Optimizing Surgeon Scheduling with Mathematical Programming

As healthcare expenditures rise, pressure is growing to increase the value of care and improve patient access. Some patients in need of care lack the resources to seek it or are geographically constrained to remote areas. To better serve these patients, many hospitals have established satellite locations regularly staffed by their physicians. There are both economic and operational challenges to scheduling surgeons across multiple clinic locations and operating rooms (ORs). Current methods for assigning surgeons to clinic dates and OR blocks are not quantitative, do not maximize patient access, and do not optimize for surgeon utilization or convenience. We developed a generalizable mathematical model for designing optimized surgeon schedules. We designed and implemented changes to the schedule for the Stanford Children’s Orthopedic and Sports Medicine Center that increase patient access to multiple surgical subspecialties and decrease surgeon drive time. We quantify the potential benefit of increased telemedicine investment.

In 2015, national healthcare expenditure in the United States totaled $3.2 trillion and accounted for 17.8% of the U.S. gross domestic product (GDP).1 Under new reimbursement procedures, the healthcare industry is faced with pressure to increase the value of care—for example, by improving access for patients. Due to the widespread incidence of burnout among physicians, burnout prevention and physician wellness have become pressing concerns.2 As hospital networks expand to cover satellite clinics, they are faced with challenges and goals that often may be at odds: minimize costs while providing high-quality care; maximize patient access to all of the tertiary referral hospital’s surgical subspecialties; and maintain surgeon satisfaction and commitment.

A surgeon’s time is incredibly valuable financially. If a surgeon spends three hours per day commuting rather than providing patient care, that time comes at an extremely high opportunity cost, which can be illustrated, although significantly underestimated, by his or her mean hourly wage.3 Research has found an inverse relationship between length of commute and physical health, anxiety, and happiness, and also between the risk for burnout and the amount of time physicians spend on the most meaningful aspects of their work.4-6 Maximally efficient daily schedules allow surgeons to spend more time on patient care rather than on commuting (Figure 1).

20210908 275-279_Hong_MarApr19_Figure1a
20210908 275-279_Hong_MarApr19_Figure1b

Figure 1. (a) Lucile Packard Children’s Hospital has satellite clinics that span the entire Bay Area (circles). (b) Daily commute times calculated based on weekday traffic patterns. The dots indicate there are additional cities and clinics not shown in this example.

Current methods of surgeon scheduling, which include assigning surgeons to OR blocks, clinic locations, and shift dates, are time-intensive and do not result in optimal schedules for either patient care or surgeon utilization. There are millions of possible schedules, and a human is not likely to find the optimal one. Mathematical optimization is a tool for finding the best possible allocation of resources (e.g., surgeon assignment to specific clinics) under specified constraints (e.g., subspecialty requirements). Optimization models provide a generalizable, quantitative approach to scheduling that is quick, customizable, and objective. Several clinical applications have been studied to optimize hospital financial planning, nurse staffing, and multiclinic assignment.7-9 However, a review of the literature shows that few of these optimization techniques have been implemented in practice. This is in marked contrast to other high-cost, highly efficient industries, such as airlines, global commerce chains, and the National Football League, that use these optimization methods.10-12

Advances in technology have made optimization more useful and accessible. We improved on prior work using tools that are free for academic use: R, Google Maps API, Python, and Gurobi. For example, previous studies use distance as a proxy for drive time in multiclinic assignment. The limitations of this method are apparent to anyone who lives in an area with significant rush hour traffic. The integration of Google Maps facilitates the calculation of time-specific commutes, rather than driving distance, to account for traffic variability and gives significantly more realistic results. The calculation of “windshield time”—the time a surgeon is behind the windshield driving a car rather than providing patient care—allows departments to effectively quantify the benefits of making changes such as implementing telemedicine and hiring a car service or app-based ride service for surgeons with long commutes.



We conducted extensive interviews within the Stanford Children’s Orthopedic and Sports Medicine Center and identified the requirements and constraints surgeons’ schedules must satisfy. These included physician seniority, the OR block schedule, where the physicians lived, each physician’s subspecialty expertise (spine, hip, sports, etc.), and the subspecialty coverage goals for each clinic. The goal of the department leadership was to achieve full subspecialty coverage across all primary clinics, maintain current coverage at the secondary clinics, and decrease the surgeons’ overall commutes. We created an integer program (IP) to create clinic and OR assignments that maximizes subspecialty coverage and minimizes the department’s collective windshield time, while respecting the specified constraints.

Quantification of Current State

We used Google Maps to calculate the drive time accounting for direction-specific and time-specific traffic patterns for each morning, afternoon, and evening. We calculated drive time between the city of each physician’s residence and each clinical location and the drive time between each pair of clinical locations.

We established a baseline of the current state by quantifying drive times, subspecialty coverage of the department, and drive time per session. We validated these results with the surgeons.

Integer Programming Mathematical Optimization Model

Mathematical optimization is a method for finding the value of decision variables in order to minimize a specified objective function—for example, drive time—while meeting certain constraints—for example, assigning exactly one surgeon to work in each OR. Integer programming is a powerful type of mathematical optimization used for scheduling problems where one or more of the decision variables must take on integer values, such as, for instance, the number of surgeons to assign to each clinic. After a problem of practical interest has been translated into an integer program, solving it may involve comparing millions or billions of combinations of variables, which may be done with a modern commercial integer program solver. Bradley et al.13 offer an introduction to integer programming and describe how to model practical problems with integer programming.

Our decision variables are whether or not to assign each surgeon to each shift and location for each week and each day. A decision variable takes on value 1 if the corresponding surgeon was assigned to the corresponding location on that week on that day on that shift and 0 otherwise. The objective function to be minimized is the total windshield time of all the surgeons. The constraints are all fixed clinic and OR sessions, subspecialty coverage requirements, surgeon location assignments, and each surgeon’s required number of clinic and OR sessions. After initial models revealed that it was mathematically impossible to meet all of the department’s stated constraints, we modified the model to allow the constraint on the number of assignments for each surgeon to be violated and added a term to the objective function to minimize the amount by which it would be violated.

We implemented the model in Python and used the Gurobi solver to numerically solve the integer program.14 The program consists of multiple decision variables and 15 to 16 constraints, depending on the iteration of the IP. There is a decision variable for each surgeon, week, day, shift, and location.

Full Optimization

Our initial optimization accounted for 11 physicians, 16 physician locations (6 primary clinic locations, 5 secondary or specialty clinic locations, and 5 OR locations), and 5 physician subspecialties. This optimization was used to estimate the best possible coverage and reduction in windshield time by omitting some of the restrictions on where physicians could be assigned.

Targeted Optimization

Our second iteration of the optimization took into account the addition of two new physician hires, and was a more operationally feasible, targeted optimization aimed toward minimizing windshield time based on predetermined locations for surgeon assignment. We restricted each physician’s potential locations to one primary clinic, two satellite clinics, and any specialty clinics and OR locations to which the physician was already assigned.


Current State

We found significant inequalities in drive time, with some surgeons driving over 40 hours per month to fulfill their clinical duties. Subspecialty coverage was not in line with the department’s stated goals. Only two of the six primary clinics achieved full subspecialty coverage every week of the month.

Full and Targeted Optimization

The solve time for each optimization was approximately 2 minutes on a computer with 2.2 GHz CPU and 8 GB of RAM.

The full optimization resulted in a total decrease in windshield time of 41 hours per month across the department and achieved full subspecialty coverage at all six primary clinics, while maintaining subspecialty coverage at the five secondary clinics. The largest decrease in individual monthly windshield time was 27 hours per month. Of the surgeons who experienced an increase in drive time per session, the largest increase was 7 minutes per session. The results of the full optimization are summarized in Table 1.

20210908 275-279_Hong_MarApr19_Table 1

The targeted optimization resulted in an overall decrease in windshield time of 34 hours per month and doubled full subspecialty coverage from one to two primary clinics, while maintaining subspecialty coverage at the five secondary clinics. The largest decrease in individual monthly windshield time was 12 hours per month. Of the surgeons who experienced an increase in drive time per session, the largest increase was 2 minutes per session. The results of the targeted optimization are summarized in Table 2.

20210908 275-279_Hong_MarApr19_Table2

Time-Specific Quantification of Drive Times

We calculated windshield time during morning, midday, and evening weekday commute hours for physicians. We also calculated drive time at 3 AM, when traffic is minimal, to show how grossly windshield time would be underestimated if our optimizations were calculated independent of traffic patterns. The comparisons are summarized in Table 3.

20210908 275-279_Hong_MarApr19_Table3


We built a generalizable mathematical model that uses real-world inputs, such as traffic-adjusted drive times, to create surgeon schedules that maximize patient access and surgeon convenience. The schedules produced for the Stanford Children’s Orthopedic and Sports Medicine Center improved surgical subspecialty coverage and decreased overall surgeon windshield time. The model may be used by any other surgical department with the appropriate modification of the input variables, such as the location and subspecialty of the surgeons; it does not depend on any information specific to our institution or department.

Optimizing and quantifying surgeon windshield time increases awareness of physician time utilization.

The fully optimized schedule gives the best possible combination of clinic subspecialty coverage and overall windshield time. The targeted optimization schedule maximizes subspecialty coverage given the operational constraints of providing consistent surgeon schedules, minimizing the number of distinct locations each surgeon visits, and meeting other surgeon preferences. A comparison of these two schedules allows the department to evaluate the potential benefits of removing some of the current constraints; to better understand the gaps in coverage to help target future hires; and to rigorously analyze how physicians use their time. Optimizing and quantifying surgeon windshield time increases awareness of physician time utilization, and allows for identification of where physicians are spending—or in this case, “wasting”—their time on daily commutes. The elucidation of drive time for each surgeon resulted in the implementation of a department-wide app-based car service for commutes estimated at 45 to 60 minutes. This has resulted in increased satisfaction, because it allows surgeons to sleep, call patients, prepare for cases, dictate charts, or work on research during their commutes.

With patient access becoming a greater concern throughout the country, the need to provide subspecialty services at all satellite offices will continue to grow. Newer locations may require longer commutes and have higher visit-to-surgery conversion ratios, so they may not be as attractive for physicians. By quantifying each physician’s drive time in a transparent way, the optimized schedules can shed light on how windshield time is being distributed throughout the department and help resolve questions of fairness. Calculating the best possible combination of clinic coverage and windshield time also allows for a quantitative cost–benefit analysis of the potential benefits of telemedicine implementation at a clinic.

As with all optimizations, limitations in implementation may arise when optimizing for certain metrics. Fully optimized schedules may pose clinical challenges. The schedules are not optimized to assign a physician to the same clinic at regular intervals or on the same day each week. This may cause uncertainty for referring providers if they do not know to whom or where they are referring their patients. Challenges also may arise in scheduling follow-up appointments for patients based on their clinical course (e.g., one-week follow-up post-surgery). Constraints for regularity could be added into the integer program, but each additional constraint reduces the effectiveness of the optimization. Overhauling physician schedules is a significant clinical and administrative task, because surgeons already have cases and clinic appointments booked out far in advance. It may be necessary for some physicians to experience an increase in windshield time in order to optimize the entire department’s schedules. These pose necessarily non-mathematical challenges of how to balance a department’s goals with individual physician satisfaction that lie outside the scope of our model.


We created a computer model that generates schedules optimized to reduce windshield time while respecting constraints such as surgeon days off, clinic coverage, OR time, and surgeon preferences. Our novel contributions include using an interface with Google Maps to determine traffic-dependent travel times. We created an optimized schedule for the Stanford Children’s Orthopedic and Sports Medicine Center that was implemented with relatively minor modifications; provided the quantitative basis for contracting with a car service for the surgeons; and revealed how the department could most efficiently achieve its goals by hiring surgeons in specific subspecialties and investing in telemedicine. This model can easily be customized for use by another department by anyone with intermediate programming experience, a skill significantly more common than knowledge of how to build a mathematical optimization.

Scheduling with mathematical optimization is a generalizable, implementable approach that allows departments to optimize schedules to meet their objectives while satisfying their constraints.



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  9. Franz LS, Baker HM, Leong GK, Rakes TR. A mathematical model for scheduling and staffing multiclinic health regions. European Journal of Operational Research. 1989;41(3):277-89.

  10. Yu G, Yang J. Optimization applications in the airline industry. In: Du DZ, Pardalos PM (eds). Handbook of Combinatorial Optimization. Vol. 2. Boston, MA: Kluwer Academic Publishers; 1998:635-726.

  11. Ahire SL, Malhotra MK, Jensen JB. Carton-mix optimization for Walmart.com distribution centers. Interfaces. 2015;45:341-357.

  12. Creating the NFL Schedule. NFL Football Operations. https://operations.nfl.com/the-game/creating-the-nfl-schedule/. Accessed August 1, 2018.

  13. Bradley SP, Hax AC, Magnanti TL. Integer programming. In: Applied Mathematical Programming. Boston, MA: Addison-Wesley; 1997:272-319.

  14. Gurobi Optimizer Reference Manual. Gurobi Optimization Inc. www. gurobi.com. Accessed May 10, 2018.


Tracey Hong, MS
Stanford University, Department of Management Science and Engineering, 475 Via Ortega, Stanford, CA 94305; traceyhong@alumni.stanford.edu.

Michael Fairley, MS
Stanford University, Department of Management Science and Engineering; Lucile Packard Children’s Hospital, Stanford, California.

David Scheinker, PhD
Stanford University, Department of Management Science and Engineering, Stanford, California.

Steven Frick, MD
Lucile Packard Children’s Hospital, Stanford, California.

This article appeared in the March/April 2019 issue of The Journal of Medical Practice Management.




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