An enhanced differential evolution based algorithm, named multiobjective differential evolution with simulated annealing algorithm (MODESA), is presented for solving multiobjective optimization problems (MOPs). The proposed algorithm utilizes the advantage of simulated annealing for guiding the algorithm to explore more regions of the search space for a better convergence to the true Paretooptimal front. In the proposed simulated annealing approach, a new acceptance probability computation function based on domination is proposed and some potential solutions are assigned a life cycle to have a priority to be selected entering the next generation. Moreover, it incorporates an efficient diversity maintenance approach, which is used to prune the obtained nondominated solutions for a good distributed Pareto front. The feasibility of the proposed algorithm is investigated on a set of five biobjective and two triobjective optimization problems and the results are compared with three other algorithms. The experimental results illustrate the effectiveness of the proposed algorithm.
Many real world problems are MOPs and it has prompted a wide research boom about the MOPs over the past few decades. A lot of multiobjective evolutionary algorithms (MOEAs) have been suggested such as these famous algorithms in [
In the present study, we emphasize on the differential evolution (DE) for solving MOPs. DE, one of the most popular evolutionary algorithms, was initially presented by Storn and Price [
Besides the DE method, in this paper, the concept of simulated annealing is utilized for controlling the acceptance of the candidate solutions. Simulated annealing, proposed by Kirkpatrick et al. [
The rest of the paper is organized as follows. In Section
We assume that the MOPs given in this paper are minimization problems, while each maximization problem can be transformed into a minimization problem. A MOP can be formally described as follows:
Suppose that there are two solutions
The proposed algorithm is an extension of MODEA algorithm proposed by Ali et al. [
In MODESA, initial population is generated in the same way as that of MODEA [
The advantage of doing this is that we can choose the better one between the solution and the opposite one as the initial solution. The procedure of initialization in the paper is given as follows. A population with
Differential Evolution (DE) presented by Storn and Price [
The mutation operation is defined as follows:
After the perturbed solution is generated, a trial solution
In the proposed MODESA, the target individual
After the mutation phase, the crossover operator is performed, and the algorithm enters the selection phase. In MODESA, we incorporate the principle of simulated annealing into the selection phase. Before introducing the proposed selection phase, we first give a concept of domination in simulated annealing.
To calculate the acceptance probability of a new solution, the amount of domination used in AMOSA [
As (
If
In this paper, given a temperature
If (
where
}// End If
The diversity maintenance mechanism adopted in the paper is presented by Kukkonen and Deb [
As mentioned before, the proposed algorithm first uses the population initialization phase to generate a population with
At the
In this paper, the parameters that need to be set a priori are listed in Table
The parameters used in the proposed algorithm.
Parameter's name  Description  Value 


The population size  100 

Number of objectives  As the test problem 

Number of variables in a solution  As the test problem 

Maximum number of generations  250 
cr  The crossover rate used in DE  0.3 

The control parameter used in DE  0.5 

Maximum temperature in simulated annealing  100 

Minimal temperature in simulated annealing 


The cooling rate in simulated annealing  0.6 

Maximum prior life cycle value of the individual  1 
Initialize the values of the parameters in Table
Generate the initial population
Generate
Select
vicinity distance based pruning method on
For (
While (
Construct a temporary population which is denoted as
For (
Randomly select three distinct individuals:
from the current individual
Select the best individual
for the next mutation operation.
Produce a trial individual
generate a random number
For each variable
}// End For
Union the two population:
If (
}// End If
Select
nondominated sorting.
For (
If (
}// End For
Apply the vicinity distance based pruning method.
}// End While
The proposed MODESA approach is a simple algorithm to be implemented. In the following, the overall computational complexity of MODESA is analyzed. Basic operations and their worst case complexities are given as follows.
The initialization phase is selecting
The procedure of mutation is
The procedure of checking the domination status of the trial solution and the current solution is
The procedure of selecting
Here,
Considering that the proposed algorithm MODESA is an improvement of MODEA, it is necessary to provide the difference between MODESA and MODEA. There are two main differences between MODESA and MODEA as follows.
When comparing the new solution and the current solution, there exist two processing ways according to two different results of comparison. If the new solution dominates the current solution, the MODEA replaces the current solution with the new solution and puts the current solution into the temporary population; otherwise, the new solution is taken into the temporary population. After each solution in the population is handled, the two populations are combined. It indicates that the new solutions are all retained. However, MODESA works in three kinds of ways according to different results. If the new solution dominates the current solution, the current solution is replaced with the new solution and the current solution is removed directly, which means that if the current solution is worse than the new solution, it would not be retained. If the new solution and the current solution are incomparable, the proposed simulated annealing algorithm is adopted to determine the acceptance of the current solution; moreover, the usage of a prior life cycle enables some potential solutions preferentially to be selected in the next generation. Otherwise, the new solution is added to the temporary population and the current solution is retained in the current population.
The diversity maintenance mechanism is different. In MODEA, the diversity preserving strategy based on crowding distance in NSGAII is utilized, while in MODESA, the vicinity distance based pruning method in [
The performance of the proposed algorithm is tested on a set of five biobjective and two triobjective optimization problems. The first five problems are ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6, which were described in [
Test problems used in the paper.
Problem 

Variable bounds  Objective functions  Optimal solutions  Comments 

ZDT1  30 



Convex 







ZDT2  30 



Nonconvex 







ZDT3  30 



Convex disconnected 

 





ZDT4  10 



Nonconvex 


 





ZDT6  10 



Nonconvex 

 





DTLZ1 





 
 
 






DTLZ2 











 

In this section, three performance measures will be introduced to evaluate the effectiveness of the proposed algorithm MODESA with the other algorithms. Here, let
The concept of inverted generational distance (IGD) is proposed by Coello and Cortés in [
van Veldhuizen and Lamont [
The spread metric
In this paper, the population size and the number of generations are set to 100 and 250, respectively. For the other parameters in NSGAII and MODEA, they have remained the same with their original studies [
Tables
The mean and standard deviation IGDmetric values of nondominated solutions in 10 runs on ZDTs where
IGD  NSGAII (mean ± std)  MODEA (mean ± std)  MODESA (mean ± std) 

ZDT1 



ZDT2 



ZDT3 



ZDT4 



ZDT6 



The mean and standard deviation
Spread  NSGAII (mean ± std)  MODEA (mean ± std)  MODESA (mean ± std) 

ZDT1 



ZDT2 



ZDT3 



ZDT4 



ZDT6 



The test problems ZDT1 and ZDT2 may be the most simple problems in the series of ZDTs. The Paretooptimal front of ZDT1 is convex while that of ZDT2 is nonconvex. It is clear from Tables
Plots of the nondominated solutions with the lowest IGDmetric values found by MODESA in 10 runs in the objective space on ZDT1 where
Plots of the nondominated solutions with the lowest IGDmetric values found by MODESA in 10 runs in the objective space on ZDT2 where
The third test problem ZDT3 is different from the above two test problems. The Paretooptimal front of ZDT3 consists of five disjoint curves. From Figure
Plots of the nondominated solutions with the lowest IGDmetric values found by MODESA in 10 runs in the objective space on ZDT3 where
The fourth test problem is ZDT4 which is perhaps the most difficult problem in the five problems. It has
Plots of the nondominated solutions with the lowest IGDmetric values found by MODESA in 10 runs in the objective space on ZDT4 where
The last test problem ZDT6 is such a problem with thin density and nonuniformed spread of solutions [
Plots of the nondominated solutions with the lowest IGDmetric values found by MODESA in 10 runs in the objective space on ZDT6 where
Plots of the nondominated solutions with the lowest IGDmetric values found by MODEA in 10 runs in the objective space on ZDT6 where
In this section, to show the effectiveness of the proposed MODESA for solving problems that have more than two objectives, we choose two triobjective optimization problems DTLZ1 and DTLZ2 for comparison. The results of NSGAII, MODEA, and MODESA are given in Table
The mean and standard deviation IGDmetric and
Metric  IGD  Spread  

Problem  DTLZ1  DTLZ2  DTLZ1  DTLZ2 
NSGAII (mean 




MODEA (mean 




MODESA (mean 




Plots of the nondominated solutions with the lowest IGDmetric values found by MODEA in 10 runs in the objective space on DTLZ1 where
Plots of the nondominated solutions with the lowest IGDmetric values found by MODESA in 10 runs in the objective space on DTLZ1 where
Plots of the nondominated solutions with the lowest IGDmetric values found by MODEA in 10 runs in the objective space on DTLZ2 where
Plots of the nondominated solutions with the lowest IGDmetric values found by MODESA in 10 runs in the objective space on DTLZ2 where
As mentioned in Section
The mean and standard deviation GDmetric values of nondominated solutions in 10 runs where
GD  MODEA (mean ± std)  MODESACD (mean ± std) 

ZDT1 


ZDT2 


ZDT3 


ZDT4 


ZDT6 


DTLZ1 


DTLZ2 


In this paper, an enhanced differential evolution based algorithm with simulated annealing, named MODESA, is presented for solving MOPs. During the selection, MODESA employs a procedure of simulated annealing to control the acceptance of every candidate solution, and a concept of the prior life circle is proposed to allow some potential solutions entering the next generation to escape from the local optimum. Finally, a fast and efficient diversity maintenance mechanism is adopted. The proposed algorithm is tested on five biobjective and two triobjective optimization problems in terms of IGDmetric and spread metric, and the experimental results show that MODESA outperforms the other two algorithms on these test problems except on ZDT4. Furthermore, the effect of convergence is also tested in comparison to MODEA, the computational results show the superiority of the proposed MODESA.
Improvement can be made further in the performance of the proposed MODESA on the test problem ZDT4. In a word, MODESA is an easy and efficient method for solving MOPs. In the near future, applying MODESA for constrained MOPs and reallife application problems are our work.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Science Foundation of China (nos. 60672018 and 40774065).