Optimization Online


Distributionally robust workforce scheduling in call centers with uncertain arrival rates

Shuangqing Liao(shuangqing.liao***at***ecp.fr)
Christian van Delft(vandelft***at***hec.fr)
Jean-Philippe Vial(jphvial***at***gmail.com)

Abstract: Call center scheduling aims to set-up the workforce so as to meet target service levels. The service level depends on the mean rate of arrival calls, which fluctuates during the day and from day to day. The staff scheduling must adjust the workforce period per period during the day, but the flexibility in so doing is limited by the workforce organization by shifts. The challenge is to balance salary costs and possible failures to meet service levels. In this paper, we consider uncertain arrival rates, that vary according to an intra-day seasonality and a global \textit{busyness} factor. Both factors (seasonal and global) are estimated from past data and are subject to errors. We propose an approach combining stochastic programming and distributionally robust optimization to minimize the total salary costs under service level constraints. The performance of the robust solution is simulated via Monte-Carlo techniques and compared to the pure stochastic programming.

Keywords: Call centers; uncertain arrival rates; robust optimization; ambiguity; staff-scheduling; totally unimodular

Category 1: Stochastic Programming

Category 2: Robust Optimization

Citation: [1] Aksin, Z., Armony, M., and Mehrotra, V. (2007). The modern call-center: a multi-disciplinary perspective on Operations Management Research. Production and Operations Management, 16:665-688. [2] Avramidis, A., Deslauriers, A., and L'Ecuyer, P. (2004). Modelling daily arrivals to a telephone call center. Management Science, 50:896-908. [3] Babonneau, F., Vial, J.-P., and Apparigliato, R. (2010). Robust optimization for environmental and energy planning. In Filar, J. and Haurie, A., editors, Handbook on “Uncertainty and Environmental Decision Making", International Series in Operations Research and Management Science, pages 79-126. Springer Verlag. [4] A. Ben-Tal and A. Nemirovski. (1998) Robust convex optimization. Mathematics of Operations Research, 23:769 - 805. [5] Ben-Tal, A., El Ghaoui, L., and Nemirovski, A. (2009). Robust Optimization. Princeton University Press. [6] A. Ben-Tal, D. den Hertog, A. de Waegenaere, B. Melenberg, and G. Rennen. (2011) Robust solutions of optimization problems affected by uncertain probabilities. Working paper, Tilburg University. [7] Birge J. and F. Louveaux (1997). Introduction to Stochastic Programming. Springer Verlag, New York. [8] Breton, M. and El Hachem, S. (1995). Algorithms for the solution of stochastic dynamic minimax problems. Computational Optimization and Applications, 4:317-345. [9] Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S., and Zhao, L. (2005). Statistical analysis of a telephone call center: a queueing-science perspective. J. Amer. Statist. Assoc., 100:36-50. [10] G. C. Calafiore. (2007) Ambiguous risk measures and optimal robust portfolios. SIAM Journal on Optimization, 18(3):853-877. [11] Calafiore, G. and El Ghaoui, L. (2006). On distributionally robust chance constrained linear programs with applications. Journal of Optimization Theory and Applications, 130(1):1-22. [12] A. Charnes, W. Cooper, and G. Symonds. (1958) Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Management Science, 4:235-263. [13] Delage, E. and Ye, Y. (2010). Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research, 58(3):595-612. [14] Dupacova, J. (1987). The minimax approach to stochastic programming and an illustrative application. Stochastics, 20:73-88. [15] L. El-Ghaoui and H. Lebret. (1997) Robust solutions to least-square problems to uncertain data matrices. SIAM Journal of Matrix Analysis and Applications, 18:1035-1064. [16] El Ghaoui, L., Oks, M., and Oustry, F. (2003). Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research, 51:543-556. [17] Gans, N., Koole, G., and Mandelbaum, A. (2003). Telephone calls centers: a tutorial and literature review and research prospects. Manufacturing and Service Operations Management, 5:79-141. [18] Green, L., Kolesar, P., and Whitt, W. (2007). Coping with time-varying demand when setting staffing requirements for a service system. Production and Operations Management, 16:13-39. [19] Gross, D. and Harris, C. (1998). Fundamentals of Queueing Theory. Wiley Series in Probability and Mathematical Statistics. 3rd Edition. [20] Harrison, J. and Zeevi, A. (2005). A method for staffing large call centers based on stochastic fluid models. Manufacturing and Service Operations Management, 7:20-36. [21] Jimenez, T. and Koole, G. (2004). Scaling and comparison of fluid limits of queues applied to call centers with time-varying parameters. OR Spectrum, 26:413-422. [22] D. Klabjan, D. Simchi-Levi, and M. Song. (2010). Robust stochastic lot-sizing by means of histograms. Working paper. [23] P. Kouvelis and G. Yu. (1997) Robust Discrete Optimization and its Applications. Kuwer Academic Publishers, London. [24] Liao, S., van Delft, C., Koole, G., and Jouini, O. (2010). Staffing a call center with uncertain non-stationary arrival rate and flexibility. Working paper. Ecole Centrale Paris. To appear in OR Spectrum. [25] L. Pardo. (2006). Statistical Inference Based on Divergence Measures. Chapman & Hall/CRC, Boca Raton. [26] Robbins, T. (2007). Managing service capacity under uncertainty. Ph.D. Dissertation, Penn State University. [27] Scarf, H. (1958). A Min-max Solution of an Inventory Problem. Studies in The Mathematical Theory of Inventory and Production, pages 201-209. [28] Wang, Z., Glynn, P., and Ye, Y. (2009). Likelihood robust optimization for data-driven newsvendor problem. Working Paper. [29] Whitt, W. (1999). Dynamic staffing in a telephone call center aiming to immediately answer all calls. Operations Research Letters, 24:205-212. [30] Whitt, W. (2006). Staffing a call center with uncertain arrival rate and absenteeism. Production and Operations Management, 15:88-102. [31] Yue, J., Chen, B., and Wang, M. (2006). Expected Value of Distribution Information for the Newsvendor Problem. Operations Research, 54:1128-113. [32] Zackova, J. (1966). On Minimax Solution of Stochastic Linear Programming Problems. Casopis pro Pestovani, 91:423-430.

Download: [PDF]

Entry Submitted: 05/25/2011
Entry Accepted: 05/25/2011
Entry Last Modified: 05/25/2011

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society