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 LastNatpreClose LastNatClose, 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 LastNatpreClose LastNatClose to handle neutrosophic optimization.A sensitivity analysis demonstrates robustness under parameter perturbations, complementing recent advances in neutrosophic linear programming and decision-making under uncertainty LastNatpreClose LastNatClose.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.