A novel dynamic reference point model for preference-based evolutionary multiobjective optimization

Abstract

In the field of preference-based evolutionary multiobjective optimization, optimization algorithms are required to search for the Pareto optimal solutions preferred by the decision maker (DM). The reference point is a type of techniques that effectively describe the preferences of DM. So far, the reference point is either static or interactive with the evolutionary process. However, the existing reference point techniques do not cover all application scenarios. A novel case, i.e., the reference point changes over time due to the environment change, has not been considered. This paper focuses on the multiobjective optimization problems with dynamic preferences of the DM. First, we propose a change model of the reference point to simulate the change of the preference by the DM over time. Then, a dynamic preference-based multiobjective evolutionary algorithm framework with a clonal selection algorithm (ĝa-NSCSA) and a genetic algorithm (ĝa-NSGA-II) is designed to solve such kind of optimization problems. In addition, in terms of practical applications, the experiments on the portfolio optimization problems with the dynamic reference point model are tested. Experimental results on the benchmark problems and the practical applications show that ĝa-NSCSA exhibits better performance among the compared optimization algorithms.

Introduction

In real-world applications, there are many optimization problems whose number of objectives is at least two, and any two objectives of them are in conflict. Such problems with multiple optimization objectives are called multiobjective optimization problems (MOPs) [1, 2]. Especially, the multiobjective optimization problems with over three objectives are often called many-objective optimization problems (MaOPs) [3, 4].

Generally, MOPs have multiple optimal solutions, called Pareto optimal solutions or non-dominated solutions. Evolutionary algorithms (EAs) are a class of population-based optimization algorithms that are well suited for solving MOPs [5]. Lots of work has been devoted to proposing efficient multiobjective evolutionary algorithms (MOEAs) to find the well-distributed and well-converged optimal solutions of MOPs, such as the non-dominated sorting genetic algorithm II (NSGA-II) [6], the multiobjective EA based on decomposition (MOEA/D) [7] and their improved versions [8, 9].

Searching for the whole well-distributed solutions may not be a good idea for the decision. A huge amount of the optimal solutions with wide distribution often overwhelms the decision maker (DM) and reduces the satisfaction of the DM [10, 11]. Specifically, as the number of the objectives increases in MOPs, the number of the Pareto non-dominated solutions grows exponentially [12], exacerbating the dissatisfaction of the DM.

To deal with the issue of dissatisfaction, the DM can add his/her preference information to an MOP, requiring MOEAs only to find the Pareto optimal solutions in the regions of interest (ROIs) rather than all the optimal solutions [13]. So far, in the field of evolutionary optimization, researchers have proposed lots of multiobjective evolutionary optimization algorithms (MOEAs) to find the optimal solutions based on preference information. These methods can be separated into the following three categories.

The first category is the prior methods. The optimization algorithm has known preference information, i.e., the decision maker’s preferences are given before the evolutionary process. Among these methods, the reference point model is a popular way to achieve the goal, which adopts a point (sometimes more than one) to express the preference of the DM [14]. For example, according to the pre-defined reference point, Refs. [15] and [16] designed different types of ROIs based on the proposed r-dominance and g-dominance relations, and proposed two preference-based MOEAs to search for the optimal solutions in the ROIs, respectively. In [17], Wang et al. proposed a generation method for the weight vector to lead solutions closer to the reference point. As for the practical applications, Tang et al. [18] compared the performances of three popular reference point-based MOEAs, i.e., R-NSGA-II [19], r-NSGA-II [15], and g-NSGA-II [16], in solving reservoir operation problems. All these comparison algorithms require reference point information at the beginning stage. In [20], Yi et al. design a novel ROI according to the distance and angle from the solutions to the reference point.

The second category is the posterior methods. In this case, the optimization algorithms first find out the whole Pareto optimal solutions, then the DM set the reference point based on the Pareto optimal solutions to select the preferred solutions. It can be seen that most of the existing multiobjective evolutionary algorithms could be regarded as the posterior methods [21].

The last category is the interactive methods, which allow the DM interactively sets his/her preference during the evolutionary process according to the domain knowledge [13]. This type does not require the DM to have prior knowledge of the multiobjective optimization problem. Instead, the DM can decide his/her preference in the process of interacting with the evolutionary algorithm. In [22], Li et al. designed an interactive multiobjective optimization algorithm based on decomposition to solve the preference-based MOP. The proposed algorithm provides a consultation module that requires the DMs to provide their preferences to generate an approximate value function to model the preferences. In [23], Gong considers that the weight vector corresponding to the selected solution is also preferred by the DM, and then constructs the ROI to track preference-based MOPs. Besides, the interactive optimization algorithm is adopted to solve bolt supporting networks [24]. In [25], Hakanen et al. embed the interactive method into the reference vector guided evolutionary algorithm to search for the optimal solutions satisfying the preference of the DM.

However, in addition to these reference point models mentioned above, there is a new type of reference point model to describe the preference change by the DM, which is called dynamic reference point. In this model, the DM usually needs to make multiple decisions based on his/her preference, and usually the next decision is modified from the previous one due to the changes in the external environment.

A typical example is the portfolio optimization problem [26, 27], which is modeled by the Markowitz mean–variance model [28]. In this model, given a set of assets, there are two conflicted objectives, one is the return which should be maximized and the other one is the risk to be minimized. Generally speaking, the DM usually fine-tunes his/her preference for the next investment according to the results of the previous investment and the current investment environment. For instance, if the finance market is booming and the previous investment was successful, the DM may move the preference towards more return and risk based on his/her previous preference. In other words, the preferences of the DM for the profit and the risk are not the same in different investments, but change over time.

The existing preference methods do not effectively handle such scenarios with dynamic change of the preferences. In detail, the reference point in the prior methods is fixed before the evolutionary process. When the reference point moves, the prior methods cannot search for the optimal solutions in the moved ROI. The posterior methods catch all the Pareto optimal solutions and select the preferred solutions for the DM, which leads to low accuracy and a waste of computing resources. Interactive methods seem to be related to the change of the preference due to the interaction between the optimization algorithm and the DM, but its purpose is to find the optimal solution most preferred by the DM. When the DM is satisfied with a solution, the decision maker can terminate the algorithm immediately [29]. However, the dynamic reference point model focuses on the change of the preference, which requires the algorithm to be able to track changes in preferences and find the Pareto optimal solution under each preference as fast as possible. In other words, instead of helping the DM find the optimal solution that best matches the preference, the algorithm finds the Pareto optimal solutions under each preference when solving the MOP with the dynamic reference point.

Based on the above descriptions, it can be found that the reference point changes dynamically, and such optimization problems are transformed into a kind of dynamic optimization problems. In each environment, the Pareto optimal solutions should be found by the algorithms. For convenience, such problems are defined as DR-MOPs in this paper, where “DR" represents the dynamic reference points.

It can be seen that the core of the dynamic reference point model is that the preference region is dynamically changed, and therefore DR-MOPs belong to a class of dynamic multiobjective optimization problems. Both of these two problems consider that the optimal solutions in the adjacent two environments are different, and the optimization algorithm is required to find all the optimal solutions in the new environment as many as possible. However, compared with the general dynamic multiobjective optimization problems, the proposed dynamic reference point model is more concerned with the dynamic change of the reference point, and requires the algorithm to track the change of ROI.

In this paper, first, to simulate the movement behavior of the preference by the DM, a change model of the reference point is proposed which is inspired by the moving peaks problem (MPB) [30] in the field of evolutionary dynamic optimization. The change direction is related to the relationship between the current reference point and the optimal solutions. Second, a dynamic preference-based evolutionary framework is designed with a clonal selection algorithm and a genetic algorithm, called ĝa-NSCSA and ĝa-NSGA-II, respectively, to solve DR-MOPs. Experimental results on the benchmark problems demonstrate ĝa-NSCSA performs better than the other compared algorithms. At last, we test the dynamic change model and the algorithms on a practical application, i.e., portfolio optimization problems, and experiment results show superior performance of the proposed framework.

The remainder of this paper is organized as follows. Section 2 describes the definition of MOPs and related work. Section 3 provides a full description of the change model of the reference points. The proposed algorithms are presented in Sect. 4. Section 5 illustrates the performance metrics and experimental settings. The performances of the different algorithms are compared in Sect. 6. Section 7 show the experiment details on the portfolio optimization problems, which is a practical application in the finance market. Finally, a brief conclusion is given in Sect. 8.

Related work

Definition of multiobjective optimization problems

In this study, without loss of generality, a multiobjective optimization problem (MOP) is considered, minimizing each objective and represented as,

$$\begin{aligned} \begin{aligned} \hbox {Minimize}~F({\varvec{x}})=(f_1 ({\varvec{x}}),\ldots ,f_M ({\varvec{x}}) )^T,\\ \hbox {Subject~to}\left\{ \begin{aligned} h_i ({\varvec{x}})=0&,&i=1,\ldots ,n_p\\ g_j ({\varvec{x}})\le 0&,&j=1,\ldots ,n_q\\ {\varvec{x}}\in {\Omega }^D&\\ \end{aligned}\ , \right. \end{aligned} \end{aligned}$$

(1)

where $f_i({\varvec{x}})$ is the ith objective of ${\varvec{x}}$, which is minimized by MOEAs, M is the number of objectives, $h_i({\varvec{x}})=0$ and $g_j({\varvec{x}}) \le 0$ are the equality and inequality constraints of the MOPs, and the corresponding numbers of equality and inequality constraints are denoted as $n_p$ and $n_q$, respectively. ${\Omega }^D$ defines the entire decision space with D dimensions.

In the field of multiobjective optimization, several frequently used concepts are given as follows.

• Pareto dominance: Given two solutions ${\varvec{x}}$ and ${\varvec{y}}$, if all the objective values of ${\varvec{x}}$ are not worse than those of ${\varvec{y}}$ and at least one objective value of ${\varvec{x}}$ is better than that of ${\varvec{y}}$, it is said that ${\varvec{x}}$ Pareto dominates ${\varvec{y}}$.

• Pareto non-dominated solutions: Given an MOP, if the solution ${\varvec{x}}$ cannot be Pareto dominated by all the other solutions in the decision space, the solution ${\varvec{x}}$ is called Pareto non-dominated solution.

• Pareto optimal set (PS): All the Pareto non-dominated solutions construct the Pareto optimal set which should be found by MOEAs.

• Pareto optimal front (PF): Pareto optimal front is composed of the objective vectors corresponding to all the solutions in the Pareto optimal set.

Regions of interest (ROIs)

Given a reference point, there are several studies that set various ROIs to express the preference of the DM. A common method integrates additional comparison rules into the Pareto optimal set to select parts of the non-dominated solutions. Typically, r-dominance [15] and g-dominance [16] are two preference relations widely adopted in preference-based evolutionary multiobjective optimization. The r-dominance relation utilizes $\delta$ to control the size of the ROI according to the distance between the solutions and the reference point. The g-dominance relation considers that the solutions whose objectives are all greater or less than the given reference point are better than others.

Recently, in [31], Zou et al. proposed a novel dominance relation called Ra-dominance to determine the quality of two Pareto optimal solutions using a reference direction vector and a reference radius. In [32], Szlapczynski and Szlapczynska proposed a dominance relation called w-dominance based on weight intervals instead of the reference point, requiring no knowledge of the preferred optimal solutions. In [20], Yi et al. designed a dominance relation to construct an ROI called ar-dominance, which adopts the distance to the reference point and the preference angle to describe the ROI.

In this work, for a reference point, ĝ-dominance [33] is adopted to represent the ROI preferred by the DM. The ĝ-dominance relation is an improved version of g-dominance. Given two solutions ${\varvec{x}}$ and ${\varvec{y}}$, if ${\varvec{x}}$ ĝ-dominates ${\varvec{y}}$, one of the following two conditions should be satisfied.

• ${\varvec{x}}$ Pareto dominates ${\varvec{y}}$ on the objective of the original problem $F({\varvec{x}})=(f_1 ({\varvec{x}}),\ldots ,f_M ({\varvec{x}}) )^T$, where $f_i({\varvec{x}})$ is the i-th objective value.

• If ${\varvec{x}}$ and ${\varvec{y}}$ are Pareto non-dominated by each other, ${\varvec{x}}$ Pareto dominates ${\varvec{y}}$ on the transformed MOP $P(F({\varvec{x}}))=((|f_1({\varvec{x}})-R_1|),\ldots ,(|f_M ({\varvec{x}})-R_M| ))^T$, where $R=(R_1,\ldots , R_M)$ is the reference point.

The forming ĝ-ROI by the ĝ-dominance is very similar to g-ROI based on the g-dominance, especially when at least one non-dominated solution dominates or is dominated by the reference point. However, when the reference point and the non-dominated solutions of the MOPs are non-dominated, there still exist several solutions in the ĝ-ROI, while g-ROI does not exist in the objective space.

Figure 1 show the domination relation in the three cases. Here, the blue star represents the reference point R, and $P_1$, $P_2$, $P_3$ and $P_4$ represents four individuals in the population. The dashed lines with double arrows represent the distance to the reference point, i.e., the purple color indicates the distance on the $f_1$ objective, while the red color indicates the distance on the $f_2$ objective. In 1(a), $P_1$ and $P_2$ are non-dominated according to the ĝ-dominance, because they are Pareto non-dominated at first, and second the distances from P1 and P2 to R are still non-dominated (the distance of $P_1$ is smaller from R on the $f_2$ objective and the one of $P_2$ is smaller from R on the $f_1$ objective). As for the points $P_2$ and $P_3$, although they are Pareto non-dominated by each other, the distances of $P_3$ are greater than those of $P_2$ on both objectives, which makes $P_2$ ĝ-dominates $P_3$. The rest individual $P_4$ is Pareto dominated by $P_1$. From these points, it can be seen that the PF points whose objectives are smaller than the reference point compose the ĝ-ROI, and ĝ-ROI is similar to the g-ROI based on the g-dominance. Figure 1(b) shows the cases where the reference point dominates parts of PF. The dominance relation is the same as in Figure 1(a). The final case is that the reference points and PF are non-dominated shown in Fig. 1(c). It can be seen that according to ĝ-dominance, the optimal solution that has the same position as the reference point is the best solution, because the distances on both objectives to the reference point are 0. Other solutions on PF have a larger distance on $f_1$ and $f_2$.

In summary, the ĝ-dominance can be regarded as an improved version of the g-dominance, which can still construct an ROI when the reference point and PF are non-dominated. In this paper, we adopt ĝ-dominance to describe the preferences of decision makers mainly because in some cases the reference point and the optimal solutions are non-dominated by each other.

Fig. 1

The domination relation in different cases

Popular multiobjective evolutionary algorithms

In the field of evolutionary multiobjective optimization, the non-dominated sorting genetic algorithm II (NSGA-II) and the multiobjective immune algorithm (MOIA) are two popular MOEAs for solving MOPs. We implement the operators of these two algorithms in the proposed framework.

The framework proposed in this study is based on the non-dominated sorting genetic algorithm II (NSGA-II) [6], one of the most popular MOEAs. It mainly focuses on the selection operation. NSGA-II selects individuals based on two techniques: non-dominated sorting and crowding distance. The non-dominated sorting method classifies the individuals according to the Pareto dominance rule. In detail, the non-dominated solutions are firstly marked as the first layer and removed from the population. Then, the non-dominated solutions in the remaining population are marked as the next layer and discarded from the population. This step is repeated until all solutions are processed. It can be seen that the individuals in the layer with a smaller number are better than those with a larger number. The crowding distance is used to select individuals from the same layer. The higher the value, the better the current individual.

Recently, MOIAs have been reaching lots of attention when solving MOPs. Inspired by the clonal selection theory [34], MOIA regards the solutions as the antibodies and the problem as the antigen [35, 36]. Each solution will be cloned several times, the higher the quality the more clones [37], which leads the population towards the promising areas.

Among these MOIAs, the non-dominated neighbor immune algorithm (NNIA) [38] is a typical immune algorithm for solving MOPs. When generating the offspring, the selected parents are cloned several times, and the number of clones is proportional to the crowding distance. Then, the recombination and hypermutation operations are executed on these clones to generate the offspring. Subsequently, the parent and offspring populations are mixed together, and the non-dominated solutions in the mixed population are selected. If the number of the non-dominated solutions is not larger than the population size N, all these solutions are directly put into the population of the next generation. Otherwise, the N non-dominated solutions with the largest crowding distances are in the population of the next generation. Finally, if the termination is satisfied, the algorithm exports the non-dominated solutions to the DM, otherwise, returns to the clonal operation.

In our proposed framework, the clone operator in MOIA is implemented. Considering that the individuals with large crowding distances have more clones, the algorithm will explore a wide objective space, which is more suitable for finding the whole preferred optimal solutions.

Change model of reference points

In this section, we mainly focus on the multiobjective optimization problems with dynamic preference. Due to the external environmental change over time, the preference of the DM varies. Back in the portfolio optimization problems, the DM changes the preference according to the situation of the finance market and the previous investment results.

Therefore, this section proposes a change model of the reference point to simulate the modification of the DM’s preference. The following assumptions are adopted to formulate the change model.

• (1) Due to the change of the external environment, the preference of the DM varies.

• (2) The Pareto optimal solutions within the previous preferences do not satisfy the changed preferences.

• (3) Considering that the goal of the DM is to search for the non-dominated solutions in ROI, the movement direction of the reference point is toward the non-dominated solutions, instead of away from them.

• (4) Considering that the changes are continuous, the moving distance of the reference point is not too large between two adjacent environments.

In the field of evolutionary dynamic optimization, there are various dynamic models to describe the change over time. As one of the popular dynamic benchmarks, the moving peaks benchmark (MPB) [30] adds small values to the core parameters every several generations to change its fitness landscape. Based on the above assumptions and the changes behavior of MPB, we propose a change model of the reference point. The main idea is that, when the environment changes, a small value is added to the current reference point. The change model is shown as follows.

$$\begin{aligned} \begin{aligned} R_i(t)=R_i(t-1)+s\times v_i(t), \end{aligned} \end{aligned}$$

(2)

where $R_i(t)$ is the i-th objective value of the reference point R in the t-th environment, s is the shift length, and $v_i(t)$ is the i-th value of a vector ${\varvec{v}}$ consisting of M values. Note that the vector ${\varvec{v}}$ is a unit vector, that is, $|{\varvec{v}}|=1$. Parameter s controls the move step which is the distance from the current position to the next position, and ${\varvec{v}}$ simulates the moving direction of the reference point.

Obviously, a randomly generated unit vector ${\varvec{v}}$ does not guarantee the reference point moves toward non-dominated solutions. Therefore, the generation of ${\varvec{v}}$ should consider the relative positions of the non-dominated solutions and the reference point. Thus, after randomly generating the direction vector ${\varvec{v}}$, the following three cases are considered to set the sign of ${\varvec{v}}$ in each objective.

Case 1. Among the non-dominated solutions, there is at least one solution which Pareto dominates the reference point.

This case illustrates that the reference point may be in the feasible regions of the objective space. Considering that the MOP is a minimized optimization problem, all values of ${\varvec{v}}$ cannot be positive values; otherwise, the reference point will move in the direction away from the non-dominated set. To handle this situation, when all the components of ${\varvec{v}}$ are positive, the values are transformed into negative values. In other words, when $v_i>0$ for all $i \in \{1,\ldots , M\}$, $v_i$ is set to $-v_i$.

Fig. 2

Parts of non-dominated solutions dominates the reference point

Figure 2 shows Case 1. In an MOP with two objectives, i.e., $f_1$ and $f_2$, the red points represent the non-dominated solutions, and the blue star point R is the reference point. It can be seen that some non-dominated solutions dominate the reference point. When the environment changes, the DM will not choose a position worse than the current reference point. That is, the point R should not move far away from the non-dominated solutions, and thus the vector v is forbidden.

Case 2. The reference point Pareto dominates one or more solutions in the non-dominated set.

Here, the reference point may be an infeasible point. To prevent the reference point from moving away from the non-dominated set, if all components of ${\varvec{v}}$ are less than 0, each component of ${\varvec{v}}$ is multiplied by $-1$.

Fig. 3

The reference point dominates parts of non-dominated solutions

Similar to the previous case, Fig. 3 illustrates that the reference point R dominates several non-dominated solutions. It can be seen that the reference point R cannot be shifted by the vector ${\varvec{v}}$, due to the domination to more non-dominated solutions.

Case 3. The reference point and non-dominated solution set are Pareto-equivalent.

Under this situation, it is reasonable to move the reference point in any direction based on the preference of the DM. In this study, the randomly generated direction vector ${\varvec{v}}$ is no longer modified and directly used to simulate the movement in any direction when the environment changes.

Fig. 4

The reference point and the non-dominated solutions are non-dominated

In Fig. 4, the reference point R (blue star) and the non-dominated solutions (red points) are non-dominated by each others. In this case, any directions of movements are possible, i.e., ${\varvec{v}}_1$ and ${\varvec{v}}_2$. Since the behavior of the DM can not be predicted currently, the ROI of DM in the next environment can either be a zoom in or out of the current ROI (the direction of movement is top right or bottom left), or a movement to the other position (the direction of movement is top left or bottom right).

Generally, different optimization algorithms find different non-dominated sets in most situations, which may result in different moving behaviors of the reference point in the next environment. Those different reference points form different ROIs, making comparisons among the optimization algorithms meaningless. In order to make relatively fair comparisons in our experiments, the moving direction of the reference point is based on the relative position between the reference point and the true PF instead of the non-dominated solutions in the current environment. Notably, it is easily implemented that the proposed algorithms adopt non-dominated solutions in the current environment to modify the change direction of the reference point. In addition, considering that the reference point may move to a position far from the true PF on parts of the objectives, each objective value of the reference point is limited to its variation range on the true PF.

After defining the change model, we embed the model into the multiobjective optimization problems and proposed the DR-MOPs. In other words, the ROI of the DR-MOP is modified when the environment changes. The goal of solving DR-MOPs is to find all the preferred optimal solutions in each environment. Figure 5 shows the movement of the reference point in the first 5 environments in our experiments. The blue star R(t) represents the reference point in the tth environment and the red line represents PF. The reference point begins with R(0), and then moves to R(1) to narrow the ROI. In the next environment, due to the change of the environment, the DM prefers the solutions in another ROI by R(2). The 3rd environment may be similar to the 1-st one, so the DM moves his/her preference back to R(3), which is near to R(1). Subsequently, the DM moves the reference point into the infeasible region R(4) when the environment changes.

Fig. 5

The dynamic reference point in the experiments

Framework and algorithms

This section gives a detailed description of the algorithm framework. Specifically, we combine the archive mechanism and the clonal selection theory with NSGA-II and design two algorithms, i.e., ĝa-NSCSA and ĝa-NSGA-II, to solve DR-MOPs.

Archive-based population updating

In the DR-MOPs, finding the optimal within the ROI corresponding to the changing reference point is not an easy task. It requires that the algorithm has the ability to quickly adapt to the new preferences when the reference point changes. That is, the algorithm should find the optimal solution as fast as possible when the environment changes. However, when the preference region changes, the population generally has converged to the previously preferred non-dominated solutions, which leads to a lack of diversity to rapidly find the optimal solutions in the changed ROIs.

To improve the ability of the algorithm to find the solutions in the ROI after moving to a new environment, some mechanisms which can introduce the diversity should be embedded into the optimization algorithms. In the field of evolutionary dynamic optimization, memory strategy is a popular method widely adopted in solving dynamic optimization problems. This strategy uses an additional space to preserve the potential optimal solutions, and uses some superior ones to assist the population to quickly adapt to the new environment when the environment changes [39].

Inspired by the memory strategy, we adopt an archive to preserve the whole Pareto non-dominated solutions. The archive mechanism helps the algorithm to quickly adapt to new environments when the reference point changes. On the one hand, because the archive holds a number of well-distributed and well-converged individuals, it can increase the diversity of the population by adding new individuals from the archive to the population. On the other hand, compared with the population, some individuals in the archive are closer to or in the ROI preferred by the changed reference point, which rapidly guides the population toward the new preference.

Given a population P and an old archive A, Algorithm 1 shows the update process of the archive, similar to [40]. First, the population P and old archive A are combined together into set Q. Then, the non-dominated solutions of Q are added to NS based on the Pareto dominance rule. If the number of non-dominated solutions does not exceed the given threshold $N_a$, all the solutions in NS are considered members of the new archive NewA. Otherwise, the parts of the NS form the new archive newA, where the solutions are selected by the crowding distance. That means, the individuals in NS with the largest crowding distance make up the new archive NewA.

When the reference point moves, the archive that saves the non-dominated individuals can guide the population toward the new ROI to adapt rapidly to the new environment. To facilitate the archive, we adopt Algorithm 2 as the dynamic response strategy, which shows the update process of the population when the reference point changes.

In Algorithm 2, first, after the reference point moves to a new position, the non-dominated solutions S in the archive are obtained according to the preference-based dominance rule with the reference point R. The details of the dominance rule are presented in Sect. 4.2. Then, the number of non-dominated solutions is denoted as num. Next, from Line 3–5, the worst num individuals are removed based on preference dominance and the calculation of crowding distance. The greater the number of dominance layers and the smaller the crowding distance of the individual, the more likely it is removed from P. Finally, the remaining individuals in P and those in S are placed together to form the initial population in the new environment, whose size returns to N.

Archive-based Non-ĝ-dominated sorting

Here, the ĝ-dominance relation and the archive strategy are embedded into non-dominated sorting to separate the solutions into multiple layers. The ĝ-dominance relation leads the population to converge into the ĝ-ROI, and an archive strategy is adopted to accelerate the population convergence to the Pareto non-dominated solutions.

Algorithm 3 shows the pseudo-code of the proposed archived-based non-dominated sorting. This algorithm is modified from [33], and the difference is that the archive is considered.

In Algorithm 3, first, parameter i is initially set to 0, which is used for numbering the iterations. Second, a loop is executed to separate individuals into multiple layers. In the loop, parameter i increases by 1. Then, the Pareto non-dominated solutions in population P are put into ND. If $i=1$ and the archive is not empty, each solution in ND is checked to determine whether it is Pareto dominated by one or more solutions in A. If the solution is Pareto dominated by one or more solutions in A, it is removed from ND. Next, the individuals in ND are separated into two parts, the solutions in the ROI and out of the ROI, based on the ĝ-dominance relation. Finally, the solutions in the ĝ-ROI are preserved in the i-th layer, i.e., $FrontNo_i$, and discarded from the population P. This step is repeated until all the solutions are assigned to a layer, and the layer set is returned.

Note that the dominance relation used in the archive update process is not the same as the one in the non-dominated sorting process. When updating the archive, the Pareto dominance rule is adopted to compare the two solutions because the archive stores the entire Pareto optimal set. For the non-dominated sorting, the Pareto dominance rule is first adopted to select the non-dominated solutions, and then the ĝ-dominance rule is applied to find the optimal solutions in the ĝ-ROI.

Framework

In this section, the proposed algorithm framework that solves the multiobjective optimization problems with the dynamic reference point is illustrated in detail. Algorithm 4 presents the pseudo-code of the framework.

The process of the algorithm framework is described as follows. First, similar to most evolutionary optimization algorithms, the random generation method is adopted to generate the initial population. An empty archive is then assigned. Next, the layer numbers and crowding distances of the solutions in P are calculated by the corresponding methods. Then, a loop is conducted to search for the optimal solutions in the ROI of each reference point. Different optimization problems have different termination conditions. In this paper, we consider the termination as the consumed number of fitness evaluations reaches a maximum value.

In the loop, first, the archive is updated by Algorithm 1. If the reference point is modified, the non-ĝ-dominated solutions are preserved and the population P should be updated by Algorithm 2. Next, the offspring population is generated. Finally, population P and its offspring Q are gathered into TempSet to select the individuals for the next generation based on the non-dominated sorting layer and crowding distance.

At the end of the optimization algorithm, all the preserved non-ĝ-dominated solutions are returned as the results required by the DM.

In Line 11, the offspring population can be generated by evolutionary operators from different evolutionary algorithms. Here, we use two implementations. One is from a clonal selection algorithm [41], and the other is from NSGA-II [6]. The corresponding implementations of the framework are called ĝa-NSCSA and ĝa-NSGA-II, respectively.

ĝa-NSCSA

ĝa-NSCSA adopts the reproduction operator of the clonal selection algorithm in [41] to generate the offspring.

First, the solutions go through the clone operator to generate the clone set. The clonal selection principle considers that the number of clones is proportional to their affinity. In MOIAs [41], the crowding distance is regarded as the affinity. Therefore, the clone number nc is calculated by the following formula.

$$\begin{aligned} nc_i = \left\lceil cs * \frac{cd_i}{\sum _{j=1}^{j=N}cd_j} \right\rceil , \end{aligned}$$

(3)

where $nc_i$ is the clone number of the i-th individual, cs is the total number of clones, $cd_i$ is the crowding distance of the i-th individual, and N is the size of the population. Parameter cs is set to $1.5*N$. It is noted that the cloned individuals are sorted from the best to the worst, and when there reaches cs clones, the remaining individuals are discarded. In addition, the crowding distance of the current solution cannot be calculated if the solution is at the boundary of a certain objective space. In this paper, this value is set to $2*\mathrm{max}(cd)$, similar to [41], which is equal to twice the maximum crowding distance.

After cloning, each clone is mutated to generate offspring. In this paper, the simulated binary crossover operator is executed to make recombination between the current clone and the random individual in the population different from the current one. Next, the polynomial mutation operator mutates individuals to generate the offspring [41].

ĝa-NSGA-II

ĝa-NSGA-II utilizes the same operators as NSGA-II. First, it uses the binary tournament selection to select parents for the following recombination and mutation. Then, the simulated binary crossover operator and the polynomial mutation operator generate the offspring [6].

Metrics and experimental settings

Performance metrics

In the field of evolutionary multiobjective optimization, several indicators evaluate the performance of MOEAs. Considering that the PF of the MOPs used in our experiments is known, the inverted generational distance (IGD) [36] is adopted to compare the qualities of different algorithms. IGD is a composite metric that measures the degree of convergence, range, and distribution uniformity of the non-dominated solutions found by the algorithm. Formula (4) shows the calculation of IGD.

$$\begin{aligned} \mathrm{IGD}(P^*, P) = \frac{\sum _{v\in P^{*}} d(v, P)}{|P^*|}, \end{aligned}$$

(4)

where P is the approximate PF found by the algorithm, $P^*$ represents the true PF, and d(v, P) presents the distance from point v in the true PF to its nearest point of P.

Owing to the change of the reference point, the optimization algorithm should catch all the preferred Pareto optimal solutions in all the environments. Therefore, two changes are added to IGD to better accommodate the DR-MOPs. The first is that P and $P^*$ are the populations of the optimization algorithm and the true PF in the ROI determined by the reference point. Second, the value of IGD needs to be recorded for each reference point, and the average value of IGD at different environments is regarded as the final performance metric of the algorithm. For convenience, the modified IGD is called IGD-DR, as described in Formula (5).

$$\begin{aligned} IGD\text {-}DR = \frac{\sum _{i=1}^{Env} \mathrm{IGD}(PP_i^*, PP_i)}{Env}, \end{aligned}$$

(5)

where Env is the number of the environments, and set to 30. $PP_i^*$ and $PP_i$ represent the approximate PF obtained by the MOEA and the true PF in the ROI corresponding to the ith reference point, respectively.

It is noted that before moving the reference point by the proposed model, the information of the current reference point will be recorded into a set. In each environment, the corresponding reference point is obtained from the set to select the Pareto optimal solutions in the preferred region and facilitate the calculation of the IGD value.

Initial reference points

It is an issue of how to set the reference point at the beginning. To test different situations, each benchmark problem is tested with three initial reference points shown as follows,

$$\begin{aligned} R_{idx, i}(0) = lb_i + (ub_i - lb_i) * idx / 4, \ \ \ 1\le idx \le 3, \end{aligned}$$

(6)

where $R_{idx, i}(0)$ represents the initial value of the ith objective of the reference point $R_{idx}$, $lb_i$ and $ub_i$ are the lower and upper bounds of the ith objective of the true PF, respectively.

For example, considering the multiobjective optimization problem DTLZ1 with two objectives, the range of the true PF for each objective is [0, 0.5]. Therefore, the three reference points $R_1$, $R_2$ and $R_3$ in the initial environment are set to (0.125, 0.125), (0.25, 0.25) and (0.375, 0.375), respectively, as shown in Fig. 6. When idx is set to 1, the reference point $R_1$ may be in the infeasible region, which indicates that the DM starts with an infeasible preference. While idx is set to 3, the reference point $R_3$ may be in the feasible region and has a long distance to the true PF, simulating the case in which the DM starts with an unsuitable reference point. If idx is equal to 2, the reference point $R_2$ may be very close to the true PF, which shows that the DM has significant knowledge about the problem and sets an appropriate reference point.

Fig. 6

Different reference points in the objective space of DTLZ1

Experimental settings

The key parameters of the experiments are set as follows.

• (1) We test the experiments on 15 MOPs, i.e., DTLZ1$\sim$6 [42] and WFG1$\sim$9 [43] with three objectives.

• (2) The population size N is set to 100 as default.

• (3) The size of archive $N_a$ is set to 100.

• (4) The termination condition of all the optimization algorithms is that the number of the consumed fitness evaluations reaches the maximum number. Each benchmark problem has 30 environments, meaning the reference point moves 29 times. Each environment has the same maximum number of fitness evaluations, which is defined as $\tau$. The parameter $\tau$ is set to 20,000 for each environment in DTLZ series and 50,000 in WFG series, respectively.

• (5) When the algorithm consumes all the fitness evaluations in the current environment, it is necessary to preserve the current non-dominated solution set for calculating the performance metric. Specifically, when the total number of consumed fitness evaluations reaches $\{\tau , 2\tau , 3\tau ,\ldots , 30\tau \}$, the non-dominated solutions are saved to calculate IGD-DR according to Formula (5). Besides, 10,000 uniform optimal points are sampled as the true PF in all the experiments.

• (6) In the change model of the reference point, the shift length s is set to 0.1. In addition, the movement direction v is modified when the two rules are satisfied, preventing the reference point from moving away from the optimal solutions. The details are described in Section 3.

• (7) Considering the precise computation, the position in Formula (6) is rounded to 4 decimal places. Table 1 shows the initial reference points set in the experiments. Each value is rounded to 4 decimal places.

Table 1 Initial reference points in different experiments

Experimental results

This section illustrates comparisons between different algorithms. First, ĝa-NSCSA is compared with other preference-based multiobjective optimization algorithms. Then, the experiments are conducted among different offspring generation methods as well as with and without archive, exploring the impacts of the performance on solving DR-MOPs. The source code of this paper can be found at https://github.com/iskcal-research/dr-MOEAs.

Comparison between ĝa-NSCSA and other algorithms

In this part, we compare the performances of ĝa-NSCSA and other popular and recent preference-based MOEAs, including the multiobjective evolutionary algorithm based ar-dominance (ar-MOEA) [20], strength Pareto evolutionary algorithm 2 (SPEA2) [44], Pareto front shape estimation based evolutionary algorithm (PeEA) [45], multiobjective evolutionary algorithm based on decomposition (MOEA/D) [7], and interactive reference vector guided evolutionary algorithm (iRVEA) [25]. All these algorithms adopt the reference point to describe the preference of the DM. The difference is listed as follows.

• (1) ar-MOEA: This method is a prior method. That is, the reference point is given before the evolutionary process. ar-MOEA searches the preferred region according the distance and angle to the reference point, and the corresponding parameters $\delta _{max}$, $\delta _{min}$, $\xi _{max}$ and $\xi _{min}$ are set to 0.7, 0.3, 0.8 and 0.2 respectively, which is the same in [20].

• (2) SPEA2: This method searches for the entire PF of the MOPs, which can be regarded as the posterior method. In other words, after consuming all fitness evaluations, the solutions in the ROI preferred by the DM are selected to calculate the IGD-DR.

• (3) PeEA: Similar to SPEA2, this method is a posterior method to find the entire PF. The experiments in [45] show the performance of PeEA is better than SPEA2 on MaOPs. It is noted that there is no user-defined parameters in PeEA.

• (4) MOEA/D: It is another posterior method to catch all the individuals in PS. MOEA/D decomposes the MOP into multiple scalar optimization subproblems instead of sorting the population by Pareto dominance. Here, the Tchebycheff approach is implemented as the decomposition method. The parameter T is set to $10\%*N$, where N is the population size.

• (5) iRVEA: This method is the interactive version of RVEA and thus it is an interactive method. The dynamic reference point model can be seen as the interaction between the DM and the optimization algorithm. When the reference point is changed by the DM, each vector is re-arranged according to the interactive mechanism. In iRVEA, parameter r controls the close degree from each vector to the new reference point, and is set to 0.15 as default in [25].

For a fair comparison, the population sizes of all these algorithms are set to 100, which is the same as in ĝa-NSCSA, other parameters in each algorithm keep in the same as the original work.

Tables 2 and 3 show the comparison between the proposed algorithm ĝa-NSCSA and the other algorithms on DTLZ and WFG benchmarks, respectively. Each cell records the average or standard deviation of IGD-DR in all runs. The value in bold font indicates that the current algorithm has the best performance among these algorithms when solving the corresponding optimization problems and the reference point.

From an overall perspective on DTLZ problems, the proposed ĝa-NSCSA is better than the other compared algorithms. In terms of the mean value of IGD-DR, ĝa-NSCSA performs the best on all DTLZ problems. For example, in the case of DTLZ3 and $idx=1$, the performance of ĝa-NSCSA is 1.079E−01, while the algorithm with second place obtains 9.593E−01, the indicator values of the rest algorithms are all greater than 1. As for the standard deviations, ĝa-NSCSA performs the most stable among the compared algorithms on most DTLZ problems except DTLZ5. On DTLZ5, MOEA/D has a more stable ability than ĝa-NSCSA. Except for the proposed algorithm, it can be seen that the posterior methods (i.e., SPEA2, PeEA and MOEA/D) are better than ar-MOEA and iRVEA obtains the worst performance on the most problems.

Table 3 shows the comparison results on WFG problems. It can be seen that ĝa-NSCSA get the best mean values in most cases, only worse than MOEA/D on WFG3 with $idx=2$, as well as worse than SPEA2 on WFG1 and WFG2 with $idx=3$. For example, on WFG1 with $idx=1$, the performance of ĝa-NSCSA is $5.323E\text {-}02$, while the performances of the rest algorithms are all greater than 0.1. From the perspective of standard deviation, ĝa-NSCSA performs stable when the initial reference point is near to the true PF. When the reference point is more likely in the infeasible regions, MOEA/D performs more stable than ĝa-NSCSA.

In addition, Table 4 shows the Wilcoxon rank test results between ĝa-NSCSA and the compared algorithms at different initial reference points by KEEL [46]. The results show that ĝa-NSCSA performs better than all the other algorithms at the significance level of 0.05.

Here, DTLZ1 and WFG1 are adopted as examples, and Fig. 7 shows the IGD values in all 30 environments with different initial reference points. The horizontal coordinates indicate the index of the all environments, i.e., from 1 to 30. The vertical coordinates indicate the average IGD values over all runs in the corresponding environments. In order to show the performance gaps among these algorithms, the performance indicators are shown in a specific range, so that some of the very poor performance cannot be shown (such as lots of results of iRVEA). Overall, from Fig. 7, it can be seen that ĝa-NSCSA performs better than other algorithms in most environments. Specifically, in the first environment, the IGD value of ĝa-NSCSA is similar to those of other compared algorithms, but the gaps of IGD values between ĝa-NSCSA and other compared algorithms are large in the following environments. A typical example is shown in Fig. 7(a), where the IGD value of ĝa-NSCSA is closer to that of PeEA with the initial reference point (around 0.02), but decreases to a low value in the subsequent environments (less than 0.01), indicating that ĝa-NSCSA performs more powerfully in the dynamic environments.

Fig. 7

IGD values in different environments of various problems and initial reference points

Table 2 IGD-DR results on DTLZ series

Table 3 IGD-DR results on WFG series

Table 4 Wilcoxon rank test between ĝa-NSCSA and other compared algorithms

Impact of offspring generation methods

This part compares the performances of ĝa-NSCSA and ĝa-NSGA-II, which generates offspring by a clonal selection algorithm and a genetic algorithms, respectively. Table 5 shows experimental results of those algorithms.

In this table, idx represents the value used to generate the initial position of the reference point according to the Formula (6). The last two rows called ‘$+/=/-$’ records the number of the means and standard deviations of IGD-DR of the experiments in which the first algorithm is better/equal/worse than the second one. In detail, The penultimate row has three groups with idx of 1, 2, and 3, respectively. The last row records the data under all the problems and all the initial reference points.

From the experiments on DTLZ and WFG problems in Table 5, it can be seen that ĝa-NSCSA performs better than ĝa-NSGA-II. From an overall perspective, in terms of mean performance, ĝa-NSCSA wins 26 times out of 45 experiments, indicating a superior performance to ĝa-NSGA-II. In terms of the standard deviation, there are 26 cases where ĝa-NSCSA produces lower results than ĝa-NSGA-II, illustrating better stability. The performances of these two algorithms on different initial positions of the reference points are different. Compared with ĝa-NSGA-II, ĝa-NSCSA obtains the more powerful performances when idx is set to 2 and 3, i.e., in the cases that the reference points are more likely close to the true PF, or in the dominated regions. Specifically, when idx is set to 2, ĝa-NSCSA gets the average IGD-DR value of 8.997E−02, while ĝa-NSGA-II gets 1.012E−01 on WFG3, showing a better performance of ĝa-NSCSA.

Table 5 Performances of the algorithms with different optimizers

Impact of archive

In this part, we compare the performance of the proposed algorithm with and without the archive method. The optimization algorithm without archive is denoted as ĝa-NSCSA-None.

To test the performance of the archive mechanism, Table 6 shows the average performances of the algorithm framework with and without the archive. Overall, from Table 6, the archive can improve the performance on the DR-MOPs based on DTLZ and WFG series, through the winners of 31 cases. In details, when idx is set to 1, 2 and 3, the mean performances of ĝa-NSCSA is better than those of ĝa-NSCSA-None on 10, 11 and 10 benchmark problems, respectively. As for the standard deviation, the number of winning cases of ĝa-NSCSA is also greater than the one of ĝa-NSCSA-None, indicating that the algorithm with archive has better stability. Besides, on the WFG problems, the archive offers significant performance improvement. For example, on WFG3 and $idx=2$, ĝa-NSCSA has a mean IGD-DR of 8.997E−02 and a standard deviation of 6.094E${-}$03, while ĝa-NSCSA-None achieves a mean IGD-DR of 1.110E−01 and a standard deviation of 8.049E−03.

Table 6 Performances of the algorithms with and without archive

Here, DTLZ1 is used as an example to illustrate the performance of the archive. Figure 8 shows the gap of IGD when changing the environment for differential initial reference points, i.e., idx is set to 1, 2 and 3, respectively. In each figure, the horizontal coordinate shows the number of the environmental changes, and the vertical coordinate shows the average gap of IGD before and after moving the reference points among the 30 runs. In details, when changing to the $(i+1)$-th environment, the IGD in the ith environment is recorded, and the IGD in the $(i+1)$-th environment is also recorded after the algorithm responds to the change. The gap between the ith and $(i+1)$-th is plotted in these figures. The blue line with ‘o’ and red line with ‘x’ depict the gap of the proposed algorithm with and without archive, respectively. It can be seen that the smaller the value is, the better the population is able to adapt to the new environment. Figure 8 shows that the archive mechanism can reduce the gap when the environment changes.

Fig. 8

The gap of IGD before and after moving reference point in DTLZ1

Experiments on a practical application

As a practical application of multiobjective optimization problems, the goal of portfolio optimization problems [26, 27] is to maximize the return and minimize the risk. Generally, the decision maker modifies his/her preference for the next investment according to the preference in the previous investment and the current investment environment. It can be found that the change model of the reference point is well adapted to this problem.

In this section, the experiments on the practical application are tested. In details, the problem definition, initial reference points, parameter settings and the experiment results are illustrated.

Problem definition

Among the multiobjective optimization problems, portfolio optimization problems have received lots of attentions from the researchers as a type of practical applications [27, 47, 48]. There are a large number of models to describe the portfolio optimization problems. Among them, the Markowitz mean-variance model [28] is selected as an example to check the dynamic reference point model and the algorithm.

The Markowitz model is widely adopted in the field of finance [49]. The model considers that the DM concerns the return and the risk during the investment, and quantifies these two factors into two conflicting objectives. That is, given a set of asset information, the DM wants to maximize the return and minimize the risk of his/her investment, and the optimization algorithm is required to find out the corresponding solutions which decide the allocation ratio for all assets.

Formula (7) shows the calculation of the return and risk in the Markowitz model [28].

$$\begin{aligned} \begin{aligned} \hbox {Minimize} F(\varvec{(}x))&= (f_1({\varvec{x}}), f_2({\varvec{x}})) \\ f_1({\varvec{x}})&= -\sum _{i=1}^{D}x_i r_i \\ f_2({\varvec{x}})&= \sum _{i=1}^{D} \sum _{j=1}^{D} x_i \sigma _{ij} x_j \\ 0&\le x_i \le 1 \\&\sum _{i=1}^{D}x_i = 1, \end{aligned} \end{aligned}$$

(7)

where $x_i$ represents the allocation ratio for the ith asset, $\sigma _{ij}$ records the covariance between the ith and jth assets, $r_i$ denotes the return of the i-th asset, and D is the number of the assets (i.e., the number of the dimension). The objective $f_1$ is negatively correlated with return for the sake of simplicity, while the objective $f_2$ represents the risk of the given allocation scheme ${\varvec{x}}$. Besides, there exists some constraints in the application. The i-th dimensional value is bounded in [0, 1], and the sum of all values in a solution must be equal to 1.

Fig. 9

Different moving directions of different occasions

Typically, a DM will fine-tune his/her investment preference for the next investment based on the reference point and the investment result in the previous environment. In terms of the investment result, three main scenarios can be concluded, which correspond to the three change cases of the reference point. Figure 9 shows the different moving directions of different occasions, where the red curve represents the non-dominated solutions and the blue star R is the reference point. The vector $v_1$, $v_2$ and $v_3$ are the three moving directions.

• (1) If the decision maker fails in the previous investment, the DM may prefer the lower risk in the next investment. This leads to a shift of the reference point towards the lower return and lower risk (i.e., $v_1$).

• (2) If the DM succeeds in the previous investment, the DM may prefer a higher return in the next one to make more profits. In this case, the reference point will move towards the upper and left direction (i.e., $v_2$).

• (3) If the DM satisfies the preference in the previous environment, the DM may consider the reference point should be closer to the non-dominated solutions for a more accurate ROI, and the reference point moves towards to the non-dominated solutions (i.e., $v_3$).

Initial reference point

Since return and risk are conflicted with each other on PF, and high return is accompanied by high risk, different decision makers have different initial preference for these two factors. For example, optimistic DMs may prefer high return and high risk for getting more money, while pessimistic DMs pay much attention on the instability associated with risk and want to choose a low-risk portfolio. Besides, there is a type of DMs endorsing moderate return and moderate risk.

Based on the above description, the three types of the initial reference points are conducted, which is slightly modified from Sect. 5.2. Considering that the problem has two objectives, Formula (8) defines the initial reference point for these three initial preferences.

$$\begin{aligned} \begin{aligned} R_{idx, 1}(0)&= lb_1 + (ub_1 - lb_1) * idx / 4, \ \ \ 1\le idx \le 3, \\ R_{idx, 2}(0)&= lb_2 + (ub_2 - lb_2) * (4-idx) / 4, \ \ \ 1\le idx \le 3, \end{aligned}\nonumber \\ \end{aligned}$$

(8)

where $R_{idx, 1}(0)$ and $R_{idx, 2}(0)$ represent the initial preference in the objectives of risk and profit, respectively. Parameters $lb_1$ and $lb_2$ are the lower bounds of the true PF in both objectives, and $ub_1$ and $ub_2$ are the corresponding upper bounds.

Figure 10 shows the three cases, where idx is 1, 2 and 3 for the initial preferences of the optimistic, pessimistic and intermediate DMs, respectively. It can be seen the reference point $R_1$ with $idx = 1$ has high risk and high return, and the one $R_3$ with $idx = 3$ has low risk and low return, while the one $R_2$ with $idx = 2$ lies between the former two points.

Fig. 10

Different initial reference point in the objective space

It is worth noting that the objectives of risk and profit may not have the same scale. When the environment changes, a move step of 0.1 may not be suitable for both objectives. Therefore, in the practical application, when moving the reference point, the moving step in the i-th objective is scaled from the range [0, 1] to the range $[lb_i, ub_i]$.

Table 7 Information of the portfolio optimization problems

Table 8 IGD-DR results on portfolio optimization problems

Parameter settings

The key parameters used in the portfolio optimization problems are listed as follows.

• (1) In our experiment, 5 portfolio optimization problems, which comes from QR library [50], will be tested. The datasets are shown in Table 7. It is noted that the number of assets is the dimension D of the problem.

• (2) Similar in the previous experiments, each problem has 30 environments, that is, the reference point will move 29 times. In each environment, the maximum number of the fitness evaluation is set to 50,000. Besides, each problem is tested in 30 independent runs.

• (3) Since the true PFs of these problems are unknown, we consider the unconstrained efficient frontiers (UCEFs) [50] as the true PFs to obtain the performances. UCEFs are widely adopted as the true PF in lots of work [51].

• (4) When changing the reference point, the shift length s may be too large in these problems. In order to adapt to the problem environment, when generating the moving step (i.e., $ms = s * v$, where ms, s and v are the moving step, shift length and the velocity, respectively), each value is scaled to the range of the PF.

• (5) To make the comparison fairer, the moving direction of the reference point is based on the relationship between the reference point and the true PF instead of the non-dominated solutions in the current population, which is the same in the previous experiments.

• (6) Similar to the previous experiments, each value of the initial reference point is rounded to 4 decimal places.

Experimental results

In this part, the posterior methods from the previous experiments are selected as the comparison algorithms. In other words, ĝa-NSCSA is compared with SPEA2, PeEA and MOEA/D. The parameters of these algorithms are the same as those within the previous experiments. It is noted that SPEA2 and MOEA/D are widely adopted in solving portfolio optimization problems [26].

Table 8 shows the results of different algorithms with different initial reference points. It can be seen that ĝa-NSCSA performs the best on the most problems, and MOEA/D get the best results on the rest problems. In particular, from the perspective of the mean results, ĝa-NSCSA performs the best on Problem 2, 3 and 4, all receiving the first place with all initial reference points. On Problem 1, MOEA/D achieves the best performance when $idx=1$, while ĝa-NSCSA performs the best in the cases with the rest initial reference points. On the last type of the problem, it can be seen ĝa-NSCSA performs worse than MOEA/D. As for the standard deviation, ĝa-NSCSA performs more stable in most cases, especially when idx is set to 1 and 3, the algorithm get the minimum indicators on 4 out of 5 problems, respectively. The last row represents the Wilcoxon rank test between ĝa-NSCSA and other compared algorithms at a level of significance of 0.05. It can be seen that ĝa-NSCSA outperforms SPEA2 and PeEA. From the data of $R^+$ and $R^-$, it can be seen ĝa-NSCSA gets better results against the other three algorithms. Overall, compared to these MOEAs, ĝa-NSCSA has a strong performance and is able to find the preferred optimal solutions of the portfolio optimization problems in the environments with changing reference points.

Here, the IGD values in all environments of Problem 1 is recorded in Fig. 11. The horizontal coordinates represent the environment number and the vertical coordinates represent the recorded IGD value when consuming all the fitness evaluation in the corresponding environment. It can be seen that the performance of SPEA2 and PeEA are both worse than ĝa-NSCSA in Problem 1 with different initial reference points. The gap between ĝa-NSCSA and MOEA/D is small. When idx is set to 1 and 2, either one of ĝa-NSCSA and MOEA/D is better than the other in some environments. When idx is set to 3, ĝa-NSCSA performs better than MOEA/D in most environments.

Fig. 11

The IGD values in all environments of Problem 1

Conclusion

This paper focuses on the influence of the external environment on the DM’s preference and models this phenomenon as a type of multiobjective optimization problem with the dynamic reference point. These problems require the algorithm to rapidly adapt to changes in preferences and to find the optimal solutions within the ROI in each environment. Therefore, we firstly develop a dynamic reference point model for the DR-MOPs, simulating the time-varying properties of preferences. Besides, a framework with the archive mechanism and two types of algorithms, i.e., ĝa-NSCSA and ĝa-NSGA-II, are designed. Experimental results on the benchmarks show ĝa-NSCSA has better performance against the compared algorithms. Finally, taking the portfolio optimization problems as an example, we embed the proposed changing model into the practical application, and verify the effectiveness of the framework. In the future, we will work on the more complex change model to better simulate the real applications with different characteristics and propose an efficient optimization algorithm.