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Quality-of-service routing (QoSR) with multiple constraints, which seeks to find a feasible path satisfying multiple constraints simultaneously, is a challenging problem of the next-generation networks. For its NP-complete complexity, we propose an adjustable heuristic based on converting multiple QoS weights to a single metric with energy functions. By applying the breadth-first search (BFS) to Dijkstra's algorithm with adjustable depth, BFS _MCP (BFS for multi-constrained paths) can adjust its time complexity according to the CPU load on a router in real time. Thus, it has an extensive adaptability. Additionally, we propose a novel approach to performance evaluation by generating QoS constraints, named weight-proportion simulation. Generating QoS constraints similar to QoS applications, this method extends the original success ratio, only used in relative performance comparison, to the evaluation of absolute performance. By this method, extensive simulations show that BFS improves the performance greatly. The main contribution of the paper includes a heuristic for multi-constrained routing and a novel approach to performance evaluation.