© 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2)1 Dipl....

© 2009 - TU Ilmenau, Department of Commercial Information Technology for Services (WI2) 1

Dipl. Wirt.-Inf. Maik Gü[email protected]

Prof. Dr. Volker [email protected]

TU IlmenauDepartment of Commercial Information Technology for Services (WI2)

Staff Scheduling with Particle Swarm Optimisation and Evolution Strategies

EVO Cop 2009, Tuebingen


Structure of Presentation

• Sub Daily / Sub Shift Staff Scheduling

• Particle Swarm Optimisation

• Evolution Strategies

• Results and Conclusion


Sub Daily / Sub Shift Staff Scheduling


Overstaffing

Understaffing

Hou

rs w

orke

d

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].


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


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


Demand for Personnel at the Nine Workstations


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!


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

• ....


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


Particle Swarm Optimisation


• 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 // 4 actionscalculate fitnessnew pBest? / new gBest?

next iuntil termination criterion holdsoutput gBest from current run


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


Evolution Strategies


Overall Outline of Evolutionary Approach

initialize the populationcalculate fitness of initial populationrepeat

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

until termination criterion holdsoutput best solution from current run

• termination of ES

• after 400.000 inspected solutions (to keep results comparable)


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


Mutation of Solutions

• self adaptive step size for mutation

• mutation creates only valid solutions (no availability and qualification errors)

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


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 (G1 and G2) is added to each element of the matrix• G1 and G2 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


Results and Conclusion


heuristicminimal

erroraverage

errornumber of jobchanges

wrong qualifications

in minutes

understaffing in minutes

overstaffing in minutes

(demand > 0)

overstaffing in minutes

(demand = 0)

number of fitness

evaluations

manual plan 411330 411330 0,0 1545,0 20130,0 14610,0 33795,0 -PSO (20) 51967 52162 1666,8 0,0 7478,5 28488,0 7265,5 400000PSO (40) 52085 52222 1730,2 0,0 7568,6 28112,1 7731,4 400000PSO (100) 52400 52591 1778,5 19,1 8136,8 27874,1 8537,7 400000PSO (200) 53467 53727 2220,3 0,0 7658,5 28017,0 7916,5 400000ES (1 , 5) 55545 55987 1616,8 0,0 7994,5 26163,0 10106,5 400001ES (1 + 5) 55575 55893 1604,2 0,0 7978,5 26181,0 10064,5 400001ES (10 , 50) 55744 56948 1677,3 0,0 8093,0 25560,0 10808,0 400010ES (10 + 50) 55701 56484 1664,8 0,0 8029,5 25819,5 10485,0 400010ES (30 , 200) 58587 63953 1536,8 0,0 8999,5 21132,5 16142,0 400030ES (30 + 200) 58449 63634 1531,7 0,0 8906,0 21165,5 16015,5 400030ES (1 , 5) Entropy 69522 61052 3583,9 0,0 11373,0 24397,7 14917,0 400001ES (1 + 5) Entropy 70145 62158 3063,8 0,0 10837,0 21980,0 16601,0 400001ES (10 , 50) Entropy 53648 53048 2149,4 0,0 7674,5 28075,2 7873,5 400010ES (10 + 50) Entropy 52864 52493 1919,2 0,0 7554,5 27422,3 7560,5 400010ES (30 , 200) Entropy 54471 53954 2451,1 0,0 7725,5 27706,5 8294,0 400030ES (30 + 200) Entropy 53824 53428 2351,1 0,0 7678,0 28111,0 7842,0 400030

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


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


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, Bristol3. Beyer H.-G., Schwefel, H.-P. (2002) Evolution strategies: a comprehensive introduction. Natural Computing 1: 3-524. Blöchlinger I. (2004) Modeling Staff Scheduling Problems. EJOR 158: 533-5425. Chu S. C., Chen Y. T., Ho J. H. (2006) Timetable Scheduling Using Particle Swarm Optimization. In: Proceedings

of ICICIC Beijing 2006, Vol. 3: 324-3276. Brodersen, O., Schumann, M. (2007) Einsatz der Particle Swarm Optimization zur Optimierung universitärer

Stundenpläne. Technical Report 05/2007, University of Göttingen7. Ernst A. T., Jiang H., Krishnamoorthy M., Owens B., Sier D. (2002) An Annotated Bibliography of Personnel

Scheduling and Rostering. Annals of OR 127: 21-1448. Garey, M.R.; Johnson, D.S. (1979) Computers and Intractability. A Guide to the Theory of NP-Completeness9. Kennedy J., Eberhart R. C., Shi Y. (2001) Swarm Intelligence. Kaufmann, San Francisco10. Kragelund, L., Kabel, T. (1998) Employee Timetabling. An Empirical Study, Master's Thesis, University of Aarhus11. Meisels A., Schaerf A. (2003) Modelling and solving employee timetabling problems. Annals of Mathematics and

Artificial Intelligence 39: 41-5912. Proudfoot Consulting (2007) Produktivitätsbericht 2007. Company Report13. 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. 866:139-14815. Tien, J; Kamiyama, A. (1982) On Manpower Scheduling Algorithms, SIAM Rev. 24(3): 275-28716. Vanden Berghe G. (2002) An Advanced Model and Novel Metaheuristic Solution Methods to Personnel Scheduling

in Healthcare. Thesis, University of Gent, Belgium

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