Modeling the Dutch Healthcare Workforce: An Integer Programming Game for Nurse Scheduling Problem
Z. Zhang (TU Delft - Technology, Policy and Management)
K. Staňková – Graduation committee member (TU Delft - Technology, Policy and Management)
P.S.A. Stokkink – Mentor (TU Delft - Technology, Policy and Management)
I. Grossmann – Mentor (TU Delft - Technology, Policy and Management)
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Abstract
The Dutch healthcare system currently operates under intensifying fiscal and workforce pressures, transforming nurse scheduling from a simple operational task into a complex strategic conflict involving divergent stakeholder priorities. Addressing this "wicked problem," this study addresses nurse scheduling as a strategic decision challenge, introducing a novel methodological framework that bridges Operations Research and Game Theory. Unlike traditional optimization models, this research introduces the "Nurse Scheduling Game" (NSG), formulating the problem as an Integer Programming Game (IPG) to capture the behaviors and interactions of stakeholders explicitly.
The interaction is analyzed through two distinct equilibrium concepts representing different governance structures. First, the uncoordinated state is modeled as a simultaneous Nash Equilibrium, formulated as a Generalized Nash Equilibrium Problem (GNEP). Second, the potential for strategic improvement is explored through the hierarchical Stackelberg Equilibrium, formulated as a Bilevel Integer Problem (BIP). In the latter, the Hospital Manager (Leader) explicitly anticipates the Nurses' (Followers) reactions. To solve this computationally intractable bilevel problem, the study implements a novel Monte Carlo Multilevel Optimization (MCMO) framework.
Applied to a representative case study of a mid-sized Dutch hospital, the computational results quantify the significant costs associated with uncoordinated planning. Under Nash dynamics, the system converges to a state of "defensive buffering," resulting in outcomes approximately twice as expensive as the coordinated alternative. Conversely, the Stackelberg Equilibrium demonstrates the value of strategic anticipation. By transitioning from volume-based to precision-based allocation, the hierarchical model achieved a 51.0% reduction in total system costs and an 11.8% reduction in patient waiting times compared to the Nash baseline.
These findings translate into actionable policy implications, suggesting that the solution to budget overruns lies in shifting from reactive to anticipatory governance. The study supports the implementation of Algorithmic Workforce Management systems that couple budget setting with schedule design. Ultimately, this research offers a unified game-theoretic optimization framework that reconciles financial constraints with workforce autonomy, providing a viable pathway toward sustainability for the Dutch healthcare system.