In this paper, we study a joint service authorization and capacity planning problem for the home health care (HHC) system with demand uncertainty. We formulate the problem as a two-stage stochastic programming with recourse which maximizes the expected total revenue. The service authorization and capacity planning are the mixed binary-and-continuous first stage decisions, and the service hour (resource) allocation is modeled as the second-stage decision which adapts to the demand realizations. The resulting sample average approximation problem (of the stochastic program) could be a large scaled mixed integer linear program. To solve the model in a more scalable fashion, we propose a supergradient-based nested decomposition algorithm that exploits the nice decomposable structure of the problem to cope with the binary and continuous variables separately. The proposed nested decomposition scheme consists of two-layer of decomposition, where the outer decomposition solves for the authorization decision (binaries) with the supergradient cuts and each supergradient can be computed iteratively in the inner decomposition scheme. The proposed nested decomposition algorithm is feasibility-cut-free and is guaranteed to reach the exact optimality in finite steps. Furthermore, we extend our stochastic HHC planning model to a more general framework of Conditional Value-at-Risk (CV@R), and by model transformation with variable change we show that the CV@R model can actually be reformulated in a structure that the proposed nested decomposition scheme can be applied almost directly. Finally, a comprehensive computational study is performed which demonstrates the effectiveness of our model and the performance of the algorithm.

在本文中，我们研究了具有需求不确定性的家庭医疗保健 (HHC) 系统的联合服务授权和容量规划问题。我们将问题表述为具有追索权的两阶段随机规划，以最大化预期总收入。服务授权和容量规划是混合二元和连续的第一阶段决策，服务时间（资源）分配被建模为适应需求实现的第二阶段决策。由此产生的样本平均逼近问题（随机程序的）可能是一个大规模混合整数线性程序。为了以更具可扩展性的方式求解模型，我们提出了一种基于超梯度的嵌套分解算法，该算法利用问题的良好可分解结构来分别处理二元变量和连续变量。所提出的嵌套分解方案由两层分解组成，其中外分解使用超梯度切割解决授权决策（二进制），并且每个超梯度可以在内分解方案中迭代计算。所提出的嵌套分解算法是无可行性切割的，并保证在有限的步骤中达到精确的最优性。此外，我们将随机 HHC 规划模型扩展到更一般的条件风险价值 (CV@R) 框架，并且通过具有变量变化的模型转换，我们表明 CV@R 模型实际上可以重新表述为几乎可以直接应用所提出的嵌套分解方案的结构。最后，进行了全面的计算研究，证明了我们模型的有效性和算法的性能。