There are, I suspect, two types of sophisticated defensive metrics: the publicly available ones, and the proprietary ones designed by individual teams' analytics departments.
Baseball Info Systems, which generates the numbers in the annual Bill James Handbook, is prominent in the world of public defensive metrics. Its work has, among other things, laid out for public viewing the rationale behind the current trend toward infield shifts,
In his essay accompanying the shift stats in the 2016 Handbook, John Dewan says that major league teams shifted 13,298 times in 2014 and 17,733 times in 2015. (Those numbers represent balls put in play when the defense is in a shift, so at-bats in which the defense is in a shift early and out of it later aren't counted, and neither shifts in which the batter strikes out, walks, etc.) That's a 33 percent increase last year.
The shifts, by BIS' calculations, prevented 266 runs over all. That's a 36 percent increase over 2014's 196, and suggests that teams are becoming more efficient at shifting.
One other general point: BIS believes that the optimal number of shifts is more than double last year's total, that the upper limit is probably around 40,000 shifts. MLB isn't there yet, but if shifting continues to increase, 40,000 is possible in a few years.
OK, that;s the general stuff. On to the Twins specific shift data:
In 2014 -- with Ron Gardenhire as manager and Paul Molitor assigned to handle shifts (an assignment widely viewed as being an override of Gardenhire's preferences) -- the Twins deployed 478 shifts. Only three teams in the American League used fewer shifts.
In 2015 -- with Molitor now the manager -- the Twins used 724 shifts, sixth most in the AL. That's an increase of 246, That's a 34 percent increase.
The Handbook does not provide a team-by-team breakdown of runs saved by shifts, so I don't have a precise number for the Twins. Across the majors, teams saved 1.5 runs per 100 shifts, so if the Twins shifted at average efficiency they saved roughly 10 runs with the shifts. Ten runs sabermetrically equates to a win, so that's hardly an insignificant matter.