  We show that transverse coupled K\"ahler-Einstein metrics on toric Sasaki
manifolds arise as a critical point of a volume functional. As a preparation
for the proof, we re-visit the transverse moment polytopes and contact moment
polytopes under the change of Reeb vector fields. Then we apply it to a coupled
version of the volume minimization by Martelli-Sparks-Yau. This is done
assuming the Calabi-Yau condition of the K\"ahler cone, and the non-coupled
case leads to a known existence result of a transverse K\"ahler-Einstein metric
and a Sasaki-Einstein metric, but the coupled case requires an assumption
related to Minkowski sum to obtain transverse coupled K\"ahler-Einstein
metrics.
