Unifying Soft Set Theory and Temporal MCDM: A Matrix-Algebraic Framework for Equilibrium Analysis and Frequency-Based Ranking in Dynamic Environments

Abstract

Modern decision systems increasingly require uncertainty models that are both parameter-dependent and temporally adaptive. Although soft set theory provides a versatile means of working with imprecise data, classical soft sets are static and unable to reflect changes in time or context. Recent development of the dynamic soft sets has provided a route through which uncertainty in adaptive settings can be modeled, but any single algebraic matrix framework has not been developed yet. This research paper forms an extensive theory of the dynamic soft set based on time or context-indexed matrix form and operator-based aggregation ( $$\max$$ , $$\min$$ , $$\text {avg}$$ ) using frequency matrix decision methods. We demonstrate closure, commutativity, and equilibrium theorems of dynamic soft matrix operations, defining strict structural stability with respect to time or context evolution. Computational experiments prove that soft matrix consistency values ( $$\epsilon \le 0.2$$ ) conserve more than 95% structural coherence between consecutive time steps. In addition, a comparative simulation of the dynamic service selection demonstrates that the offered Frequency Matrix Decision Making (FMDM) has a balanced trade-off between stability in ranking ( $$\rho = 0.66$$ ) and robustness to noise ( $$\tau = 0.59$$ ) as well as a unique provision of authenticity measures to check the consensus. These findings address an open gap in the temporal matrix-base uncertainty modelling for adaptive decision support.