Grey Number Theory Research Articles

Uncertainty analysis of large structures using universal grey number theory

In this paper, a new uncertainty-based approach, termed the universal grey number-based Gaussian elimination method, is presented for the analysis of large structures that require the solution of systems of linear interval algebraic equations. Although exact ranges of the solution can be found using the enumeration method, it is known to be computationally expensive as it requires a large number of analyses. Although interval analysis has been used by some researchers, it is found to lead to wider ranges of the response quantities due to the dependency problem. Hence, modified procedures such as the truncation-based interval analysis have been suggested in the literature to overcome the dependency problem. In fact, no interval analysis-based method is available in the literature for solving large number of interval linear equations accurately. The present method is expected to overcome the limitations associated with the available methods in terms of accuracy and computational effort. To demonstrate the accuracy of the proposed method, the stress analysis of several truss structures under specified interval values of input parameters is considered. It is shown that the proposed method yields accurate results more efficiently with less computational effort compared to the truncation-based interval analysis and enumeration method.

Improved uncertain method for safety analysis of aircraft landing gear

There are various uncertain factors in most practical engineering applications, such as input loads, structural sizes, manufacturing tolerance, and initial and boundary conditions. The interval method and grey number theory are common methods to deal with uncertainty. In this article, the interval truncation method and grey number theory are improved. And a mixed method is proposed to represent the confidence interval of output result based on the improved interval truncation method and improved grey number theory. The proposed methods’ feasibility is verified by a stepped bar; the methods are applied to the analysis of aircraft landing gear safety uncertainty.

Universal grey number theory for the uncertainty presence of wiper structural system

PurposeThis study aims to accurately evaluate the influence of various error intervals on the performance of the wiper.Design/methodology/approachThe wiper structural system is decomposed into classical four-link planar for kinematics analysis, and it was modeled respectively by using interval method, universal grey number theory and enumeration approach depending on the nature of uncertainty.FindingsThe universal grey number theory is a viable methodology for the accurate analysis of uncertain structural system.Originality/value(1) The model of uncertain wiper structural system is established. (2) Universal grey number theory and new parameters are adopted to analyze the presence of uncertain wiper structural system. (3) Comparative analysis of response quantities is obtained by interval method, universal grey number theory and enumeration method.

Method of rock mass quality classification in ancient underground engineering based on three-parameter interval grey number

Rock mass quality classification is an important basis for long-term protection and tourism development of ancient underground engineering, however, the existing rock mass quality classification methods are difficult to apply to the special project of ancient underground engineering. Based on the in-depth analysis of the characteristics of ancient underground engineering and the influencing factors of rock mass quality, an evaluation index system for rock mass quality classification of ancient underground engineering is established by taking the long-term tensile strength of rock and the weathering velocity of rock mass as indicators. Secondly, for the evaluation index value has the character of bounded-but-unknown uncertainty, and in order to avoid information loss in interval numbers, the three-parameter interval grey number theory is introduced, based on the concept of three-parameter interval grey number distance measure, a rock mass quality classification calculation model for ancient underground engineering is established. Then, because of the complexity of ancient underground engineering, the idea of group decision-making is introduced, and based on the variation coefficient method, the weight calculation method of evaluation index is established to improve the reliability of weight calculation. Finally, a rock mass quality classification method for ancient underground engineering based on three-parameter interval grey number is established. The method is used to analysis the practical engineering, it shows that the method proposed here is feasible and reasonable.

Tolerance analysis and evaluation of uncertain automatic battery replacement system

Considering manufacturing errors and installation errors in the automatic battery replacement system, it is necessary to analyze the uncertainty of the automatic battery replacement system to improve the efficiency and accuracy of the system. There are a few universal approaches for uncertainty analysis, such as interval analysis, probability analysis, and combination approach. This paper uses grey number theory, a relatively new method, to analyze. The analysis on the coordinates of the support points finds that the grey number theory can output a range smaller than that of the other three methods, and its results are in line with the actual situation. The grey number theory is used to analyze the coordinate position change of the support point of the automatic battery replacement device with the input error change and the driving rod angle change. The bar tolerance and error circle (installation error) existing in the system are also taken into consideration. Comparing the results produced by the case where error circle is considered and that is not, it can be seen that the influence of the error circle on the results is relatively insignificant compared with that of rod tolerance, but the influence cannot be ignored for structures requiring high precision. Therefore, the analysis of the input error percentage can yield a reasonable control range of the input error based on the grey number theory, which not only ensures the system is safe and reliable, but also is cost-efficient.

Uncertainty-Based Structural Optimization Using Universal Grey Number Theory

Several types of uncertainties are encountered in the design of machines and structures. The geometric parameters or dimensions are specified in terms of nominal values and tolerances, the experimentally determined material properties exhibit scatter over a range, and many types of loads are inherently variable (loads due to wind, floods, and earthquakes). It is difficult to develop probability distributions to all such uncertain parameters. Even if probability distributions are developed, their confidence levels need to be established using a t- or chi-squared distribution. However, it is easier and more accurate to model the uncertain quantities/parameters as intervals. Thus, for example, in the interval parameter-based optimization, only the known or expected ranges of the design parameters are used with no need to establish the confidence levels of their intervals. This work presents an interval parameter-based optimum design of structural components. Although the field of interval analysis deals with computation of interval numbers, it suffers from the so-called dependency problem, whereby the computations get more and more inaccurate, resulting in wider intervals of the computed response quantities as compared to the true intervals if multiple arithmetic operations are performed using the same interval number or if complicated expressions are evaluated using interval numbers. Hence, a more accurate method, known as the universal grey number theory, is used in this work for the analysis and computations involved in the optimization procedure. Two numerical optimization problems, which are the design of a stepped bar subjected to an axial load and the design of a welded beam subjected to a transverse load, are presented to illustrate the procedure.

Reliability Evolution of Elevators Based on the Rough Set and the Improved TOPSIS Method

Focusing on reliability evaluation of elevator equipment operation, this paper builds a model for reliability evaluation of elevator equipment. First, we analyze the factors influencing the reliability of elevators and extract the initial evaluation indicators from four macro-perspectives: people, management, equipment, and environment, to build the index system for reliability evaluation of elevator equipment. And, we reduce the indicator set to remove the unnecessary indicators by using the rough set and weight the reduced indicators according to expert opinions. Finally, on this basis, we combine the grey number theory with the TOPSIS method to build an improved TOPSIS evaluation model, to evaluate the reliability of elevators and determine the optimal ordering.

Universal Grey System Theory for Analysis of Uncertain Structural Systems

The uncertainty present in many structural systems is commonly modeled using probabilistic, fuzzy, or interval approaches depending on the nature of uncertainty present. The probabilistic approach is based on the probability distributions of the uncertain parameters, which are not known for most practical systems. The fuzzy analysis is applicable to systems whose parameters are described in terms of linguistic and imprecise or vague statements. In most practical systems, particularly structural systems, the parameters such as the geometry (sizes of members), material properties, and loads are available as ranges or intervals. For such systems, the use of interval analysis, coupled with the finite element analysis, appears to be appropriate for predicting the ranges of the response quantities such as nodal displacements and element stresses. However, the accuracy of the results given by the interval analysis suffers from the so-called dependency problem, which can lead to undesirable expansions of the ranges of the computed results. Depending on the nature of expressions involved in the computation and the width of the ranges (intervals) of the uncertain parameters, the computed response quantities may sometimes violate the physical laws. Although several approaches, such as numerical truncation during interval arithmetic operations, parameterization of intervals, and the use of subintervals within an interval, have been suggested to limit the growth of the intervals of the results predicted, there has not been a simple approach that can improve the accuracy of the basic interval analysis. This work presents a new methodology, called universal grey number (or system) theory, for the analysis of systems whose parameters are described as intervals or ranges. The computational feasibility and improved accuracy (compared to the interval analysis) of the methodology is demonstrated by considering three examples: the stress analysis of a stepped bar, the stress analysis of a planar 10-bar truss, and the rigid-body (vertical) response of an airplane taxiing on a wavy runway based on a single-degree-of-freedom model. This work demonstrates that the universal grey system approach is a viable methodology for the accurate analysis of structural and other engineering problems involving uncertain parameters that are described in terms of ranges or intervals.

The grey linear programming approach and its application to multi-objective multi-stage solid transportation problem

The multi-objective solid transportation problem (MOSTP) constitutes one of the foremost areas of application for linear programming problem. The aim of this problem is to obtain the optimum distribution of goods from different sources to different destinations with different mode of conveyances which minimizes the total transportation cost and time. But it may contain one or more stage to transport the commodities with different mode of transport. In this paper, two new multi-objective multi-stage solid transportation problems (MOMSSTP) are investigated under grey uncertainty. Since using interval grey number theory we can absorb stochastic and interval uncertainty at a time, for this reason we developed two multi-stage STP under interval grey environment. The goal programming approach and fuzzy goal programming approach are used to reduce the multi-objective programming problem into a single-objective programming problem. Finally, the equivalent crisp models are solved using generalized reduced gradient technique (LINGO.13.0 optimization software) and the nature of the results is discussed.

Malmquist Productivity Index with Grey Data

Malmquist Productivity Index (MPI) is widely used method to measure the productivity changes of Decision Making Units (DMUs) between two time periods. Although the conventional MPI requires accurate data, in many real life conditions the input and output data of DMUs usually involve uncertainty and only lower and upper bounds of data could be obtained. Grey (number) theory is one of the theories which are used for describing uncertainty. A grey number, with both a lower and upper bounds, is called an interval grey number. The purpose of this paper is to measure the productivity changes under uncertainty conditions based on the interval grey number theory. In the paper, new grey MPI models are proposed to measure productivity changes of DMUs which have interval data. A numerical example is provided to illustrate the application of the proposed models. Results of the numerical example show us that the proposed models are easy to handle and applicable for real life problems.

Study and Prediction on the Development Trend of Chinese Sports Venues on the Basis of Grey Prediction Model

The grey forecasting method in most of the existing prediction, there are little research on the problem of interval prediction. This paper presents the concept of synthesis of grey number and grey theory, describes the properties of the synthetic ash gray because the grey prediction model is proposed the importance to build the stadium for the first time. On this basis, the thesis analyze the research background and significance, then describes some scene at home and abroad, and then analyzes the important characteristics of large-scale sports events and sports stadium construction, at last, the paper does some prediction about the construction of the stadium. This paper mainly uses the literature investigation method combined with a survey method.

A grey STA-GERT quality evaluation model for complex products based on manufacture-service dual-network

Purpose – The purpose of this paper is to find out a scientific method to evaluate quality of complex products, whose quality is different from general products. Design/methodology/approach – Based on interval grey number theory, reliability analysis method and stochastic network theory, authors have established grey success tree analysis-graph evaluation and review technique (GSTA-GERT) model in this paper. Findings – Comparing complex products and general products, authors have found that complex products have two characters, i.e. quality of manufacture and quality of service. Furthermore, this paper has proved the GSTA-GERT model is a scientific and reasonable approach to estimate quality of complex products from the sight of manufacture-service network. Originality/value – This paper has established GSTA-GERT model, which surmounts the defect of traditional estimation method, such as lacking logic analysis in the method of analytic hierarchy process.

Research on grey function analysis model (1,1) based on regional economy impact analysis

PurposeThe purpose of this paper is to realize scientific reasoning and prediction in economic catastrophe, which occurs in the short‐term and leads to invalidation of most classical prediction models through lacking basic sample data.Design/methodology/approachBased on functional theory, grey number algebra theory, Bayesian network theory and interval grey number theory, the authors established GFAM (1,1), which is grey function analysis model (1,1), to excavate and utilize the existing data sufficiently.FindingsThis paper proved least squares parameters theorem and prediction theorem and the process of GFAM (1,1). A case was established and demonstrated the utility and good prediction of this model.Originality/valueThis paper established GFAM (1,1), which overcomes the hysteretic defect of classical prediction model and provides a preferable solution in system prediction in economic catastrophe.

A conceptual grey rainfall-runoff model for simulation with uncertainty

This paper presents an approach based on grey numbers to represent the total uncertainty of a conceptual rainfall-runoff model. Using this approach, once the grey numbers representing the model parameters have been properly defined, it is possible to obtain simulated discharges in the form of intervals (grey numbers) whose envelope defines a band which represents the total model uncertainty. The application to a real case showed that the construction of this band, according to a rigorous application of grey number theory, involves long computational times. However, these times can be significantly reduced using a simplified computing procedure with minimal approximations in the quantification of the simulated grey discharge. Relying on this simplified procedure, the conceptual rainfall-runoff grey model was then calibrated in order to respect a predefined level of model uncertainty, i.e. the band obtained from the envelope of simulated grey discharges had to include an assigned percentage of observed discharges and was at the same time as narrow as possible. Finally, the uncertainty bands were compared with the ones obtained using a well-established approach for characterising uncertainty, the Generalised Likelihood Uncertainty Estimation (GLUE) method. The results of the comparison showed that the proposed approach may represent a valid tool for characterising the total uncertainty of a rainfall-runoff model.

A Soft- Computing Approach for Multi Criteria Project Selection Problem with Grey Number

Multi-criteria decision support systems are used in various fields of human activities. In every alternative multi-criteria decision making problem can be represented by a set of properties or constraints. The properties can be qualitative & quantitative. For measurement of these properties, there are different unit, as well as there are different optimization techniques. Depending upon the desired goal, the normalization aims for obtaining reference scales of values of these properties. This paper deals with the multi-attribute Complex Proportional Assessment of alternative. In order to make the appropriate decision and to make a proper comparison among the available alternatives Analytic Hierarchy Process (AHP) under fuzziness and COPRAS method with grey numbers has been used. The uses of AHP is for analysis the structure of the project selection problem is used under fuzziness and to assign the weights of the properties and the COPRAS-G method is used to obtain the final ranking and select the best one among the projects. To illustrate the above mention methods survey data on the expansion of optical fiber for a telecommunication sector is reused. The decision maker can also used different weight combination in the decision making process according to the demand of the system.COPRAS-G method is used to evaluate the overall efficiency of a project with the criterion values expressed in terms of intervals. It is based on the real conditions of decision making and applications of the grey number theory.

Application of grey numerical model to groundwater resource evaluation

Based on interval grey number theory, a grey numerical model was developed to simulate groundwater flow in two-dimensional porous-medium aquifer. The solution method proposed in this paper ensures the integrity and validity of the grey data transportation in the calculation processes. Because most of the parameters characterizing groundwater flow are of grey nature, a grey model can better describe the groundwater flow than the traditional “white” model. The method was applied to evaluate the sustainable groundwater resource in a groundwater basin of north China. The results from the case study help develop the most appropriate approaches to utilize the water resources in the area.