The random-fuzzy variables: a new approach to the expression of uncertainty in measurement

The good measurement practice requires that the measurement uncertainty is estimated and provided together with the measurement result. The practice today, which is reflected in the reference standard provided by the IEC-ISO "Guide to the expression of uncertainty in measurement," adopts a statistical approach for the expression and estimation of the uncertainty, since the probability theory is the most known and used mathematical tool to deal with distributions of values. However, the probability theory is not the only tool to deal with distributions of values and is not the most suitable one when the values do not distribute in a totally random way. In this case, a more general theory, the theory of the evidence, should be considered. This paper recalls the fundamentals of the theory of the evidence and frames the random-fuzzy variables within this theory, showing how they can usefully be employed to represent the result of a measurement together with its associated uncertainty. The mathematics is defined on the random-fuzzy variables, so that the uncertainty can be processed, and simple examples are given.     

Modified analytic hierarchy process to incorporate uncertainty and managerial aspects

The analytic hierarchy process (AHP) is a powerful multiple-criteria decision analysis technique for dealing with complex problems. Traditional AHP forces decision-makers to converge vague judgements to single numeric preferences in order to estimate the pairwise comparisons of all pairs of objectives and decision alternatives required in the AHP. The resultant rankings of alternatives cannot be tested for statistical significance and it lacks a systematic approach that addresses managerial/soft aspects. To overcome the above limitations, the present paper presents a modified analytic hierarchy process, which incorporates probabilistic distributions to include uncertainty in the judgements. The vector of priorities is calculated using Monte Carlo simulation. The final rankings are analysed for rank reversal using analysis of variance, and managerial aspects (stake holder analysis, soft system methods, etc.) are introduced systematically. The focus is on the actual methodology of the modified analytic hierarchy process, which is illustrated by a brief account of a case study.     

Correct evaluation of type-B standard uncertainty

The evaluation of uncertainty components “by means other than the statistical analysis of series of observations” (type-B evaluation) is considered. The problem comes down to selection of an appropriate a priori probability distribution of a random quantity within specified limits on the basis of all available information. Typical situations are indicated that lead to simple model distributions: rectangular (uniform), triangular, normal, and arcsine. Attention is drawn to the unjustified use of the iniform distribution model in examples of measurement-uncertainty evaluation. Translated from Izmeritel'naya Tekhnika, No. 5, pp. 33–34, May, 2000.     

A methodology for augmenting sparse pairwise comparison matrices in AHP: applications to energy systems

Multiple-attribute decision making (MADM) techniques can be used to provide a systematic approach to selection problems in energy engineering and management. They may be used for selecting the best technologies or policies based on environmental, technical, and socio-economic criteria. Among the many available MADM techniques, the analytic hierarchy process (AHP) has become one of the most widely used due to its effective hierarchical decomposition of complex problems. However, AHP may be tedious due to the large number of pairwise comparisons needed in large problems. Furthermore, in many cases, relevant information may also be available for determining criteria weights based on past decisions that have proven satisfactory in retrospect. Thus, we propose a simple methodology for augmenting sparse pairwise comparisons in AHP through a non-linear programming model that extracts a set of consistent weights from a priori ranking of a subset of alternatives. Two case studies on the ranking of bioethanol feedstocks and of CO₂ storage sites are then shown to illustrate this technique.     

Propagation of uncertainty by the possibility theory in Choquet integral-based decision making: application to an E-commerce website choice support

This paper is concerned with some aspects of uncertainty evaluations in multicriteria decision making (MCDM) in the framework of e-commerce website recommendation. The emphasis is on the interest of handling uncertainty with possibility distributions in the MCDM process where evaluations coming from the users present variability. Thus, the authors consider the propagation of possibility distributions through the multicriteria aggregation made by a Choquet integral that takes into account the interactions between the decision-making criteria. To support the recommendation process, location and uncertainty indicators of possibility distributions are defined, as well as their marginal contributions to the aggregated result. The proposed approach is applied here to the problem of the choice of an e-commerce website for purchase purposes, but it can also be used for dealing with uncertainty in other complex problems.     

Comparing algorithms for evaluating key comparisons with linear drift in the reference standard

Three algorithms are compared for evaluating the data from key comparisons with linear drift in the value of the comparison standard (use of weighted least squares and two modifications of it). If some simplifying assumptions are met, the first two algorithms produce less uncertainty in the degree of equivalence. Conditions are defined under which the third algorithm provides less uncertainty in estimating the slope coefficient. Paper written from materials of a report at the 4th International Seminar on “Mathematical, statistical, and computer support to measurement quality” (see the selection of papers in Measurement Techniques, Nos. 2–7, 2007). __________ Translated from Izmeritel’naya Tekhnika, No. 10, pp. 3–7, October, 2007.     

Pre-selection of suppliers through an integrated fuzzy analytic hierarchy process and max-min methodology

There are a variety of analytical models for supplier selection ranging from simple weighted techniques to complex mathematical programming approaches. However, these models are specifically aimed at supporting a decision maker in a single phase, especially in the final selection phase and they have failed to consider the supplier selection process from a holistic point of view. Although the methodology presented in this paper primarily focused on the prequalification of potential suppliers, the outputs of the previous phases, namely problem definition and formulation of criteria, are used as inputs in this methodology. The methodology utilises a fuzzy analytic hierarchy process (AHP) method to determine the weights of the pre-selected decision criteria, a max-min approach to maximise and minimise the supplier performances against these weighted criteria, and a non-parametric statistical test to identify an effective supplier set. This information supports decision makers in making the final selection with effective alternative choices. Potential application of the proposed methodology is demonstrated in Audio Electronics in Turkey's electronics industry.     

GUM法和MCM法评定测量不确定度对比分析

以气象检定领域中的风速测量不确定度评定为例,分别在输入量正态分布以及其他分布两种假设情况下,对比分析了GUM法与MCM法评定风速简化模型测量不确定度结果,并对简化模型所忽略的水汽修正项带来的测量不确定度进行了量化。实例对比表明,MCM法较GUM法适用范围广,特别是针对一些复杂模型,MCM法不需要繁琐的求导过程,不需要舍弃一些小影响量的不确定度计算,使得不确定度评定结果更完整合理,可操作性强。     

Automation of the process of testing statistical hypotheses by means of the von Mises and Pearson criteria

A compound method of testing of statistical hypotheses by means of the von Mises and Pearson criteria is considered. An evaluation of the selection of the level of significance of criteria for critical cases is given and a reliability factor is introduced. A computer program for automatic testing of statistical hypotheses created in accordance with newly developed recommendations is described. Translated from Izmeritel'naya Tekhnika, No. 9, pp. 42–44, September, 2008.     

Estimating measurement uncertainty on the basis of observed and logical correlation

An algorithm has been devised for simulating the joint distributions of inputs. The Monte Carlo method has given models for compositions of input distributions. Coverage factors have been determined for fiducial estimation of the uncertainties in measurements in the absence of and in the presence of observed and logical correlations between the inputs. Paper written from the materials of a report at the 4th International Seminar “Mathematical, statistical, and computer support to measurement performance” (see this journal, Nos. 2–7, 2007). __________ Translated from Izmeritel’naya Tekhnika, No. 8, pp. 6–11, August, 2007.     

Fuzzy α-discounting method for multi-criteria decision-making

The α-Discounting Method was developed to be an alternative to and extension of the Analytical Hierarchy Process (AHP) to solve multi-criteria decision-making (MCDM) problems with non-commensurable and conflicting criteria. In contrast to the AHP, this method works not only for pairwise comparisons but also for n-wise comparisons if relative importance of criteria can be expressed in a system of linear homogenous equations. This method also has a comparative advantage as it can transform those MCDM problems, classified as inconsistent by the AHP, into a consistent form. This study briefly compares the two methods and then develops the Fuzzy α-Discounting Method for Multi-Criteria Decision Making (Fα-DM MCDM). Two illustrative fuzzy MCDM problems from the literature have been solved to show how the Fα-DM MCDM works.     

Measurement uncertainty in the determination of total petroleum hydrocarbons (TPH) in soil by GC-FID

In this investigation two methods were used for estimating the measurement uncertainty due to sampling and analysis of petroleum hydrocarbon contaminated soil. Analysis of variance (ANOVA) was used for type A evaluation of the measurement uncertainty. The results showed that the statistical evaluation of measurement uncertainty can be complicated by the log-normality and heteroscedasticity of the data. Although mathematical transformation of raw data is widely suggested for overcoming the discrepancy between data and ANOVA assumptions, its use results in problems with the interpretation of the ANOVA results at the original scale.

The measurement uncertainty was also estimated from the calculated precision equations for sampling and analysis. Comparison of measurement uncertainty values with the equivalent values obtained with ANOVA revealed that ANOVA overestimates the expanded uncertainty at both low and high TPH concentrations. Consequently, correct selection of the statistical analysis method needs comprehensive knowledge of the assumptions and limitations of statistical methods and careful consideration of the special characteristics (distribution, constancy of measurement variance) of the raw data as these may affect the validity of the estimated uncertainty. The expanded uncertainty obtained in this study for the results of TPH determinations with linear measurement precision modelling was moderate, ranging from 21% at a TPH concentration of 895 mg/kg to 9% at a TPH concentration of 10 019 mg/kg. If a single sample taken in a survey is analyzed only once, then the analytical variance contributes the most to the measurement variance, ranging from 68– 80% at a TPH concentration of 100–10 000 mg/kg.     

A hybrid approach to machine-tool selection through AHP and simulation

The selection process of a machine tool has been a critical issue for companies for years, because the improper selection of a machine tool might cause many problems having a negative effect on productivity, precision, flexibility, and a company's responsive manufacturing capabilities. Therefore, in this paper, to determine the best machine tool satisfying the needs and expectations of a manufacturing organization among a set of possible alternatives in the market, a hybrid approach is proposed, which integrates an analytic hierarchy process (AHP) with simulation techniques. The AHP as one of the most commonly used multiple criteria decision-making methods is used to narrow down all possible machine tool alternatives in the market by eliminating those whose scores (or weights) are smaller than a determined value obtained under certain circumstances. Then, a simulation generator is used first to automatically model a manufacturing organization, where the ultimate machine tool will be used, and second to try each alternative remaining from the AHP as a scenario on the generated model. Finally, the final alternative is selected by using the unit investment cost ratio, which is calculated by dividing the investment cost per year of each alternative by the additional number of produced units obtained from the simulation experiment of the relevant alternative.     

测量不确定度自动评定的研究

通过分析Met/Cal中进行测量不确定度评定时默认考虑的3种不确定度来源,介绍测量不确定度评定所用到的7个计算公式,研究Met/Cal中如何设置测量不确定度评定,以系统地说明Met/Cal中实现测量不确定度自动评定的原理和方法。最后给出Met/Cal默认测量不确定度评定的流程以及自定义测量不确定度评定的编程实例,以指导使用者正确地使用其测量不确定度评定功能。     

Using AHP and the IPMA Competence Baseline in the project managers selection process

This paper proposes the joint use of the AHP (Analytic Hierarchy Process) and the ICB (IPMA Competence Baseline), as a tool for the decision-making process of selecting the most suitable managers for projects. A hierarchical structure, comprising the IPMA’s ICB 3.0 contextual, behavioural and technical competence elements, is constructed for the selection of project managers. It also describes the AHP implementation, illustrating the whole process with an example using all the 46 ICB competence elements as model criteria. This tool can be of high interest to decision-makers because it allows comparing the candidates for managing a project using a systematic and rigorous process with a rich set of proven criteria.     

Determination of stray light at the PTB goniophotometer facility

For improved luminous flux measurements, the Physikalisch-Technische Bundesanstalt (PTB) was equipped with a new robotic goniophotometer. In order to characterize this instrument, to reduce the combined measurement uncertainty, and to determine the respective uncertainty components, the influence of stray light on the measurement results was investigated. Considering the specific structure of the robotic goniophotometer, stray light was measured on-line by means of a purpose-built “back-looking” photometer. By this means, the influence of the luminous intensity distribution of each individual light source with regard to its stray light could be considered. Before the first experimental application, the invented procedure was tested by means of calculations using mathematical models. In this paper, the mathematical model for the calculation of stray light in the goniophotometer room is presented together with the calculation results and the results of the first measurements. The achieved results indicate that the recommended stray light measurements can decrease the total relative standard measurement uncertainty of luminous flux measurements down to 2 × 10⁻³.     

Estimation of flux distributions with Monte Carlo functional expansion tallies

Monte Carlo methods provide a powerful technique for estimating the average radiation flux in a volume (or across a surface) in cases where analytical solutions may not be possible. Unfortunately, Monte Carlo simulations typically provide only integral results and do not offer any further details about the distribution of the flux with respect to space, angle, time or energy. In the functional expansion tally (FET) a Monte Carlo simulation is used to estimate the functional expansion coefficients for flux distributions with respect to an orthogonal set of basis functions. The expansion coefficients are then used in post-processing to reconstruct a series approximation to the true distribution. Discrete event FET estimators are derived and their application in estimating radiation flux or current distributions is demonstrated. Sources of uncertainty in the FET are quantified and estimators for the statistical and truncation errors are derived. Numerical results are presented to support the theoretical development.     

Components of a Variance Model for the Analysis of Repeated Measurements in Fixed-Point Comparisons

Experimental design is a statistical tool concerned with the planning of experiments to obtain the maximum amount of information from the available resources. This tool may be applied to metrology, especially for the analysis of a large number of repeated measurements (replicates) of short-term repeatability and the medium-term and long-term reproducibilities, enabling the inclusion of these “time-dependent sources of variability” in the uncertainty budget. The realization of the International Temperature Scale of 1990 (ITS-90) scale requires that laboratories usually have more than one cell for each fixed point, for comparison on a regular basis. The calculation of the uncertainty of such comparisons is considered here, taking into account these time-dependent sources of variability. These components of the uncertainty evaluated by a Type A method are obtained by the statistical analysis of the experimental results using the components of a variance model for designs consisting of nested or hierarchical sequences of measurements, as foreseen by the mainstream GUM. An application example of a balanced nested structure in the comparison of two fixed-point cells is presented.     

Ply level uncertainty effects on failure curves and optimal design of laminated composites using directional bat algorithm

The present study investigates the effect of both ply level material uncertainty and ply angle uncertainty on the failure envelope, strength characteristics and design of laminated composite. Multiple failure envelopes and distributions of the strength parameters are obtained for Tsai-Wu and maximum stress criteria using Monte Carlo simulation. A newly developed directional bat algorithm (dBA) is then used to perform the constrained design optimization of laminated composite for the first time while considering uncertainty effects. The effect of ply level uncertainty on failure envelopes and the corresponding optimal design of laminated composite structures is thus quantified.     

Standard stochastic pulse signals: Applications, reproduction, certification

We consider notions of application, modelling, and certification for stochastic pulse signals with normalized statistical characteristics as a means for monitoring metrological parameters of information measurement systems. We describe a signal model, present examples of its application, and consider a generalized functional scheme for a generator and principles for algorithmic certification of the model. Translated from Izmeritel'naya Tekhnika, No. 6, pp. 24–33, June, 1997.