Mixed Linear Equation–Inequality Systems over the Pura Vida Neutrosophic Algebra.

This paper proposes a neutrosophic extension of max-plus algebra for solving mixed systems of linear equations and inequalities. Classical max-plus algebra is a powerful tool for modeling synchronization in discrete-event systems [1, 2], but it assumes fully deterministic data. To incorporate uncertainty and indeterminacy, we reformulate the framework so that coefficients and variables are expressed as neutrosophic numbers γ + λI, γ, λ ∈ R, I ∈ [0, 1], where I quantifies the degree of indeterminacy. We redefine the max-plus semiring in this neutrosophic setting, extend solvability and uniqueness results, and adapt the ONEMLP-EI algorithm [3] to handle neutrosophic optimization. A sensitivity analysis demonstrates robustness under parameter perturbations, complementing recent advances in neutrosophic linear programming and decision-making under uncertainty [4–6]. Through a numerical example and a compact case study, we illustrate the feasibility, non-uniqueness, and optimization capabilities of the framework. Applications in scheduling and production systems, under incomplete information, highlighting the novelty and practical value of neutrosophic max-plus modeling. [ABSTRACT FROM AUTHOR]