BIL APCH CTLR FAILED TO COMPLETE VISUAL SEPARATION APPLICATION RESULTING IN OPERROR BTWN 1 DEP AND 2 ARR ACFT.

Narrative

I WAS WORKING APCH CTL AT BILLINGS ATCT. ACR X WAS INBOUND NW OF THE FIELD. ACR A WAS SW OF THE FIELD ON A VECTOR FOR THE VISUAL APCH. ACFT Z DEPARTED RWY HDG. ACFT Z IS A TWIN CESSNA. ACR B DEPARTED BEHIND THE TWIN HDG 310 DEGS INTO THE ONCOMING ACR X. I TURNED ACR X TO A HDG OF 140 DEGS; VECTOR FOR OPPOSITE DIRECTION TFC. I THEN TOLD ACFT Z ABOUT ACR X. THEIR FLT PATHS WERE CONVERGING; SO I TURNED ACFT Z TO A HDG OF 360 DEGS AND ISSUED TFC. ACR Y DEPARTED RWY 28R RWY HDG. I TURNED ACFT Z NBOUND FOR THE FASTER TFC FOLLOWING ACFT Z. ACFT Z AND ACR X WERE PARALLEL EACH OTHER; OPPOSITE DIRECTION. I STOPPED ACR Y AT 6000 FT FOR TFC. ACR X WAS STOPPED AT 7000 FT. I THEN DSNDED ACR X TO 6000 FT ANTICIPATING A TURN AND CLB FOR ACR Y. I CLRED ACR A FOR THE VISUAL APCH AND TOLD HIM OF THE RJ. I THEN TOLD HIM TO EXPEDITE HIS DSCNT. ACR Y RPTED BOTH ACR X AND ACR A IN SIGHT. I CLBED ACR Y WITHOUT TELLING HIM TO MAINTAIN VISUAL SEPARATION FROM ACR X. WITHIN 1 MIN OR SO THE CONFLICT ALERT WENT OFF BTWN ACR Y AND ACR X. WHEN ACR Y RPTED BOTH ACFT; I BELIEVED HE HAD THEM BOTH IN SIGHT AND THAT SEPARATION WAS MAINTAINED. THAT WAS UNTIL THE CONFLICT ALERT WENT OFF.