  Ratings are frequently used to evaluate and compare subjects in various
applications, from education to healthcare, because ratings provide succinct
yet credible measures for comparing subjects. However, when multiple rating
lists are combined or considered together, subjects often have missing ratings,
because most rating lists do not rate every subject in the combined list. In
this study, we propose analyses on missing value patterns using six real-world
data sets in various applications, as well as the conditions for applicability
of imputation algorithms. Based on the special structures and properties
derived from the analyses, we propose optimization models and algorithms that
minimize the total rating discordance across rating providers to impute missing
ratings in the combined rating lists, using only the known rating information.
The total rating discordance is defined as the sum of the pairwise discordance
metric, which can be written as a quadratic function. Computational experiments
based on real-world and synthetic rating data sets show that the proposed
methods outperform the state-of-the-art general imputation methods in the
literature in terms of imputation accuracy.
