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A Comparison of Metaheuristics on a Practical Staff Scheduling Problem

A Comparison of Metaheuristics on a Practical Staff Scheduling Problem TU Ilmenau Department of Commercial Information Technology for Services (WI2). 1. Dipl. Wirt.-Inf. Maik Günther maik.guenther@gmx.de Prof. Dr. Volker Nissen volker.nissen@tu-ilmenau.de.

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A Comparison of Metaheuristics on a Practical Staff Scheduling Problem

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  1. A Comparison of Metaheuristics on a Practical Staff Scheduling Problem 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 • Particle Swarm Optimization • Evolution Strategies • 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) • 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 3 Description of the application problem

  4. 4 Demand for personnel at the 9 workstations

  5. 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! 5 Current planning

  6. 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 • ... 6 Input and constraints

  7. numbers • 0: employee is not working • 1-9: correspond to workstations • based on two-dimensional matrix • time is viewed as discrete • 65 rows and 560 columns = 36.400 dimensions • Garey and Johnson demonstrate that even simple versions of staff scheduling problems are NP-hard [7] • Kragelund and Kabel show the NP-hardness of the general employee timetabling problem [9] 7 Problem representation for PSO and ES

  8. Description of the Application Problem • Particle Swarm Optimization • Evolution Strategies • Results and Conclusion 8 Structure of presentation

  9. 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 9 Overall outline of PSO approach

  10. for each element (> 0) of the matrix • probability to chose one of the 4 actions • no change • random workstation • workstation from pBest at the same position • workstation from gBest at the same position • PSO (and ES) can be improved with a repair • repair in the following order (descending error point size) • qualifications • overstaffing and understaffing • rotations of workstations 10 Calculate the new position with4 actions & repair particles

  11. Description of the Application Problem • Particle Swarm Optimization • Evolution Strategies • Results and Conclusion 11 Structure of presentation

  12. each individual of the population embodies a solution to the problem • they search for optima with the aid of a fitness function • primarily search operator is mutation • self-adaption of mutation step size • each individual has a strategic parameter which will be mutated and recombined • higher probability for individuals with a good strategic parameter to survive • 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 • repair offspring • calculate fitness for offspring • select the new population • until termination criterion holds • output best solution from current run 12 Overall outline of evolutionary approach

  13. selection • deterministic, non-elitist comma- and plus-selection • following suggestions in the literature [2] [3], the ratio μ/λ is set to 1/5 – 1/7 • (1,5), (1+5), (10,50), (10+50), (30,200) and (30+200) • best solution kept in “golden cage” (not part of population) • recombination • recombination of two parent solutions ((10,50), (10+50), (30,200), (30+200)) • two parents have a random crossover point for all employees parent 1 parent 2 offspring 13 Draw and recombine parent solutions & select the new population

  14. self adaptive step size for mutation • σ = strategic parameter • 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 14 Mutate offspring – with classical Gaussian mutation

  15. the principle of maximum entropy is used in [9] to construct a mutation distribution for unbounded integer search spaces • the difference (Z) of two independent geometricallydistributed random numbers (G1 and G2) is addedto each element of the matrix • G1 and G2 have the parameter p which iscontrolled by the strategic parameter • the problem of the logistics service provider is bounded(9 workstations), many 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 15 Mutate offspring – with the principle of maximum entropy [9]

  16. Description of the Application Problem • Particle Swarm Optimization • Evolution Strategies • Results and Conclusion 16 Structure of presentation

  17. 17 Results for the application problem

  18. 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) • mutation with the concept of maximum entropy better fits the combinatorial domain than classical Gaussian mutations • make small changes in one iteration/generation • future research • create further test problems with the aid of cooperating companies • adapt other heuristics from roughly comparable problems in the literature 18 Conclusions

  19. 19 Data sets and benchmarks

  20. ATOSS Software AG, FH Heidelberg (2006) (ed.) Standort Deutschland 2006. Zukunftssicherung durch intelligentes Personalmanagement. München Bäck T. (2002) (ed.) Handbook of Evolutionary Computation. Institute of Phys. Publ., Bristol Beyer H.-G., Schwefel, H.-P. (2002) Evolution strategies: a comprehensive introduction. Nat. Comp. 1: 3-52 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) Einsatz der Particle Swarm Optimization zur Optimierung universitärer Stundenpläne. 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 Rudolph, G. (1994) An evolutionary algorithm for integer programming. PPSN III, Jerusalem, Israel, Proceedings, LNCS, Vol. 866: 139-148 Tien J., Kamiyama A. (1982) On Manpower Scheduling Algorithms, SIAM 24(3): 275-287 Proudfoot Consulting (2007) Produktivitätsbericht 2007. Company Report 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

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