European Journal of Operational Research

European Journal of Operational Researc 239 (2014) 227 236 Contents lists available at ScienceDirect European Journal of Operational Researc ournal omepage: www.elsevier.com/locate/eor Innovative Applications of O.R. Master surgery sceduling wit consideratiof multiple downstream units Andreas Fügener a, Erwin W. Hans b, Rainer Kolisc c,, Nikky Kortbeek d, Peter T. Vanberkel e a Universitäres Zentrum für Gesundeitswissenscaften am Klinikum Augsburg (UNIKA-T), Scool of Business and Economics, Universität Augsburg, Germany b Center for Healtcare Operations Improvement and Researc (CHOIR), Scool of Management and Governance, University of Twente, Te Neterlands c TUM Scool of Management, Tecnisce Universität Müncen, Germany d Department of Applied Matematics, University of Twente, Te Neterlands e Department of Industrial Engineering, Dalousie University, Canada article info abstract Article istory: Received 5 November 2012 Accepted 8 May 2014 Available online 20 May 2014 Keywords: OR in ealt services Resource allocation Master surgery sceduling Ward and ICU occupancy We consider a master surgery sceduling (MSS) problem in wic block operating room (OR) time is assigned to different surgical specialties. Wile many MSS approaces in te literature consider only te impact of te MSS operating teater and operating staff, we enlarge te scope to downstream resources, suc as te intensive care unit (ICU) and te general wards required by te patients once tey leave te OR. We first propose a stocastic analytical approac, wic calculates for a given MSS te exact demand distribution for te downstream resources. We ten discuss measures to define downstream costs resulting from te MSS and propose exact and euristic algoritms to minimize tese costs. Ó 2014 Elsevier B.V. All rigts reserved. 1. Introduction Due to an aging society and tecnological progress, te demand for ealt care services is rising in industrialized countries (Hay, 2003; OECD Indicators, 2011). At te same time, cost cuts and uman resource sortages lead to increasing pressure ospital resources. Terefore, te importance of optimizing te usage of scarce resources ispitals is self-evident. Te most expensive resource in most ospitals is te operating room (OR) (Guerriero & Guido, 2011). ORs are clearly connected wit oter downstream resources, for example, te post-anestesia care unit (PACU), te intensive care unit (ICU), and te general patient wards, ereafter referred to as wards. Anderson, Price, Golden, Jank, and Wasil (2011) sow tat a ig level of utilization ispital wards leads to a iger discarge rate of patients, wic migt reduce te quality of care. On days wit ig patient inflow to te ICU te danger of readmissions (Baker, Pronovost, Morlock, Geocadin, & Holzmueller, 2009) and te probability of reected ICU requests (McManus et al., 2003) strongly increases. Terefore, downstream units sould also be considered in surgery planning for medical reasons. Wen planning te operating rooms and te downstream units, decision makers face a trade-off between te Corresponding autor. Tel.: +49 8928925160. E-mail addresses: andreas.fuegener@wiwi.uni-augsburg.de (A. Fügener), e.w. ans@utwente.nl (E.W. Hans), rainer.kolisc@tum.de (R. Kolisc), n.kortbeek@ utwente.nl (N. Kortbeek), pvanberk@dal.ca (P.T. Vanberkel). ig complexity of a olistic view and te danger of suboptimal solutions resulting from focusing on isolated units (Vanberkel, Boucerie, Hans, Hurink, & Litvak, 2010). Many ospitals use a so-called block-booking system wen planning surgeries. In tis system a medical specialty, e.g. urology, is assigned to blocks denoting a specific amount of time, e.g. a day, ine OR. Tese blocks can be combined into cyclical master surgery scedules (MSS), were every block is repeated after a fixed cycle, e.g. every two weeks. In planning and sceduling, problems can be categorized according to levels of a decision ierarcy (Hans, van Houdenoven, & Hulsof, 2011): Te strategic, tactical, offline-operational (i.e. planning in advance) and te online-operational (i.e. reacting/monitoring) level. In block-booking systems, decisions are made on all ierarcical levels. At te strategic level te number of blocks assigned to te specialties during a MSS cycle is determined. At te tactical level, OR-days are allocated to specialties in an MSS, suc tat te strategic allocation is met. At te operational level, patients are sceduled (offline) and resceduled in case of emergencies or unexpected canges (online). An overview of OR planning may be found in Hans and Vanberkel (2011). In te paper at and, we discuss te tactical MSS problem, concentrating on te effect te MSS as on te patient flow to downstream inpatient care units. Surgeries performed in eac block of te MSS create a flow of patients troug te ICU to te ward, or directly from te OR to te wards, before tey ttp://dx.doi.org/10.1016/.eor.2014.05.009 0377-2217/Ó 2014 Elsevier B.V. All rigts reserved.

228 A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 leave te ospital. As te PACU is part of te OR department in many ospitals, we exclude tis unit iur tactical problem and denote te ICU and te ward as downstream units. Tis paper concentrates on te inpatient flow because outpatients leave te ospital te day of surgery and tus require only OR capacities. We define a model to calculate te distributions of recovering patients in te downstream units expected from te MSS. Based on tis, we propose an approac for planning te MSS wit te obective to minimize downstream costs by leveling bed demand and reducing weekend bed requests. Te remainder of tis paper is organized as follows: Section 2 provides a brief overview of te relevant literature. Section 3 presents an algoritm for calculating te distributiof recovering patients in te downstream units ICU and multiple wards. Section 4 offers a generic model to determine optimal MSSs and a discussion of relevant obective functions to determine downstream costs. In Section 5 we present a branc-and-bound algoritm and different euristics to minimize tese costs. We test te algoritms in Section 6 in an experimental investigation using data from a Dutc ospital. Finally, we discuss managerial implications, limitations, and potential extensions of our study. 2. Literature review Operating rooms are among te most expensive resources in ospitals and is a focus of a large number of sceduling studies (Cardoen, Demeulemeester, & Beliën, 2010). For recent literature reviews on OR sceduling, see Cardoen et al. (2010) and Guerriero and Guido (2011). Articles about ealt care models tat include bot te OR and downstream units are reviewed in Vanberkel et al. (2010). In tis section, we focus on articles tat combine OR sceduling wit te effect on downstream units, suc as ICUs or wards. Adan and Vissers (2002) present a deterministic integer programming approac to scedule patients based on fixed capacities in te OR, te ICU, and te ward. Te ICU and ward capacities are te number of beds available for eac specialty, wile te OR capacity is te total available operating time per day. Additionally, te capacity of te nursing staff is considered. Based on tis, a daily admission profile for different specialties tat minimizes te deviation from resource utilization targets is obtained. Gartner and Kolisc (2014) propose a binary program wic decides for eac patient wat day te patient is admitted, wat day eac clinical activity is undertaken and wat day te patient is released. Te obective is to maximize te sum of te contribution margins of all patients taking into account limited availability of clinical resources. Santibanez, Beliën, and Atkins (2007) discuss various trade-offs in tactical OR planning. Tey also apply a deterministic mixed-integer program and compare different obectives, e.g. maximizing trougput of patients or leveling te bed requests of downstream units. Teir study differentiates between beds and nursing levels as well as between ORs and surgeons. An integer linear program (ILP) model to construct an MSS were patient types are assigned to blocks is formulated by van Oostrum et al. (2008). Tey seek to minimize te required OR capacity and to level ospital bed requirements. To incorporate te uncertainty of OR durations, tey introduce probabilistic constraints. Tey solve te model in two steps. First, OR capacities are optimized witout consideratiof ospital-beds using socalled Operating Room Day Scedules (ORDSs), i.e. lists of surgery types tat are assigned to one OR day. Ten, te ORDSs are assigned to OR days irder to level ospital-bed demand. Terefore, leveling ospital-bed demand is only possible using te precomputed set of ORDSs. All four aforementioned papers model multiple downstream units wit mainly deterministic approaces, wile our study employs a stocastic approac. Models for creating MSSs wit leveled bed occupancy in downstream units are presented in Beliën and Demeulemeester (2007). Contrary to te articles presented above, bot te number of patients and te lengt of stay in te ospital are assumed to be stocastic. A multinomial distribution is used to model te lengt of stay. Te autors aim to minimize te expected bed sortage and employ a mixed-integer programming and a simulated annealing approac. Te approac of Beliën and Demeulemeester (2007) differs from our approac inly allowing one downstream resource (ward), wile we model te patient flow including te ICU and wards and tus consider multiple downstream units. Min and Yi (2010) propose operational sceduling of elective surgeries tat considers bot uncertainty and downstream capacity constraints. Tey formulate a stocastic surgery sceduling problem minimizing te sum of costs directly related to patients and expected overtime costs. Te downstream capacities are modeled as constraints. In contrast to teir approac, wic considers te operational surgery planning level, we focus on te tactical level. Our study is based on te approac of Vanberkel et al. (2011b) were binomial distributions and discrete convolutions are used to calculate te exact distributiof recovering patients in te ward resulting from a given MSS. Vanberkel et al. (2011b) propose a set of equations to determine te distributions of ward occupancy, patient admissions, patient discarges, and te number of patients on eac day of teir recovery period. A case study were te algoritm is implemented in a Dutc ospital is presented in Vanberkel et al. (2011a). Te autors use teir approac to construct several MSSs and to coose one wit a favorable ward occupancy pattern. We build upon teir study by extending it in te following ways: First, Vanberkel et al. (2011b) only include one ward as a single downstream unit. As te ICU is an important bottleneck ispitals (Litvak, van Risbergen, Boucerie, & Houdenoven, 2008), we incorporate ICU bed requests as well as multiple wards iur model as a valuable extension. Second, Vanberkel et al. (2011b) do not determine te costs resulting from an MSS. As different downstream costs exist, e.g. costs for providing fixed capacities or costs for weekend staffing, we develop an approac to assign costs to specific MSSs. Tird, we introduce several exact and euristic algoritms to minimize tese downstream costs. To te best of our knowledge, te current study presents te first exact stocastic MSS approac to calculating patient occupancy distributions in te ICU and multiple wards. In addition, we present exact and euristic algoritms to minimize costs resulting from patients in downstream units. 3. Recovering patients in downstream units In tis section, we describe a model tat calculates te exact distributiof post-operative inpatients in te ICU and multiple wards resulting from a given MSS cycle. We do not furter distinguis between different ICUs in tis study. However, te presented approac can be extended to include several ICUs. We now present te general underlying assumptions regarding te process, te required data, and te detailed model. After aperation several patient pats exist. In most cases, patients are admitted to a ward. In more severe cases, patients are sent to te ICU. Alternatively, patients migt be discarged witout being sent to a ward (e.g. due to mortality). Patients in te wards will be transferred to te ICU if teir condition becomes unstable. Most patients leave te system only after recovering in a ward, but tey migt also leave te ospital directly from te ICU

A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 229 (e.g. in case of deat or if transferred to anoter ospital). Te patient pats are outlined in Fig. 1. In studying data from a large University Hospital in Munic, Germany, we found tat more tan 98% of inpatients follow one of tree pats. Te vast maority (92%) follow te pat OR! ward! discarge. About 5% follow OR! ICU! ward! discarge. Just above 1% follow te pat OR! ICU! discarge, i.e. te previous pat wit a zero day stay in te ward. It is very rare for patients to return to te ICU after being transferred from te ICU to a ward (ust above 1%). Based on tis data we simplified te modeled patient patway as depicted in Fig. 2. Te number of patients sent to te ICU or te wards after one surgery block is modeled by a discrete empirical distribution. Tis distribution may also include emergency patients wo were operated on during tis block. A stay in te ICU is denoted by I, a stay in a ward of a patient wo directly came from te operating room by WO, and a stay in a ward of a patient wo was transferred from te ICU by WI. Te lengts of stay (in days) in te ICU or a ward, after being transferred from te OR or from te ICU, are also modeled by discrete empirical distributions. Suc distributions are easily obtained from istorical records. Te main sets and indices used in te following model are sown in Table 1. Te index n is used to determine days after surgery, were 1 denotes te day of surgery. Days after a transfer to a ward from te ICU will be denoted by u. We differentiate between multiple wards 2H. Eac specialty corresponds to one specific ward, wereas eac ward may accommodate more tane specialty. Te set of specialties accommodated by ward is denoted by J. Required istorical or estimated data for every specialty 2J are as follows: a ðpþ represents te probability tat p 2f0;...; P g patients are operated on during a surgery block of specialty. b represents te probability tat a patient of specialty is admitted to te ICU immediately after surgery. 1 b is te probability tat te patient is admitted to te ward. c I ðnþ represents te probability tat a patient from surgery of specialty stays n 2 1;...; N I days in te ICU after surgery. c WO ðnþ represents te probability tat a patient from surgery of specialty stays n 2 1;...; N WO days in te ward after surgery. c WI ðuþ represents te probability tat a patient from surgery of specialty stays u 2 0;...; N WI days in te ward after being released from te ICU. A stay of zero days implies a direct release from te ICU. Te approac works in tree steps (see Fig. 3). First, we calculate for a single surgery block te distributions of recovering patients in te ICU and te specific ward. Tis step is carried out for eac surgical specialty. In te next step we calculate te distributions for a single cyclical block. It is important to note tat we assume te MSS to be cyclical. Terefore, eac block will be repeated for eac new MSS cycle. In te tird step we combine all blocks from a cyclical MSS to calculate te occupancy levels for te ICU and eac ward 2H. Te first two steps do not depend on te specific MSS, we only need information about te definition of surgery blocks (e.g. lengt of a block) and te lengt of te MSS cycle. Terefore, tese steps can be calculated during preprocessing. Due to te structure of te problem, te tird step as to be calculated for eac MSS we want to evaluate. 3.1. Calculatiof te distributions of patients resulting from a single OR block (Step 1) In te following, we present te algoritm to derive te distributions of number of patients resulting from a single OR block. First, using te probability of an ICU admission and te empirical lengt of stay distributions, we analyze te patway of a single patient troug te ospital (see Fig. 4). After surgery, a patient of specialty can be admitted eiter to te ICU or to te specific ward. On eac day n, a patient in te ICU may eiter stay or be transferred to tis ward. A patient in te ward may eiter stay or be released from te ospital. We assume tat te probability for a patient to be discarged from te ward after being transferred from te ICU only depends on te time since te transfer from te ICU. Eq. (1) calculates te conditional probability d I ;n for a patient of specialty in te ICU to be transferred to te ward on day n, given tat e was not released before. Analogously, Eq. (2) calculates te probability d WO ;n tat a patient wo is in te ward n days after surgery is discarged on tat day. Eq. (3) calculates te probability d WI ;u for a discarge u days after te transfer from te ICU to te ward. Patients wo leave te ospital after staying in te ICU are modeled to ave a stay of zero days in te ward. Calculations (1) (3) follow te logic of Vanberkel et al. (2011b). d I ;n ¼ PN I ci ðnþ n 2J; n 2 1;...; N k¼n ciðkþ ðnþ d WO ;n ¼ cwo PN WO k¼n cwo d WI ;u ¼ PN WI cwi ðuþ k¼u cwi ðkþ ðkþ I o ; ð1þ 2J; n 2 1;...; N WO ; ð2þ 2J; u 2 0;...; N WI : ð3þ We denote te latest possible day wit a positive probability of a patient staying in te ICU and in te ward as N I and N W ¼ max N WO ; N I þ NWI, respectively. Now, we calculate in Eq. (4) for all specialties 2J and eac day n 2 1;...; N I te probabilities e I ;n tat a patient of specialty wo ad surgery on day 1 is in te ICU. Accordingly, te same is done for patients staying in te ward e W ;n for n 2 1;...; NW. For te probability tat a patient of specialty is in te ICU on day n we get 8 b ; n ¼ 1 >< e I ;n ¼ 1 d I ;n 1 e I ;n 1 ; n 2 2;...; NI >: 0; oterwise: ð4þ Fig. 1. Patient pats.

230 A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 Fig. 2. Simplified patient pats. Table 1 Sets and indices. Description Index 2 set Wards 2H Operating rooms (ORs) i 2I Surgery specialties 2J Surgery specialties connected to ward 2J Patients p 2f0;...; P g Days in te ICU after surgery n 2 1;...; N I Days in a ward after surgery n 2 1;...; N WO Days in a ward after ICU u 2 0;...; N WI Days in te MSS cycle 2L Weekdays in te MSS cycle q 2Q Weekend days in te MSS cycle 2LnQ probability tat te patient came directly from te OR and is in te ward on day n is denoted by e WO ;n, wereas te probability tat te patient is in te ward on day n after staying m days in te ICU is e WI ;m;n. 8 1 b ; n ¼ 1 >< e WO ;n ¼ 1 d WO ;n 1 e WO ;n 1 ; n 2 2;...; NWO ð5þ >: 0; oterwise: e WI ;m;n ¼ 8 1 d WI ;0 d I ;m ei ;m ; m 2 1...NI ;n ¼ m þ 1 >< n o 1 d WI ;n m 1 e WI ;m;n 1 ; m 2 1...NI ;n 2 m þ 2;...;mþN WI >: 0; oterwise: ð6þ Te calculatiof e WO ;n in Eq. (5) is analogous to ei ;n. To calculate ewi ;m;n in Eq. (6), te different transfer times from te ICU are taken into account. After staying m days in te ICU (n ¼ m þ 1), e WI ;m;n equals te product of (a) te probability 1 d WI ;0 tat te patient did not leave te ospital immediately, (b) te probability d I ;m tat e Fig. 3. Process steps. On day 1, tis probability equals b, i.e. te probability tat te patient is directly transferred to te ICU after surgery. For te following days, te probability decreases as patients migt be transferred to te ward. Irder to calculate e W ;n, we differentiate between patients wo were directly transferred to te ward after leaving te OR and tose wo were transferred via te ICU. Te was transferred to te ward tat day, and (c) te probability e I ;m tat te patient was in te ICU on day m. Terefore, te probability e W ;n tat a patient is in te ward on day n is calculated in Eq. (7) by adding te probability e WO ;n tat e came directly from te OR and te probabilities e WI ;m;n tat e stayed m days in te ICU before for all possible number of days m < n. 8 ewo ;1 >< ; n ¼ 1 e W ;n ¼ e WO ;n þ P n 1 m¼1 ewi ;m;n ; n 2 2;...; NW ð7þ >: 0; oterwise: Now, we calculate for eac day n te probability distribution for te number of patients in te ICU, f I ;nðpþ, in Eq. (8) and in te ward, f W ;n ðpþ, in Eq. (9). Te probability tat out of k patients wo ad surgery, p patients are in te ICU or te ward on day n can be determined using a binomial distribution (Vanberkel et al., 2011b). Fig. 4. Patient pats.

A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 231 Next, we ave to sum tese probabilities weigted by a ðkþ for all possible k (number of patients tat ad surgery) tat could lead to p patients on day n. f I ;n ðpþ¼xp k¼p f W ;n ðpþ¼xp k¼p k p k pa n e I ;n 1 e I ;n ðkþ 2J; n 2 1;...;N I p k p k pa n e W ;n 1 e W ;n ðkþ 2J; n 2 1;...;N W p o : ð8þ o : ð9þ 3.2. Calculatiof te distributions of patients resulting from a cyclical OR block (Step 2) As te MSS scedule is cyclical, eac block will be repeated in every cycle. For example, consider a weekly cycle in wic a urological block on Monday takes place on every Monday. As te maximum recovery time of patients usually exceeds te cycle time, patients aving teir surgery in different cycles migt be recovering at te same time. Te number of overlapping cycles depends on te cycle lengt L ¼ L and te maximum lengt of stay N I for patients in te ICU and N W for patients in te wards. To obtain te distributions of patients on te days of one cycle, we perform discrete convolutions, see (10) and (11), of te patient distributions of all overlapping cycles for te ICU and te specific ward, respectively. We use te symbol for te discrete convolution. F I ; F W ; represents te distribution te t day of a cycle of te number of recovering patients of specialty in te ICU (ward) wic results from a cyclical surgery block on day 1 of all previous cycles including te current cycle. F I ; ¼ f I ; f I ; þl f I ; þ F W ; ¼ f W ; f W ; þl... f W ; þ ð ð N I Þ=L L N W Þ=L L 2J; 2L ð10þ 2J; 2L ð11þ 3.3. Calculatiof te distributions of patients resulting from a cyclical MSS (Step 3) To calculate a cyclical MSS, we obtain te patient distributions coming from eac block ði; qþ, were i denotes te operating room and q te day of te cycle. We assume tat surgery blocks are only provided on weekdays. For a given MSS, x is set and eac x i;q; as a value of 1 if specialty is assigned to block ði; qþ and a value of 0 oterwise. F I in (12) i;q;l (FW ;i;q;l in (13)) is te distributiof te number of recovering patients in te ICU (ward ) on day of te MSS cycle coming from surgery in OR i on day q of te MSS cycle. F I i;q; ¼ ( P 2J FI ; qþ1 x i;q;; P 2J FI ; qþ1þl x i;q;; P q oterwise: ( P F W ;i;q; ¼ 2J F W ; qþ1 x i;q;; P q P 2J F W ; qþ1þl x i;q;; oterwise: i 2I; q 2Q; 2L; ð12þ 2H; i 2I; q 2Q; 2L ð13þ Now we ave to convolve te distributions of all blocks to obtain te patient distribution resulting from te MSS. F I in (14) (F W ; in (15)) denotes te distributiof recovering patients in te ICU (ward ) on day of te MSS cycle. supfig denotes te last operating room, supfqg te last weekday wit an active surgery slot. F I ¼ FI 1;1; FI 1;2; FI supfig;supfqg; 2L ð14þ F W ; ¼ FW ;1;1; FW ;1;2; FW ;supfig;supfqg; 2H; 2L ð15þ Te steps presented in tis section calculate for a given MSS te distributiof patients for every day in te MSS cycle for te ICU and every ward 2H. Note tat x i;q; is assumed to be set for now but will become a variable wen we are searcing for a good MSS. In te next sections we present metods to minimize downstream costs of an MSS using tese distributions. 4. Generic model and discussiof obectives In Section 4.1 we present a generic model tat minimizes downstream costs using a general assignment problem. We ten discuss different downstream cost functions for tis model in Section 4.2. 4.1. Generic model We define a generic assignment problem tat minimizes te downstream costs cðxþ. cðxþ is a functiof te distributiof patients in te downstream units calculated in Steps 1 3 in te previous section resulting from te MSS x, i.e. te assignment of all blocks ði; qþ to a specialty. Min cðxþ ð16þ s:t: X x i;q; 6 1 i 2I; q 2Q ð17þ 2J XX x i;q; P d 2J ð18þ i2i q2q X x i;q; 6 s q q 2Q; 2J ð19þ i2i x i;q; 2f0; 1g i 2Iq 2Q; 2J: ð20þ Eqs. (17) and (18) are te assignment problem constraints. Eq. (17) ensures tat at most one specialty is assigned to eac block, wile (18) ensures tat te number of blocks assigned to eac specialty at least equals te number of required blocks d obtained from strategic planning. Te maximum number of blocks s q assigned to eac specialty per day is modeled in (19). Eq. (16) represents a generic obective function. Tis model can easily be adusted to deal wit specific constraints, e.g. some specialties ave to operate in specific ORs. 4.2. Discussiof downstream cost functions Discussions wit operating room managers indicated tat tere are four cost components tat drive downstream costs: fixed costs, overcapacity costs, staffing costs, and additional weekend staffing costs. Fixed costs. We consider te costs for creating and maintaining fixed capacities. We define c f ;I and c f ;W as te costs for creating and maintaining te capacity for one patient in te ICU and ward per cycle, respectively. An example of fixed costs is te costs associated wit an ICU bed. Te model determines te required capacity of tese resources to ensure certain service levels a I and a W. We denote Q I ðai Þ as te a I -quantile of te distribution F I of te number of patients in te ICU on day. Q W ; a W is te a W -quantile for te distribution F W ; of patients in ward. For example, Qð:99Þ denotes te capacity tat will not be exceeded wit a probability of 99%. Te number of beds we need to provide in te ICU and in eac ward are terefore cap I ða I Þ¼max 2L Q I ðai Þ and

232 A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 cap W a W ¼ max 2L Q W ; a W, respectively. We obtain total fixed costs of c f ;W cap W a W : ð21þ cost f ¼ c f ;I cap I ða I Þþ X 2H Overcapacity costs. Overcapacity costs are costs tat incur due to requiring capacity beyond cap I and cap W. Tis situatioccurs, for example, wen patients must be transferred to ICUs or wards in oter ospitals, or to a wrong ICU or ward, as capacity limits (depending on te service levels a I and a W ) are reaced. We assign costs of c o;i and c o;w for eac patient above existing capacities per day. Te expected number of tese patients per day is exc I ða I Þ¼ P Q P UB I ¼1 p¼cap I ða I Þþ1 pfi ðpþ in te ICU and excw a W ¼ P Q UB W ¼1P p¼cap W ðaw Þþ1pFW ; ðpþ in ward. UBI and UB W is an upper bound on te number of patients tat request a bed in te ICU and ward, respectively. A simple upper bound is te product of te number of overlapping cycles and te maximum number of patients per cycle. We obtaivercapacity costs of c o;w exc W a W : ð22þ cost o ¼ c o;i exc I ða I Þþ X 2H Bot, te fixed costs and te overcapacity costs depend on te service levels a I and a W. Te iger te service level is, te iger te fixed costs and te lower te overcapacity costs are. Setting te appropriate service levels a I and a W sould be done on a strategic level and is terefore outside te scope of tis study. Staffing costs. Staffing costs are dependent on te number of patients, i.e. occupied beds. Te staffing decision for every bed is made in advance. Terefore, we assume a service level of b I and b W for staffing beds. For example, a ospital migt staff te.75 quantile of demand to be understaffed no more tan 25% of te time. For simplicity, we assume te costs for staffing one bed per day are constant wit c s;i for te ICU and c s;w for te wards. Te total number of beds to be staffed during one cycle in te ICU and in ward are terefore sta I ðb I Þ¼ P 2L Q I ðbi Þ and sta W b W P ¼ 2L Q W ; bw, respectively. Te staffing costs, cost s, assuming constant wages, are cost s ¼ c s;i sta I ðb I Þþ X c s;w sta W b W : ð23þ 2H Weekend staffing costs. Usually, tere are additional costs for staffing beds on weekends. Te additional costs for one bed per day are c we;i and c we;w. Te total number of beds to be staffed on te weekends of one cycle in te ICU and in ward are terefore sta we;i ðb I Þ¼ P 2LnQ Q I ðbi Þ and sta we;w b W P ¼ 2LnQ Q W ; bw, respectively. Te additional costs on weekends are cost we ¼ c we;i sta we;i ðb I Þþ X c we;w sta we;w b W : ð24þ 2H Many combinations of downstream costs are possible. Iur case study, we employ downstream costs of cðxþ ¼cost f þ cost we. Te resulting obective function is Min c f ;I cap I þ X cap W þ cwe;i sta we;i þ X 2H 2H c f ;W 5. Solution approaces c we;w sta we;w : ð25þ Te generic model presented in te previous section is a classical assignment problem. Altoug te generalized assignment problem is well-known to be NP-ard, tere are efficient procedures, suc as branc-and-bound (Ross & Soland, 1975), to solve even large instances to optimality. However, te calculatiof te obective function value is, due to te convolutiof distributions, quite extensive. Hence, in addition to aptimal brancand-bound procedure, we discuss te following two euristic strategies to solve te master surgery sceduling problem: 1. Exact obective function and euristic solution metod. 2. Approximated obective function and exact solution metod. For strategy 1, we apply an incremental improvement euristic, a 2-Opt euristic, and simulated annealing. For strategy 2, we consider two approximated obective functions: te first uses expected values only, wile te second employs a combinatiof expected values and variances. Te last two approaces sow some similarities to Beliën and Demeulemeester (2007), wo minimize expected sortage of ward beds by linearizatiof teir model. 5.1. Straigtforward branc-and-bound Te straigtforward branc-and-bound (SBB) algoritm is based on complete enumeration but avoids redundant solutions. Tese redundant solutions could be caused by aving different combinations of te same specialties on te same day in different ORs. Te algoritm fills up block after block of te MSS using a dept-first searc. It assigns all blocks, i.e. combinations of days q and operating rooms i, to specialties starting wit te specialty wit te lowest index. After eac block of a day is assigned to a specialty, te next day is started. To avoid redundant solutions, remaining blocks on te same day will only be filled wit specialties wit te same or a iger index. An example for te solutions is presented in Fig. 5. Here, we sow for 5 blocks (1 day wit 1 OR, 2 days wit 2 ORs) and 3 specialties (specialty 1 and 2 wit 2 blocks eac, specialty 3 wit one block) all 11 possible non-redundant solutions (compared to a total of 5! ¼ 30 solutions to assign 2!2! tese 3 specialties to 5 blocks). As an example, te solution 3 assigns te two required blocks of specialty 1 to OR 1 on days 1 and 3. Te two required blocks of specialty 2 are assigned to operating rooms 1 and 2 on day 2, and te block of specialty 3 is assigned to operating room 2 on day 3. After assigning a specialty to a block, te algoritm updates te distributions of patients in te ICU and te wards. A lower bound of te obective function is calculated by considering all blocks already planned leaving all blocks not yet planned empty. An upper bound is te best feasible solutiobtained so far. A good first upper bound may be obtained by simulated annealing, as detailed in te following section. A partial solution is fatomed as soon as its lower bound is not strictly smaller tan te upper bound. If a new feasible solution is obtained tat is below te current upper bound, te upper bound is updated. We present te example of Fig. 5 wit upper and lower bounds, fatoming of non-optimal solutions and te optimal solution (dark nodes wit wite numbers) in Fig. 6. Wile tis metod is exact, it may only be applied to small problem instances due to ig computation times. 5.2. Exact obective function and euristic solution metod Incremental improvement euristic. Te incremental improvement euristic (IIH) is motivated by te way MSSs are altered in practice. Usually, an MSS already exists and te ospital is not willing to allow many canges, since te MSS affects many departments suc as outpatient clinics. Te proposed euristic will seek te best option if only one swap of two blocks is allowed. Terefore, it will realize te swap wit te maximum incremental improvement. Tis metod may be used to sow improvements for a defined maximum number of swaps. 2-Opt euristic. We repeat te incremental improvement euristic until no furter improvement of te obective function

A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 233 Fig. 5. All non-redundant solutions for an example wit 5 blocks and 3 specialties. Fig. 6. Optimal solution for an example wit 5 blocks and 3 specialties. is observed. In tis case it is equivalent to te 2-Opt (2OH) approac known from te traveling salesman literature (Lin & Kernigan, 1973). Simulated annealing. IIH and 2OH presented above are likely to get stuck in a local optimum and te quality of te obtained solution is sensitive to te starting solution. To overcome tis weakness we propose a simulated annealing approac (SA) (Aarts, Korst, & Miciels, 2005) wit te same neigborood as IIH and 2OH. In contrast to IIH, proximity to te given initial solution cannot be controlled wit SA. SA will accept every move, i.e. swap of two blocks, tat improves te obective function. A swap causing an increase in te obective function will be accepted wit a probability wic decreases over time. We implement te SA using a geometric cooling scedule wit t k ¼ cf t k 1, were cf denotes te cooling factor and t k te temperature level in round k. Te lower te cooling factor is, te faster te cool dowccurs and tus te faster te SA terminates. More details of te algoritm are given in Section 6. 5.3. Approximated obective function and exact solution metod Te following two solution approaces approximate te obective function, suc tat it can be solved wit standard optimization software. Approximated obective function based on expected values. We approximate te quantiles Q I and Q W ; used in te exact obective function by teir expected values, E F I ;l for te ICU and E F W ;;l for ward, multiplied wit parameters a I and a W for fixed capacities and b I and b W for weekend staffing, respectively. We calculate tese parameters as te average quotient of te quantiles QðÞ and te expected values EðÞ of te given initial solution. Table 2 states te approximated quantiles. We denote te euristic using expected values as EV. As defined in te previous section, te obective function is Min c f ;I cap I þ X cap W þ cwe;i sta we;i þ X 2H 2H c f ;W c we;w sta we;w : ð26þ were cap I and cap W denote te capacity levels, sta we;i and sta we;w te cumulated beds to be staffed on weekends during one MSS cycle for te ICU and ward, respectively. For te approximatiof te obective function, Constraints (27) (29) need to be added to te generic model presented in Section 4.1. We only sow te constraints for te ICU, te ones for eac ward are formulated analogously. E F I ¼ X XX E F I ; qþ1 x i;q; þ X X X L E F I ; qþ1þl x i;q; 2L ð27þ i2i 2J q¼1 i2i 2J q¼ þ1 a I E F I 6 cap I 2L ð28þ X b I EðF I Þ¼stawe;I ð29þ 2LnQ In (27), te values for te expected number of patients in te ICU are determined for eac day. In (28) te required capacity for te ICU is calculated. Finally, in (29) te number of patients per weekend day relevant for staffing is determined. Approximated obective based on expected values and variances. Te algoritm EV neglects te distributiof patients as it only considers te expected values. Wit te approximated obective function based on expected values and variances (EVV), we assume te distributions of patients to be normally distributed

234 A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 Table 2 Approximations for EV euristic. Quantile Exact model EV euristic Patients in te ICU relevant for fixed capacities Q I ðai Þ a I E F I Patients in te ward relevant for fixed Q W ; a W a W capacities ; Patients in te ICU relevant for weekend staffing Q I ðbi Þ b I E F I Patients in te ward relevant for weekend Q W ; b W b W staffing ; Table 3 Approximations for EVV euristic. and approximate te quantiles using te expected value and te approximated standard deviation. To avoid a square root function, te standard deviations SD F I and SD F W ; are approximated by a linear functiof te variance V F I ; for te ICU and V F W ; for te ward. We employ one linear factor for te ICU, sr I in Constraint (31), and one for eac ward, sr W. We use te factors tat minimize te squared errors for te variances of te given initial solutions. Te z-values for te quantiles are z cap;i and z cap;w for te capacity levels and z sta;i and z sta;w for te weekend staffing levels. Table 3 states te approximated quantiles. Te obective function (26), te assignment problem constraints (17) (20), and te constraints to determine te expected values (27) stay uncanged. Constraints to determine te variances (30) and te approximated standard deviations (31) need to be added. Te constraints determining te capacities (32) and te beds to be staffed at weekends (33) ave to be canged. Te approximatiof te square root function to determine te standard deviation in (31) can be carried out in many ways. Te most simple one is to use a linear function. To account for differences in te variances, a piecewise linear function as described in van Essen, Bosc, and van der Veen (2011) may also be applied. Again, we only present te constraints for te ICU (30) (33), te constraints for eac ward are formulated analogously. VðF I Þ¼X XX V F I ; qþ1 x i;q; þ X X X L VðF I ; qþ1þl Þx i;q; 2L ð30þ i2i 2J q¼1 i2i 2J q¼ þ1 sr I V F I ¼ SD F I 2L ð31þ E F I þ z cap;i SD F I 6 cap I 2L ð32þ X E F I þ z sta;i SD F I ¼ sta we;i ð33þ 2LnQ Quantile Patients in te ICU relevant for fixed capacities Patients in te ward relevant for fixed capacities Patients in te ICU relevant for weekend staffing Patients in te ward relevant for weekend staffing 6. Numerical study Exact EVV euristic model Q I ðai Þ E F I þ z cap;i SD F I Q W ; a W E F W ; þ z cap;w SD F W ; Q I ðbi Þ E F I þ z sta;i SD F I Q W ; b W E F W ; þ z sta;w SD F W ; We tested all solution approaces for tree scenarios wit an MSS cycle of two weeks, seven specialties, one ICU and two wards. Te results of tese numerical experiments are described in tis section. Te starting point for te data collection was te data employed in Vanberkel et al. (2011b), wic considers te OR and a single ward only. Irder to acquire te missing data for te ICU and a second ward, we proceeded as follows. We interviewed a Dutc ospital manager responsible for patient logistics. Te required values for ICU probability and te lengt of stay distributions for te ICU and te wards after a stay in te ICU were derived from data locally available in te ICUs and wards, respectively. We cross-cecked te values wit data from a Germaspital wit similar specialties. A summary of te data for eac specialty can be found in Table 4. Te downstream costs to be minimized are te fixed costs cost f and te additional weekend staffing costs cost we. As a result of discussions wit te Dutc ospital manager, we set te values of te cost parameters as presented in Table 5. Te downstream units in te case study are one ICU and two wards. Te specialties Hypertermic Intraperitoneal Cemoterapy (HIPEC), General Surgery and Breast Surgery sare one ward, te remaining specialties sare te oter ward. Tere is no limit on te number of blocks of any specialty on any given day oter tan te number of ORs. To test te performance of our solution approaces on instances of varying size, we build tree scenarios. In building tese scenarios, we scale te MSS te ospital currently uses. By doing tis, we keep te percentage of required blocks per specialty approximately constant for all tree scenarios. We denote tese MSSs as ospital MSS. We distinguis te following tree examples iur case study: A small MSS wit only one OR on every weekday of te two weeks and a second OR on Wednesdays. Te number of OR blocks is terefore 12. A medium MSS wit tree ORs available during weekdays (30 blocks). A large MSS wit nine ORs available during weekdays (90 blocks). As a starting solution, we use te ospital MSS for all scenarios, Fig. 7 sows te ospital MSS for te medium scenario. As we cannot compute te optimal solution for te medium and te large scenario, we use te ospital MSS as a reference point. We compare te simple branc-and-bound (SBB), te incremental improvement euristic (IIH), te 2-Opt euristic (2OH), simulated annealing (SA), te approximated obective function based on expected values (EV), and te approximated obective based on expected values and variances (EVV). For te IIH we continue swapping blocks until a maximum of one tird of all blocks are swapped. For te simulated annealing (SA), a cooling factor of cf ¼ 0:9 is cosen as proposed by Aarts et al. (2005). Te number of iterations for eac temperature level is 5 times te number of OR blocks for eac case. Te initial temperature is t 0 ¼ 9; 000, and te SA stops wen te temperature falls below t ¼ 1; 000. EV and EVV are solved using CPLEX solver employing a branc-andcut algoritm using IBM ILOG CPLEX Optimization Studio Version 12.2. All remaining euristics and SBB were solved wit MATLAB R2013a. All computations were run a Windows-based Intel(R) Core 2 Duo CPU wit 3.16 gigaertz. For eac scenario we compare computation time, total cost, relative improvement of te starting solution, and te percentage of canged blocks. Irder to analyze te sensitivity of computation times wit respect to te cost parameter settings, we set up a ceteris paribus experimental design to evaluate variations in cost parameters. As a result we found computation times varying only sligtly for realistic canges in te parameter settings. A summary of te results of te 12 block example is provided in Table 6. In tis example te IIH consists of swapping four blocks

A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 235 Table 4 Specialty, expected number of patients per OR day E a ðpþ, probability for ICU b, expected LoS ICU E c I ðnþ, ward (after OR) E c WO ðnþ and ward (after ICU) E c WI ðnþ. Specialty E a ðpþ b (%) E c I ðnþ E c WO ðnþ E c WI ðnþ Urology 3.2 5 5.4 3.7 6.4 HIPEC 1.0 100 20.3 11.2 21.3 Gynecology 4.3 3 3.5 2.7 4.4 General surgery 4.3 18 8.0 5.0 9.0 Breast surgery 5.5 3 2.5 2.2 3.5 Ortopedics 4.1 5 6.1 4.0 7.1 Plastic surgery 4.2 2 3.4 2.7 4.4 Table 6 Results small scenario. Algoritm Computation time (seconds) Total costs Improvement (%) Hospital 68,420 MSS SBB 19,816 61,400 10.3 92 IIH 8 66,420 2.9 33 2OH 15 65,800 3.8 50 SA 26 62,780 8.2 92 EV 1 67,400 1.5 83 EVV 1 61,780 9.7 92 Canged blocks (%) Table 5 Input parameters. Description Notation Value Service level for fixed capacities a I ; a W 0.99 Fixed costs ICU bed/two weeks c f ;I 5,000 Fixed costs ward bed/two weeks c f ;W 500 Service level for staffing b I ; b W 0.75 Additional costs staffing ICU bed at weekends/day c we;i 700 Additional costs staffing ward bed at weekends/day c we;w 120 and results in an improvement of 2.9% compared to te ospital MSS. 2OH swaps alf te blocks and acieves an improvement of 3.8%. Te SA takes nearly twice te computing time, canges nearly every block, but obtains a muc better solution tan te 2OH (8.2% instead of 3.8% in cost savings are gained). Te two euristics using an approximated obective function are very fast (around 1 second), but acieve quite different results. EV suffers from a poor approximation and only yields improvements of about 2%, wile EVV acieves te igest improvement of more tan 9%. EV uses 242 variables and 119 constraints, EVV 328 variables and 205 constraints. Te optimal SBB demonstrates tat a maximum improvement of 10.3% is possible. Table 7 provides te results for te example wit 30 blocks were all euristics acieve comparable improvements between 6% and 7%. We stopped te optimal SBB after 25,000 seconds and acieved a comparably low improvement of 3.6%. EV and EVV are muc faster tan te oter euristics; EV uses 340 variables and 133 constraints, EVV 426 variables and 219 constraints. However, tey cange 90% and 67% of te blocks of te initial MSS. Te results of te large example wit 90 blocks (see Table 8) sow large computing times. EV, wic employs aptimal Table 7 Results medium scenario. Algoritm Computation time (seconds) Total costs Improvement (%) Hospital 130,860 MSS SBB 25,000 126,120 3.6 70 IIH 187 123,000 6.0 33 2OH 336 121,800 6.9 50 SA 299 122,420 6.4 80 EV 7 122,600 6.3 90 EVV 5 122,360 6.5 67 Table 8 Results large scenario. Algoritm Computation time (seconds) Total costs Improvement (%) Hospital 341,020 MSS SBB 25,000 341,020 0.0 0 IIH 10,490 322,780 5.3 18 2OH 10,490 322,780 5.3 18 SA 2,827 315,740 7.4 81 EV 12,242 309,480 9.2 84 EVV 2,987 313,260 8.1 86 Canged blocks (%) Canged blocks (%) solution approac, sows te igest computational times. However, at te same time it acieves te greatest improvements. IIH and 2OH suffer from te large problem size as well, since for eac swap all possibilities ave to be calculated. In fact, in tis Fig. 7. Starting solution medium MSS.

236 A. Fügener et al. / European Journal of Operational Researc 239 (2014) 227 236 example for eac iteration more tan 4,000 swaps ave to be evaluated. Still, IIH and 2OH sow te lowest improvement of only 5.3%. SA and EVV sow similar computation times of less tan one our, wit improvements of 7.5% and 8.1%, respectively. EV uses 929 variables and 217 constraints, EVV 1,014 variables and 303 constraints. SBB was unable to find a solution tat improves te starting solution witin te time limit of 25,000 seconds. Wit regard to tese results tree conclusions can be drawn. First, te euristics acieve improvements of around 6 9% in te tree examples. Second, wile IIH and 2OH acieve moderate improvements by maintaining most parts of te current scedule, tey are outperformed by te SA, te EV and te EVV euristics in most cases. Tird, a maority of te approaces discussed in tis paper require long computation times for te large example. Only te SA and te EVV require computation times of less tane our. EV and EVV could be run using a time limit. Even if te optimal solutiof te approximated obective function migt not been found witin te time limit, good results can be acieved after a relatively sort time period. However, as we are discussing a tactical problem, relatively ig computation times of up to a few ours can be reasonably tolerated in practical settings. 7. Conclusion In tis paper we presented an algoritm for calculating te exact distributions of patients bot in te ICU and te wards resulting from a given cyclical MSS. We furter discussed measures as fixed capacities and staffing levels to estimate te downstream costs of an MSS and proposed algoritms to find an MSS wit te obective to minimize costs. We considered several euristics. Two simple euristics tat swap MSS blocks, a simulated annealing algoritm tat finds good solutions in a reasonable period of time, and a simple branc-and-bound, tat canly be applied for small problems. We furter tested solution metods approximating te obective function and solving te resulting model to optimality by using off-te-self solver. Tese sowed excellent results for medium and large instances, but required long computation times for large instances. For large instances tere is furter room for researc on euristics. For example, one could investigate on a combinatiof te approximated obective function, suc as EV or EVV, wit a nonoptimal solution metod to gain satisfactory results witin sort time. Furtermore, tere are many possible modeling extensions. Upstream units like te outpatient clinic can be incorporated as surgeons work tere too, so sceduling in bot departments could be coordinated. Effects on te post-anestesia care unit could also be incorporated. Operations on weekends (for emergency patients only) as well as pre-operative stays in ICUs and wards or patients wit no surgery could be included. 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