Building Master Surgery Schedules with Leveled Bed Occupancy and Nurse Workloads

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1 Buiding Master Surgery Shedues with Leveed Bed Ouany and Nurse Workoads Z. Yahia a, N. Harraz 2b mr B. Etawi 3 Deartment of Industria Engineering and Systems Management, Egyt - Jaan University of Siene and Tehnoogy (E-JUST), exandria, Egyt 2 rodution Engineering Deartment, Fauty of Engineering, exandria University, exandria, Egyt 3 Deartment of Industria Engineering and Systems Management, Egyt-Jaan University of Siene and Tehnoogy (EJUST). On eave from the rodution Engineering Deartment, Fauty of Engineering, exandria University, Egyt a zakaria.abdegawad@ejust.edu.eg, b nharraz@gmai.om, etawi@ejust.edu.eg bstrat - The master surgery sheduing robem invoves the deveoment of a master surgia shedue, a yi timetabe that determines the atient ategory assoiated with eah bok of oerating room time. myriad of variants of the robem has been addressed in iterature. Here we fous on two major variants, arising during ooeration with armoze hosita, a non-rofit hosita oated in exandria, Egyt. The first variant asks for baaning both beds and nurses daiy requirements, whereas the seond for onsidering surgeons referenes. To oe with these robems we introdue a new mixed integer formuation. The objetive funtion minimizes the weighted sum of eaks in the daiy bed ouany and nurse daiy workoads. The resuts show that our mode rovides a more eveed daiy bed ouany and nurse requirements. Furthermore, our mode redues both the daiy bed ouany and nurse workoads. eywords - Heath-are otimization, Master surgery sheduing, Mixed-integer rogramming, Oerating room anning, Sheduing I. INTODUCTION ording to Heathare Finania Management ssoiation (HFM), Oerating ooms (Os) resent a arge share of hosita are servies and exenditures. Os resut in an estimated 40% of hosita revenue []. t the same time, this faiity is one of the hosita s argest ost and revenue enters with a ost-to-harge ratio of 0.44 [2] and it has a major imat on the erformane of the hosita as a whoe. This makes the sheduing of Os a ritia robem to study in order to meet the hosita goas. In this onern, it is neessary to shedue the Os, in suh a way that the oerations/surgeries are arried out with maxima effiieny. The inrease in effiieny of Os shedue has a bearing on the overa resoures effiieny, redution and eveing of staff workoads and utiization of resoures. The snag of Os sheduing an be divided into three different and reated robems, namey (i) the Case-Mix robem (CM), (ii) the Master Surgery Sheduing robem (MSS), and (iii) the Surgery Sheduing robem (SS) [3]. The CM refers to the strategi deision of distributing the Os times between the different atients ategories in order to otimize erformane, in other words, how the avaiabe Os time is distributed among the different seiaties. In the MSS robem, the Os time is aoated to these surgia seiaties over the sheduing window (tyiay, one week) in order to maximize and eve resoures utiization. This deision takes ae on the tatia anning eve of hosita management with the target of reating a yi timetabe, Master Surgery Shedue (MSS). Finay, at the oerationa eve, the SS refers to assigning eah surgia ase a start time, a day, and an Oerating oom (O) with the target of minimizing the waiting time and maximizing resoures utiization. In this aer, we are onsidering the tatia MSS whih is onerned with deveoing a MSS, whih usuay is yiay onstruted for a given anning eriod (usuay 3 months to one year). n MSS defines the aoated Time Boks (TBs) on eah O for eah atient ategory overa the working day. The target of this aer is to buid a baaned MSS with eveed resuting bed ouany and nurse workoads whie onsidering surgeon referenes (e.g. a boks at maxima two different days or ertain days). We assume that the surgeons refer to work ertain number of days er week due to other engagements (teahing ativity, rivate work, et). The objetive funtion minimizes the weighted sum of eaks in the daiy bed ouany and nurse workoads. The aer is organized as foows: Setion 2 resents an overview of the reated iterature. The roosed mode is rovided in Setion 3. Finay, resuts and disussions are resented in Setion 4, and we draw onusions in Setion 5. II. ELTED LITETUE The MSS iterature is rih with many soving aroahes. There are three main aroahes aied to sove this robem; Mathematia modes, simuation modes and ombined aroahes. The reader is referred to [4], [5] and [6] for more detais on the MSS iterature. Many authors aied mathematia modes in order to get a osed form soution. Originay, Bake et a. [7] roosed an integer inear rogramming mode to take the MSS with the target of minimizing the shortfa /4/$ IEEE

2 roeedings of the 204 IEEE IEEM between eah atient ategory s target and atua assignment of O time. Testi et a. [8] firsty soved the CM in order to seet the otima ase-mix. Then they aied bok-sheduing strategy for determining otima time tabes. eenty, Mannino et a. [9] taked the MSS and foused on baaning atient queue engths and minimizing resort to overtime. To oe with these robems they introdued a new Mixed Integer Linear rogramming (MIL) formuation and showed its benefiia roerties. Furthermore, they deveo a ight robustness otimization aroah in order to onsider surgery demand unertainty. Most reenty, Ma and Demeuemeester [0] first hoose the otima atient mix then they onstruted a baaned MSS in order to imrove the atient servie eve by minimizing the tota exeted bed shortage. The objetive funtion of minimizing and eveing bed ouany has reeived a ot of attention in the MSS iterature. Beiën and Demeuemeester [] deveoed a number of mixed integer rogramming based heuristis and a meta-heuristi to buid yi MSS with eveed resuting bed ouany with the target of minimizing the exeted tota bed shortage. They reommended that nurse staff aaity onstraints and surgeons referene onstraints shoud be onsidered in future studies, simiary [0]. Oostrum et a. [2] modeed the MSS as a mathematia rogram and roosed a oumn generation aroah that maximizes the oeration room utiization and eves the requirements for subsequent hosita beds. However many authors suggested to onsider Surgeons referenes [], ony Testi et a. [8] who have onsidered this issue in their MSS mode. They assumed that the surgeon utiity funtion deends on the exusion of artiuar days due to other engagements (teahing ativity, et). In our study, the definition of surgeon referenes is sighty deferent. We assumed that surgeons refer to work ertain number of days er week due to other engagements (teahing ativity, rivate work, et). On the basis of the exored iterature, there is a need to onsider minimizing and eveing the daiy nurses workoads and to math the surgeon referenes whie buiding the MSS. III. THE OOSED MTHEMTICL MODEL In this researh, a new MIL mode is resented. The roosed mode addresses the MSS with the objetive of eveing the daiy beds and nurses workoads whie onsidering the surgeons referenes. In this study, we assume that surgeons refer working for ertain number of days er week. The soutions TH of the first hase (CM) are used as a onstraint in the urrent mode (MSS). The MSS aims to define the assignment between atient ategory, day of week, surgery room and time bok. The objetive funtion of the MSS minimizes both the maxima of the required daiy beds and the maxima of the daiy nurses workoads. This hoefuy resuts in a fat distribution of the bed ouany and nurses requirements over a days of the week. In other words, the aim is to eve the daiy bed and nurse resoures onsumtion as muh as ossibe. In order to buid a baaned master surgery shedue with eveed resuting bed ouany and nurse requirements, a mathematia MIL mode is roosed as beow:. Indies a r k The foowing are the indies for the mode: atient ategory index, ( =,..., ). Working day index, (a =,, ). Oerating room index, (r =,, ). Time bok index, (k =,, ). Week day index, ( =,, 2L). B. Data arameters The data arameters for the mode are dened as foow: The weight for minimizing the maxima daiy beds needed ( > > 0). - The weight for minimizing the maxima daiy nurses workoads. T The surgery duration time for atients beong to atient ategory (hours). BL k The ength of time bok k (hours). NB The maximum number of avaiabe arae boks in a day (boks). TH The number of surgery ases er week for atient ategory (surgery ases). NT The number of surgery teams that may work in arae for atient ategory. LOS The ength of stay for atients beong to atient SNN ategory (days). The standard number of nurses required to serve one atient in Os (nurse/atient). ND The maximum number of working days that surgeon grou of atient ategory refer to work in a week (days/week). M arge enough number. C. Deision Variabes The foowing deision variabes are onsidered in the mode: ) Main deision variabes: n integer variabe refers to the number of surgery ases of tye whih are assigned to a working day a on surgery room r and time bok k. b binary variabe with if a time bok k on surgery room r at working day a is assigned for atient ategory, and 0 otherwise. w a binary variabe whih equas if atient ategory is assigned to work at working day a, and 0 otherwise.

3 roeedings of the 204 IEEE IEEM 2) Summary variabes (derived from main deision variabes): MDB The maxima daiy beds needed. MDN The maxima daiy nurses workoads. NBD a variabe that auates the required number of beds on day that aused by surgeries of working day a for atient ategory. DB The number of required beds at day. TDB The tota number of daiy required beds in day aong one week or for ony L (to onsider atients who stay after the week end unti the next week days). N ak The nurses workoads aong time bok k on working day a for a surgery rooms. D. Mode Formuation Our robem an now be formuated as the foowing MIL mode: The objetive funtion is: Min α * MDB + ( α) * MDN () Subjet to: T * b * BLk, (2) a= r= k= = b = r= r= b r= k= a= w NBD DB a a b NT = TH NB M * w ND = r= k= = = a=, a NBD a, a, a a + LOS (3) (4) (5) (6) (7) (8) (9) (0) TDB = DB + DB + L () L TDB MDB L (2) N ak N ak = SNN * = r= (3) MDN (4) b w a {,}, r 0, (5) {,} a 0, (6) Ζ +, (7) The objetive funtion () aims at minimizing the weighted eaks in the daiy bed ouany and in the daiy nurse workoads. It onsists of two terms; the first term aims at minimizing the maxima daiy number of beds, whie the seond one aims at minimizing the maxima daiy nurses workoads. Fators and (- ) are weights that are used to refet the riority of eah term. Constraint (2) guarantees that, for eah assigned bok to atient ategory, the bok ength is suffiient for the surgery ases assigned to this bok. Constraint (3) is a aaity onstraint guarantees that the tota number of weeky surgery ases (TH ) seeted in the ase-mix anning hase is onsidered. The eft term auates the tota number of surgery ases assigned for eah atient ategory. TH denotes the atient throughut of atient ategory, whih is reeived from the ase-mix anning eve. The reader is referred to Yahia et a. [3] for more detais on how the vaues TH are auated. Constraint (4) is a we-known assignment onstraint foring eah time bok k on eah surgery room r to be assigned ony to one atient ategory during the a th working day. Constraint (5) ensures that the number of assigned boks does not exeed the maximum avaiabe number of boks in eah eriod. Constraint (6) is an assignment onstraint foring eah atient ategory to be assigned to a number of arae boks that must be inferior or equa to the maximum avaiabe number of surgery teams beong to atient ategory. Constraints (7-8) are reated to the surgeons referenes. In these onstraints the maximum working days for eah surgeon grou er week (ND ) are guaranteed. Constraint (7) defines the working days assigned for eah atient ategory by the binary variabe w a. In this onstraint, M reresents a arge enough number i.e. TH, we used this formuation to guarantee the mode inearity. Constraint (8) guarantees that the number of working days for hysiians reated to atient ategory does not exeed the referred maximum weeky working days ND. Constraints (9-2) are reated to the daiy and the maxima daiy beds auation. Constraint (9) auates the required beds in eah week day as a funtion of the MSS. It auates the number of beds required in day resuting from surgeries of atient ategory on day a. Constraint (0) sums the beds requirements for a atient ategories at eah day. Constraint () auates the required daiy beds for atients who stay after the week end days. Obviousy, when the LOS exeeds the ye time L, the orresonding number of beds has to be added to the beds required on the starting days of the next week. For exame, a surgery is arried out at Wednesday with LOS=5 (days), the staying eriod for this ase wi be

4 roeedings of the 204 IEEE IEEM extended to the next week days, i.e. Saturday and Sunday, whih exains the formua of onstraint (). In overa, it defines the tota number of ouied beds on eah week day over the ye time L, i.e. one week. Constraint (2) rovides the ink with the objetive funtion by imosing for eah week day that the bed ouany on eah day annot exeed the eak bed ouany (the first term). It imies that MDB exeeds eah TDB whih ensures that the objetive funtion minimizes the maxima beds ouany MDB. Constraints (3-4) are reated to the daiy and the maxima daiy nurse auation. Constraint (3) auates the nurses workoad on working day a over bok k. Constraint (4) rovides the ink with the objetive funtion by imosing for eah working day a, the nurses workoads on that day annot exeed the eak nurses workoads, whih is used in the seond term of the objetive funtion. It imies that MDN exeeds eah N ak whih ensures that the objetive funtion minimizes the maxima nurse ouany MDN. Finay, onstraints (5-6) and (7) refet the binary and integer roerties of the deision variabes resetivey. IV. NUMEICL ESULTS The aroah for generating the MSS was tested with data from the armoze hosita, a non-rofit hosita oated in exandria, Egyt. roximatey 4,000 atients annuay undergo surgery in the O deartments of armoze hosita. The data used in this aer is oeted based on database reords and information derived through interviews with hosita s surgia suites managers and staff. There are nine atient ategories (=9); Gyneoogy, Uroogy, Maxiofaia, Genera surgery- (GS-), Genera surgery-b (GS-B), Genera surgery-c (GS-C), Orthoedis, Otoaryngoogy, and Vasuar. The surgery deartment in armoze hosita onsists of three surgia suites, eah of them onsists of three Os, with a tota of nine Os. The hosita inudes many wards with 302 beds, but ony around a haf of this aaity is dediated to the surgery deartment wards. The surgery rooms are avaiabe Saturday through Thursday for 5 hours daiy (from 9:00 M to 2:00 M). The historia data for years were used for auation of LOS. The Os database was used to quantify the amount of time sent in the Os for the erformed surgia roedures. In this study, the surgery duration for eah atient is reorded and the average for eah atient ategory is auated. so it is assumed that the surgery time T for a ases within the same atient ategory is the same. The number of surgery teams avaiabe for eah atient ategory NT varies from 2 to 6. The MSS mode is soved near otimay with the ommeria MIL sover LINGO (LINDO Systems In.). runs were erformed on a Core i5 roessor, with 4 GB memory and 2.5 GHz. The mode runs are terminated after around 6 hours with an otimaity ga ess than 0.5 %. The mode is used to sove the origina ase with 9 Os and one bok er day with BL k =5 hours. reiminary anaysis is onduted to investigate the mode erformane at different vaues for the weighting fator (). We tested the mode at three different vaues i.e. ( 0.5), ( = 0.5) and ( 0.5). From the reiminary anaysis resuts, we found that the mode erformane is better with = 0.5. Thus, a runs are erformed at this vaue of. Firsty; we verified the mode by soving a sma size instane robem and heking the obtained soutions. Then; to investigate the mode erformane, we soved the roosed mode at two vaue grous of TH. The first grou refers to the urrent ase-mix an on our ase study hosita, whie the seond one refers to our roosed ase-mix an in revious study [3]. The MSS for the first grou is denoted as (roosed ) whie the MSS for the seond one is denoted as (roosed 2). Comarisons are onduted between the urrent Os MSS denoted as Current and the two roosed MSS ans. The resuts of the roosed mode and omarisons with the atua urrent erformane are summarized in Figures and 2. Figure omares between the three MSS in terms of daiy beds ouany, whie Figure 2 omares between them in terms of daiy nurses requirements. Daiy Beds (Beds) Sat. Sun. Mon. Tue. Wed. Thu. Fri. Current roosed roosed Fig.. omarison between the daiy required beds in the urrent and the roosed MSSs. Daiy Nurses (Nurses) Sat. Sun. Mon. Tue. Wed. Thu. Current roosed roosed Fig. 2. omarison between the daiy required nurses in the urrent and the roosed MSSs. Figure shows the omarison between the three MSS in terms of daiy beds ouany. It an be seen eary the

5 roeedings of the 204 IEEE IEEM unbaaned beds ouany in the urrent MSS. s we an see, beds ouany goes down to around 50 near the week end days and it imbs shary to reah the eak, with 29, at the midde of the week. On the other hand, we an find the eveed and smoothed beds ouany eve in either the roosed MSS an for the urrent asemix (roosed ) or the roosed MSS an for the roosed ase-mix in our revious study (roosed 2). The differene between the ouany eve in the roosed and roosed 2 an be justified by the differene in the overa number of surgery ases for eah an. In regards to roosed, the weeky number of surgery ases is 257, whie it is 287 surgery ases in roosed 2. This inrease in the number of surgery ases an be justifying the shift in the beds ouany eve. Figure 2 shows the omarison between the three MSS in terms of daiy nurses requirements. It an be eary seen the futuation in the daiy nurses requirements in the urrent MSS. s we an see, nurse s requirements go u and down with a eak of 25 at the midde of the week. On the other hand, we an find the reativey eveed and smoothed nurses requirements eve in either the roosed MSS an for the urrent asemix (roosed ) or the roosed MSS an for the roosed ase-mix in our revious study (roosed 2). Simiary, the differene between the roosed and roosed 2 an be justified by the differene in the overa number of surgery ases between the two ans. V. CONCLUSION The master surgery sheduing robem invoves the deveoment of a master surgia shedue, a yi timetabe that determines the atient ategory assoiated with eah bok of oerating room time. myriad of variants of the robem has been addressed in iterature. In this aer we foused on two major variants. The first variant asked for baaning both beds and nurses daiy requirements, whereas the seond for onsidering surgeons referenes. To oe with these robems we introdued a new mixed integer inear formuation. The objetive funtion minimizes the weighted sum of eaks in the daiy bed ouany and nurse workoads. In order to find the best vaues for the weighting fators, we onduted a reiminary anaysis by soving the mode at different weighting vaues. The resuts showed that our mode rovides a more eveed and smoothed daiy bed ouany and nurse workoads, and in the same time it redues both the daiy bed ouany and nurse workoads. Future work wi be done on enhaning the soution quaity and omutationa time by deveoing a heuristi or meta-heuristi agorithm to sove this mode. CNOWLEDGMENT Internationa Cooeration geny (JIC) in the soe of the Egyt Jaan University of Siene and Tehnoogy (E-JUST). The authors wish to thank a members of the Oerating ooms staff of the armoze Hosita, exandria, Egyt for their suort to this work and for roviding the ase study data. EFEENCES [] hieving oerating room effiieny through roess integration (2005), Tehnia reort, Heath Care Finania Management ssoiation eort. [2]. Maario, T. S. Vitez, B. Dunn, T. MDonad, Where are the osts in erioerative are?: naysis of hosita osts and harges for inatient surgia are, nesthesioogy, vo. 83, no. 6, , 995. [3] Z. Y. bderaso, N. Harraz,. Etawi, roosed soution framework for the oerating room sheduing robems, roeedings of the Word Congress on Engineering and Comuter Siene, vo. 2, San Franiso, US, Otober 203. [4] Z. Y. bderaso, N. Harraz,. Etawi, Oerating room sheduing robems: a survey and a roosed soution framework, Transations on Engineering Tehnoogies, H.. im et a. (eds.), Sringer, [5] F. Guerriero,. Guido, Oerationa researh in the management of the oerating theatre: a survey, Heath Care Management Siene, vo. 4,. 89 4, 20. [6] B. Cardoen, E. Demeuemeester, J. Beiën, Oerating room anning and sheduing: iterature review, Euroean Journa of Oerationa esearh, vo. 20, , 200. [7] J. Bake, F. Dexter, J. Donad, Oerating room managers use of integer rogramming for assigning bok time to surgia grous: a ase study, nesthesia and nagesia, vo. 94, , [8]. Testi, E. Tànfani, G. Torre, three-hase aroah for oerating theatre shedues, Heath Care Management Siene, vo. 0, , [9] C. Mannino, E. Nissen, T. Nordander, attern based, robust aroah to yi master surgery sheduing, Journa of Sheduing, vo. 5, , 202. [0] G. Ma, E. Demeuemeester, mutieve integrative aroah to hosita ase mix and aaity anning, Comuters & Oerations esearh, vo. 40, , 203. [] J. Beiën, E. Demeuemeester, Buiding yi master surgery shedues with eveed resuting bed ouany, Euroean Journa for Oerationa esearh, vo. 76, , [2] J. V. Oostrum, M. V. Houdenhoven, J. Hurink, E. Hans, G. Wuink, G. azemier, master surgia sheduing aroah for yi sheduing in oerating room deartments, O Setrum, vo. 30, , [3] Z. Yahia, N. Harraz,. B. Etawi, The oerating room strategi ase-mix robem with demand unertainty and nurses aaity onstraint, roeedings of the Internationa Conferene on Comuter and Industria Engineering, vo. 44, Istanbu, Turkey, 4-6 Otober 204. This researh rojet is sonsored by the Egytian ministry of higher eduation grant and the Jaanese