We treat uncertain linear programming problems by utilizing the notion of weighted analytic centers and notions from the area of multi-criteria decision making. In addition to many practical advantages, due to the flexibility of our approach, we are able to prove that the robust optimal solutions generated by our algorithms are at least as desirable to the decision maker as any solution generated by many other robust optimization algorithms. We then develop interactive cutting-plane algorithms for robust optimization, based on concave and quasi-concave utility functions. We present some probabilistic bounds for feasibility of robust solutions and evaluate our approach by means of computational experiments.