Call center capacity allocation with random workload S.Liao 1 , Ch.Van Delft 2 , G. Koole 3 ,Y. Dallery 1 ,O. Jouini 1 1 Ecole Centrale Paris, France (shuangqing.liao@ecp.fr) 2 Groupe HEC, France (vandelft@hec.fr) 3 VU University Amsterdam,The Netherlands ABSTRACT We consider a call center staffing problem with two types of costumers of which the arrival rates are allowed to be a random and non-stationary. In order to efficiently cope with such random workload fluctuations, the workforce presents some flexi- bility: the agents can be, in real-time, affected to each type of customers according to the instantaneously observed workload and the associated/relative cost criteria. We model this staffing problem as a cost optimization-based newsboy-type model. We then show how to numerically solve this model. In order to deal with the randomness characterizing the workloads of the call processes, we consider several solution tracks. First, we solve the model under the assumption that the workloads are deterministic and equal to their average values. In the second approach, we explicitly formulate in the optimization model the stochastic nature of the workloads. As a third approach, we develop a robust-type solution. Via several numerical analyzes we show the impact of the arrival rates randomness on the optimal staffing policy and on the operating costs. Keywords: Call centers, staffing, non-stationary arrival rates, newsboy-type model, stochastic approach, robust approach. 1. Introduction 1.1. The general setting and literature review Telephone call centers have become more and more im- portant for many large organizations to communicate with their customers. For example, it is estimated that in 2002 more than 70% of all customer-business interactions were handled in call centers, and the U.S call center industry em- ployed more than 3.5 million people, or 2.6% of the work- force (see[3]). Managing call center operations more effi- ciently is of significant economic interest. Up to now, the focus on call center research has been mainly on forecasting, the translation of forecasts into staffing levels (using the Erlang C or some other queu- ing formula), and agent scheduling (see [1, 2]). A third planning activity, after forecasting and agent planning, con- cerns intra-day performance management. This has rarely been studied in the literature, except for a few papers that study intra-day forecasting where the consequences of hav- ing the realization of the first part of the day on the forecast for the rest of the day are studied (see [5, 6, 14, 13, 12]). Workforce management is a critical component of call cen- ter operations. Since labor costs are a major component of the total cost of operation, efficient staff scheduling is criti- cal. But when uncertainty plagues the arrival rates, efficient staffing is difficult. The variability has several origins. A cause that has attracted some attention in the literature is the forecasting error. It is not uncommon that forecasts are regularly off by 5% to 10%. The realization of the volume on a day can be modelled as a random variable with expec- tation equal to the forecast. This is not all the randomness related to the call demand: for a given average workload, the arrival rates as well as the handling times are random. For fixed staffing levels, this leads to a considerable vari- ability in the service level, even for extended periods such as an entire day. A final cause of variability, that we do not consider in this paper, is absence of agents due to illness or 2009-CIE39-FR other reasons. Most call center models in the literature fo- cus on the stochastic variability of inter-arrival times for a given known arrival rate and ignore the issue of arrival rate uncertainty. Only a few papers have addressed this issue. In the what follows, we highlight some of these works. T. Robbins (see [6, 7, 8, 9]) focused several research pa- pers on this specific topic. Its research considers the issue of arrival rate uncertainty and its impact on scheduling. He reviews empirical data from call centers that demonstrates the level of uncertainty present in many applications, and proposes heuristic scheduling and staffing models that con- sider arrival rate uncertainty. In ([7, 8]), the authors de- velop a solution method consisting of applying a stochas- tic programming approach (without recourse) based on a set of scenarios for the arrival process. In ([6, 9, 10, 11]), the authors evaluate by simulation the impact of fluctuat- ing arrival rates on the call center performance indices, for different types of call centers. In [14], W. Whitt proposes simple methods for staffing a single-class call center with uncertain arrival rate and uncertain staffing due to employee absenteeism. The ba- sic model is a multi-server queue with customer abandon- ment, allowing non-exponential service-time and time-to- abandon distributions. The goal is to maximize the ex- pected net return. Two approximations are simultaneously used for the conditional performance measures: a deter- ministic fluid approximation and a Markovian birth-and- death model, having state-dependent death rates. In [15], this author considers the staffing problem for staffing a single-class call center with uncertain arrival rates. The solution method relies on exploiting detailed knowledge of system state in order to obtain reliable estimates of the mean and variance of the demand in the near future. The necessary staff is then accordingly adapted. 1.2. The considered problem In this paper, we focus on a relatively simple model in this random arrival rates setting. We consider a call cen- ter staffing problem with two types of costumers and with 863 978-1-4244-4136-5/09/$25.00©2009IEEE