Industry: Healthcare and Epidemiology Product: ModelRisk Application: Hospital Resource Allocation — Stochastic LP for Beds, OR-Time, and Nurses
The hospital's annual operating-plan spreadsheet — built on a deterministic linear program — said the 480-bed tertiary facility would run at 88% utilization and comfortably within capacity. The same exposure data fed through a Monte Carlo simulation said census exceeded 480 beds on roughly 3% of days, and the ICU spilled over its 28 staffed beds on 10% of days. The deterministic LP is a beautiful answer to the wrong question — it allocates against the average day. Hospitals do not live on the average day; they live on the joint draw of a flu spike, a multi-vehicle trauma, and a weekend staffing dip, and any plan that does not price that joint draw is mis-pricing the resource base.
The deterministic LP optimizes against the green line — the mean census of 420. The simulation says the 95th-percentile day sits at roughly 473 patients, and the 99th-percentile day at roughly 497 patients (a 17-bed overflow that must be absorbed by hallway boarding, transfer-out, or elective cancellation). The probability of any-day overflow is ~3% — roughly one day in a month, on average, the hospital is over its staffed bed count, a tail the deterministic plan does not show at all.
A 480-bed tertiary hospital with five service lines (ICU, Cardiac, Surgical, Medical, Maternity) rebuilt its allocation model in ModelRisk. The deterministic LP still runs nightly — it remains the right tool for the daily duty-roster — but its constraints are now stress-tested against a 100,000-iteration demand simulation before the annual capacity plan is signed.
Three years of hospital census data show variance-to-mean ratios between 1.0 (Maternity) and 2.6 (Surgical). Treating each service's daily census as Poisson — which the deterministic LP implicitly does whenever it uses a single mean — understates the tail. The simulation uses a Negative Binomial per service, calibrated to the historical mean and variance:
These tails are not academic. The simulation says the total inpatient demand reaches 497 patients on the worst 1% of days — 17 above the staffed bed count — even though the mean total is 420 and the deterministic plan would call that day "well within capacity."
The single biggest mover is Medical service mean LOS: shaving the average length of stay from 5.0 to 4.2 days drops the overflow probability by ~2.6 percentage points — a large relative move against a ~3% baseline. Total staffed beds is second — opening 10 additional beds buys roughly 2 percentage points. Surgical demand surge sits third, ahead of ICU mean census. That ranking is what told the operations COO that the highest-leverage spend was a discharge-acceleration program for the Medical service, not a capital project to add ICU beds.
The simulation re-evaluates allocation under common random numbers — every strategy is tested against the same 100,000 demand draws, so the comparison reflects strategy difference, not sampling noise.
Strategy A (today) leaves ~24 patients per day at the 95th percentile without a designated service-line bed. Flexing 8 beds from Surgical to Medical (Strategy B) barely moves the P95 — it actually edges it up to ~26, because Surgical demand has a fat tail of its own that the flex exposes. Opening 12 step-down beds (Strategy C) cuts the P95 shortage to about 12 patients/day and the P99 from 39 to 27. The step-down approach is more expensive at standing capacity but materially better in the tail, which is exactly where overflow drives the worst patient-safety incidents.
The same simulation evaluated a forecast-triggered flex-staffing rule for the ICU: when the next-day forecast exceeds 30 patients, open up to 6 additional ICU beds with a pre-credentialed flex-pool nurse roster.
ICU overflow days fall from a mean of ~37 days/year to ~17 days/year, and the probability of exceeding the 20-day quality-of-care threshold drops from roughly 100% to about 21%. The marginal cost of the flex roster is $540,000/year; the alternative — staffing six additional ICU beds permanently — was costed at $2.1M/year and produced essentially the same risk-reduction profile. The simulation paid for itself in the first quarter of the next budget cycle.
A hospital running at the deterministic mean is a hospital one bad week from a capacity crisis. ModelRisk turns that hidden tail into a number the COO can buy down with the cheapest available unit of capacity — which is rarely the unit the deterministic LP would have spent on first.