Modeling of Multi-Level Planning of Shifting Bottleneck Resources Integrated with Downstream Wards in a Hospital

Featured Application

The proposed model in this research is significant for hospitals for effective and efficient planning and scheduling of their resources. It has practical significance in a hospital where the whole system’s performance is affected by one bottleneck (i.e., either the operating room, ICU, or ward), and this bottleneck keeps changing in different planning horizons.

Abstract

Planning and scheduling critical resources in hospitals is significant for better service and profit generation. The current research investigates an integrated planning and scheduling problem at different levels of operating rooms, intensive care units, and wards. The theory of constraints is applied to make plans and schedules for operating rooms based on the capacity constraints of the operating room itself and downstream wards. A mixed integer linear programming model is developed considering shifting bottleneck resources among the operating room, intensive care unit, and hospital wards to maximize the utilization of resources at all levels of planning. Different sizes of planning and scheduling problems of the hospital, including small, medium, and large sizes, are created with variable arrivals and surgery durations and solved using a CPLEX solver for validating the developed models. Later, the application of the proposed models in the real world to develop planning systems for hospitals is discussed, and future extensions are suggested.

1. Introduction

Providing high-quality and affordable health care is one of the greatest challenges faced by the world’s nations. In developed countries, the aging populations are straining the healthcare budget, while many third-world countries face constant hardships in the form of low life expectancy, high infant mortality due to harsh environmental conditions, and scarce medical resources. The healthcare system must become more efficient at delivering care and preventing disease to solve these human and economic health problems [1]. A very important way to improve the efficiency of a healthcare system is by improving the planning and scheduling procedures in healthcare facilities for the efficient use of the critical resources of hospitals, including operating rooms, intensive care units (ICU), wards, patients, doctors, nurses, etc. Specifically, the operating room is considered one of the most critical units of hospitals in terms of cost and profit generation.

Hence, the current study is focused on the planning and scheduling problems of operating rooms.

Planning and scheduling problems in healthcare facilities are divided into three different levels including strategic, tactical, and operational level planning [2]. Strategic level planning is also called higher-level planning and is based on forecast and surgical demands, where the operating room time is allocated to different surgical specialties. Tactical level planning is also termed medium-level planning, and it involves the division of allocated time for different surgical specialties and preparing a master surgery schedule. Operational level planning is also termed lower-level planning, where the daily plans of the operating room are made and patients are sequenced within the allocated operating rooms. Some articles investigate decision-making regarding operating room planning and scheduling in hospitals at four hierarchical decision levels: strategic, tactical, offline operational, and online operational. The offline operational level deals with assigning patients to dates and sequencing the patients in the operating rooms and involves monitoring and controlling the schedule execution [3,4].

The literature contains studies of the different levels of planning and scheduling problems to improve the performance of healthcare facilities [5,6,7]. For example, McRae and Brunner [8] studied case mix planning decisions for higher-level planning and proposed a mixed integer linear programming model to solve this problem. Freeman et al. [9] proposed an iterative approach for case mix planning decisions. They developed mathematical models to generate a pool of candidate solutions and used a simulation method to evaluate the effectiveness of each candidate solution for higher-level and lower-level performance measures, i.e., the overutilization of operating rooms and utilization of downstream wards. The case mix planning problem was also investigated by Yahia et al. [10]. They developed a stochastic model to find the optimal case mix of patients considering surgery duration, length of stay, and arrival time as uncertain variables.

Medium-level planning problems have also been studied in the literature. For example, Santos and Marques [11] developed a master surgery schedule and proposed a stochastic programming model to integrate the medium-level decision-making of operating rooms with downstream departments such as intensive care units and wards. Spratt and Kozan [12] studied the master surgery scheduling problem on a medium level along with the surgical case assignment problem. They proposed a mixed integer nonlinear programming model and hybrid metaheuristics to solve the problem. Additionally, Fügener et al. [13] studied the medium-level planning problem to calculate the demand distribution of downstream departments for a given master surgery schedule in a hospital.

Lower-level planning has also been investigated in the literature. For example, Pham and Klinkert [14] studied operation room planning and the scheduling problem as a generalized job shop scheduling problem. They developed a mixed integer linear programming model to minimize the makespan of surgeries at a lower-level planning horizon. Oliveira et al. [15] studied the problem of elective patient scheduling at a lower-level planning horizon to maximize operating room utilization based on patient prioritization. They developed an integer programming model and tested it on case hospital data. Younespour et al. [16] considered the scheduling problem of patients at a lower level to minimize the overtime costs, makespan, and completion time costs of surgeons considering parallel surgeries. They proposed a mixed integer programming model and constraint programming models for the scheduling problem.

All the planning level decisions are interrelated, and the solution of one level influences the solution of other levels. In the literature, all these planning levels are dealt with in a hierarchical manner where the output solution of one planning level is moved to the next lower planning level as input. All three decision levels are linked to each other so that decisions made at the strategic level influence the quality of decisions made at the tactical level. Both the higher-level decisions influence further operational-level decisions. Such interdependence of these decision levels provides the basis for research to investigate all these levels concurrently and solve the problem of multi-level planning and scheduling.

Some researchers have focused on multi-level planning in healthcare facilities. For example, Ma and Demeulemeester [17] investigated all levels of decision-making, such as case mix planning, master surgery scheduling, and patient sequencing, at the operational level in a hierarchical manner. All three planning levels were integrated iteratively in three phases with different objective functions for each phase. In addition, Fügener [18] integrated higher-level planning and medium-level planning decisions. They proposed a method to calculate the distribution of patient demand in downstream departments and a model to generate a master surgery schedule with an assumption of fixed capacities. Guido and Conforti [19] studied planning and scheduling problems at medium and lower levels. They proposed an algorithm for allocating operating rooms to specialty groups for a pre-determined period and scheduling patients in the allocated operating rooms. They developed a cyclic master surgery schedule where the patients were allocated to operating rooms based on the developed master surgery schedule.

Reviewing the literature on operating room planning and scheduling shows that the importance of all levels of decision-making can be inferred. In the literature, however, when efforts to integrate any of two or three levels are made, and integration occurs in a hierarchical manner, it proceeds that the results of one planning level are used to make decisions at the lower planning level. Hierarchically, the quality of decisions made at the lower level is greatly determined and influenced by the higher-level decisions. All levels’ performance can be increased if all decision-making levels are integrated. This integration can help to improve the robustness and flexibility of operating room schedules [3]. To increase the performance of the healthcare system, the integration of all three decision-making levels is important, and, hence, the present study is focused on integrating such levels.

In addition, most of the literature has considered the capacity of constraint resources of the hospital during planning and scheduling decisions. For example, the planning and scheduling of the operating theatre complex have gained much attention, as it is considered one of the hospital’s critical resources and its performance has a significant impact on other departments [20,21]. An in-depth review of the literature on operating theatre planning and scheduling is available in many articles [3,22,23]. The literature on operating theatre planning and scheduling has covered many perspectives to improve the operations of the healthcare facility, considering maximizing the utilization of operating theatre [9,15,24,25] and minimizing the patient waiting time [26], surgeon overtime [16], the cost of operating theatre [27], the makespan [28], the overtime [29], etc. Roshanaei and Naderi [30] integrated the problem of patient allocation to a day in an operating room and sequenced the allocated patients within the operating rooms to maximize the total scheduled surgical time. They developed mixed integer and constraint programming models and various bender decomposition algorithms to solve the developed models to optimality. The authors of [31] studied the operating room planning and scheduling problem integrated with downstream wards. They proposed a two-stage artificial bee colony algorithm for solving the problem. In [32], researchers developed a robust optimization model that combines staffing and scheduling decisions to minimize the impact of variations in surgery duration, staff availability, and emergency arrivals.

In most of the literature, the planning and scheduling of surgeries focus on the optimal use of the available operating theatre capacity. However, the performance of healthcare facilities is constrained not only by the capacity of the operation theatre complex but also by the capacity constraint of other critical resources of hospitals [33]. For example, in hospitals, scheduled surgeries can be canceled in large numbers due to the unavailability of beds for post-operation recovery [34]. Furthermore, in most hospitals, the scheduling systems consider the available capacity of beds, while most surgery planning systems used in hospitals consider the capacity limit of the operating theatre. Since there is an interconnection between the operation theatre complex, the available number of beds in wards, and the available number of beds in the ICU, it is, therefore, significant to study the capacity consideration of different interlinked resources, including the capacity of the operating theatre, the number of available beds in the ICU, and the number of available beds in the wards, etc. Therefore, the capacity consideration of critical resources without consideration of their interconnection with other resources may lead to their suboptimal use [35]. Thus, independent optimization of the operating theatre’s resources or the ward’s optimization may not lead to global optimization of the healthcare system. Therefore, some researchers have addressed the operating theatre planning problem with other critical resources, such as the ICU and wards, at a different level of the planning problem [11,36,37].

Testi et al. [38] developed a three-phase hierarchical approach for the weekly scheduling of operating rooms where the optimal case mix is identified in the first phase to maximize the overall benefit. In the second phase, they developed a master surgery schedule. In the third phase, they performed a simulation to evaluate the operational performance of the master surgery schedule and the utilization of downstream departments. Chow et al. [39] proposed a simulation and mixed integer programming model to integrate and improve surgical scheduling and predict ward and leveling bed occupancy, respectively. Fügener et al. [13] investigated a medium-level planning problem to calculate the demand distribution of downstream departments for a given master surgery schedule. They developed a method to optimize the master surgery schedule and analyzed the impact of resulting block allocation on the bed requirements in the ICU and general wards. Fügener [18] integrated the strategic and tactical master surgery scheduling while considering the impact on downstream resources such as the ICU and general patient wards. The study aimed to increase hospital earnings by optimizing the master surgery schedule. Freeman et al. [9] studied the case mix planning problem with consideration of the resources of downstream wards. They developed mathematical models to generate a pool of solutions with different case mix plans and employed a simulation to evaluate each solution for the overutilization of the operating theatre and variability in bed usage in downstream wards. In their multi-phase solution approach, they developed four mixed integer programming models for case mix planning, block allocation, and master surgery scheduling, and used simulations to assess the quality of the generated solution. In another article, Santos and Marques [11] developed a master surgery schedule and proposed a stochastic programming model to integrate the medium-level decision-making of operating rooms with downstream departments such as intensive care units and wards. In their approach, they estimated the bed requirements at the operational level and developed master surgery schedules based on these estimates of bed requirements.

In most of the literature, the planning and scheduling problems did not consider the interconnection of the critical resources with the other interlinked wards. For a feasible and realistic solution to the health care system, the interconnection of the operating room with the ICU and ward cannot be ignored, and the optimization of the operating theatre alone may result in the underutilization or congestion of other resources and may lead to infeasible schedules. Hence, it is proposed in the present study that the optimal utilization of operating rooms integrated with the ICU and ward can lead to feasible and optimal solutions and enhance the performance of the healthcare facility as a whole because the operating rooms also pace the activities of the linked units. Optimizing the utilization of the operating room alone without considering the capacity constraints of downstream units may result, in some cases, in the underutilization of these units and, in other cases, congestion in these downstream units, which ultimately leads to early discharge or even surgery cancellations. For the balanced use of all the hospital resources, it is necessary to consider the planning and scheduling of operating rooms in combination with interconnected downstream wards. This work, therefore, aims to integrate higher-level, medium-level, and operational-level planning and scheduling along with the integration of the planning and scheduling of operating rooms with downstream units such as beds in the ICU and ward. Such a problem of the simultaneous consideration of all decision levels and critical resources of the healthcare system is new and, to the best of the authors’ knowledge, has not been addressed so far in the literature.

In the literature, the theory of constraints was used to improve the overall system performance while focusing on only the critical resource [40,41]. The theory of constraints applies the drum buffer rope method to identify the resource with limiting capacity, called the drum, and a rope mechanism to provide the planning information to upstream resources of the drum for effective planning and execution. To fully utilize the drum resource, buffers in the form of time and material are provided to prevent the drum from starvation. The drum buffer rope method has been applied successfully in different planning and scheduling problems [42,43,44,45]. For example, Ronen et al. [46] applied the drum buffer rope concept for aircraft scheduling, Gilland [43] used the DBR method in a serial production line for production planning and control, Pegels and Watrous [47] applied the DBR method for the optimization of assembly shops, Sirikrai and Yenradee [45] used the DBR method on production planning and control, and Georgiadis and Politou [42] used the DBR method to solve production planning problems; they introduced a dynamic DBR approach for a time buffer for production planning and control in two machine-capacitated flow shops. Saif et al. [44] recently applied DBR concepts for multi-level integrated production planning and scheduling problems in a flow shop considering industry 4.0 concepts. However, in the literature for planning and scheduling on multiple planning levels in the healthcare industry, the DBR method of the theory of constraints is rarely found; for example, [45,48] used the theory of constraints and identified the bottlenecks among the human resources, such as anesthesiologists, doctors, and nurses, which has limited its application in operating room planning. In the literature, no study is reported on the method in which planning problems either on a single decision level or in an integrated manner are solved using the theory of constraints. The current research used the theory of constraints method to solve multi-level planning and scheduling problems in hospitals, integrating all decision levels and critical resources of the healthcare system. To the best of the authors’ knowledge, the theory of constraints has not yet been applied to the multi-level planning and scheduling of critical resources of hospitals while considering the capacity constraints of downstream wards. It is considered for the first time in the literature in the current study.

In the literature, mathematical programming, particularly mixed integer linear programming, has been widely used for planning and scheduling in healthcare facilities [3]. In mathematical programming, minimizing, or maximizing objectives are formulated subject to the constraints related to the considered problems. The developed model is then solved using standard software such as CPLEX [49,50] or by developing exact solution methods [17,51]. Further, heuristic methods, metaheuristics [12,52], and mathematical modeling and analytical procedures such as the Markov Decision process and queuing theory have also been used for the planning and scheduling of operating rooms [53,54,55,56]. Careful analysis of the research methodologies employed by various researchers reveals that most studies focus on obtaining the optimal solution to the formulated problems using heuristics or exact solutions. In the case mix level, the authors developed mixed integer programming and stochastic models in a study and used a sample average approximation and simulation to solve the models [8]. Many others developed mathematical models and heuristics to solve a case mix planning problem [9,10]. On the tactical level, mathematical modeling and exact method or heuristics are used to solve the developed models [11,12,13]. On the operational level, mathematical modeling and heuristics are employed to solve operating room allocation and sequencing problems [14,15,16]. On integrated levels, the methodology employed is similar [17,18]. For example, Roshanaei and Naderi [30] solved the integrated problem of patient allocation to the operating room, day, and sequencing of patients using mixed integer programming and constraint programming and developed a heuristic algorithm to solve the developed models. However, heuristics and metaheuristics give near-optimal solutions to the considered problem and use the constraints presented in the proposed model of the problem. The exact methods for linear programming models provide an accurate solution to the problem, which is significant for validating new mathematical models. Since the current problem is new in the literature and our study proposes a mathematical model for it, the present study uses CPLEX to solve the problems used for its validation. The CPLEX solver can be used to code and solve integer programming, mixed integer programming, multi-objective optimization, and quadratic programming problems. However, for large-sized problems, it requires more computational time, data handling from various data sources, and a greater number of parameters to define the problem.

The objective of the current study is to develop and solve a mathematical model using the theory of constraints for operating room planning and scheduling considering all the decision levels, such as the strategic, tactical, and operational levels, and integrating them with downstream wards.

The current study contributes to the planning and scheduling literature in the following ways.

• The current research is new in integrating all the planning levels of the hospital, considering the higher-level, medium-level, and lower-level planning considering constraints of the interlinked resources, including the operation theatre, ICU, and wards;
• The current research is new to applying the theory of constraints for multi-level planning in hospitals;
• The current research proposes a new mixed integer linear programming model for multi-level planning and scheduling in hospitals considering the theory of constraints concept;
• The current research develops a new mixed integer linear programming model considering the capacity constraints of the operating room, ICU, and wards.

The rest of the paper is organized as follows: Section 2 presents the problem description and mathematical model. Section 3 and Section 4 present the solution approach, computational experiments, and results. Finally, Section 5 shows the conclusion of the research with important findings and highlights the limitations and future extensions of the work.

2. Problem Description

In a healthcare facility, elective and emergency patients flow through different departments. The general flow of patients moving through critical units of a hospital is illustrated in Figure 1. Figure 1 indicates that elective patients are added to the waiting list while emergency patients are sent to the planning process and assigned to the required resources. It further explains the flow of patients from one resource such as the operating room to the other resources such as the ICU and ward. For capacity planning during higher-level planning decisions, the forecasted patients are considered. It is considered that the patients can directly come for treatment in the operating room (OR), intensive care units (ICU), or wards.

It can be seen from Figure 1 that the patients arriving in the OR have the possibility to move into the ICU or the wards or can be discharged after treatment in the OR; the patients arriving in the ICU can move into a ward or can be discharged after treatment in the ICU. Current research considered the multi-level planning of patients on critical resources. The process considered in the current research for the multi-level planning of critical resources is presented in Figure 2.

Figure 2 illustrates that the higher-level planning involves the planning of patients on each critical resource of the hospital. At a higher level, the patients are allocated to the planning horizons based on their arrival, considering the infinite capacity of all critical resources. At the medium level, the patients from higher levels are allocated to their required critical resources, including the operating room, ICU, and ward. Then, according to the theory of constraints, the bottleneck resource is identified. The bottleneck resource is identified based on patients’ workload on all resources and utilization of the resources. After identifying the bottleneck resources, at the lower level, the respective planning and scheduling model is performed, and a plan is released which is communicated to the other upstream departments and resources. All other resources follow the released plan and scheduling of the bottleneck resources. Figure 2 further explains the patients for each resource; for example, the patients in the ICU include the patients from the operating room and randomly arriving patients. The mathematical relations for the multi-level planning and scheduling of patients on each critical resource are presented in this section.

2.1. Higher-Level Planning

At a higher planning level, the randomly arriving patients are inserted on different days of the planning horizon. The surgery patients and patients needing the ICU and ward are inserted into the planning horizon as explained in this section.

2.1.1. Allocation of Patients to Operating Room

On the higher level, the available information of arriving surgery patients, such as arrival date, due date, preoperative time, surgery duration, etc., is used to insert the patients into different days of the planning horizon. The capacity of all resources is considered infinite at a higher level, and all patients are placed in suitable planning horizons. Equation (1) explains the objective function of higher-level planning.

$$obj=\mathrm{maximize}\left({\displaystyle \sum _{p=1}^P{H}_{pd\tau }}\right) \forall 0<d\le D \forall \tau$$

(1)

$${\displaystyle \sum _{d=1}^D{H}_{pd\tau }=1 \forall p\in P} \forall \tau$$

(2)

$$H_{pd\tau }=1 \forall p\in P \forall D{D}_p\le d\ge EPS{T}_p \forall \tau$$

(3)

$${\mathrm{EPST}}_p=A_p+t_p^{pre-op} \forall p\in P$$

(4)

The objective function (see Equation (1)) is aimed to maximize the number of patients inserted into each day of the planning horizon. Equation (2) gives the constraint that each patient can be inserted for one day of the planning horizon only. Equation (3) gives the constraint that a patient can be assigned to a planning horizon only if the patient’s earliest possible start time (EPST) is the day d of the planning horizon. It is worth mentioning that $(D{D}_p\le d\ge EPS{T}_p)$ in Equation (3) means that a patient can be inserted into a day equal to EPST or greater than EPST in the planning horizon. It also states that the patient’s due date is considered while assigning a patient to a day. This equation has practical significance when there is a larger number of patients and limited capacity. Equation (4) shows the method to calculate the EPST of a patient. The earliest possible start time of a patient is calculated from the arrival time of the patient and after the time required for the preoperative activities of the patient. For the patients arriving at the ICU and ward, the preoperative time is zero, and patients are inserted into the planning horizon based on their arrival date using Equation (4).

2.1.2. Allocation of Patients to the ICU

The patients who require the ICU in a planning horizon are already current patients receiving care in the ICU, patients from operating rooms, and randomly arriving patients during a planning horizon. The number of patients who need ICU after surgery is determined using Equation (5), where $N{P}_{d\tau }^{oi}$ is the number of surgery patients who require the ICU on day d of the planning horizon $\tau$. The total number of patients who require the ICU on day d in a planning horizon $\tau$ is calculated using Equation (6), where $N{P}_{d\tau }^{i\exp }$ is the expected number of randomly arriving patients in the ICU while $N{P}_{d\tau }^{iocc}$ represents the number of patients already occupying a bed in the ICU.

$$N{P}_{d\tau }^{oi}={\displaystyle \sum _{e=1}^E{\mathrm{Pr}}_{N{P}_e}\times N{P}_e} \forall 0<d\le D, \forall \tau$$

(5)

$$N{P}_{d\tau }^i=N{P}_{d\tau }^{oi}+N{P}_{d\tau }^{i\exp }+N{P}_{d\tau }^{iocc} \forall 0<d\le D, \forall \tau$$

(6)

The surgery patients based on their date of surgery and the randomly arriving patients based on their date of arrival are inserted into the days of the planning horizon.

2.1.3. Allocation of Patients to the Ward

The patients in the ward are those patients who are already receiving care in the ward, patients who will arrive at the ward after surgery, and randomly arriving patients. The number of patients who require a ward bed after surgery is determined using Equation (7), where $N{P}_{d\tau }^{ow}$ is the number of patients who require a ward bed after surgery. The total number of patients that require a ward in a day of the planning horizon $\tau$ is calculated using Equation (8), where $N{P}_{d\tau }^{w\exp }$ is the expected number of patients randomly arriving at the ward, $N{P}_{d\tau }^{wocc}$ is the number of patients already occupying beds in the ward, and $N{P}_{d\tau }^{iw}$ is the number of patients that require a ward bed after the ICU.

$$N{P}_{d\tau }^{ow}={\displaystyle \sum _{e=1}^E{\mathrm{Pr}}_{N{P}_e}\times N{P}_e} \forall 0<d\le D, \forall \tau$$

(7)

$$N{P}_{d\tau }^w=N{P}_{d\tau }^{ow}+N{P}_{d\tau }^{iw}+N{P}_{d\tau }^{w\exp }+N{P}_{d\tau }^{occ} \forall 0<d\le D \forall \tau$$

(8)

The output of higher-level planning is a list of patients for the operating room, ICU, and ward for each day d of the planning horizon $\tau$.

2.2. Medium-Level Planning

Based on the lists of patients from the higher-level planning for each operating room, ICU, and ward, the bottleneck is identified from these three units based on the load of each planning horizon. The percentage utilization of all three units, such as the operating rooms, ICU, and ward, is calculated to identify the system’s bottleneck. The unit with the highest percentage utilization is identified as the bottleneck. The workload of surgery patients in operating rooms can be calculated using Equation (9). The total workload of all operating rooms is calculated by adding the workload of all patients on the day of the planning horizon as given by Equation (9).

$${\mathrm{TWL}}_{od\tau }={\displaystyle \sum _{p=1}^P\left(\left(t_p^{pre}+t_p^{set}+t_p^{sur}+t_p^{cl}\right)\times H_{pd\tau }\right)} \forall 0<d\le D \forall \tau$$

(9)

$${\mathrm{UTL}}_{od\tau }^{\%}=\frac{{\mathrm{TWL}}_{od\tau }}{A{t}_{od\tau }\times N_{od\tau }\times {\eta }_o^{avg}} \forall o\in O_{d\tau } \forall 0<d\le D \forall \tau$$

(10)

$${\mathrm{UTL}}_{id\tau }^{\%}=\frac{N_{id\tau }^{wl}}{N_{id\tau }\times {\eta }_i^{avg}} \forall i\in I_{d\tau } \forall 0<d\le D \forall \tau$$

(11)

$${\mathrm{UTL}}_{wd\tau }^{\%}=\frac{N_{wd\tau }^{wl}}{N_{wd\tau }\times {\eta }_w^{avg}} \forall w\in W_{d\tau } \forall 0<d\le D, \forall \tau$$

(12)

$$N_{id\tau }^{wl}=N{P}_{d\tau }^i \forall 0<d\le D \forall \tau$$

(13)

$$N_{wd\tau }^{wl}=N{P}_{d\tau }^w \forall 0<d\le D \forall \tau$$

(14)

The percentage utilization of operating rooms is calculated as the ratio of the total workload of operating rooms to the total available capacity of all operating rooms, and is given by Equation (10), where $N_{od\tau }$ is the total number of operating rooms on the day $d$ of the planning horizon $\tau$ and ${\eta }_o^{avg}$ is the average efficiency of the operating rooms. Similarly, the percentage utilization of the ICU and the downstream ward is calculated in Equation (11) and Equation (12), respectively. Where $N_{id\tau }^{}$ and $N_{wd\tau }^{}$ are the total number of beds in the ICU and ward, respectively, on the day $d$ of the planning horizon $\tau$; $N_{id\tau }^{wl}$ and $N_{wd\tau }^{wl}$ are the number of occupied beds in the ICU and downstream ward, respectively; and ${\eta }_i^{avg}$ and ${\eta }_w^{avg}$ are the average efficiency of the ICU and ward, respectively. Equations (13) and (14) provide the method to calculate workload in the ICU and ward, respectively.

After calculating the percentage utilization of the operating room, ICU, and ward, the bottleneck is identified. The resource with the highest percentage utilization is identified as a bottleneck on that day of the planning horizon. The following is the condition for a resource to be considered a bottleneck. Consider that R is a set of critical resources consisting of the OT, ICU, and ward, and R* is the resource with maximum utilization and is the bottleneck resource as given in Equation (15)

$$R*=resource with \max {\left\{UTL\%\right\}}_{rd\tau } \forall r\in R=[OT,ICU,Ward]$$

(15)

The condition for any resource to be bottlenecked is represented by the expression given below:

$$\mathrm{if}\{\begin{array}{l}UTL{\%}_r>UTL{\%}_r,where r\in R, r^{\prime }\in [R-r] r=R* for any r\cup r^{\prime }=R\\ UTL{\%}_r<UTL{\%}_r,where r\in R, r^{\prime }\in [R-r] r\ne R* r\cap r^{\prime }=\varphi \end{array}$$

After identifying the bottleneck, the respective planning and scheduling model is triggered, as explained here.

2.2.1. Operating Room Is the Bottleneck Resource (R*)

As a result of the allocation of patients and utilization of the operating room, ICU, and ward, if the operating room is identified as the bottleneck, the model for the operating room planning operates. Figure 3 explains the procedure of multi-level integrated planning when the operating room is identified as the bottleneck.

After identifying the bottleneck, the patients are allocated to the operating rooms considering the available capacity of operating rooms on each day $d$ of the medium-level planning horizon $\tau$. It is worth mentioning that if the available capacity of the operating rooms is less than the required capacity of the operating room, the patients are moved to the next day by considering the patients’ due dates. The patients are then allocated to the operating rooms to maximize the workload of the operating rooms such that the patients of the same specialty are allocated to the same operating room to minimize the operating room turnover time when a patient of one specialty is operated in the same operating room after a patient of a different specialty. The objective function of medium-level planning when the operating room is identified as bottleneck resource R* is given by Equation (16).

$$obj=\mathrm{maximize}{\left(WL\right)}_{od\tau } \forall o\in O \forall o<d\le D \forall \tau$$

(16)

$$W{L}_{od\tau }={\displaystyle \sum _{p=1}^P\left(\left(t_p^{pre}+t_p^{set}+t_p^{sur}+t_p^{cl}\right)\times X_{po}^{d\tau }\right)} \forall o\in O \forall 0<d\le D, \forall \tau$$

(17)

$$W{L}_{od\tau }\le A{t}_{od\tau } \forall o\in O \forall 0<d\le D \forall \tau$$

(18)

$${\displaystyle \sum _{d=1}^D{\displaystyle \sum _{o=1}^O{X}_{po}^{d\tau }}=1 \forall p\in P \forall \tau }$$

(19)

$${\displaystyle \sum _{s=1}^S{U}_{spod\tau }}=1 \forall p\in P \forall o\in O \forall 0<d\le D \forall t$$

(20)

$${\displaystyle \sum _{p=1}^P{U}_{spod\tau }}\le N_{sd\tau }^{all} \forall s\in S \forall 0<d\le D, \forall \tau$$

(21)

$$U_{spod\tau }=1 \forall p\in P_e \forall s\in S_e \forall 0<d\le D, \forall \tau$$

(22)

$${\displaystyle \sum _{o=1}^O{U}_{spod\tau }}=1 \forall s\in S \forall 0<d\le D, \forall \tau$$

(23)

The objective function of the medium-level planning of operating rooms is to maximize the workload of all operating rooms. Equation (17) provides the expression to calculate the workload of each operating room. The objective function of the medium level is significant to increase the utilization of the operating rooms. Greater operating room utilization leads to less idle time and hence reduced costs. Equation (18) puts an upper bound on an operating room’s workload such that the operating room’s workload should be less than or equal to the available capacity on the day $d$ of the planning horizon $\tau$. Equation (19) ensures that each patient is allocated to one operating room and only one day of the planning horizon. Equation (20) ensures that each patient is assigned to one surgeon only. Equation (21) puts an upper bound to the number of surgeries a surgeon performs on a day $d$ of the planning horizon $\tau$. Equation (22) states that only a surgeon of the same specialty can be assigned for the surgery of a patient from a specialty. Equation (23) states that each surgeon can be assigned to one operating room on a day $d$ of the planning horizon $\tau$.

The output of medium-level planning is the allocation of patients to operating rooms and surgeons’ assignment for patients’ surgery. As a result of the allocation of surgery patients to operating rooms, the corresponding number of patients that require beds in the ICU and ward is calculated using Equations (7) and (8), respectively. The number of beds required by the surgery patients in the ICU and ward is fixed, and the information is sent to both the ICU and ward to freeze the capacity for patients in the operating room, which is the bottleneck in the considered planning horizon.

The medium-level plan based on the allocation of patients to operating rooms is released for downstream departments, such as the ICU and ward, to reserve capacity for planned patients and to upstream departments for the preparation of the surgery of planned patients before the actual start of surgery.

2.2.2. ICU Is the Bottleneck Resource (R*)

The planning is performed based on the capacity of the ICU when it is identified as the bottleneck resource. Figure 4 represents when the intensive care unit becomes the system’s bottleneck. In this case, the ICU’s capacity is fixed, and this capacity information is sent through a rope to the operating room. Operating room planning and scheduling now take the capacity information of the ICU, and the patient allocation to the operating rooms is performed based on the capacity of the bottleneck of the ICU.

$$obj=\mathrm{maximize}{\displaystyle \sum _{p=1}^P{Y}_{pi}^{d\tau }} \forall 0<d\le D \forall \tau$$

(24)

$${\displaystyle \sum _{i=1}^I{\displaystyle \sum _{d=1}^D{Y}_{pi}^{d\tau }}}=1 \forall p\in P \forall t$$

(25)

$${\displaystyle \sum _{p=1}^P{Y}_{pi}^{d\tau }}\le N_{id\tau } \forall 0<d\le D \forall t$$

(26)

$$N_{id\tau }^{occ}=N_{i(d-1)\tau }^{occ}-N_{i(d-1)\tau }^{disc}+N_{id\tau }^o+N_{idt}^{\exp } \forall i\in I \forall 0<d\le D \forall \tau$$

(27)

$$N_{id\tau }^{occ}\le N_{id\tau } \forall i\in I_{d\tau } \forall 0<d\le D \forall \tau$$

(28)

$$t_{pi}^{disc}=t_{pi}^{arr}+LO{S}_p^i \forall i\in I \forall p\in P$$

(29)

The following are the objective function and constraints of medium-level planning when the ICU becomes a bottleneck. The objective function given by Equation (24) aims to maximize the number of patients requiring a bed in the ICU.

Equation (25) ensures that on a given day $d$ of the planning horizon $\tau$, a patient can be assigned to one bed only in ICU. Equation (26) ensures that the number of patients requiring the ICU is less than the total number of ICU beds. Equation (27) provides the method of calculating the number of occupied beds in the ICU, where $N_{id\tau }^{disc}$ is the number of beds released after the discharge of patients from the ICU. $N_{id\tau }^o$ and $N_{id\tau }^{\exp }$ are the number of beds required in the ICU for the patients from the operating room and other expected patients. Similarly, based on the information on the available capacity of the ICU, Equation (28) puts an upper bound on the number of beds occupied in the ICU that should be less than or equal to the total number of beds in the ICU. Where $N_{id\tau }$ is the total number of beds in the ICU on the day $d$ of the planning horizon $\tau$. $N_{id\tau }^{occ}$ is the number of occupied beds in the ICU on $d$ of planning horizon $\tau$. Equation (29) provides the patient’s discharge time from the ICU. Based on the available capacity of the ICU, the ICU patients’ schedule is released to the operating room, and operating room planning is performed based on the released schedule of the ICU. The information on the bed occupancy in the ICU is then sent to the operating rooms to plan surgeries as per the available capacity of the ICU.

2.2.3. Ward Is the Bottleneck Resource (R*)

In the integrated planning of the operating room with the ICU and ward, when the percentage utilization of beds in the ward is maximized, the ward becomes the bottleneck. Figure 5 explains the procedure for multi-level planning when the ward becomes the system’s bottleneck. In such a case, the capacity information of the ward beds is provided through the rope mechanism to the ICU and operating rooms, and some portion of the capacity allocation of the operating room becomes fixed. The operating room allocation of patients is then performed based on this capacity constraint of ward beds. The objective function given by Equation (30) is aimed at maximizing the number of patients that require a bed in the ward.

$$obj=\mathrm{maximize}{\displaystyle \sum _{p=1}^P{Z}_{pw}^{d\tau }} \forall 0<d\le D \forall \tau$$

(30)

$${\displaystyle \sum Z_{pw}^{d\tau }}=1 \forall p\in P,0<d\le D \forall \tau$$

(31)

$${\displaystyle \sum _{p=1}^P{\displaystyle \sum _{d=1}^D{Z}_{pw}^{d\tau }}}\le N_w^{d\tau } \forall p\in P \forall \tau$$

(32)

$$N_{wd\tau }^{occ}=N_{w(d-1)\tau }^{occ}-N_{w(d-1)\tau }^{disc}+N_{wd\tau }^o+N_{wdt}^{icu} \forall i\in I \forall 0<d\le D \forall \tau$$

(33)

$$N_{wd\tau }^{occ}\le N_{wd\tau } \forall i\in W_{d\tau } \forall 0<d\le D \forall \tau$$

(34)

$$t_{pw}^{disc}=t_{pw}^{arr}+LO{S}_p^w \forall w\in W \forall p\in P$$

(35)

Equation (31) ensures that on a given day $d$ of the planning horizon $\tau$, a patient can be assigned to one bed only in the ward. Equation (32) ensures that the number of patients requiring a bed in the ward is less than the total number of beds in the ward. Equation (33) provides the method to calculate the number of occupied beds in wards, where $N_{wd\tau }^{disc}$ is the number of beds in wards from where patients are discharged on the day $d$. $N_{wd\tau }^o$ and $N_{wd\tau }^{icu}$ are the number of beds required in wards for the patients from the operating room and ICU, respectively.

Equation (34) puts an upper bound on the wards’ workload such that the number of beds occupied should be less than or equal to the total number of beds in the ward. Where $N_{wd\tau }$ is the total number of beds in the ward on the day $d$ of the planning horizon $\tau$. $N_{wd\tau }^{occ}$ is the number of occupied beds in downstream wards on $d$ of planning horizon $\tau$. Equation (35) provides the patient’s discharge time from the ward. The information on the bed occupancy in the ward is then sent to the operating rooms and ICU to plan surgeries and to make a schedule of the ICU as per the available capacity of the ward beds.

2.3. Lower-Level Planning

At the lower level of planning, the sequencing of patients is performed. Based on the load at the medium-level planning, any unit such as the operating room, ICU, and ward can become the bottleneck. The lower-level planning is performed based on the bottleneck resource at the medium-level planning.

2.3.1. Lower-Level Planning when the Operating Room Is the Bottleneck Resource (R*)

When the operating room becomes the bottleneck, the lower-level planning model for the operating room is triggered. The objective function of operating room planning at the lower level is given by Equation (36).

$${\left(obj\right)}_{od\tau }=\mathrm{minimize}{\left(makespan\right)}_{od\tau } \forall o\in O_{d\tau } \forall 0<d\le D, \forall \tau$$

(36)

$${\left(makespan\right)}_{od\tau }={\displaystyle \sum _{p=1}^P{t}_p^{comp}}\times X_{po}^{d\tau } \forall o\in O \forall 0<d\le D \forall \tau$$

(37)

$$t_p^{comp}\ge (t_{sp}^{arr}+t_p^{set}+t_p^{sur}+t_p^{cl})\times W_{pod\tau } \forall p\in P$$

(38)

$$t_p^{comp}\ge (t_{sp}^{arr}+t_{p{p}^{\prime }}^{sds}+t_p^{sur}+t_p^{cl})\times S_{p{p}^{\prime }od\tau } \forall p\in P$$

(39)

$${\displaystyle \sum _{p=1}^P{S}_{p{p}^{\prime }od\tau }}=1-V_{podt} \forall o\in O 0<d\le D \forall \tau$$

(40)

$${\displaystyle \sum _{p=1}^P{S}_{p{p}^{\prime }od\tau }}=1-W_{podt} \forall o\in O 0<d\le D \forall \tau$$

(41)

$${\displaystyle \sum _{p=1}^O{V}_{pod\tau }}=1 \forall o\in O,0<d\le D \forall \tau$$

(42)

$${\displaystyle \sum _{p=1}^P{W}_{pod\tau }}=1 \forall o\in O \forall 0<d\le D \forall \tau$$

(43)

$$t_{p^{\prime }}^{comp}\ge t_p^{comp}+t_{p{p}^{\prime }}^{sds}+t_p^{sur}+t_p^{cl}-M\left(1-S_{p{p}^{\prime }od\tau }\right) \forall p,p^{\prime }\in P \forall 0<d\le D \forall \tau$$

(44)

$${\left(makespan\right)}_{od\tau }\ge 0 \forall o\in O \forall 0<d\le D, \forall \tau$$

(45)

$$S{t}_p^i\ge D{t}_p^o \forall p\in P^{o\to i}$$

(46)

$$S{t}_p^w\ge D{t}_p^w \forall p\in P^{o\to w}$$

(47)

The objective function given by Equation (36) aims to minimize the makespan of the operating room. Equation (37) represents the method of calculating the makespan of an operating room. The objective function considered at lower-level planning is significant to minimize the completion time of all surgeries in the allocated operating rooms. The minimum makespan leaves time to accommodate more surgeries, thus leading to increased utilization of operating rooms and cost savings. Equation (38) represents the method of calculating the completion time of the surgery of a patient if they were operated on at the first position in the operating room o. Equation (39) represents the method of calculating the completion time of the surgery of patient p’ if they were operated on immediately after patient p in the operating room o. Equations (38) and (39) also state that each patient can be operated on after the arrival of the surgeon for the patient p. Equation (40) states that each patient must precede another patient unless they are operated on last. Equation (41) states that each patient must be preceded by another patient unless they are operated on first. Equations (42) and (43) impose the constraint that only one patient can be operated on as first and last in the operating room o. Equation (44) imposes that each patient’s surgery must be started before the surgery of the previous patient is completed. Equation (45) states that the makespan of an operating room must be non-negative. Equations (46) and (47) provide constraints for the ICU and ward, respectively, when the operating room is the bottleneck resource (R*). These constraints limit the starting time of surgery patients in the ICU and ward. The output of the lower-level plan is the optimal schedule of patients in the allocated operating rooms. The optimal schedule is released to the upstream units of the operating room as well as to the ICU and ward to accommodate the patients as per the capacity of the operating room, which is bottleneck resource (R*) on day d of the planning horizon $\tau$.

2.3.2. Lower-Level Planning When the Intensive Care Unit Is the Bottleneck Resource (R*)

As a result of medium-level planning, if the ICU is identified as the bottleneck, the lower-level planning of the ICU is performed, the schedule is released for ICU, and information is sent through the rope mechanism to the operating room to follow the release schedule of the ICU. Operating room planning and scheduling now take the capacity information of the ICU, and patient allocation and sequencing to the operating rooms are performed based on the capacity of the bottleneck of the ICU.

The patients arriving at the ICU are admitted ICU based on their priority as per the procedure given below, where Q is the set of patients arriving at the ICU. ${\alpha }^q$ is used to represent the relative priorities of the ICU patients in set Q. A patient q is assigned to the ICU if ${\alpha }^q$ is the highest.

$$\mathrm{if}\{\begin{array}{l}{\alpha }^q>{\alpha }^{q^{\prime }} \forall q\in Q, q^{\prime }\in [Q-q] \mathrm{Assign\; ICU}\\ {\alpha }^q<{\alpha }^{q^{\prime }} \forall q\in Q, q^{\prime }\in [Q-q] \mathrm{Do\; not\; assign\; ICU}\end{array}$$

$$\alpha =\frac{c^p}{LO{S}_p^i}$$

(48)

$$D{t}_p^o\le S{t}_p^i \forall p\in P^{o\to i}$$

(49)

$$S{t}_p^w\le D{t}_p^i \forall p\in P^{i\to w}$$

(50)

The relative priority of a patient to the ICU is calculated by Equation (48), where $c^p$ is the criticality weight of the patient assigned by the doctors and $LO{S}_p^i$ is the length of the patient’s stay in the ICU. Equations (49) and (50) represent the extra constraints for the operating room and ward when the ICU is identified as the bottleneck.

Based on the available capacity of the ICU, the ICU patients’ schedule is released to the operating room, and operating room planning is performed based on the released schedule of the ICU.

2.3.3. Lower-Level Planning When the Ward Is the Bottleneck Resource (R*)

In the integrated planning of operating rooms with the ICU and ward, when the percentage utilization of beds in wards is maximized, the ward becomes the bottleneck. Figure 5 explains the procedure when the ward becomes the system’s bottleneck. In such a case, the capacity information of ward beds is provided through the rope mechanism to the ICU and operating rooms, and some portion of the capacity allocation of the operating room becomes fixed. The operating room allocation of patients is then performed based on this capacity constraint of ward beds. The following procedure is adopted for lower-level planning when the ward becomes the bottleneck. The patients arriving for a bed in the ward are allocated a ward bed based on their relative priority as given below.

$$\mathrm{if}\{\begin{array}{l}{\alpha }^q>{\alpha }^{q^{\prime }} \forall q\in Q, q^{\prime }\in [Q-q] \mathrm{Assign\; ward}\\ {\alpha }^q<{\alpha }^{q^{\prime }} \forall q\in Q, q^{\prime }\in [Q-q] \mathrm{Do\; not\; assign\; ward}\end{array}$$

$$\alpha =\frac{c^p}{LO{S}_p^w}$$

(51)

$$D{t}_p^o\le S{t}_p^w \forall p\in P^{o\to w}$$

(52)

$$D{t}_p^i\le S{t}_p^w \forall p\in P^{i\to w}$$

(53)

The relative priority of a patient for the ward beds is calculated by Equation (51), where $c^p$ is the criticality weight of the patient assigned by the doctors and $LO{S}_p^w$ is the length of stay of the patient in the ward. Equations (52) and (53) represent the extra constraints for the operating room and ICU when the ward is identified as the bottleneck.

The optimal sequence of patients in the operating room when the ward is the bottleneck is determined based on the release schedule of the ward, and the optimal schedule of operating rooms is released. Finally, Equations (54) and (55) give the binary domain decision variables and non-negativity conditions of parameters, respectively.

$$\begin{array}{l}{H}_{pd\tau },X_{po}^{d\tau },Y_{pi}^{d\tau },Z_{pw}^{d\tau },W_{pod\tau },R_{pod\tau },U_{spod\tau },S_{p{p}^{\prime }od\tau }\in \left\{0,1\right\}\\ \forall p\in P,o\in O,0<d<D, \forall \tau \end{array}$$

(54)

$$W{L}_{od\tau }\ge 0$$

(55)

3. Solution Methods

In hospitals, any resource among the operating rooms, ICUs, and wards can become bottlenecks in the considered planning horizon. Random samples of different problem sizes with varying lengths of stay in the operating room and arrival patterns of patients for the ICU and ward were created to solve the developed mathematical models. Three different sets of problems were generated considering the operating room, ICU, and ward as the bottleneck resource. The surgery duration data from a real hospital was collected, and more samples were created using a lognormal distribution with the parameters of the real data. The data collected from the hospital contained the data of surgery patients for one year.

Table 1. Characteristics of the different-sized problems.

Problem Size	No. of OR	No. of Beds in ICU	No. of Beds in Ward
Small	3–4	7–10	25–35
Medium	6–8	11–15	45–55
Large	10–12	16–18	80–100

represents the characteristics of the three different-sized problems.

4. Computational Experiments

Various experiments were designed to evaluate the performance of the developed method and validate the models. The three major problem categories were small, medium, and large-sized problems. These problems were further divided based on the patients’ length of stay in the operating room and the identification of the bottleneck resource.

Table 2. Matrix of experiments.

Sr. No.	Problem Size	Surgery Duration	Bottleneck Resource	Number of Instances
1	Small	Small	OR	10
2	Small	Medium	OR	10
3	Small	Large	OR	10
4	Small	Small	ICU	10
5	Small	Medium	ICU	10
6	Small	Large	ICU	10
7	Small	Small	Ward	10
8	Small	Medium	Ward	10
9	Small	Large	Ward	10
10	Medium	Small	OR	10
11	Medium	Medium	OR	10
12	Medium	Large	OR	10
13	Medium	Small	ICU	10
14	Medium	Medium	ICU	10
15	Medium	Large	ICU	10
16	Medium	Small	Ward	10
17	Medium	Medium	Ward	10
18	Medium	Large	Ward	10
19	Large	Small	OR	10
20	Large	Medium	OR	10
21	Large	Large	OR	10
22	Large	Small	ICU	10
23	Large	Medium	ICU	10
24	Large	Large	ICU	10
25	Large	Small	Ward	10
26	Large	Medium	Ward	10
27	Large	Large	Ward	10

represents the experiments conducted to evaluate the developed method and models. A total of 270 experiments were conducted to test the developed models.

The numerical experiments in this section were carried out on a laptop Intel^® Core™ i3-5005U CPU @ 2.00 GHz with a memory of 8.0 GB. The developed models were implemented in the IBM ILOG CPLEX 12.10 optimization studio.

4.1. The Data

For the arrival of elective surgery patients, it is assumed that the elective patients follow a discrete uniform distribution derived from historical data. The data collected from the case hospital consisted of 10,664 patients in operating rooms. Out of these patients, 95% require a ward after surgery and 5% require the ICU after surgery. For each type of patient, the underlying distribution is derived, and the number of patients for each type for the planning horizon is randomly generated using Equation (56) based on the relative frequencies of each type of patient in the historical data where $N{P}_e$ is the total number of surgery patients of specialty, e $NP$ is the total number of surgery patients for all specialties in the considered planning horizon, and $f_e$ is the frequency of patients for each specialty in the historical data.

$$N{P}_e=f_e\times NP \forall e\in E$$

(56)

Patient arrivals in the ICU and ward are assumed to follow a Poisson distribution with a known mean [1]. Patients’ length of stay in the operating room, ICU, and ward follows lognormal distribution [9,36]. The binomial distribution is used to find the number of patients visiting the ICU and ward after surgery. The probability that out of total surgery P patients who had surgery on a given day, the p patients in the ICU or ward can be calculated using the binomial distribution [57]. The formula for binomial distribution is given by Equation (57), where p is the probability that a patient will go to the ICU or ward after surgery. Using Equation (57), the number of patients from each specialty who will move to the ICU after surgery is determined.

$${\mathrm{Pr}}_{N{P}_e}=\left(\begin{array}{c}NP\\ N{P}_e\end{array}\right)p^{N{P}_e}{q}^{NP-N{P}_e}$$

(57)

It is considered that 98% of the patients might follow one of the following paths after surgery. About 95% of patients are admitted to the ward before discharge and around 5% go to the ICU and ward before discharge. Only about 1% are discharged directly from the operating room [13,37]. After generating the number of patients arriving and length of stay data in the operating room, ICU, and ward, the elective surgery patient list is generated by assigning random arrivals and due dates to the patients within the planning horizon. Moreover, the preoperative time, operating room setup time, and cleaning time after the surgery of patients are known parameters and are generated randomly. The patients are inserted into the operating rooms, ICUs, and wards in the higher-level planning horizon. For experimentation using patients’ arrival information, it is assumed that 95% of patients operated in operating rooms are moved to the ward for post-operative recovery, and 5% of patients are admitted to ICU after surgery. Further, for experimentation, the efficiency of all the resources is considered to be 100% while performing the experiments. In practice, the efficiency of a resource can be limited due to the expertise level of deputed staff and the efficiency of the equipment in the operating room, ICU, and ward. Regarding the arrival of patients into the ICU and ward, it is considered that patient arrival follows a Poisson distribution. The arrival patterns considered for small, medium, and large problems are represented in

Table 3. Value of $\lambda$ for different problem sizes.

Problem Size	Poisson $\mathit{\lambda }$ for ICU	Poisson $\mathit{\lambda }$ for ward
Small	2–3	5–6
Medium	5–7	10–12
Large	10–12	15–17

with the mean ($\lambda$) values for the ICU and ward.

4.2. Results

To validate our developed method, computational experiments are performed with different problem sizes and surgery durations. The arrival pattern for the ICU and ward is kept variable to generate scenarios of bottleneck resources among the operating room, ICU, and ward. As a first step, the patients are inserted into higher-level planning horizons based on their planned start time for the operating room and their arrival time for both the ICU and ward. After the higher-level of planning, the patients are allocated to different resources based on their requirements during their stay in the hospital. After the allocation of patients to all resources, based on the workload and utilization of the resources such as the operating room, ICU, and ward, the bottleneck resource with the highest utilization is identified. After identifying the bottleneck resource, the lower-level planning model for the bottleneck resource is run, and the schedule obtained is sent through a rope to be followed by the upstream departments.

Table 4. Summary of results.

Problem Size	Surgery Duration	No. of Patients in the OR	Workload of OR	Utilization of OR	No. of Patients in the ICU	Utilization of ICU	No. of Patients in the Ward	Utilization of Ward
The operating room becomes a Bottleneck
Small	Small	16	1695	93%	5	60%	22	72%
Small	Medium	11	1452	84%	5	59%	10	66%
Small	Large	9	1413	76%	5	58%	18	61%
Medium	Small	33	3542	92%	8	69%	41	82%
Medium	Medium	22	3017	83%	8	68%	36	73%
Medium	Large	21	3190	84%	9	72%	36	71%
Large	Small	43	4905	87%	11	71%	61	68%
Large	Medium	35	5067	88%	12	72%	59	66%
Large	Large	28	5188	90%	11	70%	67	75%
ICU becomes Bottleneck
Small	Small	12	1397	79%	7	87%	21	71%
Small	Medium	9	1224	71%	7	90%	19	63%
Small	Large	9	1302	74%	7	88%	21	69%
Medium	Small	26	2933	76%	11	92%	37	75%
Medium	Medium	20	2882	76%	11	91%	34	68%
Medium	Large	17	3155	82%	11	91%	34	68%
Large	Small	33	3843	68%	15	91%	68	76%
Large	Medium	28	3947	71%	15	95%	63	70%
Large	Large	22	4023	70%	15	91%	63	70%
Ward becomes Bottleneck
Small	Small	14	1550	85%	6	70%	28	94%
Small	Medium	10	1360	71%	5	59%	28	92%
Small	Large	7	1313	68%	5	68%	27	91%
Medium	Small	23	2617	71%	8	66%	45	90%
Medium	Medium	20	2901	76%	9	71%	43	87%
Medium	Large	16	2877	75%	8	70%	43	86%
Large	Small	34	3919	73%	12	76%	82	91%
Large	Medium	28	4017	74%	11	72%	82	91%
Large	Large	23	4151	74%	12	77%	85	94%

represents the summary of results with average values for all samples based on the percentage utilization when the operating room, ICU, and ward are identified as a bottleneck.

The resource with the highest percentage utilization is the bottleneck resource. The percentage utilization is related to the idle time as given in Equation (58) where Idle% is the percentage idleness of the resource. As the percentage utilization increases, the idle time decreases, which in turn leads to cost savings.

$$Idle\%=1-UTL\%$$

(58)

When the operating room is the bottleneck resource, the operating room’s lower-level plan is operated, patients are allocated to operating rooms, and the optimal sequence of operating rooms is identified to minimize the makespan for all operating rooms.

Table 5. Summary results of the lower-level plan when the operating room becomes the bottleneck.

PS	SD	Operating Room		ICU			Ward
		PP	Makespan	POR	OB	EP	POR	PICU	OB	EP
Small	Small	16	1347	1	2	2	15	0	4	3
Small	Medium	11	1242	1	2	2	10	1	5	4
Small	Large	9	1386	0	2	2	8	1	5	4
Medium	Small	33	2858	2	1	5	31	0	0	10
Medium	Medium	22	2624	1	4	4	21	1	4	11
Medium	Large	21	2620	1	3	5	20	1	4	11
Large	Small	43	3930	2	1	8	41	2	2	16
Large	Medium	35	4710	2	1	9	33	1	11	14
Large	Large	28	4521	1	2	8	27	2	23	15

PS = Problem Size; SD = Surgery Duration; PP = Patients Planned; POR = Patient from OR; OB = Occupied Beds; EP = Expected Patients; and PICU = Patients from ICU.

represents the summary results of the lower-level plan when the operating room is the bottleneck resource.

When the optimal sequence of patients in operating rooms is obtained, this sequence is followed by the ICU and ward such that the arrival times of patients in the ICU and ward are scheduled after the discharge times of patients from the operating room as represented by Equations (46) and (47).

Table 6. Summary results of the lower-level plan when the ICU becomes the bottleneck.

PS	SD	Operating Room		ICU			Ward
		PP	Makespan	POR	OB	EP	POR	PICU	OB	EP
Small	Small	12	1063	1	3	4	13	1	2	5
Small	Medium	9	1119	0	5	2	10	1	4	4
Small	Large	9	1221	0	5	2	10	1	4	6
Medium	Small	26	2542	1	5	5	25	2	4	7
Medium	Medium	20	2572	1	5	5	19	2	2	11
Medium	Large	17	2923	1	4	6	16	1	5	12
Large	Small	33	3307	2	5	8	32	2	19	15
Large	Medium	28	3621	1	6	8	26	3	21	13
Large	Large	22	3685	1	5	9	21	4	24	14

PS = Problem Size; SD = Surgery Duration; PP = Patients Planned; POR = Patient from OR; OB = Occupied Beds; EP = Expected Patients; and PICU = Patients from ICU.

represents the summary results of the lower-level plan when the ICU is identified as the bottleneck resource. When ICU becomes the bottleneck resource, the lower-level plan for the ICU is made, and it is followed by both the operating room and ward such that for the operating room, the discharge time of patients from the operating room is planned before the arrival time of the patients in the ICU as explained in Equation (49).

In the same way, the patient’s arrival time in the ward is planned after the patient’s discharge from the ICU as given by Equation (50).

Table 7. Summary results of the lower-level plan when the ward becomes the bottleneck.

PS	SD	Operating Room		ICU			Ward
		PP	Makespan	POR	OB	EP	POR	PICU	OB	EP
Small	Small	14	1220	1	3	2	13	1	8	6
Small	Medium	10	1212	0	3	2	9	2	12	5
Small	Large	7	1238	0	3	2	7	1	15	4
Medium	Small	23	2273	1	2	5	22	2	11	11
Medium	Medium	20	2557	1	3	5	19	2	13	9
Medium	Large	16	2630	1	3	5	15	2	16	10
Large	Small	34	3156	2	1	9	32	2	35	13
Large	Medium	28	3691	1	1	9	27	5	36	15
Large	Large	23	3524	1	1	10	21	6	43	15

PS = Problem Size; SD = Surgery Duration; PP = Patients Planned; POR = Patient from OR; OB = Occupied Beds; EP = Expected Patients; and PICU = Patients from ICU.

represents the summary results of the lower-level plan when the ward is the bottleneck resource.

When the ward becomes the bottleneck resource, the lower-level plan for the ward is made, and it is followed by both the operating room and ICU such that for the operating room, the discharge time of patients from the operating room is planned before the arrival time of the patients in the ward, as explained in Equation (52). In the same way, the patient’s discharge time from ICU is planned before the patient’s arrival time in the ward as given by Equation (53).

CPU Time

The average CPU time in seconds for each category of problem for one planning horizon is shown in Figure 6. The CPU time is proportional to the size of the problem in cases when the operating room and ward are the bottlenecks. While in the scenario where the ICU is the bottleneck, the trend in the CPU time for the medium-sized problem is random. It is proposed that the CPU time can be reduced using heuristic algorithms without compromising the solution’s optimality.

5. Conclusions, Limitations, and Future Research

The planning and scheduling of hospital resources are significant for better service and profit generation. The current research proposed an integrated planning and scheduling problem at different levels for operating rooms, intensive care units, and recovery wards. The theory of constraints was applied, and a mixed integer linear programming model was developed for the integrated planning and scheduling of the operating room, considering the constraints of the downstream ward. Different models for planning and scheduling were made considering the shifting of the bottleneck resources of hospitals. The integrated models for planning at different levels when any one resource such as the operating room, ICU, or ward becomes a bottleneck were proposed for the first time in the literature. The proposed model for integrated planning and scheduling was solved using the CPLEX solver on different sizes of test problems. The small, medium, and large-sized test problems were made for each scenario where the operating room, ICU, or ward became the bottlenecked resource.

The proposed model is significant for hospitals for effective and efficient planning and scheduling of their resources. This approach has practical significance in a hospital where the whole system’s performance is affected by one bottleneck which keeps changing in different planning horizons. Planning based on the capacity of the bottleneck resource subordinates the rest of the resources to the bottleneck resource to achieve optimal hospital performance. The theory of constraints is significant to identify the bottleneck resource among all the critical resources of the hospital. Therefore, the mathematical models developed using the theory of constraints in the current study can be used to develop hospital planning systems in which planning is based on the capacity of bottleneck resources.

The current study has its limitation in using the CPLEX solver, which provides an exact method to solve the mathematical models. In the future, constraint-based heuristics can be developed to reduce the CPU time of the different sizes of the considered problems, particularly for larger-sized instances. In addition, the current study uses deterministic surgery durations and lengths of stay in downstream units. Further extension of the work includes the use of data mining and machine learning methods to predict patient arrivals and length of stay in the operating room and downstream units and using the predicted data as the inputs in the models developed in the current study.