There is a lot of talk about the end of 4 engine jets. They just aren't economical people are saying. Kinda sort maybe I guess common sense would say "they" are right, in some ways, but this is aviation, and things are complicated.
The thinking is two engine operations is more efficient than 4 engine operation. Yes, running 4 engines will use more fuel than running two engines, for most aircraft. The thing is, 4 engine aircraft haul more, so thinking of cost per available seat, things aren't black and white. Wikipedia has a great chart comparing various modern aircraft. http://en.wikipedia.org/wiki/Fuel_economy_in_aircraft
This page has some good comparisons. Looking at the column to the far right labeled "Fuel efficiency per seat" you will see, especially in the jets built since 1997, most of the jets in the medium haul (transatlantic) will get about 90+ miles per gallon per seat (mpg). Even in the transpacific (long haul chart) the jets are all pushing 70-100 mpg per seat. The reason the economy goes down for the longer flights is because the jets must carry the fuel longer.
(the chart has a couple outliers, any jet that hasn't flown yet (IE first flight > 2015) you can't really count, the numbers are estimates). Newer jets are much more efficient than older jets.
Comparing things, is more complicated than the charts may indicate. Thinking about a New York to London flight, and using a 747-8 with 467 seats at 91mpg or a 787-9 with 304 seats at 99mpg it would seem that the 787-9 wins every time. 2 engines means less maintenance, and less fuel, so it should win. To buy a 787-9 will set you back about $250million, where the 747-8 will cost about $357million. The 787 still wins, right. Well again it is complicated.
No one sells all the seats on the aircraft, so assuming a 95% load factor, 3400 miles (JFK-LHR) in big round numbers, the 787 will use 10440 gallons of fuel, where the 747 will use 17450 gallons. With only 289 folks on the 787, the mpg drops to 94mpg and the 747 with 444 folks on the 747 the jet only gets 87 mpg per seat.
The 747 is hauling about 35% more passengers per flight, so 3 flights of the 747 would equal 4 flights of the 787, This is where the numbers aren't very general, but you can see that on popular routes where the load factors are high, the 747 may actually win. Less boarding time to load 3 flights, less taxi time. The fuel efficiency that I have shown here only counts the cruise portion of the flight, climb is very expensive. There are landing fees, and gate fees that need to be accounted for.
Fuzzy things though include maintenance. The aluminium 747 can be maintained by most maintenance facilities, where the composite 787 may need special maintenance facilities. More engines may mean more expense, but they are smaller engines, so there could be less cost handling them.
My goal here isn't to show the 747 is more efficient, my goal here was to show it may not be the end of the line for the four engine jets, just yet. There is still time for them, they aren't hugely inefficient, and may offer economies that may not be completely obvious.