Two Presses, One Jam, and a Suspiciously Round Cost
A print shop owns two presses. Press A, running alone, completes a standard job in 6 hours and costs $45 per hour to operate. Press B, running alone, completes the same job in 9 hours and costs $30 per hour. Both presses work at constant, independent rates and can run simultaneously on the same job without interfering. Fixed costs (rent, salaried labor, depreciation) are excluded from these figures; the shop bills each press only for the hours it actually runs, prorated to the minute, and a press that is not running costs nothing. A job is started with both presses running together. After exactly 2 hours, press B jams and is out of service for the rest of that job; press A continues alone until the job is complete.
1. How long does the job take in total, measured from the start?
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Work in job-fractions per hour.
- Press A: 1/6 job per hour
- Press B: 1/9 job per hour
- Together: 1/6 + 1/9 = 3/18 + 2/18 = 5/18 job per hour
First 2 hours (both running): 2 x 5/18 = 10/18 = 5/9 of the job done. Remaining: 4/9.
Remainder on A alone: time = (4/9) / (1/6) = (4/9) x 6 = 24/9 = 8/3 hours = 2 hours 40 minutes.
Total: 2 h + 2 h 40 min = 4 hours 40 minutes (4.667 h).
Sanity check: the whole job would have taken 18/5 = 3.6 h with both presses. Losing B for the back half pushes it to 4.67 h, a delay of about 1.07 h - plausible, since B is the slower press and half the job was already banked.
2. Compute the operating cost of this job, then compute what the job would have cost with no jam at all. Explain the coincidence.
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With the jam:
- Press A ran 4.667 h: 24/3 h x $45 = $210 (exactly; 8/3 x 45 = 120, plus 2 x 45 = 90)
- Press B ran 2 h: 2 x $30 = $60
- Total: $270
Without the jam: both run 3.6 h.
- A: 3.6 x $45 = $162
- B: 3.6 x $30 = $108
- Total: $270
The coincidence is structural, not luck. Compute each press's cost to complete a job by itself:
- A alone: 6 h x $45 = $270
- B alone: 9 h x $30 = $270
Each press costs exactly $270 per job, so any schedule that finishes the job with these two presses costs $270 - the jam changed when the money was spent and the split between presses, not the total. Equivalently, cost per job is invariant because cost per hour is inversely proportional to speed for both machines. Whenever two resources have equal cost-per-unit-output, scheduling between them is a pure time decision, not a cost decision.
That reframes the jam: it cost the shop 1.07 hours of throughput and any rush or late-delivery penalty, but no extra direct operating dollars.
3. A supplier offers a rebuilt Press B rated at 7 hours per job, priced at $38 per hour. Using the same accounting, should the shop buy it? Quantify the cost case and state what additional factors decide the question.
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Quantify: rebuilt B alone would cost 7 h x $38 = $266 per job, versus $270 for the current B and $270 for A. Savings: $4 per job, about 1.5%.
Check whether pairing changes the answer: rebuilt B runs at 1/7 job/h and A at 1/6, so together 1/6 + 1/7 = 13/42 job/h, finishing in 42/13 = 3.23 h. Cost = 3.23 x ($45 + $38) = 3.23 x $83 = $268. Still within a dollar or two of $270. The invariance from the previous task is broken only slightly, because $38 x 7 = $266 is only barely below the $270 benchmark.
Verdict: the cost case alone is too thin to justify the purchase. A $4-per-job margin is swamped by any of the following, each of which should be weighed explicitly:
- Reliability: the whole scenario exists because a press jammed. If the rebuilt unit is more failure-prone than the current B, one lost hour at $45-plus-rush-penalty erases dozens of jobs' worth of savings.
- Capital cost and payback: at $4 per job, a purchase price of P requires P/4 jobs just to break even on operating savings; state that number before deciding.
- Capacity, not unit cost, is the real constraint. The jam cost 1.07 hours of throughput, and throughput is what limits revenue. A press that is 22% faster (7 h vs 9 h) adds capacity worth far more than $4 per job if there is backlog.
- Rounding and proration: with per-minute billing and real job mix, a $4 theoretical edge can vanish.
The defensible answer is: buy only if the throughput gain and reliability record clear the capital hurdle - not because of the $4.