الگوریتم ژنتیک چند هدفه برای حل مسائل بهینه سازی سبد سرمایه گذاری در بازار برق

The multi-objective portfolio optimization problem is not easy to solve because of (i) challenges from the complexity that arises due to conflicting objectives, (ii) high occurrence of non-dominance of solutions based on the dominance relation, and (iii) optimization solutions that often result in under-diversification. This paper experiments the use of multi-objective genetic algorithms (MOGAs), namely, the non-dominated sorting genetic algorithm II (NSGA-II), strength Pareto evolutionary algorithm II (SPEA-II) and newly proposed compressed objective genetic algorithm II (COGA-II) for solving the portfolio optimization problem for a power generation company (GenCo) faced with different trading choices. To avoid under-diversification, an additional objective to enhance the diversification benefit is proposed alongside with the three original objectives of the mean–variance–skewness (MVS) portfolio framework. The results show that MOGAs have made possible the inclusion of the fourth objective within the optimization framework that produces Pareto fronts that also cover those based on the traditional MVS framework, thereby offering better trade-off solutions while promoting investment diversification benefits for power generation companies.

The Markowitz mean–variance (MV) approach [1] is widely regarded as a ground theory in portfolio selection. This framework assumes that investors make an investment decision in asset allocation in order to maximize their utility by maximizing portfolio return and minimizing portfolio risk subject to a given budget constraint. However, assumptions underlying the MV model such as the quadratic utility function and the normal distribution of returns are often violated, both theoretically [2], [3] and [4] and empirically [5], [6] and [7]. In addition, the skewness preference theory and its relevance for applications are widely documented [8], [9], [10] and [11]. The introduction of skewness in portfolio decision-making brings about a new research direction in portfolio selection. In the mean–variance–skewness (MVS) model [12], [13] and [14], the mean and skewness of portfolio returns are to be maximized and portfolio risk is to be minimized simultaneously. From the viewpoint of optimization, a solution that simultaneously optimizes all objectives does not exist. Nevertheless a set of compromising solutions can be explored. Besides, the MVS portfolio optimization problem is not easy to solve because the objectives compete and conflict with each other. As a result, the optimal Pareto fronts seem to be non-smooth and discontinuous. In the literature, both the MV [15] and [16] and MVS frameworks [17] had been used to set up portfolio optimization problems for electricity generation companies. However, given the fact that electricity spot prices are not normally distributed but skewed, asset allocation based on the MVS framework is more suitable than the MV framework for a generation company (GenCo). In spite of the framework, the shape of the Pareto front presented in previous studies [17], for example, does not reflect the nature of a problem that has competing and conflicting objectives.1 Further, as observed in previous work [20], the number of assets included in most of portfolio optimization solutions was limited, and had greatly reduced the diversification benefit. In view of the weakness, this paper proposes to include a diversification enhancing objective into the MVS portfolio model. Therefore, a four-objective portfolio optimization problem (MVS-D) is formulated for a GenCo that produces and trades electricity in a deregulated electricity market. This inclusion adds to the complexity of the optimization problem by increasing the number of objectives from three to four. In optimization problems, an increase in the number of conflicting objectives significantly raises the difficulty in the use of an algorithm to find the optimal solution [21]. In conventional multi-objective optimization algorithms (MOOAs), when two candidate solutions are compared, solution a does not dominate solution b unless all objectives from a satisfy the domination condition. With a large number of objectives, the chance that no one solution can dominate the other is expectably high. Therefore, in order for algorithms to provide a good approximation of the true Pareto front, a large number of non-dominated solutions have to be screened using suitable techniques [22] and [23]. During the past decade, genetic algorithms (GAs) had been successfully applied for solving multi-objective portfolio optimization problems (MOPOPs) in finance subject to different constraints [24] and [25]. Their applications are also common in multi-objective optimization problems in the power systems [26], [27], [28] and [29] and other resource allocation problems [30] and [31]. However, the ability of MOGAs for solving MOPOPs with more than three objectives to be optimized has been rarely investigated. Therefore, the first objective of this paper is to explore if GAs can efficiently and reliably solve MOPOPs with a high number of objectives. The second objective is to conduct a cross-algorithm performance comparison. To achieve these objectives, two well established GAs, namely, non-dominated sorting genetic algorithm II (NSGA-II) [32] and strength Pareto evolutionary algorithm II (SPEA-II) [33], and the newly developed compressed objective genetic algorithm II (COGA-II) [34] were utilized and compared in this study. The paper is organized as follows. The proposed MOPOP is discussed in Section 2. Section 3 explains the portfolio selection problem in electricity market. A description of the three MOGAs together with the performance comparison criteria are given in Section 4. Section 5 exhibits the numerical experiments and parameter setting. The results and discussions are presented in Section 6, while Section 7 states our conclusions.

This paper proposes the use of MOGAs for solving multi-objective optimization problems in the MVS framework widely applied in finance. The problem is extended to the trading of electricity in a deregulated market by a power generation company. To overcome the potential weakness of the MVS framework where optimized solutions have a tendency to limit the scope of investment, an additional objective is proposed to increase diversification benefits. The shift from the three-objective problem to one with four objectives increased the complexity of the optimization process. To deal with this, we suggest the use of the newly proposed COGA-II, of which the results were compared to the widely accepted algorithms of NSGA-II and SPEA-II. The results suggest the superiority of performance in COGA-II in dealing with high dimensional multi-objective optimization problems in terms of not only proximity to the Pareto-optimal solutions but also diversity of its solution. In addition, COGA-II also produces solutions where the non-dominated fronts of a problem with more objectives envelope those of a problem with a lesser number of objectives. In the context of the application, solving the electricity allocation problem based on the MVS framework together with the proposed additional objective (MVS-D) is particularly useful because COGA-II can provide solutions with Pareto fronts that also cover those based on the traditional MVS framework. As a result, the approach avoided over-concentration of investment in a few trading choices. The electricity was more uniformly allocated among a larger number of trading choices that promote diversification benefit for the power generation companies.