I would probably call \$$\mathbf{Cost^{op}}\$$ \$$\mathbf{Gross}\$$, or \$$\mathbf{Reward}\$$. It's useful when you are interested in the maximum of alternate paths. What's the most you can get out of going from a to b.

Unfortunately, even though this feels like it should work to describe how much energy you can get by running a heat engine, or how much money you can make on a paper route, any non zero loops force the co distance to infinity.