  Ben-Tal and Teboulle \cite{BTT86} introduce the concept of optimized
certainty equivalent (OCE) of an uncertain outcome as the maximum present value
of a combination of the cash to be taken out from the uncertain income at
present and the expected utility value of the remaining uncertain income. In
this paper, we consider two variations of the OCE. First, we introduce a
modified OCE by maximizing the combination of the utility of the cash and the
expected utility of the remaining uncertain income so that the combined
quantity is in a unified utility value. Second, we consider a situation where
the true utility function is unknown but it is possible to use partially
available information to construct a set of plausible utility functions. To
mitigate the risk arising from the ambiguity, we introduce a robust model where
the modified OCE is based on the worst-case utility function from the ambiguity
set. In the case when the ambiguity set of utility functions is constructed by
a Kantorovich ball centered at a nominal utility function, we show how the
modified OCE and the corresponding worst case utility function can be
identified by solving two linear programs alternatively. We also show the
robust modified OCE is statistically robust in a data-driven environment where
the underlying data are potentially contaminated. Some preliminary numerical
results are reported to demonstrate the performance of the modified OCE and the
robust modified OCE model.
