Interesting post 172_driver, and somewhat contradicts what I wrote in the previous post. But then not quite anyway...
As such the entire "checklist instruction" section is well worth a careful read. In the end everything will be blamed on the captain:
In all situations, the captain must assess the situation and use good judgment to determine the safest course of action.
A good systems knowledge is hence required. For the case of single pack, my
opinion is that the captain didn't use good judgement climbing to FL410:
Above FL370 the pressure controller changes schedule from max differential pressure of 7.8 to 8.35 psi. Hence, above FL370 the demand of bleed-air becomes higher. The bleed air demand is satisfied by taking bleed-air from stage 5 and 9 of the compressor. With their low weight, and after leveling off, the cruise thrust would have been low as well, causing an increased demand of (hot) stage 9 bleed-air. So the bleed trip off would be due to a bleed air overheat:
Soon after the aircraft levelled out at the predetermined altitude, the right-side system activated a “Bleed Trip Off” warning.
A level off at FL370 would have made more sense as the last 4000 feet brings little to nothing in operational value (at most a few kg fuel saved on the short flight they were doing), and hence flying at this lower level would not have caused any disruption in the schedule. Also an increase in speed to increase the thrust output, and lower the demand for (hot) stage 9 bleed-air. Since the report mentions the stage 9 valve for the operating side was regulating out of limits (the other side outright broken, causing the first bleed trip off), maybe the bleed would have tripped anyway, but it would probably have reduced the risk staying in FL370.
Not to go into a discussion about that particular flight in general, but a couple of other points speaking in favor of FL370. It is not mentioned that the crew tried to restore the bleed to the remaining pack, which I find strange. This is a slow decompression scenario. As did the commander manually and unnecessarily deploy the cabin oxygen masks (cabin altitude never exceeded 14.000 feet). Also, the crew to donned oxygenmasks despite the cabin altitude was below FL100 (no cabin altitude warning yet).
It seems to me it got hectic, and being at FL410 probably didn't contribute to reduce the stress level (time of useful continuousness at that level probably being on their minds too). Had they been at FL370 they would have bought themselves valuable time (1-2 minutes), enabling the whole process to go slower and less stressed. E.g. to start a descent and reset the trip off, before going into a full blown emergency descent.
While every attempt is made to supply needed non–normal checklists, it is not possible to develop checklists for all conceivable situations.
It's always easy to look in hindsight, and it may have been a top crew that in this one occasion put themselves in a bad position. But with pressurization problems I wouldn't go above FL370. Maybe it due to experience, I had fluctuating climb rate in FL390 some years ago. The same considerations (higher differential pressure) led to the decision to descent to FL370, which caused the fluctuations to stabilize. Afterwards we learned that there had been some sort of problem with the regulating of the outflow valve. Less stress on a system = less likelihood that it will fail.
The whole topic here could be reduced to "airmanship", which includes knowing the equipment you operate thoroughly... It's not addressed in the report, instead the Swedes put all the blame on Boeing, and none pilots operating "by the book". And surprisingly none either on their state owned airline (despite both both stage 9 valves were found to be broken and outside regulating limits respectively). I sincerely hope that Boeing will not impose further limits based on this, I think it's enough that we have to brief cabin altitude warning before each flight of the day because of Helios.
Thanks for an interesting link, which gives cause for consideration on how to handle such a malfunction.