The Sickness Shortfall
A station has 32 constables available for patrol duties. Four are on approved leave and two are at a training course, leaving 26 deployable. The station must mount three two-person patrols on each of three 8-hour shifts, every day of a seven-day week. Each officer can work a maximum of five shifts per week without overtime. The inspector also wants to keep a reserve of at least 10% of the weekly officer-shifts to cover sickness, court appearances, and emergencies. Overtime is expensive and the station has already used most of its overtime budget. On Monday morning, two more officers call in sick for the week. The inspector asks the duty sergeant: 'Can we still cover every patrol without overtime, and if not, what is the smallest change we can make?' The sergeant must show the arithmetic, not just give an opinion, and must recommend a change that protects both officer welfare and public safety.
1. Calculate the weekly officer-shifts required and the number available before and after the two officers call in sick. Show each step.
Read the worked answer
Required patrol shifts per day: 3 patrols × 3 shifts = 9 patrols per day. Each patrol needs 2 officers, so 9 × 2 = 18 officer-shifts per day. Over 7 days: 18 × 7 = 126 officer-shifts per week.
Before sickness: 26 deployable officers × 5 shifts = 130 officer-shifts. Surplus over patrol requirement = 130 − 126 = 4 shifts.
Reserve requirement: 10% of 126 = 12.6, which must be rounded up to 13 shifts. So the station needs 126 + 13 = 139 officer-shifts to cover patrols and keep the reserve. Before sickness it has 130, so it is already 9 short of the reserve target.
After two more officers call in sick: 24 × 5 = 120 officer-shifts. Shortfall against patrols alone = 126 − 120 = 6. Shortfall against patrols plus reserve = 139 − 120 = 19.
The key point: covering every patrol and keeping a 10% reserve are two different tests. The station fails both after the sickness calls.
2. The inspector proposes to run one patrol short on two shifts. Evaluate the proposal with the numbers.
Read the worked answer
Running one patrol short on two shifts saves 2 officers × 2 shifts = 4 officer-shifts. After the sickness calls the station is 6 short of the patrol requirement, so the proposal still leaves a 2-shift hole in patrol coverage. It also leaves no reserve at all.
Measured against the 139-shift target, the proposal is far short: 120 available − 4 saved = 116 effective shifts against 139 needed, a gap of 23. Even if the inspector ignores the reserve, the proposal does not meet the patrol requirement. It is not a solution; it is a partial cut that hides the shortfall.
The arithmetic also shows the proposal is poorly targeted: cutting two patrols may remove coverage from the same high-risk period, while a planned reduction across low-demand shifts would be safer. The inspector should not describe the plan as 'covered' when the numbers say otherwise.
3. Recommend the smallest practical change that restores patrol coverage, and explain the trade-off with the reserve.
Read the worked answer
First, cover the 6-shift hole. The smallest numerical fix is 6 overtime shifts, because 120 + 6 = 126. That restores every patrol but leaves the reserve at zero, so any court appearance or emergency will reopen the hole.
Second, meet the reserve. The station needs 139 shifts, so it needs 19 overtime shifts in total (6 to cover patrols + 13 for reserve), or 4 more officers (4 × 5 = 20 shifts, giving 140). The smallest change that meets both tests is therefore either 19 overtime shifts or the recall of 4 officers, whichever is actually available and lawful under the roster rules.
Recommendation: ask the divisional office for 4 officers from a reserve unit, or authorize 19 overtime shifts if the budget can bear it. If neither is possible, reduce patrols in a planned way - for example, cut one patrol on the three lowest-demand shifts - and record the reduced coverage and the risk. Do not pretend the reserve exists. The trade-off is explicit: overtime protects coverage but costs money and risks fatigue; cutting patrols protects the budget but reduces visible presence and response capacity.