A datadriven hierarchical MILP approach for scheduling clinical pathways: a realworld case study from a German university hospital
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Abstract
Facing economic pressure and casebased compensation systems, hospitals strive for effectively planning patient hospitalization and making efficient use of their resources. To support this endeavor, this paper proposes a flexible hierarchical mixedinteger linear programming (MILP)based approach for the daylevel scheduling of clinical pathways (CP). CP form sequences of ward stays and treatments to be performed during a patient’s hospitalization under consideration of all relevant resources such as beds, operating rooms and clinical staff. Since in most hospitals CPrelated information needed for planning is not readily available, we propose a datadriven approach in which the structure of the CP to be scheduled including all CPrelated constraints is automatically extracted from standardized hospital billing data available in every German hospital. The approach uses a flexible multicriteria objective function considering several patient and hospitalrelated aspects which makes our approach applicable in various scenarios. Furthermore, in contrast to other approaches, it considers several practically relevant aspects ensuring the implementability of the scheduling results such as multiple ward stays per hospitalization and genderseparated room assignments. Regarding the treatment resources such as operation rooms and clinical staff, it considers the eligibility of resources for treatments based on information such as special equipment or qualification and represents complex resources individually to avoid disaggregation problems. To allow solving the resulting complex and largescale scheduling problem for realistically dimensioned problem instances, we propose a hierarchical twostage MILP approach involving carefully designed anticipation components in the firststage model. We evaluate our approach in a case study with realworld data from a German university hospital showing that our approach is able to solve instances with a planning horizon of 1 month exhibiting 1088 treatments and 302 ward stays of 286 patients. In addition to comparing our approach to a monolithic MILP approach, we provide a detailed discussion of the scheduling results for two practically motivated scenarios.
Keywords
Clinical pathways Scheduling MILP Health care planning Patient flow Data driven Case study1 Introduction
In Germany, hospitals cause around 25% of the total costs of the health care system (Destatis 2014). To reduce these costs of about 76bn Euro per year, the German DiagnosisRelated Groups (DRG) system was established in 2003. In contrast to the earlier lengthofstaybased reimbursement system, hospitals now receive a diagnosisdependent lump sum per case. As a consequence, hospitals have a strong incentive to reduce their patients’ lengths of stay and to effectively manage clinical processes and resources.
At the time of writing, however, every third hospital is still unprofitable (Augurzky 2015) which, together with demographic change, yields a great need for increasing the efficiency of hospitals. According to Villa et al. (2009), clinical pathways (CP) are among the most promising instruments for achieving an efficient utilization of both human and physical hospital resources. CP are specific sets of timeconstrained treatments and ward stays to be performed between a patient’s admission and discharge to cure a certain disease. Besides increasing the transparency and the standardization of medical processes, CP are instrumental in hospitalization scheduling and controlling (Jacobs 2007). Despite this oftenstated potential, however, CP and patient flow scheduling are still waiting to form an integral part of daytoday practice in many hospitals.
According to Roeder et al. (2003), Kirschner et al. (2007), Küttner and Roeder (2007), and Salfeld et al. (2009), an important obstacle for a widespread utilization of CP is the fact that manually creating CP forms a tedious endeavor. In addition, since CP are mostly used in a descriptive way, only few CP formalisms discussed in the literature enable both the automatic extraction of CP structures from hospital data and their employment for scheduling. Furthermore, many important parameters affecting pathway schedules such as emergency patient arrivals, (exact) treatment durations, resource availability or complications arising in surgeries are subject to uncertainty. Moreover, given that the pathwayscheduling problem has to consider complex constraints as well as all relevant resources, it constitutes a complex largescale optimization problem. As a consequence, many pathwayscheduling approaches rely on simplifications and operate on a high level of aggregation which makes putting the resulting schedules into practice difficult: The approach from Saadani et al. (2014), for example, completely ignores bed resources. Finally, while hospital scheduling typically considers multiple objectives related to economics, patients and staff members, many approaches focus on onedimensional performance indicators, typically based on cost minimization or profit maximization (Hulshof et al. 2012).
Motivated from a case study with a German university hospital, this paper addresses most of the issues discussed above and proposes a tactical pathwayscheduling approach intended to support hospital case managers responsible for the coordination and scheduling of treatments and ward stays. The main contributions of this paper can be summarized as follows.
First, we propose a datadriven approach to pathway scheduling. Our approach is based on scheduling relevant pathway information automatically extracted from standardized hospital billing data available for every hospital in Germany using the pathwaymining approach introduced by Helbig et al. (2015). As a result, one of the most timeconsuming data preparation steps necessary for pathway scheduling—defining pathways and their constraints—is automated to a large extent. To the best of our knowledge, the resulting approach is the first to combine process mining techniques with optimizationbased decision support for scheduling clinical pathways.
Second, we propose a daylevel pathwayscheduling approach considering all pathway constraints and hospital resources on an adequate level of detail which is flexible enough to consider multiple scheduling criteria: it considers multiple ward stays during hospitalization and genderseparated room assignments. Moreover, it explicitly accounts for resource eligibility constraints for treatments, e.g., based on specific feature and skill requirements. In addition, it allows considering important resources individually while aggregating identical resources if appropriate. Finally, to support scheduling pathways for a broad range of possible objective structures, the objective function in our approach involves a weighted combination of multiple hospital and patientrelated criteria including fairness and workloadbalancing criteria.
Third, to tackle the resulting complex largescale scheduling problem for realistic problem instances involving up to 300 patients with a planning horizon of a full month, we propose a twostage mixedinteger linear programming (MILP) approach based on a hierarchical decomposition: The firststage model determines the admission dates and schedules complex treatments with a high duration. Given instructions from the first model, the second model schedules the ward stays, assigns all patients to genderseparated rooms and schedules the remaining treatments. To align the firststage instructions with the secondstage requirements, the firststage model involves anticipation components to account for the gender separation requirement and to consider pathway constraints involving the full set of treatments.
Fourth, we evaluate the overall approach including the automatic CP extraction in a set of experiments conducted with realworld data from a German university hospital involving 286 elective patients and 1088 treatments. On the one hand, we evaluate our hierarchical approach in comparison with a monolithic mixed integer linear programming (MILP) approach: we compare solution times as well as solution quality of both approaches; furthermore, we investigate the effect of employing different degrees of anticipation within the hierarchical approach. On the other hand, we provide a detailed evaluation of two scheduling scenarios: in the first scenario, our approach is used to establish balanced resource utilization throughout the planning horizon; in the second scenario, the main goal is to obtain a high resource utilization at the beginning of the planning horizon.
Note that in this paper, we do not explicitly address the stochastic character of the pathwayscheduling problem: on the one hand, our approach only deals with scheduling elective patients for which most planningrelevant information is known ahead. Nonetheless, one of the goals of the planning approach is to establish balanced resource utilization and resource buffers to ensure that emergency cases can be handled adequately. On the other hand, our approach employs conservative point estimates for uncertain treatment times to obtain schedules which are robust with regard to treatment time variations. Like the other authors dealing with clinical pathway planning (see e.g., Gartner and Kolisch 2014 and Burdett and Kozan 2018), we assume that additional uncertainties such as absence of medical staff, resource breakdowns or stochastic patient arrivals are addressed by embedding our pathwayscheduling approach in a rolling planning process.
The remainder of this paper is structured as follows. Section 2 provides an overview of the rich literature dealing with offline operational scheduling in hospitals and identifies possible objectives, efficient modeling techniques and aspects to be considered in clinical pathway scheduling. Section 3 introduces the constraintbased CP concept according to which the scheduling is carried out and sketches the automated CP mining approach mentioned above. In Sect. 4, we introduce our hierarchical MILP approach consisting of an aggregated firststage model for scheduling complex treatments and determining the admission day and a detailed secondstage model for scheduling the remaining treatment and assigning patients to rooms. In Sect. 5, we explain the realworld dataset from a university hospital used in our computational experiments. In addition, we illustrate the results from mining CP for the hospital’s largest department. Section 6 presents the results from using our MILP approach for scheduling the extracted CP for all elective patients of the department of urology. To demonstrate the effects of varying the weights in the multicriteria objective function, we discuss two scenarios: in the first scenario, the resource allocation is balanced across the planning horizon; in the second scenario, resources are preferably allocated at the beginning of the horizon. Section 7 concludes the paper and points to further research opportunities.
2 Related work
Overview of related work
Resources  Related articles 

Operating Room  Jebali et al. (2006), Denton et al. (2007), Lamiri et al. (2007), Chaabane et al. (2008), Lamiri et al. (2008a, b), Cardoen et al. (2009a), Fei et al. (2009), Testi and Tànfani (2009), Denton et al. (2010), Roland et al. (2010), Batun et al. (2011), Cardoen and Demeulemeester (2011), Marques et al. (2012) and Clavel et al. (2017) 
Beds  Demeester et al. (2010), Ceschia and Schaerf (2011), Schmidt et al. (2013), Helm and Van Oyen (2014) and Vancroonenburg et al. (2014) 
Treatments  Vlah Jerić and Figueira (2010, 2012) and Schimmelpfeng et al. (2012) 
Operating room and beds  Pham and Klinkert (2008), Cardoen et al. (2009b), Augusto et al. (2010), Fei et al. (2010), Chow et al. (2011), Banditori et al. (2013), Sun et al. (2013), Vancroonenburg et al. (2013), Ceschia and Schaerf (2014) and Li et al. (2015) 
Clinical pathways  Vissers (2005), Conforti et al. (2011), Helbig (2011), Gartner and Kolisch (2014) and Saadani et al. (2014) 
Since surgeries form the greatest source of income for most hospitals (Denton et al. 2007) and at the same time operating rooms (OR) constitute the most expensive type of resource using more than 10% of a typical hospital budget (Jebali et al. 2006; Chaabane et al. 2008), there is a rich body of literature dealing with surgery scheduling. The main goals of surgery scheduling are to reduce costs (overtime costs, penalties for idle time or fixed costs for opening an OR) and to increase the utilization of the available OR. This is achieved by either allocating surgeries to a given set of OR (Lamiri et al. 2007; Lamiri et al. 2008a, b; Fei et al. 2009; Denton et al. 2010; Riise and Burke 2011; Meskens et al. 2013), sequencing a set of given surgeries (Denton et al. 2007; Cardoen et al. 2009a; Cardoen and Demeulemeester 2011) or both (Jebali et al. 2006; Testi and Tànfani 2009; Roland et al. 2010; Batun et al. 2011; Marques et al. 2012; Clavel et al. 2017). In most cases, mathematical programming, heuristics, column generation or combinations of these methods are used to find a good or even optimal solution with respect to a typically multicriteria objective function. With regard to the scope of the present article, the main insights from the surgery scheduling literature can be summarized as follows: Overtime should be avoided and the wellbeing of the medical staff should be considered (Roland et al. 2010; Meskens et al. 2013), a flexible pooling strategy of OR has significant benefits (Batun et al. 2011), the eligibility of operating rooms for surgeries may depend on special operating room equipment (Denton et al. 2007) and the patients’ welfare should be taken into account (Testi and Tànfani 2009). Regarding the solution methods, carefully decomposing the problem into solving multiple small problems often yields a good performance in terms of solution time without having a significantly negative impact on solution quality (Jebali et al. 2006; Lamiri et al. 2007; Lamiri et al. 2008a, b; Fei et al. 2009; Denton et al. 2010; Batun et al. 2011; Marques et al. 2012). For a recent survey on OR scheduling, see Samudra et al. (2016).
As stated by Helm and Van Oyen (2014), a proper bed management is necessary to avoid having to dismiss patients because of blocked beds, surgical cancelations and operational chaos. In the literature, two kinds of bed management problems are discussed both of which typically consider medical needs as well as patient preferences: elective admission scheduling and the patienttoroom assignment. Elective admission scheduling problems are solved, e.g., by local search heuristics (Ceschia and Schaerf 2011), MILP (Helm and Van Oyen 2014) or random forest models (Schmidt et al. 2013). Demeester et al. (2010) solve the patienttoroom assignment problem using a tabu search heuristic. Vancroonenburg et al. (2014) develop an integrated model to solve both problems at once to increase both planning flexibility and operational efficiency. Genderseparated room assignment is typically considered as a crucial issue, see Demeester et al. (2010, Ceschia and Schaerf (2011), Schmidt et al. (2013), and Vancroonenburg et al. (2014).
According to Schimmelpfeng et al. (2012), scheduling treatments (encompassing not only surgeries but all types of medical activities) manually often leads to an inefficient resource allocation and can have negative effects on quality of care and patient satisfaction. To avoid this, Vlah Jerić and Figueira (2010) develop a decision support system (DSS) to support the construction of a daily schedule of medical treatments considering available resources and other criteria. They use a scatter search heuristic to construct a set of Paretooptimal solutions among which the user can interactively choose. Another approach by the same authors presented in Vlah Jerić and Figueira (2012) determines daily treatment schedules considering the availability of medical equipment and physicians. Their approach is based on a multiobjective binary programming formulation for which different solution approaches such as variable neighborhood search, scatter search and nondominated sorting genetic algorithms are compared. Schimmelpfeng et al. (2012) outline a conceptual framework for a DSS to support the scheduling process of treatments in a rehabilitation hospital. To handle realistic problem dimensions, a hierarchical approach is developed using a daily model for aggregated longterm scheduling and an intraday model (either time or resourceoriented) for shortterm scheduling.
As argued by Chow et al. (2011), trying to achieve a high OR utilization without considering further surgeryrelated resources often results in issues such as staff overtime, surgical cancelations and long surgical waiting times. In particular, scheduling surgeries without taking the availability of recovery beds into account often leads to problems. To avoid this issue, several authors propose to simultaneously schedule both the surgery and the following stay in a recovery bed or on a ward. For instance, Pham and Klinkert (2008) propose to allocate hospital resources to individual surgical cases divided into preoperative, perioperative (= surgery), postoperative intensive care unit (ICU) modes and formulate this problem as a MILP multimode blocking job shop model. Cardoen et al. (2009b) solve a surgical case scheduling problem in a daycare facility. They formulate a multiobjective model to minimize peaks in recovery bed usage, the occurrence of recovery overtime and violations of staff and patients’ preferences. Their approach is based on a combination of column generation and dynamic programming. To analyze the impact of recovery in OR when no recovery bed is available, Augusto et al. (2010) use a Lagrangian relaxationbased approach. Their results show a high benefit of this improved flexibility in resource usage, in particular in case of high demands for recovery beds. Fei et al. (2010) construct weekly surgery schedules for OR considering recovery beds to maximize resource utilization while minimizing overtime and idle time between surgeries. They use a hierarchical twostage solution approach which first computes the date of the surgeries based on a set partitioning formulation solved by a column generation technique. In the second step, taking into account the availability of recovery beds the daily surgery scheduling problem in which the sequence of surgeries is determined is formulated as a flowshop problem and solved by a hybrid genetic heuristic. The authors show that their overall approach allows reducing idle and overtime increasing resource utilization. Chow et al. (2011) use Monte Carlo Simulation to predict bed requirements from historical data. Based on this, they formulate a MILP to reduce peaks in bed occupancy by scheduling surgeon blocks and patient types. Another simulation optimization approach for the surgery scheduling problem aiming at a maximal patient throughput is developed by Banditori et al. (2013). The authors first use a MILP model to determine the number of cases to be treated by surgery groups in each time slot of a month. The model is then finetuned with a simulation approach to obtain both robust and easytoimplement schedules. Sun et al. (2013) propose a weekly scheduling approach to achieve a high OR utilization. The authors formulate a MILP taking into account limited key resources like ICUbeds and workload of surgeons. Vancroonenburg et al. (2013) incorporate OR and gender separation constraints in their approach for the elective patient admission scheduling problem. Their goal is to determine how scheduling surgical and nonsurgical admissions impacts room assignments at wards. Ceschia and Schaerf (2014) address a similar patient admission problem considering OR utilization constraints, genderseparated room assignment, a flexible planning horizon and the notion of patient delay using a local search heuristic. Li et al. (2015) develop a MILPbased lexicographic goal programming approach of scheduling OR and beds considering competing resources such as surgical and nursing staff, anesthesiologists and recovery beds.
The most promising way to avoid bottlenecks, idle time and overtime during hospital patient flow is to schedule not only surgeries, possibly augmented with induced bed resources, but each patient’s full CP. Vissers (2005) proposes a master surgery scheduling approach for CP involving cardiothoracic surgeries based on computing the patient mix with a MILP model. He divides the care process into four successive steps (medium care unit (MCU), surgery, ICU, MCU) and computes the master schedule for a planning cycle of 4 weeks taking OR, MCUbeds, ICUbeds and nursing staff for ICUbeds into account. An approach to schedule full CP of elective patients for a week hospital^{1} is introduced by Conforti et al. (2011). The authors show their results from scheduling clinical services (diagnostics and surgeries) as well as patient admissions to maximize the patient flow using a MILP model in a case study with 20 patients. Helbig (2011) presents a MILP model for the operational hourbased scheduling of interdisciplinary CP containing treatments, surgeries, order constraints and ward stays. The aim of the model is to minimize waiting time and to finish each CP as early as possible within the planning horizon under consideration of practical constraints like gender separation in rooms and a fair distribution of waiting time among all patients. The paper reports results from experiments with artificial smallscale problem instances with up to four patients. A patient flow planning approach to maximize the length of stay (LOS)based contribution margin for DRGbased reimbursement policies is developed by Gartner and Kolisch (2014). The authors formulate two MILP models to schedule CP, one with fixed and one with variable patient admission dates. They experiment with these models in a static and in a dynamic setting in which the MILP is embedded in a rolling horizon approach. The results dealing with a planning horizon of 28 days and 150 patients show the potential to achieve a higher contribution margin and a significantly reduced time between admission and surgery compared to given manual solutions. Saadani et al. (2014) propose a MILPbased CP scheduling approach in which treatments using different resources are scheduled to minimize the patients’ length of stay. Their CP contain multiple treatments but no bed requirements. The authors present results for an instance of 20 patients and a planning horizon of 14 days. Recently, Burdett and Kozan (2018) proposed an approach using constructive heuristics and metaheuristics for scheduling CP operating on a very high level of detail: They schedule not only the day, but also the time of day of the treatments. Their approach takes into account all major treatment resources under consideration of resource eligibility constraints; in addition, it considers multiple ward stays and bed resources on wards.

Given the multitude of relevant objectives, the approach should allow to include multiple objective criteria related to both patients and resources:

Important patientrelated objectives encompass the consideration of LOS (the main driver of schedulerelated costs in a DRG system), admission day preferences and hospitalization without delays.

Among the resourcerelated objectives, minimization of overtime and idle time, establishing a balanced workload and achieving a high level of resource utilization are the most important.


To obtain implementable pathway schedules, patients should be assigned to concrete rooms taking gender separation into account.

For an adequate representation of resource requirements, scheduling should consider the eligibility of resources for treatments which depends on resource properties such as special qualifications or equipment. Furthermore, to avoid disaggregation problems as sketched in the introduction, important treatment resources such as OR should be considered individually instead of in an aggregated way.

It is unlikely to find a monolithic approach for exactly solving a CP scheduling problem considering all of the aspects described above. However, decomposing the problem into multiple stages, if carried out carefully, can be a promising approach to achieve good results within acceptable solution times also for largescale instances involving more than 200 patients, a 30day planning horizon and 1000 treatments.
We are not aware of any mathematical programmingbased CP scheduling approach taking into account all the aspects mentioned above. As a consequence, this paper proposes a novel hierarchical MILP approach to schedule whole CP with hospitalwide ward stays able to handle the mentioned aspects for realworld problem instances.
An additional important feature of our approach is its datadriven nature: by employing the pathwaymining approach and the constraintbased representation of CP introduced by Helbig et al. (2015), the structure of the CP to be scheduled can be automatically extracted from standardized data available in every German hospital. It becomes clear from the recent survey articles Yang and Su (2014) and Rojas et al. (2016) that process mining for clinical pathways and in healthcare in general forms a very active field of research. As described in Rojas et al. (2016), however, the results of the process mining are typically used for analyzing and improving healthcare processes using techniques from business process management. Our approach in contrast is, to the best of our knowledge, the first one tightly coupling process mining with mathematical optimization for scheduling clinical pathways.
3 Constraintbased CP representation
Even though the general concept of CP is wellknown and widely accepted, there is no standard formalism for representing CP (Vanhaecht et al. 2006). While the formal representation of a CP is of secondary importance if it serves merely descriptive purposes, it is crucial when CP form the basis of scheduling patient flow. Conforti et al. (2011), for example, consider a CP as a set of treatments to be scheduled without imposing intertreatment dependencies. Gartner and Kolisch (2014) represent the possible schedules for each CP on a directed graph in which the arcs form minimum time lags between the clinical activities. The CP scheduling approach proposed in this work relies on the constraintbased CP representation introduced in (Helbig et al. 2015). We give a brief overview and example of this representation in this section; for details regarding the representation as well as regarding the pathwaymining approach, we refer the reader to (Helbig et al. 2015).
In the CP representation, a ward stay \( s \in {\mathcal{S}} \) is associated with a set of eligible wards \( {\mathcal{W}}_{s} \); this means that for stay \( s \), the patient may be assigned to one of the wards in \( w_{s} \). In Fig. 1, for example, the ward stay \( s_{2} \) may either be assigned to ward \( w_{2} \) or to ward \( w_{3} \). A ward stay is characterized by an admission event \( v_{s}^{\text{ad}} \) and a discharge event \( v_{s}^{\text{dis}} \). The admission event \( v_{{s_{1} }}^{\text{ad}} \) of the first stay \( s_{1} \) corresponds to the hospital admission and always takes place at the first day of the CP. All other ward stay events have feasible time intervals indicating when they can be scheduled (e.g., the discharge \( v_{{s_{1} }}^{\text{dis}} \) can be scheduled from day 3 to day 6, see Fig. 1). The number of days between admission and discharge on a ward forms the length of stay (LOS) on that ward. For every ward stay, the LOS is constrained by a minimum and a maximum number of days. During hospitalization, each ward stay requires a bed resource and it is assumed that there are no gap days between two adjacent ward stays.
On the second level of the CP representation, a treatment t corresponds to a surgery or any other clinical service to be performed to cure a disease. A CP contains a set \( {\mathcal{T}} \) of different treatments. Each of these treatments lasts at most 1 day and can only be performed if the required amount of all necessary resource types (e.g., surgeons, special rooms, theater nurses or the patient) is available. To facilitate scheduling, treatments are clustered into treatment groups; the treatments within the same group \( g \) have to be scheduled at the same day (see e.g., \( t_{2} \) and \( t_{3} \) in Fig. 1). Some treatment types may occur in multiple groups which means that they have to be performed once per group (\( t_{2} \), for example, occurs in two groups). To ensure medically correct schedules, each treatment group \( g \) has a feasible time interval \( \left[ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{\delta }_{g} ;\bar{\delta }_{g} } \right] \) relative to the first admission event. In Fig. 1, this interval corresponds to the length of the bar associated with the treatment group (e.g., \( g_{2} \) has the feasible interval [2;3]). In addition, a medically correct order can be enforced by precedence constraints imposing a minimum and a maximum time lag \( \left[ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{\gamma }_{{g,g^{\prime} }} ;\bar{\gamma }_{{g,g^{\prime}}} } \right] \) between two related treatment groups \( g \) and \( g^{\prime} \) (e.g., in Fig. 1, \( g_{5} \) needs to fall into the interval [1;3] relative to the scheduled day of \( g_{4} \)). Moreover, a group can be associated with a certain ward stay \( s \) which means that the group can only be scheduled during that ward stay (e.g., g_{1} and g_{2} from Fig. 1 can only be scheduled during the first ward stay). Note that by grouping treatments as described, many CPrelated constraints are aggregated and in some cases even considered implicitly. On the one hand, this contributes to a tractable size of the MILP models presented in the next section, on the other hand, the modeling itself is facilitated by the grouping.
4 A hierarchical MILP approach for scheduling clinical pathways
The pathwayscheduling problem considered in this paper consists of determining the patient’s admission and discharge days, the start and end days of the ward stays and the days on which treatments are performed. The scheduling accounts for precedence relations between treatments and considers all critical hospital resources such as beds, medical staff as well as treatment and operating rooms. Bed resources are considered on the level of rooms, accounting for the requirement of gender separation. Treatments are associated with resource requirements. It is assumed that for each requirement there is a set of eligible resources and a given amount of resource time needed for the treatment. Each resource, in turn, is associated with a time capacity for each given day. Identical resources may be considered in an aggregated way; however, to avoid disaggregation problems, important complex resources such as operating rooms are considered individually.
With respect to the level of detail considered, the pathwayscheduling problem addressed in this paper is situated between the problems considered in Gartner and Kolisch (2014) and in Burdett and Kozan (2018): both the problems considered in Gartner and Kolisch (2014) and in this paper schedule treatments on the timeaggregated level of days, but in contrast to the problem addressed here, Gartner and Kolisch (2014) do not consider multiple ward stays, bed assignment to genderseparated rooms and resource eligibility constraints. Burdett and Kozan (2018), in contrast, consider multiple ward stays and resource eligibility constraints such that the level of detail for representing resources is similar to our work. However, the problem addressed in Burdett and Kozan (2018) deals with the exact timing of treatments instead of only determining the day on which a treatment is performed.
Gartner and Kolisch (2014) point out that the pathwayscheduling problem they address is related to the resourceconstrained project scheduling problem (RCPSP), with the important difference that treatments are assumed to take a fraction of a the time unit (in this case: a day) while in project scheduling, activities are typically assumed to take multiple units of time. Using the terminology from Artigues et al. (2015), the model presented by Gartner and Kolisch (2014) can be characterized as a timeindexed formulation, that is, the main decision variables for scheduling activities are binary variables with a day index. The authors assume that each resource requirement is associated with a single set of aggregated resources. As a result, no treatmenttoresource assignment is needed and the resource capacity constraint can be formulated using the timeindexed treatment scheduling variables. The resulting model instances are fairly compact and can be solved to optimality by standard MILP solvers even for realistically sized instances.
Using the analogy to RCPSP, extending the problem addressed in Gartner and Kolisch (2014) by the consideration of treatmentspecific resource eligibility is very similar to the extension of the RCPSP to a project staffing and scheduling problem in which resources are considered as flexible or multiskilled and each task is associated with a set of eligible resources. As discussed in Correia and SaldanhadaGama (2015), this setting adds a whole new dimension to the RCPSP since tasks are not only scheduled but also explicitly assigned to resources. Incorporating resource eligibility in MILP formulations for the RCPSP typically results in additional variables and constraints, see, e.g., the formulation proposed in Correia and SaldanhadaGama (2015) and, as also discussed therein, makes the problems much harder to solve. Note that the modeling framework for project staffing and scheduling problems proposed in Correia and SaldanhadaGama (2015) is based on naturaldate variables, that is, on variables representing start times of tasks. Our formulation presented below employs both naturaldate variables (e.g., for the start day of ward stays) and timeindexed variables (e.g., for resource assignments). Note that Burdett and Kozan (2018) do not provide a MILP formulation of their very detailed CP scheduling problem—their solution approach uses constructive heuristics and metaheuristics.
Instead of resorting to metaheuristic approaches to deal with the computational complexity of the problem considered in this paper, similar to Schimmelpfeng et al. (2012), we employ a twostage MILP approach which can be viewed as a constructional hierarchical decomposition in the terminology of hierarchical planning, see, e.g., (Schneeweiss 1998). The first stage determines the “cornerstones” of the clinical pathway, that is, the patient admission and discharge dates as well as the dates for the complex key treatments of each clinical pathway. Considering the firststage decisions as fixed instructions, the second stage solves the full problem sketched above.
To obtain a highquality solution for the full problem, the firststage model anticipates major aspects of the full planning problem using anticipation model components obtained by the means of aggregation and relaxation. While the first stage does not determine the ward stays, it anticipates ward stay scheduling under consideration of the gender separation requirement on the level of aggregated room types. Furthermore, while it only schedules complex treatments (under consideration of all required treatment resources), to make this scheduling consistent with the pathway constraints involving all treatments, the firststage model also involves all remaining treatment groups (neglecting the respective resource requirements). Before presenting the MILP formulations for the firststage and the secondstage model, we give a short overview of the notational conventions used in the presentation. A full list of all symbols used in the models is given in the Appendix; in addition, all symbols are explained in the description of the models.
4.1 Notation conventions and variable domains
We use calligraphic symbols for representing sets, in some cases refined with descriptive superscripts. As an example, \( {\mathcal{P}} \) denotes the set of patients and \( {\mathcal{P}}^{\text{m}} \) denotes the set of male patients. Descriptive superscripts are also used for variables and parameters; subscripts are reserved for indices. Indices and parameters are in lowercase. Greek letters are used for pathrelative time information (see Sect. 3).
Decision variables are capitalized and we utilize the following conventions to enhance the readability of the models: the letter \( X \) is used for variables for which the value corresponds to a day in the planning period, that is, the domain of these variables is the set \( {\mathcal{D}} = \left\{ {1, \ldots ,\left {\mathcal{D}} \right} \right\} \). Superscripts are used to refer to specific groups of variables, e.g., \( X_{p,s}^{\text{ws}} \) denotes the start day of ward stay \( s \) of patient \( p \). Dayindexed variables are mostly denoted with \( Y \). For example, the binary variable \( Y_{p,w,d}^{\text{ward}} \) indicates whether patient \( p \) stays at ward \( w \) at day \( d \) or not. Bookkeeping variables representing a duration, a slack or a violation of a soft constraint are denoted with \( U \); again specified with a descriptive superscript and possibly with an additional superscript signifying the direction of a violation. As an example, the variable \( U_{p}^{\text{ad + }} \) represents a positive violation from the desired admission date of patient \( p \). If not specified otherwise, the bookkeeping variables are continuous nonnegative variables. The letter \( Z \) is used for integer variables mostly representing assignment decisions—for example, the indicator variable \( Z_{p,s,q}^{\text{room}} \) takes the value of 1 if patient \( p \) stays in room \( q \) throughout her ward stay \( s \).
4.2 Firststage model
In the following exposition, we present the firststage model used to determine the admission and discharge dates of the patients as well as the dates of the (complex) key treatments in the clinical pathways. Note that all other decisions considered in the following model such as the start and end days of the ward stays, the aggregated genderseparated room assignments and the scheduled dates of the noncomplex treatment groups are not transferred to the secondstage model as instructions but merely serve the purpose of anticipation.
4.2.1 Scheduling admission, ward stays and discharge
4.2.2 Aggregated consideration of bed resources and gender separation
4.2.3 Scheduling treatments considering pathway constraints
As noted in Sect. 3, our approach aggregates treatments occurring on the same day form a treatment group. In the mathematical model, the integer decision variable \( X_{p,g }^{\text{treat}} \) represents the day on which the treatment group \( g \) from the set \( {\mathcal{G}}_{p} \) of all treatment groups of a patient \( p \) is scheduled. To anticipate the pathway constraints in the firststage model, the set \( {\mathcal{G}}_{p} \) encompasses all treatment groups (not only those containing key treatments to be scheduled in the first stage).
4.2.4 Modelling treatment resource requirements
To perform the treatments in the aggregated treatment groups, certain resources such as operating rooms, physicians and nurses are needed. As mentioned above, the firststage model only considers the resource requirement for a subset of treatment groups. In the following exposition, this subset is denoted with \( {\mathcal{G}}_{p}^{\text{res}} \).
When it comes to the representation of these resource requirements, most publications dealing with pathway scheduling rely on a simple representation of resources and requirements: resources have a unique resource type (or a unique identifier) and the resource requirements are formulated in terms of time needed per resource type (or per specific resource). In reality, however, the eligibility of resources for treatment requirements depends on a set of resource properties such as qualifications (e.g., the capability of conducting a certain surgery) or features (e.g., special equipment only available in certain operating rooms). In this work, this more complex type of representing resource eligibility is modelled as follows:
We represent the resource requirements on the level of treatment groups. For each group, the requirements of all treatments in the group are aggregated. The set of resource requirements for treatment group \( g \) of patient \( p \) is denoted with \( {\mathcal{K}}_{p,g} \); the amount of resource time needed for the requirement \( k \in {\mathcal{K}}_{p,g} \) is given by the parameter \( a_{p,g,k} \). A resource requirement \( k \) is further associated with a set \( {\mathcal{R}}_{k} \) of eligible resources.
Note that our formulation of resource requirements and capacities given by the constraints (16–19) is valid both for individual and aggregated resources. Resources can be aggregated if they have the same properties, e.g., the same qualifications or features. For example, in our case study, the ORnurses can be considered as an aggregated resource. In case of an aggregated resource \( r \in {\mathcal{R}} \), the resource capacity \( \bar{b}_{r,d}^{\text{res}} \) per day corresponds to the sum of the capacity of all individual resources on that day.
Note, however, that in addition to adequately modelling resources with different features and skills, there is another reason for individually considering (complex) resources: the requirements for these resources often comprise multiple hours and it is not possible to switch the assigned resource during a treatment. Aggregating these resources thus easily leads to a situation where there is no feasible disaggregated allocation to individual resources for an aggregated solution.
4.2.5 Objective function
4.3 Secondstage model
Considering the instructions from the firststage model, that is, the patient admission and discharge dates as well as the dates of the complex treatments, the secondstage model schedules the remaining treatments and performs the detailed allocation of patients to rooms on the wards respecting gender separation constraints.
Besides the fact that in the secondstage model, certain decisions are considered as fixed, the main differences between the firststage and the secondstage model are the level of detail on which the bed resources are modeled and the fact that in the secondstage model, all treatment groups are scheduled under consideration of their resource requirements. As a result, assuming that in the secondstage model, the sets of treatment groups \( {\mathcal{G}}_{p}^{\text{res}} \) contains all relevant treatment groups instead of only the subset of complex key treatments, the constraint sets (1–4) dealing with scheduling ward stays (implying admission and discharge days) and (13–23) mainly dealing with treatment scheduling and the allocation treatment resources also appear in the secondstage model. In the following, we present the structurally different parts of the secondstage model, that is, the constraints for fixing the firstlevel decisions, the model component for representing bed resources on a high level of detail and the objective function in which the bedrelated terms make use of the variables introduced in the secondstage model.
4.3.1 Enforcing the instructions from the firststage model
4.3.2 Bed resources, room assignment and gender separation based on individual rooms
4.3.3 Objective function
5 Realworld case study from a German university hospital
We agreed with the partner hospital to base our analysis and our experiments on the peak month March 2011 exhibiting the highest number of cases. In close collaboration with the case manager, we prepared and validated the input data required for our approach including two sets objective function weights representing two different planning scenarios. The remainder of this section describes and illustrates these input data: Sect. 5.1 deals with the cases and their CP information automatically extracted by the pathwaymining approach sketched in Sect. 3. Section 5.2 describes how the data regarding resource requirements and resource capacities were obtained. Finally, Sect. 5.3 explains the planning scenarios used for the computational experiments.
5.1 Case data and automated pathway extraction
As explained above, we conduct our investigation with the realworld data for cases admitted in March 2011 to the department of urology of a German university hospital. For these cases, we applied the CP mining approach proposed in Helbig et al. (2015) and shortly sketched in Sect. 3 of this paper. Since this mining approach operates with standardized hospital billing data according to the §21KHEntG^{2} which has to be reported by each hospital in Germany, it is readily applicable for each German hospital. The pathwaymining approach identifies similar treatment profiles for each primary diagnosis and creates corresponding homogenous case groups. Based on these groups, feasible intervals for treatments and ward stays as well as order constraints are determined. Note that if only a single case with a certain treatment profile occurs in the database, the mining approach returns treatment chains without scheduling flexibility for treatments and ward stays.
In total, the data in the considered month encompass 286 cases (217 male and 69 female patients) with 90 different primary diagnoses and 69 different DRGs. Applying the mining approach to the described dataset resulted in 229 different CP among which 136 are based on unique treatment profiles and 93 represent profiles occurring more than once. In total, the cases involve 1088 treatment groups and 302 ward stays; around 6% of all cases involve more than one ward stay.
5.2 Resource requirements and availability
Resource time required for OPSCodes (excerpt)
OPScode  Required time on resource type (h)  

Physician  Nurse  OR  ORnurse  Deputy  Urography^{a}  MRT  Anesthetist  ..  
12750  0.5  0.5  ..  
13341  2  2  2  ..  
557840  2 × 0.75  0.75  0.75  0.75  ..  
560401  2 × 4  4  4  4  4  ..  
81372  0.5  1  0.5  0.5  ..  
..  ..  ..  ..  ..  ..  ..  ..  ..  .. 
Since OPSCodes are very detailed and surgeries usually involve multiple codes, considering each individual set of codes would typically result in a unique type of pathway for each surgery. To avoid this, all surgery codes (starting with a “5”) appearing at the same day of a CP are aggregated to a single artificial code. To obtain a robust approximation the resource requirements for such a surgery code, we consider the maximum time a single resource unit was required for any of the OPSCodes aggregated to the surgery code.
This approximation is based on the assumption provided by the case manager that all required resources need to be available for the full surgery duration. This assumption is reflected by all of her resource requirement estimates for individual surgery OPSCodes. Using the maximum duration over all involved OPSCodes, however, tends to overestimate the actual surgery duration: Taking all cases from March 2011 considered in our study, on average the duration of a surgery is overestimated by 4.6 min. In accordance with the case manager from the involved department, we chose this slightly pessimistic estimation procedure for the surgery duration and resource consumption instead of a more optimistic estimate to make the scheduling results more robust against uncertainty with respect to surgery durations.
Resource time required for surgeries (excerpt)
Diagnosis (ICDcode)  CPvariant  Required Resource Time (h)  

Physician  Deputy  ORnurse  OR  Anesthetist  .  
N40  22  2  2  4  2  –  .. 
N40  24  0,75  0,75  0,75  0,75  –  .. 
C61  1  8  4  4  4  4  .. 
..  ..  ..  ..  ..  ..  ..  .. 
The case study department has a capacity of 88 patients during workdays and 73 at weekends. It has 7 single, 5 two, 9 three and 11 fourbed rooms. We assume that small rooms are not used for scheduling elective patients because of their flexibility in handling emergencies. Based on this assumption, we reduced the amount of available beds to 53 by keeping only the 11 fourbed rooms and 3 threebed rooms. With regard to the CP to be scheduled, there are four additional departments in which beds are needed for ward stays. For these departments, we assume a bed capacity based on the observed number patient cases available in form of twobed rooms.
Regarding the treatment resources, we consider scarce and special resources such as operating rooms and the urography individually. On the one hand, this accurately models resource eligibility for treatments requiring special equipment: as an example, the central OR is eligible for 139 treatments whereas the OR of the urology department is only eligible for 66 of these treatments. A more complex example arises in the Urography: There are ten treatments for which both the Uro1 and the Uro2 resource is eligible; however, there are 19 treatments for which only Uro1 and 13 treatments for which only Uro2 is eligible.
On the other hand, considering complex resources individually avoids disaggregation issues such as scheduling three 4h surgeries on a day while there are only two OR available for 6 h on the same day. In addition, complex resources often exhibit different daily capacities since they may be shared among different departments. Other resources such as human resources with identical skills (e.g., ORnurses) are aggregated.
Resource capacity for elective patients (excerpt)
Resource type  Resource  Amount of available time (h)  

Mon  Tue  Wed  Thu  Fri  Sat  Sun  
Physician  58  58  58  58  58  15  10  
OR  Central OR  10  10  10  6  10  0  0 
ORHA2200  5  5  5  5  5  5  5  
ORnurse  24  24  24  24  24  5  5  
Deputy  20  20  20  20  20  0  0  
Urography  Uro1  5  5  5  5  5  5  5 
Uro2  10  10  10  10  10  10  10  
Uro3  8  8  8  8  8  8  8  
..  ..  ..  ..  ..  ..  ..  ..  .. 
5.3 Scheduling scenarios considered in the computational experiments
Objective function weights in the two experimental scenarios
Scenario  \( c^{\text{maxDel}} \)  \( c^{\text{del}} \)  \( c^{\text{ad}} \)  \( c^{\text{bed + }} \)  \( c^{\text{maxOt}} \)  \( c^{\text{maxIt}} \)  \( c_{d}^{\text{ot}} \)  \( {\text{c}}_{d}^{\text{it}} \) 

Smooth allocation  10  2  1  5  10  10  10  2 
Early allocation  10  2  0  5  10  0  10  \( 5 + \frac{{\left( {31  {\text{d}}} \right)^{2} }}{100} \) 
In the second scenario referred to as “early allocation”, the hospital aims at establishing a high level of resource utilization at the beginning of the planning horizon. The scenario can be motivated by the observation from the case manager that the level of uncertainty affecting schedulingrelevant information regarding patient cases and resource availability increases towards the end of the planning horizon. This scenario is particularly useful in a setting where our approach is embedded in a rolling horizon approach. Establishing a high level of resource utilization at the beginning of the planning period aims at providing more flexibility for coping with the higher amount of uncertainty towards the end of the planning period. In addition, freeing future resource capacity can be instrumental for admitting a higher number of elective patients. The set of objective function coefficients for this scenario is depicted in the second row of Table 5; note that in particular, this scenario involves a quadratic decreasing daydependent penalty for the idle time of resources and zero penalties for the maximum idle time and for the deviation from admission day penalties.
6 Experimental results
The realworld data described in the previous section forms the basis of a series of computational experiments with our hierarchical approach. The first set of experiments, discussed in Sect. 6.1, aims at assessing the hierarchical approach proposed in this paper. The second sets of experiments aim at discussing the results of the two scenarios in detail.
For the experiments, the MILP models presented in Sect. 4 were implemented AMPL; all treatments exhibiting a duration of more than 30 min (about 50% of the treatments) were considered to form the set \( {\mathcal{G}}_{p}^{\text{res}} \) of treatment groups to be scheduled in the firststage model. The model instances were solved with Gurobi 6.5 on an Intel i73770 CPU with 3.40 GHz and 8.00 GB RAM using 6 threads. For solving the aggregated firststage model, the relative MIP gap tolerance was set to 1%, the time limit was set to 2 h and the MIP search strategy was set to focus on improving bounds. The secondstage model was solved using default Gurobi parameter settings.
6.1 Evaluating the hierarchical approach
Compared to approaches presented in the literature, our approach addresses a very detailed pathwayscheduling problem considering multiple ward stays, genderseparated room assignment and treatmentspecific resource eligibility. In this section, we first compare the hierarchical twostage approach to a monolithic MILP formulation. Next, we evaluate the effect of ignoring the gender separation requirement on the quality of the solution, and finally, we present experimental results showing the importance of carefully anticipating the secondstage problem in the first stage of our hierarchical solution approach.
6.1.1 Monolithic model vs hierarchical approach
Integrated vs. hierarchical approach
Scenario  Monolithic  Hierarchical  Monolithic vs hierarchical  

\( \bar{b}^{\text{devAd}} \)  Integer (h)  Total (h)  Gap (%)  Objective  Total (m)  Objective  Objective gap (%)  
Smooth allocation  3  7.25  12  1.03  7856  5,2  8344  6.21 
5  11.6  12  4.14  8096  9,1  8299  2.50  
10  –  12  ∞  72,6  8101  –  
Early allocation  3  3.5  3.9  1  25160  0,6  26,030  3.46 
5  –  12  ∞  3,2  25,549  – 
The results in Table 6 show that within the desired solution time of 2 h, the MILP solver was unable to find a feasible solution to any of the instances of the monolithic model. For instances permitting a higher deviation from the desired admission day, this even holds after 12 h of solution time: it was only possible to find integer feasible solutions of allowed admission day deviations of up to 5 days in the smooth allocation scenario and for up to 3 days in the early allocation scenario. Furthermore, only for a single instance, a solution meeting the predefined 1% optimality gap is found within 12 h. For the hierarchical approach, the solution time is less than 10 min for all but one instance for which the solution time is less than 73 min.
The last column in Table 6 aims at evaluating the price at which this considerable decrease in solving time comes: For those instances for which a feasible solution could be found for the monolithic model, it depicts the percentage objective function gap between the solution obtained with the hierarchical approach and the best integer solution found for the monolithic model. It turns out that for this gap is between 2.5 and 6.2%; the highest gap is observed in the smooth allocation scenario with the smallest flexibility regarding the deviation from the desired admission day. The objective function gaps between the monolithic and the hierarchical approach can mainly be attributed to the fact that the first stage model determining the cornerstones of the CP schedule is based on an aggregate representation of room assignments and only considers the resource requirements for complex treatments. However, as will become clear in the next section, in contrast to the monolithic model, the hierarchical approach allows obtaining highquality feasible solutions with a fully flexible admission date within a time limit of 2 h.
6.1.2 The effect of ignoring gender separation
Compared to other pathwayscheduling approaches, our model operates on a higher level of detail. With regard to treatments, our model considers the eligibility of resources based on resource properties (qualifications, available equipment) instead of characterizing resources using a single resource type (nurse, operating room) and stating requirements in terms of these resource types. While the advantage of this more realistic representation for matching treatments and resources cannot be evaluated experimentally in a straightforward way, in the following, we discuss results from comparing different levels of detail of representing the allocation of bed resources. To evaluate the relevance of considering genderseparated rooms, we compare the solution obtained with our approach (a priori consideration of gender separation) to a solution obtained by first scheduling pathways ignoring the gender separation requirement and afterwards determining a genderseparated room assignment is performed based on these schedules (a posteriori consideration of gender separation).
Amount of additional beds respecting and ignoring gender restrictions
Instance  Considering gender a priori  Considering gender a posteriori  

A priori  A priori  A posteriori  
Extra beds  Objective  Extra beds  Objective  Extra beds  Objective  
Given capacity  0  24,284  1  24,194  3  24,294 
Reduced capacity  19  25,414  16  25,181  28  25,781 
6.1.3 The role of anticipation in the hierarchical approach
Comparing the amount of extra beds
\( \bar{b}^{\text{devAd}} \)  With gender anticipation  Without gender anticipation  

Extra beds  Objective value  Extra beds  Objective value  
10  26  26,019  37  26,577 
30  19  25,414  29  26,053 
6.2 Smooth resource allocation scenario: detailed results
Solving time, gaps and objectives in the smooth resource allocation scenario
\( \bar{b}^{\text{devAd}} \)  Solution time 1st model (s)  Gap (%)  Objective  Solving time 2nd model (s)  Gap (%)  Objective 

0  4  1  9228  20  0  9066 
3  310  1  8493  1  0  8212 
5  543  1  8435  1  0  8150 
10  4352  1  8264  1  0  7919 
30  3069  1  8253  1  0  7898 
Key indicators in the smooth resource allocation scenario
Indicator  \( \bar{b}^{\text{devAd}} \)  

0  3  10  30  
Maximum patient delay  3  5  0  0 
Total patient delay  46  23  0  0 
Maximum overtime  
OR  6  1.8  0.7  0.7 
ORnurse  2  0  0  0 
Uro  3.5  0.75  0.2  0 
Maximum idle time  
OR  12  7.8  4.2  4.7 
ORnurse  19.2  10.2  8.7  7.7 
Uro  11.5  10.3  10.5  9.5 
Total overtime  
OR  36.8  2  0.85  1.75 
ORnurse  3.6  0  0  0 
Uro  4.2  1.25  0.23  0 
Total idle time  
OP  74.9  40.5  39  39.9 
ORnurse  135.7  132.8  132  132 
Uro  192.7  189.9  188.8  188.6 
Extra male beds  1  0  0  0 
Extra female beds  1  0  0  0 
For each resource, the allocation obtained by our approach is very smooth compared to the empirical data. The resource allocation of the physicians shows an even workload distribution. Given the large blue area, considering only elective patients, physicians and ORnurses obviously do not form scarce resources. Regarding the combined allocation profiles of all operating rooms, the results obtained by our approach exhibit a better fit to the available capacity profile with less overtime occurring during the month compared to the empirical data. Since there is no idle time for the central OR, the visible available capacity can be attributed exclusively to the wardowned OR.
To quantify the smoothing effect obtained by our scheduling approach, let us consider the standard deviation of the daily allocation of each resource: For physicians, the standard deviation change from 12.7 to 7.1, for operating rooms from 7 to 4, for ORnurses from 8.4 to 5.2 and for the urography from 4.6 to 2.2. Given these values, it can be summarized that the resource allocation is successfully smoothed in this scenario. Furthermore, even under consideration of gender separation, no extra beds are required to hospitalize all 286 elective patients.
6.3 Early resource allocation scenario: detailed results
While the scenario presented in the previous section aimed at obtaining a smooth allocation of resources, the present section focuses on a scenario in which scheduling favors a high resource utilization at the beginning of the planning horizon. As in the first scenario, we vary the maximum allowed deviation from the preferred admission day by changing the parameter
Solving time, gaps and objectives aiming on early high resource allocation
\( \bar{b}^{\text{devAd}} \)  Solving time 1st model (s)  Gap (%)  Objective  Solving time 2nd model (s)  Gap (%)  Objective 

0  1  1  27,982  13  0  27,384 
3  34  1  26,490  1  0  26,030 
5  190  1  26,062  4  0  25,549 
10  870  1  25,160  24  0  24,679 
30  4488  1  24,726  506  0  24,186 
Key indicators in the high early allocation scenario
Indicators  \( \bar{b}^{{{\text{dev}}Ad}} \)  

0  3  10  30  
Maximum patient delay  6  1  0  0 
Total patient delay  45  2  0  0 
Maximum overtime  
OP  7  3.8  2.4  4.7 
ORnurse  1.3  0.4  0.1  0.3 
Uro  3.5  0.2  0.4  0.7 
Total overtime  
OP  39.1  13.1  8.6  14.8 
ORnurse  3.6  0.8  0.1  0.3 
Uro  4.5  0.3  0.2  0.3 
Total idle time  
OP  77.2  51.3  46.9  52.9 
ORnurse  135.7  132.8  132.2  132.4 
Uro  193.2  189  189.6  190 
Extra male beds  1  0  4  2 
Extra female beds  1  3  1  0 
Summarizing the results of this scenario, the desired maximization of the resource allocation at the beginning of the planning horizon can be achieved. However, putting the described results into practice may turn out to be risky since in case of a rate of arriving emergency patients higher than the expected ratio of 40%, capacity bottlenecks will arise at least during the first week. Nevertheless, our scheduling approach can be used to identify such bottlenecks and to take measures to avoid them or to mitigate their consequences. In the current case, operating rooms and ORnurses seem to be the scarcest resource types. This becomes particularly obvious after scheduling, showing noticeable available capacity only during the last days of the planning horizon.
7 Conclusion
This paper presents a hierarchical MILP approach for CP scheduling aiming at practical applicability in several ways. The CP information including the pathway constraints can be automatically extracted from standardized hospital billing data. The datadriven character of our approach substantially reduces the amount of work to be performed to obtain the CP information needed for scheduling. Moreover, the CP scheduling approach considers all relevant resources on an adequate level of detail; in particular, it models bed assignment on the level of rooms and allows considering treatment resources individually to avoid disaggregation problems of complex resources such as operating rooms. Furthermore, it considers several practically relevant aspects such as multiple ward stays per hospitalization, genderseparated rooms and eligibility of resources for treatments (e.g., special equipment in operating rooms needed for a certain surgery or special qualifications of nurses). To support a broad range of application scenarios and hospitalspecific objective structures, our approach employs a multicriteria objective function accounting for both patient and hospitalrelated objectives. Since the detailed representation of practically relevant aspects yields complex largescale model instances, we propose a carefully designed hierarchical twostage approach involving anticipation components aligning the firststage solution with aspects relevant to the secondstage problem.
The experimental results based a case study with realworld data involving a planning horizon of 30 days, 1088 treatments and 302 ward stays of 286 patients show that our approach is capable of obtaining highquality solutions within a reasonable amount of time: even the most complex instances in which the admission days of the patients can be freely chosen can be solved within less than 90 min. Furthermore, the detailed evaluation shows that our model is capable of considering different scenarios such as achieving a smooth allocation of resources over the planning horizon and establishing a high resource allocation in the beginning of the horizon.
The results presented in this paper offer multiple opportunities for future research: The robustness of the schedules obtained with our approach could be assessed by a simulation study accounting for sources of uncertainty such as such as emergency patients, stochastic treatment times and shortterm resource unavailability. Moreover, evaluating our approach in additional settings would be instrumental for understanding in how far the results presented in this paper can be generalized. From a practical perspective, the main goal of our efforts is to embed our model into the information systems of our partner university hospital to provide decision support for hospitalwide scheduling of clinical pathways. While we are aware that this goal is presently out of reach mainly due to the present organizational structure in hospitals and due to a lack of integrated information systems, we believe that providing an approach that is datadriven and considers many practically relevant aspects is a step in the right direction.
Footnotes
 1.
A week hospital is a hospital where the duration of all hospitalizations is at most 1 week.
 2.
More details about structure and containing data as well as an example dataset can be found at: http://www.gdrg.de/cms/Datenveroeffentlichung_gem._21_KHEntgG.
Notes
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