1. Scheduling Surgery Residents New rules limit work hours Scheduling problem Interesting context

2. Medical Education in the US Bachelor (4 years) Medical School (4 years) –$42,974/year tuition at Northwestern Residency –3-7y, 5y for general surgery –paid (first years: $48,270.96 at Northwestern) Fellowship (subspecialties)

3. Residency Training of doctors Cheap labor –Funded by Medicare –“If you have one less nurse, then you've saved $40,000. If you have one less resident, you've saved $40,000, but you've lost $100,000 in reimbursement.” Limited supply –The “Match” and the “Scramble” Working conditions

4. Work Rules NY State Libby Zion law (1989) From accreditation body (ACGME, 2003) –Work week ≤ 80 hrs (over averaged over 4wks) –≤1 on call night every 3 days –30 hr max shift –10 hr between shifts Routinely flouted Does not apply to fellows Controversy –Reducing medical errors –Getting necessary medical training

5. Scheduling Problem Hospital’s (cheap labor) perspective How to cover various areas –Rotations in Duke hospital, and –Nearby hospitals Without violating the law

6. Formulation Seeking a schedule for a week MIP feasibility problem I = set of residents J = services to cover –Includes other hospitals –Includes time-off K = set of 4 hr time slots x ijk = 1 if resident i covers service j at time k –binary

7. Formulation z ij =1 if resident i does rotation j (binary) ∑ j z ij ≤ 1 (at most one rotation in a week) ∑ k x ijk ≤ 20z ij (80 hrs/rotation limit) ∑ j x ijk ≤ 1 (one thing at a time) ∑ ik x ijk ≥ 1 (at least one available at each loc) 2x i,11,k +x i,11,k+1 +x i,11,k+2 ≥ 2x i,11,k-1 (breaks ≥ 12hr) …

8. Sample Result 1st year resident in trauma rotation Schedule –Monday 6am to Tuesday 6am –Tuesday 6pm to Wednesday 6am –Thursday 10am to Friday 6am –Saturday 6am to Sunday 2am 76 hours total

9. Criticism A year is scheduled at a time Educational component? Ensuring coverage not a compelling constraint Hospital could hire more –Residents –Doctors