Set Theory-Based Operator Design in Evolutionary Algorithms for Solving Knapsack Problems

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.