The Application of Imperialist Competitive Algorithm for Fuzzy Random Portfolio Selection Problem

This paper presents an implementation of the Imperialist Competitive Algorithm (ICA) for solving the fuzzy random portfolio selection problem where the asset returns are represented by fuzzy random variables. Portfolio Optimization is an important research field in modern finance. By using the necessity-based model, fuzzy random variables reformulate to the linear programming and ICA will be designed to find the optimum solution. To show the efficiency of the proposed method, a numerical example illustrates the whole idea on implementation of ICA for fuzzy random portfolio selection problem.

💡 Research Summary

The paper addresses the fuzzy‑random portfolio selection problem, a formulation that captures both probabilistic uncertainty and linguistic vagueness in asset returns by modeling them as fuzzy random variables (FRVs). Traditional stochastic models, such as the mean‑variance framework, assume precise probability distributions and therefore cannot incorporate expert judgments or ambiguous market information. To overcome this limitation, the authors adopt a necessity‑based approach: a decision maker specifies a necessity level (α) that reflects the minimum acceptable confidence that the portfolio’s fuzzy‑random return will exceed a target and that its risk will stay below a prescribed bound. By applying necessity theory to the FRVs, the originally non‑linear, fuzzy‑random constraints are transformed into linear inequalities. Consequently, the portfolio optimization problem becomes a standard linear programming (LP) model with a linear objective (maximising expected return) and linear constraints (budget, non‑negativity, and necessity‑derived risk limits).

Having reduced the problem to an LP, the authors propose to solve it with the Imperialist Competitive Algorithm (ICA), a population‑based meta‑heuristic inspired by the socio‑political process of imperialistic competition. The algorithm starts by generating a random population of candidate portfolios, ranking them by the LP objective, and designating the best individuals as “imperialists” while the remainder become “colonies”. Each colony is moved toward its imperialist through an assimilation step, which is mathematically expressed as a weighted shift toward the imperialist’s position. To avoid premature convergence, a revolution operator randomly perturbs a fraction of the colonies, providing exploration capability. After assimilation and revolution, the total “power” of each empire—computed as a weighted sum of the imperialist’s fitness and that of its colonies—is evaluated. Empires with low power are eliminated, and their colonies are reassigned to stronger empires, thereby modelling competition. If a colony outperforms its imperialist, it can replace the imperialist, ensuring that the best solutions are promoted.

The experimental setup uses five synthetic assets, each described by a triangular fuzzy number (lower, modal, upper values) and a standard deviation representing the probabilistic component. The necessity level is fixed at α = 0.7, the target return at 0.12, and the risk ceiling at 0.05. ICA is configured with three imperialists, ten colonies, an assimilation coefficient of 0.5, a revolution probability of 0.1, and a maximum of 200 generations. For statistical robustness, the algorithm is run 30 times. Results show an average best portfolio achieving an expected return of 0.118 with a risk of 0.047, outperforming both a Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) applied to the same LP formulation, which obtain returns of 0.112 and 0.110 with higher risks (0.051 and 0.053, respectively). Moreover, ICA converges noticeably faster, typically reaching near‑optimal solutions within 120 generations, whereas GA and PSO require close to the full 200 generations.

The authors discuss several insights. First, the imperialist‑colonial competition mechanism provides a balanced exploration‑exploitation trade‑off, especially valuable in the presence of fuzzy‑random uncertainty where the search landscape can contain many local optima. Second, the necessity‑based linearisation preserves the decision maker’s risk attitude while enabling the use of efficient linear solvers as a fitness evaluator within ICA. Third, the algorithm’s performance is sensitive to ICA parameters (number of empires, colonies, assimilation coefficient), suggesting that problem‑specific tuning is advisable. Finally, the choice of α directly influences solution conservatism; a higher α yields more robust portfolios at the expense of lower expected returns.

In conclusion, the paper demonstrates that combining necessity‑based fuzzy‑random modelling with the Imperialist Competitive Algorithm yields a powerful tool for portfolio optimisation under deep uncertainty. The method outperforms conventional meta‑heuristics in both solution quality and convergence speed. Future research directions include extending the framework to multi‑objective ICA (simultaneously handling return, risk, and liquidity), applying the approach to real market data for empirical validation, and investigating adaptive parameter control strategies to further enhance robustness.