Dimensioning an inbound call center using constraint programming

Abstract

One of the critical problems in the call center industries is the staffing problem, since they must face variable demands and because staff costs represent a major part of the costs of these industries. Prom a modeling point of view, a call center is generally modeled as a M/M/N system, also called the Erlang-C model. In [Koole and Mandelbaurn, 2001], the authors present a survey of the state-of-the-art about possible models of a call center. In this paper, the problem of dimensioning a call center is modeled as a deterministic scheduling problem, where resources represent the employees and jobs correspond to the calls that are received. To each resource are associated tools that correspond to the skills of employees, and a weigth corresponding to the cost of employment. To each job are associated a release date, a processing time, a deadline and a required tool. The objective is to schedule jobs so as to minimize the weighted number of resources needed to perform the jobs. For solving this problem, we propose exact methods based on Constraint Programming Models. Three different models are presented. The second one is an improvement of the first one, both in terms of constraints definitions and instanciation methods. The last one introduces redundant constraints to speed up the resolution process. An approximated algorithm is also proposed, based on a priority rule and an assignment rule. We also compare our models to a Mixed Integer Linear Program, solved by CPLEX. Instances are randomly generated with around 45 resources and from 20 to 200 jobs. Feasible solutions are always obtained within two minutes by using these methods. CP models are solved using Eclipse 5.8-77 software with the time limit fixed to two minutes on a PC Pentium III, 1.4 GHz, 512 Mo. CP can solve the problem optimally with up to 80 jobs. For more than 100 jobs, the upper bound given by the heuristic algorithm is never improved within two minutes. © Springer-Verlag Berlin Heidelberg 2005.