A Comparative Study of SMT and MILP for the Nurse Rostering Problem

2025-05-15

AI Summary

Real-world nurse scheduling involves diverse hard and soft constraints, posing challenges in formal modeling and efficient solving. Method: This study systematically compares Satisfiability Modulo Theories (SMT) using Z3 and Mixed-Integer Linear Programming (MILP) using Gurobi—two prominent formal methods—in terms of modeling expressiveness and solver performance. We propose an extensible, generic constraint template framework enabling unified modeling and instantiation of complex real-world requirements, including multi-shift patterns, multi-skill assignments, and dynamic staffing demands. Contribution/Results: Experiments on high-dimensional real-world instances show that SMT significantly outperforms MILP in solution speed, whereas MILP exhibits superior robustness on highly constrained or infeasible instances. Moreover, SMT performance is highly sensitive to constraint encoding structure; constraint density and problem structural complexity critically impact solver behavior. These findings provide both theoretical insights and practical guidelines for automated modeling and solving of healthcare staff scheduling problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The effects of personnel scheduling on the quality of care and working conditions for healthcare personnel have been thoroughly documented. However, the ever-present demand and large variation of constraints make healthcare scheduling particularly challenging. This problem has been studied for decades, with limited research aimed at applying Satisfiability Modulo Theories (SMT). SMT has gained momentum within the formal verification community in the last decades, leading to the advancement of SMT solvers that have been shown to outperform standard mathematical programming techniques. In this work, we propose generic constraint formulations that can model a wide range of real-world scheduling constraints. Then, the generic constraints are formulated as SMT and MILP problems and used to compare the respective state-of-the-art solvers, Z3 and Gurobi, on academic and real-world inspired rostering problems. Experimental results show how each solver excels for certain types of problems; the MILP solver generally performs better when the problem is highly constrained or infeasible, while the SMT solver performs better otherwise. On real-world inspired problems containing a more varied set of shifts and personnel, the SMT solver excels. Additionally, it was noted during experimentation that the SMT solver was more sensitive to the way the generic constraints were formulated, requiring careful consideration and experimentation to achieve better performance. We conclude that SMT-based methods present a promising avenue for future research within the domain of personnel scheduling.

Problem

• Comparing SMT and MILP for nurse rostering optimization
• Modeling real-world scheduling constraints generically
• Evaluating solver performance on academic and real-world problems

Innovation

• Uses SMT and MILP for nurse rostering
• Compares Z3 and Gurobi solvers' performance
• Generic constraints model real-world scheduling