Mean-Entropy Model of Uncertain Portfolio Selection Problem

Portfolio selection problem is a single period investment model, where an investor has to select and distribute available capital among various securities to achieve the target investment. The conventional portfolio selection models are generally obtained by probability theory based on precise historical data. However, in real situation, there are many input parameters in the securities, such as market forces of supply and demand, political factors and company performance, which are associated with non-statistical uncertainty and cannot be assessed using probability theory. Under such circumstances, it becomes essential to invite certain domain experts and to evaluate their belief degrees. To deal rationally with such human beliefs rationally, Liu (2007) introduced the uncertainty theory. In this study, a bi-objective portfolio selection model has been proposed, which maximizes the average return and minimizes the investment risk of the securities. Here, security returns are considered as uncertain and defined under the framework of uncertainty theory. In the proposed model, the average return and the risk are represented respectively by mean and entropy of the uncertain securities. The expected value and the triangular entropy of the uncertain securities are determined to represent the mean and entropy, respectively. The proposed model is solved with two different classical multi-objective solution techniques: (i) weighted sum method, and (ii) weighted metric method. Both the methods generate a single compromise solution. To generate a set of nondominated solutions, for the problem, two different multi-objective genetic algorithms (MOGAs): (i) nondominated sorting genetic algorithm II (NSGA-II) and (ii) multi-objective evolutionary algorithm based on decomposition (MOEA/D) are used. Finally, the performances of the MOGAs are analysed and compared based on different performance metrics.