Multi-Objective Particle Swarm Optimization with Dynamic Crowding Entropy-Based Diversity Measure

A multi-objective particle swarm optimization with dynamic crowding entropy-based diversity measure is proposed in this paper. Firstly, the elitist strategy is used in external archive in order to improve the convergence of this algorithm. Then the new diversity strategy called dynamic crowding entropy strategy and the global optimization update strategy are used to ensure sufficient diversity and uniform distribution amongst the solution of the non-dominated fronts. The results show that the proposed algorithm is able to find better spread of solutions with the better convergence to the Pareto front and preserve diversity of Pareto optimal solutions the more efficiently.


Introduction
Evolutionary algorithms have been successfully applied to multi-objective optimization problems (EMO). In 1985, Schaffer proposed vector evaluated genetic algorithms (VEGA) [1], it is seen as the pioneering work for solving multi-objective optimization by evolutionary algorithm. Scholars from various countries developed different evolutionary multi-objective optimization algorithms after 1990.Fonseca and Fleming proposed Multi-objective Genetic Algorithm (MOGA) [2], Sriniva and Deb proposed Non-dominated Sorting Genetic Algorithm (NSGA) [3], and Horn and Nafpliotis proposed Niched Pareto Genetic Algorithm (NPGA) [4] respectively in 1993.The corresponding improved versions with keeping elitists appeared soon after, i.e. NPGA2 [5],NSGA-II [6]and so on. Scholars have proposed multi-objective algorithms based on new pattern evolutionary algorithms in recent years.Multi-objective particle swarm optimization algorithm based on dynamic crowding distance and its application [7], and multi-objective self-adaptive differential evolution with elitist archive and crowding entropy-based diversity measure [8].A multi-objective particle swarm optimization with dynamic crowding entropy strategy (MOPSO-DCE),which combines the elitist archive strategy, dynamic crowding entropy strategy and the update of global optimal strategy, is introduced in this paper.

Multi-objective optimization problem and related concepts
Owing to that minimization and maximization are essentially the same optimization problems, we only consider the minimization problem. Definition 1 [9] (Multi-objective optimization problem,MOP) A general multi-objective optimization problem with k conflicting objectives can be described as follows: where x is decision vector and X is the decision space, y is the objective vector and Y is the objective space.

Multi-Objective Particle Swarm Optimization with Dynamic Crowding Entropy Strategy
Basic PSO The velocity of particle and its new position will be assigned according to the following two equations [11]: where the superscript t denotes the th t iteration; 1 c and 2 c are positive constants, called the cognitive and social parameter respectively, 1 r and 2 r are random numbers uniformly distributed in the range (0,1). This paper adapts linearly decreasing inertia weight [12] w .

External elitist archive strategy
An external elitist archive is used to store non-dominated solutions found so far in the whole evolution process. Initially, An external elitist archiveis empty. Table 1 will give the pseudo-code of external elitist archive strategy. Where A is the set of non-dominate solutions in the current archive; x is new non-dominate solutions. Table 1 The pseudo-code of external elitist archive strategy If x is dominated by any member of A in external elitist archive discard x The size of external elitist archive increase gradually as the evolution process, and its computational complexity is As the evolution process, if there is no control of external elitist archive, computational complexity will greatly increase. Therefore, when the external archived population reaches its maximum capacity, the crowding entropy measure is used in [8]. In this paper, the dynamic crowding entropy strategy based on crowding entropy measure is proposed in next section.

Dynamic Crowding Entropy
In this paper, we present a dynamic crowding entropy strategy to remain the size of external elitist archive, which can assure the spread of solutions with the better convergence to the Pareto front and the uniformity of Pareto optimal solutions. We give the definition of crowding entropy [8]as following: where the parameters max

Update of global optimal strategy
In generally, archive strategy was used in multi-objective particle swarm optimization. Firstly, the non-dominated solutions generated in iterative process were stored in an external archive, then randomly selected a particle from the external archive as the global optimal position, this selection Advanced Engineering Forum Vol. 1 strategy lose the opportunity to get non-dominated solutions in dense regions so that the population loss diversity. We should make particles in scattered region search relative, in order to ensure that the diversity of population and the uniformly distribution of Pareto front. Therefore, we use the following strategy to update the global optimal: 1) If the crowding entropy value of each individual is infinite in the elitist external archive, which includes only a small number of boundary individual, then randomly select one as g p . 2) If the crowding entropy value of each individual is not infinite in the elitist external archive, roulette wheel selection method is used, namely the greater probability selected the individual as g p . Computation formula is:

Description of MOPSO-DCE algorithm
Step 1 Initialization populations. The maximum iteration max t ，let the size of internal population x is N ，and generate randomly the position of each particle i x in feasible decision space, let the initial velocity i v of each particle is 0, and the optimal of each individual i p , the size of external population A is M ,and [ ] = A ,then calculate the fitness of each particle.
Step 2 Update the external elitist archive A according to Table 1and calculate the crowding entropy of each particle in the external elitist archive A .
Step 3 According to the update of global optimal value strategy to update new g p ． Step 4 The velocity and position of the internal population are updated according to Eq.(2) and Eq.(3). The extreme of each individual is updated according to the domination.
Step 5 Update the external elitist archive A .
Step 6 If the maximum iteration is reached, stop and output Pareto optimal solution set, otherwise return Step3.

Experimental results
To validate our approach, we adopted the test problems [8](ZDT1,ZDT2,ZDT3,ZDT6) and the methodology normally adopted in the evolutionary multi-objective optimization literature, where the convergence metric γ and diversity metric ∆ proposed in [10] are applied.

The Results of Corresponding Comparision
In order to know how competitive the proposed approach was, it is compared with the NSGA-II, one of the most classical evolutionary multi-objective algorithms are given as follows:

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Emerging Engineering Approaches and Applications To further confirm the efficiency and feasibility of algorithm, we will compared the result of MOPSO-DCE algorithm and the six classical algorithms [8,[13][14].    As can be seen from Table 2-5, whether from the Convergence degree of Pareto-optimal set or the uniform of the Pareto-optimal set, the new algorithm is better than other algorithms. It shows that the proposed MOPSO-DCE is feasible and effective, and it can be used for solving multi-objective optimization problem.

Conclusion
This paper proposed MOPSO-DCE, which combines the elitist archive strategy, dynamic crowding entropy strategy and the update of global optimal strategy. The results show that the proposed algorithm generally outperforms in convergence and diversity performance.. The work is supported by the National Natural Science Foundation of China (60962006).