The Bees Algorithm has been successfully applied for over a decade to a large number of optimisation problems. However, a mathematical analysis of its search capabilities, the effects of different parameters used, and various design choices has not been carried out. As a consequence, optimisation of the Bees Algorithm has so far relied on trial-and-error experimentation. This paper formalises the Bees Algorithm in a rigorous mathematical description, beyond the qualitative biological metaphor. A review of the literature is presented, highlighting the main variants of the Bees Algorithm, and its analogies and differences compared with other optimisation methods. The local search procedure of the Bees Algorithm is analysed, and the results experimentally checked. The analysis shows that the progress of local search is mainly influenced by the size of the neighbourhood and the stagnation limit in the site abandonment procedure, rather than the number of recruited foragers. In particular, the analysis underlines the trade-off between the step size of local search (a large neighbourhood size favours quick progress) and the likelihood of stagnation (a small neighbourhood size prevents premature site abandonment). For the first time, the implications of the choice of neighbourhood shape on the character of the local search are clarified. The paper reveals that, particularly in high-dimensional spaces, hyperspherical neighbourhoods allow greater search intensification than hypercubic neighbourhoods. The theoretical results obtained in this paper are in good agreement with the findings of several experimental studies. It is hoped that the new mathematical formalism here introduced will foster further understanding and analysis of the Bees Algorithm, and that the theoretical results obtained will provide useful parameterisation guidelines for applied studies.

Bees算法已经成功应用了十多年，解决了许多优化问题。但是，尚未对其搜索能力，所使用的不同参数的影响以及各种设计选择进行数学分析。结果，迄今为止，对Bees算法的优化一直依赖于反复试验实验。除了定性的生物隐喻之外，本文还通过严格的数学描述形式将Bees算法形式化。本文对文献进行了回顾，重点介绍了Bees算法的主要变体以及与其他优化方法相比的类比和差异。分析了Bees算法的本地搜索过程，并通过实验检查了结果。分析表明，本地搜索的进度主要受站点规模和停滞过程中停滞极限的影响，而不是受雇的觅食者的数量。尤其是，该分析强调了局部搜索的步长（较大的邻域大小有利于快速进步）和停滞的可能性（较小的邻域大小可防止站点过早放弃）之间的权衡。首次阐明了邻域形状选择对本地搜索特征的影响。该论文表明，特别是在高维空间中，与高三次邻域相比，超球形邻域允许更大的搜索强度。本文获得的理论结果与一些实验研究的结果非常吻合。希望这里介绍的新的数学形式主义将促进对Bees算法的进一步理解和分析，并且所获得的理论结果将为应用研究提供有用的参数化指南。