The optimality of an (s, S) hiring policy on a workforce planning problem with fixed recruitment costs and binomial turnover

Abstract

We study a finite-horizon nonstationary 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; this represents explicit (e.g. administration costs) or implicit (e.g. team disruption) factors that are commonplace in workforce planning. 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 program 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 in presence of endogenous uncertainty. 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. To compute approximate policy parameters, we develop a mixed-integer linear programming heuristic based on a piecewise approximation. In our computational study, we demonstrate that our approach achieves fast computation and small optimality gaps on the test instances.