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Adaptive Multi-Objective Optimization Artificial Immune Algorithm

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Adaptive Multi-Objective Optimization Artificial Immune Algorithm IAENG International Journal of Applied Mathematics, 49:1, IJAM_49_1_03 ______________________________________________________________________________________ AMOAIA: Adaptive Multi-objective Optimization Artificial Immune Algorithm Zhongda Tian, Gang Wang, and Yi Ren individual corresponds to the antigen [10]. This antigen is Abstract—An adaptive multi-objective optimization artificial kept in the antigen group. The new clustering algorithm is immune algorithm (AMOAIA) is presented in this paper. An used to update the antigen in the antigen group constantly, and innovating sorting mechanism based on its Pareto ratio is used then a large number of Pareto optimal sets are obtained. to sort individuals in the antibody population. The selection and According to the experimental results of Coello in his cloning scheme is improved by using a neighborhood-based literature [11], it is known that MOAIA can obtain the fitness assessment. An adaptive clone selection mechanism is introduced to preserve the diversity of the antibody. A new satisfactory solution, even better when solving the non hybrid mutation operator using chaos random series for globally constrained multi-objective optimization problem. In recent optimization solution has been proposed to maintain the years, the application of artificial immune algorithm in diversity of the antibody population. A multi-objective multi-objective optimization has been widely studied. The optimization clustering algorithm based on the distribution of research shows that the artificial immune algorithm has some distributed Pareto frontiers is proposed. In addition, the advantages in maintaining the diversity of the solutions effectiveness of the proposed algorithm is verified under many [12-16]. Artificial immune algorithm and its improved difficult conditions such as local optimality, non-uniformity, algorithms showed many advantages on many optimization discontinuity, non-convexity, high-dimension, and constraints. problems, such as having a good individual diversity The comparative study shows the effectiveness of the proposed algorithm, which produces solution sets that are highly maintaining mechanism, strength of multi-modal function superiority in terms of global convergence, diversity and optimization ability, strong get rid of local extreme value, and distribution. global search ability. But it also reflects some deficiencies of the artificial immune algorithm, such as the existence of Index Terms—Multi-objective optimization, artificial premature convergence, local search ability is not strong [17, immune algorithm, Pareto sort, adaptive clone selection, chaos 18], etc. Therefore, it is necessary to improve the artificial mutation immune algorithm to make up the deficiency of the algorithm, to enhance the ability of optimization, to make it better I. INTRODUCTION applied in multi-objective optimization problems. In view of the mentioned shortcomings, this paper proposes an improved adaptive multi-objective artificial immune PTIMAZTION problem is one of the hot topics in O algorithm, called adaptive multi-objective optimization engineering practice and scientific research [1, 2]. Only one artificial immune algorithm (AMOAIA). In the AMOAIA, a objective function of the optimization problem is a single kind of improved Pareto individual ranking mechanism is objective optimization problem. The objective function is introduced. It makes the ranking is carried out in the current more than one and has to be handled at the same time is called population, but do not need to compare the many existing the multi-objective optimization problem [3-5]. For ranking methods. It only needs to solve the maximum and multi-objective optimization problems, a solution to a certain minimum values, and simplify the comparison process. At the goal may be good, but other goals may be poor, so there is a same time, the fitness evaluation function is improved. compromise solution set, known as the Pareto-optimal set [6, Meanwhile, the adaptive clone selection is introduced. These 7]. two improvements can make the algorithm to keep good Artificial immune system (AIS) is an adaptive system which diversity. The chaos mutation mechanism can ensure the is inspired by immunology. AIS simulates the immune convergence of the algorithm, and enhance the optimization function, principle and model to solve the complex problems capability of the algorithm. Simulation results on some typical [8, 9]. In the multi-objective optimization artificial immune test functions show the effectiveness of the proposed algorithm (MOAIA), the feasible solution of the optimization algorithm. problem corresponds to the antibody, and the Pareto optimal The main contents of this paper are as follows. Section 2 introduces the preliminaries of multi-objective optimization Manuscript received Apr 09, 2018; revised September 12, 2018. This problem and artificial immune algorithm. Section 3 proposes work was supported in part by the Science Research Project of Liaoning AMOAIA algorithm. The simulation results are presented in Education Department under Grant LGD2016009, and the Natural Science Section 4. The summary and prospects of the paper are Foundation of Liaoning Province of China under Grant 20170540686. Zhongda Tian is with the College of Information Science and summarized in Section 5. Engineering, Shenyang University of Technology, China, E-mail: [email protected]. Gang Wang is with the College of Information Science and Engineering, Shenyang University of Technology, China, E-mail: [email protected]. Yi Ren is with the College of Information Science and Engineering, Shenyang University of Technology, China, E-mail: [email protected]. (Advance online publication: 1 February 2019) IAENG International Journal of Applied Mathematics, 49:1, IJAM_49_1_03 ______________________________________________________________________________________ II. THE PRELIMINARIES Step 7: Judge whether the optimization algorithm satisfies the termination condition. If the condition is satisfied, then the A. Multi-objective optimization problem algorithm terminates and outputs the optimal results; The general multi-objective optimization problem consists otherwise, go to Step 3 to continue. of a set of objective functions, some related equations and inequality constraints, which can be described as the III. AMOAIA ALGORITHM following [19]. Although artificial immune algorithm has better x min F( )= ( f1 (), x f 2 ( x ),..., fk ( x )) performance than genetic algorithm, particle swarm optimization algorithm and other optimization algorithms in s. t . g(x )≤ 0, i = 1,2,⋯ , p (1) i the multi-objective optimization problems, and achieve a x ⋯ hj ( )= 0, j = 1,2, , q better accuracy and success rate [20]. But artificial immune algorithm also has premature convergence; local search x T where, = [x1 , x 2 ,..., xn ] is a n-dimensional vector in capability is not strong and other problems. It is necessary to vector space ℝn . f( x ),( i= 1,2,..., k ) is the i-th sub improve the performance of artificial immune algorithm. This i paper will improve artificial immune algorithm in the next 5 objective function. f1( x ), f 2 ( x ),..., fk ( x ) with aspects. k-dimensional vector is called target spatial of the problem, A. Artificial immune algorithm g ()x is the i-th inequality constraint, h ()x is the j-th i j In multi-objective optimization algorithm, since the equality constraint. ranking method reflects the preference relationship of Different from the single objective optimization problems individual sets, this paper introduces a Pareto frontier which have the only optimum solution, the optimum solutions selection strategy based on non-dominant individual ranking. of multi-objective optimization problems are a group of At present, it is difficult and inefficient to sort multiple targets trade-off solutions, namely Pareto optimum solutions set, and high dimensional variables based on the non-dominated non-dominated solutions or non-inferior solutions. Optimal sorting method of Pareto. In this paper, a non-dominated solutions set means there is at least one goal solution better sorting method based on Pareto coefficients is proposed. This than all the other solutions. method does not need to distinguish the effective area, Let x∗ ∈ D is Pareto optimal solutions, if ∀x ∈ D , then determine and select the individual directly, reduce the meets the conditions are as the following. complexity of the algorithm, so that the convergence rate and ∗ the degree of approximation are improved. ∩(fi ( x )≥ f i ( x )) (2) Definition 1: Pareto coefficient. i∈ I For the multi-objective optimization problem in Equation where, I= {1,2,⋯ , k } , k as the number of the objective m× k (1), if f ∈ ℝ , its Pareto sort can be expressed as next. function. And there is at least one j∈ I that makes m 1 1 2 2 k k ∗ ⋯ µ(xi )= 1 − max{min{fw i − fw p , fw i − fw p , , fw i − fw p }} (4) fj ()() x> fj x (3) p=1 In general, the optimal solution is not only one, but also an p≠ i j j j j j ⋯ optimal solution set. The goal of the multi-objective where, fwi=()/() f i − fmin f max − f min , i= 1,2, , m optimization algorithm is to construct the non-dominated set, j j j= 1,2,⋯ , k ， f=min{ f , i = 1,2,⋯ , m } ， and make the non-dominated set approach to the Pareto min i optimal solution set. j j ⋯ fmax =max{ fi , i = 1,2, , m }. B. Artificial immune algorithm Theorem 1. Let inffi ( x ) ≠ −∞ , supfi ( x ) ≠ +∞ , The standard artificial immune algorithm can be described x∈X x∈X ⋯ as follows. (i= 1, 2,
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