Constrained Optimization of Workforce Allocation under Time-Varying Operational Uncertainty

Abstract

Workforce planning under the simultaneous uncertainty and stochasticity of demand requires a balancing act between reliability, labour costs, and nimbleness of operations. Conventional robust workforce planning problems model the uncertainty radius as a parameter, specified externally to the model; however, operational uncertainty propagates as forecast errors unfold. A parsimonious adaptive robust workforce planning model is constructed in which the uncertainty radius, when applied to the current planning horizon, is updated based on the absolute forecast error realized during the immediately preceding period. The proposed mechanism is defined by ????????+1 = ????0 +???? ∣ ???????? −???????? ∣, where ????0is the baseline uncertainty level and ???? controls feedback responsiveness. The resulting workforce problem aims to minimize the costs of the typical workforce, the penalties for shortages, and the costs of transitioning across periods, within workforce capacity constraints. A further contribution to the structure is an exact explicit representation of the solution to the continuous one period problem; since the cost function is convex and piecewise linear, the optimal workforce level is obtained by evaluating a finite set of candidates: the feasible boundary points, the previous period's workforce, and the demand breakpoint of the robust demand. A contraction result is also established, indicating that the feedback workforce allocation map is contractive.0 ≤ ???? < 1 under stationary conditions. Numerical experiments over 120 periods and 500 replications compare low-uncertainty, time-varying-uncertainty, and shock environments. Results show that feedback has little value under stable low uncertainty, but substantially reduces shortage cost and service violations under dynamic and shock-prone uncertainty. In the time-varying environment, ???? = 0.60reduces mean total cost by 3.31%, shortage cost by 58.41%, and service violations by 55.05% relative to the nonadaptive benchmark. In the shock environment, the strongest tested feedback level ???? = 0.95, reduces total cost by 9.73%, shortage cost by 34.44%, and service violations by 46.04%. The results demonstrate that forecast-error feedback provides a mathematically simple and operationally interpretable extension of robust workforce allocation.