“参考文献 A double-module immune algorithm for multi-objective optimizationproblems Applied Soft Computing 35 (2015) 161–174

- Multi-objective optimization problems (MOPs) have become a research hotspot, as they are commonlyencountered in scientific and engineering applications. When solving some complex MOPs, it is quitedifficult to locate the entire Pareto-optimal front. To better settle this problem, a novel double-moduleimmune algorithm named DMMO is presented, where two evolutionary modules are embedded to simul-taneously improve the convergence speed and population diversity. The first module is designed tooptimize each objective independently by using a sub-population composed with the competitive indi-viduals in this objective. Differential evolution crossover is performed here to enhance the correspondingobjective. The second one follows the traditional procedures of immune algorithm, where proportionalcloning, recombination and hyper-mutation operators are operated to concurrently strengthen the multi-ple objectives. The performance of DMMO is validated by 16 benchmark problems, and further comparedwith several multi-objective algorithms, such as NSGA-II, SPEA2, SMSEMOA, MOEA/D, SMPSO, NNIA andMIMO. Experimental studies indicate that DMMO performs better than the compared targets on most oftest problems and the advantages of double modules in DMMO are also analyzed.

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- Artificial immune system is an interesting bio-inspired intel-ligent approach that simulates the information processingprocedures of biologic immune system [45,46]. When foreign anti-gens are detected in biologic immune system, its B-cell is adaptedcorrespondingly to eliminate the intruders, which is realized bythe processes known as clonal selection and affinity maturationthrough hyper-mutation. Antibodies that can better recognize anantigen will be selected to proliferate by cloning, which is known asthe process of clonal selection. Then, hyper-mutation implementsan affinity maturation process proportional to the fitness valuesin order to generate the matured population. At last, some anti-bodies with better affinities will be remained as memory cells toprevent the re-intrusion of the previous antigens. This informa-tion processing principle gives some inspirations to design artificial immune algorithm, which improves the convergence speed andmaintain the diversity of the antibody population.
- In multi-objective immune algorithm, the problems and thecorresponding constraints can be treated as the antigen while apotential solution can be seen as an antibody. That is to say, as theMOPs defined in Eq. (1), a solution vector x = (x1, x2, . . ., xn) ∈ ˝is considered as an antibody and f(x) = {f1(x), f2(x), . . ., fm(x)} isregarded as an antigen. An antibody population is made up by a setof antibodies. When an antibody is called a non-dominated anti-body, it indicates that there does not exist another antibody in theantibody population that can dominate it. 人工免疫系统是一种有趣的仿生智能方法，它模拟了生物免疫系统的信息处理过程[45,46]。当生物免疫系统中检测到外来抗原时，其b细胞会相应地进行相应的适应以消除入侵者，这是通过超突变的克隆选择和亲和成熟过程来实现的。能够更好识别抗原的抗体将通过克隆选择增殖，这一过程被称为克隆选择。然后，超突变实现与适应度值成比例的亲和成熟过程，以产生成熟群体。最后，一些亲和性更好的抗体将作为记忆细胞保留下来，以防止以前抗原的再次入侵。该信息处理原理为设计人工免疫算法提供了一些启示，提高了算法的收敛速度，保持了抗体种群的多样性。

- To solve optimization problems, most of immune algorithmsare designed by mimicking the two important immune principles,such as clonal selection and affinity maturation by hyper-mutation.The first immune optimization algorithm [47] was developed forsolving single-objective optimization problem with an abstractclonal selection concept. Then, De Castro and Timmis proposedan artificial immune network algorithm named opt-aiNet [45] formultimodal optimization, in which the immune network decideswhich antibody will be cloned, repressed or maintained. After that,the learning and optimization algorithm using the clonal selectionprinciple (CLONALG) [46], which takes the affinity maturation ofimmune response into account, was adapted to solve multimodaland combinatorial optimization problems.
- Motivated by the promising performance of AIS in single-objective optimization problems, immune algorithms are extendedto tackle MOPs. The multi-objective immune system algorithm(MISA) [34], as an early multi-objective immune algorithm, waspresented based on clonal selection principle, in which an exter-nal memory is used to preserve the non-dominated antibodies andonly the highest fitness antibodies will be selected to proliferate. InRefs. [48,49], a vector artificial immune system (VAIS) was reported,which is extended from opt-aiNet [45] by few adjustments to dealwith MOPs. Similar antibodies are inhibited in VAIS, and a numberof new antibodies are added by random sampling to increase thepopulation diversity. Furthermore, an immune dominance clonalmulti-objective algorithm (IDCMA) was designed in Ref. [35], whichadopts Pareto domination relationship to determine the proce-dures of clonal selection with binary string representation. Theimproved version of IDCMA is a non-dominated neighbor-basedimmune algorithm (NNIA) [38], which designs a novel neighbor-based selection technique and population retention strategy toguarantee the population diversity using real-coded representa-tion. A dynamic multi-objective immune algorithm was presentedfor solving constrained nonlinear MOPs [50], which is realized bysimulating the simple interactive metaphors between antibodypopulation and multiple antigens. A novel simplified metaphor ofimmune response is implemented to obtain multiple excellent fea-sible solutions so as to explore the whole feasible PF. This work isfurther extended to the application of greenhouse control [51]. More recently, there are many competitive immune algorithmsstill presented to further enhance the performance. In Ref. [39],a hybrid immune multi-objective optimization algorithm (HIMO)was designed by us, which presents a hybrid mutation operatorcombining Gaussian and polynomial mutations. The performanceof HIMO was also enhanced by us [40] with an adaptive mutationoperator for local search and a fine-grained selection mechanismfor archive update. The adaptive mutation operator is executedaccording to the crowding-distance values, which promotes to useadaptive steps respectively for crowded and less-crowded individ-uals. A novel multi-objective immune algorithm was implementedby using a multiple-affinity model [52], which uses six measuresfor affinity assignment. Based on the selected affinity measures immune operators such as clonal proliferation, hyper-mutationand immune suppression are executed accordingly, which pro-liferate the superiors and suppress the inferiors. Afterwards, anovel immune clonal algorithm (NICA) [41] was presented withan improved clonal selection strategy, which can overcome theshortcomings of simple immune algorithm and well handle somecomplicated MOPs; in Ref. [53], a multi-objective immune algo-rithm with Baldwinian learning (MIAB) was designed, in which aBaldwinian learning strategy is presented to improve the searchcapability of NNIA [38]. The environment information and theevolving history of the parent solution are exploited to generatea predictive improving direction. All the above-mentioned immune algorithms treat the multipleobjectives as a whole and are aimed at optimizing them simulta-neously. However, as the conflicts exist among the objectives, thesimultaneous optimization on all the objectives may decline theconvergence speed on each objective, which is very evident in thecase with many local PFs. For example, some of the above immunealgorithms [38–41] cannot solve very well within the limited gener-ations for the test problems characterized with many local PFs, e.g.,WFG1, DTLZ1 and DTLZ3 [54,55]. Thus, this paper embeds a single-objective optimization process into traditional immune algorithm,which can accelerate the convergence speed for each objective andresultantly enhance the performance. Compared with the aboveimmune algorithms, the distinct feature of our algorithm is thedouble modules cooperatively evolved for solving MOPs and itssuperior performance is also confirmed by the experimental studiesdescribed in Section 4.

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原始发表时间：2020-07-30

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