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A Comparison of Metaheuristics on a Practical

Staff Scheduling ProblemTU 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

• Particle Swarm Optimization

• Evolution Strategies

• Results and Conclusion

Structure of presentation

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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

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Demand for personnel at the 9 workstations

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• 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)

Current planning

Demand driven staff scheduling cannot be realised today!

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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

• timesheet balances

• qualifications

• no unnecessary workstation-rotations

• one employee can only assigned to one workstation at a time

• ...

Input and constraints

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• 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]

Problem representation for PSO and ES

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• Description of the Application Problem

• Particle Swarm Optimization

• Evolution Strategies

• Results and Conclusion

Structure of presentation

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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

• 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

Calculate the new position with4 actions & repair particles

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• Description of the Application Problem

• Particle Swarm Optimization

• Evolution Strategies

• Results and Conclusion

Structure of presentation

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• 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)

Overall outline of evolutionary approach

initialize the populationcalculate fitness of initial populationrepeat

draw and recombine parent solutionsmutate offspringrepair offspringcalculate fitness for offspringselect the new population

until termination criterion holdsoutput best solution from current run

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• 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

Draw and recombine parent solutions & select the new population

parent 1 parent 2 offspring

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• self adaptive step size for mutation

• σ = strategic parameter

Mutate offspring – with classical Gaussian mutation

τ = 0,1σ‘ = σ * exp(τ * N(0,1))Count = round│N(0,σ‘)│if Count < 1 then Count = 1for i = 1 to Count

random employee e random time interval trandom workstation change value at matrix element (e,t)

next i

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• 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 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 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

Mutate offspring – with the principle of maximum entropy [9]

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• Description of the Application Problem

• Particle Swarm Optimization

• Evolution Strategies

• Results and Conclusion

Structure of presentation

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Results for the application problem

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• 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

Conclusions

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Data sets and benchmarks

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1. ATOSS Software AG, FH Heidelberg (2006) (ed.) Standort Deutschland 2006. Zukunftssicherung durch intelligentes Personalmanagement. München

2. Bäck T. (2002) (ed.) Handbook of Evolutionary Computation. Institute of Phys. Publ., Bristol

3. Beyer H.-G., Schwefel, H.-P. (2002) Evolution strategies: a comprehensive introduction. Nat. Comp. 1: 3-52

4. 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

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

6. 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

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

8. Kennedy J., Eberhart R.C., Shi Y. (2001) Swarm Intelligence. Kaufmann, San Francisco

9. Kragelund L., Kabel T. (1998) Employee Timetabling. An Empirical Study, Master's Thesis, Univ. of Aarhus

10. Rudolph, G. (1994) An evolutionary algorithm for integer programming. PPSN III, Jerusalem, Israel, Proceedings, LNCS, Vol. 866: 139-148

11. Tien J., Kamiyama A. (1982) On Manpower Scheduling Algorithms, SIAM 24(3): 275-287

12. Proudfoot Consulting (2007) Produktivitätsbericht 2007. Company Report

13. 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