Defining the goal function in an optimization problem
If you know the minimum and maximum values for $a,b,c$, you can use a weighted linear combination.  e.g., if $a,b,c \in [0,1000]$ (they are all between 0 and 1000), then you can use the objective function

$$\Phi(a,b,c) = 10^6 a + 10^3 b + c.$$

Minimizing this function is guaranteed to be optimal under your partial order.

Sometimes, even when you don't have upper and lower bounds, you can still use an objective function like the one above, as a heuristic.  It's not guaranteed to give optimal results, but it might be good enough in your particular application.