  Utility preference robust optimization (PRO) has recently been proposed to
deal with optimal decision making problems where the decision maker's (DM)
preference over gains and losses is ambiguous. In this paper, we take a step
further to investigate the case that the DM's preference is not only ambiguous
but also potentially inconsistent or even displaying some kind of randomness.
We propose a distributionally preference robust optimization (DPRO) approach
where the DM's preference is represented by a random utility function and the
ambiguity is described by a set of probability distributions of the random
utility. An obvious advantage of the new DPRO model is that it no longer
concerns the DM's preference inconsistency. In the case when the random utility
functions are of piecewise linear structure, we propose two statistical methods
for constructing the ambiguity set: an ellipsoidal method and a bootstrap
method both of which are fundamentally based on the idea of confidence region
with the sample mean of the random parameters, and demonstrate how the
resulting DPRO can be solved by a cutting surface algorithm and an MISOCP
respectively. We also show how the DPRO models with general random utility
functions may be approximated by those with piecewise linear random utility
functions. Finally, we apply the proposed DPRO model in car manufacturing and
facility location planning and show how the random samples may be extracted by
multinomial logit method and conjoint analysis/machine learning. The paper is
the first attempt to use distributionally robust optimization methods for PRO.
