Staff Scheduling with Particle Swarm Optimisation and Evolution Strategies

Staff Scheduling with Particle

Swarm Optimisation and Evolution Strategies

Volker Nissen and Maik Gunther

Technical University of Ilmenau, Chair of Information Systems in Services,D-98684 Ilmenau, Germany

[email protected], [email protected]

Abstract. The current paper uses a scenario from logistics to showthat modern heuristics, and in particular particle swarm optimization(PSO) can significantly add to the improvement of staff scheduling inpractice. Rapid, sub-daily planning, which is the focus of our researchoffers considerable productivity reserves for companies but also createscomplex challenges for the planning software.

Keywords: staff scheduling, sub-daily planning, particle swarm opti-mization, combinatorial optimization, evolution strategy.

1 Introduction to the Problem of Staff Scheduling

Staff scheduling involves the assignment of an appropriate employee to the appro-priate workstation at the appropriate time while considering various constraints.This work describes a method for solving the problem of subdaily staff schedulingwith individual workstations. According to current research employees spend upto 36% of their working time unproductively, depending on the branch [17]. Ma-jor reasons include a lack of planning and controlling. The problem can be facedwith demand-oriented staff scheduling. Key planning goals are increased produc-tivity, reduction of staff costs, prevention of overtime, motivation of employeeswith positive results for sales and service [19].

In practice, the application of a system for staff scheduling has not been veryprevalent up to now. Most often planning takes place based on prior experienceor with the aid of spreadsheets [1]. It is obvious that the afore-mentioned goals ofdemand-oriented staff scheduling cannot be realised with these planning tools.Even with popular staff planning software employees are regularly scheduledfor one workstation per day. However, in many branches, such as trade andlogistics, the one-employee-one-station concept does not correspond to the actualrequirements and sacrifices potential resources. Therefore, sub-daily planningshould be an integral component of demand-oriented staff scheduling.

In the following section, the application problem is stated more formally.Then, we discuss work related to our own research before developing approachesbased on PSO and evolution strategies (ES) for sub-daily staff scheduling insection 4. Section 5 describes the practical planning scenario and experimentalsetup before the empirical results are presented and discussed in section 6. Thepaper concludes with a short summary and some indications for future work.

C. Cotta and P. Cowling (Eds.): EvoCOP 2009, LNCS 5482, pp. 228–239, 2009.c© Springer-Verlag Berlin Heidelberg 2009

Staff Scheduling with Particle Swarm Optimisation 229

2 Formal Statement of the Problem

The problem starts out assuming a set of employees E = {1, . . . , E}, a set ofworkstations W = {1, . . . , W} and a discrete timeframe T with the index t =0, . . . , T − 1, where each period t of the range has a length lt greater than zero.The demand dwt of employees per workstation and period cannot be negative.

lt > 0 ∀t ∈ Tdwt ≥ 0 ∀w ∈ W and ∀t ∈ T (1)

The availability of employees is known at the beginning of the sub-daily plan-ning and is determined using the binary variable aet.

aet ={

1 if employee e is available at period t0 otherwise (2)

The assignment of an employee to a workstation is controlled using the binaryvariable xewt.

xewt ={

1 if employee e is assigned to workstation w at period t0 otherwise (3)

An employee can only be associated with a workstation w in the period t ifhe or she is actually present.

W∑w=1

xewt ≤ aet ∀e ∈ E and ∀t ∈ T (4)

Additionally, an employee can only be designated to one workstation at atime.

W∑w=1

xewt ≤ 1 ∀e ∈ E and ∀t ∈ T (5)

Any workstation can require a set of qualifications Qw, and employees havea set of qualifications Qe at their disposal. If an employee is planned for aworkstation but does not meet all necessary qualifications, error points Pq

are generated for the duration of the assignment according to the error pointsize cq.

Pq =T−1∑t=0

W∑w=1

E∑e=1

cqltxewt

cq > 0 if employee e is not qualifiedfor workstation w,

cq = 0 else(6)

If a discrepancy arises from the workstation staffing target dwt, error points Pd

are generated for the duration and size of the erroneous assignment according tothe error point size. Different types of errors can be distinguished: cdo represents

230 V. Nissen and M. Gunther

overstaffing when the demand dwt > 0, cdn signals overstaffing when the demanddwt = 0, cdu signals cases of understaffing.

Pd =T−1∑t=0

W∑w=1

(cdn + cdo + cdu)lt

∣∣∣∣∣(

E∑e=1

xewt

)− dwt

∣∣∣∣∣ , with: (7)

cdn > if workstation w is overstaffed at period t and dwt = 0, else cdn = 0cdo > if workstation w is overstaffed at period t and dwt > 0, else cdo = 0cdu > if workstation w is overstaffed at period t and dwt = 0, else cdu = 0

To avoid an excessive number re of sub-daily workstation (job) rotations for anyemployee cr error points arise for such (job) rotations.

Pr = cr

E∑e=1

re (8)

Therefore, the objective function to be minimised becomes:

minP = Pq + Pd + Pr. (9)

3 Related Work

The basic principles of particle swarm optimisation (PSO) were developed byKennedy and Eberhart among others [10] [11]. Swarm members are assumed tobe massless, collision-free particles that search for optima with the aid of a fitnessfunction within a solution space. In this process each single particle togetherwith its position embodies a solution to the problem [23]. While looking for theoptimum, a particle does not simply orient itself using its own experience butalso using the experience of its neighbours [8]. This means that the particlesexchange information, which can then positively influence the development ofthe population in the social system as a whole [15].

Modifications of standard real-valued PSO exist for binary variables, wherethe speed of a particle is used as the probability for the change of the binaryvalue [11]. This approach, however, has several limitations and was changed frombinary to decimal variables in [24]. Another PSO-variant was developed for se-quence planning tasks [20]. In 2007 Poli analysed the IEEE Xplore database forthe thematic grouping of PSO applications [16]. Of approximately 1100 publi-cations only one work is focused specifically on timetabling [6] which is relatedto our own application problem. In [6], the authors adjust PSO to the combi-natorial domain. No longer is the position of a particle determined by its speedbut rather by using permutation operators. In [5] university timetabling was alsoapproached with PSO.

In [7] Ernst et al. offer a summary of papers related to the issue of staffscheduling - about 700 papers between the years 1954 and 2004 have been in-cluded. They identify certain categories of problems, such as the category flexible


demand. This category is characterised by little available information on sched-ules and upcoming orders. A demand per time interval is given as well as arequired qualification. Thus, the application problem discussed here can be clas-sified in the group flexible demand schemes. It can additionally be classed undertask assignment. Task assignment is used to generate assignments requiring cer-tain qualifications and needing to be completed in a certain period of time, whichare then distributed amongst the employees. The employees have already beenassigned shifts.

As work related to our research Vanden Berghe [22] presents an interesting,though non-PSO heuristic to sub-daily planning. Here, demand is marked bysub-daily time periods , which allows the decoupling of staff demand from fixedshifts resulting in fewer idle times. However, scheduling ist not performed at thedetailed level of individual workstations as in our research.

In [13] Schaerf and Meisels provide a universal definition of an employeetimetabling problem. Both the concepts of shifts and of tasks are included,whereby a shift may include several tasks. Employees are assigned to the shiftsand assume task for which they are qualified. Since the task is valid for the du-ration of a complete shift, no sub-daily changes of tasks (or rather workstations)are made. Blochlinger [4] introduces timetabling blocks (TTBs) with individuallengths. In this model employees may be assigned to several sequential TTBs, bywhich subdaily time intervals could be represented within a shift. Blochlinger’swork also considers tasks; however, a task is always fixed to a TTB. Essentially,our problem of the logistics service provider represents a combination of [13](assignment of staff to tasks) and [4] (sub-daily time intervals), but with theassignment periods (shifts) of the employees already being set.

Staff scheduling is a hard optimization problem. In [9] Garey and Johnsondemonstrate that even simple versions of staff scheduling problems are NP-hard. Kragelund and Kabel [12] show the NP-hardness of the general employeetimetabling problem. Moreover, Tien and Kamiyama prove in [21] that practicalpersonnel scheduling problems are generally more complex than the TSP whichis itself NP-hard. Thus, heuristic approaches appear justified for our application.Apparently, there exists no off-the-shelf solution approach to the kind of detailedsub-daily staff planning problem considered here. A PSO approach and evolutionstrategy for this application are outlined in the following section. We assume thereader is familiar with standard-PSO [11] [8] and standard-ES [2] [3].

4 PSO Approach and Evolution Strategy

4.1 Problem Representation

To apply PSO and the evolution strategy, the sub-daily staff scheduling problemneeds to be conveniently represented. A two-dimensional matrix is applied. Eachparticle in the swarm (for PSO) has an own matrix that determines its position.Also, each individual in the ES-population uses a matrix to represent it’s solu-tion to the application problem. The rows of the matrix signify employees and thecolumns signify each time period of the length lt > 0. To mark times in which an


Table 1. Assignment of workstations in a two-dimensional matrix

periodemployee

0 1 2 3 4 5 6 ...

1 1 1 1 1 1 1 1

2 0 0 2 2 2 2 2

3 0 0 1 1 2 2 2

4 0 0 6 6 6 6 2

5 3 3 3 2 2 0 0

...

employee is not present due to his work-time model, a dummy workstation is intro-duced (in table 1: workstation 0). For example, employee two is absent in the firsttwo periods and then is assigned to workstation 2. Assignment changes can onlybe made on non-dummy workstations, so that no absent employee is included.

To lower the complexity the number of dimensions should be reduced. Thiscan be realised via a suitable depiction of time. Within the planned day, time isviewed with a time-discrete model. An event point (at which a new time intervalbegins) occurs when the allocation requirement for one or more workstations oremployee availability change. With this method, however, the periods are notequally long any more, so that their lengths need to be stored.

4.2 Outline of Combinatorial PSO for This Application

At the start of PSO the initialisation of the particle position does not takeplace randomly. Rather, valid assignments w.r.t. the hard constraints are madethat use information from the company’s current full-day staff schedule to setwhich employee works at which station. Therefore, valuable foreknowledge isnot wasted. Based on this plan, improved solutions can now be determined thatinclude plausible workstation changes.

In each iteration the new particle position is determined by traversing all di-mensions and executing one of the following actions with predefined probability.The probability distribution was heuristically determined in prior tests:

– No change: The workstation already assigned remains. (prob. p1)– Random workstation: A workstation is randomly determined and assigned.

Only those assignments are made for which the employee is qualified. Theprobability function is uniformly distributed. (prob. p2)

– pBest workstation: The corresponding workstation is assigned to the particledimension from pBest, the best position found so far by the particle. Throughthis, the individual PSO component is taken into account. (prob. p3)

– gBest workstation: The corresponding workstation is assigned to the particledimension from gBest (or rather lBest if a gBest neighbourhood topology isnot being used). gBest (lBest) represents the best position of all particlesglobally (in the local neighbourhood). The social behaviour of the swarm iscontrolled with these types of assignments. (prob. p4)


By considering the best position of all particles, the experience of the swarm isincluded in the calculation of the new position. Premature convergence on a sub-optimal position can be avoided by using the lBest topology, in which a particleis only linked to its neighbour. The extent to which the swarm acts individu-ally or socially is determined by the probability with which the workstation isassigned from pBest, gBest or lBest. The behaviour of the PSO-heuristic is rela-tively insensitive to changes of p1, p3, and p4. The optimal value for p2 dependson the problem size. Pre-tests revealed that a value of 0.3% for p2 works bestfor the problem investigated here. The other probabilities were set at p1=9.7%,p3=30%, and p4=60% with a gBest topology.

The characteristics of PSO have not been changed with these modifications.There are merely changes in the way to determine a new particle position, sothat the calculation of the velocity is not needed. The current form of posi-tion determination makes it unnecessary to deal with dimension overruns. Allother peculiarities of PSO regarding social or global behaviour remain. Evenall neighbourhood topologies established as part of continuous parameter opti-misation in standard-PSO remain and can be used without restrictions. In ourimplementation, PSO terminates after 400,000 inspected solutions. In the future,other convergence-based termination criteria could be employed. The followingpseudocode presents an overview of the implemented PSO.

1: initialise the swarm2: determine pBest for each particle and gBest3: loop4: for i = 1 to number of particles5: calculate new position // use the 4 alternative actions6: if f(new position)<f(pBest) then pBest=new position // new pBest7: if f(pBest)<f(gBest) then gBest=pBest // new gBest8: next i9: until termination

4.3 Outline of Evolution Strategy for This Application

PSO-results are compared to several variants of the evolution strategy, origi-nally developed by Rechenberg and Schwefel [2] [3]. In our application, it wasan objective to preserve the basic characteristics of evolution strategies, eventhough they are most often used for continuous parameter optimization. How-ever, evolution strategies have proved to be powerful heuristics in combinatorialoptimization, see for instance [14].

The ES-population is initialized with valid solutions w.r.t the hard problemconstraints. Again, information from the company’s current full-day staff sched-ule is used. (μ, λ)-selection (comma-selection) as well as (μ + λ)-selection (plus-selection) are used as well as different population sizes. The best solution foundduring an experimental run is always stored and updated in a ”golden cage”. Itrepresents the final solution of the run. Following suggestions in the literature[2] [3], the ratio μ/λ is set to 1/5 - 1/7 during the practical experiments. Therecombination of parents to create an offspring solution works as follows:


Fig. 1. Recombination operator employed

A crossover point is determined independently and at random for each em-ployee (row) of a solution and the associated parts of the parents are exchanged(see fig. 1).

Mutation of an offspring is carried out by picking an employee at randomand changing the workstation assignment for a time interval chosen at random.It must be ensured, though, that valid assignments are made w.r.t. the hardproblem constraints. The number of employees selected for mutation follows a(0, σ)-normal distribution so that small changes are more frequent than largeones. Results are rounded and converted to positive integer numbers. Otherapproaches are possible here [18]. The mutation stepsize sigma is controlledself-adaptively using a log-normal distribution and intermediary recombination,following the standard scheme of evolution strategies [3].

In prior tests, other recombination and mutation schemes as well as selectionpressures were tried, but they performed worse than the current approach. TheES terminates when 400,000 solutions have been inspected to allow for a faircomparison with PSO. The following pseudocode presents an overview of theimplemented ES.

1: initialise the population with μ individuals2: evaluate the μ individuals3: loop4: copy and recombine parents to generate λ offspring5: mutate the λ offspring6: evaluate the λ offspring7: select ((μ + λ) or (μ, λ)) best individuals as the new generation8: until termination

5 Test Problem and Experimental Setup

The present problem originates from a German logistics service provider. Thiscompany operates in a spatially limited area 7 days a week almost 24 hoursa day. The tasks of employees concern loading and unloading, short distancetransportation and other logistic services. The employees are quite flexible interms of their working hours, which results in a variety of working-time models.There are strict regulations especially with regard to qualifications because theassignment of unqualified employees might lead to significant material damageand personnel injury. The employer regularly invests a lot of time and moneyin qualification measures so that many different employees can work at several


different workstations. The personnel demand is known for each workstation,resulting in high planning security. Currently, monthly staff scheduling is carriedout manually within MS EXCELTM. Employees are assigned a working-timemodel and a fixed workstation each day. Several considerations are included, suchas presence and absence, timesheet balances, qualifications and resting times etc.

The personnel demand for the workstations is subject to large variations dur-ing the day. However, employees are generally scheduled to work at the sameworkstation all day, causing large phases of over- and understaffing. This lowersthe quality of service and the motivation of employees and leads to unnecessarypersonnel costs as well as downtime. At this time, sub-daily workstation rotationis only rarely used in the planning. Usually, department managers intervene di-rectly on site and reassign the employees manually. Obviously, demand-orientedstaff scheduling cannot be realised with this approach.

The planning problem covers seven days (20 hours each), divided into 15-minute intervals. It includes 65 employees and, thus, an uncompressed totalof 36,400 dimensions for the optimization problem to be solved. The generalavailability of the employees is known for each interval from the previous full-day planning. Employee shift planning was done for 13 possible shifts plus aplanned off-shift. Nine different workstations need to be filled, with seven havingqualification requirements. The variety of qualifications was summarised in fourqualification groups.

A staff schedule is only valid if any one employee is only assigned to oneworkstation at a time and if absent employees are not included in the plan. Thesehard constraints can be contrasted with soft constraints, which are penalisedwith error points. The exact determination of error point counts is in practicean iterative process and will not be covered in detail here. The error points usedhere are from an interview with the logistics service provider and reflect thatcompanys requirements.

All test runs were conducted on a PC with an Intel 4 x 2.67 GHz processorand 4 GB of RAM. Thirty independent runs were conducted each time for eachof the experiments to allow for statistical testing. The runtime of ca. 25 minutesfor a single run is similar for all solution methods tested in the next section.

6 Results and Discussion

The full-day manual staff schedule without sub-daily workstation changes resultsin 411,330 error points after an evaluation that included the penalties arisingfrom the afore-mentioned constraints.

The results of the methods are shown in table 2. All heuristics for sub-dailystaff scheduling significantly outperform the manual full-day schedule in termsof total error points. This demonstrates the value of sub-daily scheduling ascompared to today’s standard staff scheduling approaches which not only wasteresources but also demotivates personnel and deteriorates quality of service.Generally, the problems of understaffing and overstaffing for periods withoutdemand are greatly reduced. On the other hand, all heuristics lead to more


Table 2. Comparison (error points) of the different sub-daily scheduling heuristics,based on 30 independent runs each. Best results are bold and underlined.

error number wrong under- overstaffing in minutes numberof qualifi- staffing of

heuristic mean min job- cations in fitnesschanges in minutes demand > 0 demand = 0 evalua-

minutes tions

manualplan

411330 411330 0,0 1545 20130.0 14610.0 33795.0 -

PSO 52162 51967 1666.8 0 7478.5 28488.0 7265.5 400000(20)

PSO 52222 52085 1730.2 0 7568.6 28112.1 7731.4 400000(40)

PSO(100)

52591 52400 1778.5 19.1 8136.8 27874.1 8537.7 400000

PSO(200)

53727 53467 2220.3 0 7658.5 28017.0 7916.5 400000

ES 55987 55545 1616.8 0 7994.5 26163.0 10106.5 400001(1,5)

ES 55893 55575 1604.2 0 7970.5 26181.0 10064.5 400001(1+5)

ES(10,50)

56948 55744 1677.3 0 8093.0 25560.0 10808.0 400010

ES(10+50)

56484 55701 1664.8 0 8029.5 25819.5 10485.0 400010

ES(30,200)

63953 58587 1536.8 0 8999.5 21132.5 16142.0 400030

ES(30+200)

63634 58449 1531.7 0 8906.0 21165.5 16015.5 400030

overstaffing in periods with demand > 0 as compared to the initial plan. Thisapproach, however, is sensible because employees can still support each otherinstead of being idle when demand = 0.

Interestingly, the PSO heuristic provides the best results with a rather smallswarm size of 20 particles. ES reacted similarly, with the (1+5)- and (1,5)-strategy providing the best results on average. There is no clear advantage forthe comma- or plus-selection scheme in the evolution strategies tested here.Moreover, recombination apparently does not guarantee an advantage for the ESon this type of problem. The (1+5)- and (1,5)-strategy with a parent populationsize of one make no use of recombination, but still perform better than the otherES-variants with larger populations and recombination applied. An increasedpopulation size without using recombination also created results worse thanthose stated in the table, though. Thus, neither a large population size nor theuse of recombination necessarily lead to success with evolution strategies.

PSO(20) and ES(1+5) provided the best mean error results in their respec-tive groups. With 30 independent runs for each heuristic it is possible to test the


Table 3. T-test results for pairwise comparison of heuristics

95% confidenceH1 T df significance mean intervall of

H0 difference differences(l-tailed) lower upper

PSO(20) < ES(1+5) -86.19 39.02 < 0.001 -3731.07 -3818.62 -3643.51

PSO(20) < ES(1,5) -100.26 42.26 < 0.001 -3825.33 -3902.31 -3748.35

PSO(200) < ES(1+5) -45.85 49.15 < 0.001 -2166.60 -2261.55 -2071.65

PSO(200) < ES(1,5) -53.07 53.45 < 0.001 -2260.87 -2346.30 -2175.44

Fig. 2. Convergence chart for PSO(20) and ES(1+5)

statistical significance of the performance difference between both solution meth-ods with a t-test (see table 3). A Levene-test revealed the heterogeniety of vari-ances between both groups (F = 21.23, p < 0.001). The corresponding t-test witha 95% confidence interval confirms the better performance of PSO(20) with avery high statistical significance (p < 0.001 for H0). A further test was conductedto compare the worst parameterisation of PSO tested here (PSO(200)) with thebest parameterisation of ES, the ES(1+5). Even here PSO outperforms the ESon a highly significant level (p < 0.001 for H0). The results are equally signifi-cant when the comparison is made between PSO and the ES(1,5) as can be seenfrom table (3). This success must be attributed to the operators of PSO since thecoding of PSO and ES are identical. A second reason concerns the fewer strategyparameters in our PSO-approach which are more easily adapted to the applica-tion domain. Fig. 2 shows the convergence behaviour of best variants from PSOand ES in comparison. Not only does PSO generate the better final solution, butit also demonstrates a more rapid convergence towards good solutions. This isgenerally a desirable characteristic, particularly when the available time for anoptimization is rather limited (not the case here).

Notwithstanding the fact that the PSO heuristic was able to provide bet-ter results for this problem, it also has one technical advantage over ES. The


PSO outlined in this paper only requires the varying of two parameters (swarmsize and p2), which can both be very easily set. ES, on the other hand, offersmore parameterisation possibilities (selection pressure, recombination scheme,plus or comma selection etc.), resulting in greater heuristic complexity from auser’s perspective. However, while our ES-approach adheres quite closely to thestandard-ES procedure, other parameterisations are certainly possible that couldpotentially improve over the performance here.

7 Conclusion and Future Work

Sub-daily staff scheduling is a meaningful practical problem area. Using an actualplanning scenario, it was demonstrated that particle swarm optimisation pro-duces far better results than traditional full day scheduling. Sub-daily schedul-ing significantly increases the value contributions of individual staff members.Because PSO in its traditional form is not suitable for the planning problem athand, the method was adapted to the combinatorial domain without sacrificingthe basic PSO-mechanism. PSO also outperformed different variants of the evo-lution strategy on this problem. The superior performance must be attributed tothe operators and parameters of PSO since the coding of PSO and ES are iden-tical. In future research, these promising results are to be expanded by creatingfurther test problems with the aid of cooperating companies. Moreover, otherheuristics from roughly comparable problems in the literature are currently beingadapted to our domain and tested to further validate the results.

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Staff Scheduling with Particle Swarm Optimisation and Evolution Strategies

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