By Bruce Levine-

(CBS) The White Sox struck quickly in the reliever market by signing left-hander Zach Duke to a three-year deal Tuesday afternoon worth $15 million. Duke’s coming off a career season in the bullpen for the Brewers and will be expected to be Chicago’s primary set-up man in 2015.

White Sox general manager Rick Hahn had talked to left-handed reliever Andrew Miller’s agent last week at the general manager meetings in Arizona, where Hahn found out that Miller would be shopping himself for a deal in the range of four years and $35 million to $40 million. After considering their options, the White Sox found greater value in Duke.

“Zach was one of the most desirable relievers on the market,” Hahn said. “He addresses some of our needs, but we are not counting on him alone. We will continue to be active in the market.”

Duke had a career season in 2014, striking out 74 hitters in 58 2/3 innings over 74 appearances while registering a 2.45 ERA. His pitching numbers were all in the top 10 of lefty relievers in the National League.

“We were proactive with Zach,” Hahn said. “He was on our short list of lefties, and we told him so early.”

Duke met with White Sox pitching coach Don Cooper recently to compare notes, as both reside in Nashville in the offseason.

“He is an interesting character,” Duke said of Cooper. “He was right on in his knowledge as to where I have been in my career down the muddy road. I hope I can pitch well and also impart some wisdom and knowledge to the young pitchers on the team.”

After amassing a 4.38 bullpen ERA that rated 28th in MLB last season, the White Sox probably need to add two more quality relievers to the mix, as well as right-handed starter, in order to contend in a competitive AL Central.

A free agent signing or a trade that makes sense are both avenues the team will consider.

“We are open-minded on all of our players,” Hahn said. “We will continue to look to build our club with 2015 and beyond in mind. We are happy with this signing but now  move on to the next (deal).”

The White Sox will continue to listen to teams interested in shortstop Alexei Ramirez and outfielder Dayan Viciedo. The price for Ramirez will be extremely high, as he’s coming off an All-Star season and on a fair contract.

Bruce Levine covers the Cubs and White Sox for 670 The Score and CBSChicago.com. Follow him on Twitter @MLBBruceLevine.