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Automatic generation of optimized working time

models in personnel planning

TU IlmenauDepartment of Commercial Information Technology

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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• bank holiday on Thursday

• workforce management is not demand driven

• high personnel costs

• loss of sales

Practical example – overstaffing and understaffing

time

em

plo

yees

requirement

personnel hours

loss sales revenues

excessivepersonnel

costs

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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

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):

• only one or no working time model per employee a day

• keep minimal/maximal allowed length of working time models

• avoid over- and understaffing

• avoid unnecessary workstation rotations

• employees should not work more than their maximal working time per week

Input and constraints

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• problems in retailing with automated generated working time models

• but only 1 workstation

• some differences in constraints

• smaller planning horizon

• Prüm [9] was not able to solve the MIP in reasonable time solved the relaxed LP and transformed the result (real values) to a solution (integer values)

• Sauer and Schumann [10] uses a constructive heuristic

Work related to the application problem

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• numbers

• 0: store is closed / employee is not available

• 1-2: correspond to workstations

• 3: dummy workstation (employee is not working)

• based on two-dimensional matrix

• time is viewed as discrete

• 8.760 rows and 15 columns = 131.400 dimensions

• Garey and Johnson demonstrate that even simple versions of staff scheduling problems are NP-hard [6].

• Kragelund and Kabel show the NP-hardness of the general employee timetabling problem [8].

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

• 4 actions

• no change

• random workstation

• workstation from pBest at the same position

• workstation from gBest at the same position

Calculate the new position with4 actions

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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))

• one 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

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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• 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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• ES-approach with (1,5)-selection and repair is the most effective heuristic for this problem

• plus-selection often get stuck in local optima

• comma-selection has a higher ability to escape from local optima explore other regions (with worse results over some generations on the way to other regions)

• PSO is easy to use (2 important parameters swarm size and probability to set a random workstation)

• make small changes in one iteration/generation

• future research

• create further test problems with the aid of cooperating companies

• adapt other heuristics from roughlycomparable problems in the literature

Conclusions

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Data sets and benchmarks

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1. Bäck T. (2002) (ed.) Handbook of Evolutionary Computation. Institute of Physics Publishing, Bristol

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

3. 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

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

5. 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

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

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

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

9. Prüm H. (2006) Entwicklung von Algorithmen zur Personaleinsatzplanung mittels ganzzahliger linearer Optimierung. Master's Thesis, FH Trier

10. Sauer J., Schumann R. (2007) Modelling and Solving Workforce Scheduling Problems. in: Sauer J., Edelkamp S., Schattenberg (ed.): Proceedings of the 21th PuK 2007: 93-101.

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

References