Multiobjective optimization problem (MOP) is an important and challenging topic in the fields of industrial design and scientific research. Multiobjective evolutionary algorithm (MOEA) has proved to be one of the most efficient algorithms solving the multiobjective optimization. In this paper, we propose an entropybased multiobjective evolutionary algorithm with an enhanced elite mechanism (EMOEA), which improves the convergence and diversity of solution set in MOPs effectively. In this algorithm, an enhanced elite mechanism is applied to guide the direction of the evolution of the population. Specifically, it accelerates the population to approach the true Pareto front at the early stage of the evolution process. A strategy based on entropy is used to maintain the diversity of population when the population is near to the Pareto front. The proposed algorithm is executed on widely used test problems, and the simulated results show that the algorithm has better or comparative performances in convergence and diversity of solutions compared with two stateoftheart evolutionary algorithms: NSGAII, SPEA2 and the MOSADE.
Optimization problems exist in all kinds of engineering and scientific areas. When there is more than one objective in an optimization problem, it is called a multiobjective optimization problem (MOP). Since these objectives are usually in conflict with each other, the goal of solving a MOP is to find a set of compromise solutions regarding all objectives rather than a best one as in singleobjective optimization problems. The solutions of MOP, also called as the Paretooptimal solutions, are optimal in the sense that there exist no other feasible solutions which would decrease some criteria without causing the increase of at least one other criterion. Evolutionary algorithm (EA) is an optimization algorithm based on the evolution of a population. As it can search for multiple solutions in parallel, it has gained great attention from researchers. In recent years, many excellent EAs [
Generally, there are two performance measures in evaluating the Paretooptimal solutions obtained by MOEA. One is the convergence measurement, which evaluates the adjacent degree between the Pareto solutions and the true optimal front. Another one is the diversity measurement, which evaluates the distribution of solutions in the objective space. In order to achieve good performance, many excellent strategies and methods have been presented in MOEA [
Since 1948, Shannon [
In this paper, we propose a new MOEA to solve the MOP more effectively, in which an enhanced elitism makes the nondominated solutions play the better guide role and an entropybased strategy is applied to preserve the diversity of the population. We call it an entropybased multiobjective evolution algorithm with an enhanced elitism, namely, EMOEA in brief. Specifically, we employ the enhanced elitism in which only the nondominated solutions in the union population are selected as the parents to ensure that the solution set converges to the optimal front more quickly. With the algorithm going on, the number of the nondominated solutions in union population will increase gradually. In order to keep the size of the elitist population (the maximum number of the elitist population in our algorithm is set as
In this paper, we consider the following continuous multiobjective optimization problem (continuous MOP):
Let
A point
In this section, we first present an enhanced elitist mechanism; then an entropybased strategy is proposed to maintain the diversity of population. Finally, the entropybased multiobjective evolutionary algorithm with the enhanced elitism is described.
Recent researches have proved that the elite mechanism is an excellent method to speed up the convergence of evolutionary algorithm. The population can produce a good offspring population through the elitism’s guide role, which achieves the rapid evolution of the population. On the basis of this idea, different forms of the elite mechanism have been proposed in some EAs [
According to the elite mechanism, it is reasonable that the better the solutions chosen as the parents are, the better the offspring solutions which are produced by these parents are. Therefore, in order to enhance the guide of the elitism in our algorithm, we just only select all the nondomination solutions as the parents of the next iteration instead of a certain number of relatively good solutions, which may include dominated solutions because the number of nondominated solutions in the union population is less than
In the enhanced elitist mechanism, the number of the nondominated solutions in union population will gradually increase with the algorithm going on. For keeping the maximum size of the elitist population
For ease of operations, we order the nondominated solutions by one objective, and then, the region taking each solution as the center is defined as
In light of the Shannon information theory [
The schematic diagram for computing the entropy of a region is shown in Figure
Computation entropy of a grid.
Selecting the worst solution in the grid.
Combing the basic evolutionary algorithm and the tradition of the method producing offspring (crossover and mutation) in genetic algorithm, we proposed the entropybased multiobjective evolutionary algorithm with an enhanced elite mechanism (EMOEA). The main steps are shown in the following.
We have the following:
Generate an initial population
Copy all the nondominated solutions in
If
If
Execute recombination and mutation operators to the
Output the current elitist population
In this section, a large number of experiments are conducted to test the performance of EMOEA on the biobjective and the 3objective problems. Specifically, our algorithm is compared with other advanced MOEAs: NSGAII and SPEA2 which have the different strategy of constructing the elitist population. And then, the comparisons of the proposed EMOEA and the MOSADE are presented.
In our experiment, the biobjective problem is from ZDT series: ZDT1, ZDT2, ZDT3, and ZDT6. The 3objective problems we selected is composed of the DTLZ family of scalable test problems [
There have been several metrics proposed for measuring the performance of the Paretooptimal obtained by MOEAs. In our work, we choose the GD metric [
The metric SP can be used to measure the diversity of obtained solutions. Here,
The another indicator which is used usually to evaluate the diversity of the solution set is
In this part, we will compare the EMOEA proposed and two stateoftheart algorithms, NSGAII and SPEA2. All three algorithms are given realvalued decision variables. Simulated binary crossover (SBX) [
Four biobjective problems, ZDT1–3 and ZDT6, and three 3objective problems DTLZ1–3 are used. For each test problem, 30 times runs are executed. Convergence metric GD and diversity metric SP are employed to evaluate the performance. The results are given in Table
Performance comparison of the three MOEAs on the test problems.
Test problems  MOEAs  GD  SP  

Average  Std. dev  Average  Std. dev  
EMOEA 





ZDT1  NSGAII  0.000682375 

0.00837093 

SPEA2  0.000440864 

0.00673195 


 
EMOEA 





ZDT2  NSGAII  0.000512546 

0.00906621 

SPEA2  0.000358487 

0.00426378 


 
EMOEA 





ZDT3  NSGAII  0.000501522 

0.00926551 

SPEA2  0.000467296 

0.00674565 


 
EMOEA 


0.00295588 


ZDT6  NSGAII 


0.00696434 

SPEA2 





 
EMOEA 





DTLZ1  NSGAII  0.0112144 

0.0254555 

SPEA2  0.00731387 

0.0134803 


 
EMOEA 





DTLZ2  NSGAII  0.021829197 

0.0281401 

SPEA2  0.0024392 

0.0258512 


 
EMOEA 





DTLZ3  NSGAII  —  —  —  — 
SPEA2  —  —  —  — 
DTLZ serial test problems are proposed by Deb K, which can be set with the different number of objectives. In here, we choose the same settings as in [
Figures
The final solutions obtained by three MOEAs on ZDT1.
EMOEA
NSGAII
SPEA2
The final solutions obtained by three MOEAs on ZDT2.
EMOEA
NSGAII
SPEA2
The final solutions obtained by three MOEAs on ZDT3.
EMOEA
NSGAII
SPEA2
The final solutions obtained by three MOEAs on ZDT6.
EMOEA
NSGAII
SPEA2
The final solutions obtained by three MOEAs on DTLZ1.
EMOEA
NSGAII
SPEA2
The final solutions obtained by three MOEAs on DTLZ2.
EMOEA
NSGAII
SPEA2
The final solutions obtained by EMOEA on DTLZ3.
Both of the EMOEA and MOSADE make use of entropy to maintain the diversity of the solution sets. In EMOEA, we select the region with the worst distribution to keep the distribution of solutions through applying the information entropy formula. In MOSADE, the improved information entropy formula is used to update the archive, which maintains the diversity of the solution set. For these two algorithms, further experiments are conducted to compare their performances. For EMOEA, all parameters are the same as the parameters of EMOEA as described in Section
Performance comparison of the EMOEA and MOSADE on the test problems.
Test problems  GD  SP  

EMOEA  MOSADE  EMOEA  MOSADE  
ZDT1 


0.19677 






ZDT2 


0.18056 






ZDT3 


0.49072 






ZDT6 


0.36294 






DTLZ1 



0.63755 





DTLZ2 



0.61323 





DTLZ3 



0.58601 





DTLZ4 



0.93230 





DTLZ5 


0.42787 






DTLZ6 


0.43264 






DTLZ7 



0.85454 




The results obtained from Table
In this paper, a novel entropybased multiobjective evolutionary algorithm with an enhanced elite mechanism (EMOEA) is proposed. The algorithm improves the elitism and presents a new strategy based on entropy to construct the elitist population. At first we only select the nondominated solutions in the population as the elitist solutions, and when the size of the nondominated solutions exceeds the size of population, we delete worse solutions one by one to preserve the diversity of the population through the entropybased strategy. Experimental results on seven widely used popular test functions show that EMOEA can obtain the solutions set with better or comparative convergence and diversity performances compared with NSGAII, SPEA2, and MOSADE.
As eliminating one solution needs to recalculate the entropies of the crowded regions, the worst time complexity of the EMOEA is
This work is supported by the NSFC major research program (60496322, 60496327), the Beijing Natural Science Foundation (4102010). The authors thank Professor Wu Lianghong very much for his guidance to the experiment of MOSADE.