LOSS OF ONE TANK FUEL PUMP COUPLED WITH REDUCED OUTPUT FROM THE OTHER PUMP CAUSES AN ENG TO FLAME OUT.

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ON CLBOUT FROM TOKYO'S NARITA ARPT; I NOTICED #3 AFT BOOST PUMP LOW PRESSURE LIGHT. OPENED ALL FUEL XFEED MANIFOLDS AND TURNED ON ALL BOOST PUMPS. CONTINUED CLB. AT TOP OF CLB WE HAD A FUEL IMBAL OF 10000 LBS BTWN TANKS #2 AND #3. #1 AND #4 WERE EVEN; #2 HAD 48000 LBS; AND #3 HAD 58000 LBS. DISCUSSED PROB WITH CAPT AND FO. WE DECIDED TO TRY AND BURN FUEL OUT OF #3 TANK TO FEED #3 ENG AND #4 ENG USING 1 GOOD BOOST PUMP IN #3 TANK. CLOSED #1 AND #2 XFEED VALVES (MAKING SURE BOOST PUMPS WERE ON IN THOSE TANKS). OPENED XFEED VALVES FOR #3 AND #4 TANKS. MADE SURE BOTH BOOST PUMPS WERE ON IN #3 (AFT PUMP LOW PRESSURE LIGHT STILL ON) THEN TURNED OFF PUMPS IN #4 TANK. MONITORED ENG FOR 1 1/2 - 2 MINS. NOTED FUEL WAS NOT DECREASING IN #3 TANK. CHKED FUEL GAUGE. TESTED OKAY. ABOUT THAT TIME; #4 ENG FLAMED OUT. IMMEDIATELY TURNED ON BOOST PUMPS IN #4 TANK AND PERFORMED ENG INFLT RESTART CHKLIST. ENG STARTED OKAY. THROUGHOUT ALL PROCS FO FLEW AIRPLANE AND HANDLED RADIOS; CAPT WORKED WITH AND BACKED ME UP. AFTER ENG RESTART; WE THEN USED THE JETTISON PUMPS IN #3 TANK AND THE FUEL DUMP VALVE IN CTR TANK TO PRESSURIZE THE XFEED MANIFOLD AND FEED FUEL FROM #3 TANK TO OTHER ENGS. SYS WORKED AS DEPICTED IN COCKPIT OPERATING MANUAL. FLT PROCEEDED UNEVENTFULLY TO DEST IN HNL. BELIEVE FUEL IMBAL CAUSED BY LOW PRESSURE IN #3 TANK FORWARD BOOST PUMP. PRESSURE WAS HIGH ENOUGH TO KEEP LOW PRESSURE LIGHT OFF FOR THE FORWARD PUMP; BUT NOT HIGH ENOUGH TO OVERCOME PRESSURE PROVIDED BY PUMPS IN #2 TANK. WITH THE #3 AFT PUMP OUT (MAINT DISCOVERED POPPED CIRCUIT BREAKER IN LOWER ELECTRONICS) AND XFEED MANIFOLD VALVES OPEN; ALL ENGS WERE BEING FED FROM #2 TANK. WHEN WE TRIED TO CORRECT FUEL IMBAL; THE 1 PUMP WORKING IN #3 TANK (FORWARD) SHOULD HAVE PROVIDED SUFFICIENT PRESSURE TO FEED BOTH ENGS #3 AND #4. IN FACT; AT FL350; THE GRAVITY FEED TO THE ENG DRIVEN PUMPS SHOULD BE SUFFICIENT. THIS WAS NOT THE CASE; HOWEVER; AND THE ENG FLAMED OUT. SUPPLEMENTAL INFORMATION FROM ACN 252000: I SUSPECTED A STUCK FUEL GAUGE; EVEN THOUGH IT TESTED OK. WE THEN SET UP THE FUEL PANEL USING THE OVERRIDE/JETTISON SYS AND THE #3 FUEL GAUGE QUANTITY STARTED TO DECREASE. THE GAUGE WAS OK. THE PUMP WAS BAD. IN RETROSPECT; I FEEL WE DID THE PROPER THING BY TROUBLE-SHOOTING OUR FUEL PANEL WHILE STILL CLOSE TO NRT. HOWEVER; HAD WE USED THE JETTISON/OVERRIDE SYS FIRST; WE WOULD NOT HAVE FLAMED OUT THE ENG. WE WOULD HAVE DISCOVERED THE QUANTITY GAUGE WAS WORKING AND FOR SOME REASON THE #3 FORWARD LOW PRESSURE LIGHT WAS NOT WORKING. I'M RELUCTANT TO USE ALTERNATE PROCS UNTIL THE PRIMARY PROCS DON'T WORK.