Stochastic joint homecare service and capacity planning with nested decomposition approaches

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.