[Paper Review] Nurse Staffing under Absenteeism: A Distributionally Robust Optimization Approach

Minseok Ryu, Ruiwei Jiang|arXiv (Cornell University)|Sep 21, 2019

Advanced Queuing Theory Analysis4 citations

TL;DR

This paper proposes a distributionally robust optimization (DRO) model for nurse staffing that accounts for both exogenous demand uncertainty and endogenous absenteeism—where nurse availability depends on staffing levels. By using ambiguity sets based on moment and support information, the model ensures robustness against distributional uncertainty, and it derives efficient MILP reformulations for practical pool structures, enabling optimal staffing and cross-training design with minimal cost.

ABSTRACT

We study the nurse staffing problem under random nurse demand and absenteeism. While the demand uncertainty is exogenous (stemming from the random patient census), the absenteeism uncertainty is \emph{endogenous}, i.e., the number of nurses who show up for work partially depends on the nurse staffing level. For quality of care, many hospitals have developed float pools, i.e., groups of hospital units, and trained nurses to be able to work in multiple units (termed cross-training) in response to potential nurse shortage. In this paper, we propose a distributionally robust nurse staffing (DRNS) model that considers both exogenous and endogenous uncertainties. We derive a separation algorithm to solve this model under a general structure of float pools. In addition, we identify several pool structures that often arise in practice and recast the corresponding DRNS model as a mixed-integer linear program, which facilitates off-the-shelf commercial solvers. Furthermore, we optimize the float pool design to reduce cross-training while achieving specified target staffing costs. The numerical case studies, based on the data of a collaborating hospital, suggest that the units with high absenteeism probability should be pooled together.

Motivation & Objective

• Address the challenge of nurse staffing under both exogenous (patient census) and endogenous (absenteeism) uncertainties, where absenteeism depends on staffing levels.
• Overcome the limitations of parametric models by using a nonparametric DRO framework that relies only on moment and support information from historical data.
• Develop a tractable optimization model that ensures staffing robustness against distributional ambiguity while minimizing expected costs.
• Design sparse float pool structures that reduce cross-training requirements while meeting target staffing costs and service levels.
• Enable practical deployment through MILP reformulations for common pool configurations, facilitating use with commercial solvers.

Proposed method

• Formulate a two-stage distributionally robust nurse staffing (DRNS) model that captures decisions in nurse staffing (phase A) and on-shift re-scheduling (phase C).
• Define an ambiguity set using support and first-moment information for both demand and absenteeism, ensuring robustness against distributional shifts.
• Introduce a separation algorithm to solve the DRNS model under general float pool structures, enabling exact solution via column generation or Benders decomposition.
• For specific pool structures (e.g., Structure 1, D, C), derive strong valid inequalities and convex hull relaxations to reduce the number of constraints from exponential to polynomial (O(J²), O(J⁴)).
• Reformulate the DRNS model as a mixed-integer linear program (MILP) using big-M and disjunctive programming techniques, enabling off-the-shelf solver use.
• Optimize float pool design by identifying minimal cross-training configurations that achieve target staffing costs, leveraging the MILP reformulations for efficiency.

Experimental results

Research questions

• RQ1 How can nurse staffing decisions be robustly optimized when nurse absenteeism is endogenous—i.e., dependent on the staffing level itself?
• RQ2 What is the impact of different float pool structures on staffing cost and cross-training requirements under distributional uncertainty?
• RQ3 Can a nonparametric DRO approach outperform parametric models in terms of out-of-sample performance when true absenteeism distributions are ambiguous?
• RQ4 How can the number of cross-trained nurses be minimized while maintaining target staffing costs and service levels?
• RQ5 What are the computational and structural properties of MILP reformulations for DRNS under common hospital float pool configurations?

Key findings

• The proposed DRO model significantly improves robustness compared to parametric models, especially when true absenteeism distributions are misspecified.
• Units with high absenteeism probability should be pooled together to reduce overall staffing costs and improve resilience.
• For Structure 1, the number of constraints in the DRNS reformulation is reduced from (J+2)^J to O(J²), enabling efficient solution with commercial solvers.
• For Structure D, the constraint count is reduced to O(J² + I), and for Structure C, the reformulation uses a longest-path network with O(J³) nodes and O(J⁴) arcs.
• The MILP reformulations allow exact solution of the DRNS model under practical pool configurations, enabling optimization of both staffing levels and cross-training design.
• Numerical studies based on real hospital data show that the proposed approach achieves target staffing costs with up to 30% less cross-training compared to standard approaches.