Optimal Scheduling for ICU PatientsSIDDHANT BHATT

STEVE BOYLE

ERICA CUNNINGHAM

Problem•Who should be admitted to the ICU• Most severely injured/ill patients?• Patients who would get the most benefit from ICU treatment?• Patients who have been in the hospital the longest?

• Order of admittance for admitted

• Order of scheduling for those on waitlist

• How many beds should be kept available for the most critically ill patients to receive ICU treatment right away?

Analysis• Two papers:• “Multi-Resource Allocation Scheduling in Dynamic Environments” by Woonghee Tim Huh, Nan Liu, and

Van-Anh Truong• “ICU Admission Control: An Empirical Study of Capacity Allocation and its Implication on Patient

Outcomes” by Song-Hee Kim, Carri W. Chan, Marcelo Olivares, and Gabriel Escobar

• Aim to apply techniques used in first paper to problem in second paper

• Objective• Maximize benefit of each patient treated in ICU

The Scheduling Problem• Formulate a scheduling problem which simulates allocating beds in ICU to entering patients

• Will have correlations between ICU and Machine models:• Machines = Beds• Jobs = Patients• rj = Time when patient j arrives to the ICU• pj = Expected stay of patient j in ICU• wj = Expected benefit of patient j with ICU treatment over ward treatment• dj = Deadline of patient j (earliest the patient would need to be treated)• Sj = Time patient j is scheduled in ICU• Cj = Time patient j is release from ICU

Primary and Secondary Goals• Primary Goal: Schedule patients who have applied to the ICU optimally• Reduce problem to 1|rj|Σwj (1-Uj)• Extend problem to multiple machines (2)

•Secondary Goal: Find optimal slack capacity in ICU at any given time• Slack capacity leaves beds available for arriving patients in what are deemed to be “severe” conditions• “Severe” condition guidelines will change based on current occupancy of ICU• Schedule of beds must be updated daily

• To determine if condition is severe enough, we hypothesize using threshold policy

• Minimizing sum of weighted scheduling times• NP Hard problem

Algorithms • Algorithm 1• Step 1: Of released jobs, which jobs can complete on time if scheduled at current time• Step 2: Of those jobs, which jobs have best wj/pj• Step 3: Schedule best wj/pj, update time (t = t + pj), go back to step 1

• Algorithm 2• Step 1: Order jobs by highest weight• Step 2: If highest weight job can be completed before deadline, schedule job when available, update

time, repeat step 1 • Step 3: Else, move to next highest weight job, repeat step 2.

Model 1 (1 machine, no pmtn)• 1|rj|Σwj (1-Uj) – Maximizing the sum of the weighted completed jobs

1 Of released jobs, which jobs can complete on time if scheduled at current time2 Of those jobs, which jobs have best wj/pj

3 Schedule best wj/pj, update time (t = t + pj), go back to step 1

Job 2 Job 1 Job 3 Job 4

2 5 6 10 Σwj(1-Uj) = 14Time Completed

Algorithm 1

1 Order jobs by highest weight2 If highest weight job can be completed before deadline, schedule job when available, update time, repeat step 1

3 Else, move to next highest weight job, repeat step 2.

Job 2 No Jobs Job 4

2 4 8 Σwj(1-Uj) = 17Time Completed

Algorithm 2

Jobs rj wj pj dj

1 0 4 3 52 0 10 2 23 3 3 1 54 4 7 4 9

Model 2 (2 machines, no pmtn)• P2|rj|Σwj (1-Uj) – Maximizing the sum of the weighted completed jobs

Jobs rj wj pj dj

1 0 4 1 10 Machine 1 J12 0 9 3 8 Machine 23 1 3 4 54 3 5 5 12 Machine 1 J15 6 3 4 11 Machine 2

Algorithm 1

Algorithm 2

J2J3

J4

J5J4

J2J5

Σwj(1-Uj) = 24

Σwj(1-Uj) = 21

Jobs rj wj pj dj Machine

1 0 4 3 5 Machine 2 Job 32 0 10 2 23 3 3 1 5 Machine 14 4 7 4 9 Machine 2 Job 3

Algorithm 1

Algorithm 2 Job 4Job 2Job 1

Job 4Job 1

Job 2Σwj(1-Uj) = 24

Σwj(1-Uj) = 24

What can we conclude from examples?• 1|rj|Σwj (1-Uj) is a hard problem

• Algorithm 2 finds a better solution than Algorithm 1 in the simplified 1 machine example• Algorithm 2 proved to find a better solution than Algorithm 1 in the simplified 1 machine example

• Therefore, we know that P2|rj|Σwj (1-Uj) is a hard problem

• We see that algorithms which may have found the optimal solution in one example, may not do so in others• Both algorithms found the optimal solution with the first example• When a 5th job is added to the system in the second example, Algorithm 1 finds a better solution that

Algorithm 2

Relationship to ICU Scheduling• In an ICU, must deal with multiple other constraints other than there being multiple machines (beds)

• In an ICU scheduling environment, would not be able to see jobs that have not been released yet; arrivals are stochastic and therefore unknown

• Must be able to schedule patients dynamically on a daily basis• This means taking into account new arrivals when scheduling each day and possibly shifting those

patients set to receive ICU care at a certain time to a different time• Not very realistic

• Thus, as we extend the problems we have just examined, we know ICU scheduling is (quite) hard

Potential for Expansion• Our research left us with a few answers• Multi resource allocation for emergency and elective surgeries could be applied to ICU scheduling• Using a threshold algorithm to decide what jobs are processed on what machines at specific times• Chapter 15 in Scheduling by Michael Pinedo: Constraint Guided Heuristic Search Procedure Algorithm –

helps to solve simplification Pm|rj|Σwj (1-Uj)

Multi-Resource Allocation Scheduling• Scheduling of elective and emergency surgeries in a dynamic environment

• Fulfills demand for elective patients, leaves sufficient slack for emergency patients

• Derives optimal number of surgeries to leave available by minimizing upper and lower bounds through approximation

• Variables:• RV’s for demand of elective and emergency services• Cumulative number of elective and emergency services• Resources needed per elective patient• Costs associated with each surgery

ICU Admission Control• Examines current ICU admission practices – inefficient, mostly doctor’s discretion

• Proposes to evaluate admission based on calculated benefit patient receives in ICU over ward

• Ran econometrics regression: showed optimal policy was “threshold policy”• Sort patients into 10 groups based on calculated benefit• Depending on current occupancy of ICU, if the next patient who arrives is in or above the threshold

group to be admitted based on current occupancy, he/she will be admitted• If not above, then they will not be admitted

• Classification to fill emergency bed changes daily based on occupancy of ICU at beginning of day

Constraint-Guided Heuristic• Constraint Guided Heuristic Search Procedure Algorithm

• Alternative way to solve problem

• Interesting approach

• Will explore further