A travel company offers a transportation service by picking up customers from certain places and sending them to the other places. Concerning traffic in time domain function for every path, the problem is how to find the quickest paths in picking up the customers. The problem may be considered as the quickest path problem. We may consider the quickest path problem as an extension of the shortest path problem. The quickest path problem optimizes length of time in visiting some places. The quickest path is not always the shortest path. In the relation to the problem, this paper proposes an algorithm for solving the quickest path problem. The algorithm is constructed by modifying a concept of transitive closure method.