Throughput at a Food Bank Packing Line
Northside Food Bank must assemble 60 emergency boxes between 9:00 and 12:00. Three volunteers are available, but their hours differ. Dana packs 12 boxes per hour and can work from 9:00 to 11:00. Eli packs 9 per hour and works 9:00 to 12:00. Farah packs 7 per hour and works 10:00 to 12:00. Every box must be sealed and labeled after packing; a single labeling station processes 20 boxes per hour and cannot start until at least 10 boxes are packed. Labels are applied by a fourth volunteer, Gus, who is available 9:00 to 12:00. Boxes cannot leave the building unlabeled. The truck arrives at 12:00. A staff member suggests adding a second labeling station at 10:30, which would raise labeling capacity to 35 boxes per hour but requires 20 minutes of setup by Gus, during which no labeling occurs.
1. Without the second labeling station, how many boxes can be packed and labeled by 12:00? Show the timeline and identify the bottleneck.
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Packing. Dana: 2 h × 12 = 24. Eli: 3 h × 9 = 27. Farah: 2 h × 7 = 14. Total packed by 12:00 = 65. Labeling. Labeling can start once 10 boxes exist. Dana alone packs 12/h, so 10 boxes are ready at 9:50. From 9:50 to 12:00 is 130 min = 2.1667 h. At 20 boxes/h, Gus labels 43.33, so 43 complete boxes. Packing is not binding; labeling is. Answer: 43 boxes, not 60. The remaining 22 packed boxes are unlabeled and cannot leave.
2. Evaluate the proposal to add the second labeling station at 10:30. Does it reach 60 boxes, and is the setup cost justified?
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With the second station. Labeling before setup: 9:50–10:30 = 40 min × 20/h = 13.33, so 13 boxes. Setup 10:30–10:50: 0 boxes. Capacity 35/h from 10:50–12:00 = 70 min × 35/60 = 40.83. Total = 13.33 + 40.83 = 54.16, so 54 boxes. Check supply: by 10:50, packed is about Dana 22 + Eli 16.5 + Farah 5.8 = 44.3; packing continues to 65 by 12:00, so supply is sufficient over the interval. The proposal raises output from 43 to 54, still 6 short of 60. The 20-minute setup is expensive: without it, the second station would add about 11 more boxes. Recommendation: the change helps but is not sufficient to meet the truck.
3. What single change would most likely close the gap to 60 boxes, and what assumption must hold?
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Binding constraint is labeling start time and capacity. The cheapest single change is to lower the labeling trigger from 10 boxes to, say, 3, so Gus can start at 9:15 instead of 9:50. That adds 35 min × 20/h = 11.7 boxes, bringing the one-station total to about 55; combined with the second station, the added early time plus eliminating idle setup could reach 60. Alternatively, have Dana pack a small first batch before 9:00. Assumption: Gus can label boxes as they are packed and the 10-box threshold is a policy, not a physical requirement. If the threshold exists because labels are printed in batches of 10, the change may not be feasible. The analysis also assumes packing rates are steady and that box quality does not fall when workers rush.