In practise, it is often desirable to provide the decision-maker with a rich set of diverse solutions of decent quality instead of just a single solution. In this paper we study evolutionary diversity optimization for the knapsack problem (KP). Our goal is to evolve a population of solutions that all have a profit of at least $(1-\varepsilon)\cdot OPT$, where OPT is the value of an optimal solution. Furthermore, they should differ in structure with respect to an entropy-based diversity measure. To this end we propose a simple $(\mu+1)$-EA with initial approximate solutions calculated by a well-known FPTAS for the KP. We investigate the effect of different standard mutation operators and introduce biased mutation and crossover which puts strong probability on flipping bits of low and/or high frequency within the population. An experimental study on different instances and settings shows that the proposed mutation operators in most cases perform slightly inferior in the long term, but show strong benefits if the number of function evaluations is severely limited.

中文翻译：

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背负多样性进化算法的背包问题繁育

实际上，通常希望为决策者提供一套丰富的，质量上乘的多样化解决方案，而不仅仅是一个解决方案。在本文中，我们研究了背包问题（KP）的进化多样性优化。我们的目标是要开发出一组全部收益至少为$（1- \ varepsilon）\ cdot OPT $的解决方案，其中OPT是最优解决方案的价值。此外，就基于熵的分集度量而言，它们在结构上应有所不同。为此，我们提出了一个简单的$（\ mu + 1）$-EA，其初始近似解是由著名的FPTAS为KP计算的。我们调查了不同标准突变算子的影响，并引入了有偏的突变和交叉，这为翻转种群中低频和/或高频位提供了很大的可能性。