Making Hosptials Run Smoothly

Operating Room Schedules

  • Hospitals have a limited number of ORs.

  • Operating room managers must determine a weekly schedule assigning ORs to different departments in the hospitals.

Difficulties

  • Creating an acceptable schedule is a highly political process within the hospital

  • Surgeons are frequently paid on a fee-for-service basis, so changing allocated OR hours directly affects their income

  • The operating room manager’s proposed schedule must strike a delicate balance between all the surgical departments in the hospital

Logistical Issues

  • Operating rooms are staffed in 8 hour blocks.

  • Each department sets their own target number of allocation hours, which may not be integer.

  • Departments may have daily and weekly requirements:
    • EX) Gynecology needs at least 1 OR per day
    • EX) Ophthalmology needs at least 2 ORs per week
    • EX) The oral surgeon is only present on Tuesdays and Thursdays.

Case study: Mount Sinai Hospital

  • Has 10 ORs which are staffed Monday - Friday
    • 10 ORs x 5 days x 8 hours/day = 400 hours to assign
  • Must divide these 400 hours between 5 departments:

Problem Data

  • Number of surgery teams from each department available each day:
  • Maximum number of ORs required by each department each day:

Additional Problem Data

  • Weekly requirement on number of ORs each department requires:

The Traditional Way

  • Before the integer optimization method was implemented at Mount Sinai in 1999, the OR manager used graph paper and a large eraser to try to assign the OR blocks

  • Any changes were incorporated by trial and error

  • Draft schedule was circulated to all surgical groups

  • Incorporating feedback from one department usually meant altering another group’s schedule, leading to many iterations of this process

Optimization Problem

  • Decisions
    • How many ORs to assign each department on each day.
    • Integer decision variables \(x_{jk}\) represent the number of operating rooms department \(j\) is allocated on day \(k\).

Objective

  • Maximize % of target allocation hours that each department is actually allocated.

  • If target allocation hours are \(t_J\) for department \(j\), then we want to maximize the sum of \((8 * x_{jk}) \div t_j\) over all departments and days of the week.
    • Ex) If otolaryngology has a target of 37.3 hours per week and we allocate them 4 ORs then their % of target allocation hours = \((8 * 4) \div 37.3 = 85.8\%\)

Constraints

  • At most 10 ORs are assigned every day

  • The number of ORs allocated to a department on a given day cannot exceed the number of surgery teams that department has available that day

  • Meet department daily minimums and maximums

  • Meet department daily weekly minimums and maximums

Constraints

  • \(x_{OP,M} + x_{GY,M} + x_{OS,M} + x_{OT,M} + x_{GY,M} \leq 10\)

  • \(0 \leq x_{GY,F} \leq 3\)

  • \(0 \leq x_{OS,W} \leq 0\)

  • \(0 \leq x_{GS,T} \leq 8\)

  • \(3 \leq x_{OP,M} + x_{OP,T} + x_{OP,W} + x_{OP,F} \leq 6\)