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The shaft for the dumbwaiter consists of stud walls. Build two stud walls for the main floor and use the corner of the closet for the other two (Image 1).
Build the walls according to the specs that came with the dumbwaiter. Build the wall with 2x4s and secure it with nails.
The second wall will be framed out with the space for the dumbwaiter door (Image 2).
Use a masonry bit to drill pilot holes for the bottom plate since there is ceramic tile on the floor (Image 1).
Add braces between the studs every 36" for additional support on the sidewalls for the guide rails (Image 2).