Unit 4 Staffing Shortfall and Search Rate
Unit 4 houses 176 inmates in 88 cells, two inmates per cell. Post orders require one officer for every 22 inmates on day watch (0600–1400, an 8-hour watch) and one officer for every 44 inmates on night watch (1400–0600, a 12-hour watch). One officer covers one post for the entire watch. The unit roster lists 14 officers; this week, 3 are on approved leave and 2 are at off-site training, all five for the full seven days. Any unfilled post is covered by overtime at $61.50 per hour, and one overtime shift fills exactly one post for one full watch. A contraband search this week recovered prohibited items in 11 of the unit's cells.
1. How many posts does Unit 4 require per 24-hour day, how many officers are actually available, and what is the shortfall? Show each calculation.
Read the worked answer
Day watch posts: 176 inmates ÷ 22 inmates per officer = 8 posts Night watch posts: 176 inmates ÷ 44 inmates per officer = 4 posts Total posts required: 8 + 4 = 12 posts
Available officers: 14 on the roster − 3 on leave − 2 at training = 9 officers
Shortfall: 12 required − 9 available = 3 posts
Reasoning note: always round up on a staffing ratio. If the unit had held 177 inmates, day watch would require 177 ÷ 22 = 8.05, which means 9 posts, not 8. Having 8 officers for a 9-post requirement leaves part of the unit uncovered.
2. All three unfilled posts must be covered. What is the lowest possible overtime cost for one 24-hour day, and how should the shifts be assigned to achieve it? Extend the figure to the full seven days.
Read the worked answer
Compare the two ways to fill three posts:
- Three day posts: 3 × 8 hours = 24 overtime hours day cost = 24 × $61.50 = $1,476.00
- Three night posts: 3 × 12 hours = 36 overtime hours night cost = 36 × $61.50 = $2,214.00
Lowest daily cost: $1,476.00, achieved by assigning the three overtime shifts to day watch (8-hour posts) rather than night watch.
Seven-day cost: $1,476.00 × 7 = $10,332.00
Reasoning note: each overtime shift covers one full watch regardless of watch length, so cost is driven by hours, not by the number of posts. Filling shorter watches costs less per post. Two cautions before acting: confirm the officers used are not already at a consecutive-hour limit, and confirm the leave and training assignments cannot be staggered, which would usually be cheaper than overtime.
3. Contraband was found in 11 of the 88 cells. Express this as a percentage. If next month's search samples 32 cells and the rate holds, how many cells should be expected to contain contraband?
Read the worked answer
Rate: 11 ÷ 88 = 0.125 = 12.5% of cells
Projection for 32 cells: 32 × 0.125 = 4 cells
Check: 32 cells is 32 ÷ 88 ≈ 36.4% of the unit, so the expected count is about 36.4% of 11 ≈ 4.0, which agrees.
Reasoning note: this is a projection, not a prediction. A sample of 32 cells taken from one tier, or taken right after a shakedown, will not reflect the unit's overall rate. Present the figure as an estimate and state how the sample was drawn. Percentages of very small bases also mislead: if only 8 cells had been sampled, one find would read as 12.5% on a single cell.