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1

Estimation of maintenance costs of a pipeline for a U-shaped hazard rate function in the imprecise setting

Romaniuk Maciej, Hryniewicz Olgierd

Eksploatacja i Niezawodność

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2020

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Vol. 22, no. 2

352--362

EN

In this paper, we discuss imprecise settings for an evaluation of the maintenance costs of a water distribution system (WDS). Moments of failures of pipes are modelled using a newly proposed three-piece convex hazard rate function (HRF) for which number of previous failures is taken into account, too. Both fuzzy sets and shadowed sets are used to model the impreciseness of important parameters of this HRF and the costs of maintenance services. Contrary to more classical and widely-used approaches to cost analysis (i.e. a constant yield or nominal value of money), a strictly stochastic process (i.e. the one-factor Vasicek model) of an interest rate is assumed in the analysis of maintenance costs. This approach models future behaviour of the interest rate (i.e. the future value of money) in a more realistic way. Respective algorithms together with exemplary results of numerical simulations for two setups, which are related to fuzzy and shadowed sets, are also provided.

PL

W niniejszym artykule omawiamy nieprecyzyjne podejścia do problemu obliczenia kosztów eksploatacji systemu dystrybucji wody (WDS). Czasy uszkodzeń rur modelowane są z wykorzystaniem nowo zaproponowanej trzyczęściowej wypukłej funkcji intensywności uszkodzeń (hazard rate function, HRF) dla której brana jest pod uwagę również liczba wcześniejszych uszkodzeń. Do modelowania nieprecyzyjności istotnych parametrów tej HRF oraz kosztów działań serwisowych są wykorzystywane zarówno zbiory rozmyte jak i zbiory cieniowane. W przeciwieństwie do bardziej klasycznych i szeroko wykorzystywanych podejść do analizy kosztów eksploatacji (tzn. stałej stopy procentowej lub wartości nominalnej pieniądza), założono ściśle stochastyczny proces (tzn. jednoczynnikowy model Vasicka) dla stopy procentowej. Podejście to modeluje przyszłe zachowanie stopy procentowej (czyli przyszłej wartości pieniądza) w bardziej realistyczny sposób. Zaprezentowano również odpowiednie algorytmy wraz z przykładowymi wynikami symulacji numerycznych dla dwóch zestawów parametrów, związanych ze zbiorami rozmytymi i cieniowanymi.

2

From Numeric to Granular Description and Interpretation of Information Granules

Pedrycz W.

Fundamenta Informaticae

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2013

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Vol. 127, nr 1-4

399--412

EN

Fuzzy sets (membership functions) are numeric constructs. In spite of the underlying semantics of fuzzy sets (which is inherently linked with the higher level of abstraction), the membership grades and processing of fuzzy sets themselves emphasize the numeric facets of all pursuits stressing the numeric nature of membership grades and in this way reducing the interpretability and transparency of results. In this study, we advocate an idea of a granular description of membership functions where instead of numeric membership grades, introduced are more interpretable granular descriptors (say, low, high membership, etc.). Granular descriptors are formalized with the aid of various formal schemes available in Granular Computing, especially sets (intervals), fuzzy sets, and shadowed sets. We formulate a problem of a design of granular descriptors as a certain optimization task, elaborate on the solutions and highlight some areas of applications.

3

Orthopairs: A Simple and Widely UsedWay to Model Uncertainty

Ciucci D.

Fundamenta Informaticae

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2011

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Vol. 108, nr 3/4

287-304

EN

The term orthopair is introduced to group under a unique definition different ways used to denote the same concept. Some orthopairmodels dealing with uncertainty are analyzed both from a mathematical and semantical point of view, outlining similarities and differences among them. Finally, lattice operations on orthopairs are studied and a survey on algebraic structures is provided.

4

Shadowed Sets and Related Algebraic Structures

Cattaneo G., Ciucci D.

Fundamenta Informaticae

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2003

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Vol. 55, nr 3,4

255-284

EN

BZMVdM algebras are introduced as an abstract environment to describe both shadowed and fuzzy sets. This structure is endowed with two unusual complementations: a fuzzy one \lnot and an intuitionistic one ~ . Further, we show how to define in any BZMVdM algebra the Boolean sub-algebra of exact elements and to give a rough approximation of fuzzy elements through a pair of exact elements using an interior and an exterior mapping. Then, we introduce the weaker notion of pre-BZMVdM algebra. This structure still have as models fuzzy and shadowed sets but with respect to a weaker notion of intuitionistic negation ~ a with a Î [0,1/2). In pre-BZMVdM algebras it is still possible to define an interior and an exterior mapping but, in this case, we have to distinguish between open and closed exact elements. Finally, we see how it is possible to define a-cuts and level fuzzy sets in the pre-BZMVdM algebraic context of fuzzy sets.

5

Architectures of Granular Information and Their Robustness Properties: a Shadowed Sets Approach

Pedrycz W.

International Journal of Applied Mathematics and Computer Science

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1999

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Vol. 9, no 2

435-455

EN

This paper addresses an important issue of information of granulation and relationships between the size of information granules and the ensuing robustness aspects. The use of shadowed sets helps identify and quantify absorption properties of set-based information granules. Discussed is also a problem of determining an optimal level of information granulation arising in the presence of noisy data. The study proposes a new architecture of granular computing involving continuous and granulated variables. Numerical examples are also included.