在最短路修复合作博弈中，当灾后运输网络规模较大时，最优成本分摊问题难以直接求解。基于拉格朗日松弛理论，提出了一种最短路修复合作博弈成本分摊算法。该算法将最短路修复合作博弈分解为两个具有特殊结构的子博弈，进而利用两个子博弈的结构特性，可以{高效地}求解出二者的最优成本分摊，将这两个成本分摊相加，可以获得原博弈的一个近乎最优的稳定成本分摊。结果部分既包含运输网络的随机仿真，也包含玉树地震灾区的现实模拟，无论数据来源于仿真还是现实，该算法都能在短时间内为最短路修复合作博弈提供稳定的成本分摊方案。
When the scale of disrupted transportation networks is large, the optimal cost allocation problem for multiple pairs of shortest paths recovery game is often intractable. In this paper, we propose a cost allocation algorithm based on Lagrangian relaxation method, which decomposes the multiple pairs of shortest paths recovery game into two subgames approximatively. We then find some properties for these two subgames which can help to solve their optimal cost allocations efficiently. According to the optimal cost allocations of the subgames, we are able to develop a near-optimal stable cost allocation for the original game. In the end, we conduct some computational experiments by checking the performance of our cost allocation algorithm on both simulated networks and the disrupted transportation network of Yushu after earthquake. The results show that our algorithm is both efficient and effective in solve optimal cost allocation problem for multiple pairs of shortest paths recovery game.
[1] 谢政. 对策论导论[M]. 北京:科学出版社, 2010.
[2] 于晓辉, 杜志平, 张强, 等. 基于T-联盟Shapley值的分配策略[J]. 运筹学学报}, 2020, (4):113-127.
[3] 于晓辉, 杜志平, 张强, 等. 一种资源投入不确定情形下的合作博弈形式及收益分配策略[J]. 运筹学学报, 2019, (4):71-85.
[4] Maschler M, Peleg B, Shapley L S. Geometric properties of the kernel, nucleolus, and related solution concepts[J]. Mathematics of Operations Research, 1979, 4(4):303-338.
[5] Kern W, Paulusma D. Matching games:The least core and the nucleolus[J]. Mathematics of Operations Research, 2003, 28(2):294-308.
[6] Schulz A S, Uhan N A. Sharing supermodular costs[J]. Operations Research, 2004, 58(4):1051-1056.
[7] Schulz A S, Uhan N A. Approximating the least core value and least core of cooperative games with supermodular costs[J]. Discrete Optimization, 2013, 10(2):163-180.
[8] Jain K, Mahdian M. Cost sharing[M]//Algorithmic Game Theory. New York:Cambridge University Press, 2007:385-410.
[9] Faigle U, Fekete S P, Hochstattler W, et al. On approximately fair cost allocation in Euclidean TSP games[J]. Operations Research-Spektrum, 1998, 20(1):29-37.
[10] Blaser M, Ram L S. Approximately fair cost allocation in metric traveling salesman games[J]. Theory of Computing Systems, 2008, 43(1):19-37.
[11] Caprara A, Letchford A N. New techniques for cost sharing in combinatorial optimization games[J]. Mathematical Programming, 2010, 124(1):93-118.
[12] Liu L, Qi X. Network disruption recovery for multiple pairs of shortest paths[C]//201411th International Conference on Service Systems and Service Management (ICSSSM). Beijing:IEEE, 2014:1-6.
[13] Andonov R, Poirriez V, Rajopadhye S. Unbounded knapsack problem:Dynamic programming revisited[J]. European Journal of Operational Research, 2000, 123(2):394-407.
[14] Held M, Karp R M. The traveling-salesman problem and minimum spanning trees[J]. Operations Research, 1970, 18(6):1138-1162.
[15] Zhang C, Gao Y, Yang L, et al. Joint optimization of train scheduling and maintenance planning in a railway network:A heuristic algorithm using Lagrangian relaxation[J]. Transportation Research Part B:Methodological, 2020, 134:64-92.
[16] Liu L, Qi X, Xu Z. Computing near-optimal stable cost allocations for cooperative games by Lagrangian relaxation[J]. INFORMS Journal on Computing, 2016, 28(4):687-702.
[17] 张惠珍, 魏欣, 马良. 求解无容量设施选址问题的半拉格朗日松弛新方法[J]. 运筹学学报, 2015, (4):37-47.
[18] 赵宇哲, 周晶淼, 匡海波, 等. 航运资产整合下海运网络的航线、路径和船舶的集成优化[J]. 系统工程理论与实践, 2018, (8):2110-2122.
[19] Held M, Wolfe P, Crowder H P. Validation of subgradient optimization[J]. Mathematical Programming, 1974, 6(1):62-88.
