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 r j = Time when patient j arrives to the ICU p j = Expected stay of patient j in ICU w j = Expected benefit of patient j with ICU treatment over ward treatment d j = Deadline of patient j (earliest the patient would need to be treated) S j = Time patient j is scheduled in ICU C j = 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|r j |Σw j (1-U j ) 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|r j |Σw j (1-U j ) – Maximizing the sum of the weighted completed jobs

8. Model 2 (2 machines, no pmtn) P 2 |r j |Σw j (1-U j ) – Maximizing the sum of the weighted completed jobs

9. What can we conclude from examples? 1|r j |Σw j (1-U j ) 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 P 2 |r j |Σw j (1-U j ) 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 5 th 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 P m |r j |Σw j (1-U j )

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