Scatter search approach for solving a home care nurses routing and scheduling problem

Transcription

1 Scatter search approach for solvng a home care nurses routng and schedulng problem Bouazza Elbenan 1, Jacques A. Ferland 2 and Vvane Gascon 3* 1 Département de mathématque et nformatque, Faculté des scences, Unversté Mohammed V-Agdal, B.P. 14, Rabat, Morocco 2 Département d nformatque et de recherche opératonnelle, Unversté de Montréal, P.O. Box 6128, Succursale Centre-vlle, Montréal, Québec, Canada, H3C 3J7 3 Département des scences de la geston, Unversté du Québec à Tros-Rvères, P.O. Box 500, Tros- Rvères, Québec, Canada, G9A 5H7 * Correspondng author. Emal: [email protected]; Phone: , #3136; Fax : Abstract The home care nurses routng and schedulng problem conssts n determnng the sequences of vsts performed by home care nurses to patents. Very often publc medcal clncs are responsble for ths plannng. The problem shows smlartes wth the vehcle routng problem wth tme wndows (VRPTW) but t also ncludes addtonal constrants related to medcal restrctons and to the contnuty of care. In ths paper, we descrbe the specfc problem. We next present n detals the soluton methods used for the global problem. They are based on a scatter search approach. We present numercal results and comparsons wth real lfe data provded by a medcal clnc (CLSC Les Forges located n Tros-Rvères, Canada). Keywords: routng and schedulng, home care nurses, mathematcal programmng, metaheurstcs (Tabu). 1. Introducton The ageng populaton and the urge to reduce costs ental health care servces to be provded at home more often than they used to. Health care managers must solve new types of problems such as nurse schedulng, dstrctng or routng problems. Routng home care nurses s a dffcult tas that we address by solvng a mathematcal programmng model usng a scatter search approach. In ths paper we model a real problem and we analyse tools to specfy nurse routes to vst ther patents. Ths approach gves rse to a varant of the vehcle routng problem wth tme wndows (VRPTW) ncludng addtonal constrants related to medcal restrctons and to the contnuty of care. These addtonal constrants mae the problem more complex and more dffcult to solve. In ths paper, we llustrate the scatter search approach by solvng a real problem faced by the CLSC Les Forges n Tros-Rvères, a publc medcal clnc coverng a terrtory ncludng several urban and rural sectors. Home care nurses vst patents each day. Each nurse and each patent are assgned to one sector. Usually a nurse can vst only patents from her sector. Hence, each sector has a set of nurses to provde servce to ts patents. Contnuty of care s mportant n home care servces. Patents prefer to always receve ther treatments by the same nurse. Nurses also thn that ths s preferable to become better acquanted wth ther patents. Thus contnuty of care s taen nto account n our model. Moreover nurses are also responsble for tang blood samples for some patents. But then, for medcal reason, each blood sample must be returned to the clnc wthn a tme nterval after beng taen. For ths reason, our model requres addtonal returns to the clnc that are more dffcult to schedule because we do not now n advance the exact tme of the day when a patent requrng blood samplng wll be vsted. + Correspondng author. Tel.: , #3136; fax : Emal address : [email protected]

2 Ths paper s organzed as follows. Frst, a lterature revew s presented n Secton 2, n order to poston our contrbuton n ths feld of research. Next, Secton 3 depcts the home care routng and schedulng problem faced by the Centre Local de Servces Communautares (CLSC) Les Forges n Tros- Rvères (Canada, provnce of Québec). Secton 4 s devoted to the mathematcal programmng model. Then n Secton 5 we ntroduce the scatter search soluton approach. Numercal results to compare the dfferent soluton approaches are presented n Secton 6. We also compare the results obtaned wth the best soluton approaches and the manual results provded by the clnc. To conclude, we summarze our wor and we ndcate possble extensons for future research. 2. Lterature revew The lterature ncludes only a few artcles on routng and schedulng home care nurses. Those artcles deal manly wth problems and cases related to specfc clncs or companes. Bertels and Fahle (2006) develop a computerzed system whch solves the nurse schedulng and the routng home care nurses problems. Eveborn et al. (2005) ntroduce a schedulng problem for a varety of home care provders whch s modeled as a set parttonng problem and solved wth a repeated matchng algorthm. Begur et al. (1997) developed a software enablng to vsualze the patents locatons. The objectves are to mnmze the total travelng tme and to balance nurse worloads. Cheng and Rch (1998) present the routng home care nurses problem as a vehcle routng problem wth tme wndows and multple depots. The problem s to determne optmal routes mnmzng the total dstance, the overtme wored by regular nurses, and the number of hours wored by part tme nurses. Several papers can be found n the lterature on the vehcle routng and schedulng problem wth tme wndows (VRPTW). Such a problem s underlyng the routng and schedulng of home care nurses. We refer the reader to the papers of Solomon and Desrosers (1988), Malandra and Dasn (1992) and Bräysy and Gendreau (2005a, 2005b) focusng on heurstc methods. Papers of Solomon and Desrosers (1988) and Malandra and Dasn (1992) nspred us n defnng our mathematcal model. However routng and schedulng home care nurses wth addtonal constrants and a dynamc aspect does not seem to have receved much attenton n the lterature. 3. Problem overvew Nowadays, more patents receve ther medcal treatments at home. The medcal publc clnc CLSC Les Forges n Tros-Rvères (coverng a terrtory dvded nto sectors) s accountable for plannng nurse vsts to patents. Each patent s assgned to a sector accordng to hs home address (note that the masculne gender s used throughout the paper). Each nurse s also assgned to one sector, each sector ncludes several nurses. Even f a nurse should vsts only patents from hs sector, some nurses may however have to vst patents from other sectors n order to balance nurse worloads. If a patent has to be vsted by a nurse from another sector, we try to choose the one from the nearest sector to reduce the total travelng tme. The mpact of ths flexblty nduces that the problem s not separable by sector. When solvng the problem, we assume that each nurse completes hs route to return to the clnc no later than 12h00 PM even though n practce some flexblty s allowed to fnsh later. The rest of the worng day s devoted to completng proper medcal reports durng the afternoon. In home care servces t s mportant to consder the human factor related to the contnuty of care. The patents prefer recevng ther treatments by the same nurse, and nurses also thn that t s preferable to become better acquanted wth ther patents. But a follow-up nurse mght not be avalable because of holdays and days off. Thus n our model, we account for the contnuty of care requrements as soft 2

3 constrants. Moreover, to replace nurses durng ther holdays and days off, the admnstrators rely on a recall lst ncludng substtute nurses. At the CLSC Les Forges, the rules specfyng the tme frames for returnng the blood samples to the clnc are formulated as follows. If a blood sample s performed before h00 AM, the nurse must turn bac the blood sample at the clnc no later than h00 AM (strctly speang). If a blood sample s performed between h00 AM and 11h00 AM, the nurse must then turn bac the blood sample no later than 11h00 AM. In our approach, we model these returns to the clnc as fcttous destnaton nodes to be vsted before h00 AM and 11h00 AM, respectvely. Hence the nurses must vst these fcttous nodes only f blood samples are completed. The objectve s to determne routes n a reasonable tme frame satsfyng all constrants and mnmzng the objectve functon. The objectve functon to be mnmzed n our model s specfed n terms of the followng components: the travelng tme of the nurses; the fxed cost (salary) for the nurses accordng to ther types (regular or from the recall lst); a penalty cost for each patent vsted by a nurse dfferent from hs follow-up nurse; a penalty cost when a patent s vsted by a nurse from a dfferent sector. Note that the thrd component s a way to account for the contnuty of care requrements as a soft constrant. The objectve functon also ncludes an addtonal artfcal component useful n formulatng the blood sample constrants. 4. Mathematcal model The home care nurse routng problem s formulated as a vehcle routng problem wth tme wndows (VRPTW) ncludng addtonal constrants. In ths model, we denote P= {1, 2,.., N}: the set of all patents. P : the set of patents requrng a blood samplng, P P. r l f r l f I I I I : the set of all nurses ncludng the sets I, I and I of regular nurses, of nurses from the recall lst, and of fcttous nurses, respectvely. Cr, Cl, and C f : the daly costs for the dfferent nurse categores where Cr Cl C f. Note that n addton to the regular nurses and to the nurses from the recall lst categores, we ntroduce a thrd category of fcttous nurses that are assgned to patents n last resort ndcatng that these patents are not vsted. Thus all patents are assgned to a route. To specfy the underlyng networ, we defne the set of nodes V as follows: a node s assocated wth each patent n P; two dfferent nodes 0 and D are assocated wth the clnc where 0 and D are used to denote the orgn and the endng node of each route, respectvely; two fcttous nodes p and p 11 are also assocated wth the clnc; these nodes have to be vsted whenever some nurse has to tae bac blood samples before h00 AM or 11h00 AM, respectvely. Wth each node V we assocate r : the tme requred to complete the treatment of patent P. r r r r 0. O D p p11 [e, f ]: the tme wndow of the arrval tme at node V. More explctly, the tme wndows for the dfferent nodes are specfed as follows: 3

4 e, f H8, H11 f P H8, H12 f V P p, p11 H8, H f p H, H11 f p 11 where H8 < H < H11 < H12 are constant values assocated wth 8h00 AM, h00 AM, 11h00 AM, and 12h00 AM, respectvely. A j V V e r t f, j A f the tme The set, ncludes the admssble arcs;.e. wndows of nodes and j allow suffcent tme to move from to j. The varables are denoted as follows: x 1 f nurse vsts successvely nodes and j j 0 otherwse b : the tme when nurse arrves at node V. b b p p 11 j j : the tme when nurse taes blood samples to the clnc before h00 AM (correspondng to the tme when nurse arrves at the fcttous node p ). : the tme when nurse taes blood samples to the clnc for turnng bac blood samples before 11h00 AM but after h00 AM (correspondng to the tme when nurse arrves at the fcttous node p 11 ). y : the addtonal modelng bnary varables, P, I Note that for all nurses I, the varables H8 b H11, H8 b H12 P, V P p, p b p H b p H b are ntalzed at the followng values: The value of b may be modfed durng the soluton procedure whenever the nurse vsts the node. To specfy the objectve functon, denote by t j, the travelng tme from node to node j. The cost assocated wth nurse movng from to j s specfed as follows: r l f tj f I I I and 0 r j r f and 0 t C I cj l tj C l f I and 0 f tj C f f I and 0 Accordng to ths notaton, the travelng tme s consdered as a cost. Also when the nurse s leavng the clnc, we add the daly cost correspondng to hs category. The cost structure s also modfed n order to account for the requrement that a patent should be vsted by a nurse of hs sector whenever possble. Denote by s() and ς(j) the sectors of the nurse and of the node j, respectvely. Note that 0, D, p and p 11 4

5 are nodes for every sector. For any par of nodes and j and for any nurse, the cost s specfed as follows: cj f ς(j) s() or f j D, p, p cj cj Ca f ( j) s( ) where Ca 11 s a postve parameters of the model. The standard constrants for the VRPTW constrants (see Solomon (1987)) are summarzed as follows: I jv jv x j x j 1, P x jv j 0 0, V \,D, I x0 j 1, I jp x jd x 0 j, I (4) jv jv b r t j b j M 1 x j, (,j) A, I e b f, V, I (6) Now addtonal constrants are requred for the blood samples and the contnuty of care requrements. 4.1 Blood sample related constrants The blood sample constrants are more complex to formulate. The constrants (7) specfy that whenever a nurse performs a blood sample on at least one patent before h00 AM, then he must vst the fcttous destnaton node p (.e. he has to return to the clnc). 1 H b M x 1 x jp j, jp jp p P, I Here M s a very large scalar. To understand ths constrant, frst note that, f any patent AM, then t follows that Hence H b 0 jp xj 1 and b H. p and 1 0. M x 1 x jp j M x jp jp jp p jp P (1) (2) (3) (5) (7) s vsted by nurse before h00 H b Also, snce 1 x j 0, t follows that jp p. 5

6 Thus constrant (7) reduces to nducng that x 0 jp jp H b M x jp 0 1 jp (.e., forcng a vst to the fcttous destnaton node p ). Note that f nurse does not vst any patent P before h00 AM, then b H for all P. Hence the constrants (7) are nactve for all n the sense that they are satsfed for any value of jp x jp. Thus jp P x 0 snce the costs assocated wth all arcs (j, p ) are postve. jp Smlar arguments can be used to verfy that the constrants (8) and (9) force a nurse performng any blood sample after h00 to vst the fcttous node p b b M x 1 x p jp11 j, (8) jp jp p11 P, I 1 b H M x 1 x jp11 j, jp jp p11 P, I (9) The constrants (8) apply when the nurse had to turn bac to the clnc before h00 and constrants (9) when he dd not. Now even f the constrants (7) guarantee that the nurse vsts the fcttous destnaton node p whenever he performs any blood sample before h00, nevertheless these constrants do not prevent the vst to p tang place later than h00 AM (.e., havng b H ), nor they prevent ths vst tang place before any blood sample has been performed. For these reasons, we nclude the followng constrants requrng also addtonal bnary modelng varables y, P, to be formulated. Note that these varables are useful to formulate the model but they have no partcular nterpretaton. Frst we ntroduce a set of constrants guaranteeng that y 1 for exactly one patent P vsted by nurse before h00 AM, f any. The constrants () y x j, P, I () jv D guarantee that only f jv D x j 1 y can tae value 1 only f nurse performs a blood sample on some patent The constrants (11) H - b M y j, P, I jp (11) P,.e. 6

7 guarantee that whenever a blood sample s performed on some patent before h00 AM (.e., H ), then y 0. b I jp j Moreover, n order to guarantee that y j 1 whenever y j 0, we ntroduce the term jp y j jp n the objectve functon to be mnmzed. Fnally, t follows that the constrants p b M (H M) y, I guarantee that bp jp j jp H, nducng that nurse returns to the clnc no later than h00 AM whenever y j 1. Furthermore, t s easy to see that the constrants (12) are nactve when nurse s not jp performng any blood sample before h00. Fnally to guarantee that nurse returns to the clnc no later than h00 AM but only after performng at least one blood sample on a patent, constrants (13) are added to the model. (13) b - b M 1 y, P, I p Indeed, snce y 1 for exactly one patent P vsted by nurse before h00 AM, t follows that the correspondng constrant (13) s actve. Hence b b p 0, and b b. It s easer to deal wth the case where the nurse has to vst p 11 (.e., returnng to the clnc before 11h00 AM) snce ths s the last return. On the one hand, to guarantee that p 11 s vsted no later than 11h00 AM, t s suffcent to fx properly the upper bound of the tme wndow assocated wth b p to the 11 value 11h00. On the other hand, the constrants (14) guarantee that p 11 s vsted after the last blood sample has been performed. b b, P, I (14) p Contnuty of care constrants In order to account for the contnuty of care requrement, a set L of patents s assgned to each regular r nurse I such that L Lh for each par of nurses and h. Also the set L ncludes only I r patents requrng a follow-up because of medcal decson based on the nd of patent treatment or because of hs general health state. Snce the contnuty of care requrement s consdered as a soft constrant, then t s modeled by ncludng a penalty cost each tme a nurse vsts a patent j requrng follow-up by nurse h. The costs n the objectve functon are then modfed accordngly: for all pars of r nodes, j P, and for all regular nurses I. P p (12) c j cj Ccc f j Lh wth h and I cj otherwse r where C cc s the penalty cost for bypassng the contnuty of care requrement. 7

8 Consequently, the objectve functon to be mnmzed s specfed as follows c x j j j I (, j) A I jp y ncludng two terms related to the nurses and to the modelng bnary varables 5. Soluton Approach to solve the problem y j, respectvely. We use a metaheurstc approach based on the scatter search method (Glover et al.(2003)) relyng on an adaptve memory (Rochat et al.(1995)) ncludng pools of solutons generated usng Tabu search. The approach s summarzed n Fgure 1. In the next sectons, we descrbe dfferent elements of the procedure. Intalzaton Generate a set of feasble solutons for the problem. Select a subset of solutons to generate a pool ncludng the best solutons. In the rest of the set, select a second pool ncludng the most dfferent ones from those n the frst pool n order to explore more extensvely the feasble doman. Step 1: Generatng an offsprng soluton Select solutons from the pools accordng to some crteron. Generate an offsprng soluton by recombnng (accordng to some operator) the selected solutons Improve the offsprng soluton usng a Tabu Search procedure. Step 2: Updatng the pools Consder the offsprng soluton: o f t s better than the worst soluton n the frst pool, then t replaces the later n the frst pool; o f not, and f t s more dfferent from those n the frst pool than any soluton n the second pool, then t replaces the later n the second pool; o otherwse, t s dscarded. Step 3: Stoppng crtera Repeat Step 1 for 200 teratons. 5.1 Intalzaton Fgure 1. Scatter Search Approach Each feasble soluton of the problem s generated wth the purpose of complyng wth the objectve that each nurse should vst patents of hs sector n prorty. Thus each sector problem s solved to determne the nurse routes, and then they are merged nto a feasble soluton for the problem. Two soluton methods based on a Tabu search approach are developed to solve the sector problem. The ultmate goals of the two methods are bascally dfferent. The frst method relyng on Lau et al. (2003) approach ams at reducng the number of nurses requred to complete the vsts n the sector. Hence, at each teraton an addtonal nurse (f needed) s ntroduced, and a Tabu search procedure s used to optmze the routng and schedulng of the current nurse. The set of vsts not completed belongs to some holdng lst. The procedure terates untl all vsts are assgned (or untl the holdng lst s not 8

9 empty), even f fcttous nurses must be ntroduced. (Note that the vsts assgned to a fcttous nurse are not completed n fact). The purpose of the second approach s to reduce the total cost of the assgnment. An ntal soluton s generated usng Solomon (1987) heurstc, and then a Tabu search method s used to mprove the soluton. A feasble soluton s ntated as the unon of the sector solutons generated wth ether Lau-Tabu method or Solomon-Tabu method. In a frst step, we try to reduce the number of nurses worng. In order to do so, we try to elmnate, f possble, nurses vstng fewer patents. These patents are then redstrbuted among the remanng nurses and nserted n ther routes. If any patent can not be nserted n one of the remanng nurse routes then the correspondng nurse route can not be elmnated. Durng ths elmnaton process, the fcttous nurses are the frst to be elmnated, followed by nurses belongng to the recall lst, and fnally, the regular nurses vstng only a few patents. Ths step may result n a worse soluton accordng to the value of the objectve functon. The process s repeated to generate an adaptve memory ncludng two pools of solutons: a pool BP of best solutons and a pool DP of dverse solutons. Startng wth a set E of solutons for the global problem, the pool BP s generated by selectng the BP best solutons chosen from E. To generate the pool DP, DP solutons are selected sequentally among the E\BP solutons. The next soluton s selected accordng to a dssmlarty measure wth respect to the solutons n BP and those already ncluded n DP. The dssmlarty used n our mplementaton s due to Rego and Leao (2001). The measure d j s equal to the number of arcs that are dfferent n solutons X and X j : dj X X j \ X X j. Therefore the DP solutons are the ones showng the greatest dversty compared to the solutons already n BP and DP. 5.2 Step 1: Generatng a new soluton from the pools In order to account for the addtonal constrants related to the contnuty of care and the blood samplng, we construct a new partal soluton by selectng sequentally for each nurse a worload assgned to hm n some soluton of the pools. More specfcally, consder each nurse ndvdually. We frst determne the pool where hs worload wll be chosen. The selecton of the pool s made randomly: the pool BP s selected wth a probablty p b, otherwse the pool DP s selected. Once the pool s specfed, we select the soluton from whch the nurse worload s chosen accordng to one of the followng operators: Roulette selecton operator of Goldberg (1989) whch conssts n choosng a nurse worload proportonally to ts ftness measured n terms of the number of patents vsted. Let f be the ftness of the nurse worload n soluton. The probablty of choosng the worload n soluton f among the nurse worloads n the set of the t solutons s equal to p t. f Tournament selecton operator of Mchalewcz (1996) whch conssts n choosng randomly a set of worloads for the nurse among all solutons and n eepng the one wth the best ftness. Random selecton operator whch conssts n choosng randomly any canddate worload for the nurse among all solutons. The partal soluton generated accordng to ths process needs to be repared, n general, to obtan a new feasble soluton. Indeed, some patents may be assgned to several nurses (and hence they wll have to be removed from some nurse worloads) and some patents may not be assgned to any nurse. We consder three dfferent ways to remove a duplcated patent from some nurse worloads: 1 9

10 Mode 1: consder the order n whch the nurse worloads are generated; leave the patent n the frst worload generated and elmnate hm from the others. Mode 2: consder the solutons of the pools ncludng the nurse worloads where the patent s assgned; leave the patent n the nurse worload belongng to the best of these solutons and elmnate hm from the others. Mode 3: leave the patent n the worload of the nurse commssoned for hs contnuty of care. Fnally, a new feasble soluton s obtaned by ncludng the remanng patents not yet assgned. They are ncluded sequentally as follows: Include the patent n the worload of the nurse commssoned for hs contnuty of care, f possble. Otherwse, nclude the patent n the worload of a nurse from hs sector. Otherwse, nclude the patent n the worload of a nurse from another sector. Otherwse, add a new nurse to vst the patent. The precedng repar process generates a new feasble soluton to be mproved accordng to the followng procedure:. Frst redefne the sectors as follows. Consder the nurses belongng to an orgnal sector. The new sector assocated wth ths set of nurses ncludes all the patents assgned to them n the new feasble soluton. Improve the current soluton of each new sector usng the Tabu search.. Startng wth the unon of the sector solutons, apply the approach defned n the Intalzaton step to generate a new mproved global soluton. 6. Numercal results A frst set of problems s generated randomly accordng to Solomon process (1987) adapted to our problem. Ths set s used to determne proper values for the parameters and to analyze the dfferent selecton strateges, the ways to remove a duplcated patent, and to nclude the remanng patents. The second set ncludes real world data problems provded by the CLSC Les Forges n Tros-Rvères. 6.1 Analyss usng the frst set of random problems We use Solomon process (1987) adapted to account for the contnuty of care and the blood samplng constrants to generate problems wth 50 and 0 patents. To generate the sectors, we used the K-means parttonng algorthm of MacQueen (1967). Problems wth 50 and 0 patents nclude 2 and 4 sectors, respectvely. Twenty four problems (twelve wth 50 and twelve wth 0 patents) are generated accordng to the same characterstcs used for the sector problem generaton. Prelmnary tests were performed to determne the parameter values. The parameters values used for the tests are C r = 400, C l = 2C r = 800, C f = 3C r = 1200, C cc = 0, C a = 60. To compare the dfferent selecton operators, the pool sze s fxed to 5, and we use the frst removng mode. The numercal results ndcate that the 3-tournament selecton operator generates better results usng smaller computatonal tme than the two other operators. Ths seems to ndcate that a more eltst selecton s benefcal. The dfferent removng modes are also compared usng a pool sze equal to 5 and the 3-tournament selecton operator. The numercal results ndcate that mode 1 largely domnates the other two modes wth respect to the qualty of the solutons generated and the computatonal tme used. Ths may be due to the

11 fact that mode 1 nduces some dversfcaton strategy to explore more extensvely the feasble domans. Indeed ths mode does not account for the qualty of the soluton ncludng the worload of the follow-up nurse. Fnally, usng the 3-tournament selecton operator and the frst removng mode, the results for three dfferent pool szes are compared: 5, and 20. On the one hand, the qualty of the solutons generated usng pools of sze 20 s slghtly better. On the other hand, the computatonal tme ncreases rapdly wth the pool sze. Therefore, we use a pool sze of 5 to complete the comparson of the 4 varants. On the one hand, we compare the soluton approach when usng the two dfferent pools of solutons generated wth Lau et al. (2003) heurstc (LMultApproach) and Solomon (1987) heurstc (STMultApproach), respectvely. But on the other hand, we would also le to see the advantage of usng a populaton based approach le Scatter Search by comparng these results wth one of the soluton generated durng the Intalzaton. Denote by LMonoApproach and STMonoApproach the solutons generated wth Lau heurstc and Solomon heurstc, respectvely. The numercal results to compare the four soluton method varants are summarzed n Table 1. Comparng the two mono soluton approaches, the results ndcate that the LMonoApproach generates solutons where the average cost s reduced by 11% at the expense of an ncrease of the computatonal tme by a factor of 38% wth respect to STMonoApproach. Furthermore the solutons generated wth the LMonoApproach requre an average number of nurses reduced by a factor of 12%, but as expected, ths nduces an ncrease of the number of patents wthout follow-up by ther assgned nurse by a factor of 21%. Fnally, note that for both approaches the average number of nurses requred reduces when the number of vsts wth blood samplng reduces. Ths s not surprsng snce the number of returns to the clnc should ncrease wth the number of vsts wth blood samplng. When we compare the two mult soluton approaches, we observe that the average cost of the solutons generated wth the LMultApproach s smaller by a factor of 5% wth respect to the STMultApproach, but the computatonal tme requred by the LMultApproach s larger. However note that the dfference between the two mult soluton approaches s not as large as between the two mono soluton approaches. In order to analyse more thoroughly the effcency of the two mult solutons approaches, we refer to the curves n Fgure 2 showng the evoluton of the average cost over tme. The average cost s evaluated every 20 seconds startng at 35 seconds after the procedures started. The curves n Fgure 3 llustrate clearly the domnance of the LMultApproach over the STMultApproach. LMonoApproach STMonoApproach LMultApproach STMultApproach Average cost Average number of nurses Average number of patents wthout follow-up Average computatonal tme Table 1. Comparson between the mono and mult soluton approaches 11

12 Objectve functon LMultAppro STMultAppro Fgure 2. Objectve functon over tme for the mult soluton approaches (Lau-Tabu and Solomon-Tabu versons) Overall the mult solutons approaches generate better solutons where the average cost s reduced by a factor of 16% wth respect to the mono soluton approaches. But ths mprovement n the qualty of the solutons s at the expense of a sgnfcant ncrease (by a factor of 117%) of the computatonal tme Numercal results wth real value data Tme (s) The followng tests are completed wth data provded by the Centre Local de Servces Communautares (CLSC) Les Forges à Tros-Rvères. The total number of patents vsted each day vares between 80 and 0. The terrtory covered by the clnc s dvded nto four sectors. 14 regular nurses and 12 nurses from the recall lst are avalable to complete the home care vsts. Each regular nurse s assgned to one sector but she can complete some vsts n other sectors, f necessary. Each regular nurse also has a lst of follow-up patents. We use data avalable for three dfferent days of September Table 2 summarzes the data for the three problems. The clnc provded us wth an estmated tme of servce for each patent accordng to hs treatment but t s worth notcng that the estmated tmes provded may be hgher n practce. For nstance, a patent may need to tal wth the nurse more than usual. Ths may have an mpact when comparng our results wth those of the CLSC. Usng each patent address, we establshed the travelng tme matrx. These travelng tmes are adjusted to account for the tme needed to fnd a parng space, to wal from the car to the house, to remove materal from the car, etc. Problem Number of patents Number of regular nurses Number of nurses from the recall lst % of blood sample vsts Day ,5 Day ,6 Day ,1 Table 2. Real value data characterstcs Analyzng the solutons generated manually, we notce that some of the constrants are not always satsfed: 12

13 Some blood samplng vsts are completed after 11h00. Some regular vsts end after noon. Ths can be explaned by the fact that n practce some flexblty wth respect to these constrants s allowed to the nurses. But recall that the blood samplng and the tme wndow constrants are consdered as hard constrants n our models. Problem Manual cost Number of blood samplng constrants unsatsfed Number of regular vsts endng after noon Number of nurses Day Day Day Mean value (over the 3 problems) Table 3. Unsatsfed constrants by manual solutons These test problems are solved 5 tmes wth each of the four approaches LMonoApproach, STMonoApproach, LMultApproach and STMultApproach. The numercal results are summarzed n Table 4 where the gap of a soluton method measures, n percentage, the devaton of ts average cost from the best average cost among all four soluton methods. The numercal results are consstent wth those obtaned for randomly generated problems. The average cost s smaller for the solutons generated wth the LMultApproach even though the computatonal tme ncreases by only a factor of 2% wth respect to the STMultApproach. Comparng the solutons generated manually wth those obtaned wth the LMultApproach, the results n Tables 3 and 4 ndcate that more nurses (12.8 % more on the average where ths percentage s evaluated by the dfference between the average number of nurses obtaned wth the LMultApproach soluton (13) and the average number of nurses obtaned wth the manual soluton (14.67) dvded by the average number of nurses obtaned wth the LMultApproach soluton (13)) are requred and that the average cost s hgher n the former. However some reserve s n order. Indeed the solutons obtaned manually are peced together for the data avalable that do not nclude all the nformaton gathered for our computerzed methods. Hence some values have been evaluated n order to mae comparsons. Moreover, the travelng tmes n our methods are only estmates, and n practce, last mnute modfcatons occur whch are dffcult to model. Another aspect to consder s that our objectve functon s modeled usng dfferent costs accordng to the prorty gven to the mnmzaton of the total tme, to the possblty for nurses to vst patents from other sectors, and to the possblty for patents to be vsted by other nurses than ther assgned one, etc. The weghts assocated wth these prortes may not reflect precsely what the head nurse has n mnd when he completes the plannng n practce. Hence usng other values for these weghts may nduce dfferent solutons not as good as those presented here. Despte all that, we feel that our results are very nterestng and mght be encouragng for head nurses to use a computerzed tool to complete ther plannng. Not only the qualty of the solutons seems better, but t taes much less tme to generate them. Moreover wth a computerzed program t becomes easer to solve the problem agan whenever last mnute changes arse. Indeed the numercal effcency of the approach would allow the head nurse to complete several runs accountng for last mnute modfcatons. Furthermore, the values of the dfferent parameters n the global model can be adjusted to the satsfacton of the head nurse n order to obtan proper results for her specfc context. The choce of these values allows accountng for the proper relatve emphass of the dfferent problem components. 13

14 Average cost Average number of nurses Average number of patents wthout follow-up Gap Average computatona l tme LMonoApproach 287,13 13,80 2,27 7,9 % 5,65 STMonoApproach 611,53 13,80 2,33 11,3 % 4,29 LMultApproach 9531,27 13,00 0,60 0,0 % 549,78 STMultApproach 9791,00 13,07 1,40 2,7 % 540,22 7. Concluson Table 4: Comparsons wth real value data In ths paper, we ntroduce soluton methods to deal wth the home care routng problem. Frst, soluton methods used for the sector problem are descrbed snce they are used to generate the ntal solutons for the global problem. Two man soluton approaches, a mono and a mult solutons approaches, are compared numercally on randomly generated data and real value data. The results are very encouragng because these procedures offer the possblty to obtan good qualty solutons n very reasonable tme. We agree that more tests nvolvng varous cost parameters could allow generatng dfferent results. Also modelng the problem over several consecutve days nstead of one day at a tme, would allow accountng more extensvely for the contnuty of care constrants. Fnally, the dstrctng problem was not addressed n ths paper but we thn that allowng a dynamc redrawng of the terrtory covered by the clnc could result n more balanced worloads for the nurses. 8. Acnowledgements We are grateful to Mrs Gnette Gélnas of CLSC Les Forges n Tros-Rvères for her constant collaboraton and for provdng us wth the data necessary to complete the tests. Support for ths project was provded by NSERC (Canada) and we are thanful for ths help. References [1] S.V., Begur, D.M. Mller and J.R. Weaver, An Integrated Spatal DSS for Schedulng and Routng Home Health Care Nurses, Interfaces 27(1997), pp [2] S. Bertels and T. Fahle, A Hybrd Setup for a Hybrd Scenaro: Combnng Heurstcs for the Home Health Care Problem, Computers & Operatons Research 33 (2006), pp [3] O. Bräysy and M. Gendreau, Vehcle Routng Problem wth Tme Wndows, Part I: Route Constructon and Local Search Algorthms, Transportaton Scence 39 (1) (2005), pp [4] O. Bräysy and M. Gendreau, Vehcle Routng Problem wth Tme Wndows, Part II: Route Constructon and Local Search Algorthms, Transportaton Scence 39 (1) (2005), pp [5] E. Cheng and J.L. Rch, A Home Health Care Routng and Schedulng Problem, CAAM, Rce Unversty, Techncal Report TR98-04, [6] B. Elbenan, Problème de planfcaton des tournées des ntervenants pour les vstes à domcle, Département d nformatque et recherche opératonnelle, Unversté de Montréal, Ph. D. Thess, [7] P. Eveborn, P. Flsberg and M. Rönnqvst, Laps Care an Operatonal System for Staff Plannng of Home Care European Journal of Operatonal Research 171 (2006), pp [8] F. Glover, M. Laguna and R. Mart, Scatter Search, n Advances n Evolutonary Computng Natural Computng Seres. Sprnger, pp , [9] D.E. Goldberg, Genetc Algorthm n Search, Optmzaton and Machne Learnng, Addson-Wesley,

15 [] H.C. Lau, M. Sm and K.M. Teo, Vehcle Routng Problem wth Tme Wndows and a Lmted Number of Vehcles, European Journal of Operatonal Research 148 (2003), pp [11] J. MacQueen, Some Methods for Classfcaton and Analyss of Multvarate Observatons, n Proceedngs of 5-th Bereley Symposum on Mathematcal Statstcs and Probablty, Unversty of Calforna, pp , [12] C. Malandra and M.S Dasn, Tme Dependent Vehcle Routng Problems: Formulatons, Propertes and Heurstc Algorthms, Transportaton Scence 26(3) (1992), pp [13] Z. Mchalewcz, Genetc Algorthm + Data Structures = Evoluton Programs, Sprnger-Verlag, [14] C. Rego and P. Leao. A Scatter Search Tutoral for Graph-Based Permutaton Problems, Unversty of Msssspp, Report HCES--00, [15] Y. Rochat and E. Tallard, Probablstc Dversfcaton and Intensfcaton n Local Search for Vehcle Routng, Journal of Heurstcs 1 (1995), pp [16] MM. Solomon, Algorthms for the Vehcle Routng and Schedulng Problems wth Tme Wndow Constrants, Operatons Research 35(2) (1987), pp [17] MM. Solomon and J. Desrosers. Tme Wndow Constraned Routng and Schedulng Problem, Transportaton Scence 22(1) (1988), pp