On Arithmetic Operations of $LR$ Fuzzy Numbers With Different Shape Functions

In this article, the common arithmetic operations of left–right (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$LR$</tex-math></inline-formula>) fuzzy numbers are extended to cases where the left and right shape functions of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$LR$</tex-math></inline-formula> fuzzy numbers are different. Extending arithmetic operations to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$LR$</tex-math></inline-formula> fuzzy numbers with different left and right shape functions offers several advantages, including a more realistic representation of uncertainty, enhanced expressiveness, flexibility in modeling, improved accuracy, compatibility with real data, and robustness in decision-making. These benefits make such extensions valuable in various applications where uncertainty plays an important role. The addition, difference, multiplication, and division of such <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$LR$</tex-math></inline-formula> fuzzy numbers are defined as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$LR$</tex-math></inline-formula> fuzzy numbers. It is shown that the proposed arithmetic operations satisfy main properties, such as the distributive property of the multiplication operator over addition. The results of these new operators are examined through a numerical evaluation.