AI 中文总结
针对含固定招聘成本和二项式人员流动的有限期劳动力规划问题，研究人员通过建立离散凸性与K-凸性证明最优（s, S）招聘策略，并提出分段近似方法生成MILP公式求解，该方法计算快且最优性差距小。
AI 中文摘要
我们研究有限期劳动力规划问题，其中各期人员流动服从二项分布，其参数取决于招聘后的劳动力水平。该模型包含固定招聘成本，无论招聘人数多少，只要发生招聘就会产生该成本。目标是最小化预期总成本，包括招聘成本、薪资成本和短缺成本，低于各期特定人员配置要求的偏差会受到惩罚。为分析具有决策依赖转移概率的随机动态规划，我们建立了单期可变成本（预期薪资与惩罚成本之和）的离散凸性，以及预期总成本的K-凸性。具体而言，我们引入二项式-K-凸性的概念，以证明K-凸性在贝尔曼函数的二项式传播下得以保留。随后，我们表明最优招聘策略呈现（s, S）型结构：当某期劳动力水平低于阈值s时，招聘人员至水平S；否则不进行招聘。此外，我们提出分段近似方法，生成混合整数线性规划（MILP）公式以求解问题并计算每期的（s, S）参数。数值结果表明，所提方法计算速度快，且最优性差距小。
英文摘要
We study a finite-horizon workforce planning problem in which staff turnover in each period follows a binomial distribution whose parameters depend on the post-hiring workforce level. The model incorporates a fixed hiring cost that is incurred whenever recruitment occurs, regardless of the number of employees hired. The objective is to minimise the expected total cost, including recruitment, salary, and shortage costs, where deviations below period-specific staffing requirements are penalised. To analyse this stochastic dynamic programme with decision-dependent transition probabilities, we establish the discrete convexity of the variable single-period cost (the sum of expected salary and penalty costs) and the K-convexity of the expected total cost. Specifically, we introduce the concept of Binomial-K-convexity to facilitate the proof that K-convexity is preserved under Binomial propagation in the Bellman function. We then show that the optimal hiring policy exhibits an (s, S)-type structure: when the workforce level in a given period falls below a threshold s, staff are hired up to level S; otherwise, no hiring occurs. Furthermore, we develop a piecewise approximation approach that yields a mixed-integer linear programming (MILP) formulation for solving the problem and computing the (s, S) parameters for each period. Numerical results demonstrate that the proposed method achieves fast computation with small optimality gaps.
