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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. [ABSTRACT FROM AUTHOR]

• Propose a new variant of the GAP and formulate a mean-standard deviation model. • Develop a Lagrangian relaxation based decomposition method. • Propose a decomposition method based on alternating direction method of multipliers. The well-known generalized assignment problem has many real-world applications. The assignment costs between agents and tasks affected by several factors could be unstable and uncertain. In this paper, we assume that the means and variances of assignment costs are known in advance. The idea is that the decision-maker aims to minimize the total assignment costs not only on average, but also to keep their variability as small as possible. Then, a reliability-oriented model with a nonlinear and concave objective function is formulated, and two decomposition-based methods are systematically developed for this challenging problem. The task allocation constraints are dualized and the Lagrangian relaxed problem is broken into many reliability-oriented knapsack subproblems. By the Lagrangian substitution technique, each subproblem is further decomposed into a standard knapsack problem and a simple univariate concave minimization problem. A lower bound is constructed and multipliers are optimized in dual problems by the subgradient method. Feasible solutions can be generated from the results of these reliability-oriented subproblems by Lagrangian heuristic. To further improve the convergence and solution quality, an Alternating Direction Method of Multipliers (ADMM) based decomposition approach is proposed. The augmented Lagrangian relaxed problem is split into a sequence of subproblems by the block coordinate descent method. To cope with the quadratic terms and the concave terms in these subproblems, linearization and Lagrangian substitution techniques are applied. Feasible solutions are produced from these subproblems, and solution quality can be evaluated by the lower bound provided by the Lagrangian relaxed problem. Numerical experiments are conducted on the test cases transformed from the standard benchmark instances, and the ADMM-based method has superior convergence and solution quality. [ABSTRACT FROM AUTHOR]