Assume that 6 people submit solutions to a puzzle, and the metric is cycles->cost->area.

The 6 submissions have as triplets (cost/cycles/area):
A) 40/67/6
B) 140/33/23
C) 80/32/33
D) 180/32/65
E) 80/32/19
F) 135/35/22

First, we rank according to the full list of tiebreakers.  Solutions with 32 cycles have ties broken by minimum cost.  The two solutions which have 80 gold and 32 cycles have the tie further broken by minimum area.

E) 80/32/19
C) 80/32/33
D) 180/32/65
B) 140/33/23
F) 135/35/22
A) 40/67/6

For scoring, we identify the best value achieved in the primary metric, which is cycles.  The best value is 32.  Everyone gets 5 * (32 / cycles) category points for their solution.

E) 5 * 32 / 32  = 5
C) 5 * 32 / 32  = 5
D) 5 * 32 / 32  = 5
B) 5 * 32 / 33  = 4.849
F) 5 * 32 / 35  = 4.571
A) 5 * 32 / 67  = 2.388

First place gets 5 rank points, and the rest get 5/6 rank points fewer than the one above them (for N entrants, this amount is 5/N instead).

E) 80/32/19   ->  5
C) 80/32/33   ->  4.167
D) 180/32/65  ->  3.333
B) 140/33/23  ->  2.5
F) 135/35/22  ->  1.667
A) 40/67/6    ->  0.833

Finally, we add these up

E) 5     + 5     = 10
C) 5     + 4.167 = 9.167
D) 5     + 3.333 = 8.333
B) 4.849 + 2.5   = 7.349
F) 4.571 + 1.667 = 6.238
A) 2.388 + 0.833 = 3.221

The winning solution will always receive 10 points.  Ties in all metrics will be both treated as the higher rank, and not have to split points.