Variational optimization for call center staffing

According to Koole and Mandelbauin (2001), almost 60 to 70 percent of the total costs for operating a call center involve wage and benefit expenses for personnel. It follows that determining the optimal amount of call center agents is of great interest to call center managers. This paper addresses both the staffing of agents and the provisioning of telephone lines by introducing a revenue and penalty structure. Our goal is to develop an approximate algorithm for designing a profit optimal staffing and provisioning schedule. Our method for determining the number of agents and telephone lines arises from variational optimization methods. First, we model the call center as a multiserver queue with additional waiting spaces and abandonment. This queueing system is a special case of a natural class of queueing network models for call centers called Markovian service networks. Now we add an economic structure to our queueing model for the call center. We assume that there is a reward for every successful service completion, a penalty for every abandoned call, and a cost for the number of agents and telephone lines used. We can then express the total profit for the call center as an integral functional of the time evolution for the number of customers in the system over a fixed time interval. We call this our profit functional. We then use variational calculus methods from the theory of optimal control to derive an optimal staffing and provisioning schedule from our analysis of the fluid approximation of the profit functional.