Optimal Scheduling for ICU Patients

1. Optimal Scheduling for ICU Patients Siddhant Bhatt Steve Boyle Erica Cunningham

2. 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?

3. 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

4. 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

5. 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

6. 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.

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

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

9. 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

10. 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

11. 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)

12. 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

13. 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

14. Constraint-Guided Heuristic • Constraint Guided Heuristic Search Procedure Algorithm • Alternative way to solve problem • Interesting approach • Will explore further