A Comparison of Metaheuristics on a Practical Staff Scheduling Problem

1. A Comparison of Metaheuristics on a Practical Staff Scheduling Problem TU IlmenauDepartment of Commercial InformationTechnology 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. 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 3 Description of the application problem

4. 4 Demand for personnel at the 9 workstations

5. 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! 5 Current planning

6. 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 • ... 6 Input and constraints

7. 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] 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 • 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 10 Calculate the new position with4 actions & repair particles

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)) • two parents have a 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 – with classical Gaussian mutation

15. 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 geometricallydistributed random numbers (G1 and G2) is addedto each element of the matrix • G1 and G2 have the parameter p which iscontrolled 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 15 Mutate offspring – with the principle of maximum entropy [9]

16. Description of the Application Problem • Particle Swarm Optimization • Evolution Strategies • Results and Conclusion 16 Structure of presentation

17. 17 Results for the application problem

18. 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 18 Conclusions

19. 19 Data sets and benchmarks

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