无能力r-分配p-轮毂最大覆盖问题的数学公式及求解方法

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Olivera Stančić , Zorica Stanimirović , Raca Todosijević , Stefan Mišković
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引用次数: 2

摘要

本文考虑无能力r分配p-hub最大覆盖问题(UrApHMCP),它是对众所周知的p-hub最大覆盖问题的推广,因为它允许每个非hub节点通过最多r个hub发送和接收流,r≤p。UrApHMCP中考虑了两种覆盖标准——二元覆盖和部分覆盖,这在文献中是第一次。针对这两种覆盖标准,提出了新的UrApHMCP数学公式。考虑到UrApHMCP是一个NP-hard优化问题,提出了两种有效的启发式方法作为求解方法。前者是通用可变邻域搜索(GVNS)的一种变体,后者是基于贪婪随机自适应搜索过程(GRASP)和可变邻域下降(VND)的结合,得到了一种混合的GRASP-VND方法。计算研究分别在CAB和AP基准实例集(最多有25个和200个节点)、TR实例(包括81个节点)以及具有挑战性的USA423和URAND hub实例(最多有423和1000个节点)上进行。对于节点数不超过50的实例,CPLEX求解器可以得到最优解或可行解,而对于更大维度的实例,CPLEX求解器则无法达到。另一方面,GVNS和grip - vnd都能在较短的CPU时间内达到最优解或提高CPLEX提供的下界。此外,这两种启发式方法都能在大维度的问题实例上快速返回解决方案,从而表明它们有可能有效地解决大的、实际规模的问题实例。所进行的非参数统计检验证实了所提出的GVNS和GRASP-VND的鲁棒性,并表明这两种元启发式算法优于其他UrApHMCP测试算法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Mathematical formulations and solution methods for the uncapacitated r-allocation p-hub maximal covering problem

This paper considers the uncapacitated r-allocation p-hub maximal covering problem (UrApHMCP), which represents a generalization of the well-known p-hub maximal covering problem, as it allows each non-hub node to send and receive flow via at most r hubs, rp. Two coverage criteria are considered in UrApHMCP — binary and, for the first time in the literature, partial coverage. Novel mathematical formulations of UrApHMCP for both coverage criteria are proposed. As the considered UrApHMCP is an NP-hard optimization problem, two efficient heuristic methods are proposed as solution approaches. The first one is a variant of General Variable Neighborhood Search (GVNS), and the second one is based on combining a Greedy Randomized Adaptive Search Procedure (GRASP) with Variable Neighborhood Descent (VND), resulting in a hybrid GRASP-VND method. Computational study is performed over the set of CAB and AP benchmark instances with up to 25 and 200 nodes, respectively, on TR instances including 81 nodes, as well as on the challenging USA423 and URAND hub instances with up 423 and 1000 nodes, respectively. Optimal or feasible solutions are obtained by CPLEX solver for instances with up to 50 nodes, while instances of larger dimensions are out of reach for CPLEX solver. On the other hand, both GVNS and GRASP-VND reach optimal solutions or improve lower bounds provided by CPLEX in short CPU times. In addition, both heuristics quickly return solutions on problem instances of large dimensions, thus indicating their potential to solve effectively large, realistic sized problem instances. The conducted non-parametric statistical tests confirm robustness of the proposed GVNS and GRASP-VND and demonstrate that the these two metaheuristics outperform other tested algorithms for UrApHMCP.

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来源期刊
Discrete Optimization
Discrete Optimization 管理科学-应用数学
CiteScore
2.10
自引率
9.10%
发文量
30
审稿时长
>12 weeks
期刊介绍: Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.
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