Statistical offices face a familiar and intensifying dilemma: rising demand for detailed regional and domain-level estimates under budgets that are fixed or shrinking. National statistical offices (NSOs) either ignore the problem of optimal sample allocation for multiple target variables when designing a multi-purpose survey, or address it incorrectly - relying on ad hoc approaches such as computing Neyman allocations separately per variable and taking the element-wise maximum, a practice that simultaneously wastes budget and fails to guarantee precision across all domains. This paper presents a practical two-stage strategy that reframes the question: not how to allocate a given sample, but how small the sample can be made while still meeting pre-defined precision targets for all target variables across all geographic domains at once. The innovation lies not in inventing new methods, but in the novel combination of two well-established techniques applied to this cost-reduction problem: (i) multivariate constrained optimisation via Bethel allocation, which finds the globally minimum sample satisfying all precision constraints simultaneously; and (ii) Hierarchical Bayes (HB) small area modelling, which borrows strength across strata and permits a further reduction of the Bethel sample. The approach is validated using a Monte Carlo study (B = 1,000 replications) based on a synthetic labour-force population of one million individuals, where known population truth allows rigorous evaluation of precision, accuracy, and credible-interval coverage. Keywords: Bethel allocation; Hierarchical Bayes; small area estimation; sample size reduction; multivariate optimisation; labour force survey; coefficient of variation.

翻译：统计机构面临着一个普遍且日益严峻的困境：在预算固定或缩减的情况下，对详细区域和领域层面估计的需求却在不断增长。国家统计机构在设计多用途调查时，要么忽略多目标变量的最优样本分配问题，要么处理不当——依赖临时性方法，例如分别对每个变量计算内曼分配并取逐元素的最大值，这种做法既浪费预算，又无法保证所有领域的估计精度。本文提出了一种实用的两阶段策略，重新定义问题：并非如何分配给定的样本，而是在同时满足所有地理领域所有目标变量预设精度目标的前提下，样本量能缩减到多小。其创新之处不在于发明新方法，而是将两种成熟技术的创新组合应用于这一成本削减问题：（i）通过贝塞尔分配进行多变量约束优化，找到同时满足所有精度约束的全局最小样本；（ii）通过分层贝叶斯小区域建模，跨层借用信息，并允许进一步缩减贝塞尔样本。该方法基于包含一百万虚拟劳动人口的综合劳动力人口进行蒙特卡洛研究（B = 1,000次重复）验证，其中已知的总体真值可对精度、准确性和可信区间覆盖率进行严格评估。关键词：贝塞尔分配；分层贝叶斯；小区域估计；样本量缩减；多变量优化；劳动力调查；变异系数。