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We study the capacitated arc routing problem in the state-space-time network. A multi-commodity network flow model is formulated in the state-space-time network. Two Lagrangian relaxation-based decomposition methods are developed to tackle this problem. By dualizing the coupling constraints to the objective function, the Lagrangian relaxed problem (LR) can be decomposed into several routing subproblems in the state-space-time network. Then we extend quadratic penalty terms to the LR and obtain an augmented Lagrangian relaxed problem (ALR). By applying the alternating direction method of multipliers and linearization techniques, we decompose the ALR into a sequence of routing subproblems. We solve these subproblems of LR and ALR by a time-dependent dynamic programming algorithm. Feasible solutions are produced according to the result of the LR or ALR. The Lagrangian multipliers are updated by the subgradient optimization method. We implement the proposed methods in three networks, showing that the ALR-based method has better performance.

Lagrangian relaxation-based decomposition approaches for the capacitated arc routing problem in the state-space-time network