Allocating Night-Shift Constables Across Six Beats
A station house officer has 18 constables available for the night shift after leave, training and court duty. Standing orders require a minimum of two constables on every beat, one constable at the station diary and one in reserve; a constable cannot be counted twice. The thana has six beats, and last quarter's crime figures are: Beat A 45 cases, B 30, C 24, D 18, E 12, F 9, total 138. The market association has asked for a permanent two-constable post at the new bus stand, which falls inside Beat A. Five women constables are on the night roster, and the sub-divisional police officer's order requires at least one woman constable in every beat that recorded more than 20 women's complaints in the quarter; that condition applies to Beats A, B and C only. The officer must submit a written allocation, and two days later the sub-divisional police officer sanctions two additional constables for the same shift.
1. Can the market association's demand for a permanent two-constable post be met from the present 18, without breaking standing orders or the women's requirement? Show the arithmetic.
Read the worked answer
No. Work from the outside in.
- Strength on the shift: 18.
- Deductions: 1 station diary + 1 reserve = 2, leaving 18 − 2 = 16 for the beats.
- The six beats at the legal minimum of 2 each already need 12, so only 4 are free above the floor.
- A new permanent post of 2 at the bus stand raises the beat requirement to 18, that is, 20 for the shift. The shortfall is 2 constables.
The only ways to create it are unacceptable: cutting a beat below 2 breaches the standing order; withdrawing the diary or reserve constable breaches it as well. The women's rule also binds, since 3 of the 5 women must be placed in Beats A, B and C, leaving only 2 for the rest, and a new fixed post would consume one of them.
So the answer to the association is a reasoned no: the request can be met only after 2 additional posts are sanctioned, or in the interim by a mobile patrol halting at the bus stand during market hours, drawn from Beat A's strength rather than added to it.
2. Prepare an allocation of the 16 constables available for the beats that satisfies the minimum and the women's rule, and justify the two cases where your figure differs from a strictly proportional share.
Read the worked answer
Density weight = cases/138, share = 16 × weight:
- A: 16 × 45/138 = 5.2
- B: 16 × 30/138 = 3.5
- C: 16 × 24/138 = 2.8
- D: 16 × 18/138 = 2.1
- E: 16 × 12/138 = 1.4
- F: 16 × 9/138 = 1.0
Taking the largest whole numbers while applying the floor of 2 where the proportional share falls below it gives: A 5, B 3, C 2, D 2, E 2, F 2 = 16. The women's rule is met by posting one woman each in A, B and C, with the remaining two in A and D.
Two deviations need defence. E and F are over-provided relative to their shares (1.4 and 1.0), because the minimum of two is a legal requirement and overrides the density weight; the officer cannot lawfully run a beat with one constable to make the arithmetic neat. C is under-provided (2 against 2.8), and this is the real cost of the floor; the officer should record that C is the first beat to be reinforced if a constable returns, because it carries the largest unmet share.
3. Two additional constables are sanctioned for the same shift. Using a shortfall table, state where they should go and what the resulting density per constable looks like.
Read the worked answer
The beats now receive 18. Proportional shares recomputed: A 18 × 45/138 = 5.9; B 3.9; C 3.1; D 2.3; E 1.6; F 1.2.
Shortfall against the present allocation of A 5, B 3, C 2, D 2, E 2, F 2:
- A: 5.9 − 5 = 0.9
- B: 3.9 − 3 = 0.9
- C: 3.1 − 2 = 1.1
- D: 2.3 − 2 = 0.3
- E and F: both already above their share
The two constables go to C and B - C first, because its shortfall is the largest, then B, which is level with A but where the gap is not offset by any surplus elsewhere. Final allocation: A 5, B 4, C 3, D 2, E 2, F 2 = 18.
Resulting cases per constable: A 9.0, B 7.5, C 8.0, D 9.0, E 6.0, F 4.5. D is high at 9 and will be the next beat examined, but D's absolute case count is modest, so the officer should note D as a watch item rather than shift a constable there now. The written order should show both the allocation and this reasoning, so that any later demand is answered from the file.