Distributional utility preference robust optimization models in multi-attribute decision making

Utility preference robust optimization (PRO) has recently been proposed to deal with optimal decision-making problems where the decision maker's (DM's) 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 random. We propose to use a random utility function to describe the DM's preference and develop distributional utility preference robust optimization (DUPRO) models when the distribution of the random utility function is ambiguous. We concentrate on data-driven problems where samples of the random parameters are obtainable but the sample size may be relatively small. In the case when the random utility functions are of piecewise linear structure, we propose a bootstrap method to construct the ambiguity set and demonstrate how the resulting DUPRO can be solved by a mixed-integer linear program. The piecewise linear structure is versatile in its ability to incorporate classical non-parametric utility assessment methods into the sample generation of a random utility function. Next, we expand the proposed DUPRO models and computational schemes to address general cases where the random utility functions are not necessarily piecewise linear. We show how the DUPRO models with piecewise linear random utility functions can serve as approximations for the DUPRO models with general random utility functions and allow us to quantify the approximation errors. Finally, we carry out some performance studies of the proposed bootstrap-based DUPRO model and report the preliminary numerical test results. This paper is the first attempt to use distributionally robust optimization methods for PRO problems.