Print Shop Rush Order with a Mid-Job Machine Jam
The university print shop has two machines. Machine A can produce one 600-page booklet in 45 minutes when working alone, and Machine B can produce one in 30 minutes. A rush order calls for 3 booklets. Both machines start together. After 10 minutes, Machine B jams and stops for 8 minutes for repairs; during that time Machine A continues alone. Once repaired, B resumes at its normal rate until the job is finished, and A also continues. The shop promises the order in 50 minutes from the start. Pages are produced continuously, and partial booklets do not count as finished. The manager wants to know whether the promise is met, how much lateness is caused by the jam, and whether the 50-minute promise would have been possible even without the jam.
1. Find each machine's rate in booklets per minute, the combined rate, and the time needed for 3 booklets if B never jams. Then compute the actual finish time with the 8-minute jam.
Read the worked answer
Machine A's rate is 1/45 booklet per minute, and Machine B's rate is 1/30 booklet per minute. The combined rate is
1/45 + 1/30 = 2/90 + 3/90 = 5/90 = 1/18 booklet per minute.
Without a jam, 3 booklets take 3 / (1/18) = 54 minutes. So the 50-minute promise is already impossible by 4 minutes.
With the jam: in the first 10 minutes the machines complete 10(1/18) = 5/9 booklet. In the next 8 minutes A alone completes 8/45 booklet. Total work so far is
5/9 + 8/45 = 25/45 + 8/45 = 33/45 = 11/15 booklet.
The remaining work is 3 - 11/15 = 34/15 booklet. At the combined rate of 1/18, this takes (34/15)(18) = 40.8 minutes. Total time is 10 + 8 + 40.8 = 58.8 minutes, or 58 minutes 48 seconds. The order is 8.8 minutes late.
2. The manager blames the jam for missing the deadline. Evaluate this claim by separating the inherent shortfall from the jam's effect.
Read the worked answer
Without any jam, the shop needs 54 minutes, which is 4 minutes beyond the 50-minute promise. So the promise is infeasible even under perfect operation.
The jam raises the finish time from 54 minutes to 58.8 minutes, a delay of 4.8 minutes. Of the total 8.8 minutes of lateness, about 4 minutes is inherent and 4.8 minutes is due to the jam.
Therefore the manager is only partly right. Fixing the jam alone would still leave the order 4 minutes late. To meet 50 minutes, the shop needs extra capacity, a smaller order, or a longer promised time.