Overtime Coverage and the Per-Diem Proposal
A patrol division must staff a 24-hour operation with a minimum of 3 officers on duty on each of three 8-hour shifts. Eleven full-time officers are assigned to the division; each works five 8-hour shifts per week and may not be scheduled for more than 40 hours of straight time. Any shift not covered by a full-time officer is filled by overtime paid in whole 8-hour shifts at $54.00 per hour. The division's overtime budget for the coming week is $3,000. This week, two officers will be on injury leave for all seven days. The captain also has a proposal to hire one per-diem officer at $46.00 per hour for exactly 40 hours per week; the per-diem carries an additional 12% in administrative and equipment cost and is available for whole 8-hour shifts only.
1. At full staffing (no one on leave), how many overtime hours will the division incur next week, what will they cost, and does the division stay within budget?
Read the worked answer
Step 1 - Required coverage. 3 posts × 3 shifts per day × 7 days = 63 officer-shifts per week.
Step 2 - Coverage from full-time staff. 11 officers × 5 shifts = 55 shifts.
Step 3 - Shortfall. 63 − 55 = 8 shifts, i.e. 8 × 8 = 64 overtime hours.
Step 4 - Cost. 64 × $54.00 = $3,456.00.
Conclusion: The division is $456 over the $3,000 weekly overtime budget (3,456 − 3,000) even before any leave is taken. The structural gap, not the leave, is the primary driver - 8 of 63 required shifts are unfunded by the roster itself.
2. With the two officers on injury leave, compute the weekly overtime cost. Then determine whether the per-diem proposal reduces total cost, and by how much.
Read the worked answer
Step 1 - Coverage with leave. 9 available officers × 5 shifts = 45 shifts.
Step 2 - Shortfall. 63 − 45 = 18 shifts = 18 × 8 = 144 overtime hours.
Step 3 - Overtime cost. 144 × $54.00 = $7,776.00.
Step 4 - Per-diem alternative. The per-diem supplies 40 hours = 5 shifts, displacing 5 × 8 × $54.00 = $2,160.00 of overtime. Per-diem cost: 40 × $46.00 = $1,840.00; plus 12% = $1,840.00 × 1.12 = $2,060.80. Remaining shortfall: 18 − 5 = 13 shifts = 104 hours × $54.00 = $5,616.00. Total = 5,616.00 + 2,060.80 = $7,676.80.
Step 5 - Comparison. $7,776.00 − $7,676.80 = $99.20 per week (about 1.3%).
Judgment: The saving is trivial relative to onboarding, supervision, and liability costs of a per-diem, so cost alone does not justify the hire. The number that should drive the decision is 144 OT hours spread over 9 officers - roughly 16 extra hours each - which is a fatigue and officer-safety risk the arithmetic does not capture.
3. Ignoring the per-diem proposal, the captain proposes cutting the minimum staffing on the midnight shift from 3 officers to 2 for the week. Compute the savings and state at least two factors the arithmetic does not capture.
Read the worked answer
Step 1 - New requirement. 7 days × (3 day + 3 evening + 2 midnight) = 56 shifts.
Step 2 - New shortfall. 56 − 45 = 11 shifts = 88 overtime hours.
Step 3 - New cost. 88 × $54.00 = $4,752.00.
Step 4 - Savings. $7,776.00 − $4,752.00 = $3,024.00 per week.
Factors outside the arithmetic:
- Two-officer calls. Many calls (domestic disturbances, building searches, prisoner transport) have a two-officer minimum; with two on duty, the first such call empties the sector and forces a callback - overtime that re-appears elsewhere in the budget.
- Officer safety and backup time. A single-officer response with backup 8–12 minutes away materially changes risk on a midnight shift.
- Coverage is a floor, not a target. Minimum staffing of 2 assumes every officer is available all shift; one sick call or one transport leaves the shift at one officer.
The savings figure is real, but it is a policy decision about acceptable risk, not a bookkeeping one.