In this discussion, it's important to remember that a Type B free offered arbitration nets the player's former club a supplemental compensation draft pick but does not cost the signing club it's own pick. Thus, unlike Type A players who come with a cost over and above the contract dollars for which they sign (a surrendered draft pick), Type B players that are offered arbitration won't see reduced demand for their services. With the technicalities out of the way, let's analyze the situations for both Koji and Millwood.
Koji Uehara: 2 year/$10 million contract expired
2010 stats: 44.0 IP, 11.25 K/9, 1.02 BB/9, 2.40 FIP, 2.91 xFIP, 1.4 fWAR, 1.2 bWAR
2009 stats: 66.2 IP, 6.48 K/9, 1.62 BB/9, 3.56 FIP, 4.51 xFIP, 1.7 fWAR, 1.2 bWAR
Despite pitching barely 110 innings over two seasons, Koji earned his $10 million keep, returning 2.5-3.0 WAR (depending on your methodology). He was especially good out of the bullpen this season, proving an effective closer for the final two months of the season. Most impressive (other than his sideburns, of course)? His walk rates. He didn't walk a single batter after August 4, a span of 26.1 IP. And, oh, yeah: he struck out 35 during that time. His overall numbers weren't BABIP aided, and both his FIP and xFIP back up the notion that he can be an excellent reliever. Now, I am far from an expert on arbitration awards, but he only had 13 saves and made $5 million in 2010. That seems like a recipe for an award in the $7-$8 million range, and I think Koji is a solid bet to worth that amount out of the 'pen. Recommendation: Offer arbitration. If he accepts, fine. If you resign him and avoid arbitration, great. If he declines, take the pick.
Kevin Millwood: 5 year/$60 million contract expired
2010 stats: 190.2 IP, 6.23 K/9, 3.07 BB/9, 4.86 FIP, 4.66 xFIP, 1.3 fWAR, 0.5 bWAR
2009 stats: 198.2 IP, 5.57 K/9, 3.22 BB/9, 4.80 FIP, 4.78 xFIP, 2.4 fWAR, 3.4 bWAR
Millwood is clearly on his last legs as a Major League starting pitcher. He's been roughly the same pitcher since 2007, striking out a bit more than 6 batters per nine while walking about 3 per nine with an ERA fluctuating between solid and poor as his BABIP moves around the league average. He's not a good bet to be worth more than a win (or maybe two), meaning that he'll almost assuredly be overpaid in arbitration. He is, however, a very durable pitcher with the ability munch a lot of innings. Plus, an arbitration award would mean just a one year commitment.
I suspect that the team would very much like to have another veteran arm to complement Guthrie and Matusz; Arrieta, Bergesen and Tillman are still wild cards in terms of their performance, and Zach Britton may or may not be ready for the big league rotation. Thus, the question becomes whether or not the club can entice a similar pitcher to Millwood to sign in Baltimore for a lower cost (for clarity's sake, I'm working on the wild assumption that Cliff Lee isn't coming to Baltimore). Looking at the list of free agent pitchers, filled with injury question marks and guys seeking longer-term deals, I'm not sure they can. Recommendation: Offer arbitration, assuming an award in the $10-$12 million range. Ideally, you'd sign a player like Millwood for about $6 million, so if he accepts, you've overpaid for stability, but haven't locked yourself into a long-term deal. If you resign him and avoid arbitration, you've still overpaid, but haven't locked yourself into a long-term deal. And if he declines because he doesn't want to return to Baltimore and he's willing to accept less money elsewhere, take the pick.
I realize my Millwood recommendation is both expensive and controversial. Disagree with my conclusion? Let me know why.