A Comparison of Neighbourhood Topologies for Staff Scheduling With Particle Swarm Optimisation

1. A Comparison of Neighbourhood Topologies for Staff Scheduling With Particle Swarm Optimisation TU IlmenauDepartment of Commercial InformationTechnology for Services (WI2) 1 Dipl. Wirt.-Inf. Maik Günther maik.guenther@gmx.de Prof. Dr. Volker Nissen volker.nissen@tu-ilmenau.de

2. Description of the Application Problem I • Description of the Application Problem II • Particle Swarm Optimization • Results and Conclusion 2 Structure of presentation

3. originates from a German logistics service provider which operates in a spatially limited area 7 days a week almost 24 hours a day • nine workstations • 65 employees on duty with different start and end times according to their work-time models • employees are quite flexible in terms of working hours (13 different working time models) • strict regulations e.g. with regard to qualifications (damage, injuries) • personnel demand is given in 15-minute intervals with large variations for individual workstations during the day • monthly staff scheduling is carried out with MS EXCEL™ • they are not able to make sub-daily workstation-rotations with MS EXCEL™ 3 Application problem I

4. 4 Demand for personnel at the 9 workstations

5. input • full-day assignment (determines availability of personnel) • demand for personnel at the nine workstations in 15-minute intervals • matrix of qualifications (employees and workstations) • relevant constraints (constraints are penalised with error points) • presence and absence • one employee can only assigned to one workstation at a time • qualifications • avoid over- and understaffing • no unnecessary workstation-rotations • ... 5 Input and constraints

6. Description of the Application Problem I • Description of the Application Problem II • Particle Swarm Optimization • Results and Conclusion 6 Structure of presentation

7. department of a store (clothes) • each day 10 hours (from Monday to Saturday) • 15 employees with different contracts(weekly working time 10, 20, 25, 30, 38 and 40 hours) • 2 workplaces (sales and cash register) • variable customer frequency during the day variable personnel demand with large variations for individual workstations during the day • demand is given in 1-hour intervals for 1 year Create a rooster with automated generated working time models! 7 Application problem II

8. hard constraints • only available employees are scheduled • only one workstation per employee at a time • working time models in 1-hour intervals • soft constraints (error points): • employees should not work more than their maximal working time per week • keep minimal/maximal allowed length of working time models • only one or no working time model per employee a day • avoid over- and understaffing • no unnecessary workstation-rotations • … 8 Input and constraints

9. Description of the Application Problem I • Description of the Application Problem II • Particle Swarm Optimization • Results and Conclusion 9 Structure of presentation

10. numbers • 0: employee is not available • all other numbers correspond toworkstations • only in problem II  3: dummyworkstation (employee is not working) • based on two-dimensional matrix and time is viewed as discrete • problem I: 65 rows * 560 columns = 36.400 dimensions • problem II: 15 rows * 8.760 columns = 131.400 dimensions • Garey and Johnson demonstrate that even simple versions of staff scheduling problems are NP-hard [4] • Kragelund and Kabel show the NP-hardness of the general employee timetabling problem [6] 10 Problem representation for PSO

11. population-based modern heuristic • swarm members are assumed to be massless particles • each particle together with its position within a solution space embodies a solution to the problem • they search for optima with the aid of a fitness function • particles exchange information, which can positively influence the development of the population as a whole (pBest, gBest/lBest) • termination of PSO after 400.000 inspected solutions (to keep results comparable) • initialize the swarm • calculate fitness of initial particles • determine pBest for each particle and gBest • repeat • for i = 1 to number of particles • calculate new position with 4 actions • repair particle • calculate fitness • new pBest and new gBest? • next i • until termination criterion holds • output gBest from current run 11 Overall outline of PSO approach

12. for each element (> 0) of the matrix • probability to chose one of the 4 actions • 1: no change • 2: random workstation • 3: workstation from pBest at the same position • 4: workstation from gBest at the same position • PSO can be improved with a repair • repair in the following order (descending error point size) • working time models  only in problem II • qualifications • overstaffing and understaffing • rotations of workstations 12 Calculate the new position with4 actions & repair particles

13. gBest • all particles are connected to each other • each particle immediately knows the best global value found up to that point • wheel • only 1 particle is full connected • circle/ring (k = 2) • every particle has 2 neighbours • lBest (k = 4) • every particle has 4 neighbours 13 Neighbourhood topologies

14. Description of the Application Problem I • Description of the Application Problem II • Particle Swarm Optimization • Results and Conclusion 14 Structure of presentation

15. mean error over 30 runs 15 Results for the application problem I – the whole week

16. week (problem I) is divided into days • PSO with 20 particles • mean error over 30 runs 16 Results for the application problem I – individual days

17. PSO with 20 particles • mean error over 30 runs 17 Results for the application problem II

18. gBest topology is mostly superior to the other topologies • if the information is not available to all particlesimmediately, there is a danger of its being lost • with action nr. 2 all particles always remain in motionand have the chance to escape from a local optimum • gBest performs not always best on easier problems • it is much easier for the swarm to find "good" assignments • the influence of maximised information exchange throughout the swarm is reduced • the loss of "good" allocations can be compensated by other topologies for avoiding premature convergence to a local optimum • this suggests a relationship between the complexity of instances from an identical class of problems and the effectiveness of PSO neighbourhood topologies • small swarm sizes yield better results than large ones • make small changes in one iteration • PSO is easy to use (2 parameters: swarm size and action nr. 2 ) 18 Conclusions

19. 19 Data sets and benchmarks

20. Chu S.C., Chen Y.T., Ho J.H. (2006) Timetable Scheduling Using Particle Swarm Optimization. In: Proceedings of ICICIC Beijing 2006, Vol. 3: 324-327 Brodersen O., Schumann M. (2007) Einsatzder Particle Swarm Optimization zurOptimierunguniversitärerStundenpläne. Technical Report 05/2007, Univ. of Göttingen Ernst A.T., Jiang H., Krishnamoorthy M., Owens B., Sier D. (2002) An Annotated Bibliography of Personnel Scheduling and Rostering. Annals of OR 127: 21-144 Garey M.R., Johnson D.S. (1979) Computers and Intractability. A Guide to the Theory of NP-Completeness Kennedy J., Eberhart R.C., Shi Y. (2001) Swarm Intelligence. Kaufmann, San Francisco Kragelund L., Kabel T. (1998) Employee Timetabling. An Empirical Study, Master's Thesis, Univ. of Aarhus Tien J., Kamiyama A. (1982) On Manpower Scheduling Algorithms, SIAM 24(3): 275-287 Nissen V., Günther M. (2009) Staff Scheduling With Particle Swarm Optimisation and Evolution Strategies, In: Proceedings of EvoCOP 2009, LNCS, Vol. 5482: 228-239 20 References