Staff Scheduling with Particle Swarm Optimisation and Evolution Strategies

1. EVO Cop 2009, Tuebingen Staff Scheduling with Particle Swarm Optimisation and Evolution Strategies Dipl. Wirt.-Inf. Maik Günther maik.guenther@gmx.de Prof. Dr. Volker Nissen volker.nissen@tu-ilmenau.de TU IlmenauDepartment of Commercial Information Technology for Services (WI2)

2. Structure of Presentation • Sub Daily / Sub Shift Staff Scheduling • Particle Swarm Optimisation • Evolution Strategies • Results and Conclusion

3. Sub Daily / Sub Shift Staff Scheduling

4. Requirement Hours worked Personnel Hours Overstaffing Understaffing Time elapsed Introduction I • „Five R‘s“: • right qualified employee • right number of employees • at the right time • at the right place • at the right (optimal) costs • Garey and Johnson demonstrate that even simple versions of staff scheduling problems are NP-hard [8]. • Kragelund and Kabel show the NP-hardness of the general employee timetabling problem [10].

5. Introduction II • employees spend 27 to 36% of their working time unproductive, depending on the branch[12] • often staff scheduling takes place based on experience or with the aid of spreadsheets [1] • even with staff planning software employees are regularly scheduled for one workstation per day • in many branches the one-employee-one-station concept does not correspond to the actual requirements and sacrifices potential resources • service industry (for instance logistics), commercial trade, etc. • sub-daily (sub-shift) planning should be an integral component of demand driven staff scheduling

6. Description of the Application Problem • 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) • many employees are qualified to work at different workstations • 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

7. Demand for Personnel at the Nine Workstations

8. Current Planning • monthly staff scheduling is carried out with MS EXCEL • they are not able to make sub-daily workstation-rotations with MS EXCEL • employees are assigned on a full-day basis  large phases of over- and understaffing • floor managers intervene on-site by relocating employees ad hoc (reacting instead of ahead-planning) Demand driven staff scheduling cannot be realised today!

9. Sub-Daily Staff Scheduling • 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 • timesheet balances • qualifications • no unnecessary workstation-rotations • one employee can only assigned to one workstation at a time • ....

10. Problem Representation for PSO and ES • numbers • 0: employee is not working • 1-9: correspond to workstations • based on two-dimensional matrix (65 rows and 560 columns = 36,400) • time is viewed as discrete

11. Particle Swarm Optimisation

12. Overall Outline of PSO Approach • 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 // 4 actions • calculate fitness • new pBest? / new gBest? • next i • until termination criterion holds • output gBest from current run

13. 4 Actions to Calculate the new Position • for each element (> 0) of the matrix • probability to chose one of the 4 actions • 4 actions • no change • random workstation (no qualification errors) • workstation from pBest at the same position • workstation from gBest at the same position

14. Evolution Strategies

15. Overall Outline of Evolutionary Approach • termination of ES • after 400.000 inspected solutions (to keep results comparable) • initialize the population • calculate fitness of initial population • repeat • draw and recombine parent solutions • mutate offspring • calculate fitness for offspring • select the new population • until termination criterion holds • output best solution from current run

16. Details of the Approach • selection • deterministic, non-elitist comma- and plus-selection • following suggestions in the literature [2] [3], the ratio / is set to 1/5 – 1/7 • (1,5)-, (1+5)-, (10,50)-, (10+50)-, (30,200)- and (30+200)-selection • best solution kept in “golden cage” (not part of population) • recombination • recombination of two parent solutions ((10,50), (10+50), (30,200), (30+200)) • random crossover point for each employee

17. Mutation of Solutions • self adaptive step size for mutation • mutation creates only valid solutions (no availability and qualification errors) • τ = 0,1 • σ‘ = σ * exp(τ * N(0,1)) • Count = round│N(0,σ‘)│ • if Count < 1 then Count = 1 • for i = 1 to Count • random employee e • random time interval t • random workstation • change value at matrix element (e,t) • next i

18. Mutation with the Principle of Maximum Entropy [14] • the principle of maximum entropy is used in [14] to construct a mutation distribution for unbounded integer search spaces • the difference (Z) of two independent geometrically distributed random numbers (G1 and G2) is added to each element of the matrix • G1 and G2 have the parameter p which is controlled by the step size • the problem of the logistics service provider is bounded (9 workstations), much more dimensions and special constraints • τ² = 17,07/n instead of τ² = 1/n • no availability and qualification errors • recombination „nr. 5“ instead of uniform crossover • Z was too small  now Z has a greater variance to reach all possible workstations

19. Results and Conclusion

20. Results for the Logistic Service Provider Problem Indication of absolute minimum: PSO with repair: 51,521 error points Results averaged over 30 runs each. All tests were conducted on a standard PC.

21. Conclusion • PSO-approach is the most effective heuristic for this problem • PSO is easy to use (2 important parameters  swarm size and probability to set a random workstation) • exchange of information (gBest and pBest) • make small changes in one interation/generation • future research • create further test problems with the aid of cooperating companies • adapt other heuristics from roughly comparable problems in the literature

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