Staffing the District: Relief Factor, Shortfall, and the Real Cost of Overtime
A district must cover 5 patrol sectors. Each sector requires 2 officers on each of 3 shifts, every day of the week. Officers are scheduled for 40 hours per week, and the department applies a relief factor of 1.65 to absorb leave, training, court, and injury. The district's authorized strength is 62 officers. Officers earn a base rate of $32.00 per hour; overtime is paid at time and a half. Last month the district recorded 410 overtime hours. A full-time officer works 2,080 hours per year. The captain has asked whether the district is staffed correctly and what the overtime is actually costing.
1. How many officers does the district actually need? Show each step.
Read the worked answer
Work from the post, not from the headcount.
- Posts per day: 5 sectors × 2 officers × 3 shifts = 30 officer-shifts per day.
- Hours per day: 30 shifts × 8 hours = 240 officer-hours per day.
- Hours per week: 240 × 7 = 1,680 officer-hours per week.
- Officers on post: 1,680 ÷ 40 = 42 officers - this is the number physically needed each week if nobody ever took leave.
- Apply the relief factor: 42 × 1.65 = 69.3, rounded up to 70 officers.
Round up, never down. Rounding 69.3 to 69 leaves a post uncovered on the days the arithmetic says it will be uncovered. The relief factor is doing real work here: without it the district looks 20 officers over-strength when it is actually short.
2. The district is authorized 62 officers. Is it adequately staffed? Quantify the gap in officers and in uncovered hours per week.
Read the worked answer
No. 70 needed against 62 authorized leaves a shortfall of 8 officers, about 11.4% of the requirement (8 ÷ 70).
Translating the gap into coverage:
- Each missing position represents 40 hours per week of scheduled work that someone else must absorb.
- 8 × 40 = 320 officer-hours per week that cannot be covered by the authorized roster without overtime, an uncovered post, or reduced service.
- That is roughly 1,387 hours per month (320 × 52 ÷ 12), which is more than three times the 410 overtime hours actually recorded last month.
That final comparison is the useful one: the district is absorbing a third of its structural gap in overtime and covering the rest by some other means - thinning sectors, holding officers over, or leaving posts short. The captain's question is not really 'is overtime expensive'; it is 'what is happening to the other 977 hours.'
3. Calculate the cost of last month's overtime and express it as full-time-equivalent hours. What does that tell the captain?
Read the worked answer
Overtime rate: $32.00 × 1.5 = $48.00 per hour - a 50% premium over base.
Monthly cost: 410 hours × $48.00 = $19,680.
Annualized: 410 × 12 = 4,920 overtime hours per year. At $48.00 that is $236,160 per year; the same hours at straight time would be 4,920 × $32.00 = $157,440. The premium alone is $78,720 per year.
FTE equivalent: 4,920 ÷ 2,080 = 2.37 full-time officers' worth of hours bought at overtime rates.
What this tells the captain: the overtime is not an anomaly to be managed down - it is the visible part of a structural shortfall of 8 positions. Cutting overtime without adding positions does not save the coverage; it moves the gap somewhere it cannot be seen, usually to thinner sectors or longer response times. The honest comparison is the cost of a position against the premium paid to cover its absence, and the premium is already $78,720 a year before counting fatigue and turnover.