Routing, Scheduling, and Sorting in Consolidated Networks
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Date
2024-04-25
Authors
Van Dyk, Madison
Journal Title
Journal ISSN
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Publisher
University of Waterloo
Abstract
Modern parcel logistic networks are designed to ship demand between given origin, destination pairs of nodes in an underlying directed network. Efficiency dictates that volume needs to be consolidated at intermediate nodes in typical hub-and-spoke fashion. In practice, such consolidation requires tracking packages in both space and time (temporal network design), as well as parcel sortation.
In the first half of the thesis, we study solution methods for temporal problems arising in consolidated networks. While many time-dependent network design problems can be formulated as time-indexed formulations, the size of these formulations depends on the discretization of the time horizon and can become prohibitively large. The recently-developed dynamic discretization discovery (DDD) method allows many time-dependent problems to become more tractable by iteratively solving instances of the problem on smaller networks where each node has its own discrete set of departure times.
There are two design elements of existing DDD algorithms that make it problematic for use in region-based hub-and-spoke networks. First, in each iteration, all arcs departing a common node share the same set of departure times. This causes DDD to be ineffective for solving problems where all near-optimal solutions require many distinct departure times at the majority of the high-degree nodes in the network, an aspect characteristic of region-based networks. A second challenge is handling static storage constraints without leading to a weak relaxation in each iteration.
To mitigate these limitations, an arc-based DDD framework is proposed in Chapter 3, where departure times are determined at the arc level instead of the node level. We apply this arc-based DDD method to instances of the service network design problem (SND). We show that an arc-based approach is particularly advantageous when instances arise from region-based networks, and when candidate paths are fixed in the base graph for each commodity. Moreover, our algorithm builds upon the existing DDD framework and achieves these improvements with only benign modifications to the original implementation. Additionally, Chapter 4 introduces bounds on additional storage required in each iteration, expanding the applicability of DDD to problems with bounded node storage, such as the universal packet routing problem. Our arguments rely solely on the structure of the standard map, μ, from the original formulation to the smaller relaxed formulations.
In order to maintain consistent operations, some models stipulate that the implemented transportation schedule must be repeated each day. In Chapter 5 we present a DDD model for solving a version of SND with cyclic constraints. We show that these cyclic constraints require new conditions on the time discretization at each node, leading to larger partial networks. We then highlight challenges in reducing the size of partial networks as they grow over each iteration of DDD. We demonstrate that certain policies for removing departure times in each iteration can cause the iterations in DDD to repeat, preventing termination.
In the second half of this thesis, we study parcel sortation, an aspect of routing that has previously been left unaddressed from a theory perspective. Warehouses have limited sort points, the physical devices tasked with handling packages destined for a particular downstream warehouse. We propose a mathematical model for the physical requirements, and limitations of parcel sortation. We then show that it is NP-hard to determine whether a feasible sortation plan exists. We consider two natural objectives: minimizing the maximum number of sort points required at a warehouse, and minimizing the total number of sort points required in the network.
In Chapter 6, we consider the problem of minimizing the maximum number of sort points required at a warehouse. We discuss several settings, where (near-)optimality of a given sortation instance can be determined efficiently. The algorithms we propose are fast and build on combinatorial witness set type lower bounds that are reminiscent and extend those used in earlier work on degree-bounded spanning trees and arborescences. In Chapter 7, we present algorithms for minimizing the total number of sort points required in a network. In contrast to the min-degree setting, it is not known if this min-cardinality setting is NP-hard. In progress towards answering this question, we present fast combinatorial algorithms for solving in-tree, out-tree, and spider instances. Our algorithms are based on reduction, decomposition, and uncrossing techniques that simplify instances.