Operating Room Scheduling Problems

Operating Room Scheduling Problems: A Survey and a Proposed Solution Framework

Zakaria Abdelrasol, Nermine Harraz and Amr Eltawil

Abstract Healthcare is becoming one of the fast growing industries in both, the developed and developing countries. Operating rooms provide a large portion of hospital revenue; hence, scheduling operation room surgeries is very important to maximize profits. This paper reviews the three operating room scheduling problems: the case mix problem, the master surgery scheduling problem and the surgery scheduling problem. Also, the paper introduces a research framework for an integrated planning method for the three problems. Keywords Case-mix, healthcare management, literature survey, master surgery scheduling, operating room scheduling, operations research, simulation, surgery scheduling.

Z. Abdelrasol Department of Industrial Engineering and Systems Management, Egypt-Japan University of Science and Technology (E-JUST), Alexandria, Egypt. On leave from Industrial Engineering Department, Faculty of Engineering, Fayoum University, Fayoum, Egypt e-mail: [email protected] N. Harraz . A. Eltawil Department of Industrial Engineering and Systems Management, Egypt-Japan University of Science and Technology (E-JUST). On leave from the Production Engineering Department, Faculty of Engineering, Alexandria University, Alexandria, Egypt N. Harraz e-mail: [email protected] A. Eltawil e-mail: [email protected]

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Introduction Healthcare is becoming one of the largest industries in the developed and developing countries. Like in many other countries all over the world, hospitals in the developing countries have faced multiple challenges in the past decade, such as the occurrence of new diseases and much more budgetary constraints. Nowadays, there are many challenges facing healthcare systems, such as the limited resources, the high cost of medical technology and medication, the high demand, the high customer expectations and the shortage in planning and management decision support tools specially with the complexity of healthcare systems. In addition to these challenges, hospitals in developing countries are facing more challenges as the limited government support. Consequently, hospitals are more and more aware of the need to use their resources as efficiently as possible, which urges healthcare organizations to increase emphasis on process optimization in order to control and minimize operating costs and improve the provided services levels. According to the "Healthcare Financial Management Association"(HFMA), Operating rooms (ORs) present a large share of hospital care services and expenditure, at the same time, ORs result in an estimated 40% of hospital revenue. This makes the scheduling of ORs an important problem to study in order to meet hospital goals. In this concern, it is necessary to schedule the ORs, in such a way that the operations/surgeries are carried out with maximal efficiency. The increase in efficiency of ORs schedule has a bearing of the number of surgery cases, the cost or profit, utilization of resources and waiting time of patients, which, in turn, are widely accepted indicators of ORs efficiency. The problem of ORs scheduling can be divided into three different and related sub-problems as shown in Fig. .1, namely (i) the Case Mix Problem (CMP), (ii) the Master Surgery Scheduling Problem (MSSP), and (iii) the Surgery Scheduling Problem (SSP). The CMP refers to the time of a resource (e.g., ORs) allocated to each surgical specialty in order to minimize the total costs or maximize the total revenues or how the available ORs time is divided over the different surgeons. This stage takes place on the strategic level of hospital management as it determines for which ailments capacity will be preserved for a long time horizon. In MSS problem, the ORs time is allocated to these surgical specialties over the scheduling window (typically, a week) in order to maximize and level resources utilization. Finally, SSP refers to assigning each surgical case a start time, a day, and an OR with the target of minimizing the waiting time and maximizing resources utilization. More details and examples about the three ORs scheduling problems and the relation between them can be found in Abdelrasol et al. [1]. There are many review papers that discussed the ORs scheduling problems [21], [14], and [13]. In an recent work for the authors, [1] the three operating room scheduling problems were reviewed. Furthermore, a research framework for an integrated planning method for the three problems was introduced. Moreover, the framework included a broad system

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Strategic Level-CMP Determining the time of ORs dedicated to each surgical specialty.

Tactical Level-MSSP Allocating surgical specialties to ORs.

Operational Level-SSP Selecting and sequencing patients to be served in each OR. Fig. .1 The hierarchy of the three ORs scheduling problems and decision levels

dynamics model to analyze the relationship between the different hospital's departments and the operating rooms. In this chapter we have extended the work, by considering more references especially in the MSSP section, supporting the work with additional figures, and making some revisions. The aim of this book chapter is threefold. First, it presents a short literature review and a classification of ORs scheduling problems. The literature review and classification is structured depending on the three stages that can be distinguished in developing ORs schedules; CMP, MSSP, and SSP followed by the integration of these three problems. Fig. .3 summarizes these four sections. Second, we present a gap analysis and define a recommended research direction. Third, the proposed research framework to tackle the defined problem is presented. The remainder of this chapter will be structured as follows. Section 2 contains a focused literature on the first ORs scheduling problem, CMP. In Section 3 the second ORs scheduling problem will be reviewed, MSSP. The subsequent section reviewed the third ORs scheduling problem, SSP, followed by a discussion of the integrated work for the three problems. Section 6 provides a gap analysis, followed by presenting the proposed approach framework. Finally conclusion section has been provided.

The Case Mix Problem (CMP) This stage takes place on the strategic level of hospital management.The literature on this level is relatively sparse. The available literature can be categorized into two groups according to demand certainty. The first category contains the papers with deterministic demand which represent the large portion of the CMP literature so it will be discussed in details; the related papers are summarized in Table .1. The second category considers the variation and uncertainty on surgical demand even from a statistical viewpoint or as a newsvendor problem.

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The CMP with Deterministic Demand The literature in this category is summarized in Table .1. Further details can be found in Abdelrasol et al. [1]. Table .1 Case mix problem references summary Ref.

Authors/Year

Objective Function

Considered Resources

Techniques

[32]

Ma and Demeulemeester (2013).

Maximize overall financial ORs’ Time, Beds. contribution.

ILP.

[41]

Testi et al. (2007).

Maximize total benefit.

ORs’ Time

BPP Binary LP.

[31]

Ma et al. (2009).

Maximize profits.

ORs’ Time, Beds.

ILP and B&P.

[12]

Blake and Carter (2002).

Minimize profit deviation. ORs’ Time, Beds.

[36]

Mulholland et al. (2005).

Maximize hospital total margins plus professional payments.

ORs’ Time, ICU LP. Beds, Wards’ Beds.

[28]

Kuo et al. (2003).

Maximize professional revenues.

ORs’ Time.

LGP.

LP.

Minimized value for hospital costs.

The CMP with Uncertain Demand In the CMP and due to the uncertainty in demand or SGs workloads, the optimally allocated time must balance the costs of allocating too much time, which typically translates to idle time for ORs and staff, with the costs of allocating too little time, which typically translates to overtime charges. This issue has been tackled by two approaches; from a statistical viewpoint [35] and by addressing the CMP as a newsvendor problem [45].

The Master Surgery Scheduling Problem This problem is addressed at the tactical level and it is concerned with the development of the Master Surgery Schedule (MSS), which usually is cyclically constructed for a given planning period (usually 1–3 months to one year). An MSS defines the allocated Time Blocks (TBs) of each OR to several SGs every day.

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Historical data and actual/forecast demand (e.g., in terms of waiting list and appointment requests for surgeries) are used as input. The decision of dedicating ORs time to SGs can be taken by one of three strategies [38]; block-scheduling, modified block scheduling and open-scheduling. In block-scheduling, a set of TBs is assigned to specific SGs, generally for some weeks or months. Surgical cases are arranged in TBs and none of these can be released. On the contrary, open-scheduling allows assigning surgical cases to an available OR, at the convenience of surgeons. An empty schedule is filled up with surgical cases by following the order of arrival time. As a fundamental difference from block-scheduling, in open-scheduling the surgeons could choose "any workday" for a case. For this reason, open-scheduling strategy is sometimes called "any workday" strategy. These points make the open-scheduling strategy more flexible than block-scheduling. In modified block scheduling, as an integration of the block and open scheduling, the block-scheduling strategy can be modified in two ways to increase its flexibility; Some TBs are booked and others are left open, or unused TBs are released at some time before surgery [16]. However block and open scheduling strategies are mentioned clearly and applied widely in the literature, no one mentioned directly that he/she applied modified block scheduling in MSSP. Next, the most recent papers for both block-scheduling and open-scheduling are discussed.

The MSS with Block-Scheduling Strategy Block-scheduling strategy has been applied to construct the MSS by many authors. Most recently, Ma and Demeulemeester [32] choose the optimal patient mix,then they constructed a balanced MSS in order to improve the patient service level by minimizing the total expected bed shortage. Originally Testi et al. [41] firstly solved a bin packing-like problem in order to select the optimal case-mix. Then they applied block-scheduling strategy for determining optimal time tables. Although, many cancellations take place because the intrinsic stochastic nature of the system, a few number of authors dealt with stochastic issues. For instance; Houdenhoven et al. [26] proved that the actual OR utilization can be increased by applying the Regret-Based Random Sampling algorithm. They aimed to minimize the planned slacks while surgical case durations vary according to a normal distribution. Furthermore, the stochastic Length Of Stay (LOS) for the Intensive Care Unit (ICU) and Medium Care Unit (MCU) are considered by Adan et al. [2]. They compared their approach with that of Vissers et al. [44] which considered deterministic LOSs for ICU and MCU. As an extension to the classical MSSP, few authors integrated the problem with either operational problems like; the Surgical Case Assignment Problem (SCAP) or the nurse scheduling problem. In the SCAP, surgeries are allocated and assigned to each session. Patients are selected from the waiting lists according to several parameters, including surgery duration, waiting time and priority class of the operations. Testi and Tànfani [42] proposed a model that determines, during a

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given planning period, the allocation of ORs' blocks to surgical sub-specialties, i.e. the so called MSSP, together with the subsets of elective patients to be operated on in each block time, i.e. the so called SCAP. Recently, Agnetis et al. [3] proposed a decomposition approach for the same integration problem. In regards to the integration with nurse scheduling problem, Beliën and Demeulemeester [6] presented a model that integrates both the nurse and the ORs scheduling process in the tactical level. The block-scheduling literature can be further classified based on the solution approaches. There are three main approaches applied to solve this problem; Mathematical models, simulation models and combined approaches as illustrated in Fig. .2.

Block-Scheduling Solving Approaches

Mathematical Modeling Approach

Simulation Modeling Approach

Combined Approaches

Fig. .3 Classification of block-scheduling solution approaches

Many authors applied mathematical models in order to get a closed form solution. Originally, Blake et al. [10] proposed an IP model to tackle the MSSP. They described a hospital’s experience using the mathematical technique of IP to solve the problem with the target of minimizing the shortfall between each group’s target and actual assignment of OR time. Recently, Mannino et al. [33] tackled the MSSP and focused on balancing patient queue lengths and minimizing resort to overtime. To cope with these problems they introduced a new MILP formulation and showed its beneficial properties. Furthermore, they develop a light robustness optimization approach in order to consider surgery demand uncertainty. Regarding the second approach, originally, Harris [24] developed a simulation model to aid decision making in the area of operating theatre time tables and the resultant hospital bed requirements. Thereafter, simulation modeling approach has not been applied as a stand a loan solving approach. Very recently Banditori et al. [5] used a simulation model in order to develop their proposed combined optimization–simulation approach. Finally, combination approaches usually gather and utilize the power of many approaches. Blake and Donald [11] proposed an IP model and a post-solution heuristic in order to allocate operating room time. The developed heuristic has been applied to improve the schedule produced by the IP model. In order to gain the power of both mathematical modeling approach and metaheuristics, Beliën and Demeulemeester [7] developed a number of MIP based heuristics and a Simulated Annealing (SA) algorithm to minimize the expected total bed shortage for cyclic MSSP. Posteriorly, this model is adopted and

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embedded in the decision support system which has been presented in Beliën et al [8]. In the same line, the column generation (CG) approach has been combined with mathematical modeling in order to tackle the MSSP with uncertainty. Oostrum et al. [37] proposed a two-phase decomposition approach to deal with the uncertain duration of procedures. First, they constructed MSS plan by MILP containing probabilistic constraints. Then, they proposed a CG heuristic to maximize the operation room utilization and levels the requirements for subsequent hospital beds. Similarly, Holte and Mannino [25] proposed an implementer/adversary algorithm by combining a MILP model with a CG algorithm. The general model has been applied to compute a most robust MSS plan. To take the advantage of simulation modeling approach especially with the existence of randomness, a combined optimization–simulation approach has been applied to fine tune the optimization model to trade-off robustness and efficiency [4]. First, they presented a MIP model considering the cases’ due dates and with the objective function of maximizing the patient throughput. Secondly, they illustrated the results of a simulation study through which they tested the model solution’s robustness against the randomness of surgery duration and the length of stay.

The MSS with Open-Scheduling Strategy With regard to open-scheduling, there are no TBs that can be reserved for a particular surgeon. Many authors applied open-scheduling strategy. Recently Liu et al. [30] applied only the open-scheduling strategy and developed a heuristic algorithm. This algorithm comes from the dynamic programming (DP) idea by aggregating states to avoid the explosion of the number of states. The objective was to maximize the operating rooms’ use efficiency and minimize the overtime cost. Fei et al. [19] modeled the problem of weekly ORs scheduling as ILP and solved it by a column-generation-based heuristic procedure. Then the daily scheduling problem has been treated as a two-stage hybrid flow-shop problem and solved by a hybrid genetic algorithm. The main objective functions were; minimizing the idle time and overtime and maximizing the ORs utilization.

The Surgery Scheduling Problem (SSP) The third and final hierarchical stage is mainly devoted to define a schedule of elective surgeries (i.e., off-line scheduling). Few papers considered on-line scheduling, aimed at modifying an existing schedule given urgent and emergency arrivals. The literature in this level is very large, it can be classified according to the solution technique to: mathematical and heuristic models and simulation

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models. Firstly very recent papers which considered mathematical or heuristics model are presented, followed by a detailed discussion about simulation studies.

Mathematical and Heuristics Models for the SSP Vijayakumar et al. [43] addressed a SSP experienced at a publicly-funded hospital and conceptualize this multi-period, multi-resource, priority-based case scheduling problem as an unequal-sized, multi-bin, multi-dimensional dual bin-packing problem. A mixed integer programming (MIP) model and a heuristic based on the first fit decreasing algorithm are presented. Fei et al. [19] and Jebali et al. [27] considered surgeon agendas and recovery beds availability in open-scheduling systems. As a trial to consider the stochastic nature of SSP, Denton et al. [15] tackled the uncertainty in surgery duration by developing a two-stage stochastic MIP model, for a daily schedule of single OR. The authors assumed that the total OR schedule duration is known in advance, whereas uncertainty in surgery duration has been represented by a set of scenarios. Hans et al. [22] presented another contribution dealing with uncertainty in surgery times by applying Simulated Annealing (SA). The emergency surgeries are not taken into account, which has been considered by Lamiri et al. [29].

Simulation Models for the SSP Due to the stochastic nature of SSP, simulation models have been increasingly recognized as a valuable tool to handle this problem. The stochastic nature stems from many factors, for example, the dynamic arrivals especially for emergency cases, and the stochastic operating time. This tendency is confirmed by the scientific contributions reviewed in what follows. Simulation models are widely used in the SSP literature. The simulation studies in this problem can be classified with regards to its purpose to many categories. The main two categories are; system performance evaluation and new heuristics evaluation. There are many papers published in those two groups. Firstly, recent papers from the first category will be presented followed by the second category contributions in more details. With regard to system performance evaluation, most recently Ma and Demeulemeester [32] have been applied a simulation model to evaluate the operational performance of the case mix and capacity decision that was obtained from CMP and MSSP under the uncertainty condition. Moreover, they studied the effect of variation on bed capacity and LOS. Another example can be shown in [39] where a discrete event simulation model is developed to evaluate the performance of ORs activity scheduling.

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The use of simulation models to evaluate the performance of sequencing rules has been considered in many references. A summary of these papers can be found in Table .2. More details can be found in Abdelrasol et al. [1]. Table .2 Simulation work with scheduling rules summary Ref.

Authors (Year) Objective Functions

Assumptions

[41]

Testi et al. (2007).

Only elective case considered. LWT, LPT, SPT.

Number of operations (Yearly). Overruns and shifted operations.

Scheduling Rules

Uncertainties demand and LOS.

Bed utilization rates.

Beds and OT resources considered.

[17]

Dexter et al. (1999).

ORs utilization.

Elective Cases.

10 rules.

[34]

Marcon and Dexter (2006).

Over utilized OR time.

Deterministic durations.

7 rules.

[4]

Arnaout and Sevag (2008).

Minimizing the makespan. Stochastic operation duration. LEPST, LEPT, SEPT. SDS.

[23]

Harper (2002). ORs occupancy level.

Completion time. Percentage days with delay.

Beds occupancy level.

Demand Uncertainty. Variability in LOS.

FCFS, LTF, STF, LTSC.

The Integration of the Three Problems As mentioned previously, the ORs scheduling problem can be divided into three separated, yet related problems called; CMP, MSSP and SSP. Every decision made at a given problem influences those of the next one. This specific issue has been explicitly studied in only a few papers. In this section, only papers which integrated the three problems will be discussed. To the best of our knowledge, there are only two references that have been integrated the three problems [32] and [41]. Originally, Testi et al. [41] develop a three-phase, hierarchical approach for the three ORs scheduling problems. Most recently, Ma and Demeulemeester [32] proposed a multilevel integrative approach to the ORs planning problem. It consists of three stages, namely the CMP, the MSSP and the operational performance evaluation phase.

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Fig. .3 OR Scheduling problems classification summary

Current Gaps in the Literature According to the literature, there are many issues that should be tackled. These issues will be classified according to the problem which they are related to.

The Case Mix Problem Related Gaps 1. Generally, the literature in the CMP is sparse and the issue of uncertain cases demand has been tackled only by statistical approaches and is still an open direction to be tackled with stochastic models. 2. Due to the nature of CMP, its solution should be integer numbers. Large number of variables results in a huge integer program even for a hospital of regular size and no commercial ILP software (e.g., ILOG CPLEX) can solve it effectively [32]. Presenting a dynamic programming and meta-heuristics models for this problem, represents an interesting field for further contributions.

The Master Surgery Scheduling Problem Related Gaps 1. Although, MSSP has attracted significant research interest; few authors introduced probabilistic constraints for tackling the uncertainty mainly in surgery demand and LOS.

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2. In open scheduling, there are two decisions to be taken. Routing and scheduling rules can be imported from production management field and adapted to ORs environment, such rules can be found in Shalaby et al. [40]. 3. Despite open-scheduling strategy is more flexible and tends to find a better assignment of the surgical cases than the block one [18], the open-scheduling systems are rarely adopted in healthcare [20]. 4. Finally, the modified block-scheduling strategy has the advantages of both block-scheduling and open-scheduling as a combination of them, nevertheless, it has been rarely applied in literature.

The Surgery Scheduling Problem Related Gaps 1. The mathematical models presented in the SSP literature are generally difficult to be solved to optimality. Consequently, in order to address instances of large size and more realistic assumptions, future research should be devoted to developing efficient heuristic solution approaches. 2. There is a need for tackling more realistic assumptions. For example, to the best of our knowledge, no one considered breakdowns and resources uncertainty. 3. Proposing new scheduling rules considering more technical factors, for example, risk based sequencing, surgeon preferences based sequencing or a decreasing order of act severity, represent interesting fields for further contributions. 4. Finally, on-line scheduling of urgent cases and the consequent rescheduling of elective surgeries represent interesting fields for further research.

The Integration of the Three Problems Related Gaps 1. To the best of our knowledge, only two papers integrated the three problems [32] and [41]. Only one of them applied feedback loop to re-optimize the three decisions [32]. The integration problem is still an open area to be solved simultaneously.

A Proposed Research Framework In this section; a proposed framework to tackle the ORs scheduling problems will be presented. Generally, this proposal aims to consider more realistic assumptions to present efficient solutions to the considered problem. The framework can be

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divided into three stages as shown in Fig. .4. In the first one, the CMP will be tackled, followed by surgeries allocation and sequencing in the second stage. Finally, investigate the results of integrating the three ORs scheduling problems by different integration approaches.

The First Stage-The Case Mix Problem This work will be the first attempt to solve the CMP with the objective of minimizing the total cost and maximizing the total profit under demand uncertainty. A dynamic programming (DP) approach will be developed to solve this problem. DP is a mathematical technique that is based on the idea of separating a decision making problem into smaller sub-problems [9]. Stochastic DP model will presented to consider the demand uncertainty. The optimal case mix solution will be considered as a constraint while solving the next stage problems.

The Second Stage-Surgeries Allocation and Sequencing In this stage, there are to sub-problems; allocation problem (Case Assignment Problem) and sequencing problem. In the first one, cases are allocated to an OR among the available ORs. Modified-block scheduling and open scheduling strategies will be used to allocate elective cases to ORs, while emergency cases will be allocated by different routing rules. In the second sub-problem, cases which allocated to ORs will be sequenced according to several sequencing rules. Taking into account many realistic assumptions like, dynamic arrivals, stochastic times and equipment breakdowns, a simulation model will be built to capture the system and to test the proposed scheduling rules.

The Third Stage- The Three ORs scheduling Problems Integration In this stage, integration models will be built to solve the three ORs scheduling problems simultaneously. The purpose of this integration models is to link the three problems in downstream by the hierarchy relationship and also in upstream by feed backing. The objective is not only to optimize the performance of single problem but is to optimize the three ones.

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Case-Mix Problem (Dynamic Programming Model)

Optimally

Heuristic

Number of ORs sessions / category to Max. Profit/ Min. Cost.

Elective

Emergency

(Simulation Model) Surgery Allocation Problem by;  Modified Block-Scheduling  Open -Scheduling  Routing Rules

Best schedule of patients to;  Max. Resources Utilization  Min. Overtime  Min. Waiting Time (WT).

Compare Results (WT) with International Standards – Feedbacks

Integrate other ORs Hospital Units and IntegrateORs thewith Three Scheduling Departments; Problems. (System Dynamics Model)

Surgery Scheduling Problem by;  Scheduling Rules

Fig. .4 Basic flow chart for the proposed research frame work

Conclusion This paper illustrated a comprehensive review for the different classe of operating room scheduling problems. Also it introduced a general research framework to solve the operating rooms scheduling problems. The proposed framework is based mainly on operations research approaches. The expected model is a generic solution that can be adapted to different cases with objective to maximize short term as well as long term profits.

Acknowledgments This research project is sponsored by the Egyptian ministry of higher education grant and the Japanese International Cooperation Agency (JICA) in the scope of the Egypt Japan University of Science and Technology (E-JUST).

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