An Exact and Efficient Algorithm for the Constrained Dynamic Operator Staffing Problem for Call Centers

Call center managers are facing increasing pressure to reduce costs while maintaining acceptable service quality. Consequently, they often face constrained stochastic optimization problems, minimizing cost subject to service-level constraints. Complicating this problem is the fact that customer-arrival rates to call centers are often time varying. Thus, to satisfy their service goals in a cost-effective manner, call centers may employ permanent operators who always provide service, and temporary operators who provide service only when the call center is busy, i.e., when the number of customers in system increases beyond a threshold level. This provides flexibility to dynamically adjust the number of operators providing service in response to the time-varying arrival rate. The constrained dynamic operator staffing (CDOS) problem involves determining the number of permanent and temporary operators, and the threshold value(s) that minimize time-average hiring and opportunity costs subject to service-level constraints. We model the CDOS problem as a constrained Markov decision process (MDP) and seek the optimal nonrandomized policy. The only exact method in the literature to obtain the optimal nonrandomized policy for a constrained MDP is enumeration, which is often computationally prohibitive. We provide a novel exact and efficient solution method, the modified balance equations disjunctive constraints (MBEDC) algorithm, yielding a mixed-integer program formulation; the computation times of this algorithm for sample problems are lower than enumeration by up to a factor of 200, and by a factor of 10 on average. Using our algorithm, we quickly solve diverse instances of the CDOS problem, generating managerial insights into the effects of temporary operators and service-level constraints.