Mixed-Integer Rounding Enhanced Benders Decomposition for Multiclass Service-System Staffing and Scheduling with Arrival Rate Uncertainty

We study server scheduling in multiclass service systems under stochastic uncertainty in the customer arrival volumes. Common practice in such systems is to first identify staffing levels, and then determine schedules for the servers that cover these targets. We propose a new stochastic integer programming model that integrates these two decisions, which can yield lower scheduling costs by exploiting the presence of alternative server configurations that yield similar quality-of-service. We find that a branch-and-cut algorithm based on Benders decomposition may fail due to the weakness of the relaxation bound. We propose a novel application of mixed-integer rounding to improve the Benders cuts used in this algorithm, a technique that is applicable to any stochastic integer program with integer first-stage decision variables. Numerical examples illustrate the computational efficiency of the proposed approach and the potential benefit of solving the integrated model compared to considering the staffing and scheduling problems separately.