We use common sence. I have done it enough to know that their is no scientific answer.
Our rule of thumb takes into account a variety of options:
We do 2 pilots up to 14 hours IF their is only one leg or less then 3 time zone changes (North/ South). Multiple legs/Multiple time zones above 12 hrs we add a 3rd pilot up until 18. Then we do 2 crews.
This is not scientific we switch crews in ANC on flights to and from Asia (based in NY)
It all comes down to common sence. I have done Narita to White Plains with 2 pilots (our third became ill and was forced to stay in Tokyo). All I did was call the CEO and tell him that he could not get the slot he wanted and that he would have to leave at 8am. Which equated to an all day light flight home with the two pilots working on a normal body clock.
I have also done ANC-Narita-Singapore on the back side of the clock, all night flying, on my circadian low, but I had 3 pilots.
Their is no right answer. It's common sence. You need to look at all of the information you have and trust the crews.
The biggest things in my opinion are:
Do as much flying during the daylight as you can.
Do as much flying on your normal body clock as you can (this requires some planning)
If you think you might need 3 pilots, take 3 pilots. Clearing customs takes longer then you think, or pax stuck traffic, or a late running meeting. Murphy's law.
If you are taking 3 pilots and have to stop for fuel or an additional pax. Preposition one pilot to the tech stop. One alert pilot is better then 3 tired pilots. Example NYC-London-Cairo. If have done this a couple of times. Depart NY at 8 pm fly all night, only to find Stanstead and/or Luton below mins, winding up in Farnbourough. Then waiting for a pax to arrive, after waiting for the limo in morning London traffic. Then departing in the morning fog to Cairo.
Flying all night with 3 tired pilots made no sence but we were "legal". The next time out we sent a pilot ahead to London. He got a solid nights sleep, met us at the airport and flew the leg to Cairo.
If you are stoping for fuel. Try to work a crew change into the tech stop. I understand that you may not be able to get a contract pilot or one of your normal line pilots may be on vacation. Things like this happen. Everytime I have been painted into a corner I have gone to the principal and asked him/her to adjust their travel times for safety. I have never been turned down. Use common sence.