<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The classic interval has precise borders <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amcs-2020-0015_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo><mrow><munder><mi>a</mi><mo>_</mo></munder><mo>,</mo><mover><mi>a</mi><mo>¯</mo></mover></mrow><mo>]</mo></mrow></mrow></math><tex-math>A = \left[ {\underline{a},\bar a} \right]</tex-math></alternatives></inline-formula>. Therefore, it can be called a type 1 interval. Because of great practical importance of such interval data, several versions of type 1 interval arithmetic have been created. However, sometimes precise borders <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amcs-2020-0015_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><munder><mi>a</mi><mo>_</mo></munder></math><tex-math>\underline{a}</tex-math></alternatives></inline-formula> and ā of intervals cannot be determined in practice. If the borders are uncertain, then we have to do with type 2 intervals. A type 2 interval can be denoted as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amcs-2020-0015_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>T</mi><mn>2</mn></mrow></msub><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msub><mrow><munder><mi>a</mi><mo>_</mo></munder></mrow><mi>L</mi></msub><mo>,</mo><msub><mrow><munder><mi>a</mi><mo>_</mo></munder></mrow><mi>R</mi></msub></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msub><mrow><mover><mi>a</mi><mo>¯</mo></mover></mrow><mi>L</mi></msub><mo>,</mo><msub><mrow><mover><mi>a</mi><mo>¯</mo></mover></mrow><mi>R</mi></msub></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></mrow></math><tex-math>{A_{T2}} = \left[ {\left[ {{\underline{a}_L},{\underline{a}_R}} \right],\left[ {{{\bar a}_L},{{\bar a}_R}} \right]} \right]</tex-math></alternatives></inline-formula>. The paper presents multidimensional decomposition RDM type 2 interval arithmetic (D-RDM-T2-I arithmetic), where RDM means relative-distance measure. The decomposition approach considerably simplifies calculations and is transparent for users. Apart from this arithmetic, examples of its applications are also presented. To the authors’ best knowledge, no papers on this arithmetic exist. D-RDM-T2-I arithmetic is necessary to create type 2 fuzzy arithmetic based on horizontal <italic>μ</italic>-cuts, which the authors aim to do.</p></abstract>

The classic interval has precise borders A=[a_,a¯]A = \left[ {\underline{a},\bar a} \right]. Therefore, it can be called a type 1 interval. Because of great practical importance of such interval data, several versions of type 1 interval arithmetic have been created. However, sometimes precise borders a_\underline{a} and ā of intervals cannot be determined in practice. If the borders are uncertain, then we have to do with type 2 intervals. A type 2 interval can be denoted as AT2=[[a_L,a_R],[a¯L,a¯R]]{A_{T2}} = \left[ {\left[ {{\underline{a}_L},{\underline{a}_R}} \right],\left[ {{{\bar a}_L},{{\bar a}_R}} \right]} \right]. The paper presents multidimensional decomposition RDM type 2 interval arithmetic (D-RDM-T2-I arithmetic), where RDM means relative-distance measure. The decomposition approach considerably simplifies calculations and is transparent for users. Apart from this arithmetic, examples of its applications are also presented. To the authors’ best knowledge, no papers on this arithmetic exist. D-RDM-T2-I arithmetic is necessary to create type 2 fuzzy arithmetic based on horizontal μ-cuts, which the authors aim to do.

A Decomposition Approach to Type 2 Interval Arithmetic

Faculty of Computer Science and Information Technology

West Pomeranian University of Technology in Szczecin

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"A Decomposition Approach to Type 2 Interval Arithmetic"}

The classic interval has precise borders A=[a_,a¯]A = \left[ {\underline{a},\bar a} \right]. Therefore, it can be called a type 1 interval. Because of great...

A Decomposition Approach to Type 2 Interval Arithmetic

Published Online: Apr 03, 2020

Page range: 185 - 201

Received: May 09, 2018

Accepted: Oct 08, 2019

DOI: https://doi.org/10.34768/amcs-2020-0015

Keywords
multi-dimensional RDM interval arithmetic, type 2 interval arithmetic, RDM type 2 interval arithmetic, decomposition type 2 interval arithmetic, interval arithmetic

© 2020 Andrzej Piegat et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

A Decomposition Approach to Type 2 Interval Arithmetic

Published Online: Apr 03, 2020

Page range: 185 - 201

Received: May 09, 2018

Accepted: Oct 08, 2019

DOI: https://doi.org/10.34768/amcs-2020-0015

Keywordsmulti-dimensional RDM interval arithmetic, type 2 interval arithmetic, RDM type 2 interval arithmetic, decomposition type 2 interval arithmetic, interval arithmetic

© 2020 Andrzej Piegat et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
multi-dimensional RDM interval arithmetic, type 2 interval arithmetic, RDM type 2 interval arithmetic, decomposition type 2 interval arithmetic, interval arithmetic