Master Physician Scheduling Problem 1

Transcription

1 Master Physcan Schedulng Problem 1 Aldy Gunawan and Hoong Chun Lau School of Informaton Systems, Sngapore Management Unversty, Sngapore Abstract We study a real-world problem arsng from the operatons of a hosptal servce provder, whch we term the master physcan schedulng problem. It s a plannng problem of assgnng physcans full range of day-to-day dutes (ncludng surgery, clncs, scopes, calls, admnstraton) to the defned tme slots/shfts over a tme horzon, ncorporatng a large number of constrants and complex physcan preferences. The goals are to satsfy as many physcans preferences and duty requrements as possble whle ensurng optmum usage of avalable resources. We propose mathematcal programmng models that represent dfferent varants of ths problem. The models were tested on a real case from the Surgery Department of a local government hosptal, as well as on randomly generated problem nstances. The computatonal results are reported together wth analyss on the optmal solutons obtaned. For large-scale nstances that could not be solved by the exact method, we propose a heurstc algorthm to generate good solutons. Keywords: schedulng, optmzaton, health servce, master physcan schedulng and rosterng problem, mathematcal programmng, preferences. Introducton There has been ncreased nterest n hosptal operatons management n terms of optmzed schedulng and allocaton of employees (e.g. physcans, nurses and admnstrators). One problem s to desgn a physcan schedule whch takes a large number of constrants and physcan preferences nto account. A physcan schedule s an assgnment of physcans to perform dfferent dutes n the hosptal tmetable. Unlke nurse rosterng whch has been extensvely studed n the lterature (e.g. Ernst et al, 2004a; Glass and Knght, 2009; Petrovc and Vanden Berghe, 2008), n physcan schedulng, maxmzng satsfacton matters prmarly, as physcan retenton s the most crtcal ssue faced by hosptal admnstratons (Carter and Laperre, 2001). In addton, whle nurse schedules must adhere to collectve unon agreements, physcan schedules are more flexble and drven by personal preferences. Carter and Laperre (2001) also provdes the fundamental dfferences between physcans and nurses 1 Pre-prnt for A. Gunawan and H. C. Lau. Master Physcan Schedulng Problem. Journal of the Operatonal Research Socety, 64 (3), , 2013.

2 schedulng problems. In general, schedulng physcans requres satsfyng a large number of conflctng constrants and preferences. To our knowledge, research on physcan schedulng has focused prmarly on a sngle type of duty, such as the emergency room (e.g. Beauleu et al., 2000; Carter and Laperre, 2001; Gendreau et al., 2007; Puente et al., 2009), the operatng room (e.g. Test et al., 2007; Burke and Rse, 2008; Roland et al., 2010; Vanberkel et al., 2011), the physotherapy and rehabltaton servces (Ogulata et al., 2008). In ths paper, we consder the problem of generatng a master schedule for the physcans wthn a hosptal servce by takng a full range of day-to-day dutes/actvtes of the physcans (ncludng surgery, clncs, scopes, calls, admnstraton) nto consderaton. Our problem, termed the Master Physcan Schedulng Problem, nvolves the assgnment of physcan actvtes to the tme slots over a tme horzon, ncorporatng rosterng and resource constrants together wth complex physcan preferences. The goals are to satsfy as many physcans preferences and duty requrements as possble. The major contrbutons/hghlghts of ths paper are as follows: (1) We take a physcan-centrc approach to solvng ths problem, snce physcan retenton s the most crtcal ssue faced by hosptal admnstratons worldwde. (2) Usng mathematcal models, we provde a comprehensve emprcal understandng of the tradeoff of constrants and preferences aganst resource capactes. The paper s organzed as follows. We frst provde a revew of the lterature, before presentng a detaled descrpton of the problem. We then address varants of the problem each wth a sngle objectve and provde mathematcal programmng models. We also extend and formulate the problem as a b-objectve mathematcal programmng model. For ths Weghted-Sum model, the varyng values of weghts are calculated by lnear nterpolaton between solutons n order to obtan a set of Pareto-optmal solutons. Our developed models are tested on a real case from the Surgery Department of a large local government hosptal, as well as on randomly generated problem nstances. Computatonal results are reported together wth our analyss. We also propose a heurstc algorthm to solve the sngle objectve problem that could not be solved optmally wthn reasonable tme by the exact method, and provde computatonal results. Fnally, we provde concludng perspectves and drectons for future research. Lterature Revew Personnel schedulng and rosterng s becomng a crtcal concern n servce organzatons such as emergency servces, hgher educaton systems, health care systems, hosptalty, and transportaton systems. Schedulng n servce organzatons s dfferent from that of manufacturng systems (Aggarwal, 1982). Some of the major dfferences are that the 2

3 output of servce systems cannot be placed nto nventory, the customer receves the servce drectly from the server and so on. The prmary objectve of the manufacturng system s to mnmze the total cost, whle the servce systems deal wth conflctng objectves, such as mnmzng total cost and maxmzng staff satsfacton as regards ther schedules. A number of revews n personnel schedulng and rosterng research have appeared n Aggarwal (1982), Burke et al. (2004) and Ernst et al. (2004b). A categorzaton of comprehensve and representatve soluton technques employed for dfferent rosterng problems are found n Ernst et al. (2004b). A number of approaches ncludng artfcal ntellgence approaches, constrant programmng, metaheurstcs, and mathematcal programmng approaches have been used for solvng the specfc problems. Beauleu et al. (2000) proposed a mxed 0-1 programmng formulaton of the physcan schedulng problem. They clamed that ther work was the frst to present a mathematcal programmng approach for schedulng physcans n the emergency room n a major hosptal of the Montréal regon. The basc rules appled at the hosptal are dstngushed nto two categores: compulsory (or hard) and flexble (or soft) rules. However, ths classfcaton depends on the preferences of the hosptal and on the physcan s flexblty. The constrants are parttoned nto four dfferent categores accordng to the types of rules to whch they correspond: compulsory constrants, ergonomc constrants, dstrbuton constrants, and goal constrants. The objectve functon s to mnmze all devatons of the goal constrants. The problem was then solved by a heurstc approach based on a partal branch-and-bound. The schedules produced were compared wth those generated by a human expert n terms of the computaton tme, the effort requred and the soluton qualty. Gendreau et al. (2007) presented several generc forms of the constrants encountered n sx dfferent hosptals n the Montréal area (Canada) as well as several possble soluton technques for solvng the problem. The constrants of the physcan schedulng problem can be classfed nto four categores: supply and demand constrants, workload constrants, farness constrants and ergonomc constrants. In ths paper, we are concerned about the physcans preferences nstead of farness constrants. Four soluton technques that can be appled to the physcan schedulng problem are categorzed nto four dfferent categores: mathematcal programmng, column generaton, tabu search and constrant programmng. A number of exact and heurstc algorthms for varous schedulng problems encountered n hosptals were also proposed by Belën (2007). The frst problem, namely the tranee schedulng problem, s solved by branch-and-prce algorthm (Belën and Demeulemeester, 2006). The second problem, the operatng room schedulng problem, s 3

4 modelled as a number of mxed nteger programmng based heurstcs and smulated annealng algorthm (Belën and Demeulemeester, 2007). These models consder stochastc number of patents for each operatng room block and a stochastc length of stay for each operated patent. The man objectve s to mnmze the expected total bed shortage. Buzon and Laperre (1999) appled tabu search to acyclc schedules. The cost of the soluton s the sum of the costs of all physcan schedules where each cost represents the sum of all penaltes assocated wth the unsatsfed constrants. Constrant programmng has been appled to the nurse schedulng problem (Bard and Purnomo, 2005). Ths soluton technque can also be appled to the physcan schedulng problem after some mnor modfcatons. Rousseau et al. (2002) and Bourdas et al. (2003) presented a hybrdzaton of a constrant programmng model and search technques wth local search as well as some deas from genetc algorthms to the physcan schedulng problem. The physcan and nurse schedulng problem are nherently mult-objectve optmzaton problem wth conflctng objectves. (Burke et al., 2009) model these as soft constrants. Some classcal methods for handlng mult-objectve optmzaton problem have been proposed n lterature. One of the most commonly used method s goal programmng snce t allows smultaneous soluton of multple objectves (Ogulata and Erol, 2003; Topaloglu, 2006; Whte et al., 2006). Burke et al. (2009) presented a Paretobased optmzaton technque based on a smulated annealng algorthm to address nurse schedulng problems n the real world. Problem Defnton The problem addressed n ths paper s to assgn dfferent physcan dutes (or actvtes) to the defned tme slots over a tme horzon ncorporatng a large number of constrants and complex physcan preferences. For smplcty, we assume the tme horzon to be one workng week (Mon-Fr), further parttoned nto 5 days and 2 shfts (AM and PM). The problem that we address s a real problem n the Surgery Department of a large government hosptal. Physcans have a fxed set of dutes to perform, and they may specfy ther respectve deal schedule n terms of the dutes they lke to perform on ther preferred days and shfts, as well as shfts-off or days off. Takng these preferences together wth resource capacty and rosterng constrants nto consderaton, our goal s to generate an actual schedule. As shown n Fgure 1 as example, the deal schedules mght not be fully satsfed n the actual schedule. That may occur n two scenaros: Some dutes have to be scheduled on dfferent shfts or days whch we term nondeal scheduled dutes (e.g. Physcan 2, Tuesday dutes). 4

5 Some dutes smply cannot be scheduled due to resource constrants whch we term unscheduled dutes (e.g. Physcan 1, Frday PM duty). Physcan Physcan 1 2 I 1 2 I Monday AM Duty 1 - Duty 3 Monday AM Duty 1 - Duty 5 PM Duty 5 Duty 4 Duty 1 PM Duty 5 Duty 4 Duty 1 Tuesday AM - Duty 1 Duty 5 Tuesday AM - Duty 5 Duty 3 PM Duty L Duty 5 Duty 2 PM Duty L Duty 1 - : : : : : : : : : : : : : : : : Frday AM Duty 4 - Duty L Frday AM Duty 4 - Duty L PM Duty 1 Duty L - PM - Duty L - Physcans Ideal Schedule Fgure 1. Example of Master Physcan Schedulng Problem Actual Schedule Although each hosptal has ts unque rosterng requrements, the followng summarzes some common requrements treated n ths paper: No physcan can perform more than one duty n any shft. The number of resources (e.g. operatng theatres, clncs) needed cannot exceed ther respectve capactes at any tme. For smplcty, we assume that each type of actvty does not share ts resources wth another type of actvtes for example, operatng theatres and clncs are used to perform surgery and out-patent dutes, respectvely. Ergonomc constrants: Some dutes are regarded as heavy dutes, such as surgery and endoscopy dutes. The followng ergonomc constrants hold: o If a physcan s assgned to a heavy duty n the mornng shft, then he cannot be assgned to another type of heavy duty n the afternoon shft on the same day. However, t s possble to assgn the same type of heavy dutes n consecutve shfts on the same day. o Smlarly, a physcan cannot also be assgned to another type of heavy duty n the mornng shft on a partcular day f he has been assgned to a heavy duty n the afternoon shft on the prevous day. Note that there are other ergonomc constrants (such as those presented by Gendreau et al. (2007) on constrants related to nght shft). In ths paper, these constrants do not apply as we consder only two dfferent shfts (AM and PM). They may be added wthout loss of generalty to our proposed models. 5

6 The number of actvtes allocated to each physcan cannot exceed hs contractual commtments, and do not conflct wth hs external commtments. In ths paper, we assume external commtments take the form of physcans request for shfts-off or days-off, and hence no duty should be assgned to these requests. We study dfferent settngs that may be nstantated from the problem. The basc problem (known as Model I) s to mnmze the total number of unscheduled dutes n an unconstraned settng (.e. wthout any physcan preferences or ergonomc constrants). From ths basc problem, we look nto two constraned problem settngs that respectvely handle physcan preferences and ergonomc constrants. The frst s the problem of satsfyng the physcans deal schedule as far as possble (or maxmzng the total number of deal scheduled dutes) whle not compromsng on havng the mnmum number of unscheduled dutes. The second s the settng where physcans do not provde ther deal schedule, but nstead ergonomc constrants are employed across all physcans n mnmzng the total number of unscheduled dutes. Both problem settngs are formulated as Models IIa and IIb, respectvely. Fnally, we also consder the problem that optmzes physcan deal schedules on one hand, and on the other, mproves the qualty of duty transton on non-deal scheduled slots through ergonomc constrants. Mathematcal Programmng Models In ths secton, followng a presentaton of the notatons used n ths paper, we wll provde the mathematcal programmng formulaton of the basc model (Model I), sngle objectve models (Models IIa and IIb) and fnally the b-objectve model (Model III). Basc notatons: I = Set of physcans, 12,,, I J = Set of days, j 12,,, J K = Set of shfts per day, k 12,,, K L = Set of dutes, l 12,,, L H L = {l L : l = heavy duty} PRA = {(, j, k)i J K : (, j, k) = physcan requests not beng assgned on day j shft k} Data parameters: R l = number of resources requred to perform duty l (ll) C = number of resources avalable for duty l on day j shft k (jj, kk, ll) jkl (.e. resource capacty) A l = number of duty l requested by physcan n a weekly schedule (I, ll) F = 1 f physcan requests duty l on day j shft k, 0 otherwse jkl Decson and auxlary varables: 6

7 X jkl = 1 f physcan s assgned to duty l on day j shft k, 0 otherwse U N S = number of unscheduled dutes of physcan = number of non-deal scheduled dutes of physcan = number of deal scheduled dutes of physcan The unconstraned problem (Model I) s one of mnmzng the number of unscheduled dutes subject to resource capacty constrants. It s formulated as follows: [Model I] Mnmze Z IU (1) Subject to: R X C j J,k K,l L (2) l I jkl jkl j J X k K jkl A l I,l L (3) X 1 l L jkl I, j J,k K (4) X 0, j,k PRA (5) l L jkl U A X I (6) ll l jj kk ll jkl X 0,1 I, j J,k K,l L (7) jkl U Z I (8) Constrant (1) s the total number of unscheduled dutes that needs to be mnmzed. Constrant (2) s the resource capacty constrant (total number of resources requred does not exceed total number of avalable resources per shft) 2. Therefore, Rl s set to zero for actvtes wthout lmted number of resources avalable. Through constrant (3), the number of dutes allocated to each physcan cannot exceed hs contractual commtments. Constrant (4) ensures that each physcan cannot be assgned more than one duty n any shft, whle constrant (5) ensures that no duty would be assgned to a physcan durng any shfts-off or days-off requested. Equaton (6) defnes the number of unscheduled dutes (whch s to be mnmzed). Constrant (7) mposes the 0-1 restrctons for the decson varables X jkl, whle constrant (8) s the nonnegatve ntegralty constrant for the decson varables U. Model I computes the mnmum number of unscheduled dutes wth only resource capacty constrants. It therefore provdes the lower bound on the number of unscheduled dutes as we constran the problem further n subsequent models below. Model IIa extends the base model by consderng the physcans deal schedule. Let * U denote the optmal soluton contanng the number of unscheduled dutes for physcan obtaned by Model I. In order to keep to the number of unscheduled dutes for each physcan to ths value, we mpose ths value as an upper bound (see constrant (10)). Model IIa seeks to then maxmze the total number of deal scheduled dutes. The total 2 In (Gunawan and Lau, 2009), we defned an addtonal notaton L c to represent a set of dutes wth lmted number of resources avalable. In ths journal verson, L s used to smplfy notatons. 7

8 numbers of unscheduled, deal and non-deal dutes for each physcan are calculated by equatons (6), (11) (12). The rest of the constrants are dentcal to those of Model I. [Model IIa] Maxmze Z X jkl F (9) I jj kk ll jkl Subject to: (2) - (8) * U U I (10) S X F I (11) j J k Kl A S N I U L jkl jkl l U,N,S Z I I (12) (13) We next consder another problem settng where constrants are mposed on duty transton, whch s formulated as Model IIb. Here, we assume that dutes are classfed nto two dfferent groups: heavy and lght dutes. A physcan assgned to a heavy duty n a partcular shft cannot be assgned to a dfferent heavy duty n the next shft on the same day, as represented by constrant (15); nor a dfferent heavy duty n the frst shft on the next day (constrant (16)). However, physcans can perform the same heavy dutes n consecutve shfts. Such ergonomc constrants apparently reduce the fatgue factor and mprove physcan productvty and hence qualty of servce. [Model IIb] Mnmze Z IU (14) Subject to: (2) (8) H X jkl Xjk 1l 1 I, j J,k 1,2,, K 1,l1 &l2 L l1 l2 (15) 1 2 H X I, j 1,2,, J 1,l &l L l l j K l l X j (16) Observaton: Model I s a basc model wthout ergonomc constrants, whle Model IIb s an extended model wth ergonomc constrants. Hence, the optmal number of unscheduled dutes obtaned by Model IIb s greater than or equal to that of Model I. In Model III, we combne the two problem settngs (.e. Model IIa and IIb) presented prevously. More precsely, we are concerned wth the b-objectve problem of maxmzng the number of deal scheduled dutes and mnmzng the number of unscheduled dutes under ergonomc constrants. Note that n Model IIa, we assume that non-deal scheduled dutes can be allocated at any tme perod wthout consderng ergonomc constrants. In ths combned problem, we mprove the qualty of the duty 8

9 transton of each physcan by mposng ergonomc constrants on non-deal scheduled dutes. In the followng, we propose an approach to solve the problem, whch s based on weghted-sum method that obtans Pareto-optmal solutons. Two addtonal sets of decson varables are defned as follows: Xˆ jkl = 1 f physcan s assgned to duty l on day j shft k wth respect to the deal schedule, 0 otherwse Yˆ jkl = 1 f physcan s assgned to duty l on day j shft k wth respect to ergonomc constrants, 0 otherwse A classcal mult-objectve weghted-sum method combnes the two objectves nto a sngle objectve by multplyng each objectve wth a user-defned weght. Ths weghtedsum model s then embedded wthn a proposed algorthm to obtan a set of Paretooptmal solutons (Fgure 2). [Model III] Mnmze Z W1 ( I S ) W2 ( I U ) (17) Subject to: R (X Yˆ ) C j J,k K,l L (18) l I jkl jkl jkl X ˆ Yˆ 1 I, j J,k K,l L (19) jkl jkl ˆ ˆ I,l L (20) jj kk (X jkl Yjkl ) Al l L l L ( Xˆ Yˆ 1 I, j J,k K (21) jkl jkl ) ( X Yˆ 0 ˆ j,k PRA jkl F jkl jkl jkl ), (22) X ˆ I, j J,k K,l L (23) U ll Al jj kk ll ( Xˆ jkl Yˆ jkl ) I (24) S Xˆ I (25) jj kk ll jkl N jj kk ll Yˆ jkl I (26) ˆ ˆ ˆ H Yjkl X 1 1 j k1 l Y 2 j k1 l I, j J,k1,2,, K 1,l 2 1&l2 L l1 l2 (27) ˆ ˆ ˆ H Yj K l X 1 1 j1 1l Y 2 j1 1l I, j 1,2,, J 1,l 2 1 &l2 L l1 l2 (28) ˆ ˆ ˆ H Y X Y 1 I, j J,l &l L l (29) j K l 1 j l2 j1l l2 ˆ ˆ ˆ H j1l X 1 1 j1 K l Y 2 j1 K l I, j 2,3,...,J,l 2 1 &l2 L l1 l2 (30) X ˆ, ˆ 0,1 I, j J,k K,l L (31) Y jkl Y jkl U, N,S Z I (32) Note that n Model III above, maxmzng the total number of deal scheduled dutes s equvalent to mnmzng the functon gven by (-1) total number of deal scheduled dutes. In ths model, the objectve functon and some of constrants are dentcal to those 9

10 of Model IIa. Constrant (21) ensures that a duty s scheduled as ether an deal or a nondeal duty. The detals of ergonomc constrants are represented by equatons (27) (30). Fnally, constrant (31) mposes the 0-1 restrctons for the decson varables Yˆ jkl Xˆ jkl and whle constrant (32) s the nonnegatve ntegralty constrant for the decson varablesu, N and S. The advantage of the weghted-sum method s that t guarantees fndng Paretooptmal solutons for convex optmzaton problems, whch can be nferred from Deb (2003) Theorem 3.1.1: Corollary: The soluton to Model III s not Pareto-optmal ff ether W 1 or W 2 s set to zero. We effcently generate a set of Pareto-optmal solutons for Model III by usng the followng algorthm. We frst generate k (constant) number of solutons wth dfferent values of W 1 unformly dstrbuted between [0, 1]. Snce not all Pareto-optmal solutons may be dscovered by the ntal set of weghts, we ntroduce an adaptve exploraton on the neghbourhood of weghts usng lnear nterpolaton,.e. we examne two dfferent Pareto-optmal solutons to derve weghts for obtanng other possble optmal solutons. The detals of the algorthm for usng Model III to obtan Pareto-optmal solutons are presented n Fgure 2. (1) Set W 1 = 1 (2) Repeat (3) Set W 2 = 1 - W 1 (4) Solve Model III (5) W 1 = W (6) Untl W 1 < 0 (7) For all solutons generated by the above, let S denote the subset of Pareto-optmal solutons (8) For a pre-set number of teratons do the followng (9) Let S 1 and S 2 ( S) wth the lowest and the second lowest total number of unscheduled dutes, respectvely (10) Set W 1 = W 1 of soluton S 1 and W 2 = W 2 of soluton S 1 (11) Set W 1 = W 1 of soluton S 2 and W 2 = W 2 of soluton S 2 (12) Calculate new weghts, denoted as W* 1 and W* 2, as follows: W* 1 = (W 1 + W 1 )/2 W* 2 = 1 - W* 1 (13) Solve Model III wth W 1 = W* 1 and W 2 = W* 2 (14) If the soluton obtaned s a new Pareto-optmal soluton then (15) Update S (16) Else f the soluton obtaned and S 1 are the same then (17) Set the soluton obtaned as S 1 and Update S (18) Else f the soluton obtaned and S 2 are the same then (19) Set the soluton obtaned as S 2 and Update S Fgure 2. Algorthm to obtan Pareto-optmal solutons Computatonal Results 10

11 In ths secton, we report a comprehensve sute of expermental results whch ams to provde computatonal perspectves on one hand, and nsghts to hosptal admnstrators on the other. All models were solved by the CPLEX 10.0 solver engne. Expermental Setup Frst we dscuss problem nstance generaton. 6 sets of random nstances (Random 1 to 6) were generated wth varyng values of the followng parameters : total percentage of heavy dutes assgned to physcans (last column of Table 1) and number of resources avalable n every shft (Table 2). Ths s to enable senstvty analyss, whch wll be descrbed n more detal after the results reported (Fgure 3). Other parameters, such as number of heavy dutes and number of dutes wth lmted resource capacty, are set to a constant. In addton to random nstances, we also provde a real case study provded by the Surgery Department of a large local government hosptal and several other quasrandom nstances (Random 7, 8 and 9) whch have smlar characterstcs to the real case, to help understand the nature of the real case study. Table 1 summarzes the characterstcs of each problem set. In order to generate suffcent hard problem nstances for the purpose of senstvty analyss, the resource capacty level has to be carefully set, and nterested reader can refer to Gunawan and Lau (2009). Table 1. Characterstcs of Problem Instances Problem Set Number of physcans shfts per day Number of days Number of dutes Number of heavy dutes dutes wth lmted capacty Total percentage of heavy dutes* Case study % Random % Random % Random % Random % Random % Random % Random % Random % Random % A / I J K H 100 % * I l L l As an llustraton, Table 2 presents three columns that gve the varyng values of C jkl for dfferent dutes defned n the nstances of the Random 1 problem sets. The total number of resources requred for Duty 1, Duty 2 and Duty 3 are 15, 28 and 22, respectvely. Here, Duty 2 s the duty wth the hghest number of resources requred and hence the value of C jkl for Duty 2 s set to and decreases by one unt untl t s equal to 1. For Duty 1 and Duty 3, we set the ntal values of C jkl to 1 11

12 and 1, respectvely and varyng the values accordng to the descrpton gven above. For nstance, for Duty 1, by varyng the values for C jkl, there are 7 dfferent problem nstances generated. Table 2. Examples of varyng values of C jkl (Random 1 nstances) Problem Set Random 1 Instances L Duty 1 Duty 2 Duty Random 1a Random 1b Random 1c Random 1d Random 1e Random 1f Random 1g In the followng, we report a sute of computatonal results and analyss obtaned from our mathematcal models descrbed above. Our mathematcal models were mplemented usng CPLEX 10.0 and executed on a Intel (R) Core (TM) 2 Duo CPU 2.33GHz wth 1.96GB RAM that runs Mcrosoft Wndows XP. Results from Model I and IIa Frst, results obtaned from Model I for Random 1 and Random 2 problems are shown n Table 3. It s nterestng to observe a two-pont phase transton n the mnmum number of unscheduled dutes (column 2) wth changng values of C jkl. It remans unchanged over a suffcently large range of values. As the value of C jkl of the duty wth the hghest requrement tends to Rl K I J A l, the number of unscheduled dutes starts to ncrease. For example, as we decrease the C jkl s value for Duty 2 from 6 to 4 for Random 1 nstances, the number of unscheduled dutes remans zero, but when ths value reaches 3, the number of unscheduled dutes ncreases to 4. Then, when the number of resources R l avalable s set to I Al 1 for each actvty l L, the number of unscheduled K J dutes ncreases drastcally from 5 to 10 unscheduled dutes. The same behavor was also observed for the other problem sets lsted n Table 1. From Table 3 agan, we observe that the number of unscheduled dutes s very low n comparson to the number of scheduled dutes. The percentage of unscheduled dutes s on average less than 3% for Random 1 and Random 2 nstances. Smlar observatons are made for other random problem sets. These results demonstrate the effectveness of the optmzaton model on random hard nstances. It s nterestng to see these results n the lght of the real-world case study problem nstance where the percentage of unscheduled 12

13 dutes obtaned s around 4.7%. Ths gves evdence to the hosptal management that ther resource capacty has reached a crtcal threshold as typfed by problem nstances 1g and 2, and consequently wll experence a drastc reducton of performance f resources cannot come up to par wth physcan dutes. Table 3. Summary of results for Models I and IIa Percentage of Percentage of scheduled dutes Problem scheduled dutes unscheduled unscheduled (%) Instances dutes Ideal Non-deal dutes (%) Ideal Non-deal Case study Random 1a Random 1b Random 1c Random 1d Random 1e Random 1f Random 1g Random 2a Random 2b Random 2c Random 2d Random 2e Random 2f Random 2g Random 2h Random Table 3 also presents results on the extent of the satsfablty of deal schedule n the actual schedule, obtaned from runnng Model IIa. These results are llustrated n Fgures 3(a) and 3(b). Note that the left axes n both fgures measure the percentage of deal, nondeal scheduled and unscheduled; and the resource avalablty/requrement gap %Gap respectvely. The resource avalablty/requrement gap s defned K L C jj kk ll jkl R I ll l Al as % Gap 100%. It represents a resource Rl Al I ll buffer,.e. the proporton of total resource avalablty that exceeds the sum of resource requrement requested. (a) Fgure 3. Parameter analyss of Random 1 Problem Set (b) 13

14 From Fgure 3(a), we can nfer that n order to ensure zero unscheduled dutes (whch s often a hard constrant, snce doctor dutes should not be unfulflled), the total number of avalable resources for each actvty l L must be above the threshold Rl A I l 1 (see nstances 1a, 1b and 1c). From Fgure 3(b), we can also nfer that K J when the %Gap s decreased below 23%, the percentage of unscheduled dutes wll be doubled (see nstances 1f vs 1g, also 2h vs 2). Smlar observatons have been made for the rest of the problem nstances. From the hosptal admnstraton standpont, the latter result shows the crtcal resource avalablty threshold below whch the degradaton of servce performance wll be keenly felt. Next, the result of the case study problem nstance s compared wth that of the actual allocaton generated manually by the hosptal, as summarzed n Table 4. Although the number of unscheduled dutes va manual allocaton s smaller than the results obtaned by Model IIa, the number of non-deal scheduled dutes s sgnfcantly hgher than that of the model soluton. Note also that the manual allocaton s strctly speakng not feasble, snce two physcans had to cancel ther days-off or shfts-off to perform ther dutes. Ths manual plan s therefore very undesrable snce physcans mght have external commtments that cannot be delayed or cancelled. Suppose that the two physcans are unwllng to fulfll these dutes on ther days-off or shfts-off, the number of unscheduled dutes would be equal to the soluton obtaned by our proposed approach. We conclude that our proposed approach performs better than the manual allocaton. Table 4. Comparson between the manual allocaton and model soluton on a real case Manual allocaton Optmal soluton unscheduled dutes 5 7 non-deal scheduled dutes 10 4 physcans assgned dutes durng days-off or shfts-off 2 0 Results from Model IIb Table 5 presents results obtaned by runnng Model IIb aganst our problem nstances. We observe that Random 6 (where the percentage of heavy dutes reaches 70%) and Random 9 problem sets could not be solved to optmalty wthn the computaton tme lmt of 6 hours. As such, we only report the best known solutons that could be obtaned for these problem nstances. Usng Model IIb, we perform senstvty analyss on the mpact of ergonomc constrants on resource utlzaton, whch s of great nterest to the hosptal admnstrator. Partcularly, for solutons wth non-zero unscheduled dutes (see nstances 1d-1g,, 5j- 5m, etc on Table 5), the queston s the resource level actually needed to brng the number 14

15 of unscheduled dutes down to 0. We conducted experments over the same set of random nstances as shown n Table 1 (see last column). Fgure 4(a) gves the aggregate results. The x-axs represents the percentages of heavy dutes, whle the y-axs measures the resource requrement. (a) The square-sold curve shows the mnmum total resource requred n order that the number of unscheduled dutes wll reman zero (due to ergonomc constrants), whle the damond-dotted curve shows the correspondng resource requrement gven by the nput (refer to left y-axs). The mpact of the ergonomc constrants s gven by the gap between these two curves. We observe that across the board, a resource buffer of about 20% of the total resource requrement s needed, n order to satsfy all ergonomc constrants. In other words, the cost of enforcng ergonomcs constrants s 20% of resource requrement. (b) Another observaton s the mpact on the number of unscheduled dutes under fxed resource constrants. The trangular-dashed curve shows the number of unscheduled dutes for problem nstance 1c (refer to rght y-axs). Note that as we ncrease the percentage of heavy dutes, the total number of unscheduled dutes ncreases gradually, untl a certan threshold of 40%, when t s observed to ncrease sharply. Ths phase transton phenomenon s also observed n other random nstances. It provdes nsghts to the hosptal admnstrator n terms of plannng the lmts of heavy dutes for the physcans. Table 5. Computatonal results of Model IIb Problem nstances unscheduled dutes scheduled dutes Problem nstances unscheduled dutes scheduled dutes Case study Random Random 1a Random 4j Random 1b Random 4k Random 1c Random 5a Random 1d Random 5b Random 1e Random 5c Random 1f Random 5d Random 1g Random 5e Random 2a Random 5f Random 2b Random 5g Random 2c Random 5h Random 2d Random Random 2e Random 5j Random 2f Random 5k Random 2g Random 5l Random 2h Random 5m Random Random 6a* Random 3a Random 6b* Random 3b Random 6c* Random 3c Random 6d* Random 3d Random 6e* Random 3e Random 6f* Random 3f Random 6g* Random 3g Random 6h* Random 3h Random 6* Random Random 6j* Random 4a Random 6k* Random 4b Random 6l* Random 4c Random 6m*

16 Problem nstances unscheduled dutes scheduled dutes Problem nstances unscheduled dutes scheduled dutes Random 4d Random 6n* Random 4e Random 6o* Random 4f Random Random 4g Random Random 4h Random 9* *CPU tme = 6 hours Fgure 4(b) gves a detaled breakdown of three selected dutes for the Random 1d nstance (where the number of unscheduled dutes s non-zero, due to resource constrants). Agan, we examne the gap between the total requrement (as gven by the nput) vs the number of resources actually requred n order to ensure zero unscheduled dutes. We see that Duty1 wth fewer resource requrements would requre less resource buffer (up to 30%). On the other hand, Duty2 and Duty3 requre more resource buffer (up to 40%) n order to ensure zero unscheduled duty. Smlar observatons can be obtaned for other random nstances. (a) Fgure 4. Senstvty analyss on percentage of heavy dutes (b) Results from Model III Recall that dfferent sets of weght vectors were generated usng the proposed algorthm presented n Fgure 2 n order to obtan a set of Pareto-optmal solutons. Table 6 represents the results obtaned by the proposed algorthm for representatve nstances 6k and 6l. The number of teratons s set to 5 teratons. We start by selectng 10 dfferent weght vectors that are unformly dstrbuted wthn [0, 1]. We notce that dfferent weght vectors need not necessarly lead to dfferent Pareto-optmal solutons, and some weght vectors lead to the same soluton. Table 6. Computatonal results of nstances 6k and 6l Random 6k Random 6l Weght Scheduled dutes Unscheduled Weght Scheduled dutes Unscheduled W 1 W 2 Ideal Non- dutes Non- dutes W Ideal 1 W 2 Ideal Ideal

17 By usng lnear nterpolaton, we can obtan other possble Pareto-optmal solutons. Here, we focus on explorng neghbourhoods of the solutons wth the lowest values of the total number of unscheduled dutes snce we vew unscheduled dutes as undesrable compared to non-deal scheduled dutes. In general, the value of W 1 should be wthn [0.1, 0.2] n order to obtan the lowest number of unscheduled dutes. The followng fgure represents the Pareto-optmal solutons obtaned for some Random 5 and Random 6 nstances. Ths can be extended and appled to other nstances of other problems. In general, we observe the general trade-off between the number of unscheduled dutes and the number of non-deal scheduled dutes. Fgure 5. Pareto optmal solutons of some nstances of Random 5 and Random 6 problem sets We also tested the proposed algorthm to the real case study. It s concluded that the value of W 1 should be wthn [0.9, 1.0] n order to obtan the lowest number of unscheduled dutes (Gunawan and Lau, 2010). The result of the real case study problem s also compared wth that of the actual allocaton generated manually by the hosptal, as summarzed n Table 7. We observe that the number of deal scheduled dutes obtaned by Weghted-Sum Model s sgnfcantly hgher than that of the manual allocaton. Although the number of unscheduled dutes obtaned by Weghted-Sum Model s slghtly worse than the number of unscheduled dutes va manual allocaton, we notce that the number of non-deal scheduled dutes s better than that of the manual allocaton. One possble reason s that n 17

18 manual allocaton, the admnstrator allocates non-deal scheduled dutes to any tme slots/shfts wthout consderng the ergonomc constrants. In our proposed model, we consder both the deal schedule and ergonomc constrants. Table 7. Comparson between the manual allocaton and model solutons on a real case Manual allocaton Weghted-Sum Model unscheduled dutes 5 8 non-deal scheduled dutes 10 2 deal scheduled dutes Local Search Algorthm In order to solve large-scale problem nstances that could not be solved optmally by CPLEX 10.0 (Model IIb), we propose a heurstc algorthm based on local search. The algorthm moves from a canddate soluton to a soluton n ts neghbourhood n the search space untl a local optmum s found. The entre algorthm comprses of two man phases: (1) constructon, and (2) mprovement. The greedy heurstc s used to ntalze a soluton n the frst phase, whle the local search algorthm wth dfferent types of neghbourhood structures s used to mprove the soluton n the second phase. Each phase s presented and descrbed n detal below. Constructon Phase Let J K L l 1, l 2,..., l be the set of dutes scheduled for physcan durng the entre week. The optmal soluton of Model I s used as the ntal soluton of Model IIb. However, ths optmal soluton mght be nfeasble for Model IIb snce some dutes volate the ergonomc constrants. Fgure 6 gves our proposed algorthm to generate the ntal feasble soluton. The man dea s to remove the heavy dutes that volate the ergonomc constrants. Fgure 7 shows an example of the algorthm trace. Here, we use the followng ndexes to dstngush the dutes: 0 for heavy dutes that do not volate the ergonomc constrants, whle 1 for heavy dutes that volate the ergonomc constrants. Dutes wth ndex 1 are then removed and have to be rescheduled. For example, Duty6 on Shft2 of Day5 s not consdered as a heavy duty. The ndex of Duty6 s stll zero snce t does not volate the ergonomc constrant when t s compared to Duty1 on Shft1 of Day5. On the other hand, Duty1 has to be set to one due to the constrant volaton wth the prevous duty, Duty2 on Shft2 of Day4 (as partally shown n Fgure 7). After conductng the above mentoned procedure, the number of unscheduled dutes can be large. A lst of physcans who have a certan number of unscheduled dutes, 18

19 denoted as the excess_lst, s generated. We then proceed to the next phase, the mprovement phase to further mprove the ntal feasble soluton generated n the constructon phase. (1) Set the ntal ndex of each duty l L scheduled on day j J shft k K for physcan I to zero (2) For each physcan I do (Checkng procedure) (3) Check whether the frst two consecutve dutes 1 2 l, duty l 1 to one m m1 (4) Check whether two consecutve dutes, l set the ndex of duty l m to one m m1 (5) Check whether two consecutve dutes, l l volate the ergonomc constrant. If yes, set the ndex of l where 2 m J K 1 l where 2 m J K 1 set the ndex of duty l m to one (6) Check whether the last two consecutve dutes J K 1 J K l, l J K ndex of duty l to one (7) Remove all dutes wth ndex one volate the ergonomc constrant. If yes, volate the ergonomc constrant. If yes, volate the ergonomc constrant. If yes, set the Fgure 6. Constructon Algorthm Day1 Day2 Day3 Day4 Day5 Physcan1 Shft1 Duty1* Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft Duty1* Duty1* Duty1* Duty2* Duty2* Duty4 Duty2* Duty1* Duty6 (1) (2) (3) (4) Index Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Physcan1 Duty1* Duty1* Duty1* - - Duty2* Duty4 - - Duty6 *heavy dutes Fgure 7. Checkng procedure Improvement Phase After the ntal soluton s bult n the constructon phase, Local Search Approach wth dfferent strateges s appled n the mprovement phase to schedule unscheduled dutes as many as possble. Here, we propose 2 dfferent strateges. Each strategy conssts of a combnaton of two neghbourhood structures (N1 wth ether N2 or N3) as represented n Fgure 8. 19

20 (1) Set outer_loop = 0 (2) Whle outer_loop < max_outer_loop (3) Set nner_loop = 0 (4) Whle nner_loop < max_nner_loop (5) Apply N1 (6) nner_loop := nner_loop + 1 (7) Set nner_loop = 0 (8) Whle nner_loop < max_nner_loop (9) Apply N2 or N3 (10) nner_loop := nner_loop + 1 (11) outer_loop := outer_loop + 1 Fgure 8. Improvement Algorthm Neghbourhood1 (N1) The dea of N1 s to re-allocate some scheduled dutes and schedule unscheduled dutes n order to mnmze the number of unscheduled dutes. The procedures and an example of N1 are descrbed n Fgures 9 and 10, respectvely. (1) Select physcan I from the excess_lst randomly (2) Fnd an empty tmeslot randomly, tme2 tme1 (3) Start from Day1 Shft1, check whether the duty allocated, l whch was ntally scheduled at tme1, can be scheduled at tme2 by consderng both ergonomc and capacty constrants tme1 (4) If l can be scheduled at tme2, fnd any unscheduled duty and check whether t can be scheduled at tme1 (5) If an unscheduled duty can be scheduled at tme1, update the current soluton. Otherwse, return to step (1) Fgure 9. Neghbourhood1 (N1) Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Physcan1 Duty3 Duty7 Duty2* - Duty3 Off duty Duty2* - Duty1* Duty6 Physcan2 Duty7 - - Duty4 - Duty3 - Duty1* Off duty Duty3 (1) (2) (3) (4) The number of unscheduled dutes Duty2*. Duty L (5) Physcan Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Physcan1 Duty3 Duty7 Duty2* - Duty3 Off duty Duty2* - Duty1* Duty6 Physcan2 Duty7 - Duty1* Duty4 - Duty3 - Duty2* Off duty Duty3 *heavy dutes Fgure 10. Example of Neghbourhood1 (N1) Neghbourhood2 (N2) The dea of N2 s to re-allocate two dfferent scheduled dutes. Our proposed neghbourhood structure s n essence a knd of ejecton chan move nvolvng one physcan and two dfferent scheduled dutes. Ths neghbourhood can be consdered as a varaton on the classcal 3-Opt move for solvng the Travelng Salesman Problem and 20

21 other network optmzaton problems. The procedures and an example of N2 are shown n Fgures 11 and 12, respectvely. (1) Select physcan I from the excess_lst randomly (2) Fnd an empty tmeslot randomly, tme3 tme1 tme2 (3) Fnd two dfferent scheduled dutes, l at tme1 and l at tme2 tme1 tme2 (4) Check whether l and l can be rescheduled at tme3 and tme1, respectvely (5) If both can be allocated, let tme2 be an empty tmeslot and update the soluton. Otherwse, return to step (1) Fgure 11. Neghbourhood2 (N2) Day1 Day2 Day3 Day4 Day5 (1) Physcan3 Shft1 Duty1* Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 (4) (4) Duty7 - Off duty Duty2* Duty6 Duty3 Duty4 Duty2 Duty2 (2) (3) (3) Shft1 (5) Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Physcan3 Duty1* Duty7 Duty6 Off duty Duty2* Duty2* Duty3 Duty4 - Duty2* *heavy dutes Fgure 12. Example of Neghbourhood2 (N2) Neghbourhood3 (N3) Ths neghbourhood s smlar to N2. Instead of re-allocatng two scheduled dutes, we focus on three scheduled dutes. The procedures are descrbed n Fgure 13. Fgure 14 shows an example of Neghbourhood3 (N3). (1) Select physcan I from the excess_lst randomly tme1 tme2 tme3 (2) Fnd three dfferent scheduled dutes, l at tme1, l at tme2 and l at tme3 tme1 tme2 tme3 (3) Check whether l, l and l can be rescheduled at tme2, tme3 and tme1, respectvely. (4) If all can be re-allocated, update the soluton. Otherwse, return to step (1) Fgure 13. Neghbourhood3 (N3) Day1 Day2 Day3 Day4 Day5 (1) Physcan3 Shft1 Duty1* Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 (3) Shft1 Shft2 (2) Duty7 - Off duty Duty2* Duty6 Duty3 (2) Duty4 Duty2* Duty2* (3) (4) Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 Shft1 Shft2 (3) Physcan3 Duty1* Duty7 - Off duty Duty2* Duty4 Duty3 Duty2* Duty6 Duty2* *heavy dutes Fgure 14. Example of Neghbourhood3 (N3) 21

22 Computatonal Results To evaluate the performance of the proposed algorthm, we compare the solutons obtaned wth the optmal/best known solutons. The entre algorthm was coded n C++ and tested on a Intel (R) Core (TM) 2 Duo CPU 2.33GHz wth 1.96GB RAM that runs Mcrosoft Wndows XP. Prelmnary expermentaton was performed to determne sutable values for the parameters of algorthm. These values were chosen to ensure a compromse between the computaton tme and the soluton qualty. Both max_nner_loop and max_outer_loop are set to I J K. The performance of our proposed algorthm s defned by calculatng the percentage of unscheduled dutes compared aganst the total number of avalable tmeslots ( I J K ) (denoted by Z algorthm ). For nstance, f the number of unscheduled dutes s 10 dutes, whle the total avalable tmeslots s =200 slots, the percentage of unscheduled dutes s only 5%. Ths percentage s then compared aganst the best known/optmal solutons (denoted by Z best ) usng equaton (33). Table 8 summarzes the dfference between the percentage of unscheduled dutes by the proposed algorthm and the best known/optmal solutons of a partcular problem set. Z algorthm Z I J K best 100% (33) We observe that the combned N1-N3 neghbourhood yelds lower devaton values compared wth the N1-N2 neghbourhood. We also observe that when we ncrease the percentage of heavy dutes, the devaton value would also be hgher. It would be more dffcult to schedule a large number of heavy dutes due to ergonomc constrants. For a stuaton when the number of physcans s less and the total percentage of heavy dutes s hgh (e.g. case study, Random 7 and 8), the devatons are hgher due to dffculty n schedulng the heavy dutes. Table 8. Comparng proposed algorthm wth best known/optmal solutons (n terms of the percentage of unscheduled dutes) Data sets Neghbourhood strateges N1-N2 (%) N1-N3 (%) Case Study Random Random Random Random Random Random Random Random Random

23 Concluson In ths paper, we study the Master Physcan Schedulng problem, motvated by our work wth a local hosptal. To our knowledge, ths s the frst attempt that looks holstcally at an entre range of physcan dutes quanttatvely that enables hosptal admnstrators to ncorporate physcan preferences n ther rosterng. Snce the partcular problem studed s representatve of the Surgery Department of a large local government hosptal, we beleve our model does not requre major customzatons for use n other hosptals wth smlar constrants and preference structures. We see many possbltes of extendng the work. Our approach n ths paper s purely optmzaton-based, ncludng the handlng of physcan preferences. It wll be nterestng to nvestgate how other preference-handlng methods (such as CP-nets) can be ncorporated to model complex physcan preferences. Smlarly, one mght also consder farness constrants commonly seen n hosptals (Gendreau et al., 2007). Algorthmcally, t would be nterestng to tackle large-scale problems (such as the Random 6 nstances) wth a large number of heavy dutes (around 70% of the total dutes) that cannot be solved effcently by exact optmzaton models wth meta-heurstc or evolutonary approaches. The problem can also be formulated and solved as a b-objectve optmzaton problem (prelmnary work appears n Gunawan and Lau, 2010). Acknowledgements We lke to thank the Department of Surgery, Tan Tock Seng Hosptal (Sngapore) for provdng valuable comments and test stuatons. References Aggarwal S (1982). A focused revew of schedulng n servces. Eur J Opl Res 9: Bard JF and Purnomo HW (2005). Preference schedulng for nurses usng column generaton. Eur J Opl Res 164: Beauleu H, Ferland JA, Gendron B and Phlppe M (2000). A mathematcal programmng approach for schedulng physcans n emergency room. Health Care Mgt Sc 3: Belën J (2007). Exact and heurstc methodologes for schedulng n hosptals: problems, formulatons and algorthms. 4OR 5: Belën J and Demeulemeester E (2006). Schedulng tranees at a hosptal department usng a branch-and-prce approach. Eur J Opl Res 175: Belën J and Demeulemeester E (2007). Buldng cyclc master surgery schedules wth leveled resultng bed occupancy. Eur J Opl Res 176: Bourdas S, Galner P and Pesant G (2003). HIBISCUS: A constrant programmng applcaton to staff schedulng n health care. In: Prncples and Practce of Constrant Programmng. Lecture Notes n Computer Scence. Vol. 2833, pp

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