When promotions meet operations: Cross-selling and its effect on call center performance

We study cross-selling operations in call centers. The following questions are addressed: How many customer-service representatives are required (staffing), and when should cross-selling opportunities be exercised (control) in a way that will maximize the expected profit of the center while maintaining a prespecified service level target? We tackle these questions by characterizing control and staffing schemes that are asymptotically optimal in the limit, as the system load grows large. Our main finding is that a threshold priority control, in which cross-selling is exercised only if the number of callers in the system is below a certain threshold, is asymptotically optimal in great generality. The asymptotic optimality of threshold priority reduces the staffing problem to a solution of a simple deterministic problem in one regime and to a simple search procedure in another. We show that our joint staffing and control scheme is nearly optimal for large systems. Furthermore, it performs extremely well, even for relatively small systems.