Recommendation systems employed on the Internet aim to serve users by recommending items which will likely be of interest to them. The recommendation problem could be cast as either a rating estimation problem which aims to predict as accurately as possible for a user the rating values of items which are yet unrated by that user, or as a ranking problem which aims to find the top-k ranked items that would be of most interest to a user, which s/he has not ranked yet. In contexts where explicit item ratings of other users may not be available, the ranking prediction could be more important than the rating prediction. Most of the existing ranking-based prediction approaches consider items as having equal weights which is not always the case. Different weights of items could be regarded as a reflection of items' importance, or desirability, to users. In this paper, we propose to integrate variable item weights with a ranking-based matrix factorization model, where learning is driven by Bayesian Personalized Ranking (BPR). Two ranking-based models utilizing different-weight learning methods are proposed and the performance of both models is confirmed as being better than the standard BPR method.