A Ternary Gamma Semiring Framework for Solving Multi-Objective Network Optimization Problems

Classical shortest-path methods rely on binary tropical semirings $(\min,+)$, whose dyadic structure limits them to pairwise cost interactions. However, many real-world systems, including logistics, supply chains, communication networks, and reliability-aware infrastructures, exhibit inherently ternary dependencies among cost, time, and risk that cannot be decomposed into pairwise components. This paper introduces the \emph{Ternary Tropical Gamma Semiring} (TTGS), a $\Gamma$-indexed algebraic structure that generalizes tropical semirings by replacing binary additive composition with a non-separable ternary operator. We establish the axioms of TTGS, prove associativity, distributivity, and monotonicity, and show that TTGS forms a well-structured foundation for multi-parameter optimization. Building on this framework, we develop TTGS-Pathfinder, a ternary analogue of the Bellman--Ford algorithm. We derive its dynamic-programming recurrence, prove correctness through an invariant-based argument, analyze convergence under the TTGS order, and obtain an $O(n^2 m)$ complexity bound. Applications demonstrate that TTGS naturally models systems whose behaviour depends on triadic cost interactions, offering a principled alternative to binary tropical, vector, or scalarized multi-objective methods.