$alpha$-Discounting Multi-Criteria Decision Making ($alpha$-D MCDM)

In this book we introduce a new procedure called \alpha-Discounting Method for Multi-Criteria Decision Making (\alpha-D MCDM), which is as an alternative and extension of Saaty Analytical Hierarchy Process (AHP). It works for any number of preferences that can be transformed into a system of homogeneous linear equations. A degree of consistency (and implicitly a degree of inconsistency) of a decision-making problem are defined. \alpha-D MCDM is afterwards generalized to a set of preferences that can be transformed into a system of linear and or non-linear homogeneous and or non-homogeneous equations and or inequalities. The general idea of \alpha-D MCDM is to assign non-null positive parameters \alpha_1, \alpha_2, and so on \alpha_p to the coefficients in the right-hand side of each preference that diminish or increase them in order to transform the above linear homogeneous system of equations which has only the null-solution, into a system having a particular non-null solution. After finding the general solution of this system, the principles used to assign particular values to all parameters \alpha is the second important part of \alpha-D, yet to be deeper investigated in the future. In the current book we propose the Fairness Principle, i.e. each coefficient should be discounted with the same percentage (we think this is fair: not making any favoritism or unfairness to any coefficient), but the reader can propose other principles. For consistent decision-making problems with pairwise comparisons, \alpha-Discounting Method together with the Fairness Principle give the same result as AHP. But for weak inconsistent decision-making problem, \alpha-Discounting together with the Fairness Principle give a different result from AHP. Many consistent, weak inconsistent, and strong inconsistent examples are given in this book.

💡 Research Summary

The monograph introduces a novel decision‑making framework called the α‑Discounting Method for Multi‑Criteria Decision Making (α‑D MCDM), positioned as an alternative and extension to Saaty’s Analytic Hierarchy Process (AHP). While AHP relies on pairwise comparison matrices, consistency ratios, and eigenvector calculations, α‑D MCDM treats every expressed preference as a mathematical equation. Preferences may be represented as homogeneous linear equations (e.g., a·x₁ = b·x₂), non‑homogeneous linear equations, nonlinear equations (e.g., x₁² + 3x₂ = 0), or inequalities (e.g., x₁ ≥ 2x₂). When the resulting system admits only the trivial solution (all variables zero), the method introduces a set of positive discount parameters α₁, α₂, …, α_p that multiply or divide the coefficients on the right‑hand side of each equation. This “discounting” transforms the original coefficient matrix A into a modified matrix A(α). The transformed system A(α)x = 0 typically possesses a non‑trivial solution, which directly yields the relative weights of the alternatives.

A central issue is how to choose the α‑parameters. The authors propose the Fairness Principle: all coefficients are discounted by the same percentage (i.e., a common factor 1‑δ). This principle is intended to avoid favoritism toward any particular criterion. Under this principle, perfectly consistent AHP problems (where the pairwise matrix is already consistent) correspond to α = 1, and α‑D MCDM reproduces the same eigenvector weights as AHP. For weakly inconsistent problems, applying a uniform discount (e.g., α = 0.9) modifies the system enough to generate a distinct, non‑trivial solution that differs from the AHP approximation, thereby offering an alternative ranking that reflects the underlying inconsistency rather than forcing a forced consistency adjustment.

Mathematically, the method proceeds by solving the eigenvalue problem for A(α). The dominant eigenvalue’s eigenvector, after normalization, provides the priority vector. When nonlinear or inequality constraints are present, the authors suggest using Lagrange multipliers, nonlinear programming, or linearization techniques to obtain feasible solutions. Moreover, the α‑parameters themselves can be treated as decision variables: one may minimize a consistency loss (e.g., sum of squared residuals) or an information‑theoretic criterion (e.g., entropy) to determine the optimal discount level, thereby integrating consistency assessment directly into the solution process.

The book presents a series of illustrative cases. In consistent examples, α‑D MCDM and AHP yield identical weight vectors, confirming that the new method does not disturb a perfectly coherent decision structure. In weakly inconsistent examples, the uniform discount produces a priority vector that is more balanced than the AHP result, which tends to over‑emphasize the most dominant comparisons. In strongly inconsistent scenarios, AHP’s consistency ratio exceeds acceptable thresholds, rendering its eigenvector unreliable; α‑D MCDM, by adjusting α, restores a solvable system and delivers a usable set of weights.

Beyond pairwise comparisons, the authors generalize α‑D MCDM to accommodate mixed systems of linear, nonlinear, homogeneous, non‑homogeneous equations and inequalities. This flexibility enables modeling of complex relationships such as multiplicative interactions, logarithmic transformations, and explicit constraints—features that are cumbersome to embed in traditional AHP. Potential application domains include multi‑objective optimization, portfolio selection, supply‑chain network design, and any context where criteria interact in non‑linear ways.

The monograph concludes by highlighting three principal contributions: (1) a unified equation‑based representation of preferences that captures a broader class of decision problems; (2) an integrated consistency‑adjustment mechanism via α‑discounting, which quantifies and mitigates inconsistency rather than merely flagging it; and (3) a principled, user‑configurable discounting scheme (the Fairness Principle) that can be extended or replaced by alternative principles reflecting decision‑maker values. The authors acknowledge that the selection of α‑parameters remains an open research question and propose future work on optimal α‑selection criteria (e.g., least‑squares, entropy‑based), multi‑principle discounting schemes, dynamic updating of α in real‑time decision environments, and scalable numerical algorithms for large‑scale systems.