Abstract A heuristic method based on column generation is presented for the nurse rostering problem. The method differs significantly from an exact column generation approach or a branch and price algorithm because it performs an incomplete search which quickly produces good solutions but does not provide valid lower bounds. It is effective on large instances for which it has produced best known solutions on benchmark data instances. Several innovations were required to produce solutions for the largest instances within acceptable computation times. These include using a fast first-order linear programming solver based on the work of Chambolle and Pock to approximately solve the restricted master problem. A low-accuracy but fast, first-order linear programming method is shown to be an effective option for this master problem. The pricing problem is modelled as a resource constrained shortest path problem with a two-phase dynamic programming method. The model requires only two resources. This enables it to be solved efficiently. A commercial integer programming solver is also tested on the instances. The commercial solver was unable to produce solutions on the largest instances whereas the heuristic method was able to. It is also compared against the state-of-the-art, previously published methods on these instances. Analysis of the branching strategy developed is presented to provide further insights. All the source code for the algorithms presented has been made available on-line for reproducibility of results and to assist other researchers.

中文翻译：

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基于列生成的护士排班问题启发式方法中的一阶线性规划

摘要 针对护士排班问题，提出了一种基于列生成的启发式方法。该方法与精确列生成方法或分支和价格算法有很大不同，因为它执行不完全搜索，可以快速生成良好的解决方案，但不提供有效的下限。它在大型实例上很有效，它已经在基准数据实例上产生了最著名的解决方案。为了在可接受的计算时间内为最大实例生成解决方案，需要进行多项创新。其中包括使用基于 Chambolle 和 Pock 工作的快速一阶线性规划求解器来近似求解受限主问题。一个低精度但快速的一阶线性规划方法被证明是解决这个主要问题的有效选择。定价问题被建模为具有两阶段动态规划方法的资源约束最短路径问题。该模型只需要两个资源。这使得它能够被有效地解决。还在实例上测试了商业整数规划求解器。商业求解器无法在最大实例上生成解决方案，而启发式方法能够。它还与最先进的、先前在这些实例上发布的方法进行了比较。对所开发的分支策略进行分析以提供进一步的见解。所提出算法的所有源代码都已在线提供，以实现结果的可重复性并协助其他研究人员。该模型只需要两个资源。这使得它能够被有效地解决。还在实例上测试了商业整数规划求解器。商业求解器无法在最大实例上生成解决方案，而启发式方法能够。它还与最先进的、先前在这些实例上发布的方法进行了比较。对所开发的分支策略进行分析以提供进一步的见解。所提出算法的所有源代码都已在线提供，以实现结果的可重复性并协助其他研究人员。该模型只需要两个资源。这使得它能够被有效地解决。还在实例上测试了商业整数规划求解器。商业求解器无法在最大实例上生成解决方案，而启发式方法能够。它还与最先进的、先前在这些实例上发布的方法进行了比较。对所开发的分支策略进行分析以提供进一步的见解。所提出算法的所有源代码都已在线提供，以实现结果的可重复性并协助其他研究人员。它还与最先进的、先前在这些实例上发布的方法进行了比较。对所开发的分支策略进行分析以提供进一步的见解。所提出算法的所有源代码都已在线提供，以实现结果的可重复性并协助其他研究人员。它还与最先进的、先前在这些实例上发布的方法进行了比较。对所开发的分支策略进行分析以提供进一步的见解。所提出算法的所有源代码都已在线提供，以实现结果的可重复性并协助其他研究人员。