Abstract
Paper aims
To find efficient operation room scheduling for residents considering several resources constraints and ensuring a minimum number of surgeries for approval in the training program.
Originality
We find no research in current literature addressing operation room resource allocation for residents training in order to meet legislation approval criteria.
Research method
The number of procedures to be performed by each resident was defined by Brazilian legislation and by the rules obtained from interviews with the chief professor of a vascular hospital. To solve the allocation of doctors to surgeries planning problem, also addressed in literature as Master Surgical Schedule (MSS), we propose a mathematical programming approach.
Main findings
The mathematical model showed that some of the current rules of resource availability bring about infeasible planning when trying to achieve the legislation quantitative rules for the residents training.
Implications for theory and practice
The model allowed the decision maker to plan and schedule vascular surgeries in a better and faster way, through an automated system, instead of allocating residents to surgeries manually, which takes many hours per month. Furthermore, changing the input data in the proposed model can allow other hospitals or specialists to get efficient results in less time.
Keywords
Scheduling; master surgical scheduling; Mathematical programming; Vascular surgery; Hospital
1. Introduction
Vascular diseases are serious illnesses – their treatment is complex and requires specialized surgical procedures, which are performed in tertiary health centers. In order to perform these procedures, sophisticated equipment, qualified teams and intensive care units (ICU) prepared for the postoperative period are necessary. The increase in the life expectancy of a population causes a higher prevalence of this type of disease, which has imposed a significative burden to the Brazilian Unified Health System (SUS) (Mendes et al., 2014Mendes, C. D. A., Martins, A. D. A., Teivelis, M. P., Kuzniec, S., & Wolosker, N. (2014). Public private partnership in vascular surgery. Einstein (Sao Paulo, Brazil), 12(3), 342346. http://dx.doi.org/10.1590/s167945082014gs3029. PMid:25295457.
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).
According to Barbagallo et al. (2015)Barbagallo, S., Corradi, L., de Ville de Goyet, J., Iannucci, M., Porro, I., Rosso, N., Tanfani, E., & Testi, A. (2015). Optimization and planning of operating theatre activities: an original definition of pathways and process modeling. BMC Medical Informatics and Decision Making, 15(1), 38. http://dx.doi.org/10.1186/s1291101501617. PMid:25982033.
http://dx.doi.org/10.1186/s12911015016...
, offering a better quality of service with limited resources, and at a lower cost, is a global health challenge. This makes the goal of optimization a central theme in modern hospital management, as it allows for increases in resource efficiency through the development of advanced methods to plan and program the hospitals’ processes. On the other hand, residents have to fulfill a number of specific procedures as part of their educational requirements facing many variabilities. This may result in a high amount of overtime hours to succeed in each stage of their educational process and bring negative effects that decrease the quality of provided care (Erhard et al., 2018Erhard, M., Schoenfelder, J., Fügener, A., & Brunner, J. O. (2018). State of the art in physician scheduling. European Journal of Operational Research, 265(1), 118. http://dx.doi.org/10.1016/j.ejor.2017.06.037.
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).
In this context, a teaching hospital deals with a challenge: increasing resource efficiency planning while subject to several rules from various regulational requirements (from the Ministry of Health, the Ministry of Education and various professional councils) (Assad et al., 2018aAssad, D. B. N., Hamacher, S., & Spiegel, T. (2018a). Vascular elective surgeries planning and scheduling: a case study at a teaching hospital. In Operations Management for Social Good. 2018 POMS International Conference in Rio. Springer Proceedings Business, Economics. USA: Springer. ). Therefore, to achieve an efficient planning of resident allocation, we propose a mathematical programming approach. This technique associates the annual minimum number of surgeries that must be realized by each resident to operating room planning, through optimal resource allocation.
2. Theoretical background: hospital management and operating room planning
In several regions of the world, health care industry is facing a dilemma due, on one hand, to the poor quality standards of services offered and, on the other hand, to strong pressure to increase efficiency and productivity, and to reduce costs (Vähätalo & Kallio, 2015Vähätalo, M., & Kallio, T. J. (2015). Organising health services through modularity. International Journal of Operations & Production Management, 35(6), 925945. http://dx.doi.org/10.1108/IJOPM1220130523.
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; Spiegel & Assad, 2018Spiegel, T., & Assad, D. B. (2018). Operations model for trauma centers: multiple case study. International Journal of Public Health Management and Ethics, 3(1), 113. http://dx.doi.org/10.4018/IJPHME.2018010101.
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).
Optimization issues in healthcare have received considerable attention for over three decades. More recently, however, with declining birth rates in almost all developed countries, and increasing average longevity overall, optimization issues in healthcare have become notoriously important, attracting research institutions and great social interest. Over the years, attention has gradually expanded from resource allocation and strategic planning to include operational issues, such as resource scheduling and treatment planning (Rais & Viana, 2011Rais, A., & Viana, A. (2011). Operations research in healthcare: a survey. International Transactions in Operational Research, 18(1), 131. http://dx.doi.org/10.1111/j.14753995.2010.00767.x.
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; Silva et al., 2018Silva, A. C., Assad, D. B., & Spiegel, T. (2018). Capacity decision in emergency hospital operation rooms: sizing using simulation. In M. Ebrahimi (Ed.), Modeling and simulation techniques for improved business processes (pp. 140155). Hershey, PA: IGI Global. http://dx.doi.org/10.4018/9781522532262.ch006.
http://dx.doi.org/10.4018/97815225322...
; Assad et al., 2018bAssad, D. B. N., Silva, A. C., Spiegel, T., & Cardoso, L. (2018b). Planejamento tático de centro cirúrgico: a decisão de alocação de salas cirúrgicas. In D. F. Andrade. (Ed.), Gestão da produção em foco (Vol. 13, pp. 229240). Belo Horizonte: Editora Poisson.).
According to Barbagallo et al. (2015)Barbagallo, S., Corradi, L., de Ville de Goyet, J., Iannucci, M., Porro, I., Rosso, N., Tanfani, E., & Testi, A. (2015). Optimization and planning of operating theatre activities: an original definition of pathways and process modeling. BMC Medical Informatics and Decision Making, 15(1), 38. http://dx.doi.org/10.1186/s1291101501617. PMid:25982033.
http://dx.doi.org/10.1186/s12911015016...
, hospital budgets account for almost half of all expenditures in most health systems, and the most important reasons for hospital admission are procedures or surgical interventions.
Van Sambeek et al. (2010)Van Sambeek, J. R. C., Cornelissen, F. A., Bakker, P. J. M., & Krabbendam, J. J. (2010). Models as instruments for optimizing hospital processes: a systematic review. International Journal of Health Care Quality Assurance, 23(4), 356377. http://dx.doi.org/10.1108/09526861011037434. PMid:20535906.
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categorize decisionmaking models related to design and control processes that involve patient flows in hospitals. The authors’ objective is to map out how hospitals try to achieve the goals of higher quality and higher resource allocation efficiency in the field of operations management (OM) and operational research (OR). The synthesis of this categorization is presented in Table 1.
Abdelrasol et al. (2014)Abdelrasol, Z., Harraz, N., & Eltawil, A. (2014). Operating room scheduling problems: a survey and a proposed solution framework. In H. K. Kim, M. A. Amouzegar, & S.I. Ao (Eds.), Transactions on engineering technologies (pp. 717731). Dordrecht: Springer. state that operating room planning can be divided into 3 decision levels: (i) case mix problem, (ii) master surgical scheduling and (iii) surgery scheduling. The last decision level can be broken down into (1) operational offline and (2) operational online (Hulshof et al., 2012Hulshof, P. J., Kortbeek, N., Boucherie, R. J., Hans, E. W., & Bakker, P. J. (2012). Taxonomic classification of planning decisions in health care: a structured review of the state of the art in OR/MS. Health Systems (Basingstoke, England), 1(2), 129175. http://dx.doi.org/10.1057/hs.2012.18.
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). According to Assad (2017)Assad, D. B. N. (2017). Planejamento e programação de cirurgias eletivas: estudo de caso em um hospital universitário (Master’s dissertation). Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro., the decision level and types of decisions taken in each are as summarized in Table 2. The relation between operating room planning and decision support techniques in the OM & OR contexts is syntetized in Figure 1.
Synthesis of operating room planning literature review. Source: adapted from Assad (2017)Assad, D. B. N. (2017). Planejamento e programação de cirurgias eletivas: estudo de caso em um hospital universitário (Master’s dissertation). Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro..
Although mathematical programming models have been widely used to solve Master Surgery Scheduling Problems (Persson & Persson, 2007Persson, M., & Persson, J. A. (2007). Optimization modelling of hospital operating room planning: Analyzing strategies and problem settings. In Operational Research for Health Policy: Making Better Decisions, 31st Annual Conference of the European Working Group on Operational Research Applied to Health Services (pp. 137). Bern, Switzerland: Peter Lang International Academic Publishers.; Van Oostrum et al., 2008Van Oostrum, J. M., Van Houdenhoven, M., Hurink, J. L., Hans, E. W., Wullink, G., & Kazemier, G. (2008). A master surgical scheduling approach for cyclic scheduling in operating room departments. ORSpektrum, 30(2), 355374. http://dx.doi.org/10.1007/s002910060068x.
http://dx.doi.org/10.1007/s00291006006...
; Beliën et al., 2009Beliën, J., Demeulemeester, E., & Cardoen, B. (2009). A decision support system for cyclic master surgery scheduling with multiple objectives. Journal of Scheduling, 12(2), 147161. http://dx.doi.org/10.1007/s1095100800864.
http://dx.doi.org/10.1007/s10951008008...
; Wang et al., 2010Wang, Y., Tang, J., & Qu, G. (2010). A genetic algorithm for solving patientprioritybased elective surgery scheduling problem. In Y. Wang, J. Tang, & G. Qu. Life system modeling and intelligent computing (pp. 297304). Berlin: Springer. http://dx.doi.org/10.1007/9783642155970_33.
http://dx.doi.org/10.1007/97836421559...
; Day et al., 2012Day, R., Garfinkel, R., & Thompson, S. (2012). Integrated block sharing: a win–win strategy for hospitals and surgeons. Manufacturing & Service Operations Management: M & SOM, 14(4), 567583. http://dx.doi.org/10.1287/msom.1110.0372.
http://dx.doi.org/10.1287/msom.1110.0372...
; He et al., 2012He, B., Dexter, F., Macario, A., & Zenios, S. (2012). The timing of staffing decisions in hospital operating rooms: incorporating workload heterogeneity into the newsvendor problem. Manufacturing & Service Operations Management: M & SOM, 14(1), 99114. http://dx.doi.org/10.1287/msom.1110.0350.
http://dx.doi.org/10.1287/msom.1110.0350...
; Paul & Jotshi, 2013Paul, J. A., & Jotshi, A. (2013). Efficient operating room redesign through process improvement and optimal management of scheduled and emergent surgeries. International Journal of Mathematics in Operational Research, 5(3), 317344. http://dx.doi.org/10.1504/IJMOR.2013.053626.
http://dx.doi.org/10.1504/IJMOR.2013.053...
; Sufahani & Ismail, 2014Sufahani, S., & Ismail, Z. (2014). A real scheduling problem for hospital operation room. Applied Mathematical Sciences, 8(114), 56815688. http://dx.doi.org/10.12988/ams.2014.46413.
http://dx.doi.org/10.12988/ams.2014.4641...
; Saadouli et al., 2015Saadouli, H., Jerbi, B., Dammak, A., Masmoudi, L., & Bouaziz, A. (2015). A stochastic optimization and simulation approach for scheduling operating rooms and recovery beds in an orthopedic surgery department. Computers & Industrial Engineering, 80, 7279. http://dx.doi.org/10.1016/j.cie.2014.11.021.
http://dx.doi.org/10.1016/j.cie.2014.11....
; Villarreal & Keskinocak, 2016Villarreal, M. C., & Keskinocak, P. (2016). Staff planning for operating rooms with different surgical services lines. Health Care Management Science, 19(2), 144169. http://dx.doi.org/10.1007/s107290149307x. PMid:25366968.
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) there is no research in which quantitative legislation criteria are considered, according to Erhard et al. (2018)Erhard, M., Schoenfelder, J., Fügener, A., & Brunner, J. O. (2018). State of the art in physician scheduling. European Journal of Operational Research, 265(1), 118. http://dx.doi.org/10.1016/j.ejor.2017.06.037.
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.
Therefore, in order to cover this literature review gap Assad et al. (2018a)Assad, D. B. N., Hamacher, S., & Spiegel, T. (2018a). Vascular elective surgeries planning and scheduling: a case study at a teaching hospital. In Operations Management for Social Good. 2018 POMS International Conference in Rio. Springer Proceedings Business, Economics. USA: Springer. proposed a mixed integer linear programming model and the current research merges it with a brief theoretical background better discussed in Assad (2017)Assad, D. B. N. (2017). Planejamento e programação de cirurgias eletivas: estudo de caso em um hospital universitário (Master’s dissertation). Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro. in an enhanced version.
3. Research method
Following the classification proposed by Yin (2001)Yin, R. K. (2001). Estudo de caso: planejamento e métodos (2. ed.). Porto Alegre: Bookman., this research is an exploratory single case study. Its objective is to deepen the comprehension regarding a problem that is not yet sufficiently studied, to suggest hypotheses and questions, or to develop the theory (Miguel, 2007Miguel, P. A. C. (2007). Estudo de caso na engenharia de produção: estruturação e recomendações para sua condução. Revista Produção, 17(1), 216229. http://dx.doi.org/10.1590/S010365132007000100015.
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). The case study is divided in three parts: literature review (results presented in Table 3 – see Assad (2017)Assad, D. B. N. (2017). Planejamento e programação de cirurgias eletivas: estudo de caso em um hospital universitário (Master’s dissertation). Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro. for more information), resource parametrization and mathematical modelling.
According to Barbagallo et al. (2015)Barbagallo, S., Corradi, L., de Ville de Goyet, J., Iannucci, M., Porro, I., Rosso, N., Tanfani, E., & Testi, A. (2015). Optimization and planning of operating theatre activities: an original definition of pathways and process modeling. BMC Medical Informatics and Decision Making, 15(1), 38. http://dx.doi.org/10.1186/s1291101501617. PMid:25982033.
http://dx.doi.org/10.1186/s12911015016...
, the management of the surgery center must follow the three steps described in Figure 2. The scope of this research will be limited to the parametrization of resources necessary for the execution of surgeries in the vascular clinic, and to the proposition of optimization models to estimate the relation between resources necessary for respecting the normative instruments that govern surgical services applied to the case of vascular surgery.
Optimized planning steps for operation room management. Source: adapted from Barbagallo et al. (2015)Barbagallo, S., Corradi, L., de Ville de Goyet, J., Iannucci, M., Porro, I., Rosso, N., Tanfani, E., & Testi, A. (2015). Optimization and planning of operating theatre activities: an original definition of pathways and process modeling. BMC Medical Informatics and Decision Making, 15(1), 38. http://dx.doi.org/10.1186/s1291101501617. PMid:25982033.
http://dx.doi.org/10.1186/s12911015016... .
4. Problem description: vascular surgeries at a teaching hospital
Teaching hospitals are subject to several rules from many regulational requirements (from the Ministry of Health, the Ministry of Education and various professional councils). Regarding the residency program, the National Comission of Medical Residency (Comissão Nacional de Residência Médica – CNRM) (The specific legislation that governs the residency programs in Brazil is available at (in Portuguese) (Brasil, 2006Brasil. Ministério da Educação. (2006, May 17). Resolução CNRM n° 02/2006, de 17 de maio de 2006. Dispõe sobre requisites mínimos dos Programas de Residência Médica e dá outras providências. Diário Oficial da República Federativa do Brasil. Retrieved in 20 May 2017, from http://portal.mec.gov.br/index.php?option=com_docman&view=download&alias=100951resolucaocnrmn2de17demaiode2006&category_slug=novembro2018pdf&Itemid=30192
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) and the National Comission of Multiprofessional Healthcare Residency (Comissão Nacional de Residência Multiprofissional em Saúde – CNRMS) stand out. Both define the minimal requirements for obtaining the surgeon license in each specialty. Currently, these minimal requirements are defined by the CNRM Resolution number 02/2006, from May 17th, 2006 (Brasil, 2006Brasil. Ministério da Educação. (2006, May 17). Resolução CNRM n° 02/2006, de 17 de maio de 2006. Dispõe sobre requisites mínimos dos Programas de Residência Médica e dá outras providências. Diário Oficial da República Federativa do Brasil. Retrieved in 20 May 2017, from http://portal.mec.gov.br/index.php?option=com_docman&view=download&alias=100951resolucaocnrmn2de17demaiode2006&category_slug=novembro2018pdf&Itemid=30192
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). For the vascular surgery specialty, this resolution demands that each resident carries out at least 150 surgeries, with a minimum of 50 arterial surgeries per year of the program, and at least 30% of small size surgeries, 40% of medium size surgeries, and 40% of large size surgeries in a oneyear timespan. Differently from other specialties, these numbers are the same for both the general surgeons on the first year of residency in vascular surgery (R1) and for those on the second year (R2). The hospital being studied also offers a subspecialty of vascular surgery named endovascular surgery that may be performed by the vascular surgeon, and that lasts for a year. Typically, after concluding the second year, the new vascular surgeons participate in this residency program (R3). Surgeries are divided in small size, medium size, and large size surgeries (Table 4).
Despite the validity of the division as a normative instrument, the chief surgeon of vascular surgery categorizes surgeries of this specialty in seven aggregated groups: varicose veins and veins surgeries, aorta surgeries, fistula surgeries, femoral surgeries, carotid surgeries, amputation surgeries, and endovascular procedures.
This additional categorization brought the necessity to build Table 5 together with the surgeon chief of vascular surgery to draw the intersection between these surgery groups and the procedures described by the normative instrument regarding “size” and group. On the second to last column, arterial surgeries are assigned with an X. This emphasizes the importance of the development of decision support models in collaboration with their future users. On the last line, the number of surgeries by size are established, following the same proportion established by the CNRM Resolution Number 02/2006 (Brasil, 2006Brasil. Ministério da Educação. (2006, May 17). Resolução CNRM n° 02/2006, de 17 de maio de 2006. Dispõe sobre requisites mínimos dos Programas de Residência Médica e dá outras providências. Diário Oficial da República Federativa do Brasil. Retrieved in 20 May 2017, from http://portal.mec.gov.br/index.php?option=com_docman&view=download&alias=100951resolucaocnrmn2de17demaiode2006&category_slug=novembro2018pdf&Itemid=30192
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).
Through Table 5, it is possible to notice that a resident may reach the minimum number of surgeries by size established by the CNRM Resolution by executing many procedures in more than one category throughout the year. Despite this, the professors (preceptors) of vascular surgery (staff) understand that a less experienced resident should execute and improve their ability in more simple procedures, while more experienced ones should improve their ability in procedures of higher complexity. This relation is presented in Table 6, where X indicates which resources are necessary to execute each type of surgery. On the columns R1 Doctor, R2 Doctor, and R3 Doctor, the X indicates that this resident doctor is enabled to execute that type of procedure. However, only one resident doctor is allocated to perform each surgery. As in the case of Table 5, Table 6 was also built together with the chief surgeon of vascular surgery. The chief surgeon also indicated that the R3 resident doctor doesn’t perform amputation surgeries, even though he/she is enabled to do amputations of any size.
This research considers that the sterile processing department (Central de Material de Esterilização – CME) cannot sterilize the surgical instruments for reuse on the same day; that is, thoughout the day, the CME only stores the surgical instruments. The stock levels of each type of surgical instruments are described on Table 7. It is also assumed that the hospital’s blood bank has the capacity to provide blood bags for up to 1 patient per day.
The relation between the amount of residents by type (R1 doctor, R2 doctor, and R3 doctor) and the minimum frequency in which they should be allocated is presented in Table 8
The operating hours of the operating room, to meet the demand for elective surgeries, are from 7 am to 7 pm, on weekdays only. However, working hours were considered to be from 8 am to 5 pm (540 minutes) in order to ensure 25% of slack time (the current literature assumes slack time between 20 and 30 percent of total time) (M’Hallah & AlRoomi, 2014M’Hallah, R., & AlRoomi, A. H. (2014). The planning and scheduling of operating rooms: a simulation approach. Computers & Industrial Engineering, 78, 235248. http://dx.doi.org/10.1016/j.cie.2014.07.022.
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). It is also assumed that the time horizon (one year) in working days is 250 days.
5. Nomenclature and mathematical problem formulation: a surgery planning and scheduling
Proposing a surgery plan and a surgery schedule that respect all rules imposed by the legislation and by the hospital is a difficult and timeconsuming task. This task is currently performed manually by the chief surgeon.
This article proposes a mathematical model to increase the efficiency of surgical planning. In other words, the model has the objective of finding the configuration that minimizes resource allocation (rooms, anesthetists and equipment), while respecting the minimum amount of surgeries. In addition, the proposed model considers as business rules: the minimum frequency of scheduling of the group of residents per week; the maximum amount of rooms available in the week; the maximum number of anesthetists available in the week; and the maximum amount of equipment available in the week.
The nomenclature and mathematical model formulation are presented in Tables 9, 10 and 11. The model was implemented in the commercial software AIMMS^{®} 3.14 with the standard solver CPLEX^{®} 12.6 (Bisschop & Roelofs, 2011Bisschop, J., & Roelofs, M. (2011). AIMMS 3.14: the user guide. Netherlands, Haarlem: Paragon Decision Technology BV.) on a computer with Intel Core i5 and 16 Gb of RAM.
Objective function:
The objective function (1) minimizes the room allocation ($Roo{m}^{r,d}$), anesthetists ($AnDa{y}^{r,d}$) and equipment ($EqDa{y}_{e}^{r,d}$) throughout the year. Prioritization is defined by the weights ($Weigh{t}_{i}$). For example, if the weight assigned to the resource room is greater than the weights assigned to the other ones, the solver will seek the lowest possible room allocation value that meets all the constraints that will be presented below.
Constraints:
Equation 2 ensures that each room is used in the operating hours of the surgery room (not allowing overtime) and Equation 3 ensures that each resident works until the limit of the working hours of the surgery room (not allowing overtime). The combination of Equations 2 and 3 prevents each resident from having workloads that are incompatible with the available room time by adding all the available rooms, what would mean, in practice, to allocate the professional in two rooms at the same time.
Equation 4 just allows any resident of type (c) to be allocated to perform the surgery (s) of size (z) in the room (r) and day (d) in case this room (r) is available for the surgery service on day (d). In order to increase the model’s efficiency (reduce resolution and interactions time), this constraint is only created when $durS{u}_{s,z}>0$ and $hClS{u}_{c,s,\text{z}}=1$, in other words, when the surgery (s) of size (z) exists and when the resident of type (c) is able to perform it.
Equation 5 accumulates the quantity of each surgery (s) of size (z) performed by the resident of type (c) and is created when $durS{u}_{s,z}>0$ and $haClS{u}_{c,s,z}=1$ following the same logic of the previous equation. The variable $AClS{u}_{c,s,z}$ is used in Equations 6, 7, 8 and 9 to ensure the minimum quantity of the surgeries.
This model doesn’t allocate the surgeon to the surgery, but rather designates the group of surgeons (in this case, R1, R2, or R3) who will perform it. Thus, it’s necessary, in Equations 6 to 10, to multiply the number of surgeries by the number of surgeons of that group.
Equation 11 associates, for each surgery (s) of size (z), the necessity of an anesthetist. It is created when $durS{u}_{s,z}>0$ and $rSuA{n}_{s,z}=1$, in other words, when the surgery (s) of size (z) exists and when it needs an anesthetist.
Equations 12 and 13 associate the daily and perroom allocation of the anesthetist (variable $AnDa{y}^{r,d}$) to the quantity of surgeries performed with an anesthetist for each surgery ($\sum}_{s,z}AnS{u}_{s,z}^{r,d$) and that are necessary to meet the term $\sum}_{r,d}Weigh{t}_{2}\text{*}AnDa{y}^{r,d$ of the objective function. Since the decision to allocate some anesthetists to the room (r) and day (d) is binary ($AnDa{y}^{r,d})$, it was necessary to create an auxiliary variable $AuxA{n}^{r,d}$in Equation 11 to receive the difference between the quantity of surgeries and the binary variable. However, it’s necessary for this allocative decision to be equal to 1 every time an anesthetist is assigned to a room while, at the same time, one is trying to minimize $\sum}_{r,d}Weigh{t}_{2}\text{*}AnDa{y}^{r,d$. In order to solve this problem, one multiplies the binary decision variable by a large number (M), which, by its turn, must be greater than the right side of the equation. This method ensures that whenever the quantity of surgeries in a day and in a room is greater than 1, the decision variable will receive the value 1.
Equations 14 and 15 associate the daily and room allocation of the resource equipment (variable $EqDa{y}_{e}^{r,d}$) to the number of surgeries performed with equipment for each surgery ($EqS{u}_{e,s,z}^{r,d}$) and that are necessary to meet the term $\sum}_{e,r,d}Weigh{t}_{3}\text{*}EqDa{y}_{e}^{r,d$ of the objective function. The terms$EqDa{y}_{e}^{r,d}$, $AuxE{q}_{e}^{r,d}$, $EqS{u}_{e,s,z}^{r,d}$, and $M$ follow the same logic presented in the explanation of Equations 11 and 12.
Equations 16 and 17 associate the allocation of resident of type (r) on day (d) (variable $ClDa{y}_{c}^{d}$) and are necessary to meet Equation 21, which refers to the minimum frequency of resident scheduling per week. The terms $ClDa{y}_{c}^{d}$, $AuxC{l}_{c}^{d}$, $ClS{u}_{c,s,l}^{r,d}$, and $M$ follow the same logic presented in the explanation of Equations 11 and 12.
Equation 18 associates the surgical material ($MatS{u}_{m,s,z}^{r,d}$) necessary for each surgery (s) of size (z). It is created when $durS{u}_{s,l}>0$ and $rSuMa{t}_{s,z,m}=1$, in other words, when the surgery (s) of size (z) exists and when surgical material (m) is required. Equation 19 limits the quantitative of surgeries performed in the day (d) due to the availability of surgical materials (m).
Equation 20 associates the blood bag ($SanC{i}_{s,z,d}$) necessary for each surgery (s) of size (z). It is created when $durSu{r}_{s,z}>0$ and $rSuBloo{d}_{s,z}=1$, in other words, when the surgery (s) of of size (z) exists, and when blood bags are necessary.
Equation 21 limits the number of surgeries performed in the day (d) depending on the availability of blood bags made available by the hemotherapy service.
Equation 22 ensures, for any 5 consecutive days (week), that the group of residents (r) will be scheduled in at least $\underset{\xaf}{fClwk}$ days. In this equation, it was necessary to define a time horizon different from the rest of the model to prevent the sum, for example, of 3 daily slots beginning on day 248 (in this case, the solver would try to add until day 251, which would be greater than the planning horizon). In other words, it is created for all periods excluding the last $\underset{\xaf}{fClwk}$ days (in this case, 250 $\underset{\xaf}{fClwk}$).
Equations 23, 24 and 25 ensure, for any 5 consecutive days (week), that the model allows an allocation of up to $\overline{nRoomwk}$ rooms, up to $\overline{nAnwk}$ anesthetists and up to ${\overline{nEqwk}}_{e}$ of equipment.
6. Tested scenarios
The model presented in the previous section has parameters that haven’t been presented so far and that have the role of assigning weights to the different scenarios and of considering business rules related to the minimum frequency of scheduling of the group of residents per week ($\underset{\xaf}{fClwk}$), the maximum amount of rooms available in the week ($\overline{nRoomwk}$), the maximum number of anesthetists available in the week ($\overline{nAnwk}$), and the maximum amount of equipment (e) available in the week (${\overline{nEqwk}}_{e}$). These factors are relevant to our problem because they add internal rules of the hospital about weekly resource availability as well as the ideal frequency stablished by the chief of vascular surgery. As we do not know if there are feasible solutions given these constrains, we do not consider these factors on the first 10 scenarios. If we obtain feasible solutions on these scenarios, we will add these factors to bring the solution closer to reality.
In our operating room planning a time horizon of 250 days was considered and the current business rules are: $\underset{\xaf}{fClwk}=3$, $\overline{nRoomwk}=8$, $\overline{nAnwk}=4$ and ${\overline{nEqwk}}_{e}=6$. Depending on its parameters, the triad combination of number of surgeons by type, minimum number of surgeries, and current business rules may not create feasible solutions, as it occurs in practice, because current rules are: $\underset{\xaf}{fClwk}=3$, $\overline{nRoomwk}=12$, $\overline{nAnwk}=4$ and ${\overline{nEqwk}}_{e}=4$.
In order to assess not only the current business rules, but also the annual availability range of each resource and their criticality, we propose eleven scenarios in Table 12. The first column specifies the scenarios. Columns 2, 3, and 4 present the weights assigned to the operating room allocation ($Roo{m}^{r,d}$), to the allocation of anesthetists ($AnS{u}^{r,d}$) and to the allocation of equipment ($EqDa{y}_{e}^{r,d}$). Columns 5, 6, 7, and 8 present the parameter values that have been adopted for each business rule.
Although only the last scenario has a “practical value”, all responses allow us to evaluate which rules have more impact, and which ones interfere little in programming. From this, we may inform the decisionmaker as to which rules could be disregarded, reinforced, adopted or negotiated with other clinics or with the head of the operating room.
In this case, taking as example scenario 1 of Table 12, the solver will allocate the team of surgeons from the same residency year in as few rooms as possible (highest weight assigned in scenario 1). Then, once the number of rooms is defined, the solver will seek the least amount of anesthetist allocation (intermediate weight in scenario 1), by reallocating the surgeries in each room and in each day. At the end, after the number of rooms and anesthetists are specified, the solver will seek the least amount of equipment allocation (lower in scenario 1). Moreover, in this scenario, there is no obligation to schedule the group of residents in every week and up to two rooms available on each of the five days of the week are considered. In other words, there are 10 rooms which may all have anesthetists and equipment in the week if necessary, even if this is not desirable.
7. Results and discussion
In Table 13, the first column designates the scenario and the second presents, nominally, the order of the adopted resource prioritization (translating “the weights” shown in Table 12). The third and fourth column present the size of the model, that is, the number of decision variables (being whole or binaries) and how many constraints were used respectively. Given that a minimum frequency value of scheduling of the group of residents week has not been established ($\underset{\xaf}{fClwk}=0$) in scenarios 1 to 8, constraint 21 is not active in these cases, and they have the same number of decision variables and constraints.
The fifth column presents the resolution times of the scenarios and the sixth column presents their respective gaps. These gaps are the percentage difference between the result found by linear relaxation (fractional solution without “practical value”) and the best whole solution (allocation of groups of residents, for example), found during the resolution time. In case the gap reaches the value of 0.00%, it means that the best whole solution found is the optimal solution.
The results of the best whole solutions found for each scenario are presented in columns 7, 8, 9, and 10. These solutions specify the time horizon of 250 days, the average use and the smaller number of rooms (# Rooms), the number of rooms with anesthetist (# Rooms with anesthetist) and the number of rooms with equipment (# Rooms with equipment), which refer, in the model, to the terms $\sum}_{r,d}Roo{m}^{r,d$, $\sum}_{r,d}AnS{u}^{r,d$ and $\sum}_{e,r,d}EqS{u}_{e}^{r,d$, respectively.
One may observe that, in order to carry out the surgeries defined by the regulation and the preceptor in the first five scenarios, the amount of allocated resources vary in the intervals from 273 to 347 for rooms, from 224 to 279 for rooms with anesthetists, and from 200 to 310 for rooms with equipment. In this case, as the surgical service in question currently has 8 operating rooms during the week (a rule defined by the direction of the surgical center), and therefore 400 rooms in the year (up to 2 surgery rooms per day up to 8 rooms per week during the 250 days), it can be stated that this constraint of rooms (Equation 22) does not limit the space of viable solutions.
However, due to the scarcity of other resources (anesthetists and equipment), the direction of the surgical center assigns, in the weekly horizon, only 3 days of rooms with anesthetists and 3 days of rooms with equipment. From the results of the model, it is concluded that these two rules make any solution impossible. Considering the annual horizon (250 days), these resources would be available only in 150 days (3 days a week over 50 weeks) whereas at least 224 rooms with anesthetist and 262 rooms with equipment (scenario 4) or 247 rooms with anesthetist (scenario 2) would be required.
So, the only resource that would not make sense to prioritize (assign greater weight) in detriment of others would be the surgery rooms. However, scenario 1 was chosen as ideal by the chief surgeon of the vascular surgery service with the argument that other surgical services receive fewer rooms in the week and this kind of solution could possibly free room spaces to be allocated according to other rules or even just to be available.
Thus, considering the priority order proposed in scenario 1, scenario 9 was built by changing the minimum frequency of scheduling of the group of residents per week to 3 ($\underset{\xaf}{fClwk}=3$), and by changing the maximum amount of rooms, anesthetics and equipment (e) assigned in the week to 8 ($\overline{nRoomwk}=\text{}\overline{nAnwk}=\text{}\overline{nAnwk}=8$).
Finally, scenario 10 was obtained by changing the previous parameters, respectively, to 3, 8, 7, and 6. Therefore, in the last scenario, the goal is to bring the last two parameters closer to the current business rules and check the impact of this change in the found solution. By the preceptor’s choice, scenario 10 was admitted as the ideal one, due to the fact it is the closest to the current set of rules (scenario 11), which has no feasible solution.
8. Conclusion
The present research proposed a model of resource allocation to meet the minimum training criteria required by current legislation and by the chief surgeon of a vascular surgery service. Thus, besides efficiently designing the planning and scheduling of the elective surgeries of this service (the research question per se), it was possible to evaluate the pertinence of current business rules of the surgical center and of business rules internal to the service, responding, at a tactical level (Master Surgical Schedule), when each group of residents should be allocated in which room, with which resources, and how many times it should be scheduled to perform any surgery in a weekly horizon.
The results, for different orders of resource prioritization, indicate that intervals of 273 to 347 surgery rooms, 224 to 279 rooms with anesthetics, and 200 to 310 rooms with equipment are necessary annually for the accomplishment of the minimum number of surgeries necessary for the approval in the residence program
To ensure a feasible and efficient surgical planning, which, for each scenario, involved different logics of resource prioritization, the usage ratio of surgery rooms ranged from 67.6% to 85.6%.
Future works could incorporate constraints that take into account not only the availability of staff and ergonomic regulations (Erhard et al., 2018Erhard, M., Schoenfelder, J., Fügener, A., & Brunner, J. O. (2018). State of the art in physician scheduling. European Journal of Operational Research, 265(1), 118. http://dx.doi.org/10.1016/j.ejor.2017.06.037.
http://dx.doi.org/10.1016/j.ejor.2017.06...
), but also criteria related to social preferences (Roland & Riane, 2011Roland, B., & Riane, F. (2011). Integrating surgeons' preferences in the operating theatre planning. European Journal of Industrial Engineering, 5(2), 232250.) among which one may mention: no major surgeries on Fridays, more specific rules for scheduling in the weekly horizon (for example: that the surgeon should be scheduled just on Mondays, Wednesdays and Fridays or just on Tuesdays, Thursdays only in the morning or only in the afternoon).

How to cite this article: Assad, D. B. N., & Spiegel, T. (2019). Maximizing the efficiency of residents operating room scheduling: a case study at a teaching hospital. Production, 29, e20190025. https://doi.org/10.1590/01036513.20190025
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Publication Dates

Publication in this collection
15 Aug 2019 
Date of issue
2019
History

Received
19 May 2019 
Accepted
09 July 2019