Staff assignment with lexicographically ordered acceptance levels



Staff assignment is a compelling exercise that affects most companies and organizations in the service industries. Here, we introduce a new real-world staff assignment problem that was reported to us by a Swiss provider of commercial employee scheduling software. The problem consists of assigning employees to work shifts subject to a large variety of critical and noncritical requests, including employees’ personal preferences. Each request has a target value, and deviations from the target value are associated with integer acceptance levels. These acceptance levels reflect the relative severity of possible deviations, e.g., for the request of an employee to have at least two weekends off, having one weekend off is preferable to having no weekend off and thus receives a higher acceptance level. The objective is to minimize the total number of deviations in lexicographical order of the acceptance levels. Staff assignment approaches from the literature are not applicable to this problem. We provide a binary linear programming formulation and propose a matheuristic for large-scale instances. The matheuristic employs effective strategies to determine the subproblems and focuses on finding good feasible solutions to the subproblems rather than proving their optimality. Our computational analysis based on real-world data shows that the matheuristic scales well and outperforms commercial employee scheduling software.


Employee scheduling Staff assignment Real-world problem Goal programming Binary linear programming Matheuristic 


  1. Aickelin, U., & Dowsland, K. (2000). Exploiting problem structure in a genetic algorithm approach to a nurse rostering problem. Journal of Scheduling, 3(3), 139–153.CrossRefGoogle Scholar
  2. Aickelin, U., & Dowsland, K. A. (2004). An indirect genetic algorithm for a nurse-scheduling problem. Computers and Operations Research, 31(5), 761–778.CrossRefGoogle Scholar
  3. Al-Yakoob, S., & Sherali, H. (2007). Mixed-integer programming models for an employee scheduling problem with multiple shifts and work locations. Annals of Operations Research, 155(1), 119–142.CrossRefGoogle Scholar
  4. Azaiez, M. N., & Al Sharif, S. (2005). A 0–1 goal programming model for nurse scheduling. Computers and Operations Research, 32(3), 491–507.CrossRefGoogle Scholar
  5. Bai, R., Burke, E. K., Kendall, G., Li, J., & McCollum, B. (2010). A hybrid evolutionary approach to the nurse rostering problem. IEEE Transactions on Evolutionary Computation, 14(4), 580–590.CrossRefGoogle Scholar
  6. Ball, M. O. (2011). Heuristics based on mathematical programming. Surveys in Operations Research and Management Science, 16(1), 21–38.CrossRefGoogle Scholar
  7. Bard, J. F., & Wan, L. (2006). The task assignment problem for unrestricted movement between workstation groups. Journal of Scheduling, 9(4), 315–341.CrossRefGoogle Scholar
  8. Beaulieu, H., Ferland, J. A., Gendron, B., & Michelon, P. (2000). A mathematical programming approach for scheduling physicians in the emergency room. Health Care Management Science, 3(3), 193–200.CrossRefGoogle Scholar
  9. Berrada, I., Ferland, J. A., & Michelon, P. (1996). A multi-objective approach to nurse scheduling with both hard and soft constraints. Socio-Economic Planning Sciences, 30(3), 183–193.CrossRefGoogle Scholar
  10. Bertels, S., & Fahle, T. (2006). A hybrid setup for a hybrid scenario: Combining heuristics for the home health care problem. Computers and Operations Research, 33(10), 2866–2890.CrossRefGoogle Scholar
  11. Bester, M., Nieuwoudt, I., & Van Vuuren, J. H. (2007). Finding good nurse duty schedules: A case study. Journal of Scheduling, 10(6), 387–405.CrossRefGoogle Scholar
  12. Bixby, R. E. (2012). A brief history of linear and mixed-integer programming computation. Documenta Mathematica, Extra Volume: Optimization Stories, 107–121.Google Scholar
  13. Boschetti, M. A., Maniezzo, V., Roffilli, M., & Bolufé Röhler, A. (2009). Matheuristics: Optimization, simulation and control. In M. J. Blesa, C. Blum, L. Di Gaspero, A. Roli, M. Sampels, & A. Schaerf (Eds.), Hybrid metaheuristics: 6th international workshop on hybrid metaheuristics (pp. 171–177). Heidelberg: Springer.Google Scholar
  14. Chang, C. T. (2006). Mixed binary interval goal programming. Journal of the Operational Research Society, 57, 469–473.CrossRefGoogle Scholar
  15. Chang, C. T., & Lin, T. C. (2009). Interval goal programming for s-shaped penalty function. European Journal of Operational Research, 199, 9–20.CrossRefGoogle Scholar
  16. Charnes, A., & Collomb, B. (1972). Optimal economic stabilization policy: Linear goal-interval programming models. Socio-Economic Planning Sciences, 6(4), 431–435.CrossRefGoogle Scholar
  17. Charnes, A., Cooper, W. W., & Ferguson, R. O. (1955). Optimal estimation of executive compensation by linear programming. Management Science, 1, 138–151.CrossRefGoogle Scholar
  18. Charnes, A., Cooper, W. W., Harrald, J., Karwan, K. R., & Wallace, W. A. (1976). A goal interval programming model for resource allocation in a marine environmental protection program. Journal of Environmental Economics and Management, 3, 347–362.CrossRefGoogle Scholar
  19. Cordeau, J. F., Gendreau, M., Laporte, G., Potvin, J. Y., & Semet, F. (2002). A guide to vehicle routing heuristics. Journal of the Operational Research Society, 53, 512–522.CrossRefGoogle Scholar
  20. Cordeau, J. F., Laporte, G., Pasin, F., & Ropke, S. (2010). Scheduling technicians and tasks in a telecommunications company. Journal of Scheduling, 13(4), 393–409.CrossRefGoogle Scholar
  21. De Bruecker, P., Van den Bergh, J., Beliën, J., & Demeulemeester, E. (2015). Workforce planning incorporating skills: State of the art. European Journal of Operational Research, 243(1), 1–16.CrossRefGoogle Scholar
  22. Della Croce, F., & Salassa, F. (2014). A variable neighborhood search based matheuristic for nurse rostering problems. Annals of Operations Research, 218(1), 185–199.CrossRefGoogle Scholar
  23. Dowsland, K. A. (1998). Nurse scheduling with tabu search and strategic oscillation. European Journal of Operational Research, 106(2), 393–407.CrossRefGoogle Scholar
  24. Eiselt, H. A., & Marianov, V. (2008). Employee positioning and workload allocation. Computers and Operations Research, 35(2), 513–524.CrossRefGoogle Scholar
  25. Ernst, A. T., Jiang, H., Krishnamoorthy, M., Owens, B., & Sier, D. (2004). An annotated bibliography of personnel scheduling and rostering. Annals of Operations Research, 127(1–4), 21–144.CrossRefGoogle Scholar
  26. Falasca, M., Zobel, C., & Ragsdale, C. (2011). Helping a small development organization manage volunteers more efficiently. Interfaces, 41(3), 254–262.CrossRefGoogle Scholar
  27. Fischetti, M., & Lodi, A. (2003). Local branching. Mathematical Programming, 98(1–3), 23–47.CrossRefGoogle Scholar
  28. Ignizio, J. (2004). Optimal maintenance headcount allocation: An application of Chebyshev goal programming. International Journal of Production Research, 42(1), 201–210.CrossRefGoogle Scholar
  29. Jones, D., & Tamiz, M. (1995). Expanding the flexibility of goal programming via preference modelling techniques. Omega, 23(1), 41–48.CrossRefGoogle Scholar
  30. Jones, D., & Tamiz, M. (2010). Goal programming variants. In Practical goal programming (pp. 11–22). Boston, MA: Springer US. doi:10.1007/978-1-4419-5771-9_2.
  31. Jones, D., & Tamiz, M. (2016). A review of goal programming. In S. Greco, M. Ehrgott, & J. R. Figueira (Eds.), Multiple criteria decision analysis: State of the art surveys (pp. 903–926). New York: Springer.CrossRefGoogle Scholar
  32. Jones, D. F., & Tamiz, M. (2002). Goal programming in the period 1990–2000. In M. Ehrgott & X. Gandibleux (Eds.), Multiple criteria optimization—State of the art annotated bibliographic surveys. Dordrecht: Kluwer Academic Publishers.Google Scholar
  33. Jones, D. F., Mirrazavi, S. K., & Tamiz, M. (2002). Multi-objective meta-heuristics: An overview of the current state-of-the-art. European Journal of Operational Research, 137(1), 1–9.CrossRefGoogle Scholar
  34. Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R. E., et al. (2011). MIPLIB 2010. Mathematical Programming Computation, 3(2), 103–163.CrossRefGoogle Scholar
  35. Kopanos, G. M., Méndez, C. A., & Puigjaner, L. (2010). MIP-based decomposition strategies for large-scale scheduling problems in multiproduct multistage batch plants: A benchmark scheduling problem of the pharmaceutical industry. European Journal of Operational Research, 207(2), 644–655.CrossRefGoogle Scholar
  36. Kvanli, A. H. (1980). Financial planning using goal programming. Omega, 8, 207–218.CrossRefGoogle Scholar
  37. Lodi, A. (2010). Mixed integer programming computation. In M. Jünger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, & L. A. Wolsey (Eds.), 50 years of integer programming 1958–2008: From the early years to the state-of-the-art (pp. 619–645). Heidelberg: Springer.Google Scholar
  38. Louly, M. A. O. (2013). A goal programming model for staff scheduling at a telecommunications center. Journal of Mathematical Modelling and Algorithms in Operations Research, 12(2), 167–178.CrossRefGoogle Scholar
  39. Maenhout, B., & Vanhoucke, M. (2008). Comparison and hybridization of crossover operators for the nurse scheduling problem. Annals of Operations Research, 159(1), 333–353.CrossRefGoogle Scholar
  40. Maniezzo, V., Stützle, T., & Voss, S. (2009). Matheuristics: Hybridizing metaheuristics and mathematical programming. New York: Springer.Google Scholar
  41. Mihaylov, M., Smet, P., Van Den Noortgate, W., & Vanden Berghe, G. (2016). Facilitating the transition from manual to automated nurse rostering. Health Systems, 5(2), 120–131.CrossRefGoogle Scholar
  42. Parr, D., & Thompson, J. M. (2007). Solving the multi-objective nurse scheduling problem with a weighted cost function. Annals of Operations Research, 155(1), 279–288.CrossRefGoogle Scholar
  43. Raidl, G. R., & Puchinger, J. (2008). Combining (integer) linear programming techniques and metaheuristics for combinatorial optimization. In C. Blum, M. J. B. Aguilera, A. Roli, & M. Sampels (Eds.), Hybrid metaheuristics: An emerging approach to optimization (pp. 31–62). Heidelberg: Springer.CrossRefGoogle Scholar
  44. Rihm, T., & Baumann, P. (2015a). Improving fairness in staff assignment: An approach for lexicographic goal programming. In T. Magnanti, K. Chai, R. Jiao, S. Chen, & M. Xie (Eds.), Proceedings of the 2015 IEEE international conference on industrial engineering and engineering management, Singapore (pp. 1247–1251).Google Scholar
  45. Rihm, T., & Baumann, P. (2015b). A lexicographic goal programming approach for staff assignment with acceptance levels. In Z. Hanzálek, G. Kendall, B. McCollum, & P. Šůcha (Eds.), Proceedings of the 7th multidisciplinary international conference on scheduling: Theory and applications, Prague (pp. 526–540).Google Scholar
  46. Romero, C. (2004). A general structure of achievement function for a goal programming model. European Journal of Operational Research, 153, 675–686.CrossRefGoogle Scholar
  47. Romero, C. (2014). Handbook of critical issues in goal programming. Oxford: Pergamon Press.Google Scholar
  48. Smet, P., & Vanden Berghe, G. (2012). A matheuristic approach to the shift minimisation personnel task scheduling problem. In D. Kjenstad, A. Riise, T. E. Nordlander, B. McCollum, & Burke (Eds.). Proceedings of the 9th international conference on the practice and theory of automated timetabling, Son (pp. 145–160).Google Scholar
  49. Smet, P., Bilgin, B., De Causmaecker, P., & Vanden Berghe, G. (2014a). Modelling and evaluation issues in nurse rostering. Annals of Operations Research, 218(1), 303–326.CrossRefGoogle Scholar
  50. Smet, P., Wauters, T., Mihaylov, M., & Vanden Berghe, G. (2014b). The shift minimisation personnel task scheduling problem: A new hybrid approach and computational insights. Omega, 46, 64–73.CrossRefGoogle Scholar
  51. Tamiz, M., Jones, D. F., & El-Darzi, E. (1995). A review of goal programming and its applications. Annals of Operations Research, 58, 39–53.CrossRefGoogle Scholar
  52. Topaloglu, S. (2006). A multi-objective programming model for scheduling emergency medicine residents. Computers and Industrial Engineering, 51(3), 375–388.CrossRefGoogle Scholar
  53. Valls, V., Pérez, Á., & Quintanilla, S. (2009). Skilled workforce scheduling in service centres. European Journal of Operational Research, 193(3), 791–804.CrossRefGoogle Scholar
  54. Van den Bergh, J., Belïen, J., De Bruecker, P., & Demeulemeester, E. (2013). Personnel scheduling: A literature review. European Journal of Operational Research, 226, 367–385.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Business AdministrationUniversity of BernBernSwitzerland

Personalised recommendations