Operating Room Scheduling in Teaching Hospitals

Operating room scheduling is an important operational problem in most hospitals. In this paper, a novel mixed integer programming MIP model is presented for minimizing Cmax and operating room idle times in hospitals. Using this model, we can determine the allocation of resources including operating rooms, surgeons, and assistant surgeons to surgeries, moreover the sequence of surgeries within operating rooms and the start time of them. The main features of the model will include the chronologic curriculum plan for training residents and the real-life constraints to be observed in teaching hospitals. The proposed model is evaluated against some real-life problems, by comparing the schedule obtained from the model and the one currently developed by the hospital staff. Numerical results indicate the efficiency of the proposed model compared to the real-life hospital scheduling, and the gap evaluations for the instances show that the results are generally satisfactory.


Introduction
Health care expenditures comprise a meaningful portion of the Gross Domestic Product in both developed and developing countries. Expenditure on healthcare in the UK as a percentage of the UK Gross Domestic Product GDP was estimated to be 8.4% in 2007, from which the public share was 69% 1 . Also, according to the statics released by the WHO World Health Organization , health care expenditures in 2007 in Iran as a developing country were estimated to be about 6.4% of its GDP, and the portion covered by the government was about 46.8%. This fact makes health systems an important research field for industrial engineering and operations research to improve their operational efficiency.
Operating rooms are simultaneously the largest cost centers and the greatest source of revenues for most hospitals. OR planning and scheduling is a key tool which can be useful to improve the productivity level of ORs and the related departments. Under a blocked scheduling strategy, individual surgeons or surgical groups are assigned times in a particular OR in a periodic typically weekly or monthly schedule. The planning within the framework of a blocked strategy consists of three stages. In the first stage, the OR capacity is divided among the surgeons, surgical groups, or departments on a strategic level. Then, a cyclic timetable called "Master Surgical Schedule" is constructed that defines the number and type of operating rooms available, the hours that ORs will be open, and the surgical groups or surgeon sessions for each OR 2 . The last stage which may be called "surgery process scheduling" splits into two subproblems called "advance scheduling" and "allocation scheduling" 3 . The first subproblem at a tactical level one weak to one month solves a planning phase by assigning an operating date to each patient over the time horizon. The second subproblem solves a scheduling phase which determines the sequence and resource allocation of cases in a given day 4, 5 . In the open strategy, the hospital does not hold operation rooms specific to a single surgeon although there is sometimes a weekly schedule for each surgeon. In this strategy, the intention is to accommodate all patients. The surgeons submit cases up until the day of the surgery, and all the cases are scheduled in ORs.
The modified strategy is similar to the blocked one except that certain slots in the master surgical schedule are left open for flexibility. In fact, this strategy is a mix of open and blocked strategies.
In this paper, we investigate the deterministic daily scheduling of surgical cases in teaching hospitals using the open scheduling strategy. A novel MIP model is presented with the objective of minimizing the total idle times of operating rooms and Cmax which represent completion time of the last patient's surgery. Using this model, we endeavor to allocate the required resources including operating rooms, surgeons, and assistant surgeons to surgeries and also determine the sequence of surgeries within operating rooms and the start time of each surgery. The major features of the model include the chronologic curriculum plan for training residents and the real-life constraints in teaching hospitals. We maintain that the problem defined here with its specific characteristics is not reported in the literature and that it finds real applications in teaching hospitals.
The rest of the paper is organized as follows. A brief review of previous work will first be presented in Section 2. Section 3 will provide a definition of the problem. The MIP model will be presented in Section 4. Section 5 is devoted to solving approach. Numerical results will be reported in Section 6. Conclusions will be drawn and presented in Section 7.

A Literature Review
Over the past 60 years, a large body of literature has evolved on the management of operating theaters. Magerlein and Martin 3 reviewed the literature on surgical demand scheduling and, as previously mentioned, focused on "surgery process scheduling" to differentiate between advance and allocation scheduling. Blake and Carter 6 extended this classification and added the domain of external resource scheduling which is defined as the process of identifying and reserving all resources external to the surgical suite necessary Advances in Operations Research 3 to ensure appropriate care for a patient before and after an instance of surgery. They further divided each domain into strategic, administrative, and operational levels 7 . Many authors Magerlein and Martin, 1987; Kennedy 8 classified the literature based on solution techniques 4 , but Cardoen et al. 7 proposed a detailed classification based on 7 areas that are related to either the problem setting e.g., performance measures or patient classes or the technical features e.g., solution technique or uncertainty incorporation and analyzed the contributions in these areas.
Deterministic and stochastic mathematical programming models, queuing models, simulation models, and heuristic approaches have all been widely used to investigate OR scheduling. In this paper, we will briefly focus on the studies that have a direct bearing on surgical process scheduling and will consider the deterministic programming models used for OR scheduling. A more comprehensive review can be found in 7 .
Ogulata and Erol 9 modeled the hierarchical multiple criteria mathematical programming to assign surgeon groups and dates to patients. A goal programming has been proposed to solve this weekly programming. Vissers et al.

Advances in Operations Research
In almost all the studies cited above, the surgeon for each case was known in advance. In this paper, the daily scheduling of surgical cases in teaching hospitals is investigated in which the training operations must be performed by postgraduate alumni, called "residents" or "fellows," under the supervision of a relevant attending surgeon while they are also qualified to do certain operations independently.
Chronologic curriculum plan for training residents is addressed in resident scheduling problem RSP repeatedly, which is different from operating room scheduling problem. This problem involves assigning residents to day and night shifts over a given planning horizon subject to numerous working regulations and staffing requirements 19 . For example, Topaloglu and Ozkarahan 19 developed a mixed-integer programming model for scheduling residents' duty hours considering the on-call night, dayoff, rest period, and total work-hour ACGME Accreditation Council for Graduate Medical Education regulations as well as the demand coverage requirements of the residency program. Sherali et al. 20 addressed the resident scheduling problem RSP at hospitals concerned with prescribed work nights for residents while considered departmental staffing and skill requirements as well as residents' preferences.
Dexter et al. 21 developed a methodology to determine rotations consisting of combinations of specialties to be paired for purposes of trainee scheduling to reduce the incidence of daily assignments off rotation.
To the best of our knowledge among the body of work on OR planning and scheduling that has appeared in the literature, no report is available in the literature on operating room scheduling in teaching hospitals that takes into account real constraints such as the chronologic curriculum plan for training residents. Constraints related with allocation of operations based on different seniority levels of residents and fellowships and providing a good balance between total hours of operations that performed by them than the average in a month. Also most of these papers assume that surgeon of each patient is already known. So it seems, assigning surgeons, residents, and assistant surgeons to each instance of surgery in literature of OR planning and scheduling is new.

Problem Definition
The process of surgical scheduling consists of two steps. The first concerns a weekly plan to assign a specific date to each patient waiting for surgery while the second involves sequencing and scheduling surgical cases on a given date. In this paper, we consider a surgical case scheduling problem the second step in a teaching hospital. In such hospitals, a list is prepared of the patients waiting for operation on the following day. Both inpatients and outpatients are scheduled each day by the OR head nurse or manager who sequences the surgeries and assigns a surgeon, an operating room, and a start time to each surgery.
Patients are prioritized and sequenced by the head nurse based on both resource availability the eligible surgeon, the required equipments and aids, and the appropriate OR and patient child or old priority.
The training operations must be performed by postgraduate alumni "residents" and "fellows" under the supervision of relevant attending surgeons. The residents and fellows are usually grouped according to their seniority levels, and each of them is qualified to do certain operations based on their experience and qualifications acquired in the course of their curriculum plan. This plan determines the type of operations that can be performed by each resident or fellow group during each period of their education.

5
Certain operations also require an assistant surgeon. The distribution of surgeries among surgeon groups, residents, and fellows should be based on equal opportunity for all to acquire experience. The nontraining operations are performed by public surgeons.
Some surgeries require special equipments which are available in particular ORs. For each operation, a group of medical staff members collectively called the surgery aid group in this paper, consisting of a scrub nurse, a circular nurse, and an anesthetic technician, is required. Also several medical instruments are needed during the surgery. After each surgery, instruments possibly need to be sterilized for some periods and, hence, are not available for subsequent surgeries.
We should also deal with the likelihood of infection spread. Therefore, special sanitary procedures must be executed to avoid the transfer of infection from patient to patient. In particular, the operating room needs additional cleaning after operation on an infected patient.
In the next section, the objectives and constraints of the problem are identified in a mathematical formulation.

Mathematical Formulation of the Problem
This section describes an MIP model for determining the daily surgical scheduling of elective patients inpatient and outpatient that calls "HORS" hospital operating room scheduling .

Notations
The following notations are used in this paper.

The Mathematical Model
This model aims at minimizing operating room idle time and Cmax, where Cmax is, as mentioned earlier, the completion time of the latest patient's surgery in operating room. The most important objective of decision maker is minimizing the operating room idle time. But considering this objective alone in the model makes some operating rooms remain empty while there may be some operating rooms which are occupied up to a maximum possible. Therefore, minimizing Cmax is considered as the primary objective, and minimizing Advances in Operations Research 7 operating room idle time is the second level objective. Here, f1 is defined Cmax and f2 is the total idle time of operating rooms. Problem-solving process will be explained in a next section.
The MIP model is then formulated as follows:

Solving Approach
As previously mentioned, this model aims at minimizing operating room idle time and Cmax. Initially the problem was solved using the Lexicograph method. Due to the high time to solve, some changes were made in solving method. Below, f1 and f2 are Cmax and idle time of operating rooms, respectively, and G i x shows constraints of the model. Solving process is described below.
Step 1. Choose an appropriate upper bound for f1.
Step 2. Solve the following problem:

5.1
Step 3. If the problem is feasible, deduce the upper bound and go to Step 2.
Step 4. If the problem has no feasible solution, then exit with the current feasible solution as the optimum solution.
Flow diagram of this method is shown in Figure 1.

Computational Experiments
We did not come across any problem as defined in this paper. Although there are different works cited in the literature, the differences exist in the constraints considered. In this paper, the real constraints of teaching hospitals have been duly taken into account.
To evaluate the proposed mathematical model, several real problems were collected from the Hasheminejad Kidney Center HKC , an academic hospital, in Iran. The collected data consisted of lists of patients for each day, start and end times of surgeries, details of the patients such as infection, the surgeon assigned, and the operating room assigned. To collect some required data that is not registered in the HIS hospital information system , we have designed an ICR intelligent character recognition form that was completed by operating room personnel for 6 months and selected the most complete forms that were filled out within a one-month period. The designed form is shown in the appendix section.
There are four different surgical services at HKC: urologic, endoscopic, laparoscopic, and vascular. The operating theatre is composed of 7 operating rooms. All the operating rooms are multifunctional, but endoscopic operations can be performed in only three. There are four resident groups of urology according to their seniority levels and two fellowship groups. Each resident group consists of three residents with the same skill, and each fellowship group consists of two fellows. There is a trainee rotation that determines the days each resident or fellow should serve in the operating theatre.
The number of technician groups varies on different work shifts. There are 35 different types of important instruments used in the operating theatre e.g., laparoscopic trolley, C-Arm, etc. , each case requiring a specific set. Furthermore, surgery durations, required instruments for each surgery, instrument inventory, and the corresponding sterilization times are known. Also the duration of setup time is non-sequence-dependent and is considered in the processing time of the operation.
Time data are expressed in 30-minute units, and the scheduling horizon comprises 25 time units.
The model was implemented in GAMS and solved using CPLEX 12.1.0. The allowed computation time for the solution procedure was limited to 400 seconds per instance on a 2, 66 GHz Pentium 4 PC with the Windows 7 operating system. The model was evaluated by solving several real instances at HKC, and the gap from optimal solution was reported. The evaluation was based on 30 real test problems with 18 to 45 patients and 8 to 13 surgeons. To study the performance of the model, the MIP gap was evaluated. Table 1 shows the resulting MIP gap for each test instance defined as the absolute gap between the values obtained for the objective function and the lower bound given by CPLEX, the number of discrete variables, constraints, patients, and surgeons.
In addition, to analyze the surgical case scheduling resulting from the proposed model, the final solution for one month of chosen instances was explored and the actual scheduling of hospital compared with the proposed scheduling.
Since  on Sunday and Tuesday. Thereby the problem instances are not chosen from these days of the week because in this model, focus is on training operations and constraints of transplant operations are not considered . Because of the complexities of this problem, no predictive scheduling considering all surgeries and all constraints can be done by the OR head nurse or manager of the operating theatre. So the comparison of the obtained schedule from HORS and the predictive schedule of the hospital is not possible. In HKC, after completing each surgery, the OR head nurse schedules the next case considering the actual system status at that moment. Hence, an online evaluation is proposed that includes the actual system status. In this validation method, the proposed model is updated each hour periodically. In addition the rescheduling is done after the end of each operation or emergency patient entry. Thereby, the Hybrid Rescheduling Policy is used.
The gap evaluations for the real instances show that the results are in general satisfactory. The smallest and greatest absolute gaps were 0 and 5 in the allowed computation time and 73% of tested cases reached optimal solutions in the specified time; even the time needed to find optimal solutions for the all of the test instances was lower than 1700 s about 30 min and could, hence, be still considered as reasonable. Figure 2 shows the frequency of the results in the range, verifying the efficiency of the proposed approach.  The result of comparison between actual scheduling and proposed model HORS based on the value of operating room idle time and Cmax are presented in Table 2.
From the results presented in Table 2, the solutions provided by "HORS" scheduling are better than those of the HKC scheduling. Table 3 Figure 5: Operating room processes time recording log form.

Conclusions and Recommendations
In this paper, the second step of the surgical process scheduling was investigated for scheduling elective patients waiting surgery. We formulated a mixed integer problem to assign a set of operations to some resources consisting of human resources surgeon, assistant surgeon and applied resource OR and sequenced them simultaneously. Taking into account the training plan of residents and fellows, real constraints in teaching hospitals and assignment of assistant surgeons to each operation form the main features of the proposed model. The proposed method was evaluated and verified through solving several real instances at a teaching hospital. Also, the final solution for one month of chosen instances was explored and the actual scheduling of hospital compared with the proposed scheduling.
Numerical results indicated the efficiency of the proposed model compared to the actual hospital scheduling, and the gap evaluations for the real instances showed that the results were generally satisfactory.
The proposed model can be extended by including other resources such as ICU bed or ward bed in this model. We are working in a deterministic context; for future work, we will consider uncertainty related to surgery time and emergency patient arrival. Another suggestion for future work is to use the constraint programming, as a viable and powerful method for modeling and solving many combinatorial problems, to solve the described problem.