Hi, gav2013!
If you consider each choice in the ranking as a multinomial observation, with one item observed out of a possible set, you can apply a multinomial-Poisson transformation and express the model as a log-linear model, where the response is the count (1 or 0) of each possible outcome within each choice. Then, the model can then be fitted using Generalized Linear Models, which is a personality available in JMP Fit Model platform.
This approach can run into problems when you have lots of items to rank (lots > 10?) because you have to set up dummy variables to represent the presence or absence of each item in each choice and set up a factor to identify each choice. The combinations explode pretty quickly and you can challenge memory capacity. You don't mention how many items you have ranked to model.
It might be easier for you to utilize JMP's ability to run R. R has several packages that model ranks, including Plackett-Luce models.
Good luck!