A Hybrid Setup for a Hybrid Scenario: Combining Heuristics for the Home Health Care Problem

Transcription

1 Proceedings CPAIOR 03 A Hybrid Setup for a Hybrid Scenario: Combining Heuristics for the Home Heath Care Probem Stefan Bertes Torsten Fahe University of Paderborn Facuty of Computer Science, Eectrica Engineering and Mathematics Fürstenaee, D-3302 Paderborn, Germany Abstract Home heath care, i.e. visiting and nursing patients in their homes, is a growing sector in the medica service business. From a staff rostering point of view, the probem is to find a feasibe working pan for a nurses that has to respect a variety of hard and soft constraints, and preferences. Additionay, home heath care probems contain a routing component: A nurse must be abe to visit her patients in a given roster using a car or pubic transport. It is desired to design rosters that consider both, the staff rostering and vehice routing components whie minimizing transportation costs and maximizing satisfaction of patients and nurses. In this paper we present the core optimization components of the PARPAP software. In the optimization kerne, a combination of inear programming, constraint programming, and (meta-)heuristics for the home heath care probem is used, and we show how to appy these different heuristics efficienty to sove home heath care probems. The overa concept is abe to adapt to various changes in the constraint structure, thus providing the fexibiity needed in a generic too for rea-word settings. Introduction Home heath care (HHC), i.e. visiting and nursing patients in their homes, is a growing sector in the medica service business. More and more private companies are now working in this area. As the nursing companies get arger, the probem of how to schedue the nursing staff arises. The chaenge of this probem is to combine aspects of vehice routing and staff rostering. Both are we known combinatoria optimization probems, and good agorithms for each of these two probems are known (see e.g. [24] and e.g. [2, 6, 9]). To obtain appicabe soutions,

2 however, it is crucia to sove the nurse scheduing probem as a whoe due to the high interdependencies of optimized routes and rostering constraints. Additionay, soft constraints and preferences require the use of speciay designed agorithms. Rostering constraints incude hard ones ike quaification requirements or work time imitations and soft ones. Soft constraints are especiay difficut to hande, but have to be considered for appicabe schedues. Typica exampes are: patients prefer certain time intervas for being served, the right chemistry between patients and staff has to be ensured, patients do not ike frequent changes of nursing staff, staff satisfaction concerning e.g. work oad and work time shoud be maximized. The vehice routing aspect of the probem has to take trave times and distances, and inhomogeneous feets (bicyces, pubic transport, cars) into account. Time windows are important in both, the rostering, as we as the routing part of the home heath care probem. Soutions with minima costs and maxima patients/staff satisfaction are of most interest for companies. Costs in this context may refect expenses for fue, etc. or costs of the staff. Respecting preferences is the second important parameter to be considered in HHC rostering. Patients wi simpy change to a different heath care company if their wishes are not met. For the staff, considering their preferences increases motivation, which on the one hand impacts on the patients, and on the other hand heps to dea with many stressfu situations (death, termina inesses, ate jobs, etc.). An important rea-word requirement in our project PARPAP is runtime imitation. Panners ike to have a good soution after at most 0 5 minutes runtime. From our experience in crew rostering [9] this excudes coumn generation approaches for the HHCP. Instead, we wi appy Tabu Search, Simuated Anneaing, or Constraint Programming, respectivey, to assign staff to jobs. For optimizing an individua work-pan, we wi use a hybrid inear and constraint programming modue. In Sec. 2 we wi describe how we mode the HHC probem. For this presentation, a more compact mode than the one in the industria prototype wi be used. The simpified mode sti refects the key characteristics of the origina probem, but aows us to ignore certain technica detais when presenting the agorithms. Section 3 is dedicated to the heuristics deveoped for the probem, and we wi show how to combine these methods in a powerfu hybrid approach in Sec. 4. Section 5 presents numerica resuts of the various methods, and finay we concude.. Literature Review To our knowedge, there are ony a few pubications on the topic of optimization and scheduing in HHC: Cheng and Rich describe a combined mixed-integer programming (MIP) and heuristics approach [8]. Numerica resuts for up-to 4 nurses and 0 patients are presented. In [3] a decision support system that is based on simpe scheduing heuristics is proposed. Two reated topics have attracted more researchers: Panning systems for hospitas mode some aspects that are aso needed for HHC. We ony mention [, 5, 7, 8] as exampes. Most of these use constraint programming techniques in order to mode and sove the nurse rostering probem. Vehice Routing with Time Windows (see e.g. [24]) refects the mobiity aspect of the probem, but ignores any further restriction.

3 2 Mathematica Mode for the HHCP Staff rostering describes the process of assigning staff to tasks. In HHC, the staff consists of empoyers with various skis, and the tasks are the services to be provided to the patients, training, etc. Timing rues, quaification rues, reationship rues, and structura rues, as we as routing information dictate the way, a good roster shoud ook. The cost of a roster depends on rea costs for wages and transportation, as we as on artificia costs introduced as penaty terms for modeing certain characteristics and preferences. 2. Parameters and Notation We are given N nurses N N, P patients P P, and a set of J jobs J J. Jobs represent either a certain service to be provided to a certain patient, or a task to be performed by a nurse during her working hours (ike training, breaks, emergency service, etc.). Thus, there is a fixed ocation for each of these jobs. For each patient p P there is a subset J p J such that these jobs invove patient p, and it hods: p P J p J, and J p J q /0 p q. Since jobs are patient reated, we can aways identify the patient corresponding to a job. In the foowing, we wi ony speak of jobs, and by preferences of a job we refer to the preferences of the patient corresponding to the job. A time window represents the time interva in which a job has to be started, or describes the working time interva of a nurse. We distinguish between soft time windows and hard time windows. Whereas a soft time window is ony a preference, which we may vioate at the expense of penaty costs, any hard time window has to be met. Waiting time before a hard time window is aowed, beginning service after the hard time window eads to an infeasibe roster. The foowing functions represent the hard and soft time windows of nurses and jobs, respectivey: hb J : J IR, he J : J IR, describe start and end, resp. of a job s hard time window. sb J : J IR, se J : J IR, describe start and end, resp. of a job s soft time window. d J : J IR is the duration of the job, i.e. the time needed to compete a job. hb N : N IR, he N : N IR, sb N : N IR, se N : N IR, represent the start and end of the hard and soft time window of a nurse. min time N : N IR and max time N : N IR give minima and maxima working time of a nurse. Obviousy, for a job j J we have hb J j sb J j se J j he J j. The same hods for time windows of nurses. Figure shows an exampe for these time windows. Quaifications are the skis possessed by a nurse, or required by a job, respectivey. The concept of quaifications offers a rather fexibe too for modeing various characteristics of the HHC probem. Let S be a set of a quaifications: quai J : J 2 S and quai N : N 2 S are the quaifications required for a job or possessed by a nurse, respectivey. Hard quaifications are those that are vitay required for the job and incude e.g. graduation. Soft constraints embrace preferences of nurses and patients, and may be ignored at the expense of penaty costs. We mode preferences of patients for certain nurses, preferences of nurses for

4 penanty 00% (waiting time) (infeasibe) 0% hard_begin soft_begin soft_end hard_end time Figure : Penaty concept for time windows. Arriving before hard begin produces waiting time, arriving after hard end is infeasibe. Penaties proportiona to eariness or ateness are used for arrivas within the hard, but before or after the soft time window. certain patients, experiences for certain jobs, and factors that guide a fair distribution of difficut jobs over a nurses (over some time period). Aso, we aow soft quaifications and requests for those. In the compact mode, a these parameters are accumuated into one function, which gives the soft constraint penaty vaue for assigning nurse n to job j: sc : N J 0. The ast function provides geographica information. The routing aspect considers trave time between two jobs j j (in the industria prototype trave distances are used as we). Since each job has a fixed ocation, we use the trave time between these ocations: tr time : J J IR, trave time between the ocations of two jobs. 2.. Core Optimization Probem In the HHC probem we are ooking for an assignment of job schedues to nurses, such that a jobs are taken care of, a hard constraints are respected, ony few soft constraints are vioated, and such that the overa cost for that assignment is minima and the number of preferences satisfied is maxima. To be more specific, we formuate the foowing: A sequence R j t j k t k, j J t IR, k is caed a roster and contains k jobs and a starting time t for each job j. We assume that the sequence is ordered in increasing times, that is t t for k. We need to find a soution S R R N consisting of N rosters, where R n j n is the roster for nurse n, such that: t n j n N k n k i i t n k n j i J () i i N k i k i : j i j i i i (2) t i d J j i hb J j i t i he J j i i N k i (3) tr time j i min time N i j i t i k i max time N i t i k i t i i N k i (4) d J j i k i t i i N (5) d J j i k i t i i N (6)

5 hb N i t i and t i k i d J j i k i he N i (7) quai J j i quai N i i N k i (8) These hard constraints can be expained as foows: We need to cover a jobs by our schedues (), but none of them more than once (2). Any starting point for a job has to respect the hard time window (3). In (4) we require enough time to provide the service and to trave to the next job before the next job starts. (5) and (6) define ower and upper bounds on the working time, and (7) ensures that no job is carried out outside the work time interva of the nurse. Finay, we insist on having a hard quaifications of a job covered by the assigned nurse (8) Cost Function In order to mode the cost function, we have to define our measure for vioating soft constraints. Any vioation of a soft time window wi be penaized by a factor proportiona to the eariness or ateness. For a jobs j J et t j hb J j he J j be the time assigned to job j. Then eary J j : ate J j : 0 sb J j hb J j sb J j t j sb J j hb J j ese 0 he J j se J j t j se J j he J j se J j ese (9) (0) p J j max eary J j ate J j 0 () Since t j is within the soft time window ( eary J j 0 and ate J j 0), before the soft time window ( eary J j 0 and ate J j 0), or after it ( eary J j 0 and ate J j 0), choosing p J j as in () gives the correct penaty (see Fig. ). Vioating nurses soft time window is treated simiary. The corresponding penaty function is p N. Let R j t j k t k, j J t IR, k, be a roster assigned to nurse n. Any vioated soft quaification is penaized by adding extra costs stemming from assigning these jobs to nurse n, and normaizing them by the number of jobs assigned: p sc n : k k sc n j k n N (2) Having defined a soution of the HHC probem and knowing the measure for soft constraints, we are now abe to construct the objective function. In order to refect both, the routing and the rostering aspects, we combine them into a weighted sum of the tota trave time needed for the schedue and the sum of a penaties: minimize obj R R N where obj R R N n α i k i tr time j i j i LB travetime α 2 N UB travetime LB travetime ε n N t n k n d J j n k n t n LB worktime (3) (4) UB worktime LB worktime ε α 3 J p J j α 4 j J N p N n (5) n N

6 α 5 N p sc n (6) n N and α α 5 α i 0 (7) Whereas (5) and (6) ony sum up a penaties, and normaize these numbers, (3) and (4) are a sighty more compicated. In (3), the doube sum accumuates the trave-times needed to trave between any two consecutive jobs for a schedues. We normaize that vaue by reating it to some upper and ower bounds for the tota trave-time of a given instance. E.g. we can set LB travetime equa to the sum of the J n smaest trave-times cacuated between jobs, and accordingy, we use the J n argest trave-times for UB travetime. The ε ensures a vaid fraction in case the ower and upper bound overap. Simiary, (4) modes the tota working time of a nurse. We can set UB worktime n N max time N n and LB worktime n N min time N n. The mode defined above can represent severa NP-hard optimization probems. E.g. we get a muti-tsp with time windows, even if we ignore quaifications (8), (2), (6), work time imits (5), (6), and soft time windows (), (5). Simiar adaptations ead to muti-processor scheduing or set covering probems (see [0]). 3 Soving Home Heath Care Probems Our mode defined above is a hybrid of a rostering mode and a routing mode. Good approaches were presented for both modes in iterature, and we wi re-use some of the ideas presented previousy to buid our heuristics. However, it is not possibe to use a two-stage approach that first generates feasibe routes, and then assigns nurses to them (vioation of hard/soft quaification and nurses time window constraints). Aso the vice versa approach assigning jobs to nurses and then generating routes wi more than ikey produce infeasibe or disproportiona expensive routes. Therefore, ony an integrated approach that considers time scheduing, rostering, and route panning simutaneousy is appropriated for the HHC probem as a whoe. Our approach interweaves two parts: (a) finding a partition of jobs to nurses, and (b) finding an optima sequencing for each such partition. The atter one contains in our case a TSP with time windows and is therefore NP-hard. Whie ooking for a vaid partition we aways evauate impied tours, and tour quaity feeds back to partitioning. We have different methods to (a): Initia heuristics that quicky generate an initia soution and two improvement heuristics which take a soution and try to improve it via oca exchanges. A of these need information regarding a good sequence of jobs within a roster (part (b)), which we determine via a combined CP and LP approach (Sec. 3.2). As a first step, however, we try to reduce the data compexity of a given instance. 3. Preprocessing During the initia data preprocessing step, we determine which job/nurse pair is compatibe with the hard quaification constraint (8) and time window constraint (7), and we store this information. Shoud there exists a job that can be done by ony one nurse, we fix this assignment (without fixing the time at which this job has to be done). Aso, we compute the soft constraint vaues that incude a preference parameters. Time windows of jobs can then be shrunk to the eariest or atest time at which a nurse is avaiabe (7). In the ast step we determine precedences impied

7 min α 2 α 4 UB worktime w 2 N y y 2 k x : s.t. t t d J j tr time j j k 2: sb J j t x sb J j hb J j 3: t se J j x he J j se J j 4: x 0 5: t hb J j 6: t he J j 7: sb N n t y sb N n hb N n 8: t k d J j k se N n y 2 he N n se N n 9: y y 2 0 0: min time N n w # : max time N n w 2: t k d J j k t w 3: t hb N n 4: t k d J j k he N n α 3 J k (8) Figure 2: LP (8): Finding optima starting times for a given sequence by the time windows. That is, for any two jobs we store if they have to be performed in parae or in an induced order or if they are not reated, respectivey. 3.2 Sequencing a Roster Sequencing consists of ordering the jobs and assigning starting times. Due to time window constraints, in the HHC ony few permutations correspond to feasibe orderings. In our approach we enumerate those orderings by a CP approach, and we use an LP to find optima start times with respect to the objective Optima Start Times for a Vaid Ordering Given a set j j k and a nurse n we need to find an optima sequencing (i.e. an ordering of these jobs and an assignment of a start time to each of the jobs) such that the sum of trave times and penaties for not hitting the soft time window is optima for nurse n. For a given ordering and a given nurse, we can easiy cacuate penaties by appying (9) (). Moreover, a simpe LP can be used to optimize the starting times t of each job j, k (see (8)). We expain LP (8) in more detai: For k variabe t corresponds to the starting time of job and x accounts for soft time window vioations at job. y y 2 is the penaty for vioating the nurse s soft time window and w accounts for working time. The first constraint cass corresponds to (4), the three ines using x refer to (9) (), ines 5,6 imit the starting time to the hard time window (3). The nurse s soft time window is respected by ines 7 9 and the corresponding hard time window (7) in ines 3,4. Finay, ines 0 2 correspond to the minima and maxima work-time constraints (5) and (6), and to work time minimization. If we use the LP in a setting where sets of jobs are generated incrementay we have to repace # by w 0 to ensure feasibiity. In the LP objective we use the weight factors α 2 α 3 and α 4 of the HHC probem. Worktime

8 w is rated by the goba bound on working time. As stated before, for a given order of jobs the starting times found by the LP are optima with respect to time window penaties and working time Generating Vaid Orderings The generation of a feasibe orderings for jobs j j k now remains. Theoreticay, this invoves soving O k! many LPs. In our case, very few permutations correspond to feasibe orderings, though. Thus the agorithm is very fast with typica input (see Sec. 5.). This is due to the fact that in HHC many jobs have to be performed in the morning, after unch and at bedtime. Hence, there is a ot of overapping at these times. Intensive care, on the other hand, is provided a day, but usuay takes much onger time and thus aso overaps with other ong running jobs. We cacuate a permutations that correspond to feasibe orderings of jobs via a recursive function and sove the LP described above. Agorithm Find best sequence for set U generatesequence (sequence S, set U) : // propagate impied precedence constraints 2: for i ength S do 3: if (U /0) then goto 2 4: for a j U do 5: if (precedence for j is: after S i and before S i ) then 6: if (inserting j between S i and S i does not vioate any time window) then 7: U U j 8: insert j between S i and S i 9: goto 4 // restart oop at first eement in U 0: ese : return // infeasibe 2: // tree traversa 3: if (U /0) then 4: sove LP (8), and eventuay update bestso 5: ese 6: seect j U 7: for i ength S do 8: if (inserting j between S i and S i does not vioate any time window) then 9: S S; 20: insert j between S i and S i 2: generatesequence (S, U j ) // recursive ca Agorithm describes the procedure: Initiay caed with S /0, U j j k, the agorithm moves an eement of U to any feasibe position in S, and recursivey checks for possibe extensions of S (ines 5 20). (To ease the notation assumed that inserting before the first or after the ast eement of S can aso be represented by an insertion between two consecutive eements). If U becomes empty, S is a vaid ordering, and (8) provides optima starting times for that ordering (ines 2,3). After termination, bestso contains the best ordering found. We use propagation to fix any job that needs to be incuded between two other jobs (ines ) and we stop the current recursion as soon as we find a job for which we cannot fufi the

9 precedence constraints in the schedue S currenty under construction (ines 6 and 0,). I.e. we have to revise an earier branching decision, if inserting a job at a required position is not possibe because some other time window on the schedue or the nurse s time window wi be vioated. Basicay, such a test requires checking (3), (4), (7) and it can be improved by appying forward and backward push information (see [2]). Additiona constraints on the roster s design may be incuded in this agorithm as we, and they wi further reduce the search space. To avoid repeated cacuations for identica job-sets, we use a cache that stores a optima schedues found. In the experiments, ony some thousand different sequences are determined, whereas some miion requests are answered by the cache (see Sec. 5.). 3.3 Initia Soutions via Constraint Programming Initia heuristics are used in obtaining a first soution quicky. They provide us with the first rosters, and if the dispatcher aows more time for optimization, they serve as a starting point for improvement heuristics. Our first approach was to adapt insertion and scheduing heuristics deveoped for the vehice routing with time windows ([2]). Though they are very fast usuay ony a spit second the soutions obtained turned out not to be appicabe, as in most cases not a jobs coud be covered by nurses. Therefore, we decided to foow a CP approach within an incompete tree search. On typica instances, we usuay obtain very good starting soutions within a few seconds. Furthermore, we may trade time vs. quaity with such an approach as we can stop at any time after having found the first soution and take the best one produced so far Formuation We use a redundant modeing for the HHC probem. We represent the roster for nurse n by a set R n, n N and each job j J by an integer variabe i j. In a soution, the set R n contains a jobs assigned to nurse n, and variabe i j is the nurse who has to serve job j. The fact that we have to cover a jobs () is impied by this mode, since each job is inked to a nurse. Initiay, pos R n pos i j which aready covers (8). j quai J j quai N n n N req R n /0 (9) n quai J j quai N n j J req i j /0 (20) Next, we state that the n rosters must not intersect (2) by a goba cardinaity constraint [20]. We force consistency between roster and job variabes by stating n N j J : j R n i j n (2) We can improve this mode by redundant information. If we know from preprocessing that two jobs j j have to be served in parae, they cannot be assigned to just one nurse. We add i j i j for a j j J that have to be in performed in parae (22) Using the sequencing cache, we can add a forward checking. Whenever the shifting of job j to the current required set of a nurse woud resut in an infeasibe ordering, we can remove that job from the nurse s domain: n N : pos R n req R n j req R n : j infeasibe pos R n pos R n A set variabe has a current domain of those vaues which are sti possibe candidates for an assignment (noted as pos()), and those that are aready fixed or required (noted as req()). Integer variabes are regarded as set variabes which have req. j (23)

10 3.3.2 Branching and Tree Traversa If domain fitering aone does not provide a soution or faiure, we have to branch. Our strategy here is to seect a job j for which the possibe set pos j is the smaest, and to first choose a nurse assignment which wi resut in the best overa improvement. Agorithm 2 Goa : Branch on job with smaest domain, and assign best nurse first : j job with minima domain, that is not yet fixed. 2: n nurse in pos j with best improvement vaue 3: eft branch: (i j n) right branch: (i j n) Having produced two new subprobems, we have to seect the next one to process. We use Limited Discrepancy Search (LDS) ([3]) to traverse the search tree. 3.4 Improvement Heuristics A soution can often be improved by appying oca changes. Metaheuristics, ike Simuated Anneaing (SA) ([6]) and Tabu Search (TS) ([, 2]) are widey used and have been shown to produce good soutions in a reasonabe amount of time. In our approach, both metaheuristics are based on a simpe -shift, i.e. removing a job from one pace and inserting it esewhere. To fexiby the operator we aow an insertion at a different set of jobs as we as an insertion at a stock Γ. Accordingy, we may remove a job from a set of rosters or from the stock. Of course, deeting a job and re-inserting it in the same set again is prohibited. Moving a job to the stock is highy penaized by adding + to the objective. The costs of a -shift are defined as the gain from deeting job j from a roster or the stock minus the oss resuting from inserting it again into a certain position in a roster or into the stock. Using the stock we can start with Γ J and appy the metaheuristics to an empty soution, or we can use a soution found earier and try to improve on that. Notice, that we aow intermediate steps with a non-empty stock (i.e. we reax ()). Any soution returned by either metaheuristics, however, requires a jobs to be assigned to nurses. Since we cacuate optima sequences via the approach presented in Sec. 3.2, it suffices to partition J into good subsets. As with the CP approach, LP (8) can ony be appied, if we repace the minimum work time constraint # for subsets under construction by w 0. (Because of space imitation we wi ony present the TS modue in more detai.) Tabu Search The genera strategy of Tabu Search is to systematicay expore a possibe moves from the current to a neighboring soution. The move eading to the best (non-tabu) neighboring soution is accepted, even if this resuts in a deterioration of the objective function. To prevent the search from cycing, a tabu ist which stores the inverse move for a certain number of iterations is used. Thus, a soutions which can be obtained by appying a move stored in the tabu ist are not considered. An aspiration criterion aows us to override this rue, if a move improves the best goba soution found so far. We perform the best -shift among a possibe ones (in the above sense) to obtain a different soution. I.e., we have to find an optima sequencing for each member of the -shift neighborhood. We set the inverse of that move tabu for the next 0 iterations and update a tabe f r counting how often a certain job is assigned to a nurse. f r is used for diversification by adding

11 a specific penaty for frequenty used moves. By doing so, we graduay manipuate the cost function into moving the search away from some hot spots. The penaty term represents the additiona costs incurred by moving job j to nurse n. It was originay proposed by [23]: p j n max f r j n f r max, where f r j n is the frequency of moving job j to nurse n, f r max is the maxima frequency over a nurses and jobs, and max is the maxima absoute difference between two consecutive moves performed so far (excuding moves from/to the stock because of their high extra penaty). Diversification penaties are added to the origina objective if no goba improving soution is found for some iterations (e.g. 2000). As soon as we find an improving soution we switch back to the origina objective function. Shoud we sta for onger time, we terminate the search. We do not consider any specific intensification within the pure TS. 4 A Hybrid Soution Approach As we wi see in the experimenta evauation, the approaches we have just presented are abe to quicky find good operationa soutions. There are two points, which can be improved further: Firsty, if time aows, we woud ike to optimize more, and achieve better pans than those found so far. And secondy, we woud ike to coect severa diverse pans: We experienced that the acceptance of computer generated soutions increases if not ony one optimized pan is presented, but aso some aternative pans, and dispatchers as we as nurses know that the fina decision is made by a human. In this section we discuss a hybridization technique which accounts for both points. The approach offers a diverse variety of very good soutions to the dispatcher and ets him/her decide which roster is best. 4. The Soution Poo Concept Appying ony one improving agorithm to ony the best initia soution is not ikey to produce the best possibe soution. This probem has been recognized in severa appication fieds and approaches based on the concept of a soution poo have turned out to be quite powerfu in practica experiments (see [22]). The idea of a soution poo is to store some intermediate soutions generated via the improvement heuristics and expoit the stored know-how for designing new soutions. In a TS context, such an approach refers to the utiization of ong term memory (see []). In our approach, we appy different heuristics to the soutions in the poo and we using statistica measures for good soutions. 4.2 Using a Good Soution to Improve the Search Let Ω be a set of soutions found by some heuristics. In the beginning Ω contains soutions found by initia heuristics (IH). In the optimization oop, we appy our improvement heuristics to each soution in Ω, and repace the od soution by the improved one (see Ag. 3). Whereas TS and SA simpy start with the soution S provided by the poo, the CP approach has to be adapted sighty. We use the genera settings described in Sec. 3.3, and change the branching scheme of goa (Agorithm 2) to consider decisions in S as we. We modify the

12 Agorithm 3 A poo Ω stores initia and improved soutions found during optimization : P probem instance 2: Ω /0 Ω 3: whie (Ω not fu) do IH 4: S IH P ; Ω Ω S SA 5: repeat 6: seect S Ω; Ω Ω S TS 7: seect a heuristic method h TS, CP, SA 8: S h S P ; Ω Ω S IH CP 9: unti (termination criterion) 0: return a soutions in Ω search order with respect to soft constraint satisfaction (provided by an input soution). Given a soution S, we can sort the sc n j vaues of that soution in a preprocessing step. In goa 3 (Ag. 4), we chose j as the first job in this order that is not yet assigned. We try to use the same nurse n assigned to j as in soution S. The ratio behind this is that in a good soution many assignments aready correspond to assignments in an optima soution, and we wi use as many of these assignments as possibe. If the nurse used in S is not avaiabe (e.g. because some propagation or earier branching has assigned her esewhere), we seect the next nurse as in goa. This resuts in a CP based tree search where vaid soutions which are simiar to S get buit first. The corresponding goa 3 is described in Ag. 4. Agorithm 4 Goa 3: Branch on job with best soft constraint vaues first, and assign the nurse that was aso used in soution S preprocessing-step: sort sc n j for a j n pairs found in S : j first job in sorted ist that is not yet fixed 2: n nurse that was aso used in soution S for job j if possibe 3: ese n nurse in pos j with best improvement vaue (as in goa ) 4: eft branch: (i j n) right branch: (i j n) 4.3 Using the Essence of A Soutions A further soution improvement is gained by using statistica information from the soution poo to guide the CP approach. The idea is to detect job/nurse pairs, that often occur in soutions with high quaity, and to consider those assignments eary in a systematic search. Vice versa, pairs occurring in ow quaity soutions ony shoud be considered ate in a systematic search. This idea eads to a modified variabe ordering and variabe assignment goa within the CP approach. Let Ω S S k be a poo of k (diverse) soutions, v S be the vaue of soution S. For any nurse n N and any job j J, et µ n j be the accumuated objective vaue of a soutions in Ω assigning job j to nurse n. Accordingy, et κ n j be a counter for the number of soutions containing such an assignment. We consider those assignments that have a ow vaue of µ n j κ n j as good. This vaue is the average quaity of soutions containing an assignment of j to n. We order the pairs n j, n N, j J according to increasing vaues µ n j κ n j. In the branching decision described in Sec. 3.3 we repace the variabe seection by a seection step

13 which takes the first pair n j for which an assignment is possibe. Then we assign j to nurse n on the eft branch, and we excude j from the possibe set of n on the right branch. Since we usuay ony perform an incompete search, it is ikey that assignments made in the first part of the search tree wi be incuded in the best soution found after interrupting the search. To prevent the statistica information gathered in µ n j and κ n j from being part of a sef-stabiizing process, we refine the coection process as foows: We increase the vaue µ n j κ n j by an additiona penaty depending on the frequency of a certain assignment. The more often an assignment is used, the higher the penaty added. We use µ n j κ n j κ n j max n j κ n j max n j µ n j min n j µ n j (24) In doing so, high quaity parts are sti assigned first, but rarey used parts are preferred to more frequenty used ones. We assume this strategy produces more diverse soutions than the pure quaity ordering aone. The agorithmic framework is presented in Ag. 5. Agorithm 5 Goa 4: Branch on job/nurse pairs that occur in good soutions found earier preprocessing-step: sort n j according to increasing vaues obtained by (24) : seect n j as the first in the previousy sorted ist that is not yet fixed. 2: if no such pair exists, seect n j as in goa 3 3: eft branch: (i j n) right branch: (i j n) 5 Numerica Evauation A agorithms were coded in C++ and compied by the GNU g compier using fu optimization. Our benchmark tests were run on a Pentium III-933 PC with 52MB RAM operating Linux kerne For soving LP (8) we use ILOG CPLEX 7.5 [4], and for the CP approach we appy ILOG SOLVER 5.2 [5]. SA was impemented using PARSA [7]. We used 0 synthetic test scenarios, containing between 20 and 50 nurses, and between and 326 jobs. The data was generated according to rea-word input. Each job asts between 6 and 72 minutes, nurses hard time windows between 5 and 9 hours. Locations were chosen randomy, and eucidean distances were cacuated between these ocations. Aso, the soft constraint factor for each job was seected randomy. The tabes and figures presented in the foowing show the quaity vaue as defined by the objective. It contains routing quaity, time window penaties as we as the soft quaification measure. A terms were equay weighted by α i 5. The run times are given in seconds. To refect a rea-word setting, a run times are imited to 600sec or 840sec, respectivey. A more detaied experimenta anaysis of our methods is given by [4]. 5. Optima Sequences The tabe in Fig. 3 shows that enumerating a possibe orderings for a given set of jobs does not drasticay boost the computing time, if we use time window constraints and job precedences to imit the enumeration tree. Agorithm buids ony few sequences and even ess LPs have to be soved. Notice, that a compete enumeration for a roster of size k woud resut in k! recursive cas. E.g. there are possibe permutations for a 3-job-roster whereas we need

14 roster size recursive cas LP cas fai rate nurses, 50 patients, 600sec time SA TS CP Figure 3: Sequencing and Cache sizes first soution after 0sec N - P - J time in sec. objective objective sizes first soution after 0sec N - P - J time in sec. objective objective Figure 4: Finding initia soutions via CP ess than 9 cas on the average. An even smaer number corresponds to vaid orderings (which have to be processed by the LP). These encouraging resuts are due to tight time windows which are common to rea-word HHCP instances. In Fig. 3 the efficiency of the sequence cache defined in Sec is shown for the basic soution methods. We can see that CP requests many identica rosters due to the systematic search. TS benefits consideraby from the cache when enumerating the neighborhood. Due to its randomness SA arbitrariy jumps around in the soution space. SA is forced to behave more ocay ony after a decrease in temperature and thus can benefit from previousy optimized rosters. The sma bend of the CP curve is ikey due to the impementation of LDS in ILOG SOLVER, see [5]. 5.2 Initia Soutions via Constraint Programming In Fig. 4 we show resuts obtained for the initia soution. That is, we appied CP (using goa ) to each benchmark set unti the first soution was found. Whenever the first soution was found in ess than 0 seconds, we aso show the best vaue found after 0sec. (To get an idea of the quaity obtained, these resuts can be compared to those in Fig. 5). As can be seen in Fig. 4, a good starting soution can be found in a short time even for arger instances. Typicay, the first soution, and the one found after 0sec are of the same quaity. The 50 nurse instances take consideraby onger than the smaer ones. There are two possibe reasons for this: The CP heuristics might be bad, making many wrong decisions before finding the first soution, or the time used in each choice point may be high. When anayzing the resuts in more detai we found that the number of choice points expored, and the number of faiures encountered were reasonabe. The time spent for sequencing sets, however dominates the overa running time. In the beginning sequencing rosters is quite expensive in terms of computing

15 objective nurses, 50 patients, 600sec time in sec TS CP CP+TS SA sizes objective after 600sec N - P - J CP SA TS CP+TS Figure 5: Resuts for CP, SA, and TS started from an empty soution, and for TS started on the first soution found by CP (CP+TS). A dash indicates that no soution was found. Each approach ran for 600sec. time, since no information is stored in the cache. Later on, many sequencing requests can be answered by the cache. Thus succeeding computations benefit from the bigger effort required in the beginning. 5.3 Comparing the Heuristics Unfortunatey, if CP continues running for 0min, the good start is not foowed by a good convergence. The first coumn in the tabe of Fig. 5 gives soution vaues which are ony sighty better than the initia ones found by CP within the first few seconds. SA (started on an empty initia soution) produces better soutions than CP, but cannot compete against TS (aso started on an empty soution) as can be seen in coumn three of that tabe. TS outdoes both of the previousy mentioned approaches if it can find a soution at a. In three cases TS seems to be too aggressive, and it is not abe to generate a feasibe soution at a within 600 seconds. Athough SA and TS are both based on the same -shift neighborhood, their soution quaity differs significanty. Such a behavior seems to be party approach immanent (see e.g. [9]). Another reason is, again, the sequencing. Whereas TS and CP systematicay check their neighborhood and thus can reuse previousy generated sequencing information, SA jumps randomy, resuting in much more cache fais and thus, much more cacuations of sequences (see Fig. 3). Motivated by the success of TS we started TS on the first soution found by CP (referred to as CP+TS ). Not ony does this approach ensure obtaining a feasibe soution, it aso appears to be the best approach in this test. Except for the smaest instances, it surpasses a other methods. Figure 5 shows a typica pot comparing our four approaches. SA when starting from scratch takes quite some time to improve the soution. SA does not reach the soution quaity of TS, or CP+TS. CP quicky finds a soution, but has difficuties improving it. TS needs some time to find a vaid starting point which it then quicky improves. Combining CP and TS brings together the advantages of both approaches. 5.4 Combining Approaches Instead of stopping the search after the termination of TS, we can aso trade our computing time differenty. In one test we ran CP for two minutes to find 0 different initia soutions which we stored in Ω. Then each of these soutions was optimized via a TS imited to one minute each

16 sizes objective N - P - J CP+TS CP+TS oop 0 * CP+TS objective nurses, 50 patients CP+TS CP+TS oop 0 * CP+TS time in sec Figure 6: Combined approaches. Resuts for CP+TS, aternating CP and TS (CP+TS oop), and appying TS to 0 initia CP soutions (0* CP+TS). Running times between 600sec and 840sec. (starting diversification after 500 non-improving rounds). Finay, a information coected in the µ n j, and κ n j tabes was used to run CP using goa 4 for additiona two minutes. The other test consists of running CP for an initia soution (goa ), foowed by aternating runs of TS and CP on the best soution found (we used goa 2 which produced a simiar soution quaity as goa 3). The time imit of a round was raised whenever the previous round did not improve the goba best soution. In Tabe 6 we present the resuts obtained by these two methods. Evidenty, appying short optimization runs to a arger set of soutions outperforms both, the simpe approach (CP+TS) considered in the previous section, as we as the aternating approach. Goa 4 was sedom abe to find even better soutions. Figure 6 compares the combined approaches for Ω 0. 6 Concusions We presented a compact mode to the HHC probem which is fexibe enough to break down most rea-word HHC probems of different characteristics. Furthermore, we deveoped severa soution approaches for the mode. The approaches find good quaity rosters that satisfy routing, quaification, time windows and soft constraints. We combined the effectiveness of LP for finding optima starting points of jobs and the effectiveness of CP for generating feasibe ordering. This modue is utiized by CP and oca search methods. We can offer severa good soutions to the end users by using a poo of soutions, and we can use information coected in the poo to improve the quaity of these soutions. Experimenta resuts show significant gains when these different methods are combined. The industria prototype is currenty undergoing intensive user evauation, where soutions generated by our optimization methods are competing in rea-word scenarios. In a second step, requests for additiona ega and company constraints, or credit point systems (contributed by researchers from ergonomics) may be integrated into our agorithmic framework. References [] S. Abdennadher and H. Schenker. Nurse scheduing using constraint ogic programming. In Procs. of the th Conf. on Innovative Appic. of AI, pages , Meno Park, Caifornia, 999. AAAI Press.

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