Choosing the optimal software architecture for cloud-based systems is a critical and complex Multi-Criteria Decision Making (MCDM) problem, characterized by multiple, often conflicting, and interdependent criteria such as performance, cost, scalability, deployment speed, security, and maintainability. This research addresses this challenge by proposing and applying an integrated MCDM methodology that leverages Evidential Reasoning (ER) and Bayesian Inference (BI). The study's primary objective is to provide a robust and transparent framework for evaluating six common architecture styles: Monolithic, Microservices, Layered, Serverless, Event-Driven, and Service-Oriented Architecture (SOA). The methods employed involved a multi-stage process. First, criteria weights were derived using the Analytic Hierarchy Process (AHP) through expert pairwise comparisons. The techniques for handling uncertainty and dependencies were central. ER was utilized to aggregate subjective and objective assessments, representing them as belief distributions to explicitly account for imprecision and ignorance. Concurrently, BI was applied to model probabilistic interdependencies between criteria (Security influencing Performance, Performance influencing Scalability and Cost) within a Bayesian Network. The Intelligent Decision System (IDS) tool facilitated the operationalization of both ER aggregation and Bayesian inference. The results of the AHP weighting revealed the priorities: Performance (0.3930), Security (0.2355), Scalability (0.1420), Maintainability (0.1160), Deployment Speed (0.0568), and Cost (0.0568). The overall evaluation, integrating these weighted criteria with ER and BI, identified Monolithic architecture as the most suitable option, achieving a utility score of 0.81. This ranking was followed by Event-Driven (0.69), SOA (0.68), Serverless (0.68), Microservices (0.65), and Layered (0.47). A comprehensive sensitivity analysis was conducted to assess the robustness of this decision. Crucially, the analysis demonstrated that while the Monolithic architecture was initially optimal, significant shifts in criteria weights could alter the ranking. Specifically, when the weight of Security was substantially increased (to ~0.32) and Performance decreased (to ~0.25), the Serverless architecture emerged as the new top-ranked alternative (83% utility score), surpassing Monolithic (78%). This finding underscores the critical influence of strategic priorities on architecture selection. Future studies may also focus on developing data-driven, adaptive, and domain-specific decision frameworks to enhance the robustness, transparency, and real-world applicability of MCDM approaches for cloud-based software architecture selection.