A B767-300 WAS OPERATED IN NON COMPLIANCE WITH THE APU DEFERRED INOP AND A CHRONIC L ENG BLEED VALVE INTERMITTENT OP.

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ACFT WAS DISPATCHED WITH APU INOP MEL 49-11-1-1 (B767 MEL). DURING PUSHBACK AND ENG START THE L ENG BLEED VALVE DID NOT OPEN. WE NOTED DURING PREFLT CHK OF LOGBOOK MULTIPLE REPEAT ENTRIES CONCERNING L ENG BLEED VALVE DIDN'T OPEN AFTER ENG START. MAINT ACTION WAS; STARTED ENG AND CHKED PRSOV FOR MOVEMENT; FINDING -- WITHIN TOLERANCES. SINCE APU BLEED WASN'T AVAILABLE AND NOW THE L BLEED OFF EICAS MESSAGE AS WELL AS LIGHTS OFF INDICATIONS WERE PRESENT; WE RAN THE CHKLIST FOR L ENG BLEED OFF. WITH MECH ON HEADSET WE CYCLED BLEED VALVE SWITCH. RAN UP ENG TO 70% N2 TWICE THEN CALLED MAINT CTL AFTER REFERRING TO MEL AND MAINT FLOW CHART. NOW MEL 36-11-1 STATED RWY LIMIT WT PENALTY FOR BLEED SOURCE PROB AND AVOID ICING CONDITIONS. WX WAS NOT A FACTOR -- VFR CLR TO/FROM DEST. WHAT CONCERNED THE CAPT AND MYSELF WAS THE WT PENALTY. AS WE DELAYED TAXI DISCUSSING WHETHER THE FLT COULD BE SAFELY CONDUCTED WITH MAINT CTL AND DISPATCH WE NOTED NEITHER THE APU OR L ENG BLEED WERE DESIGNATED AS TURNBACK ITEMS WITH MAINT. WHAT WE DIDN'T KNOW WAS HOW MUCH PENALTY WE'D TAKE FOR THE LOSS OF 2 BLEED SOURCES; AS DISPATCH WAS CHKING THE FIGURES. A THIRD RUN UP TO 70% N2 WAS ATTEMPTED AND THE L ENG BLEED VALVE OPENED -- AWAY WE WENT ON OUR FLT. LOGBOOK ENTRIES WERE MADE STATING L ENG BLEED LIGHT AND EICAS MESSAGE WERE RECEIVED AND WENT OUT AFTER MULTIPLE ATTEMPTS OF RUN-UPS. AT THE DEST THE L ENG BLEED VALVE WAS DEFERRED. WE WERE ADVISED BY A NEW DISPATCHER THAT A 47000 LB WT PENALTY WAS REQUIRED TO TAKE OFF RWY DATA. GIVEN THESE NUMBERS; AT THE FIRST DEP; A TKOF WOULD BE NEAR RWY STRUCTURAL WTS FOR THE RWY USED. BESIDE REQUESTING EXTRA FUEL FOR THE RETURN LEG; NOW THE L ENG BLEED AND APU WERE DEFERRED LEAVING ONLY 1 BLEED SOURCE FOR PRESSURIZATION.