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The Inverse and General Inverse of Trapezoidal Fuzzy Numbers with Modified Elementary Row Operations

Author

Listed:
  • Mashadi

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Riau, Kota Pekanbaru 28292, Indonesia)

  • Yuliana Safitri

    (Department of Mathematics, Faculty of Science and Technology, University of Jambi, Jambi City 36361, Indonesia)

  • Sukono

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jatinangor, Sumedang 45363, Indonesia)

  • Igif Gimin Prihanto

    (Research Center for Testing Technology and Standards, National Research and Innovation Agency, Jakarta 10340, Indonesia)

  • Muhamad Deni Johansyah

    (Doctoral Program of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang 45363, Indonesia)

  • Moch Panji Agung Saputra

    (Doctoral Program of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang 45363, Indonesia)

Abstract

Trapezoidal positive/negative fuzzy numbers have no single definition; instead, various authors define them in relation to different concepts. This means that arithmetic operations for trapezoidal fuzzy numbers also differ. For the operations of addition, subtraction, and scalar multiplication, there are not many differences; for multiplication, however, there are many differences. In general, multiplication is divided into various cases. For the inverse operation, there is not much to define; in general, for any trapezoidal fuzzy number u ~ , u ~ ⊗ 1 u ~ = i ~ = ( 1 , 1 , 0 , 0 ) does not necessarily apply. As a result of the different arithmetic operations for multiplication and division employed by various authors, several researchers have tackled the same problem and reached different solutions, meaning that the application will also produce different results. To date, many authors have proposed various alternatives for the algebra of the trapezoidal fuzzy number. In this paper, using the parametric form approach to trapezoidal fuzzy numbers, an alternative to multiplication with only one formula is constructed for various cases. Furthermore, based on the definition of multiplication for any trapezoidal fuzzy number, u ~ is constructed 1 u ~ so that u ~ ⊗ 1 u ~ = i ~ = ( 1 , 1 , 0 , 0 ) . Based on these conditions, we show that various properties that apply to real numbers also apply to any trapezoidal fuzzy number. Furthermore, we modify the elementary row operational steps for the trapezoidal fuzzy number matrix, which can be used to determine the inverse of a trapezoidal fuzzy number matrix with the order m × m . We also give the steps and examples necessary to determine the general inverse for a trapezoidal fuzzy number matrix of the order m × n with m ≠ n . This ability to easily determine the inverse and general inverse of a trapezoidal fuzzy number matrix has a number of applications, such as solving fully trapezoidal fuzzy number linear systems and fuzzy transportation problems, especially in applications in fields outside of mathematics; for example, the application of triangular fuzzy numbers in medical problems is a topic currently receiving a significant amount of attention.

Suggested Citation

  • Mashadi & Yuliana Safitri & Sukono & Igif Gimin Prihanto & Muhamad Deni Johansyah & Moch Panji Agung Saputra, 2024. "The Inverse and General Inverse of Trapezoidal Fuzzy Numbers with Modified Elementary Row Operations," Mathematics, MDPI, vol. 12(7), pages 1-14, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:946-:d:1362311
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    References listed on IDEAS

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    1. Tofigh Allahviranloo & Rasoul Saneifard & Rahim Saneifard & Farzad Kiani & Samad Noeiaghdam & Vediyappan Govindan & Mehdi Ghatee, 2022. "The Best Approximation of Generalized Fuzzy Numbers Based on Scaled Metric," Journal of Mathematics, Hindawi, vol. 2022, pages 1-8, November.
    2. Haidong Zhang & Lan Shu & Shilong Liao, 2014. "Generalized Trapezoidal Fuzzy Soft Set and Its Application in Medical Diagnosis," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-12, April.
    3. K. Ganesan & P. Veeramani, 2006. "Fuzzy linear programs with trapezoidal fuzzy numbers," Annals of Operations Research, Springer, vol. 143(1), pages 305-315, March.
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