Earlier quoted context omitted.
Most optimization problems use Linear Programming if the problem has linear/continuous variables and Mixed Integer Programming if binary values (turning something on or off) is part of the solution. These are used in production pretty much everywhere. LP (linear programming) problems are extremely fast too. My company has a HUGE LP & MIP problems and the LPs are still only a few minutes to solve on nice hardware. MIP…
I don't think it's that common to solve, say, shortest paths by solving the LP/MIP formulation (unless maybe the problem at hand isn't strictly a shortest paths problem). Is it?
For something as well known as shortest path, no. One thing about LP/MIP formulations though, is that it's arguably easier to extend. Not to say there aren't tricks to modifying existing algorithms, but most of the time, there is no need to spend all that time, when adding in a few variables or constraints will do the trick for this one problem that you will probably not need to revisit for quite a while.