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We introduce an effective and efficient iterative algorithm for solving the continuous-time service network design problem. The algorithm achieves its efficiency by carefully and dynamically refining partially time-expanded network models so that only a small number of small integer programs, defined over these networks, need to be solved. An extensive computational study shows that the algorithm performs well in practice, often using time-expanded network models with size much less than 1% (in terms of number of variables and constraints) of a full time-expanded network model. The algorithm is inspired by and has many similarities to the dynamic discretization discovery algorithm introduced in Boland et al. [Boland N, Hewitt M, Marshall L, Savelsbergh M (2017) The continuous-time service network design problem. Oper. Res. 65(5):1303–1321.], but generates smaller partially time-expanded models, produces high-quality solutions more quickly, and converges more quickly. 

We consider a continuous time variant of the Inventory Routing Problem in which the maximum quantity that can delivered at a customer depends on the customer's storage capacity and product inventory at the time of the delivery. We investigate critical components of a dynamic discretization discovery algorithm and demonstrate in an extensive computational study that these components are sufficient to produce provably high-quality, often optimal, solutions. 

Waiting at the right location at the right time can be critically important in certain variants of time-dependent shortest path problems. We investigate the computational complexity of time-dependent shortest path problems in which there is either a penalty on waiting or a limit on the total time spent waiting at a given subset of the nodes. We show that some cases are nondeterministic polynomial-time hard, and others can be solved in polynomial time, depending on the choice of the subset of nodes, on whether waiting is penalized or constrained, and on the magnitude of the penalty/waiting limit parameter. Summary of Contributions: This paper addresses simple yet relevant extensions of a fundamental problem in Operations Research: the Shortest Path Problem (SPP). It considers time-dependent variants of SPP, which can account for changing traffic and/or weather conditions. The first variant that is tackled allows for waiting at certain nodes but at a cost. The second variant instead places a limit on the total waiting. Both variants have applications in transportation, e.g., when it is possible to wait at certain locations if the benefits outweigh the costs. The paper investigates these problems using complexity analysis and algorithm design, both tools from the field of computing. Different cases are considered depending on which of the nodes contribute to the waiting cost or waiting limit (all nodes, all nodes except the origin, a subset of nodes…). The computational complexity of all cases is determined, providing complexity proofs for the variants that are NP-Hard and polynomial time algorithms for the variants that are in P. 

We present an exact algorithm for the Minimum Duration Time-Dependent Shortest Path Problem with piecewise linear arc travel time functions. The algorithm iteratively refines a time-expanded network model, which allows for the computation of a lower and an upper bound, until - in a finite number of iterations - an optimal solution is obtained.