As we are looking for knowledge of stock future returns in portfolio optimization, we are practically faced with two principal concepts: Uncertainty and Information about variables. This paper attempts to introduce a pragmatic bi-objective investment model based on uncertainty, instead of probability space and information theory, instead of variance and other moments as a risk measure for portfolio optimization. Not only is uncertainty space expected to be more in line with investment theory, but also, applying and learning this approach seems more straightforward and practical for novice investors. The proposed model simultaneously maximizes the uncertain mean of stock returns and minimizes uncertain entropy as a measure of portfolio risk. The uncertain zigzag distribution has been used for variables to avoid the complexity of fitting distributions for data. This uncertain mean-entropy portfolio optimization (UMEPO) has been solved by three meta-heuristic methods of multi-objective optimization: NSGA-II, MOPS, and MOICA. Finally, it was observed that the optimal portfolio obtained from the proposed model has a higher return and a lower entropy as a risk measure compared to the same model in the probability space. Manuscript profile

In this paper we introduce concepts of pseudo-triangular entropy as a supplement measure of uncertainty in the uncertain portfolio optimization. We first prove that logarithm entropy and triangular entropy for uncertain variables sometimes may fail to measure the uncertainty of an uncertain variable. Then, we propose a definition of pseudo-triangular entropy as a supplement measure to characterize the uncertainty of uncertain variables and we derive its mathematical properties. We also give a formula to calculate the pseudo-triangular entropy of uncertain variables via inverse uncertainty distribution. Moreover, we use the pseudo-triangular entropy to characterize portfolio risk and establish some uncertain portfolio optimization models based on different types of entropy. A genetic algorithm (GA) is implemented in MATLAB software to solve the corresponding problem. Numerical results show that pseudo-triangular entropy as a quantifier of portfolio risk outperforms logarithm entropy and triangular entropy in the uncertain portfolio optimization. Manuscript profile