A MOIA with dynamic population strategy
“参考文献 A multi-objective immune algorithm with dynamic population strategy, Swarm and Evolutionary Computation 50 (2019) 100477
- In this paper, we propose a multi-objective immune algorithm with dynamic population strategy, named MOIADPS, which introduces a control strategy of dynamic population size into multi-objective immune algorithm (MOIA). This scheme helps to compensate the lack of diversity due to the clonal principle in MOIA and adequately exploits the computational resource during the evolutionary progress. (该方案有助于弥补MOIA中克隆原理导致的多样性不足，并在进化过程中充分利用计算资源) In MOIA-DPS, the status of external archive (full or not full) is used to decide the enlargement or the reduction of population size, so as to adaptively adjust the computational resource. Moreover, in order to further enhance the robustness of MOIADPS, we present an effective DE operator with two search models, called TDE, in which two search models such as rand/2/bin and rand/1/bin are alternatively exchanged according to a probability. When compared to four state-of-the-art heuristic algorithms, i.e., ISDE+, MOEA/D-GRA, AbYSS, CMPSO, and four MOIAs, i.e., IMADE, DMMO, HEIA, and AIMA, MOIA-DPS was shown to present several advantages in solving different sets of benchmark problems.
- In this paper, we propose a novel dynamic population strategy (DPS) for MOIAs (MOIADPS), in order to properly balance the diversity and the convergence. Meanwhile, a novel DE mutation operator with two search models (TDE) is designed to enhance the exploration capability and the robustness of MOIA-DPS. To summarize, the main features of MOIA-DPS are listed as follows.
- A dynamic population strategy and a cloning operator for constructing the mating population are carried out. DPS dynamically adjusts the size of mating population according to the status of the external archive. When the external archive is full, the population size is gradually decreased to save the computational resource; otherwise, the population size is gradually increased to diversify the population. After that, the superior solutions in the archive are proliferated by the cloning operator to constitute the mating population, whose size is controlled by DPS. This helps to avoid the premature convergence and to speed up the searching progress.
- A new DE mutation operator, called TDE, is executed on the mating population to enhance the robustness of MOIA-DPS. The TDE operator owns two search models, i.e., rand/1/bin and rand/2/bin. They are alternately executed and controlled by a probability parameter p. Such operator combining two search models is expected to improve the diversity of MOIA-DPS.
- Artificial immune system is a kind of heuristic algorithms imitating the information processing mechanism of biological immune system, which has found numbers of applications in handling material in industries , solving fault detection and isolation problem [39,40] and character recognition system . Especially immune algorithm has been successfully applied for MOPs and shown pretty promising performance in solving 3D terrain delayment of wireless sensor network , nonlinear interval-valued programming , design recommendation system  and scheduling problems .
- Most of immune optimization algorithms are designed by simulating the two important immune principles, i.e., clonal selection and affinity maturation by hyper-mutation. Based on the clonal selection principle, a multi-objective immune system algorithm  was proposed. In this approach, only the antibodies with highest affinity corresponding to the antigen will go for proliferation. An external memory is used to collect the found non-dominated antibodies and an adaptive grid is executed to keep a uniform distribution of the antibody population. An immune dominance clone multi-objective algorithm was designed in Ref. . This approach adopts binary string representation and uses the concept of the antibody-antibody affinity to reflect their similarity. A novel MOIA using a multiple-affinity model was presented in Ref. . This approach uses six measures for affinity assignment. Based on the specific affinity measures, immune operators such as clonal proliferation, hyper-mutation and immune suppression are performed correspondingly, which proliferate the superiors and suppress the inferiors. In Ref. , a multi-objective optimization immune algorithm using clustering (CMOIA) was reported. A clustering-based clonal selection strategy is performed in CMOIA to maintain the balance between exploration and exploitation.
- Considering the hybrid approaches for MOIAs, a hybrid immune algorithm for MOPs (HIMO) was designed by combining the advantages of Gaussian and polynomial mutations. The running of two mutations is adjusted by an adaptive switch control factor. The performance of HIMO was further enhanced in a micro-population immune multiobjection algorithm  with an efficient adaptive mutation strategy and a fine-grained selection method, and in Ref.  with a novel adaptive DE operator. Then, a double-module MOIA named DMMO  was designed by combining two evolutionary modules. The first module attempts to optimize each objective independently by using a sub-population composed with the competitive individuals in this objective, while the second one follows the traditional procedures of MOIAs to guarantee the simultaneous optimization of all objectives. More recently, a novel hybrid MOIA was presented in Ref. , which uses multiple evolutionary strategies for the cloned population. Such that, the potential weakness of certain evolutionary strategy can be avoided and the advantages of different evolutionary strategies can be combined to enhance the overall performance of MOIAs. More recently, multiple DE strategies with distinct advantages were adaptively selected for MOIA in Ref. 
- However, in the above-mentioned MOIAs, the impact of population size is seldomly studied. Generally, the population size is always fixed in MOIAs during the evolutionary process. Due to the impact of cloning operator in MOIAs, the population diversity may be harmed as only some of non-dominated solutions are cloned to speed up the convergence. Based on our preliminary study , a large population size in MOIAs may induce the waste of computational resource in one generation, while a small size may easily lead to premature convergence or stagnation on some cases. The proper allocation of computational resources at different stages is important to balance well between convergence and diversity. Thus, in this paper, we present a dynamic population strategy for MOIAs, in order to adaptively allocate the computational resources in each generation and avoid premature convergence. Compared to all the above MOIAs, the distinct features of our algorithm include the proposed DPS and the TDE operator. The superior performance of MOIA-DPS is also confirmed by our experimental studies, as provided in Section 4.
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