Most of the engineering problems are modeled as evolutionary multiobjective optimization problems, but they always ask for only one best solution, not a set of Pareto optimal solutions. The decision maker's subjective information plays an important role in choosing the best solution from several Pareto optimal solutions. Generally, the decision-making processing is implemented after Pareto optimality. In this paper, we attempted to incorporate the decider's subjective sense with Pareto optimality for chromosomes ranking. A new ranking method based on subjective probability theory was thus proposed in order to explore and comprehend the true nature of the chromosomes on the Pareto optimal front. The properties of the ranking rule were proven, and its transitivity was presented as well. Simulation results compared the performance of the proposed ranking approach with the Pareto-based ranking method for two multiobjective optimization cases, which demonstrated the effectiveness of the new ranking approach.

Evolutionary multiobjective optimization (EMO) is widely used in various engineering fields for analyzing the complex criteria [

The paper is organized as follows. The next section introduces a biobjective optimization case where the proposed method is initiated from. Based on the simple case, the new ranking method based on subjective probability is proposed in Section

In this part, I would like to give an example to introduce the idea of ranking chromosomes based on subjective probability theory as following.

Consider that we would wait for a line to buy a ticket, and there are two strategies: the ticket costs 1 but we have to wait for 100 minutes; or we can only spend 70 minutes waiting and cost 2. Which choice will you make if considering the absolute lost/earn amounts of money and time? On the other hand, which choice will you make if considering the lost/earn percentage of money and time?

Obviously, it is a biobjective minimization problem, and time and money are two objectives. Assume

First, assume that a decision maker’s attitude is influenced by the absolute difference between values of two strategies for each objective. That is,

Table

The decision table for the example without considering the decision maker’s current assets.

Actions | Consequences | |

Criterion 1: time ( | Criterion 2: money ( | |

Change from | −1 | |

No change from | −30 | |

Change from | −30 | |

No change from | −1 |

But it is limited to measure the decision maker’s evaluation only according to the absolute difference of two consequences. Always, the decision maker will change the evaluation as his possessive assets change. This is the basis of the proposed method to comprehend the true nature of the chromosomes on the Pareto optimal front. For example, when the decision maker holds the first strategy, his current assets are 100 minutes and 1, if he changes to the second strategy, his time asset will obtain

The decision table for the example considering the decision maker’s current assets.

Actions | Consequences | |

Criterion 1: time ( | Criterion 2: money ( | |

Change from | ||

No change from | ||

Change from | ||

No change from |

In Table

It can be seen that even though the absolute differences of time criterion of four actions are the same, the degree of belief and the degree of unbelief differ too much, when the current asset of the decision maker changes. In Table

From the above example, we acknowledge an implicit assumption that a decision maker’s degrees of belief/unbelief are always conditional upon his current situation. Thus, we define a decision maker’s subjective probability

Assume that there are two actions

If

The decision maker’s subjective probabilities,

Since different criteria are considered to be in the different priority level, we define

The same derivation also works for the situation when the decision maker stands at the point

If

If

If

This part illustrates the analysis of ranking two tradeoffs

When

Since

For the multiobjective optimization problem, one term is positive and another is negative. They would yield to the same solution. Hence, here we assume

As a conclusion, when

When

According to

When

When

if

if

if

Transitivity property: when

Since

In the case with

Similarly, assume that

Hence,

In the simulation part, as an extension, we applied the proposed approach ranking hundreds of candidates to a general EMO approach and discussed the improvement. Genetic programming (GP) and particle swarm optimization (PSO), two popular evolutionary algorithms for multiobjective optimization, are used to combine with the proposed ranking method in the following simulations.

From the previous researchers [

Assume an unknown system expressed in the form of a general regression model as

Thereby, the Pareto-based EMO methods would generate more Pareto optimal solutions in the first step. Here, we first used a popular Pareto-based EMO method, the nondominated sorting genetic algorithms (NSGA-II) described in [

We used 2000 records in the training data. All of the genetic programming algorithms defined the same simulation parameters: population size = 100, generation = 20, maximum depth of trees = 5, crossover probability = 0.7, and mutation probability = 0.3. Table

Comparison of the new GP algorithm with the proposed ranking approach and NRGAII-GP for a classical nonlinear system design problem: (a) model results of the new MOGP algorithm with the proposed ranking approach for 10 trials; (b) model results of NRGAII-GP for 10 trials.

Trials | Results of the MOGP algorithm with the proposed ranking approach | ||

Structure | Number of features | MSE | |

1 | 2 | 1.8479e-4 | |

2 | 2 | 1.8265e-4 | |

3 | 2 | 1.8143e-4 | |

4 | 2 | 1.7979e-4 | |

5 | 2 | 1.7688e-4 | |

6 | 2 | 1.7720e-4 | |

7 | 2 | 1.8062e-4 | |

8 | 2 | 1.7902e-4 | |

9 | 2 | 1.8638e-4 | |

10 | 2 | 1.7611e-4 |

Trials | Results of the MOGP algorithm with NSGA-II | ||

Structure | Number of features | MSE | |

1 | 4 | 1.8479e-4 | |

2 | 3 | 1.8265e-4 | |

3 | 3 | 1.8143e-4 | |

4 | 3 | 1.7979e-4 | |

5 | 4 | 1.7688e-4 | |

6 | 4 | 1.7720e-4 | |

7 | 3 | 1.8062e-4 | |

8 | 3 | 1.7902e-4 | |

9 | 4 | 1.8638e-4 | |

10 | 4 | 1.7611e-4 |

Through the comparison of the results of (a) and (b) in Table

Multiobjective particle swarm optimization (MOPSO) is widely used in a variety of applications, such as, neural network, with the outstanding advantages of simple implementation and low computational cost [

As we know, the test function T4 contains 21^{9} local Pareto optimal sets, as the red triangles in Figure

The Pareto optimal sets for the test function T4 include the global Pareto optimal sets (circles) and the local Pareto optimal sets (triangles).

However, when the proposed ranking method with the MOPSO is applied to solve this problem, it works very well, more importantly, there is no risk about missing the global optimal solution and only one step is implemented to obtain the final unique result. Assume that the first objective

The convergence curve in terms of the objective

A new ranking approach for evolutionary multiobjective optimization was presented. Compared with the Pareto-based EMO algorithms, the main advantage of this proposed ranking approach is to conduct the final solution which consists with the subjective information in one step without Pareto optimality. More importantly, this approach used the belief/unbelief degree to express a subjective choice of the user instead of the conventional absolute difference of two consequences, because the belief/unbelief degree takes the current asset into account instead of measuring different species on the same scale. Therefore, the proposed function can comprehend the true nature of the chromosomes ranking better than a simple weighted sum of the objective. The validity of this approach for ranking large numbers of candidate solution in EMO is demonstrated by two simulations. First simulation test is about nonlinear system design. Simulation results present that the results of the MOGP with the proposed ranking approach show higher solution accuracy and faster convergence than a popular multiobjective GP algorithm, NSGAII-GP. Furthermore, the proposed ranking approach is applied to the MOPSO to present its performance to deal with the multimodality optimization problem. After being compared with the general Paretooptimality method, it is found that the proposed ranking approach with the subjective preference information performs better to obtain the final global optimal solution.