  The decision forest model is a recently proposed nonparametric choice model
that is capable of representing any discrete choice model and in particular,
can be used to represent non-rational customer behavior. In this paper, we
study the problem of finding the assortment that maximizes expected revenue
under the decision forest model. This problem is of practical importance
because it allows a firm to tailor its product offerings to profitably exploit
deviations from rational customer behavior, but at the same time is challenging
due to the extremely general nature of the decision forest model. We approach
this problem from a mixed-integer optimization perspective and propose three
different formulations. We theoretically compare these formulations in
strength, and analyze when these formulations are integral in the special case
of a single tree. We propose a methodology for solving these problems at a
large-scale based on Benders decomposition, and show that the Benders
subproblem can be solved efficiently by primal-dual greedy algorithms when the
master solution is fractional for two of our formulations, and in closed form
when the master solution is binary for all of our formulations. Using
synthetically generated instances, we demonstrate the practical tractability of
our formulations and our Benders decomposition approach, and their edge over
heuristic approaches.
