Optimal scheduling in call centers with a callback option

We consider a call center model with a callback option, which allows to transform an inbound call into an outbound one. A delayed call, with a long anticipated waiting time, receives the option to be called back. We assume a probabilistic customer reaction to the callback offer (option). The objective of the system manager is to characterize the optimal call scheduling that minimizes the expected waiting and abandonment costs. For the single-server case, we prove that non-idling is optimal. Using a Markov decision process approach, we prove for the two-server case that a threshold policy on the number of queued outbound calls is optimal. For the multi-server case, we numerically characterize a switching curve of the number of agents reserved for inbound calls. It is a function of the number of queued outbound calls, the number of busy agents and the identity of jobs in service. We also develop a Markov chain method to evaluate the system performance measures under the optimal policy.We next conduct a numerical study to examine the impact of the policy parameters on the system performance. We observe that the value of the callback offer is especially important for congested situations. It also appears that the benefits of a reservation policy are more apparent in large call centers, while they almost disappear in the extreme situations of light or heavy workloads. We moreover observe in most cases that the callback offer should be given upon arrival to any delayed call. However, if balking and abandonment are very high (which helps to reduce the workload) or if the overall treatment time spent to serve an outbound call is too large compared to that of an inbound one, there is a value in delaying the proposition of the callback offer.