The freight routing problem of time-definite common carriers is to minimize the sum of handling and transportation costs, while meeting service commitments and operational restrictions. There are two types of operational restrictions, capacity and directed in-tree rooted at each destination. Directed in-tree implicitly implies that there is a singular path for each origin-destination pair. The routing problem is an integrality constrained multi-commodity problem with side constraints. In this research, we study two approaches, the Lagrangian relaxation (LR) and implicit enumeration algorithm with ε-optimality (IE-ε). We use the third largest time-definite freight delivery common carrier in Taiwan for our numerical test. The result shows that the IE-ε outperforms the LR, both quantitatively and qualitatively. In addition, two major shortcomings of the LR approach are shown: It may fail to find any feasible solutions even though they exist, and it cannot determine whether the feasible set is empty or not.
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