Workload arithmetic: are four rooms enough, and what does quality control buy?
The radiology department of a 300-bed district hospital is reviewing its workload before buying new equipment. On a typical working day 240 patients attend for radiography. Each patient receives an average of 1.4 radiographic examinations. Audit shows that 12% of all exposures are repeats caused by positioning errors, motion and incorrect exposure factors. Each examination, including patient transfer, positioning, exposure and image check, occupies a room for 7.5 minutes on average. The department has four digital radiography rooms, and each room is available for 6.5 effective hours per day after deducting warm-up, quality control, cleaning and staff breaks. Two of the four rooms are more than ten years old and are due for replacement. The hospital administrator argues that four rooms are enough because "240 patients in six and a half hours is only about 37 patients per room". The chief radiographer disagrees and wants a calculation to support a request for additional rooms and a quality-control programme. Staffing is not part of this calculation.
1. Calculate the room-hours required per day and compare them with the available capacity.
Read the worked answer
Step 1 - examinations. 240 patients × 1.4 examinations = 336 examinations.
Step 2 - add repeats. Repeats are extra exposures, so they add to room time: 336 × 1.12 = 376.3, i.e. about 377 examinations.
Step 3 - convert to time. 377 × 7.5 minutes = 2,827.5 minutes = 47.1 hours of room time.
Step 4 - available capacity. 4 rooms × 6.5 hours = 26 hours.
Step 5 - compare. 47.1 hours required against 26 hours available is a shortfall of about 21.1 hours per day. The department is working at roughly 181% of its stated capacity, which in practice means queues, shortcuts and overtime.
Step 6 - rooms needed. 47.1 ÷ 6.5 = 7.25, so eight rooms would be needed at the current repeat rate and examination time. Four rooms cannot do the work even if the repeats were eliminated entirely (336 × 7.5 = 2,520 minutes = 42 hours, still above 26 hours).
2. Identify the error in the administrator's reasoning and explain why it matters.
Read the worked answer
The administrator divided 240 patients by 6.5 hours and obtained about 37. That figure is patients per hour, not patients per room per day. To get patients per room he would need to divide by four rooms as well: 240 ÷ 4 = 60 patients per room per day, or about 9.2 patients per room per hour.
Even the corrected figure understates the load for three reasons:
- It counts patients, not examinations; the average patient needs 1.4 examinations.
- It ignores the 12% repeat rate, which consumes room time without producing a new patient.
- It treats 6.5 hours as pure throughput, but each examination takes 7.5 minutes of room occupancy, including transfer and checking, so the true service rate is about 8 examinations per room per hour at best.
The unit error matters because a decision on capital equipment is being justified by a number that is off by a factor of four and by a further factor for repeats and multiple examinations.
3. If a quality-control programme reduces repeats from 12% to 6%, what is the new room-hour requirement and how much capacity does that free?
Read the worked answer
New total. 336 × 1.06 = 356.2, i.e. about 357 examinations.
357 × 7.5 minutes = 2,677.5 minutes = 44.6 hours.
Saving. 47.1 − 44.6 = about 2.5 room-hours per day, roughly half a room's daily capacity. Alternatively, 19 fewer repeat examinations per day.
Interpretation. A 6-percentage-point reduction in repeats is worth about 2.5 hours of room time a day, which is real but does not close a 21-hour gap. Quality control should therefore be presented as a productivity and dose measure, not as a substitute for adequate rooms. The case for additional rooms must rest on the 47-hour requirement; the QC programme strengthens it by showing that the department is already trying to use its existing capacity efficiently.
It is also worth noting the bottlenecks the arithmetic does not capture: porter transport, patient changing, reporting delays, power cuts and equipment downtime. These can make real capacity lower than the nominal 26 hours.