Understanding the Interval Valued Neutrosophic Set
The Interval Valued Neutrosophic Set provides a robust mathematical framework for modeling real-world data that contains incomplete or inconsistent information. Unlike traditional fuzzy logic, this approach characterizes an object using three independent membership functions: Truth (T), Indeterminacy (I), and Falsity (F). Furthermore, each of these components utilizes an interval range instead of a single point value. Consequently, researchers gain the flexibility to express their belief degrees with far greater precision when facing vague environments.
- T(x) = [T_L, T_U]: The range of truth-membership.
- I(x) = [I_L, I_U]: The range of indeterminacy-membership.
- F(x) = [F_L, F_U]: The range of falsity-membership.
Numerical Setup for Decision Analysis using Interval Valued Neutrosophic Set
To demonstrate how this logic works, let us consider two expert evaluations for a investment project. Specifically, the experts use an Interval Valued Neutrosophic Set to record their confidence, their hesitation, and their rejection levels. Furthermore, we define two sets, A and B, representing two different project criteria.
Set B: T=[0.6, 0.7], I=[0.2, 0.3], F=[0.1, 0.2]
Consequently, we now have a structured basis to calculate the combined influence of these criteria using standard neutrosophic operators.
Step-by-Step Numerical Operations of Interval Valued Neutrosophic Set
Furthermore, an Interval Valued Neutrosophic Set requires specific rules for aggregation. Specifically, we apply maximum logic for truth and minimum logic for both indeterminacy and falsity during a union operation.
1. The Union Operation (OR)
This operation reflects the most optimistic view by prioritizing the highest truth-membership and the lowest possible falsity or indeterminacy levels. As a result, we calculate the union as follows:
A: T=[0.4, 0.5], I=[0.1, 0.2], F=[0.3, 0.4]
B: T=[0.6, 0.7], I=[0.2, 0.3], F=[0.1, 0.2]
1. Truth: [max(0.4, 0.6), max(0.5, 0.7)] = [0.6, 0.7]
2. Indeterminacy: [min(0.1, 0.2), min(0.2, 0.3)] = [0.1, 0.2]
3. Falsity: [min(0.3, 0.1), min(0.4, 0.2)] = [0.1, 0.2]
Result: { [0.6, 0.7], [0.1, 0.2], [0.1, 0.2] }
2. The Intersection Operation (AND)
In contrast, the intersection adopts a more conservative stance. Specifically, it selects the minimum truth and the maximum levels for both indeterminacy and falsity. Consequently, the user identifies the common ground between the two data points.
A: T=[0.4, 0.5], I=[0.1, 0.2], F=[0.3, 0.4]
B: T=[0.6, 0.7], I=[0.2, 0.3], F=[0.1, 0.2]
1. Truth: [min(0.4, 0.6), min(0.5, 0.7)] = [0.4, 0.5]
2. Indeterminacy: [max(0.1, 0.2), max(0.2, 0.3)] = [0.2, 0.3]
3. Falsity: [max(0.3, 0.1), max(0.4, 0.2)] = [0.3, 0.4]
Result: { [0.4, 0.5], [0.2, 0.3], [0.3, 0.4] }
Practical Applications and Benefits
Moreover, the Interval Valued Neutrosophic Set excels in multi-criteria decision-making scenarios where evidence is often contradictory. For instance, in medical diagnostics, a test result might suggest a disease (Truth) while another suggests health (Falsity), all while a third factor remains unclear (Indeterminacy). Consequently, the interval approach captures these nuances without forcing an artificial average. Finally, this method provides a much more transparent roadmap for solving supply chain and engineering problems.
| Feature | Standard Fuzzy Set | Neutrosophic Interval Set |
|---|---|---|
| Dimensions | Single (Membership) | Triple (T, I, F) |
| Measurement | Fixed Point Value | Dynamic Interval Range |
| Conflict Handling | Very Limited | High (Independent Falsity/Indeterminacy) |
In summary, this mathematical tool transforms how we process ambiguous information. By utilizing independent ranges for doubt and belief, it offers a superior level of data integrity for complex computational systems.
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator