A HyFle Module for the Personnel Schedulin Problem Tim Curtois, Gabriela Ochoa, Matthew Hyde, José Antonio Vázquez-Rodríuez Automated Schedulin, Optimisation and Plannin (ASAP) Group, School of Computer Science, University of Nottinham, Jubilee Campus, Wollaton Road, Nottinham. NG8 BB. UK. Problem Formulation The personnel schedulin problem basically involves decidin at which times and on which days (i.e. which shifts) each employee should work over a specific plannin period. However, the personnel schedulin problem is actually a title for a roup of very similar problems. There is no eneral personnel schedulin problem. Instead there is a roup of problems with a common structure but which differ in their constraints and objectives. This creates an additional challene in implementin a problem domain module for personnel schedulin. To overcome this we have desined a data file format for which each instance can select a combination of a objectives and constraints from a wide choice. We then implemented a software framework containin all the functions for these constraints and objectives. The framework also contains methods for parsin these data files, data structures which can be used by heuristic alorithms (such as neihbourhood searches) and libraries for visualisations of instances and solutions. As mentioned, there is a lare and diverse collection of constraints and objectives that can appear in personnel schedulin problems. For eample, in one problem there may be a constraint on the maimum number of hours a nurse can work in the plannin period. In another problem thouh this constraint may be an objective. That is, the nurse is allowed to eceed a certain number of hours but the ecess should be minimised. The objective may then also be iven a priority (relative to other objectives) usin a weiht. To be able to handle all these variations yet at the same time minimise the compleity of the file format and the amount of prorammin necessary, a key desin decision was made: All constraints are modelled as weihted objectives. When modellin a problem which contains constraints, the constraints are transformed to objectives with very hih weihts. As the weiht is very hih it is simple to tell if a solution is feasible or not just by eaminin the objective function value. The overall objective function is a weihted sum of all the sub-objectives. The objectives for nurse rosterin problems can be cateorised into two roups: Coverae objectives. These objectives aim to ensure that the preferred number of employees (possibly with skills) are workin durin each shift. Minimum and maimum levels of cover can also be set.
Employee workin objectives. This roup of objectives relates to the individual work patterns (schedules) for each employee. They aim to maimise the employees satisfaction with their work schedules. Eample objectives within this roup include: Minimum/maimum number of hours worked. Minimum/maimum number of days on or off. Minimum/maimum number of consecutive workin days. Minimum/maimum number of consecutive days off. Minimum/maimum number of consecutive workin weekends. Minimum/maimum number of consecutive weekends off. Minimum/maimum number of shifts of a certain type. For eample niht shifts. Minimum/maimum number of consecutive shifts of a certain type. For eample consecutive niht shifts. Shift rotations. For eample early shifts after late shifts should be avoided. Satisfyin requests for specific shifts/days on or off. To provide a more formal eplanation of the problem we have included an IP model for the ORTEC0 instance. This problem was oriinally eamined in [4] and subsequently in [8]. As accordin to our format, the constraints in the oriinal formulation have been chaned to objectives. The only constraint is that each employee can only work one shift per day. Parameters: I = Set of nurses available. I t t {,2,3} Subset of nurses that work 20, 32, 36 hours per week respectively, I = I + I 2 + I 3. J = Set of indices of the last day of each week within the schedulin period = {7, 4, 2, 28, 35}. K = Set of shift types = {(early), 2(day), 3(late), 4(niht)}. K = Set of undesirable shift type successions = {(2,), (3,), (3,2), (,4), (4,), (4,2), (4,3)}. d jk = Coverae requirement of shift type k on day j, j {,...,7 J }. m i = Maimum number of workin days for nurse i. n = Maimum number of consecutive niht shifts. n 2 = Maimum number of consecutive workin days. c k = Desirable upper bound of consecutive assinments of shift type k. t = Desirable upper bound of weekly workin days for the t-th subset of nurses. h t = Desirable lower bound of weekly workin days for the t-th subset of nurses. Decision variables: = if nurse i is assined shift type k for day j, 0 otherwise The constraints are:. A nurse may not cover more than one shift each day. 2
k K, i I, j {,...,7 J } The objectives are formulated as (weihted w ) oals. The overall objective function is: i 6 Min G( ) w i i i, Where the oals are:. Complete weekends (i.e. Saturday and Sunday are both workin days or both off). i ] [ ( j ) k i I j J k K 2. Minimum of two consecutive non-workin days 2 ma 0, [ i ( j ) k i ( j ) k] i I j 2 k K 3. A minimum number of days off after a series of shifts. 3( ) ma 0, [ i ( j ) k i ( j ) k] i I j 2 k K 4. A maimum number of consecutive shifts of type early and late. ck r ck 4 ma 0, i I r k {,3 } j r 5. A minimum number of consecutive shifts of type early and late. 5 ma 0, i I j 2 k {,3 } i( j ) k 6. A maimum and minimum number of workin days per week. 3 J 7w 6 ma 0, t ma 0, ht t i It w j 7w 6 k K c k i( j ) k 7w j 7w 6 k K 7. A maimum of three consecutive workin days for part time nurses. 3 r 3 7( ) ma 0, 3 i I r j r k K 8. Avoidin certain shift successions (e.. an early shift after a day shift). 8 i I 9. Maimum number of workin days. j ( k, k ) K 2 ma 0, i 7 J 9 ma 0, i I j k K 0. Maimum of three workin weekends.. Maimum of three niht shifts. 0 ma 0, ma{ i ( j ) k i I j J k K ( j ) k 2, } 3 m i 3
i I ma 2. A minimum of two consecutive niht shifts. 2 ma 0, i I j 2 0, 7 J j i( j )4 ij4 ij4 3 i( j )4 3. A minimum of two days off after a series of consecutive niht shifts. This is equivalent to avoidin the pattern: niht shift day off day on. 3( ) ma 0, i ( j )4 i ( j ) k i I j 2 k K k K 4. Maimum number of consecutive niht shifts. n r n 4 ma 0, i I r j r 5. Maimum number of consecutive workin days. n 2 r n2 5 ma 0, i I r j r k K 6. Shift cover requirements 6 2. Solution Initialisation j k K i I ij n 4 d jk n In the HyFle framework it is necessary to implement a method for initialisin a new solution. For the personnel schedulin domain the solution is initialised usin local search heuristic 3 described in section 3. The pseudocode for this heuristic is shown in Fiure 9. It is a hill climbin heuristic which uses a neihbourhood operator which adds new shifts to the roster. 3. Heuristics In HyFle, the heuristics are classified into four cateories: Mutation : A heuristic which randomly mutates a solution. This usually returns a solution which is worse than the oriinal solution. Crossover : A heuristic which combines two solutions to produce a new solution. Usually aimin to keep some of the ood qualities of both parent solutions in the new solution. Ruin and recreate : A perturbative heuristic which chanes part of a solution (usually makin the solution worse) and then attempts to recreate/repair it. 2 4
Local search : A heuristic which attempts to improve the objective function value of the solution it is applied to. The new solution will not have a worse objective function value but it may be the same as the oriinal solution s value. There are also two eneral parameters which can be set (in the rane 0.0 to.0) by the hyperheuristic. They are Depth of search and Intensity of mutation. One or both of these parameters may have an effect on each heuristic. We will now describe the heuristics in each cateory currently implemented in the personnel schedulin domain. Mutation Heuristic Mutation heuristic : randomly un-assins a number of shifts, keepin a feasible solution. The number of shifts the operator un-assins is proportional to the intensity of mutation parameter. Crossover Heuristics There are currently three crossover heuristics. Crossover heuristic was presented in [3]. It operates by identifyin the best assinments in each parent and makin these assinments in the offsprin. The best assinments are identified by measurin the chane in objective function when each shift is temporarily unassined in the roster. The best assinments are those that cause the larest increase in the objective function value when they are unassined. The parameter ranes from 4-20 and is calculated usin the intensity of mutation parameter as below: = 4 + round(( - intensityofmutation) * 6) Crossover heuristic 2 was published in [5]. It creates a new roster by usin all the assinments made in the parents. It makes those that are common to both parents first and then alternately selects an assinment from each parent and makes it in the offsprin unless the cover objective is already satisfied. Crossover heuristic 3 creates the new roster by makin assinments which are only common to both parents. Local Search Heuristics A number of neihbourhood operators have been previously proposed for the personnel schedulin problem. They can be classified into three roups: Vertical swaps, Horizontal swaps and New swaps. The New swaps are so called because they introduce new shifts into the roster (or oppositely delete shifts). Eamples of these swaps are iven in Fiure and Fiure 2. 5
Fiure Fiure 2 Horizontal swaps move shifts in sinle employee s work pattern hence the shifts move horizontally in the roster. Eamples are iven in Fiure 3 and Fiure 4. Fiure 3 Fiure 4 Vertical swaps move shifts between two employees hence the shifts move vertically in the roster. Eamples are iven in Fiure 5 and Fiure 6. 6
Fiure 5 Fiure 6 For all of these neihbourhood operators it is possible to move/swap shifts over a block of adjacent days. For eample the swaps/moves shown in Fiure, Fiure 3 and Fiure 5 can be considered moves over a block of days of lenth one day. Fiure 2 and Fiure 6 show moves over blocks of lenth three days and Fiure 4 illustrates a swap usin blocks of lenth two days. The lenth of the block to test is a parameter which can be set for each neihbourhood operator. Local search heuristics -3 are hill climbers which each use one of these types of neihbourhood operator. Note that, technically they are actually hill descenders as we are minimisin the objective function, but we will use the term hill climber as it tends to be more familiar. Heuristic uses the vertical neihbourhood operator, heuristic 2 uses the horizontal neihbourhood operator and heuristic 3 uses the new neihbourhood operator. The pseudocode for these heuristics is shown in Fiure 7, Fiure 8 and Fiure 9 respectively. For each heuristic, the moves/swaps that are tested rane in size of block lenth one up to a maimum block lenth (MaBlockLenth). This parameter is set to a value of 5. Previous investiations suest that this is a ood choice with reard to the balance between solution quality and computation time (see for eample [6]).. WHILE there are untried swaps 2. FOR BlockLenth= up to MaBlockLenth 3. FOR each employee (E) in the roster 4. FOR each day (D) in the plannin period 5. FOR each employee (E2) in the roster 6. Swap all assinments between E and E2 on D up to D+BlockLenth 7. IF an improvement in roster penalty THEN 8. Break from this loop and move on to the net day 9. ELSE 0. Reverse the swap. ENDIF 2. ENDFOR 3. ENDFOR 4. ENDFOR 5. ENDFOR 6. ENDWHILE 7
Fiure 7 Pseudocode for 'vertical' hill climber. WHILE there are untried swaps 2. FOR BlockLenth= up to MaBlockLenth 3. FOR each employee (E) in the roster 4. FOR each day (D) in the plannin period 5. FOR D2 := D+BlockLenth, up to num days in plannin period 6. Swap all assinments for E on D up to D+BlockLenth with all assinments for E on D2 up to D2+BlockLenth 7. IF an improvement in roster penalty THEN 8. Break from this loop and move on to the net day (D+) 9. ELSE 0. Reverse the swap. ENDIF 2. ENDFOR 3. ENDFOR 4. ENDFOR 5. ENDFOR 6. ENDWHILE Fiure 8 Pseudocode for 'horizontal' hill climber. WHILE there are untried swaps 2. FOR BlockLenth= up to MaBlockLenth 3. FOR each employee (E) in the roster 4. FOR each day (D) in the plannin period 5. FOR each shift type (S) (includin day off) 6. Remove all assinments for E on D up to D+BlockLenth and assin shifts of type S to E on D up to D+BlockLenth 7. IF an improvement in roster penalty THEN 8. Break from this loop and move on to the net day (D+) 9. ELSE 0. Reverse the swap. ENDIF 2. ENDFOR 3. ENDFOR 4. ENDFOR 6. ENDFOR 7. ENDWHILE Fiure 9 Pseudocode for 'new shifts hill climber Local search heuristics 4 and 5 are variants of the variable depth search described in [6]. The first version is similar to the one presented in that paper. The main difference is that it also uses new moves such as the ones shown in Fiure and Fiure 2 as links in the ejection chain (the oriinal version only tests vertical swaps as it was oriinally desined for instances where the cover requirements were a hard constraint rather than an objective). In the second version, as well as usin the moves shown in Fiure and Fiure 2, as potential links in the ejection chains, it also tests replacin an entire work pattern for a sinle employee as a link in the chain. These patterns are enerated usin a reedy heuristic method. The maimum search time for these two heuristics is set as: the depth of search parameter multiplied by a maimum time of 5 seconds. Ruin and Recreate Heuristics The ruin and recreate heuristics implemented are based on the one presented in [4]. 8
Ruin and Recreate Heuristic : The heuristic works by un-assinin all the shifts in one or more randomly selected employees schedules before heuristically rebuildin them. They are rebuilt by firstly satisfyin objectives related to requests to work certain days or shifts and then by satisfyin objectives related to weekends. For eample min/ma weekends on/off, min/ma consecutive workin or non workin weekends, both days of the weekend on or off etc. Other shifts are then added to the employee s schedule in a reedy fashion, attemptin to satisfy the rest of the objectives. Finally, shifts in this schedule are then swapped/moved by hill climber heuristics which use the horizontal and new neihbourhood operators. In [4] it was observed that it was best to un-assin and rebuild only 2-6 work patterns at a time (for instances of all sizes). For this reason the first ruin and recreate heuristic unassins schedules where is calculated usin the intensity of mutation parameter as follows: = Round(intensityOfMutation * 4) + 2 Ruin and Recreate Heuristic 2: The second heuristic provides a larer chane to the solution by settin usin: = Round(intensityOfMutation * Number of employees in roster) Ruin and Recreate Heuristic 3: The third variant of the heuristic creates a small perturbation in the solution by usin =. 4. Instances The instances have been collected from a number of sources. Some of the instances are from industrial collaborators. These include: ORTEC an international consultancy and software company who specialise in workforce plannin solutions and SINTEF, the larest independent research oranisation in Scandinavia. Other instances have been provided by other researchers or taken from various publications. The collection is a very diverse data set drawn from eleven different countries. The majority of the instances are real world scenarios. Table lists the instances. As can be seen, they vary in the lenth of the plannin horizon, the number of employees and the number of shift types. Each instance also varies in the number and priority of objectives present. Instance Best known Staff Shift types Lenth (days) Ref. BCV-.8. 252 8 5 28 [2, 7] BCV-.8.2 853 8 5 28 [2, 7] BCV-.8.3 232 8 5 28 [2, 7] BCV-.8.4 29 8 5 28 [2, 7] BCV-2.46. 572 46 4 28 [2, 7] BCV-3.46. 3280 46 3 26 [2, 7] BCV-3.46.2 894 46 3 26 [2, 7] 9
BCV-4.3. 0 3 4 29 [2, 7] BCV-4.3.2 0 3 4 28 [2, 7] BCV-5.4. 48 4 4 28 [2, 7] BCV-6.3. 768 3 5 30 [2, 7] BCV-6.3.2 392 3 5 30 [2, 7] BCV-7.0. 38 0 6 28 [2, 7] BCV-8.3. 48 3 5 28 [2, 7] BCV-8.3.2 48 3 5 28 [2, 7] BCV-A.2. 294 2 5 3 [2, 7] BCV-A.2.2 953 2 5 3 [2, 7] ORTEC0 270 6 4 3 [4] ORTEC02 270 6 4 3 [4] GPost 5 8 2 28 GPost-B 3 8 2 28 QMC- 3 9 3 28 QMC-2 29 9 3 28 Ikeami-2Shift-DATA 0 28 2 30 [9] Ikeami-3Shift-DATA 2 25 3 30 [9] Ikeami-3Shift-DATA. 3 25 3 30 [9] Ikeami-3Shift-DATA.2 3 25 3 30 [9] Millar-2Shift-DATA 0 8 2 4 [9] Millar-2Shift-DATA. 0 8 2 4 [9] Valouis- 20 6 3 28 [3] WHPP 5 30 3 4 [4] LLR 30 27 3 7 [0] Musa 75 4 [] Ozkarahan 0 4 2 7 [2] Azaiez 0 3 2 28 [] SINTEF 0 24 5 2 CHILD-A2 4 5 42 ERMGH-A 795 4 4 48 ERMGH-B 459 4 4 48 ERRVH-A 297 5 8 48 ERRVH-B 6859 5 8 48 MER-A 995 54 2 48 Table Problem Instances 0
The instances can be downloaded from http:///www.cs.nott.ac.uk/~tec/nrp/. New results, instances and other related information and software are also be published at this location. 5. Conclusion We have described the implementation of a personnel schedulin problem domain for the hyperheuristic software framework HyFle. Within this domain we have implemented a number of heuristics for this problem. These heuristics have appeared in various publications on personnel schedulin and have be shown to be successful methods. The benchmark instances available are diverse and challenin. The majority are real world and have been taken from scenarios worldwide. References. Azaiez, M.N. and S.S. Al Sharif, A 0- oal prorammin model for nurse schedulin. Computers and Operations Research, 2005. 32(3): pp. 49-507. 2. Brucker, P., E.K. Burke, T. Curtois, R. Qu, and G. Vanden Berhe, A Shift Sequence Based Approach for Nurse Schedulin and a New Benchmark Dataset. Journal of Heuristics, Accepted for publication, 2009. 3. Burke, E.K., P. Cowlin, P. De Causmaecker, and G. Vanden Berhe, A Memetic Approach to the Nurse Rosterin Problem. Applied Intellience, 200. 5(3): pp. 99-24. 4. Burke, E.K., T. Curtois, G. Post, R. Qu, and B. Veltman, A Hybrid Heuristic Orderin and Variable Neihbourhood Search for the Nurse Rosterin Problem. European Journal of Operational Research, 2008. 88(2): pp. 330-34. 5. Burke, E.K., T. Curtois, R. Qu, and G. Vanden Berhe, A Scatter Search for the Nurse Rosterin Problem. 200, Journal of the Operational Research Society, 6: pp. 667-679. 6. Burke, E.K., T. Curtois, R. Qu, and G. Vanden Berhe, A Time Predefined Variable Depth Search for Nurse Rosterin. 2007, School of Computer Science and IT, University of Nottinham. Technical Report. Available from: http://www.cs.nott.ac.uk/tr/2007/2007-6.pdf 7. Burke, E.K., T. Curtois, R. Qu, and G. Vanden Berhe. Problem Model for Nurse Rosterin Benchmark Instances. 2008; Available from: http://www.cs.nott.ac.uk/~tec/nrp/papers/anrom.pdf. 8. Burke, E.K., J. Li, and R. Qu, A Hybrid Model of Inteer Prorammin and Variable Neihbourhood Search for Hihly-constrained Nurses Rosterin Problems. European Journal of Operational Research, 2008 (accepted for publication). 9. Ikeami, A. and A. Niwa, A Subproblem-centric Model and Approach to the Nurse Schedulin Problem. Mathematical Prorammin, 2003. 97(3): pp. 57-54.
0. Li, H., A. Lim, and B. Rodriues. A Hybrid AI Approach for Nurse Rosterin Problem. in Proceedins of the 2003 ACM symposium on Applied computin. 2003. pp. 730-735.. Musa, A. and U. Saena, Schedulin nurses usin oal-prorammin techniques. IIE transactions, 984. 6: pp. 26-22. 2. Ozkarahan, I. The Zero-One Goal Prorammin Model of a Fleible Nurse Schedulin Support System, in Proceedins of International Industrial Enineerin Conference. in Proceedins of International Industrial Enineerin Conference. 989. pp. 436-44. 3. Valouis, C. and E. Housos, Hybrid optimization techniques for the workshift and rest assinment of nursin personnel. Artificial Intellience in Medicine, 2000. 20: pp. 55-75. 4. Weil, G., K. Heus, P. Francois, and M. Poujade, Constraint prorammin for nurse schedulin. IEEE Enineerin in Medicine and Bioloy Maazine, 995. 4(4): pp. 47-422. 2