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A Comparison of Neighbourhood Topologies for Staff Scheduling

With Particle Swarm OptimisationTU Ilmenau

Department of Commercial InformationTechnology for Services (WI2)

Dipl. Wirt.-Inf. Maik Günther maik.guenther@gmx.de

Prof. Dr. Volker Nissen volker.nissen@tu-ilmenau.de

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• Description of the Application Problem I

• Description of the Application Problem II

• Particle Swarm Optimization

• Results and Conclusion

Structure of presentation

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Application problem I

• 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™

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Demand for personnel at the 9 workstations

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• 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• ...

Input and constraints

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• Description of the Application Problem I

• Description of the Application Problem II

• Particle Swarm Optimization

• Results and Conclusion

Structure of presentation

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• 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

Application problem II

Create a rooster with automated generated working time models!

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• 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 • …

Input and constraints

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• Description of the Application Problem I

• Description of the Application Problem II

• Particle Swarm Optimization

• Results and Conclusion

Structure of presentation

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• numbers• 0: employee is not available• all other numbers correspond to

workstations• only in problem II 3: dummy

workstation (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]

Problem representation for PSO

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• 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)

Overall outline of PSO approach

initialize the swarmcalculate fitness of initial particlesdetermine pBest for each particle and gBestrepeat

for i = 1 to number of particlescalculate new position with 4 actionsrepair particlecalculate fitnessnew pBest and new gBest?

next iuntil termination criterion holdsoutput gBest from current run

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• 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

Calculate the new position with4 actions & repair particles

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Neighbourhood topologies

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• Description of the Application Problem I

• Description of the Application Problem II

• Particle Swarm Optimization

• Results and Conclusion

Structure of presentation

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Results for the application problem I – the whole week

• mean error over 30 runs

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Results for the application problem I – individual days

• week (problem I) is divided into days• PSO with 20 particles• mean error over 30 runs

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Results for the application problem II

• PSO with 20 particles• mean error over 30 runs

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• gBest topology is mostly superior to the other topologies• if the information is not available to all particles

immediately, there is a danger of its being lost• with action nr. 2 all particles always remain in motion

and 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 )

Conclusions

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Data sets and benchmarks

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1. 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

2. Brodersen O., Schumann M. (2007) Einsatz der Particle Swarm Optimization zur Optimierung universitärer Stundenpläne. Technical Report 05/2007, Univ. of Göttingen

3. 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

4. Garey M.R., Johnson D.S. (1979) Computers and Intractability. A Guide to the Theory of NP-Completeness

5. Kennedy J., Eberhart R.C., Shi Y. (2001) Swarm Intelligence. Kaufmann, San Francisco6. Kragelund L., Kabel T. (1998) Employee Timetabling. An Empirical Study, Master's Thesis,

Univ. of Aarhus7. Tien J., Kamiyama A. (1982) On Manpower Scheduling Algorithms, SIAM 24(3): 275-2878. Nissen V., Günther M. (2009) Staff Scheduling With Particle Swarm Optimisation and

Evolution Strategies, In: Proceedings of EvoCOP 2009, LNCS, Vol. 5482: 228-239

References