Automatic generation of optimized working time models in personnel planning

1. Automatic generation of optimized working time models in personnel planning TU IlmenauDepartment 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

2. Description of the Application Problem • Particle Swarm Optimization • Evolution Strategies • Results and Conclusion 2 Structure of presentation

3. bank holiday on Thursday • workforce management is not demand driven • high personnel costs • loss of sales requirement personnel hours loss sales revenues excessivepersonnelcosts employees time 3 Practical example – overstaffing and understaffing

4. 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 Create a rooster with automated generated working time models! 4 Application problem

5. 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 5 Input and constraints

6. 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 6 Work related to the application problem

7. 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]. 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 • 4 actions • no change • random workstation • workstation from pBest at the same position • workstation from gBest at the same position 10 Calculate the new position with4 actions

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)) • one 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

15. Description of the Application Problem • Particle Swarm Optimization • Evolution Strategies • Results and Conclusion 15 Structure of presentation

16. 16 Results for the application problem

17. 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 17 Conclusions

18. 18 Data sets and benchmarks

19. Bäck T. (2002) (ed.) Handbook of Evolutionary Computation. Institute of Physics Publishing, 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) Einsatzder Particle Swarm Optimization zurOptimierunguniversitärerStundenpläne. 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 Prüm H. (2006) Entwicklung von Algorithmen zur Personaleinsatzplanung mittels ganzzahliger linearer Optimierung. Master's Thesis, FH Trier Sauer J., Schumann R. (2007) Modelling and Solving Workforce Scheduling Problems. in: Sauer J., Edelkamp S., Schattenberg (ed.): Proceedingsofthe 21th PuK 2007: 93-101. Tien J., Kamiyama A. (1982) On Manpower Scheduling Algorithms, SIAM Rev. 24(3): 275-287 19 References