Date: 201605
Local time of day: 0601-1200
Location: BFI.Airport, WA
Flight conditions: VMC
Light: Daylight
Aircraft: Commercial Fixed Wing
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Flight phase: Final Approach
Airspace: Class E S46
ATC / advisory: Tower BFI
Second aircraft: Small Aircraft
Reporter organization: Air Carrier
Reporter function: Captain; Pilot Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP)
Human factors: Fatigue
Second reporter function: Pilot Not Flying; First Officer
Anomaly: ATC Issue All Types; Conflict NMAC; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Automation Aircraft RA; Automation Aircraft TA; Person Air Traffic Control
When detected: In-flight
Result: General None Reported / Taken
Contributing factors: Human Factors; Airspace Structure
Primary problem: Airspace Structure

Narrative: We were on the 3rd leg of our trip flying into Boeing Field. We were on the LOC for [13R] and approximately 15 miles from the runway. We were told to descend to 2600 ft then re-cleared to 3500 ft due to VFR traffic below us. We spotted two aircraft below us at our 2 o'clock and I said that was a close one as they passed near each other. Shortly after that we were re-cleared to 2600 ft. During our descent we got another traffic call from Approach and we saw a red plane below us at 10-11 o'clock. At around 2600 ft Approach said VFR traffic 12 O'clock and a mile; switch to Tower. I tried to reference him on our VSI/TCAS; but I soon realized trying to judge distance on that was useless. About that time we got the Traffic call so I shifted my scan to outside the aircraft. Then came the RA Climb. I was slow to process this and by the time I was ready to perform the maneuver I spotted a white Cessna going down our left side. Figuring this was our traffic I decided not to respond to the RA thinking that might create more problems for us given the busy skies. We reported the incident to Tower and continued the approach to an uneventful landing.In hind sight I am disappointed with my performance and how the event unfolded. First of all Approach put us on a collision course with another airplane and then switched us off to Tower. I think in some way I was a little numb to the situation with all of the traffic calls we had received; it was almost like flying an obstacle course. I think the biggest contributor however was a lack of sleep. We arrived in the morning at XA30 am local time. We got to the hotel at XB:30 and I went to sleep at XE:00. I slept soundly until XM:00 and got up feeling great. As the evening came upon me I knew that I was going to need some rest before my next day's flight so I laid down around XV:00 and tossed and turned until YB:30 when I got up. I may have had 2 hours sleep during that time. When I got up I took a shower and prepared for work. I felt fine and looked forward to getting this last day of flying behind us so we could get home. I flew the first leg and felt fine. On the second leg; the F/Os; could tell I was slowing down a bit with fatigue starting to catch up with me. The third leg was mine and I think I got a bit of a second wind; but as the third flight progressed though I could feel the fatigue taking its toll. In hind sight and with a fresh mind; I am disappointed with my lack of performance. The TCAS training I have received in the sim has been more than adequate and I have never had a problem performing the appropriate maneuver in response to an RA. Similarly I also realize that the aircraft that passed down our left side may not have been the one that TCAS was looking at and I should have done the Climb maneuver anyway. In the future when I get into a high density traffic situation like this one was; I am going to bring up the topic of TCAS with my FO and discuss what each of us are going to do if we get an RA.


Second narrative: [Report narrative contained no additional information.]


Synopsis: Air carrier flight crew reported a NMAC with VFR traffic while on the ILS 13R approach to BFI.  The flight crew received an RA/TA; but did not take evasive action because they had the VFR traffic in sight.