Multi-Week Nurse Scheduling in Labor and Delivery Units: A Data-Driven Integer Programming Framework

Why nurse scheduling is hard

Labor and Delivery units face substantial variation in births across days, nights, and weekends. Schedules must maintain coverage and charge-nurse presence while controlling overtime, respecting workload and consecutive-shift limits, and distributing weekend assignments fairly.

Decision variables

The core binary assignment variable is:

x[n, t] ∈ {0, 1}

n ∈ N  : set of nurses (24 baseline: 16 FT, 4 CQ, 4 PT)
t ∈ T  : set of 56 day/night shifts over four weeks

x[n, t] = 1 if nurse n is assigned to shift t, else 0

Operational constraints

1. Shift-level coverage derived from expected birth workload.
2. Minimum and maximum assignments by employment status.
3. Day/night consistency across the planning horizon.
4. Limits on consecutive workdays and consecutive night shifts.
5. At least one charge-qualified nurse on every shift.
6. Overtime thresholds and premium compensation.
7. Equitable weekend assignments and consecutive-weekend limits.
8. Minimum weekend-off requirements.
9. Configurable leave and training availability restrictions.
10. Penalized shortages to quantify coverage risk.

Objective function

The objective minimizes total staffing cost while penalizing coverage shortages and unequal weekend assignments:

minimize  regular wages + overtime compensation
        + weekend premiums + shortage penalties
        + weekend-fairness penalties

Solver setup

The model was implemented in Python with PuLP and solved with the open-source CBC (COIN-OR Branch-and-Cut) solver with a 1% optimality gap tolerance. The study evaluated 100 scenarios combining five annual birth-volume levels with 20 staffing configurations, then tested 35 one-at-a-time sensitivity cases around the baseline.

Results and takeaways

• The baseline 24-nurse pool covered all 56 shifts with zero shortage.
• Average weekly cost was $39,488; the four-week total was $157,952.
• Overtime was limited to 64 hours across the four-week horizon.
• All 35 sensitivity cases maintained zero aggregate shortage.
• Lower-volume units favored leaner staffing; higher volumes favored larger pools.

When to use ILP/MIP for scheduling

ILP/MIP shines when constraints are crisp, the planning horizon is short-to-medium, and the team values explainable, auditable schedules. For very large hospitals or rolling-horizon planning, consider column generation, Lagrangian relaxation, or metaheuristics (simulated annealing, tabu search) as complementary techniques.