Night Sort at the Parcel Hub: Clearing the 06:00 Cut-Off
At 22:00 the parcel hub still has 6,480 parcels to sort before the 06:00 carrier cut-off, and no further trailers are due. Line A runs at a steady 540 parcels an hour; Line B at 420. Line B is booked for 40 minutes of planned maintenance starting at 02:00. Line C, rated at 300 parcels an hour, sits idle: a replacement part arrives at 23:00 and a 45-minute changeover must follow before it can handle this parcel type. The supervisor can call in an agency crew of four for Line C at £19.40 per person per hour, with a four-hour minimum call, or authorise overtime on Lines A and B at £26.10 per hour for permanent staff, payable in two-hour blocks. Historical data shows that 4% of sorted parcels fail the outbound scan and must be re-run through the same lines. The site manager wants a decision by 22:30 and has asked for the arithmetic first and the contingency second.
1. Working only from the stated rates, at what time would Lines A and B alone clear the 6,480 parcels, and how much slack is left before 06:00? Then restate the finish time allowing for the 4% re-run rate.
Read the worked answer
Stage 1 - 22:00 to 02:00 (4.0 h, both lines running) Combined rate 540 + 420 = 960 parcels/h → 4 × 960 = 3,840 sorted. Remaining: 6,480 − 3,840 = 2,640.
Stage 2 - 02:00 to 02:40 (0.667 h, Line A only) 540 × 0.667 = 360 sorted. Remaining at 02:40: 2,640 − 360 = 2,280.
Stage 3 - 02:40 onward (both lines back) 2,280 ÷ 960 = 2.375 h = 2 h 22 min 30 s → finish 05:02:30. Slack before 06:00 = 57.5 minutes.
Re-runs 4% of 6,480 = 259.2, so round up to 260 parcels. At 960/h that adds 260 ÷ 960 = 0.27 h ≈ 16 minutes. Revised finish ≈ 05:19, leaving roughly 41 minutes of slack.
Assumptions to state explicitly: the 4% is a rate for this parcel type and this period; re-runs re-enter the same flow at a steady rate rather than arriving in one batch; the maintenance window is honoured exactly; and the quoted rates are sustained averages, not peak. Each assumption is doing real work - the 41-minute buffer is only 4% of the 8-hour window.
2. Should the supervisor call in the agency crew for Line C? Give the cost, the capacity it buys, and the condition that would reverse the decision.
Read the worked answer
Cost: 4 people × £19.40 × 4-hour minimum = £310.40, payable whether or not the crew is used for the full four hours.
Capacity bought: part at 23:00 + 45-minute changeover → Line C live at 23:45. To 06:00 that is 6.25 h × 300 = 1,875 parcels.
The mismatch: the demand genuinely at risk is the ~260 re-run parcels plus the 41-minute buffer - roughly 660 parcels' worth. Line C offers 1,875. It is buying almost three times the exposure, and it cannot help the constraint, which is the throughput of Lines A and B combined.
Decision at 22:30: do not call. Hold the crew as a contingency.
Reversal condition (must be a measurable trigger, not a feeling): take a count at 00:30. From 00:30 to 06:00 the lines can output 5.5 × 960 − (420 × 0.667) = 5,280 − 280 = 5,000 parcels, allowing for the maintenance window. If the 00:30 count is above 5,000 - i.e. more than about 920 parcels behind plan - call the crew then; a one-hour call-out still leaves 5.5 hours of useful Line C capacity. Below 5,000, the arithmetic does not support the spend.
A defensible answer here is not 'no' - it is 'no, with a named trigger and a named time to test it'.
3. Of the figures in the brief, which single one would you verify before 22:30, and why that one rather than the others?
Read the worked answer
Verify the sustained throughput rates (540 and 420), not the maintenance window or the re-run percentage.
Show the sensitivity:
- The 57.5-minute buffer at 960/h is worth about 552 parcels of combined output (0.958 h × 960 ≈ 920 parcels' worth of remaining volume - i.e. roughly 9.6% of the 6,480 total).
- A sustained rate 10% below plan (864/h combined) removes ~96 parcels an hour. Over the ~7.3 productive hours to 06:00 that is roughly 700 parcels - more than the buffer holds. The plan fails silently and is discovered at about 05:40, when nothing can be recovered.
- A 40-minute overrun of the maintenance window costs only 420 × 0.667 ≈ 280 parcels of Line B output, about 17 minutes. Material, but an order of magnitude smaller.
- The re-run rate is already the basis of the restated finish time; a doubling of it to 8% adds only another 16 minutes.
So the rate figure carries the most risk per unit of uncertainty. The question to ask is narrow: is 540/420 a measured average over comparable nights including jams and changeovers, or a rated maximum from the equipment spec? A rated-maximum answer converts the plan from 'comfortable' to 'unproven' and should push the supervisor toward earlier escalation.