Details

Snapshot 1: the objective function of liquid volume, which is the volume of a cylinder , where is the radius and is height

Snapshot 2: the constrained function of the mug volume, which is the sum of the outer shell and the bottom

Snapshot 3: the optimal solution, which always occurs when the mug's inner radius equals the inner depth (); the mug holds the most liquid possible with the given amount of material

Snapshot 4: the current and optimal design shown in a 2D contour plot where the liquid volume is the target function and the pink line is the constrained function, the mug material; the yellow point indicates the liquid volume for the current mug parameters and the red point is the optimal design

(For a closed container like an oil tank, the constrained function is slightly different. In that case the optimum occurs when .)