Can a four-engine aircraft with limited fuel gain extra range by shutting down one of its engines? I was reading the book Vulcan 607 by Rowland White who examines the details of a British bomber's mission during the Falklands War in 1982. At some point, the author wrote about the fuel shortage experienced by the crew of the four-engine Vulcan bomber during their return to their base. The author wrote that the pilot "considered shutting down one of the four Rolls-Royce Conway engines to get a little more range out of the remaining fuel...".  

I have not heard of this type of procedure before, so I am wondering if the decision to shut down one of the engines to maximize fuel efficiency makes sense.
Assuming the aircraft is always flying at its optimal Lift/Drag ratio (= optimal angle of attack = optimal speed for a given weight and altitude), then the question is:

Is it more efficient to generate a given amount of thrust T with 4 engines x T/4, or 3 engines x T/3?

I would think that in the general case, it is not the case. Specific Fuel Consumption tends to increase with thrust (at constant speed):



Note: the above diagram is for Turbofans (Source: Airplane Aerodynamics and Performance, Roskam). It would be interesting to have similar trends for other types of engines (turboprops, turbojets, piston engines), but I could not find any!

On the other hand, it is possible that, in this particular case, the engines were simply working too far off their design point, so increasing thrust was improving their SFC.

Also, there is a problem with cutting one engine: The pilot will need to counteract the thrust asymmetry with the rudder, and this will have a significant cost in drag. Of course, the pilot could just cut 2 engines instead of one! :-)