Bi-Objective Receding-Horizon Scheduling of Nurses and Hospital Service Robots

• The notation includes periods grouped by day, night periods, and incompatible ordered shift pairs such as night followed by morning.Source: §2 · p. 3
• Battery states include a terminal state after the final scheduling period.Source: §2 · p. 3
• Nurses have availability, skill qualifications, workload limits, and temporal rules, including a baseline limit of one shift per day.Source: §2 · p. 3
• The model selects among acceptable skill-coverage patterns, such as fewer experts or more experienced nurses.Source: §2 · p. 3
• Robots can serve in one ward per period, charge, or idle; eligible task execution consumes battery, while charging replenishes it.Source: §2 · p. 3

3.1 Decision variables

• Nurse-assignment, robot-activity, executed-service, coverage-mode, shortage, battery, workload, and supervision variables represent the joint decisions and their effects.Source: §3.1 · p. 4

3.2 Nurse assignment and skill coverage

• Nurse assignments respect availability and the baseline one-shift-per-day policy, while a separate rule can require expert presence.Source: §3.2 · pp. 5–6
• For each ward-period-skill combination, the model selects exactly one acceptable coverage mode and enforces its corresponding coverage requirement.Source: §3.2 · p. 6
• Robot relief is tied to executed services rather than presence, and coefficients are restricted to avoid double-crediting displacement or sensitive clinical skills.Source: §3.2 · p. 6

3.3 Robot task assignment, charging, and battery dynamics

• Robot task assignment is linked to capability and capacity, while charging and work update battery states bounded by the battery limits.Source: §3.3 · p. 7
• Supervision is constrained by both a headcount condition and a workload fraction that reduces a supervising nurse’s clinical contribution.Source: §3.3 · pp. 7–8
• Supervision uses an aggregate nurse-equivalent approximation and assumes any assigned nurse is eligible to supervise robots.Source: §3.3 · p. 8

3.4 Objective functions

• The two objectives keep service inadequacy distinct from operational burden and fairness because they represent different managerial concerns.Source: §3.4 · p. 8

4.1 Multi-objective solution strategy

• The model uses normalized weighted sums and epsilon constraints to explore the service-adequacy and operational-burden trade-off, with lexicographic refinement for efficient solutions.Source: §4 · p. 8, §4.1 · p. 9
• The benchmark operational reference fixes assignment and robot decisions, then minimizes shortage to obtain a reproducible shortage measure before applying its shortage budget.Source: §4.1 · p. 9

4.2 Receding-horizon predictive control

• The rolling-horizon procedure solves over a finite window, implements only selected periods, and propagates state before re-optimization.Source: §4.2 · pp. 9–10
• The carried state includes battery, cumulative nurse work and night duty, and the immediately preceding night-shift history; future demand and availability are supplied over the look-ahead window.Source: §4.2 · p. 10
• A cross-window rule prevents an implemented night assignment from being followed by an incompatible morning shift in the next window.Source: §4.2 · p. 10
• A terminal battery reserve is imposed to mitigate artificial depletion at the prediction-window boundary.Source: §4.2 · p. 10

5.1 Reference numerical verification

• The eight-nurse reference case gives Z1 = 0.78 and Z2 = 3.164, with zero robot-service shortage and human-coverage shortage from the supervision deduction.The resulting schedule checks task execution, supervision, charging, and battery constraints; it is not a clinical validation exercise.Source: §5.1 · p. 11

5.2 Synthetic benchmark design

• The synthetic benchmark reports five seeds each for B20, B30, and B45, plus one B60 stress case; each instance uses one day and three shifts.The generator represents heterogeneous wards, skills, and robot capabilities; multi-day behavior is evaluated separately.Source: §5.2 · p. 11

5.3 Benchmark and epsilon-budget analysis

• For B20–B45, the mean canonical shortage reduction relative to Zop 1 is approximately 41.8–42.8%; the B60 seed-1 incumbent gives 40.5%.These are achieved feasible outcomes, not universal optima; several larger runs timed out, with a largest reported MIP gap of 27.99%.Source: §5.3 · p. 12
• For B30 seed 1, tighter budgets increase Z2 once canonical shortage falls below approximately 96, making this an epsilon-budget sensitivity study rather than a complete Pareto frontier.The 0.70-budget run is a time-limited feasible incumbent, and reported schedules at 0.85, 0.70, and 0.60 have positive canonical slack.Source: §5.3 · p. 12, Figure 1 · p. 13

5.4 Direct baseline comparison

• Joint optimization reduces total weighted shortage by 9.68% relative to nurse-only scheduling and by 7.47% relative to the sequential approach on B20 seed 1, while keeping Z2 comparable.The comparison uses a common service standard; the reported gain comes primarily from lower robot-service shortage, not uniformly lower human shortage.Source: §5.4 · p. 13, Table 7 · p. 13

5.5 Parameter and objective-weight sensitivity

• Changing objective weights materially changes schedules, while local changes to robot-relief and supervision coefficients produce shortage changes small and of the same order as solver gaps.The experiments support calibrating parameters from observed task and supervision times; weight responses are not monotone in every component, and no universal weight ranking is inferred.Source: §5.5 · pp. 13–14

5.6 Prediction-horizon comparison and seven-day stress path

• On the seven-day deterministic path, H = 1 has the lowest cumulative Z1, while H = 2 preserves more terminal battery and uses less robot energy; the static full-horizon solution has the highest cumulative shortage.The path exercises state propagation and re-optimization after disturbances, but does not establish a universally optimal horizon or probabilistic robustness.Source: §5.6 · pp. 14–15, Figure 2 · p. 15

5.7 Computational complexity and scalability

• All five B45 runs and the B60 stress case reach the time limit, so the one-day benchmark is preliminary resource/ward scaling rather than proof of multi-week scalability.The largest reported time-limited MIP optimality gap is 27.99%.Source: §5.7 · p. 15

6. DISCUSSION

• Z1 measures inadequate clinical and support coverage, while Z2 summarizes workload spread, overtime, night assignments, preferences, robot energy use, and charging.The epsilon-constraint procedure lets managers specify a service-shortage tolerance without setting a universal exchange rate between shortage and operational burden.Source: §6 · p. 16
• The integrated model’s lower B20 shortage is mainly due to reduced robot-service shortage, not replacement of specialized nursing work.Robot presence alone earns no staffing credit; relief is tied to executed tasks, while supervision is charged against nurse capacity.Source: §6 · p. 16
• Relief and supervision coefficients should be estimated from observed task completion and supervision time; small local sensitivity changes are cautious to interpret relative to solver gaps.Objective-weight changes produce different schedules and component values, so deployment would require scaling, stakeholder elicitation, and sensitivity checks rather than unchanged synthetic coefficients.Source: §6 · p. 16
• The model uses deterministic rolling-horizon re-optimization with exogenous look-ahead values, not learned forecasting; two-day horizons preserved more battery but did not reduce cumulative shortage.Controlled disturbances do not establish stochastic robustness; that would require Monte Carlo testing, scenario-based stochastic programming, or a robust uncertainty set.Source: §6 · p. 16
• Battery and mobility are planning-level approximations that omit inter-ward routing, congestion, collision avoidance, travel-time uncertainty, and charging-station occupancy.Constant ward-period energy use and charging increments can make schedules optimistic when travel or charger competition consumes a material fraction of a period; higher-fidelity studies would need telemetry-based battery models and explicit routing.Source: §6 · p. 16
• The reported benchmark omits several multi-week rostering rules, and evidence remains synthetic rather than clinical.Adding longer-term rules changes model size and validation needs; larger B45/B60 runs were time-limited, while deployment requires hospital data, calibrated rules and weights, telemetry, and prospective stakeholder validation.Source: §6 · p. 16

7. CONCLUSION

• The bi-objective MILP combines heterogeneous nursing skills, alternative coverage modes, task-linked robot assistance, battery dynamics, and explicit human-supervision workload.A deterministic receding-horizon layer carries workload, night-history, and battery states across successive optimization windows.Source: §7 · p. 17
• On B20 seed 1, integrated joint optimization obtains total shortage 53.70, versus 58.03 for sequential nurse–robot scheduling and 59.46 for nurse-only scheduling.The reference solution gives Z1 = 0.78 and Z2 = 3.164; the study does not claim exact large-scale optimality or practical hospital validation.Source: §7 · p. 17
• The results support internal consistency and comparative value on controlled synthetic tests, while identifying deployment evidence still needed.Future work includes de-identified hospital data, explicit routing and charging constraints, telemetry-calibrated batteries, richer multi-week nurse rules, uncertainty models, and scalable methods.Source: §7 · p. 17

APPENDIX A. REFERENCE NUMERICAL DATA

• Tables 11–13 provide the reference case’s nurse skills, human coverage requirements, and robot energy and capability data.The reference case is separate from the synthetic benchmark generator and is used to check formulation and computational-result consistency.Source: Appendix A · p. 17, Table 12 · p. 18, Table 13 · p. 18
• The reference robot services are ICU transport in morning and evening and General-ward disinfection/support at night; R1 transports, idles, then charges, while R2 charges, transports, then disinfects.Source: Appendix A · p. 17
• The reported reference objective value is Z2 = 3.164.Source: Appendix A · p. 18