Differences between Moore and RDM Interval Arithmetic

Landowski, Marek

doi:10.1007/978-3-319-11313-5_30

Cited by 22 publications

(20 citation statements)

References 4 publications

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“…Definition 1 (Landowski 2015) In notation RDM, the interval X = x, x is described as a set of numbers given by (2).…”

Section: Horizontal Fuzzy Numbermentioning

confidence: 99%

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Method with horizontal fuzzy numbers for solving real fuzzy linear systems

Landowski

2018

Soft Comput

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The paper presents a method for solving real fuzzy systems of linear equations with the usage of horizontal fuzzy numbers (HFNs). Based on the multidimensional RDM interval arithmetic and a fuzzy number in a parametric form, the definition of a horizontal fuzzy number with nonlinear left and right borders was given. Additionally, the paper presents the properties of basic algebraic operations on these numbers. The usage of horizontal fuzzy numbers with linear and nonlinear borders was illustrated in the examples with n × n fuzzy linear systems. The obtained results are multidimensional and satisfy the fuzzy linear systems. Calculated solutions of fuzzy linear systems were compared with the results of other methods. The solution obtained with horizontal fuzzy numbers satisfies any equivalent form of fuzzy linear system, whereas the results of existing methods do not satisfy the equivalent forms of the system. The presented examples also show that the method with HFN delivers a full multidimensional solution (direct solution). The analyzed results of standard methods, which are only a part of span (indicator, indirect solution) of a full solution and/or include values that do not satisfy the fuzzy linear systems, are underestimated or overestimated. The proposed method gives a possibility to obtain a crisp solution together with a crisp system of equations; other methods do not possess these properties.

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“…Definition 1 (Landowski 2015) In notation RDM, the interval X = x, x is described as a set of numbers given by (2).…”

Section: Horizontal Fuzzy Numbermentioning

confidence: 99%

“…In Example 1, the differences between the results obtained by standard interval arithmetic (S-IA) and multidimensional RDM interval arithmetic were shown. The concept of the multidimensional Relative Distance Measure interval arithmetic (RDM-IA) was developed by A. Piegat Landowski 2013, 2017b;Landowski 2015Landowski , 2017.…”

Section: Introductionmentioning

confidence: 99%

Method with horizontal fuzzy numbers for solving real fuzzy linear systems

Landowski

2018

Soft Comput

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“…Moore's arithmetic detects extrema only if they lie at the boundary of the solution space as in the case of all monotonic operations such as addition, subtraction, multiplication and division. RDM arithmetic has almost the same mathematical properties as the conventional one (Landowski, 2014). Let A, B, C be intervals.…”

Section: Idea Of Multidimensional Rdm Interval Arithmeticmentioning

confidence: 99%

“…Checking whether particular laws 1-7 hold for RDM arithmetic is easy. In the formulas for the laws, RDM models of intervals A, B, C should be inserted and the law equations should be examined (Landowski, 2014). Frequently, the question is asked: "What are practical benefits of RDM arithmetic?"…”

Section: Idea Of Multidimensional Rdm Interval Arithmeticmentioning

confidence: 99%

Computing with words with the use of inverse RDM models of membership functions

Piegat

Pluciński

2015

International Journal of Applied Mathematics and Computer Science

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Computing with words is a way to artificial, human-like thinking. The paper shows some new possibilities of solving difficult problems of computing with words which are offered by relative-distance-measure RDM models of fuzzy membership functions. Such models are based on RDM interval arithmetic. The way of calculation with words was shown using a specific problem of flight delay formulated by Lotfi Zadeh. The problem seems easy at first sight, but according to the authors' knowledge it has not been solved yet. Results produced with the achieved solution were tested. The investigations also showed that computing with words sometimes offers possibilities of achieving better problem solutions than with the human mind.

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“…Its concept was elaborated by Andrzej Piegat. Some basic concepts of this arithmetic were partly presented in 29,30,31,32,33,34,35 and in this paper we will focus on a division of IPVs.…”

Section: Introductionmentioning

confidence: 99%

Some Advantages of the RDM-arithmetic of Intervally-Precisiated Values

Piegat¹,

Pluciński²

2015

IJCIS

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Moore's interval arithmetic always provides the same results of arithmetic operations, e.g. [1, 3] + [5, 9] = [6, 12]. But in real life problems, the operation result can be different, e.g. equal to [4, 7]. Therefore, real problems require more advanced arithmetic. The paper presents (on example of the division) an arithmetic of intervally-precisiated values (IPV-arithmetic) and its main advantages. Thanks to it, it is possible to process different tasks that people solve intuitively. The most important advantages are: existence of inverse elements of addition and multiplication, holding the distributivity law and the cancellation law of multiplication, possibility of achieving not only the solution span [x, x] but also the full, multidimensional solution and its cardinality distribution without using Monte Carlo method, possibility of achieving unique and complete solution sets of equations with unknowns, possibility of calculations with uncorrelated IPVs, possibility of calculations with fully correlated and partly correlated IPVs, possibility of uncertainty decreasing of original data items occurring in problems. All these advantages are illustrated and visualised by examples.

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Differences between Moore and RDM Interval Arithmetic

Cited by 22 publications

References 4 publications

Method with horizontal fuzzy numbers for solving real fuzzy linear systems

Method with horizontal fuzzy numbers for solving real fuzzy linear systems

Computing with words with the use of inverse RDM models of membership functions

Some Advantages of the RDM-arithmetic of Intervally-Precisiated Values

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