THE PIC OF AN SF340 CLBS 400 FT ABOVE HIS REASSIGNED ALT WHEN THE FO FAILS TO SET THE ALERTER. THE PIC CITED THAT AS THE PIC'S DUTY AS THE PF.

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EVENT OCCURRED ON THIRD DAY OF A 4-DAY TRIP. FIRST DAY'S OVERNIGHT WAS SCHEDULED REDUCED REST (8 HRS) WHICH CAUSED NOTICEABLE FATIGUE AND DECREASED SYNERGY BTWN MYSELF AND MY FO. THE LEG BEING OPERATED WAS THE 6TH LEG OF A 6 LEG DAY. THE CREW HAD RPTED AT XA30. FOR THE PAST 3 DAYS; ALL FLTS PRECEDING INCIDENT WERE CONDUCTED IN DENSE AIR TFC AND TSTMS -- INCREASING OUR WORKLOAD MORE THAN NORMAL. THE FO (PNF) WAS RELATIVELY NEW WITH APPROX 70 HRS IN TYPE AND 5 MONTHS AT OUR COMPANY. ON CLBOUT FROM PIT; CREW WAS CLRED BY ZOB TO 'CLB AND MAINTAIN 16000 FT.' DUE TO A FULL LOAD OF PAX AND CARGO; IN ADDITION TO WARM OUTSIDE AIR TEMPS; ACFT CLB PERFORMANCE WAS LIMITED TO 700 FPM. BECAUSE OUR ACFT WAS NOT CLBING FAST ENOUGH; CTR INSTRUCTED US TO 'STOP CLB AT 12000 FT' SO HE COULD CLB AN F100 ABOVE US. PNF FO READ BACK INSTRUCTIONS; HOWEVER; DID NOT VERIFY NEW ALT WAS SET IN ALT PRESELECT BY PF PIC; AS IS OUR STANDARD COMPANY PROC. PF NEGLECTED TO SET 12000 FT INTO ALT PRESELECT. PNF FO WAS THEN DISTR BY A CALL FROM THE FLT ATTENDANT; A CHKLIST AND THE NEED TO OBTAIN A CLB TORQUE SETTING FROM PERFORMANCE CHARTS. PF FLEW THROUGH 12000 FT AND DID NOT NOTICE ALTDEV UNTIL 12300 FT. HE INITIATED AN IMMEDIATE DSCNT; HOWEVER; ACFT DID NOT BEGIN DSNDING UNTIL 12400 FT (400 FT ABOVE ASSIGNED ALT). FLT CONTINUED TO TOL WITH NO FURTHER INCIDENT.