Knapsack problems (KPs) are famous combinatorial optimization problems that can be solved by evolutionary algorithms (EAs). In such methods, a key step is to produce new solutions for each generation. However, traditional EAs cannot guarantee the feasibility of the new solutions, causing ineffectiveness or inefficiency of the methods. Owing to the fact that directly generating a feasible solution is difficult, a practical way is to generate a potential solution and transform it to a feasible one if necessary; more ideally, transform it to the local-optimal solution. Essentially, this transforming process is to map an element of the solution set to an element of the feasible solution set, which can be analyzed and optimized from a new perspective, i.e., the set theory (ST). In this article, we provide new explanations for the transforming process of solutions based on ST and summarize the properties that the transforming process should satisfy. Furthermore, based on the proposed concepts and theories, we put forward the ideas for improving the transforming process. Consequently, some new operators for KPs are proposed. Experimental results demonstrate the superiority of the proposed operators.

中文翻译：

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求解背包问题的进化算法中基于集合论的算子设计


背包问题（KP）是著名的组合优化问题，可以通过进化算法（EA）解决。在此类方法中，关键步骤是为每一代产生新的解决方案。然而，传统的EA无法保证新解决方案的可行性，导致方法无效或效率低下。由于直接生成可行解比较困难，一种实用的方法是生成潜在解，必要时将其转化为可行解；更理想的是，将其转化为局部最优解。本质上，这个转换过程就是将解集的一个元素映射到可行解集的一个元素，可以从一个新的角度，即集合论（ST）来分析和优化。在本文中，我们对基于ST的解的转换过程提供了新的解释，并总结了转换过程应满足的性质。此外，基于所提出的概念和理论，我们提出了改进转型过程的想法。因此，提出了一些新的 KP 算子。实验结果证明了所提出算子的优越性。