Seventy Constables and Five Beats
A metropolitan police range has 96 constables sanctioned for patrol. Each day 12 are on fixed guard duty at two VIP residences, an average of 8 are on leave or training, and 6 work in the administrative section. Patrol runs in three eight-hour shifts; every beat needs 2 constables per shift and one vehicle per beat. Last month's reported incidents by beat were: A 42, B 18, C 27, D 9, E 33. Between 20:00 and 04:00 the range records 60 per cent of all incidents; the remainder fall in the two daytime shifts. The range commander must publish next month's beat deployment by Friday. The ward chair of beat D, the lowest-crime beat, has written demanding four additional constables, citing two recent motorcycle thefts. The traffic office has separately asked for 6 constables for a week-long road-safety drive in the same month.
1. How many constables are actually deployable on patrol each day, what is the minimum needed to hold all five beats for 24 hours, and how many beats of that standard could the pool sustain?
Read the worked answer
Deployable pool. 96 − 12 (fixed guard) − 8 (leave/training average) − 6 (administrative) = 70 constables per day.
Minimum for five beats. 5 beats × 2 constables × 3 shifts = 30 constable-days per day. Surplus = 70 − 30 = 40 constable-days.
Maximum beats at the full standard. One beat costs 2 × 3 = 6 constable-days. 70 ÷ 6 = 11.67, so 11 complete beats, with 4 constables left over - not enough for a twelfth 24-hour beat, but enough to double up one shift or to provide relief cover when someone reports sick.
The 4-constable remainder explains why the range cannot simply "open more beats": below a full multiple of six, a beat has holes in its shift chart.
2. Allocate the 40 surplus constable-days across A–E in proportion to last month's incidents, using the largest-remainder method, and convert the result into something a shift commander can actually use.
Read the worked answer
Total incidents = 42 + 18 + 27 + 9 + 33 = 129. Surplus share = incidents ÷ 129 × 40.
| Beat | Incidents | Exact share | Rounded | Final (base 6 + surplus) |
|---|---|---|---|---|
| A | 42 | 13.02 | 13 | 6 + 13 = 19 |
| B | 18 | 5.58 | 6 | 6 + 6 = 12 |
| C | 27 | 8.37 | 8 | 6 + 8 = 14 |
| D | 9 | 2.79 | 3 | 6 + 3 = 9 |
| E | 33 | 10.23 | 10 | 6 + 10 = 16 |
| Total | 129 | 40.00 | 40 | 70 |
Floors (13, 5, 8, 2, 10) sum to 38; the two remaining units go to the largest remainders, D (0.79) and B (0.58).
Shift-level conversion. Half a pair cannot be deployed, so re-express the 40 as 20 surplus pairs distributed over the 15 beat-shifts: exact shares A 6.51, B 2.79, C 4.19, D 1.40, E 5.12; floors sum to 18, and the two leftovers go to B (0.79) and A (0.51). That gives surplus pairs A 7, B 3, C 4, D 1, E 5 - beat totals in pairs: A 10, B 6, C 7, D 4, E 8, i.e. 35 pairs = 70 constables.
Night weighting. With 60 per cent of incidents between 20:00 and 04:00, the night shift should take the larger share of each beat's surplus pairs; A and E should receive their extra pairs almost entirely at night, D's single pair at night only.
3. How should the commander answer the ward chair of beat D and the traffic office's request for six constables?
Read the worked answer
Beat D. The chair is asking for 4 constables; the data gives D 9 constable-days in total, against 16 for E and 19 for A. Refusing outright is unnecessary - the honest answer is: (i) publish the allocation formula (base 6 + share by recorded incidents) so the decision is auditable rather than personal; (ii) offer a targeted response to the two motorcycle thefts - D's one surplus pair placed on the night shift, plus a two-week surge of one pair drawn from the 4-constable remainder, reviewed against the theft pattern afterwards; (iii) ask the ward for what the police cannot supply alone - lighting, parking arrangements, a community watch group.
Traffic office. Six constables for a week is 42 constable-days, roughly one full day's deployable pool. Options: absorb it from the 4-constable daily remainder over the week (4 × 7 = 28 constable-days), and cover the balance by shifting the road-safety drive to the two daytime shifts, when incident load is only 40 per cent. Do not quietly strip beats A and E to fund it; that would cancel the risk-based allocation the whole exercise is built on.
What to communicate. The criteria, the numbers and the review date, in writing to both parties, so that a refusal of one request is not read as favouritism.