A multi-objective immune algorithm with dynamic population strategy

Abstract

In this paper, we propose a multi-objective immune algorithm with dynamic population strategy, named MOIA-DPS, 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. 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 MOIA-DPS, 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., I_SDE+, 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.

Introduction

Multi-objective optimization problems (MOPs) have received considerable attention over the past several decades as they widely exist in many domains of scientific research and engineering applications, such as vehicle routing problem [1], cloud computing resource scheduling [2], image processing [3], environmental/economic dispatch problems [4] and distribution transformers [5]. Generally, MOPs aim to optimize several (often conflicting) objectives simultaneously. Due to the conflicts among the objectives, no any solution is best with respect to all the objectives. Instead, a set of equally-optimal solutions is found when considering all the objectives, which is often termed Pareto-optimal solutions. All Pareto-optimal solutions make up a Pareto-optimal set, whose projection in the objective space is termed Pareto-optimal front (PF). The aim of MOPs is to achieve a subset of solutions that are close to the true PF and distributed uniformly along the true PF, which can be provided to the decision maker as the available choices for various application cases.

It is difficult for traditional computation methods to solve the above MOPs due to the complex solution space, uncertain optimization processing and huge calculations. Therefore, the natural-inspired metaheuristics, such as simulated annealing, evolutionary algorithms, ant colony optimization, particle swarm optimization and artificial immune systems, were proposed for complex computation problems. Due to the population-based searching nature of evolutionary algorithms (EAs), they can obtain multiple Pareto-optimal solutions in a single run [6,7]. Thus, more and more research efforts were made to extend for tackling MOPs, called MOEAs. Generally, most of MOEAs can be classified into three kinds, i.e., Pareto-based MOEAs [8], indicator-based MOEAs [9] and decomposition-based MOEAs [10]. The representatives of these three types of MOEAs include NSGA-II [6], HyPE [11] and MOEA/D [12], respectively. NSGA-II adopts a fast non-dominated sorting approach combined with a crowded-comparison operator and an elitist strategy, while HyPE uses hypervolume indicator to rank solutions. MOEA/D decomposes MOPs into a set of single-objective optimization problems (SOPs) and then optimizes them collaboratively using genetic search method. Many competitive strategies were reported to further enhance the performance of these MOEAs [[13], [14], [15]]. Reference point based dominance was proposed in Ref. [16] to substitute the Pareto dominance in NSGA-II for handling many objective optimization problems. New strategies were developed in Ref. [17] to effectively compute and update hypervolume contributions. For MOEA/D, adaptive weight vector generation methods were proposed to enhance the robustness for problems with different PF shapes [18].

As different evolutionary stages of EAs may have different characteristics, a fixed evolutionary operator with pre-set parameter setting may not suit for each stage of evolutionary process. Thus, numbers of adaptive strategies were reported to solve the above problem. For example, when solving SOPs, since the scaling factor and crossover rate in differential evolution (DE) are very sensitive to the performance of EAs, a number of DE variants were designed, including SaDE [19], JADE [20], MDE-pBX [21] and ZEPDE [22]. The mutation strategies and their associated parameter settings in SaDE are gradually self-adapted by learning from their previous searching experience to generate the promising solution. The scaling factor and crossover rate in JADE and MDE-pBX are adaptively sampled from the Cauchy and Gaussian distributions, respectively. The mutation strategies of ZEPDE are also dynamically adjusted with the evolutionary process, while the systematic parameters in ZEPDE are self-adapted in their own zoning by evolution. When tackling MOPs, numbers of adaptive strategies for MOEAs were also designed. For example, adaptive cross-generation DE operators [23] and adaptive DE for multi-objective immune algorithm [24] were presented to adaptively adjust the scaling factor and crossover rate in DE operator. A brief review in adaptive DE schemes can be found in Ref. [25].

Actually, the population size is also an important factor for the performance of nature-inspired heuristic algorithms. A large size may induce the waste of computational resource in one generation, while a small size may easily lead to the premature convergence or stagnation. In Ref. [26], this reference provides a study of the population size in distributed evolutionary algorithm. To solve the problem of premature convergence in particle swarm optimization, a dynamic population scheme for multi-objective particle swarm optimization was proposed in Ref. [27] to enhance the population diversity. Similarly, for multi-objective immune algorithms (MOIAs), the clonal principle presents some advantages related to the convergence speed; however, they also face the problem regarding the lack of diversity [28]. In this paper, we propose a novel dynamic population strategy (DPS) for MOIAs (MOIA-DPS), 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.

The effectiveness of the proposed MOIA-DPS is validated by the experimental studies to solve the three suites of test MOPs, i.e., ZDT [29], WFG [30] and DTLZ [31]. When compared to four state-of-the-art heuristic multi-objective algorithms, i.e., I_SDE+ [9], AbYSS [32], MOEA/D-GRA [10] and CMPSO [33], and four current MOIAs, i.e., IMADE [34], DMMO [35], HEIA [36] and AIMA [37], MOIA-DPS shows some advantages on the convergence speed and population diversity.

The rest of this paper is organized as follows. In Section 2, some related work is introduced, such as MOPs and MOIAs. Then, the details of the proposed MOIA-DPS are presented in Section 3, such as DPS, cloning operator, TDE operator and archive update operator. The complete algorithm of MOIA-DPS is also summarized in Section 3. The comparison results of MOIA-DPS and various nature-inspired multi-objective algorithms are provided in Section 4. Moreover, the effectiveness of our proposed operators is also validated in Section 4. At last, the conclusions of this paper are summarized in Section 5.