is efficient because it decouples the optimization process from the process of reliability and interval analyses. The new method is applied to the RBDO of a slider mechanism with joint clearances, which are known as dependent interval variables. Copyright © 2012 John Wiley & Sons, Ltd.

A market segmentation approach to esteem and efficacy in information search

PurposeBuilding on consumer purchase behavior and self‐confidence theories combined with insights from marketing segmentation literature, this study aims to investigate the influence of self‐esteem and self‐efficacy on consumers' selection of sources of information when making a product buying decision.Design/methodology/approachMultivariate analysis of variance was used to analyze the main and interaction effects of the independent categorical variables on multiple dependent interval variables. An online survey was distributed using search engine marketing.FindingsThe results indicate that sources of information are used differently among individuals with differing levels of self‐esteem and self‐efficacy. For example, individuals with higher self‐efficacy relied on themselves for information, while those with lower self‐efficacy relied on impersonal sources of information, such as the media or internet. Individuals with higher self‐esteem relied on themselves and impersonal sources (media and internet) as important sources of information, compared with those respondents with low self‐esteem.Research limitations/implicationsMeasuring situational influence using brief descriptions of hypothetical consumption situations was required. These descriptions could not include all possible facets of a natural setting resulting in subjective interpretations by respondents of what constitutes socially acceptable behavior.Practical implicationsConsumer experiences and expectations will vary during a purchase situation. Understanding how individuals seek varied sources of information based on self‐esteem and self‐efficacy is pertinent to marketing professionals when creating advertising programs particularly for products which offer potential socially risky situations, such as gift‐giving.Originality/valueThis research contributes by expanding the understanding of search behavior and the influence of self‐esteem and self‐efficacy on different purchase situations.

Gender differences in information search: implications for retailing

PurposeThe purpose of this paper is to examine the influence on search behavior of gender, purchase confidence, and internal knowledge in different purchase situations. It is expected that there will be gender differences on search behavior, particularly given different purchase situations.Design/methodology/approachMultivariate analysis of variance was used to analyze the main and interaction effects of the independent categorical variables on multiple dependent interval variables. An online survey was distributed to employees in different geographic locations in the USA.FindingsThe results of situational use indicate that sources of information are perceived differently by males and females depending on their levels of purchase confidence and internal knowledge, suggesting that, when consumers consider sources of information, such as retail clerk, family/friends or themselves, the purchase situation influences that decision.Research limitations/implicationsThe measure of the situational influence through brief descriptions of hypothetical consumption situations was required. Such descriptions could not include every possible feature of a natural setting, resulting in subjective interpretation by respondents of what are socially acceptable, possibly confounding results.Practical implicationsConsumers bring to the buying decision different types of experiences and expectations. Understanding how males and females seek varied sources of external information is relevant to the service industry in designing promotional plans, whether the product of choice is a restaurant, vacation resort, and hotel or tourism destination such as a winery.Originality/valueThe contribution of the research is to broaden the understanding of search behavior and the role gender plays, particularly in different purchase situations.