IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 58, NO. 2, FEBRUARY2009 365 The Construction of Random–Fuzzy Variables From the Available Relevant Metrological Information Alessandro Ferrero, Fellow, IEEE, and Simona Salicone, Member, IEEE Abstract—Approaches other than the probabilistic approach recommended by the Guide to the Expression of Uncertainty in Measurement (GUM) have been proposed during the recent past for uncertainty expression and estimation. The approach based on random–fuzzy variables (RFVs) appears to be the most promising approach since it is based on the theory of evidence, which en- compasses probability theory as a particular case. The correctness of the final uncertainty estimation quite directly depends on the way the RFVs are built, which depends on the available relevant metrological information. After briefly recalling the fundamentals of the RFV approach, this paper discusses how the available relevant information should be exploited to attain correct results. Index Terms—Measurement uncertainty, random–fuzzy vari- ables (RFVs). I. I NTRODUCTION S EVERAL proposals have been published in the recent past to include uncertainty affecting the experimental data in fuzzy logic systems [1]–[8]. Some of them [3]–[8] are specifi- cally aimed at overcoming the limitations of the presently em- ployed approach, which is fully based on probability theory and recommended by the Guide to the Expression of Uncertainty in Measurement (GUM) [9] and its Supplement 1 [10], by using fuzzy variables (FVs) and random–fuzzy variables (RFVs) to express measurement uncertainty. The advantages of this new approach are evident when nonrandom effects contribute to measurement uncertainty in a nonnegligible way [3], [4] or when very limited knowledge is available about the different contributions to uncertainty [7]. These situations are currently met, for instance, whenever all significant systematic effects are not identified and corrected for, as prescribed by the GUM [9], for technical or economical reasons. The economical reasons become particularly important in many industrial applications, where the uncertainty reduction attained through the compensation of the error sources does not repay its extra costs. Under these situations, the approach in terms of RFVs seems to be the most suitable approach, both because of its generality, being based on the mathematical theory of evidence, which encompasses the probability theory as a particular case [11], and because it takes into account and processes random and nonrandom contributions to uncertainty in a single mathemati- cal object [3], [4], [11], [12]. Manuscript received September 21, 2007; revised July 11, 2008. First published August 12, 2008; current version published January 5, 2009. The Associate Editor coordinating the review process for this paper was Dr. Ganesh Venayagamoorthy. The authors are with the Dipartimento di Elettrotecnica, Politecnico di Milano, 20133 Milano, Italy (e-mail: alessandro.ferrero@polimi.it; simona. salicone@polimi.it). Digital Object Identifier 10.1109/TIM.2008.928873 Up to now, a complete mathematical framework has been available to handle, build, and process RFVs [4], [11]–[13] so that they could already be usefully employed in some particular applications, for which the GUM’s method fails to correctly evaluate the measurement uncertainty [14]–[16]. Due to this mathematical framework, processing RFVs does not represent a critical problem, because it involves all mathe- matical operations implemented as well-defined algebraic op- erations on each α-cut of the RFVs [11], [12]. On the other hand, building the RFVs related to each considered measure- ment result is a more critical issue since it not only involves well-defined operations [11] but also depends on the available relevant information about the metrological performance of the employed instruments and measurement procedure. The incor- rect interpretation or exploitation of this information might lead to incorrect results, and it is therefore worth trying to define some general recommendations to derive the most suitable RFVs. This paper, after shortly recalling the fundamentals of the RFV method, reconsiders the approaches followed in [17] to provide a general method to exploit the available rele- vant metrological information in building RFVs. The proposed method also leads to a more general formulation of the RFV mathematics defined in [11] and [12], which also considers some possible dependence between RFVs. The method is then applied to the dc measurement of a resistance using a simple voltamperometric (V –I ) procedure [18] to prove its validity under different assumptions for the available information and its capability to attain a better uncertainty evaluation than that provided by the GUM and its Supplement 1 [9], [10]. II. RFV METHOD RFVs are a particular kind of type-2 fuzzy variables, whose α-cuts are confidence intervals of type 2, which are defined as [11], [12] B α = [[b α 1 ,b α 2 ] , [b α 3 ,b α 4 ]] , α ∈ [0, 1] (1) and that obey the following constraints [11], [12]: • b α 1 ≤ b α 2 ≤ b α 3 ≤ b α 4 , ∀α; • the sequences of intervals of confidence of type 1 [b α 1 ,b α 4 ] and [b α 2 ,b α 3 ] generate two membership functions (MFs) that are normal and convex; • ∀α, α ′ in the range [0, 1] α ′ >α ⇒  b α ′ 1 ,b α ′ 3  ⊂ [b α 1 ,b α 3 ]  b α ′ 2 ,b α ′ 4  ⊂ [b α 2 ,b α 4 ] • [b α=1 2 ,b α=1 3 ] ≡ [b α=1 1 ,b α=1 4 ]. 0018-9456/$25.00 © 2009 IEEE