1. © 2009 - 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 TU Ilmenau Department of Commercial Information Technology for Services (WI2) Staff Scheduling with Particle Swarm Optimisation and Evolution Strategies EVO Cop 2009, Tuebingen

2. © 2009 - TU Ilmenau, Department 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. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2)3 Sub Daily / Sub Shift Staff Scheduling

4. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 4 Overstaffing Understaffing Hours worked Personnel Hours Time elapsed Requirement 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. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 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. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 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. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 7 Demand for Personnel at the Nine Workstations

8. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 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. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 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. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 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. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 11 Work Related to the Concept of Sub-Daily Planning Vanden Berghe [16] demand is marked by sub-daily time periods (floating requirements) decoupling of staff demand from fixed shifts scheduling is made for shifts and not at the level of individual workstations Blöchlinger [4] introduces, timetabling blocks (TTBs) with individual beginning and end a task is always fixed to a TTB Schaerf and Meisels [11] a shift may include several tasks task is valid for the duration of a complete shift (no sub-daily changes) problem is depicted in a two-dimensional matrix

12. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2)12 Particle Swarm Optimisation

13. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 13 termination of PSO after 400.000 inspected solutions (to keep results comparable) Overall Outline of PSO Approach 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

14. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 14 Details of Approach 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 PSO performed well in a similar timetabling problem [5] new position  in each iteration, for each particle: change two elements at random copy a random element from the same position in pBest copy a random element from the same position in gBest

15. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 15 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

16. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2)16 Evolution Strategies

17. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 17 Overall Outline of Evolutionary Approach 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 termination of ES after 400.000 inspected solutions (to keep results comparable)

18. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 18 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

19. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 19 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

20. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 20 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 (G 1 and G 2 ) is added to each element of the matrix G 1 and G 2 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

21. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2)21 Results and Conclusion

22. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 22 heuristic minimal error average error number of jobchanges wrong qualifications in minutes understaffing in minutes overstaffing in minutes (demand > 0) overstaffing in minutes (demand = 0) number of fitness evaluations manual plan411330 0,01545,020130,014610,033795,0- PSO (20)51967521621666,80,07478,528488,07265,5400000 PSO (40)52085522221730,20,07568,628112,17731,4400000 PSO (100)52400525911778,519,18136,827874,18537,7400000 PSO (200)53467537272220,30,07658,528017,07916,5400000 ES (1, 5)55545559871616,80,07994,526163,010106,5400001 ES (1 + 5)55575558931604,20,07978,526181,010064,5400001 ES (10, 50)55744569481677,30,08093,025560,010808,0400010 ES (10 + 50)55701564841664,80,08029,525819,510485,0400010 ES (30, 200)58587639531536,80,08999,521132,516142,0400030 ES (30 + 200)58449636341531,70,08906,021165,516015,5400030 ES (1, 5) Entropy69522610523583,90,011373,024397,714917,0400001 ES (1 + 5) Entropy70145621583063,80,010837,021980,016601,0400001 ES (10, 50) Entropy53648530482149,40,07674,528075,27873,5400010 ES (10 + 50) Entropy52864524931919,20,07554,527422,37560,5400010 ES (30, 200) Entropy54471539542451,10,07725,527706,58294,0400030 ES (30 + 200) Entropy53824534282351,10,07678,028111,07842,0400030 Results for the Logistic Service Provider Problem Results averaged over 30 runs each. All tests were conducted on a standard PC. Indication of absolute minimum: PSO with repair: 51,521 error points

23. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 23 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

24. © 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 24 References 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 Physics Publishing, Bristol 3.Beyer H.-G., Schwefel, H.-P. (2002) Evolution strategies: a comprehensive introduction. Natural Computing 1: 3-52 4.Blöchlinger I. (2004) Modeling Staff Scheduling Problems. EJOR 158: 533-542 5.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 6.Brodersen, O., Schumann, M. (2007) Einsatz der Particle Swarm Optimization zur Optimierung universitärer Stundenpläne. Technical Report 05/2007, University of Göttingen 7.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 8.Garey, M.R.; Johnson, D.S. (1979) Computers and Intractability. A Guide to the Theory of NP-Completeness 9.Kennedy J., Eberhart R. C., Shi Y. (2001) Swarm Intelligence. Kaufmann, San Francisco 10.Kragelund, L., Kabel, T. (1998) Employee Timetabling. An Empirical Study, Master's Thesis, University of Aarhus 11.Meisels A., Schaerf A. (2003) Modelling and solving employee timetabling problems. Annals of Mathematics and Artificial Intelligence 39: 41-59 12.Proudfoot Consulting (2007) Produktivitätsbericht 2007. Company Report 13.ROADEF Challenge (2007) Technicians and Interventions Scheduling for Telecommunications. http://www.g-scop. inpg.fr/ChallengeROADEF2007 (2008-06-22) 14.Rudolph, G. (1994) An evolutionary algorithm for integer programming. PPSN III, Jerusalem, Israel, Proceedings, LNCS, Vol. 866:139-148 15.Tien, J; Kamiyama, A. (1982) On Manpower Scheduling Algorithms, SIAM Rev. 24(3): 275-287 16.Vanden Berghe G. (2002) An Advanced Model and Novel Metaheuristic Solution Methods to Personnel Scheduling in Healthcare. Thesis, University of Gent, Belgium