In the normal order of operations,
you should not be manually updating the Model. The
JSF Controllers do that automatically at Update Model time. In particular, valueChangeListeners shouldn't be used to update the Model in place of the normal update mechanisms.
ValueChangeListeners also shouldn't contain business logic. Put that in the Action method, instead. At the time the action method is invoked, the Model will be stable with all updates applied.
I can't tell why you seem to have a problem with the Managed Property. However, technically speaking a bean doesn't "call" a bean, and in particular, the proper terminology here is that "tripBean" is
injected into the target bean.
A very common mistake that JSF newbies make is that they want their JSF code to "do" things and to "get" stuff to do it with. JSF is architected around the concept of Inversion of Control (IoC), where you don't go after good things, the good things come to you without being asked for. If you're not familiar with IoC and how it interacts with POJOs, I recommend studying it. Things will make a lot more sense.