Facility location models are observed in many diverse areas such as communication networks, transportation, and distribution systems planning. They play significant role in supply chain and operations management and are one of the main well-known topics in strategic agenda of contemporary manufacturing and service companies accompanied by long-lasting effects. We define a new approach for solving stochastic single source capacitated facility location problem (SSSCFLP). Customers with stochastic demand are assigned to set of capacitated facilities that are selected to serve them. It is demonstrated that problem can be transformed to deterministic Single Source Capacitated Facility Location Problem (SSCFLP) for Poisson demand distribution. A hybrid algorithm which combines Lagrangian heuristic with adjusted mixture of Ant colony and Genetic optimization is proposed to find lower and upper bounds for this problem. Computational results of various instances with distinct properties indicate that proposed solving approach is efficient.
In critical node problems, the task is to identify a small subset of so-called critical nodes whose deletion maximally degrades a network’s “connectivity” (however that is measured). Problems of this type have been widely studied, for example, for limiting the spread of infectious diseases. However, existing approaches for solving them have typically been limited to networks having fewer than 1,000 nodes. In this paper, we consider a variant of this problem in which the task is to delete b nodes so as to minimize the number of node pairs that remain connected by a path of length at most k. With the techniques developed in this paper, instances with up to 17,000 nodes can be solved exactly. We introduce two integer programming formulations for this problem (thin and path-like) and compare them with an existing recursive formulation. Although the thin formulation generally has an exponential number of constraints, it admits an efficient separation routine. Also helpful is a new, more general preprocessing procedure that, on average, fixes three times as many variables than before. Summary of Contribution: In this paper, we consider a distance-based variant of the critical node problem in which the task is to delete b nodes so as to minimize the number of node pairs that remain connected by a path of length at most k. This problem is motivated by applications in social networks, telecommunications, and transportation networks. In our paper, we aim to solve large-scale instances of this problem. Standard out-of-the-box approaches are unable to solve such instances, requiring new integer programming models, methodological contributions, and other computational insights. For example, we propose an algorithm for finding a maximum independent set of simplicial nodes that runs in time O(nm) that we use in a preprocessing procedure; we also prove that the separation problem associated with one of our integer programming models is NP-hard. We apply our branch-and-cut implementation to real-life networks from a variety of domains and observe speedups over previous approaches.
Despite the successful applications of decision diagrams (DDs) to solve various classes of integer programs in the literature, the question of which mixed-integer structures admit a DD representation remains open. The present work addresses this question by developing both necessary and sufficient conditions for a mixed-integer program to be DD-representable through identification of certain rectangular formations in the underlying sets. This so-called rectangularization framework is applicable to all bounded mixed-integer linear programs, providing a notable extension of the DD domain to continuous problems. As an application, the paper uses the developed methods to solve stochastic unit commitment problems in energy systems. Computational experiments conducted on benchmark instances show that the DD approach uniformly and significantly outperforms the existing solution methods and modern solvers. The proposed methodology opens new pathways to solving challenging mixed-integer programs in energy systems more efficiently.
In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the incoming arc flows cannot be split or merged when routed through outgoing arcs. This so-called no-split no-merge requirement arises in unit train scheduling where train consists should remain intact at stations that lack necessary equipment and manpower to attach/detach them. Solving the unsplittable network flow problems with standard mixed-integer programming formulations is computationally difficult due to the large number of binary variables needed to determine matching pairs between incoming and outgoing arcs of nodes with no-split no-merge constraint. In this paper, we study a stochastic variant of the unit train scheduling problem where the demand is uncertain. We develop a novel decision diagram (DD)-based framework that decomposes the underlying two-stage formulation into a master problem that contains the combinatorial requirements and a subproblem that models a continuous network flow problem. The master problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a substantial improvement in solution time. Similar to the Benders decomposition technique, the subproblems output cutting planes that are used to refine the master DD. Computational experiments show a significant improvement in solution time of the DD framework compared with that of standard methods. Funding: This work was supported by the Iowa Energy Center [Grant 20IEC005]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2022.1194 .
Cliques and their generalizations are frequently used to model “tightly knit” clusters in graphs and identifying such clusters is a popular technique used in graph-based data mining. One such model is the s-club, which is a vertex subset that induces a subgraph of diameter at most s. This model has found use in a variety of fields because low-diameter clusters have practical significance in many applications. As this property is not hereditary on vertex-induced subgraphs, the diameter of a subgraph could increase upon the removal of some vertices and the subgraph could even become disconnected. For example, star graphs have diameter two but can be disconnected by removing the central vertex. The pursuit of a fault-tolerant extension of the s-club model has spawned two variants that we study in this article: robust s-clubs and hereditary s-clubs. We analyze the complexity of the verification and optimization problems associated with these variants. Then, we propose cut-like integer programming formulations for both variants whenever possible and investigate the separation complexity of the cut-like constraints. We demonstrate through our extensive computational experiments that the algorithmic ideas we introduce enable us to solve the problems to optimality on benchmark instances with several thousand vertices. This work lays the foundations for effective mathematical programming approaches for finding fault-tolerant s-clubs in large-scale networks.
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