In real-world optimisation, it is common to face several sub-problems interacting and forming the main problem. There is an inter-dependency between the sub-problems, making it impossible to solve such a problem by focusing on only one component. The traveling thief problem~(TTP) belongs to this category and is formed by the integration of the traveling salesperson problem~(TSP) and the knapsack problem~(KP). In this paper, we investigate the inter-dependency of the TSP and the KP by means of quality diversity~(QD) approaches. QD algorithms provide a powerful tool not only to obtain high-quality solutions but also to illustrate the distribution of high-performing solutions in the behavioural space. We introduce a MAP-Elite based evolutionary algorithm using well-known TSP and KP search operators, taking the TSP and KP score as behavioural descriptor. Afterwards, we conduct comprehensive experimental studies that show the usefulness of using the QD approach applied to the TTP. First, we provide insights regarding high-quality TTP solutions in the TSP/KP behavioural space. Afterwards, we show that better solutions for the TTP can be obtained by using our QD approach and show that it can improve the best-known solution for a wide range of TTP instances used for benchmarking in the literature.
翻译:在现实世界的优化中,常见的情况是面对几个小问题相互作用和形成主要问题。子问题之间存在相互依存关系,因此不可能通过只注重一个组成部分来解决该问题。旅行小偷问题~(TTP)属于这一类别,由旅行推销员问题~(TSP)和 knapsack问题~(KP)的整合构成。在本文件中,我们通过高质量的多样性~(QD) 方法调查TSP和KP的相互依存性。QD算法不仅提供了获得高质量解决方案的有力工具,而且还展示了行为空间中高性能解决方案的分布情况。我们采用了基于MAP-Elite的演化算法,使用众所周知的TSP和KP搜索操作员,将TSP和KP的评分作为行为描述符。随后,我们进行了全面的实验研究,表明采用适用于TTP的QD方法是否有用。首先,我们为TSP/KP的高质量TTP的解决方案提供了深刻的洞察理解,然后用广度空间展示了我们所了解的TTP的高质量TTP文献解决方案。我们随后可以展示了一种更好的解决办法。