Advertisement

Journal of Scheduling

, Volume 10, Issue 1, pp 5–23 | Cite as

Cyclic preference scheduling of nurses using a Lagrangian-based heuristic

  • Jonathan F. BardEmail author
  • Hadi W. Purnomo
Article

Abstract

This paper addresses the problem of developing cyclic schedules for nurses while taking into account the quality of individual rosters. In this context, quality is gauged by the absence of certain undesirable shift patterns. The problem is formulated as an integer program (IP) and then decomposed using Lagrangian relaxation. Two approaches were explored, the first based on the relaxation of the preference constraints and the second based on the relaxation of the demand constraints. A theoretical examination of the first approach indicated that it was not likely to yield good bounds. The second approach showed more promise and was subsequently used to develop a solution methodology that combined subgradient optimization, the bundle method, heuristics, and variable fixing. After the Lagrangian dual problem was solved, though, there was no obvious way to perform branch and bound when a duality gap existed between the lower bound and the best objective function value provided by an IP-based feasibility heuristic. This led to the introduction of a variable fixing scheme to speed convergence. The full algorithm was tested on data provided by a medium-size U.S. hospital. Computational results showed that in most cases, problem instances with up to 100 nurses and 20 rotational profiles could be solved to near-optimality in less than 20 min.

Keywords

Cyclic scheduling Preference scheduling Nurse rostering Lagrangian relaxation Bundle method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aickelin, U. and K. Dowsland, “Exploiting problem structure in a genetic algorithm approach to a nurse rostering problem,” Journal of Scheduling, 3(3), 139–153 (2000).CrossRefGoogle Scholar
  2. Aickelin, U. and K. Dowsland, “An indirect algorithm for a nurse-scheduling problem,” Computers & Operations Research, 31(5), 761–778 (2004).CrossRefGoogle Scholar
  3. Bard, J. F. and H. W. Purnomo, “Preference scheduling for nurses using column generation,” European Journal of Operational Research, 164(2), 510–534 (2005a).CrossRefGoogle Scholar
  4. Bard, J. F. and H. W. Purnomo, “Hospital-wide reactive scheduling of nurses with preference considerations,” IIE Transactions on Operations Engineering, 37(7), 589–608 (2005b).Google Scholar
  5. Bard, J. F. and H. W. Purnomo, “A column generation-based approach to solve the preference scheduling problem for nurses with downgrading,” Socio-Economic Planning Sciences, 39(3), 193–213 (2005c).CrossRefGoogle Scholar
  6. Berrada, I., J. A. Ferland, and P. Michelon, “A multi-objective approach to nurse scheduling with both hard and soft constraints,” Socio-Economic Planning Sciences, 30(3), 183–193 (1996).CrossRefGoogle Scholar
  7. Brusco, M. J. and L. W. Jacobs, “Cost analysis of alternative formulations for personnel scheduling in continuously operating organisations,” European Journal of Operational Research, 86(2), 249–261 (1995).CrossRefGoogle Scholar
  8. Burke, E. K., P. De Causmaecker, and G. Vanden Berghe, “A hybrid tabu search algorithm for the nurse rostering problem,” in: B. McKay et al. (Eds.), Simulated Evolution and Learning, Lecture Notes in Artificial Intelligence. Springer, Berlin (1999), Vol. 1585, pp. 187–194.Google Scholar
  9. Burke, E. K., P. I. Cowling, P. De Causmaecker, and G. Vanden Berghe, “A memetic approach to the nurse rostering problem,” Applied Intelligence, 15(3), 199–214 (2001).CrossRefGoogle Scholar
  10. Burke, E. K., P. De Causmaecker, and G. Vanden Berghe, “Novel meta-heuristic approaches to nurse rostering problems in Belgian hospitals, Chap. 44, in J. Leung (Ed.), Handbook of Scheduling: Algorithms, Models, and Performance Analysis, CRC Press, Boca Raton, FL (2004a), pp. 44:1–44:18.Google Scholar
  11. Burke, E. K., P. De Causmaecker, G. Vanden Berghe, and H. Van Landeghem, “The state of the art of nurse rostering,” Journal of Scheduling, 7(6), 441–499 (2004b).CrossRefGoogle Scholar
  12. Caprara, A., M. Fischeti, and P. Toth. “A heuristic method for the set covering problem,” Operations Research, 47(5), 730–743 (1999).Google Scholar
  13. Caprara, A., M. Monaci, and P. Toth, “Models and algorithms for a staff scheduling problem,” Mathematical Programming, Series B, 98, 445–476 (2003).CrossRefGoogle Scholar
  14. Cheng, B. M. W., J. H. M. Lee, and J. C. K. Wu, “A nurse rostering system using constraint programming and redundant modeling,” IEEE Transactions in Information Technology in Biomedicine, 1(1), 44–54 (1997).CrossRefGoogle Scholar
  15. Crainic, T. G., A. Frangioni, and B. Gendron, “Bundle-based relaxation methods for multicommodity capacitated fixed charge network design,” Discrete Applied Mathematics, 112, 73–99 (2001).CrossRefGoogle Scholar
  16. De Causmaecker, P. and G. Vanden Berghe, “Relaxation of coverage constraints in hospital personnel rostering,” in: E. K. Burke and P. De Causmaecker (Eds.), Practice and Theory of Automated Timetabling, Vol. IV, 4th International Conference, PATAT 2002, Gent, Belgium, LNCS, Springer, Berlin (2003), Vol. 2740, pp. 129–147.Google Scholar
  17. Dowsland, K. A., “Nurse scheduling with tabu search and strategic oscillation,” European Journal of Operational Research, 106(2–3), 393–407 (1998).CrossRefGoogle Scholar
  18. Emmons, H., “Work-force scheduling with cyclic requirements and constraints on days off, weekends off, and work stretch,” IIE Transactions, 17(1), 8–15 (1985).Google Scholar
  19. Ernst, A.T., H. Jiang, M. Krishnamoorthy, and D. Sier, “Staff scheduling and rostering: a review of applications, methods and models,” European Journal of Operational Research, 153, 3–27 (2004).CrossRefGoogle Scholar
  20. Ferland, J.A., I. Berrada, I. Nabli, B. Ahiod, P. Michelon, V. Gascon, and E. Gagné, “Generalized assignment type goal programming problem: application to nurse scheduling,” Journal of Heuristics, 7, 391–413 (2001).CrossRefGoogle Scholar
  21. Frangioni, A. and G. Gallo, “A bundle dual-ascent approach to linear multicommodity min-cost flow problems,” INFORMS Journal on Computing, 11, 370–393 (1999).Google Scholar
  22. Griesmer, H., “Self-scheduling turned us into a winning team,” Management Decisions, 56(12), 21–23 (1993).Google Scholar
  23. Howell, J. P., “Cyclical scheduling of nursing personnel,” Hospital J.A.H.A., 40, 77–85 (1998).Google Scholar
  24. Isken, M., “An implicit tour scheduling problem with application in healthcare,” Annals of Operations Research, 128, 91–109 (2004).CrossRefGoogle Scholar
  25. Jaumard, B., F. Semet, and T. Vovor, “A generalized linear programming model for nurse scheduling,” European Journal of Operational Research, 107, 1–18 (1998).CrossRefGoogle Scholar
  26. Kawanaka, H., K. Yamamoto, T. Yoshikawa, T. Shinogi, and S. Tsuruoka, “Genetic algorithm with constraints for the nurse scheduling problem,” in: Proceedings of Congress on Evolutionary Computation, IEEE Press, Seoul, South Korea (2001), Vol. 2, pp. 1123–1130.Google Scholar
  27. Kimball, B. and E. O’Neil, “The American nursing shortage,” The Robert Wood Johnson Foundation, Princeton, NJ (2002).Google Scholar
  28. Lau, H.C., “On the complexity of manpower shift scheduling,” Computers & Operations Research, 23(1), 93–102 (1996).CrossRefGoogle Scholar
  29. Lemarechal, C., “Nondifferentiable optimization,” in: G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd (Eds.), Handbooks in Operations Research and Management Science, Vol. 1: Optimization, North-Holland, Amsterdam, The Netherlands (1989), pp. 529–572.Google Scholar
  30. Meyer auf’ m Hofe, H. “ComPlan/SIEDAPlan: personnel assignment as a problem of hierarchical constraint satisfaction,” in: Proceedings of the 3rd International Conference on the Practical Application of Constraint Technology, London, (1997), pp. 257–271.Google Scholar
  31. Meyer auf’ m Hofe, H. “Solving rostering tasks as constraint optimisation, in E.K Burke and W. Erben (Eds.), Practice and Theory of Automated Timetabling, Vol. III, 3rd International Conference, PATAT 2000, Konstanz, Germany, LNCS, Vol. 2079, pp. 191–212, Springer, Berlin (2001).Google Scholar
  32. Millar, H.H. and M. Kiragu, “Cyclic and non-cyclic scheduling of 12-hour shift nurses by network programming,” European Journal of Operational Research, 104, 582–592 (1998).CrossRefGoogle Scholar
  33. Miller, H.E., W.P. Pierskalla, and G. J. Rath, “Nurse scheduling using mathematical programming,” Operations Research, 24(5), 857–870 (1976).Google Scholar
  34. Nemhauser, G.L. and L.A. Wolsey, Integer and Combinatorial Optimization, Wiley, New York (1988).Google Scholar
  35. Nonobe, K. and T. Ibaraki, “A tabu search approach to the constraint satisfaction problem as a general problem solver,” European Journal of Operational Research, 106, 599–623 (1998).CrossRefGoogle Scholar
  36. Petrovic, S., G. Beddoe, and G. Vanden Berghe, “Storing and adapting repair experiences in employee rostering,” in: E. K. Burke and P. De Causmaecker (Eds.), Practice and Theory of Automated Timetabling, Vol. IV, 4th International Conference, PATAT 2002, Gent, Belgium, LNCS, (2003), Vol. 2740, pp. 148–165.Google Scholar
  37. Pierskalla, W.P. and D.J. Brailer, “Applications of operations research in health care delivery,” Handbooks in Operations Research and Management Science, North Holland, Amsterdam, The Netherlands (1994), Vol. 6, pp. 469–505.Google Scholar
  38. Randhawa, S.U. and D. Sitompul, “A heuristic-based computerized nurse scheduling system,” Computer & Operations Research, 20(8), 837–844 (1993).CrossRefGoogle Scholar
  39. Spratley, E., A. Johnson, J. Sochalski, M. Fritz, and W. Spencer, “The registered nurse population,” Findings from the National Sample Survey of Registered Nurses,” U.S. Department of Health and Human Services (2000).Google Scholar
  40. Topaloglu, S. and I. Ozkarahan, “An implicit goal programming model for the tour scheduling problem considering the employee work preferences,” Annals of Operations Research, 128, 135–158 (2004).CrossRefGoogle Scholar
  41. Valouxis, C. and E. Housos, “Hybrid optimization techniques for the workshift and rest assignment of nursing personnel,” Artificial Intelligence in Medicine, 20, 155–175 (2000).CrossRefGoogle Scholar
  42. Warner, D.M., “Scheduling nursing personnel according to nursing preference: a mathematical programming approach,” Operations Research, 24(5), 842–856 (1976).CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Graduate Program in Operations Research & Industrial Engineering, 1 University Station C2200The University of TexasAustinUSA
  2. 2.American AirlinesAMR Corp Headquarter HDQ1Fort WorthUSA

Personalised recommendations