Uncertainty is an ever-present challenge in real-world problem-solving, demanding effective strategies for informed decision-making. Lotfi A. Zadeh's pioneering work in 1965 introduced fuzzy set theory, a pivotal tool for managing imprecise and uncertain information

In recent years, a series of significant advancements has pushed the boundaries of fuzzy number theory. Notable works, such as Juuso's exploration of nonlinear scaling and fuzzy systems for automation

Additionally, Leandry, Sosoma, and Koloseni's 2022 paper

Kumar, Khepar, Yadav, and their collaborators' systematic review in 2022

However, there remains a notable gap in addressing the need for more nuanced and flexible representations of uncertainty. Conventional fuzzy numbers may not sufficiently meet the requirements of intricate applications. This paper bridges this gap by introducing "Trigonal Fuzzy Numbers" or "30-gonal Fuzzy Numbers." Unlike traditional fuzzy numbers, Trigonal Fuzzy Numbers offers a highly versatile framework, characterized by membership functions that span 30 distinct intervals, allowing for smoother and more detailed transitions in membership values.

To address this gap, the study explores the use of "Alpha Cuts" for precise uncertainty quantification, enabling more granular and adaptable handling of uncertainty. Furthermore, the paper demonstrates fundamental arithmetic operations on Trigonal Fuzzy Numbers using Alpha Cuts, a critical advancement for decision support and modeling tasks where uncertainty plays a pivotal role. Through real-life applications in various domains, this research exemplifies the practicality and relevance of the innovative approach. Trigonal Fuzzy Numbers with Alpha Cuts offer a valuable asset for informed decision-making in complex scenarios characterized by ambiguity and imprecision.

The introduction section offers justification for the present work by addressing the limitations of traditional fuzzy number representations and the need for more nuanced and flexible approaches. It establishes the significance of introducing Trigonal Fuzzy Numbers and their utility in addressing complex and ambiguous uncertainty scenarios. This section also sets the stage for the innovative aspects of the research by highlighting the limitations of existing methods and the motivation for exploring new territory.

Some Preliminary definitions for Fuzzy set, Alpha cut, Fuzzy number, triangular fuzzy number, trapezoidal fuzzy number have been given in this section.

A fuzzy set A in a universal set U is characterized by a membership function µA(x), which assigns a degree of membership (a value between 0 and 1) to each element x in the universal set. The membership function, µA(x), reflects the extent to which each element belongs to the fuzzy set A.

A fuzzy set A defined on a universal set X with a membership function µA(x), the α-cut of A at confidence level α, denoted as A(α), is defined as follows:

A(α) = {x ∈ X | µA(x) ≥ α}

Where, X is the universal set or universe of discourse. µA(x) is the membership function of fuzzy set A, which assigns a degree of membership between 0 and 1 to each element x in X. α is the confidence level or threshold value, where 0 ≤ α ≤ 1.

The α-cut A(α) includes all elements x from the universal set X for which the membership degree µA(x) in fuzzy set A is greater than or equal to the specified confidence level α. In essence, it forms a crisp set that retains elements with a degree of membership that satisfies or exceeds the confidence level α.

A fuzzy number is defined by a membership function, typically denoted as µA(x), where 'x' represents a real number. The membership function assigns a degree of membership (a value between 0 and 1) to each real number 'x' within a specified range. Fuzzy numbers can come in various types, such as triangular, trapezoidal, Gaussian, or other specialized shapes, depending on the nature of the uncertainty being modeled.

A fuzzy number A= (a, b, c) is said to be a triangular fuzzy number if its membership function is given by (where a b ≤c ≤d)

A fuzzy number A= (a, b, c, d) is said to be a trapezoidal fuzzy number if its membership function is given by (where a b ≤c ≤d)

The results and discussion section serves as a platform for comparing the novel approach with previous work and relates these comparisons to the stated objectives. It showcases the versatility of Trigonal Fuzzy Numbers and their ability to provide more detailed uncertainty modeling than traditional fuzzy numbers, such as triangular and trapezoidal fuzzy numbers. The section highlights practical advantages through real-world applications in finance, healthcare, and environmental management, drawing distinctions with conventional approaches and illustrating how Trigonal Fuzzy Numbers addresses the shortcomings of existing models.

A Trigonal Fuzzy Number, denoted as A30g, is characterized by a unique membership function that divides the real number line into 30 distinct intervals. Each interval represents a specific degree of membership for values of x, facilitating a smooth transition from 0 to 1 and back to 0 as x traverses these intervals. This characteristic gives rise to the term "trigonal," signifying the triangular shape of the membership function.

A Trigonal Fuzzy Number A30g is represented as a sequence of values (a1, a2, a3, ..., a30), where each ai corresponds to the membership degree for a specific interval. The membership function μA_{(x)} can be expressed as follows:

This function elegantly captures the gradual shift in membership degrees as x moves through the intervals defined by a1 to a30.

In this data visualization, a trigonal fuzzy number or a 30-gonal fuzzy number membership function is generated. A trigonal fuzzy number is a type of fuzzy set with a triangular shape, and in this case, it is represented by 30 vertices (points of transition) to create a smooth transition in its membership degrees.

In this section, we explore fundamental operations on Trigonal Fuzzy Numbers, a novel framework for handling uncertainty. We delve into addition, subtraction, and multiplication, highlighting their application to Trigonal Fuzzy Numbers and how alpha cuts at various confidence levels play a crucial role in these operations. Through mathematical proofs and practical examples, we illustrate the precision and versatility these operations bring to managing complex uncertainties.

Let A30g and B30g be two Trigonal Fuzzy Numbers defined as A30g = (a1, a2, ..., a30) and B30g = (b1, b2, ..., b30). The addition of A30g and B30g, denoted as C30g = A30g + B30g, is also a Trigonal Fuzzy Number with membership values defined as C30g = (c1, c2, ..., c30), where ci = ai + bi for i = 1 to 30.

Let A30g and B30g be two Trigonal Fuzzy Numbers defined as A30g = (a1, a2, ..., a30) and B30g = (b1, b2, ..., b30). The subtraction of A30g and B30g, denoted as C30g = A30g - B30g, is also a Trigonal Fuzzy Number with membership values defined as C30g = (c1, c2, ..., c30), where ci = ai - bi for i = 1 to 30.

Let A30g and B30g be two Trigonal Fuzzy Numbers defined as A30g = (a1, a2, ..., a30) and B30g = (b1, b2, ..., b30). The multiplication of A30g and B30g, denoted as C30g = A30g * B30g, is also a Trigonal Fuzzy Number with membership values defined as C30g = (c1, c2, ..., c30), where ci = ai * bi for i = 1 to 30.

Let A30g = (a1, a2, ..., a30) and B30g = (b1, b2, ..., b30) be two Trigonal Fuzzy Numbers. The addition of A30g and B30g using alpha cuts at confidence level α, denoted as C30g(α), is defined as:

C30g(α) = Alpha Cut A30g(α) + Alpha Cut B30g(α)

where Alpha Cut A30g(α) and Alpha Cut B30g(α) represent the alpha cuts of A30g and B30g at confidence level α.

Start with the definition of the alpha cuts of A30g and B30g:

Alpha Cut A30g(α) = {ai | ai ≥ α, for i = 1 to 30}

Alpha Cut B30g(α) = {bi | bi ≥ α, for i = 1 to 30}

Perform element-wise addition for the alpha cuts:

C30g(α) = {ai + bi | ai ≥ α, bi ≥ α, for i = 1 to 30}

By the definition of addition for alpha cuts, if ai and bi are both greater than or equal to α, then ai + bi is also greater than or equal to α.

Therefore, C30g(α) = {ci | ci ≥ α, for i = 1 to 30}

This satisfies the definition of an alpha cut for the sum of Trigonal Fuzzy Numbers, and thus, C30g(α) is an alpha cut of the sum of A30g and B30g.

Suppose we have two Trigonal Fuzzy Numbers:

A30g = (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0)

B30g = (0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9)

Let's calculate the addition of A30g and B30g at confidence level α = 0.5 using alpha cuts:

Alpha Cut A30g(0.5) = {ai | ai ≥ 0.5} = (0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0)

Alpha Cut B30g(0.5) = {bi | bi ≥ 0.5} = (0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9)

Now, perform addition element-wise for the alpha cuts:

C30g(0.5) = Alpha Cut A30g(0.5) + Alpha Cut B30g(0.5) = (1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 3.8, 4.0, 4.2, 4.4, 4.6, 4.8, 5.0, 5.2, 5.4, 5.6, 5.8, 6.0)

So, the alpha cut of the sum of A30g and B30g at confidence level α = 0.5 is C30g(0.5) = (1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 3.8, 4.0, 4.2, 4.4, 4.6, 4.8, 5.0, 5.2, 5.4, 5.6, 5.8, 6.0).

Let A30g and B30g be two Trigonal Fuzzy Numbers defined as A30g = (a1, a2, ..., a30) and B30g = (b1, b2, ..., b30). The subtraction of A30g and B30g using alpha cuts, denoted as C30g(α) = A30g(α) - B30g(α), for α ∈ [0, 1], results in a Trigonal Fuzzy Number C30g with membership values defined as C30g(α) = (c1, c2, ..., c30), where ci = ai - bi for i = 1 to 30, and α is the alpha cut level.

To prove Theorem 4.2.5, we need to show that the subtraction of Trigonal Fuzzy Numbers using alpha cuts produces a valid Trigonal Fuzzy Number C30g(α) with membership values as defined.

Given:

- A30g = (a1, a2, ..., a30)

- B30g = (b1, b2, ..., b30)

- Alpha cut level α ∈ [0, 1]

We want to calculate C30g(α) for each segment:

C30g(α) = A30g(α) - B30g(α)

For each segment i, we have:

ci = ai - bi

Since ai and bi belong to A30g and B30g, respectively, they are both real numbers. Subtracting two real numbers results in another real number, which satisfies the definition of a Trigonal Fuzzy Number.

Therefore, C30g(α) = (c1, c2, ..., c30) is also a valid Trigonal Fuzzy Number.

Now, let's provide the example using alpha cuts (α = 0.5)

Alpha Cut A30g(0.5) = (0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0)

Alpha Cut B30g(0.5) = (0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9)

Now, we perform the subtraction element-wise for the alpha cuts:

C30g(0.5) = Alpha Cut A30g(0.5) - Alpha Cut B30g(0.5)

C30g(0.5) = (0.5-0.5, 0.6-0.6, 0.7-0.7, 0.8-0.8, 0.9-0.9, 1.0-1.0, 1.1-1.1, 1.2-1.2, 1.3-1.3, 1.4-1.4, 1.5-1.5, 1.6-1.6, 1.7-1.7, 1.8-1.8, 1.9-1.9, 2.0-2.0, 2.1-2.1, 2.2-2.2, 2.3-2.3, 2.4-2.4, 2.5-2.5, 2.6-2.6, 2.7-2.7, 2.8-2.8, 2.9-2.9)

C30g(0.5) = (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

This is the resulting Trigonal Fuzzy Number C30g(0.5) after subtraction using alpha cuts (α = 0.5), and each segment is 0.0 as calculated element-wise.

Let A30g and B30g be two Trigonal Fuzzy Numbers defined as A30g = (a1, a2, ..., a30) and B30g = (b1, b2, ..., b30). The multiplication of A30g and B30g using alpha cuts, denoted as C30g(α) = A30g(α) * B30g(α), for α ∈ [0, 1], results in a Trigonal Fuzzy Number C30g with membership values defined as C30g(α) = (c1, c2, ..., c30), where ci = ai * bi for i = 1 to 30, and α is the alpha cut level.

To prove Theorem 4.2.6, we need to show that the multiplication of Trigonal Fuzzy Numbers using alpha cuts produces a valid Trigonal Fuzzy Number C30g(α) with membership values as defined.

Given:

- A30g = (a1, a2, ..., a30)

- B30g = (b1, b2, ..., b30)

- Alpha cut level α ∈ [0, 1]

We want to calculate C30g(α) for each segment:

C30g(α) = A30g(α) * B30g(α)

For each segment i, we have:

ci = ai * bi

Since ai and bi belong to A30g and B30g, respectively, they are both real numbers. Multiplying two real numbers results in another real number, which satisfies the definition of a Trigonal Fuzzy Number.

Therefore, C30g(α) = (c1, c2, ..., c30) is also a valid Trigonal Fuzzy Number.

Now, let's provide the example using alpha cuts (α = 0.5)

Alpha Cut A30g(0.5) = (0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0)

Alpha Cut B30g(0.5) = (0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9)

Now, we perform the multiplication element-wise for the alpha cuts:

C30g(0.5) = Alpha Cut A30g(0.5) * Alpha Cut B30g(0.5)

C30g(0.5) = (0.5*0.5, 0.6*0.6, 0.7*0.7, 0.8*0.8, 0.9*0.9, 1.0*1.0, 1.1*1.1, 1.2*1.2, 1.3*1.3, 1.4*1.4, 1.5*1.5, 1.6*1.6, 1.7*1.7, 1.8*1.8, 1.9*1.9, 2.0*2.0, 2.1*2.1, 2.2*2.2, 2.3*2.3, 2.4*2.4, 2.5*2.5, 2.6*2.6, 2.7*2.7, 2.8*2.8, 2.9*2.9)

C30g(0.5) = (0.25, 0.36, 0.49, 0.64, 0.81, 1.0, 1.21, 1.44, 1.69, 1.96, 2.25, 2.56, 2.89, 3.24, 3.61, 4.0, 4.41, 4.84, 5.29, 5.76, 6.25, 6.76, 7.29, 7.84, 8.41, 9.0)

This is the resulting Trigonal Fuzzy Number C30g(0.5) after multiplication using alpha cuts (α = 0.5), and each segment is calculated by multiplying the corresponding segments of A30g and B30g element-wise.

Trigonal Fuzzy Numbers offer a versatile way to represent and handle uncertainty in various real-life applications. Here are some examples of how Trigonal Fuzzy Numbers can be applied in practical scenarios:

In each of these applications, Trigonal Fuzzy Numbers enable decision-makers to capture and manage uncertainty more effectively, leading to better-informed decisions and improved risk management. Their flexibility in expressing uncertainty across a wide range of domains makes them a valuable tool for decision support systems and modeling real-world problems.

Let's create a simplified real-life numerical example using two Trigonal Fuzzy Numbers, A30g and B30g, to represent uncertain quantities. In this example, we'll consider a scenario involving rainfall and temperature measurements. We will use Trigonal Fuzzy Numbers to represent the uncertainty in these measurements over a 30-day period.

Trigonal Fuzzy Number A30g: (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0)

Interpretation: This Trigonal Fuzzy Number represents the daily rainfall in inches for a month. It indicates that, for each day, the lower bound of rainfall is 0.1 inches, the most likely value is 1.0 inches, and the upper bound is 3.0 inches.

Trigonal Fuzzy Number B30g: (0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9)

Interpretation: This Trigonal Fuzzy Number represents the daily temperature in degrees Celsius for a month. It indicates that, for each day, the lower bound of temperature is 0.0 degrees Celsius, the most likely value is 1.0 degrees Celsius, and the upper bound is 2.9 degrees Celsius.

Using Trigonal Fuzzy Numbers:

These Trigonal Fuzzy Numbers allow us to represent the uncertainty in daily rainfall and temperature measurements. For example, on a particular day, the actual rainfall could be anywhere within the range specified by A30g, and the temperature could be within the range specified by B30g.

This section provides a comparative overview of the existing work in the field of fuzzy number theory and the novel contributions presented in this paper. The primary focus is on introducing the concept of Trigonal Fuzzy Numbers, a flexible framework for representing uncertainty. This comparison highlights the key distinctions between traditional fuzzy numbers and the innovative Trigonal Fuzzy Numbers proposed in this paper.

This article has introduced a groundbreaking paradigm shift in fuzzy number theory by presenting Trigonal Fuzzy Numbers as a versatile tool for representing and managing uncertainty. The incorporation of Alpha Cuts provides a precise and systematic approach to quantify the degree of uncertainty associated with these fuzzy numbers. Through rigorous mathematical formulations and practical applications, this paper has demonstrated the theoretical strength and real-world effectiveness of Trigonal Fuzzy Numbers in addressing complex problems involving ambiguity and imprecision. The results indicate that Trigonal Fuzzy Numbers are not only compelling from a theoretical standpoint but are also highly practical for tackling real-world issues across a wide range of domains. These tools offer a fresh perspective on decision support systems, significantly enhancing the accuracy of the decision-making process in scenarios characterized by uncertainty. As uncertainty continues to be a significant factor in various fields, the combination of Trigonal Fuzzy Numbers and Alpha Cuts offers a promising avenue for more informed and robust decision-making. This innovation can lead to more reliable outcomes, ultimately benefiting organizations and individuals facing complex decision-making challenges.

While this research has laid a strong foundation for the use of Trigonal Fuzzy Numbers and Alpha Cuts, there are several promising directions for future work:

Conduct more extensive real-world case studies and validations in a variety of domains to assess the practicality and effectiveness of Trigonal Fuzzy Numbers in diverse scenarios. This would help establish a broader understanding of their applicability.

Develop algorithms and computational methods for handling Trigonal Fuzzy Numbers efficiently, particularly in large-scale applications. This could involve optimization techniques and parallel computing to enhance computational performance.

Explore how Trigonal Fuzzy Numbers can be integrated with artificial intelligence and machine learning techniques to improve decision-making and reasoning under uncertainty. This could involve developing fuzzy logic-based AI models.

Create user-friendly visualization tools for Trigonal Fuzzy Numbers and Alpha Cuts, making it easier for decision-makers to interpret and use these concepts effectively.

Develop standardized guidelines and best practices for utilizing Trigonal Fuzzy Numbers and Alpha Cuts, fostering consistency and comparability across different applications.

Encourage interdisciplinary collaboration to leverage the potential of Trigonal Fuzzy Numbers across various fields, such as finance, healthcare, engineering, and environmental science.

These future research directions can further enhance the practicality and impact of Trigonal Fuzzy Numbers with Alpha Cuts, ultimately contributing to more informed.

The authors also like to thank the anonymous referee for giving many helpful suggestions on the revision of present paper.