Automatic generation of
optimized working time
models in personnel planning
TU Ilmenau
Department of Commercial Information Technology
for Services (WI2)
1
Dipl. Wirt.-Inf. Maik Günther
[email protected]
Prof. Dr. Volker Nissen
[email protected]
• Description of the Application Problem
• Particle Swarm Optimization
• Evolution Strategies
• Results and Conclusion
2
Structure of presentation
• bank holiday on Thursday
• workforce management is not demand driven
• high personnel costs
• loss of sales
requirement
employees
personnel hours
3
excessive
personnel
costs
loss sales
revenues
time
Practical example – overstaffing and
understaffing
• 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
• 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
• 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
• 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].
7
• Kragelund and Kabel show the NP-hardness of the general
employee timetabling problem [8].
Problem representation for PSO and ES
• Description of the Application Problem
• Particle Swarm Optimization
• Evolution Strategies
• Results and Conclusion
8
Structure of presentation
• population-based modern heuristic
initialize the swarm
•calculate
swarm members
are assumed
to be massless
particles
fitness
of initial
particles
each
particle
and gBest
•determine
each particle pBest
together for
with its
position
within a solution
space
repeat
embodies a solution to the problem
for ifor= optima
1 to with
number
• they search
the aidof
of aparticles
fitness function
calculate new position with 4 actions
• particles exchange information, which can positively influence the
repair particle
development of the population as a whole (pBest, gBest/lBest)
calculate fitness
new pBest and new gBest?
next i
until termination criterion holds
output gBest from current run
• termination of PSO after 400.000 inspected solutions (to keep
results comparable)
9
Overall outline of PSO approach
• 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 with
4 actions
• Description of the Application Problem
• Particle Swarm Optimization
• Evolution Strategies
• Results and Conclusion
11
Structure of presentation
• each individual of the population embodies a solution to the
initialize the population
problem
calculate fitness of initial population
•repeat
they search for optima with the aid of a fitness function
draw
andoperator
recombine
parent solutions
• primarily
search
is mutation
mutateofoffspring
• self-adaption
mutation step size
repair offspring
• each individual has a strategic parameter which will be mutated
calculate fitness for offspring
and recombined
select the new population
• higher probability for individuals with a good strategic
until termination criterion holds
parameter to survive
output best solution from current run
12
• termination of ES after 400.000 inspected solutions (to keep results
comparable)
Overall outline of evolutionary approach
• 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
• 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
• Description of the Application Problem
• Particle Swarm Optimization
• Evolution Strategies
• Results and Conclusion
15
Structure of presentation
16
Results for the application problem
• 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
17
• adapt other heuristics from roughly
comparable problems in the literature
Conclusions
18
Data sets and benchmarks
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 NPCompleteness
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.
19
11. Tien J., Kamiyama A. (1982) On Manpower Scheduling Algorithms, SIAM Rev. 24(3): 275287
References