Simultaneous perturbation methods for adaptive labor staffing in service systems

Service systems are labor intensive due to the large variation in the tasks required to address service requests from multiple customers. Aligning the staffing levels to the forecasted workloads adaptively in such systems is nontrivial because of a large number of parameters and operational variations leading to a huge search space. A challenging problem here is to optimize the staffing while maintaining the system in steady-state and compliant to aggregate service level agreement (SLA) constraints. Further, because these parameters change on a weekly basis, the optimization should not take longer than a few hours. We formulate this problem as a constrained Markov cost process parameterized by the (discrete) staffing levels. We propose novel simultaneous perturbation stochastic approximation (SPSA) based SASOC (Staff Allocation using Stochastic Optimization with Constraints) algorithms for solving the above problem. The algorithms include both first order as well as second order methods and incorporate SPSA based gradient estimates in the primal, with dual ascent for the Lagrange multipliers. Both the algorithms that we propose are online, incremental and easy to implement. Further, they involve a certain generalized smooth projection operator, which is essential to project the continuous-valued worker parameter tuned by SASOC algorithms onto the discrete set. We validated our algorithms on five real-life service systems and compared them with a state-of-the-art optimization tool-kit OptQuest. Being 25 times faster than OptQuest, our algorithms are particularly suitable for adaptive labor staffing. Also, we observe that our algorithms guarantee convergence and find better solutions than OptQuest in many cases.