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We propose a new efficient approximation algorithm for evaluating the reliability of a traffic-path-based network model. This algorithm effectively combines the methods of `reduction,' `partial factoring,' and `partial enumeration' to evaluate reliability, where reduction and partial factoring are first applied to the traffic-path-based network model. This proposed algorithm reduces the complications in achieving high-speed computing approximation reliability, while guaranteeing the exact value of reliability between upper and lower bounding values. We emphasize that these two values are very close to each other, under the reasonable assumption that network equipment and cables are highly reliable in practice. We present numerical examples showing that the computing time is short, that this algorithm is more practical, and its approximation errors are smaller than those of the existing approximation method.