The {\alpha} -Discounting ({\alpha}-DMCDM) as an extension of AHP, TOPSIS, VIKOR, PROMETHEE, and Weighted Sum

Technical Summary: The $\alpha$-Discounting ($\alpha$-DMCDM) as an Extension of AHP, TOPSIS, VIKOR, PROMETHEE, and Weighted Sum

Problem Statement
Traditional Multi-Criteria Decision Making (MCDM) techniques, including the Analytic Hierarchy Process (AHP), TOPSIS, VIKOR, PROMETHEE, and Weighted Sum models, rely on the assumption that decision-makers can provide consistent, pairwise (or fully compatible) preference information. In real-world scenarios, however, preference structures are often incomplete, non-linear, or contradictory. For instance, cyclic inconsistencies (e.g., $C_1 > C_2$, $C_2 > C_3$, but $C_3 > C_1$) render the systems of equations used by these methods unsolvable, yielding only trivial solutions. While AHP detects inconsistency via the Consistency Ratio (CR), it offers no principled mechanism to repair the judgment set or derive weights from inconsistent data. Other methods simply require pre-computed weights and cannot process ill-posed preference structures. Existing extensions using fuzzy or neutrosophic sets often still depend on a consistent weight extraction step.

Methodology: $\alpha$-Discounting MCDM
The paper proposes the $\alpha$-Discounting MCDM ($\alpha$-DMCDM) as a unified algebraic framework that embraces inconsistent information rather than rejecting it. The core methodology transforms preference statements—whether linear, non-linear, pairwise, or $n$-wise—into a system of algebraic equations or inequalities.

1. Axiomatic Basis: The method rests on seven axioms, most notably:

   • Axiom A3 (Inconsistency Admissibility): The system may be inconsistent (i.e., $\det(I - A) \neq 0$), yielding only the trivial solution $x=0$.
   • Axiom A4 ($\alpha$-Discounting Principle): A global scalar discount factor $\alpha > 0$ is introduced such that the discounted system $(I - \alpha A)x = 0$ admits a non-trivial solution.
   • Axiom A5 (Consistency Measure): The value of $\alpha$ quantifies global consistency. $\alpha = 1$ implies full consistency; $0 < \alpha < 1$ indicates inconsistency, with smaller values reflecting stronger contradictions.

2. Existence Theorem: The paper proves that for any finite set of linear (or linearized) preference equations where the original system is inconsistent, there exists at least one positive real scalar $\alpha$ such that $\det(I - \alpha A) = 0$. This ensures the existence of a non-trivial eigenvector that can be normalized to yield priority weights.

3. Algorithmic Procedure:

   • Model preferences as a system of equations/inequalities to form matrix $A$.
   • Introduce a single discount factor $\alpha$ (adhering to the Fairness Principle).
   • Solve $\det(I - \alpha A) = 0$ for the admissible $\alpha$ (typically the smallest positive root).
   • Compute the eigenvector of $(I - \alpha A)$ associated with the zero eigenvalue.
   • Normalize the eigenvector to obtain the final weight vector $w$.

4. Extensions: The framework is modular and supports:

   • Interval $\alpha$-Discounting: Handling vague coefficients as intervals $[a^-, a^+]$, resulting in an interval of admissible $\alpha$ values and interval-valued weights.
   • Non-Linear $\alpha$-Discounting: Accommodating arbitrary functional relationships (e.g., quadratic, logarithmic) by reducing the system to a transcendental equation solvable via numerical root-finding.
   • Group Decision Making: Aggregating multiple experts' matrices via averaging, weighted averaging, or minimum $\alpha$ selection, preserving the quantitative treatment of inconsistency.
   • Neutrosophic Logic: Mapping the discount factor $\alpha$ to the "Truth" component ($T$) of neutrosophic triplets $(T, I, F)$, allowing the method to handle indeterminacy and falsity explicitly.

Key Contributions

• Theoretical Unification: Provides a self-contained exposition of $\alpha$-DMCDM, establishing a rigorous axiomatic base and an existence theorem for the discount factor.
• Inconsistency Management: Offers a principled mechanism to quantify and resolve inconsistency algebraically without discarding contradictory judgments. The parameter $\alpha$ serves as a direct metric of the degree of conflict.
• Expressiveness: Allows decision-makers to articulate preferences in natural mathematical forms (linear, non-linear, interval) without forcing them into a pairwise matrix structure.
• Practical Implementation: Includes illustrative numerical examples (linear cyclic, interval-based, and non-linear) and provides Python code implementations for solving these systems.
• Comparative Analysis: Systematically contrasts $\alpha$-DMCDM with AHP, TOPSIS, VIKOR, PROMETHEE, and Weighted Sum, highlighting its unique ability to handle non-linear preferences, interval uncertainty, and inconsistent data while maintaining computational tractability comparable to eigen-based methods.

Results and Illustrative Examples
The paper demonstrates the method through three specific cases:

1. Linear Cyclic Case: For a system where $x_1 = 2x_2$, $x_2 = 3x_3$, and $x_3 = 4x_1$ (a cyclic inconsistency), the method calculates $\alpha \approx 0.346$. The resulting weights are $(0.26, 0.37, 0.37)$, providing a solution where traditional AHP would fail to produce weights due to inconsistency.
2. Interval Case: For environmental criteria with vague importance ranges, the method derives an admissible $\alpha$-interval $[0.346, 0.550]$. Selecting a representative $\alpha$ yields interval-valued weights, allowing decision-makers to explore risk-averse or risk-seeking stances.
3. Non-Linear Case: For a technology adoption study with quadratic and logarithmic relationships, the method solves a transcendental equation to find $\alpha \approx 0.45$, yielding weights that reflect the specific non-linear behavioral insights (e.g., diminishing returns).

Significance and Claims
The paper claims that $\alpha$-Discounting bridges the gap between classical decision theory and modern uncertainty-aware reasoning. Its primary significance lies in transforming inconsistency from a negative diagnostic (a reason to reject data) into actionable information (a quantifiable parameter $\alpha$). By retaining the algebraic simplicity of linear methods while expanding expressive power to include non-linear and uncertain data, the method is presented as a robust alternative for complex engineering, strategic planning, and AI-driven preference aggregation. The authors position $\alpha$-DMCDM not as a replacement for all existing methods, but as a necessary extension for scenarios where preference structures are inherently contradictory, incomplete, or non-linear.

Limitations and Future Directions
The paper modestly acknowledges limitations, including the potential for multiple positive roots in polynomial equations (requiring domain knowledge for selection), scalability challenges for very large systems, and the need for practitioner training to interpret the $\alpha$ metric. Future research directions suggested include integrating machine learning for preference elicitation, extending the framework to dynamic time-varying $\alpha$, and developing graphical interfaces to visualize the geometric deformation of hyperplanes caused by $\alpha$.