AN ACR DA20 FREIGHTER ENRTE TO YIP FAILED TO FLY THE PUBLISHED ARR AS INSTRUCTED BY ATC.

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AS WE WERE DSNDING INTO WILLOW RUN ARPT (YIP) (ABOUT 120 MI W); ZAU INSTRUCTED US TO FLY A 090 DEG HDG TO JOIN THE CRUXX 4 ARR; FWA TRANSITION. WE WERE IMC AT THE TIME AND THROUGHOUT THE DSCNT FROM FL310 DOWN TO 6000 FT MSL. THERE WERE NUMEROUS TSTM CELLS AHEAD OF US; SO I ASKED FOR A 15 DEG TURN TO THE L (N) TO CIRCUMNAV THESE STORMS. ZAU APPROVED THE TURN AND ASKED TO RPT WHEN WE WERE ABLE TO TURN BACK TO A 090 DEG HDG TO JOIN THE ARR. AFTER WE RPTED BACK ON A 090 DEG HDG; CTR INSTRUCTED US TO JOIN THE ARR. JOINING THE ARR REQUIRED US TO INTERCEPT THE 016 DEG RADIAL FROM FWA AND FLY IT OUTBOUND. THIS IS WHERE THE PROB HAPPENED. ATC BROUGHT TO OUR ATTN THAT WE FLEW THROUGH THE 016 DEG RADIAL. ATC THEN GAVE US A HDG OF 350 DEGS TO REJOIN; SO WE DID THAT AND REJOINED. AT THIS TIME; THE WORKLOAD WAS VERY HIGH (CHKLIST; ATIS; ETC) AND THE WX WAS BAD. I FEEL THAT THIS SHOULD HAVE NEVER HAPPENED! MY FO WAS THE PF; AND SHOULD HAVE THE ABILITY TO DO SOMETHING AS SIMPLE AS INTERCEPTING AND TRACKING A RADIAL. BUT IT IS MY RESPONSIBILITY TO MAKE SURE IT HAPPENS. ULTIMATELY THIS WAS A CASE OF LACK OF SUPERVISION. AFTER WE FLEW THROUGH THE 016 DEG RADIAL; I BROUGHT TO MY FO'S ATTN; AND MENTIONED TO START THE TURN BACK TO REJOIN IMMEDIATELY; AT WHICH TIME ATC ALSO NOTICED THIS AND GAVE US A HDG TO REJOIN.