Ordinal logistic regression - How to get a single scoring function?

Hi all,

I'm running ordinal logistic regressions, with adjacent-category logits.
The dependent variable has four ordered levels.
So I have 3 logit equations, i.e. 3 linear predictors.
I'm wondering whether it is possible to get a single scoring function by aggregation of the 3 linear predictors. If yes, how to proceed to this aggregation in a rigorous way ?
Just to say, my eventual aim is to compute the Volume Under the ROC Surface (VUS), by using this single scoring function, so as to evaluate and compare my models' performance.

Thank you in advance,
Best regards.