Complexity results for the basic residency scheduling problem

Abstract

Upon graduation from medical school, medical students join residency programs to complete their clinical training and fulfill specialty board certification requirements. During residency, they are assigned several years of clinical rotations, where they work under the supervision of physician faculty in a variety of different settings, to ensure that they gain the requisite training prior to beginning independent practice. These rotations typically last a short period of time, and the problem of determining a schedule for all the residents in a program can be quite tedious. In this paper, a basic residency scheduling problem that produces a 1-year schedule is defined, and a proof of NP-completeness is presented. Furthermore, a specific model of the residency scheduling program for the internal medicine residency program at the University of Illinois College of Medicine at Urbana-Champaign is studied. Finally, a method for determining alternate optima is presented.

Appendix

1.1 A List of rotation types and additional notation for UIUC-COM

Carle Foundation Hospital (CFH)

Provena Covenant Medical Center (PCMC)

Veteran’s Administration Hospital (VA)

• – CFH adult medicine (CAM)

• – PCMC adult medicine (PAM)

• – CFH/PCMC night float (NF)

• – CFH critical care (CCC)

• – VA Nephrology (VAN) *

• – PCMC Gastroenterology (PG) *

• – Carle Cardiology (CC) *

• – VA Geriatrics (VAG) *

• – VA Triage (VAT) *

• – Carle Infectious Disease (CID) *

• – Ambulatory (AMB) *

• – Elective rotation for PGY2/3 residents (ER)

• – Introduction to clinical research/vacation (ICR-VAC)

In the above list, starred problems indicate “backup rotations,” that is, rotations that provide additional coverage in case of illness or other schedule disruptions. The following list provides the notation used in the model for the UIUC-COM residency program:

• – $\mathcal {R}{}X, X \in \{p,1,2,3\}$: the set of prelim, PGY1, PGY2, and PGY3 residents, respectively

• – $n_X, X \in \{p, 1, 2, 3\}$: the number of prelim, PGY1, PGY2, and PGY3 residents, respectively

• – $\mathcal {P}= \{1,\ldots ,13\}$: the set of 4-week rotation periods, ordered chronologically

• – $\mathcal {T}$: the set of specific rotation types

• – $\mathcal {S}$: the set of shift rotations, that is, CAM, PAM, NF, and CCC. The hours for the shift rotations are provided in Table 6.

• – $DP = \{1,\ldots ,28\}$: the days of a rotation period

• – $D = \{1,\ldots ,7\}$: the days of the week

Table 6 A list of working hours for all rotation types at UIUC-COM

1.2 B Additional requirements for UIUC-COM

The following list all of the additional side constraints present in the UIUC-COM residency scheduling problem (ACGME 2009).

• 1. Additional educational requirements

  • (a) PGY2 residents prefer to have elective in rotation periods 10–13. This model treats the preference as a hard constraint.

  • (b) A resident cannot take NF for two consecutive rotation periods in each year; a resident cannot take more than two NF rotations in a year.

  • (c) Every resident must take at least three and no more than six critical care rotations over 3 years. †

  • (d) Residents must be assigned to emergency medicine rotations for at least 4 weeks of direct experience in blocks of not less than 2 weeks over the course of 3 years. †

• 2. Additional teaching service demands

  • (a) If a resident takes a shift, then he/she must take the rotation for that shift.

  • (b) There must be at least one PGY1/prelim and one PGY2/3 covering the CAM/PAM/NF $\times$ 2 rotations 24 h, 6 days a week. Note that the last day in a week is covered by resident hospitalists from the hospitalist rotations.

  • (c) It is preferable to have one PGY2/3 providing coverage for CCC 24 h, 7 days per week. This model treats the preference as a hard constraint.

  • (d) PGY2/3 residents must be scheduled on a CCC shift no more frequently than every third night (when averaged over a 4-week period).

• 3. ACGME Rules

  • (a) Residents must not be scheduled for more than 80 h per week, averaged over a 4-week period.

  • (b) Residents must have at least one full (24 h) day out of 7, free of patient care duties, averaged over 4 weeks.

  • (c) Residents must have a minimum rest period of 10 h between duty periods.

  • (d) PGY1/Prelim residents cannot work more than 16 h straight.

  • (e) In-house calls must occur no more frequently than every third night, averaged over a 4-week period.

  • (f) The residency program must provide opportunities for experience in geriatric medicine, neurology, psychiatry, allergy/immunology, dermatology, medical ophthalmology, office gynecology, otorhinolaryngology, non-operative orthopedics, palliative medicine, sleep medicine, and rehabilitation medicine in 3 years. †

  • (g) The residency program must provide opportunities for experience in all internal medicine subspecialties; that is, cardiology, critical care, endocrinology, hematology, gastroenterology, infectious disease, nephrology, oncology, pulmonary disease, and rheumatology. †

Requirements marked with † are requirements that must be satisfied over the course of the 3-year residency program, and thus are not considered by this model.

Note that the ACGME duty hour rules can be eliminated, since they can always be satisfied, given the additional teaching service demands (2b)–(2d). To see this, note that if a resident takes a rotation which is not a shift rotation, then the resident only works from 7am to 5pm, Mon–Fri (see Table 6), which is a 50-h work week. Moreover, he has two weekend days to rest. In this case, all conditions (3a)–(3e) are satisfied.

Additionally, if a resident is assigned to CAM, PAM, or NF, in the worst case, he works from 7am to 7pm 6 days in a week (by condition (2b)), which means he will work 72 h in that week, but will have a full day of rest in between. Thus again, all ACGME rules are satisfied.

In the final case, if a PGY2/3 resident is assigned to the CCC shift rotation, then there must be at least three such residents assigned, by condition (2c) and (2d). Then, by rotating through each of the three residents, each resident gets one day of shift work from 7am to 11am (next day), followed a regular work day from 7am to 5pm on the third day, followed by another shift work day, and then take a rest on the sixth day. Therefore, the average working hour is 77 hours a week, which clearly satisfies conditions (3a)–(3e). Therefore, the ACGME rules are redundant for this model, and are not considered further.