Developing Fair and Feasible Schedules for Residents On-CallSarah RootUniversity of MichiganDepartment of Industrial and Operations EngineeringINFORMS Pittsburgh – November 5, 2006Joint work with Amy Cohn

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Outline

What is on-call resident scheduling? Boston University School of Medicine

Psychiatry Program Literature Modeling approach Algorithm

Importance of collaboration Conclusions and future research Questions?

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What is On-Call Resident Scheduling? Residency – period of formal medical

education designed to facilitate the transition from licensed medical doctor to practicing physician Typically lasts 3-4 yearsAdvanced training in specialty areaResidents complete rotations through

several different areasResidents complete a series of calls or

overnight shifts as part of their residency

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Boston University School of Medicine Psychiatry Residency Program Calls completed at three sites

Sites have different characteristics and must be scheduled differently

Different program requirements and availability for students in each class Schedule primarily PGY2s and PGY3s; other

miscellaneous residents with makeup requirements

Typically 6-8 residents per class Must complete a pre-specified number of

weekday and weekend calls at each site Different day of week availability for each class

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Boston University School of Medicine Psychiatry Residency Program Feasibility rules

All calls must be coveredAll residents must satisfy program

requirementsSpacing between calls, maximum calls

per monthResidents can only work on days they

are available

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Boston University School of Medicine Psychiatry Residency Program Call schedule spans a year

Developed manually by chief residents Typically created by scheduling one resident at a time Difficult to satisfy all feasibility rules

Schedules implemented that did not satisfy feasibility rules Want a good schedule that is fair

What is a “good schedule” and how do you assess whether or not it is fair? Must satisfy feasibility rules Should incorporate desirable characteristics No resident should get a high-quality solution at the

expense of a significantly poorer solution for another resident

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Literature Vast literature on medical scheduling

Burke, 2004; Cheang, 2003; Siferd, 1992 – Nurse scheduling survey papers

Beaulieu, 2000; Carter, 2001; Valouxis, 2000 – Emergency room physician scheduling

Ozkarahan, 1994; Franz and Miller, 1993 – Schedule residents for daytime locations

Sherali, 2002 – On-call schedules produced for 4-5 week periods

BUSM Problem Schedule horizon is one year Must schedule three sites simultaneously Difficulty in defining objective function

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On-Call Formulation Notation

R – set of residents to be scheduled RPGY2 – set of PGY2 residents RPGY3 – set of PGY3 residents

D – set of days to be scheduled Hospital A

Covered by two residency programs V – set of days to be covered by BUSM

Must schedule primary and backup each day xrd = 1 if resident r is scheduled for a primary call at

Hospital A on day d 0 otherwise for all r in R, d in V yrd = 1 if resident r is scheduled for a backup call at

Hospital A on day d 0 otherwise for all r in R, d in V

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On-Call Formulation Hospital B

Can be covered by a resident or as an “extra on-call” (EOC) zrd = 1 if resident r is scheduled for primary call at Hosptial B

on day d 0 otherwise for all r in R, d in D ed = 1 if day d is scheduled as an EOC 0 otherwise for all d in D

Hospital C Each PGY2 must be scheduled for six consecutive weeks Scheduled by chief resident prior to creation of the call

schedule qrd = 1 if resident r is scheduled at Hospital C on day d 0 otherwise

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On-Call Formulation

Hospitals A and B must be staffed:

Σ xrd = 1 d in V

Σ yrd = 1 d in V

Σ zrd + ed = 1 d in D

xrd + yrd + zrd ≤ 1 - qrd d in D, r in RPGY2

r in R

r in R

r in R

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On-Call Formulation

Training requirements for Hospitals A and B must be fulfilled:

Σ xrd = vrPW r in R

Σ xrd = vrPE r in R

Σ yrd = vrBW r in R

Σ yrd = vrBE r in R

Σ zrd = brW r in R

Σ zrd = brE r in R

d in V

d in V

d in V

d in D

d in V

d in D

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On-Call Formulation

PGY2s cannot work Tuesdays:

Σ xrd + Σ yrd + Σ zrd = 0 r in RPGY2

PGY3s cannot work Tuesdays or Wednesdays:

Σ xrd + Σ yrd + Σ zrd = 0 r in RPGY3

Each resident can work no more often than every fourth day:

Σ xrd + Σ yrd + Σ zrd ≤ 1 r in R, d’ in {1…(|D|-3)}

Each resident can work no more five calls each calendar month:

Σ xrd + Σ yrd + Σ zrd ≤ 5 r in R, m in M

d in T d in T d in T

d in {TUW} d in {TUW} d in {TUW}

d in M d in M d in M

d in {d’…d’+3}d in {d’…d’+3}d in {d’…d’+3}

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On-Call Formulation What about the objective function?

Desirable characteristics Vacation requests satisfied Alternating primary and backup calls at Hospital A PGY2 daytime-nighttime location match PGY3 blocking Day of week preferences honored

Many stakeholders with differing objectives Individual residents want good schedules, but each have

differing preferences Chief residents want to ensure equity among residents

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On-Call Formulation Often handled by assigning weights for desired

characteristics wir = weight assigned by resident r to characteristic i cir = amount of characteristic i present in resident r’s schedule

max ΣΣ wir cir

How do we assign these weights? How are we ensured that the weights are correct?

People don’t know what they want!! Similar problems in ensuring equity Capturing metrics to measure each characteristics

results in a larger, harder to solve mathematical program

i r

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On-Call Formulation

Alternating Primary and Backup Calls Ir – total number of calls resident r completes at

Hospital A crid = 1 if resident r’s ith Hospital A call is assigned on

day d prid = 1 if resident r’s ith Hospital A call is a primary call

and takes place on day d arid = 1 if resident r’s ith Hospital A call is a primary call

and takes place on day d gri = 1 if resident r’s ith Hospital A call is followed by an

i+1th call of the same type Gr = total number of non-alternated calls by resident r

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On-Call Formulation

Alternating Primary and Backup Calls, cont.Σ crid = 1 i in {1…Ir}, r in R

Σ drid ≥ Σ d’cr(i-1)d’ i in {2…Ir}, r in R

crid ≤ xrd + yrd i in {1…Ir}, r in R, d in V

prid ≤ xrd i in {1…Ir}, r in R, d in V

arid ≤ yrd i in {1…Ir}, r in R, d in V

Σ prid + Σ pr(i+1)d ≤ 1 + gri i in {1…(Ir-1)}, r in R

Σ arid + Σ ar(i+1)d ≤ 1 + gri i in {1…(Ir-1)}, r in R

Gr = Σ gri r in R

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On-Call Formulation

KEY INSIGHT:

People have a hard time translating their preferences into weights, but they can easily identify what they don’t like about potential schedules

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Algorithm to Solve On-Call Scheduling Problem

1. Find best case scenario for each characteristic in isolation.

Vacation requestsTotal: 2 requests deniedMax for individual: 1 request denied

Sarah Root : 1 req. deniedShervin Williams: 0 req. deniedAmy Smith: 0 req. deniedRichard Chang: 1 req. deniedAda Johnson: 0 req. deniedYihan Clinton: 0 req. denied

1 2 3 4 5 6

STOP

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Algorithm to Solve On-Call Scheduling Problem

2. Work with chief residents to establish an initial, generalized ordering of desirable characteristics.

1. Vacation requests satisfied2. PGY2 daytime-nighttime location match3. Alternating primary and backup calls at Hospital A4. Day of week preferences honored5. PGY3 blocking

1 2 3 4 5 6

STOP

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Algorithm to Solve On-Call Scheduling Problem

3. For each criterion in sequence, find optimal solution while restricting the set of solutions to remain optimal or near-optimal with respect to previous characteristics.

1. Minimize unsatisfied vacation requests 2 unsatisfied vacation requests

2. Maximize PGY2 daytime-nighttime location match s.t. unsatisfied vacation requests ≤ 2 3 unmatched daytime-nighttime locations

1 2 3 4 5 6

STOP

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Algorithm to Solve On-Call Scheduling Problem

4. If the chief resident is happy with the solution, terminate with a feasible schedule. Otherwise proceed to the Step 5.

1 2 3 4 5 6

STOP

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Algorithm to Solve On-Call Scheduling Problem

5. Identify undesirable attributes in the schedule and introduce additional constraints to prevent them.

Inequity in number of Fridays assigned to each resident s.t. number of Fridays for each resident ≤ 8

1 2 3 4 5 6

STOP

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Algorithm to Solve On-Call Scheduling Problem

6. Resolve problem with additional constraints to generate schedule. Return to Step 4.

1 2 3 4 5 6

STOP

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Conclusions

Solution implemented in July 2006 and currently in use No schedule modifications other than individual

ad-hoc swaps Defining a single, accurate objective function

very difficult Involvement of chief residents critical

Understanding tradeoffs in solution characteristics

Credibility and solution buy-in

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Conclusions

“We are using the schedule in its entirety, exactly as your program

generated. There have been surprisingly few requests for call

swaps so far. There were no errors that we could detect, and the residents had

no complaints about it. It has been great, and greatly decreased our

headaches as Chiefs!”

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Future Research Directions

Understand other related problemsResident scheduling problems for other

schools and specialtiesMedical scheduling problems

Generalize approach to solve related problems with unclear objective functions

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Questions?

Sarah Rootseroot@umich.edu