Stochastic Operating Room Planning with Recovery Flow€¦ · new = argmin j W ij else i new = argmax j:W ij 0 W ij end exclude from consideration the row and column corresponding

Stochastic Operating Room Planning

with Recovery Flow

Maya Bam1

([email protected]),

Brian Denton1, Mark Van Oyen1, Mark Cowen, MD2

1University of Michigan

2St. Joseph Mercy Health System

July 30, 2015

Funded in part by the National Science Foundation under grant CMMI-0844511.

Bam, Denton, Van Oyen, Cowen Surgery Scheduling with Recovery Resources July 30, 2015 1 / 27

Importance

“At the present rate, the average American can expect to undergoseven operations during his or her lifetime... [This] has broughtnew societal concerns... [including] how to make certain thatpatients have access to needed surgical care and how to managethe immense costs.”1

1Atul Gawande. “Two Hundred Years of Surgery”. In: New England Journal of Medicine 366.18 (2012), pp. 1716–1723.


Introduction

Stages of the Surgical Process

1 Check-in

2 Preop

3 Surgery

4 Post-anesthesiacare unit (PACU)

5 Transfer

OR1

OR2

...

ORn

PACUn

Our previous research suggests that it is important to considersupporting resources.


Introduction

Challenges of Surgery Planning and Scheduling

Resource constraints

I ORs, PACU beds• OR boarding

I patient-surgeon assignmentI surgery blocks

Uncertainty in

I surgery durationI recovery duration


Introduction

Research Question

How can we generate surgery schedules within reasonable time that

account for resources directly supporting surgery

ORssurgeons

resources indirectly supporting surgery

PACU

balance utilization and overtime of ORs

and perform well in a stochastic setting?


Modeling

Modeling Assumptions

(1) Single day schedule

(2) Surgery, recovery and turnover times are deterministic

(3) The surgical team is available whenever patient is ready

(4) Time discretized into time slots


Modeling

Model Features

(1) OR boarding not allowed - recovery in PACU starts rightafter surgery

I delay surgery start time instead

(2) Patient-surgeon assignments are respected

(3) Consecutive surgeries for each surgeon (surgery block)


Modeling Mixed Integer Program: MIP[OR,PACU]

Mixed Integer Program: MIP[OR,PACU]

Objective: minimize total of

fixed cost of opening the ORs (c f )

variable cost of OR overtime (cv )

variable cost of surgeon elapsed time (c s)

Decisions:

for each patient, in which OR and at what time slot to startsurgery (αijt)

I what time to start recovery in PACU (βit)I when surgeons are busy and with which patient (uikt)I which ORs are open (xj)I overtime for each OR (oj)


Modeling Mixed Integer Program: MIP[OR,PACU]

Minimize costmin

∑Rj=1

(c f xj + cv oj

)+∑K

k=1 cs (∆k − (T − δk ) + 1− n)

Determine number of ORs to open

s.t.∑I

i=1 αijt ≤ xj ∀j, t∑Ii=1

∑Rj=1 qijt ≤

∑Rj=1 xj ∀t∑R

j=1 xj ≤ R

Determine when patient is in the OR∑Rj=1

∑Tt=1 αijt = 1 ∀i

qijt ≥ αijt ∀i, j, t∑Ii=1 qijt ≤ 1 ∀j, t∑Rj=1

∑Tt=1 qijt = di ∀i∑t+di−1

t′=tqijt′ ≥ diαijt ∀i, j, t

Calculate surgeon elapsed time∑Ii=1 tuikt ≤ ∆k ∀k, t∑Ii=1(T − t)uikt ≤ δk ∀k, t

Calculate OR overtimetqijt ≤ Sj xj + oj ∀i, j, t

Patient-surgeon assignment is respected∑Rj=1 qijt =

∑Kk=1 uikt ∀i, t∑T

t=1 uikt = di sik ∀i, k∑Ii=1 uikt ≤ 1 ∀k, t

Recovery in PACU starts right after surgery

βi,t+di−n ≤∑R

j=1 αijt ∀i, t∑Tt=1 βit = 1 ∀i

Account for recovery duration

zit ≥ βit ∀i, t∑Tt=1 zit = ri ∀i∑t+ri−1

t′=tzit′ ≥ ri βit ∀i, t

PACU bed capacity constraint∑Ii=1 zit ≤ B ∀t

αijt , qijt , uijk , βit , zit ∈ {0, 1}; δk ,∆k , oj ≥ 0

∀i, j, k, t


Heuristics

Heuristics

Realistically sized problem instances are computationallychallenging.

Heuristics that separate surgeon-to-OR assignments andsequencing decisions can decompose the MIP into smallersubproblems.


Heuristics 2-Phase Heuristic

2-Phase Heuristic

Phase 1 Objective:

determine the number of ORs to opendetermine surgeon-to-OR assignment

Heuristic:

Longest Processing Time First (LPT)

Phase 2 Objective:

determine the sequence of surgeries within a surgeon’s blockdetermine the sequence of surgeons within an OR

Heuristic:

Difference Heuristic (DH)


Heuristics 2-Phase Heuristic: Phase 1

Surgeon-to-OR Assignment

Standard Bin PackingProblem: given bins of fixedsize and a list of items of varyingsize, what is the least number ofbins needed to pack all items?

Extensible Bin PackingProblem: bins can be extendedif necessary at an additionalcost.



Longest Processing Time First (LPT)

Surgeon-to-OR Assignment:

ORs = 1

while there is overtime and K > ORs do

Cost(ORs) = LPT heuristic cost;

ORs = ORs + 1;

end

Cost=minORs{Cost(ORs)}

Output: number of ORs to open, surgeon-to-OR assignment.

Paolo Dell’Olmo et al. “A 13/12 approximation algorithm for bin packing with extendablebins”. In: Information Processing Letters 65.5 (1998)



Worst-Case Performance Ratio of LPT

Theorem 1.

For any instance, we have

C LPT

C ∗≤ 1 +

Scv

12c f

where c f is the fixed cost of opening an OR, cv is the variable cost ofOR overtime, and S is the planned session length of the ORs.Moreover, this bound is tight for every even number of ORs.

Tested on 270 random instances,measuring performance by:

C LPT − C ∗

C ∗· 100%

average performance 0.42%worst-case performance 6.99%

% of instances optimal solution found 77.41%



Surgery Sequencing

Problem: given a single surgeon’s case list for a day, how tosequence the given surgeries with

one OR,

one PACU bed available

Once sequence is determined, assign start times to surgeries avoidingOR boarding.

The goal is to minimize OR idling under the constraint that no ORboarding is allowed.



Difference Heuristic

Setup: let Wij = ri − dj for i 6= j , and Wii =∞,where r is recovery duration and d is surgery duration.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

OR PT 1 PT 2

PACU PT 1 PT 2

W12 < 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14

OR PT 1 PT 2

PACU PT 1 PT 2

W12 > 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14

OR PT 1 PT 2

PACU PT 1 PT 2

W12 = 0



Difference Heuristic (cont.)

Setup: let Wij = ri − dj for i 6= j , and Wii =∞.

Pick first patient: i∗ = argmaxi minj Wij

while ∃i ∈ I that has not been sequenced do

if minj Wi∗j > 0 theni∗new = argminj Wi∗j

elsei∗new = argmax

j :Wi∗j≤0Wi∗j

endexclude from consideration the row and column

corresponding to patient i∗

i∗ = i∗new

end

Set surgery start times so that there would be no boarding, i.e.,insert OR idling if necessary.



Worst-Case Performance of DHTheorem 2.Letting

Di = maxj :i 6=j{(ri − dj)

+} −minj :i 6=j{(ri − dj)

+},

then for any instance we have

CDH − C ∗ ≤ c s

(I∑

i=1

Di −mini

Di

),

where c s is the variable cost of surgeon elapsed time. Moreover, thisbound is tight.

Tested on 270 random instances,measuring performance by:

CDH − C ∗

C ∗· 100%

average performance 0.70%worst-case performance 30.30%

% of instances optimal solution found 95.19%Bam, Denton, Van Oyen, Cowen Surgery Scheduling with Recovery Resources July 30, 2015 18 / 27

Heuristics Decomposition Heuristic

Decomposition Heuristic

(1) First assign surgeons to ORsuse an extensible bin packing formulation for this (MIP[OR]),with the objective of minimizing

I the fixed cost of opening the ORs (c f )I the variable cost of OR overtime (cv )

(2) Using the surgeon-to-OR assignments obtained from MIP[OR],sequencing the surgeries with MIP[OR,PACU].

(3) Obtain lower bound from

c f x∗ + cvR∑j=1

o∗j︸︷︷︸MIP[OR]

+ csI∑

i=1

di︸︷︷︸input


Discrete Event Simulation

Discrete Event Simulation

Purpose: to evaluate a surgery schedule

Characteristics:

Spans a single day

Cost as defined in the heuristics

OR1

OR2

...

ORn

PACUn

I Durations are stochastic: sampled from a lognormal distribution

I surgery and recovery duration distributions are caseand surgeon specific

I Measures OR boarding


Heuristic Comparison

Comparison of Two Heuristic Approaches

LPTSurgeon

Assignmentto ORs

MIP[OR]

Difference Heuristic MIP[OR,PACU]

Simulation

Comparison ofMean Total Cost

SequencingDecisions

EvaluateSchedule

2-Phase HeuristicDecomposition

Heuristic


Case Study

Case Study:

General, Orthopedic,

and Urology Surgical Services


Case Study

Case Study Specifics

Our previous research suggests the following percentiles for durations:

surgery - 60th recovery - 70th

Compared two types of schedules:

1 2-Phase Heuristic

2 Decomposition Heuristic

Case Study characteristics:

Average

# ORs # patients # surgeonsSurgery Recovery

duration (min) duration (min)

6 18 8 166 133


Case Study

Simulation Objective Comparison

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Test Instance

2-Phase Heuristic Decomposition Heuristic


Case Study


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Test Instance


The 2-phase heuristic generates schedules that perform extremelywell when compared to the decomposition heuristic.


Case Study


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Test Instance


The decomposition heuristic took 4 hrs to solve on average, witha maximum of 34 hours, while the 2-phase heuristic takes seconds.


Case Study

Case Study Results1 The 2-phase heuristic was

within 10% in 93% within 5% in 74%of the instances compared to the decomposition heuristicbenchmark.

2 Comparison based on the lower bound to MIP[OR,PACU]


HeuristicAverage performance 6% 0.7%

Worst-case performance 27% 9%% of time optimal solution found 26% 86%

3 Comparison based on OR boarding


HeuristicAvg OR time used for boarding 0.05% 0.27%Max OR time used for boarding 0.34% 3.16%


Conclusions

The problem of elective surgery scheduling considering

ORs

surgeons

the PACU

is computationally challenging for practical problem instances.

We can tackle this problem by

1 generating schedules with fast heuristics that perform well

2 and a simulation that evaluates the generated schedules.


Maya BamUniversity of Michigan

[email protected]


Documents

Stochastic Operating Room Planning with Recovery Flow€¦ · new = argmin j W ij else i new = argmax j:W ij 0 W ij end exclude from consideration the row and column corresponding