Dynamic Daily Surgery Scheduling - MIM 2016mim2016.utt.fr/MIM2016.pdf · 2016-07-01 · Dynamic...

Dynamic Daily Surgery Scheduling

Centre for Biomedical & Healthcare Engineering

CNRS UMR 6158 LIMOS

Ecole des Mines de Saint Etienne, France

xie@emse.fr

Centre for Healthcare Engineering

Dept. Industrial Engr. & Management

Shanghai Jiao Tong University, China

xie@sjtu.edu.cn

Xiaolan XIE

- 2 -

Healthcare engineering lab

At

EMSE & SJTU

- 3 -

People

Xiaolan Xie, professor

Saint Etienne Vincent Augusto, CR, HDR en 2015 Thierry Garaix, MA Ramesky Pham, Engineer (2016) 4-8 Ph.D. students Associate members: Marianne Sarazin, MD, Hopital Firminy Bruno Salgue, IMT

Shanghai Zhibin Jiang, professor Andrea Matta, professor Na Geng, Asso prof. Ran Liu, Assist prof. Feng Chen, asso. Prof. Na Li, asso. Prof. About 10 PhD students and 10 Msc students Zheng Zhang, SJTU-Univ. Michigan Siqiao Li, SJTU-Free Univ. Amsterdam

- 4 -

Eco-system

Campus for Health & Innovation @ Saint Etienne • EMSE-Centre for Biomedical & Healthcare Engineering • Medical school • Teaching hospital CHU-SE • Incubator for spin-offs in med tech PTM

Shanghai Jiao Tong University

• 13 affiliated hospitals including 6 LARGE ones

• Strong incentives for medicine-engineering collaboration

- 5 -

Smoothing demand

Triangle: Quality of service, Quality of Work, Cost

Extra-beds at ED, 2013.07 Outpatient queue, 6h AM,11/15/2011

Matching capacity & demand Improving service quality

- 6 -

Mission statement

Develop quantitative methods for modeling, simulation and

optimization of health care systems & health services

Explore the integration of medical knowledge and patient

health information in operations management of health care

systems

in close collaboration with hospitals

Stochastic modeling and optimization in the face of random events and changing system dynamics

- 7 -

Theme I : Engineering health care systems & services

To develop scientific methods for performance evaluation, capacity planning and process engineering.

Examples of work done : • Patient flow analysis with UML and Petri nets • Simulation & capacity planning of Emergency departments

• Process improvement of hospital supply chains by RFID • Health care logistics with mobile service robots • Permance evaluation of Hospital Information Systems

• Designing home healthcare networks • Design and operations of perinatal care networks • Care pathway for elderly people

• Blood collection optimization

- 8 -

Dynamic perinatal network reconfiguration

Context • 3 types of neonatal cares (OB = obstetrics care,

Neo = basic Neonatal Care, NICU) • 3 types of maternity services (OB, OB+Neo,

OB+Neo+NICU) • Demographic evolution • Immediate admission of random arrivals

Dynamic capacity planning and location of hierarchical service networks under service level constraints, IEEE Transactions on Automation Science and Engineering, 2014.

Perinatal Network of North Hauts-de-Seine

(Type-3) H. Louis Mourier

H. Beaujon

(Type-1)

H. FOCH

(Type-2)

CH Neuilly (Type-2)

H. Franco Britan (Type-2)

H. Nanterre (Type-1)

Challenge: • Determine optimum reconfiguration of perinatal

networks to meet demographic changes with equal service level of care

Solution & results: • Erlang loss-queueing model for admission probability evaluation; • Original hierarchical service network with nested hierarchy of patients and maternity services • Network reconfiguration by opening/closing services, capacity transfers, hiring/firing • Large-scale nonlinear optimization models solved with original linearization techniques • 5% increase of admissions at the 1st choice hospital.

- 9 -

Process mining of cardiovacular patients (funded by HEVA company)

Goal • Extract the process model of

hospitalization events • From what patients actually endured

instead what the « experts » think

“An Optimization Approach for Process Discovery of Complex Event Logs”, on going.

Challenge: • Huge number of hospitalization events • Delicate balance between details and

readability (avoid spaghetti diagrams)

Key treatment: Implantable Cardioverter Defibrillators

Solution & results: • Hierarchical structure of event classes to

capture event relations • Formal mathematical modelling of the process

mining optimization • Application of efficient optimization algorithms

- 10 -

Traceability in biobanks

Research questions

Performance evaluation of traceability technologies

Design supply chains of drugs and medical devices with RFID

New operation management problems (re-warehousing of bio-banks, skill/quality monitoring, ...)

Info errors

Inventory error Current situation

Samples stored in nitrogen tanks (77°K) “Cold Chain” constraints Resistance of the tags?

Hand-made inventories, data-base updates, cryotube numbering or label edition…

Problems: Error probabilities (Hand-copy, inventory, picking, computerization…)

Impacts of Radio-Identification on Cryo-Conservation Centers, TOMACS, 2011.

- 11 -

Theme II: Planning and logistics of health care delivery

To develop optimization methods for operations management of healthcare delivery and its supply chains.

Example of work :

• Planning and scheduling operating theatres subject to uncertainties • Capacity planning control MRI examinations of stroke patients • Stochastic optimization for hospital bed allocation

• Inpatient admission control • Dynamic outpatient appointment scheduling

• Operation management of outpatient chemotherapry • Capacity planning and patient admission for radiotherapy

• Robust home healthcare planning • Home healthcare admission planning&control

• Management of winter epidemics (flu, bronchitis, gastroenteritis) • Long-term care planning & scheduling

- 12 -

Optimization of outpatient chemotherapy

ICL Loire Cancer Institute

Major challenges of further research: • Integration of decisions different levels and different time scales

(medical planning, patient assignment, appointment scheduling) • Modeling treatment protocols with rich medical knowledge • Modeling the dynamics of health conditions based on rich patient data • High uncertainties of patient flow and patient's health care requirement

Large variation in bed capacity requirement in actual planning

20% reduction of peak bed requirement in the optimized planning

bed requirement

Planning oncologists of ambulatory care units. Decision Support Systems. 2013

- 13 -

Capacity planning of diagnostic equipment (MRI)

MRI examination of stroke patients

Expensive (over 1 million $) -> high utilization

Demand uncertainties and demand diversity (both elective and emergency)

Goal: Reduce waiting time for stroke patients without degrading MRI utilization

Actual waiting times of 30-40 days for MRI examination

2 - 10 days with the optimized reservation and control strategy。

Monte Carlo optimization and dynamic programming approach for managing MRI examinations of stroke patients. IEEE Transactions on Automatic Control, 2011

- 14 -

Some funded projects

• Management of winter epidemics (flu, bronchitis, gastroenteritis) (ANR-TECSAN project HOST)

• Engineering home health care logistics (Rhone-alps Region, Labex IMOBS 3, St Etienne metropole)

• Performance modeling & evaluation of HIS (DGOS-PREPS e-SIS)

• CIFRE-Heva : Patient pathway mining with national database

• Care pathway of elderly people (Fondation Caisse d’Epargne)

• Spare care management of family caregivers (Fondation MSD-Avenir)

• CIFRE-Lomaco : Ambubalance network optimization

Past:

• FP6-IST6-IWARD on mobile & reconfigurable robots for hospital logistics.

- 15 -

Planning and optimisation of hospital resources

5-year project funded by Natural Science Foundation of China (2012-2016)

Consortium: IE, B. School, Ruijin hospital all from SJTU

Four major research tasks: Planning / scheduling of key clinical resources (human +

beds) Capacity planning / preventive maintenance of diagnostic &

treatment equipment Coordination / cooperation mechanism design Modelling / simulation of hospital emergency responses

- 16 -

Dynamic Daily Surgery Scheduling

Centre for Biomedical & Healthcare Engineering

Ecole des Mines de Saint Etienne, France

xie@emse.fr

Centre for Healthcare Engineering

Dept. Industrial Engr. & Management

Shanghai Jiao Tong University, China

xie@sjtu.edu.cn

Xiaolan XIE

- 17 -

Basics of surgery scheduling

- 18 -

Importance of efficient surgery planning/scheduling

• Heart of a hospital involving nearly all medical specialties/units

• Relied on expensive skilled human resources and material resources

• About 10% of hospital budget

• Efficiency in terms of Cost-Quality-Delay is a must

• Mutation from a monospecialty with ad hoc organization to a multi-specialities with better organisation due to budget constraints and more strict safety regulations

• Health system reforms impose efficient management that the health professionals are not prepared and trained to

- 19 -

Overview of surgery patient journey

Patient arrivals Waiting

lists

Transfer

Leave the hospital

Surgery & Recovery

- 20 -

Patient perspective

• Elective patients = regular patients that can be planned

• Non elective patients = patients that arrive unexpectedly and have to be operated urgently

• Emergency patients = patients to be operated as soon as possible

• Urgent patients = patients to be operated in a short period

• Inpatients = patients requiring at least one-night hospitalization

• Outpatients = patients arriving & departing the same day

• Patient classification by DRG (Diagnosis Related Groups)

- 21 -

Health service perspective

• Presurgery: consultation, medical examination, …

• Surgery operations

• Post surgery: recovery and monitoring in wards

T1 T2 T3 T4 T5

Patient preparation Anaesthesia Surgery Bandage Cleaning

Patient arrival in OR

induction incision end of surgery OR available

Patient departure Surgeon time

Medical time

Patient sojourn time in OR

Total OR occupation time = Surgery time

- 22 -

Material resources perspective

Other material resources

Recovery rooms

Induction rooms

Stretchers

Obstetric labor rooms

Interventional radiology ORs

Emergency department

Sterilisations

Wards

Operating room (OR)

Operating theatre = set of ORs of a hospital

- 23 -

Human resources perspective

Surgeon = main operator

Anaesthesist

Surgery team = nurses of various skills assigned to an OR

Stretchers

Hospital attendants

Secretaries

Operating room (OR)

- 24 -

Performance mesure perspective

Resource utilization

• OR occupation

• Overtime

• Hospital revenue

Service quality

• Access time

• Waiting time

- 25 -

Operation decision perspective

- 26 -

Field observations of surgery scheduling

- 27 -

Ruijin Hospital (since 1907 by French missionaries)

Teaching hospital of the medical school of the Shanghai Jiao Tong University

Top 1 hospital in Shanghai

+12000 outpatient visits / day

A 23-floor outpatient consultation building

- 28 -

Field observation of the operating theatre of Ruijin Hospital

An integrated operating theatre of 21 OR and a second one recently constructed

60-70 elective surgery interventions + 10 emergency surgeries / day

No integrated surgery planning but each surgery speciality is given an amount of total OR time

Each speciality decides the surgeries to perform the next day

The operating theatre (OT) is responsible for daily OR assignment and the OR program execution.

- 29 -

Field observation of the operating theatre of Ruijin Hospital

Special features of the Ruijin Hospital

Queue of elective patients never empty

Availability of patients to be operated in short notice

Availability of surgeons to operate each day

Large variety of surgeons : top surgeons, senior surgeons, ordinary surgeons

Strong demand to operate at the OT opening in the morning to avoid endless waiting

Strong concern of OT personal overtime

- 30 -

Field observation of the operating theatre of Ruijin Hospital

Issues to be addressed

Promising surgery starting times to meet surgeon's demand for reliable surgery starting

(Tell me early enough when I start my surgery)

Surgery team overtime management

(How to guarantee the on-time end of duty of surgery teams?)

Outpatient surgery appointment when servers respond to congestion

- 31 -

Managing surgeon appointment times

- 32 -

Why surgeon appointments not used in practice

• Not used in practice to avoid potential OR capacity loss

Research question

How to provide surgeon appointment guarantee while ensuring appropriate OR capacity usage?

Observed Daily OR utilization

• But OR capacity usage is not always high over the day

- 33 -

Related work

Static scheduling for a single OR

Surgeon appointment scheduling (AS):

Two surgeries: AS solved by a newsvendor model (Weiss, 1990)

A fixed sequence of surgeries: stochastic linear program solved by SAA and L-shape algo to determine the allowance of each surgery, or equivalently, the arrival time (Denton 2003).

Others: discrete appointment (Begen et al, 2011), robust appointment (Kong et al, 2011)

Sequence scheduling: The problem is to jointly determine the position and arrival time of each surgery (Denton 2007; Mancilla 2012).

- 34 -

Related work

Dynamic scheduling for a single OR

Arrival scheduling: The demand of surgeries is uncertain, surgeries are processed as FCFS rule. The problem is to dynamically determine the arrival time upon each application(Erdogan 2011).

Sequence scheduling: The demand of surgeries is also uncertain. The problem is to jointly determine the position and arrival time of each surgery upon each application (Erdogan 2012).

- 35 -

Our focus

Multi-OR setting

- 36 -

Our focus

Multi-OR setting

Single-OR

Multi-OR

A1 A2 A3 An

A1/A2 A3 A4 An

No OR assignment

Dynamic OR assignment

- 37 -

Our focus

Two inter-related problems:

• Determining surgeon arrival times by taking into account OR capacities and random surgery durations.

• Dynamic surgeon-to-OR assignment of during the course of a day as surgeries progress by taking into account planned surgeon arrival times.

- 38 -

Assumptions of our work

A1: Emergency surgeries in dedicated ORs and hence neglected.

A2: Identical ORs and surgeries assignable to any OR.

A3: At most one surgery per surgeon each day.

A4: Promised starting or appointment time informed at the end of day D-1 (Surgeon appointment scheduling or proactive problem).

A5: Surgeons not available before the promised times.

A6: Dynamic surgery-to-OR assignment during the course of the day upon the surgery completion events.

- 39 -

Dilemma of promising surgery starting time

Promise too early

Surgery 1

promised start of surgeon 2

Surgery 2

Surgery 1

promised start of surgeon 2

Surgery 2

Promise too late

surgeon waiting

OR idle OR overtime

Easy if known OR time but OR times are uncerain

- 40 -

Data

J set of surgery interventions or surgeons

N number of identical ORs

T length of OR session

pi(ω) random duration of surgery i in scenario ω

bi unit time waiting cost of surgeon i

c1 unit OR idle time cost

c2 unit OR overtime cost

Similar to parallel machine scheduling but with planned job release dates and random service time.

- 41 -

Dynamic Surgery Assignment of Multiple Operating Rooms with Planned Surgeon Arrival Times

Zheng Zhang, Xiaolan Xie, Na Geng

In IEEE Trans. Automation Science and Engineering

- 42 -

Plan

Approximate optimal surgery start promising

Real time OR assignment strategies

Some numerical results

Conclusion and perspective

- 43 -

Decision variables

si promised surgery starting time of surgeon i Approximation assumption: fixed assignment & sequencing xir = 1/0 assignment of surgery i to OR r yij = 1 if surgery i precedes j in the same OR = 0 if not Auxiliary scenario-based random variables Cir(ω) completion time of surgery i on OR r Ir(ω) idle time of OR r Or(ω) overtime of OR r Wi(ω) waiting time of surgeon i

- 44 -

Model for promising surgery starting times

Assign each surgery to an OR ∑r xir = 1

Relation between assignment & sequencing yij + yji ≥ xir + xjr -1

Promised start before the end of the session si ≤ T

Scenario-dependent completion time xir pi(ω) ≤ Cir (ω)

Cir (ω) ≤ M xir

Cjr (ω) ≥ Cir (ω) + pj(ω) - M (1- yij) - M(2- xir - xjr )

Scenario-dependent OR idle time Cir (ω) ≤ Ir (ω) + ∑i∈J xir pi(ω)

Scenario-dependent OR overtime Or (ω) ≥ Cir (ω) - T

Scenario-dependent surgeon waiting time ∑r∈E Cir(ω) = si + Wi(ω) + pi(ω)

OR idle cost OR overtime cost

surgeon waiting cost

min Eω{c1 ∑r Ir(ω) + c2 ∑r Or(ω) + ∑i biIi(ω)}

- 45 -

Proposed solution

1. Convertion into mixed-integer linear programming model by Sample Average Approximation by using a given number of randomly generated samples

2. Heuristic for large size problem based on a) Local search for surgery-to-OR assignment

optimization b) Surgery sequencing rule based on optimal

sequencing of the two-surgery case c) Optimal promised start time by SAA and MIP

- 46 -

Plan

Approximate optimal surgery start promising

Real time OR assignment strategies

Some numerical results

Conclusion and perspective

- 47 -

Dynamic surgery assignment optimization

At time 0, start surgeries planned at time 0

At the completion time t* of a surgery in OR r*, select a surgery i* to be the next surgery in OR r* among all remaining ones J*

Surgery i* starts at time max{ t*, si* } in OR r* after the arrival of the surgeon at time si*

An Event-Based Framework

- 48 -

Dynamic surgery assignment optimization

Surgery i* is selected in order to minimize E[ TC(t*, i*, J*)] where E[ TC(t*, i*, J*)] is the minimal total cost similar to promised time planning model by conditioning on all completed surgeries and ages of

all on-going surgeries by scheduling i* as the next surgery on OR r*

- 49 -

Two-stage stochastic programming approximation

• At k-th surgery completion event at time tk

where J\J(k-1) is the set of remaining surgeries

• The first stage cost is the OR-idle or surgeon waiting cost induced by surgery l

• Θlk is the second stage cost, i.e. the total cost induced by remaining surgeries plus OR overtimes.

( )\ 1mink lk

l J J klkV g

∈ −= + Θ

( ) ( )ˆlk l k l k ls t t sg β+ += − + −

- 50 -

The second stage cost

( ) { }\ 1 \minlk jlk

j J J k lθ

∈ −Θ =

where • θjlk is the expected stage cost induced by surgery j

• if surgery l is selected at event k and surgery j at event k+1 Jensen's inequality is used to speedup the OPLA rule.

One-period look-ahead (OPLA) approximation

- 51 -

The second stage cost (cont'd)

Min. cost of two dynamic assignment rules: • Rule 1 (minimal stage cost first): Remaining surgeries assigned

in the scenario-independent order of minimal expected first stage cost, i.e. the surgery in selected at event n > k minimizes the stage n cost induced by in.

• Rule 2 (FCFS): Remaining surgeries are selected in non-

decreasing order of their surgeon arrival times si Jensen's inequality and another valide inequality are used to speedup the MPLA rule.

Multi-period look-ahead (MPLA) approximation

- 52 -

Lower bound of the dynamic surgery assignment

• Based on perfect information, i.e. all surgery duration realizations pj(ω) are known at the beginning of the day, i.e. randomness known at time 0+

• The lower bound problem is similar to the proactive problem but with

o given promised surgery start times

o scenario-dependent surgery assignment xir(ω) and sequencing yij(ω)

- 53 -

Dynamic surgery assignment policies

Policy Static: No real time rescheduling OR assignment / sequencing decisions of promised time

planning model are followed Policy FIFO: Dynamic surgery assignment in FIFO order of surgeon

arrival times Policy I: Dynamic surgery assignment optimization with OPLA Policy II: Dynamic surgery assignment optimization with MPLA

- 54 -

Plan

Approximate optimal surgery start promising

Real time OR assignment strategies

Some numerical results

Conclusion and perspective

- 55 -

Optimality gap

Observations

• Optimality gap is relatively small

• High surgery duration variation degrades the optimality gap

• High workload reduces the optimality gap

• MPLA better than OPLA

GAP = (costX- LB) / LB

(η,ρ%) GAPI(%) GAPII(%)

Ave. Min. Max. Ave. Min. Max. (0.3,0.75) 7.4 0.1 14.7 6.3 0.1 12.8 (0.7,0.75) 8.5 5.1 14.8 7.7 3.8 18.4 (0.3,1.25) 5.6 1.3 11.2 4.1 1.0 8.3 (0.7,1.25) 7.8 1.9 17.3 6.0 1.6 9.6

(80 3-OR instances)

- 56 -

Value of dynamic scheduling

OR# (η,ρ%) VDS (%)

Ave. Min. Max. 3 (0.3,75) 10.6 2.6 22.9

(0.7,75) 14.8 5.5 26.9 (0.3,125) 7.4 3.9 14.1 (0.7,125) 11.1 5.7 15.5

Ave. 11.0 4.4 19.9 6 (0.3,75) 25.4 18.7 31.6

(0.7,75) 29.2 24.7 39.9 (0.3,125) 11.1 7.1 15.5 (0.7,125) 19.1 12.8 24.1

Ave. 21.2 15.8 27.8 12 (0.3,75) 33.6 30.1 37.9

(0.7,75) 36.0 28.9 42.1 (0.3,125) 18.6 17.2 20.4 (0.7,125) 26.1 23.9 30.1

Ave. 28.6 25.0 32.6

Observations • Dynamic surgery scheduling always

helps.

• The benefit is more important for larger OT.

• Dynamic surgery scheduling is able to cope efficiently with surgery uncertainties.

• VDS decreases as the workload of OT increases.

η : variation parameter of surgery time ρ : workload

VDS = (costStatic - costDyna) / costStatic

- 57 -

Value of dynamic scheduling optimization

Observations • VOS increases as OR# increases.

• VOS increases as η increases, i.e. the variance of surgery durations increases.

• VOS decreases as ρ increases, i.e. the workload of OT increases.

OR# (η,ρ%) VOS (%)

Ave. Min. Max.

3 (0.3,75) 2.8 0.0 14.4

(0.7,75) 5.4 0.0 26.5

(0.3,125) 2.3 0.0 7.0

(0.7,125) 3.1 0.0 10.2

Ave. 3.4 0.0 14.5

6 (0.3,75) 5.4 -0.1 13.6

(0.7,75) 6.0 -0.1 11.3

(0.3,125) 2.9 0.0 5.0

(0.7,125) 5.0 0.6 8.7

Ave. 4.8 0.1 9.6

12 (0.3,75) 7.0 5.8 7.8

(0.7,75) 9.3 6.1 11.8

(0.3,125) 5.0 3.4 6.8

(0.7,125) 6.4 4.7 9.2

Ave. 6.9 5.0 8.9

η : variation parameter of surgery time ρ : workload

VOS = (costFIFO - costDynaOpt) / costFIFO

- 58 -

Value of proactive decisions

Observations • Proactive decision is very important to dynamic assignment scheduling.

• The arrival times that optimize the proactive model may not be adjustable to the dynamic assignment scheduling.

• Joint optimization of promised start times and dynamic assignment policies is an open research issue.

VOS = (costX - costX) / costX

where costX is the average cost of the strategy X but with promised start times determined with deterministic surgery duration.

(η,ρ%) VPSI(%) VPSII(%)

Ave. Min. Max. Ave. Min. Max. (0.3,0.75) 7.2 -15.2 23.3 7.0 -20.9 22.6 (0.7,0.75) 6.8 -11.1 20.4 6.4 -14.4 20.4 (0.3,1.25) 9.8 1.1 23.1 10.0 0.9 21.6 (0.7,1.25) 10.1 1.1 19.2 10.1 3.2 17.9

- 59 -

Plan

Approximate optimal surgery start promising

Real time OR assignment strategies

Some numerical results

Conclusion and perspective

- 60 -

Optimal surgery promised starting times for a given OR assignment / sequencing?

Features of surgeries planned to start at OR opening?

Time slacks in promised times vs surgery OR time and waiting cost?

Design of efficient optimization algorithms for promised time planning and real time rescheduling?

Promising time planning under starting time reliability constraints?

Open issues

- 61 -

Simulation-based Optimization of Surgery Appointment Scheduling

Zheng Zhang, Xiaolan Xie

in IIE Transactions

- 62 -

Outline

• BACKGROUND AND MOTIVATION

• SURGERY APPOINTMENT SCHEDULING PROBLEM

• SAMPLE PATH ANALYSIS

• STOCHASTIC APPROXIMATION

• NUMERICAL EXPERIMENTS

• CONCLUSION AND PERSPECTIVE

- 63 -

Our focus

Example :

1st released OR allocated to surgeon 3,

2nd released OR to surgeon 4, ....

Multi-OR

A1/A2 A3 r1

An

FCFS assignment

r2 A4

Surgeon appointment optimization for a given sequence of surgeries assigned to ORs on a FIFO basis.

- 64 -

Outline

• BACKGROUND AND MOTIVATION

• SURGERY APPOINTMENT SCHEDULING PROBLEM

• SAMPLE PATH ANALYSIS

• STOCHASTIC APPROXIMATION

• NUMERICAL EXPERIMENTS

• CONCLUSION AND PERSPECTIVE

- 65 -

Modeling

Parameters

n surgeries\surgeons

m ORs with regular capacity T for each OR

pi(ξ): surgery duration with known distribution

1 / α /βi: unit OR idling cost / overtime cost / surgeon waiting cost

Decisions

Surgeon arrival time A = [Ai] such that:

A1 = A2 = … Am = 0 ≤ Am+1 ≤ Am+2 ≤ … ≤ An

- 66 -

Modeling

Sample-path cost function

C[i](ω): i-th surgery completion event time.

C[i](ω) depends on A and ω and can be solved using a simple recursion.

[ ] ( )( ) [ ] ( )( ) [ ] ( )( )1

1 0( , )

n m

i i ii m i m n pi m p

f A C A A C C Tω β ω ω α ω−+ + +

− − −= + =

= − + − + − ∑ ∑

Waiting cost Idling cost Overtime cost

- 67 -

Modeling

Expected cost function

Objective

( )( ) ,g A E f Aξ ω=

1

min ( )

0, 1,...,, ,..., 1

A

i

i i

g A

A i mA

A A i m n

∈Θ

+

= = Θ = ≤ = −

- 68 -

Outline

• BACKGROUND AND MOTIVATION

• SURGERY APPOINTMENT SCHEDULING PROBLEM

• SAMPLE PATH ANALYSIS

• STOCHASTIC APPROXIMATION

• NUMERICAL EXPERIMENTS

• CONCLUSION AND PERSPECTIVE

- 69 -

Sample path analysis

LEMMA . The sample path cost function f(A,ω) is

• differentiable with probability 1 and

• Lipschitz-continuous throughout Θ with finite Lipschitz constant

1 2 1 2 1 2( , ) ( , ) , ,f A f A K A A A Aω ω− ≤ − ∀ ∈Θ

- 70 -

Sample path analysis

THEOREM 1 (unbiasednes of sample path gradient). The objective function g(A) is continuously differentiable on Θ, and the gradient of g(A) exists for all A∈Θ with

( ) ( ), ,A AE f A E f Aξ ξω ω∇ = ∇

The noisy sample-path gradient is on average correct!

- 71 -

Sample path analysis : partial derivative at interior point

\{ }

\{ }

\{ }

A:B:

C: 1

D: 1

i

i

i

i

jj BP i

ji j BP i

jj BP i

fA

β

β

β

α β

∈

∈

∈

−∂ = +∂ + +

∑∑

∑

Ai

i BP2(i) j

A.

B.

i

Ai waiting

i BP2(i) BP3(i)C.

Ai

i BP2(i) BP3(i)D.

Ai

waiting

waiting waiting

waiting waiting overtime

[i-m]1 …

[i-m]1 …

[i-m]1 …

[i-m]1 … BP4(i)

waiting

1 = unit OR idling cost

α = overtime cost

βi = surgeon waiting cost

Busy Period approach

A. i does not initiate BP(i)

B. i initiates BP(i) but not the last BP of the OR

C. i initiates the last BP of the OR without overtime

D. i initiates the last BP of the OR with overtime

- 72 -

Sample path analysis : directional derivative at boundary point

Boundary point A with Ak = Ak+1 = … = Al

( ) ( ) ( )

( ) ( ) ( )

( )( )

[ ][ ] [ ] ( ){ }[ ] ( ){ }

0

0

... , ,, lim

... , ,, lim

, if 0

, if 0

1 1 , if

1 0 , if

i

i

lk i

v jj i

ii l

u jj k

i i ii

i i

j j

j

j m j mj

f A e e f Af A

f A e e f Af A

x W

W

C T j n mx

W x j n m

ω ωω γ

ω ωω γ

β ωγ

β ω

β α ω

β ω

− ∆→=

∆→=

+ +

− ∆ − − ∆ −∇ = = −

∆

+ ∆ + + ∆ −∇ = =

∆

− == − > + + > > −=

+ > ⋅ ≤ −

∑

∑

Left-hand directional derivative

Right-hand directional derivative

- 73 -

Sample path analysis : improving direction

At an interior point, i.e. Ai-1 < Ai < Ai+1 At a boundary point A with Ak = Ak+1 = … = Al Select two surgeries i < j such that Determine the improving direction

( ),f A ω= −∇d

( ) ( ), 0, , 0i jv uf A f Aω ω−∇ < ∇ <

,..., ,0,...,0, ,...,i i j jv v u ud f f f f− −

= ∇ ∇ −∇ −∇

- 74 -

Outline

• BACKGROUND AND MOTIVATION

• SURGERY APPOINTMENT SCHEDULING PROBLEM

• SAMPLE PATH ANALYSIS

• STOCHASTIC APPROXIMATION

• NUMERICAL EXPERIMENTS

• CONCLUSION AND PERSPECTIVE

- 75 -

Stochastic approximation

( )1k k k kA A s d+Θ= Π +

( )

( )

where

is an improving direction according to sample-path gradient ,

= is a converging step-size

min is the orthogonal projection into the feasible set

k k

k

d f A

ask

ω

Θ

∇

Π = − Θy

x y x

Hill-climbing with noisy sample-path gradient

- 76 -

Outline

• BACKGROUND AND MOTIVATION

• SURGERY APPOINTMENT SCHEDULING PROBLEM

• SAMPLE PATH ANALYSIS

• STOCHASTIC APPROXIMATION

• NUMERICAL EXPERIMENTS

• CONCLUSION AND PERSPECTIVE

- 77 -

Convergence of stochastic approximation

BAD NEWS: The sample path cost function is not quasiconvex. Counter-example: p(ξ) = {9, 4, 4, 1}; 2 ORs, OR session T=10; idle time cost = 1; no overtime cost; Unit waiting cost β3=1, β4=3. Three arrival time vectors: A1=(0, 0, 4, 7.5) A2=(0, 0, 6, 8.5) A = αA1 + (1-α)A2

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 0,2 0,4 0,6 0,8 1

f(A, ω)

α

- 78 -

Convergence of stochastic approximation

By randomly perturbing p around {9, 4, 4, 1}, we implement the stochastic approximation algorithm.

Evolution of arrival times visited by the stochastic approximation algorithm in Example 1, when applying it over 200 sample paths.

- 79 -

Convergence of stochastic approximation

Hopeful news: The sample path cost fuction f(A,ω) is strongly unimodal.

Properties verified experimentally:

• Unimodality of the expected cost function

• Convergence of the stochastic approximation algorithm.

- 80 -

Convergence of stochastic approximation: numerical evidence

Log normal distribution Uniform distribution

var, wkload 0.3,0.75 0.7,0.75 0.3,1.25 0.7,1.25 0.3,0.75 0.7,0.75 0.3,1.25 0.7,1.25

Initial dispersion

3-OR 5.0 4.9 6.5 7.0 5.4 4.8 6.6 6.8

6-OR 6.5 6.7 8.5 9.5 6.5 6.6 10.3 9.8

9-OR 8.0 7.4 11.2 10.5 7.9 7.7 10.5 10.5

Final dispersion

3-OR 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

6-OR 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

9-OR 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Final grad

3-OR 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

6-OR 0.0 0.0 0.1 0.1 0.0 0.0 0.1 0.1

9-OR 0.0 0.2 0.1 0.3 0.0 0.2 0.2 0.3

- 81 -

Allowances of Multi-OR vs single OR settings

Optimal allowance shape dome shape in 1-OR, zigzag shape in 2-OR 2-OR vs 1-OR smaller allowances, half total allowance, highly uneven Increasing surgery duration variability (o vs o) smoothing 2-OR allowances, increasing 1-OR allowance variability Higher waiting cost (o vs o) larger allowances in both settings but rather insensitive in the 2-OR setting

- 82 -

Allowances vs OR#

Zigzag shape

1 large allowance followed by m-1 small allowances

Total m-OR allowance = 1/m of total-1-OR allowance

Higher OR# and higher duration variation → smoother allowances

- 83 -

Allowances vs OR#

Two-parameter heuristic

Larger 1st allowance followed by constant allowances

- 84 -

Value of dynamic assignment and proactive solution

Three strategies Strategy I : no dynamic surgery-to-OR assignment Strategy II : same appointment times, FIFO surgery-to-OR assignment Strategy III : same surgeon arrival sequence, FIFO surgery-to-OR assignment, simulation-based optimized appointment times Value of dynamic assignment (VDA) percentage improvement of strategy II over strategy I Value of proactive anticipation and dynamic assignment (VPD) percentage improvement of strategy III over strategy I

- 85 -

Value of dynamic assignment and proactive solution

VDA > 0, VPD > 0 , VPD > VDA : dynamic assignment and the proactive anticipation of dynamic assignments always pay

Higher OR number : increasing VDA and VPD due to scale effect and benefit of well planned arrivals. Higher duration variability: increasing VDA and VPD implying the importance of careful appointment planning and dynamic scheduling. Higher waiting costs: higher VPD but smaller VDA implying the importance of appointment time optimization. Higher workload: smaller VPD and VDA due to unimprovability of overloaded syst Impact of case-mix: • larger VPD when surgeries are identical due to their interchangeability. • smaller VDA when surgeries are identical due to suboptimal appointment times

Value of dynamic assignment (VDA) Value of proactive anticipation and dynamic assignment (VPD)

- 86 -

Outline

• BACKGROUND AND MOTIVATION

• SURGERY APPOINTMENT SCHEDULING PROBLEM

• SAMPLE PATH ANALYSIS

• STOCHASTIC APPROXIMATION

• NUMERICAL EXPERIMENTS

• CONCLUSION AND PERSPECTIVE

- 87 -

Summary

A more realistic model of AS which has m servers; patients are served in a pre-determined order but are flexible to any server.

Our aim is to proactively optimize the arrival times under the FCFS dynamic assignment strategy.

We formulate a simulation-based optimization model to smooth integer

assignments, and derivate a continuous and differentiable cost function. The proposed stochastic approximation algorithm is able to solve

realistic-sized instances and significantly improve the initial solution.

- 88 -

Managing surgery team overtime

“Branch and Price for Chance Constrained Bin Packing”

Zhang, Denton, Xie

submitted

- 89 -

Motivation

ORs: critical resources that require high utilization

Unpredictable overtime causes high nurse turnover rate

Nurses ask for ...

• Some ORs to have low overtime

• Predictable completion times

Challenges: • Fixed number of ORs • Uncertain service time • High cost of overtime

- 90 -

A chance constrained OR scheduling setting

Chance constraint (δr , α) of an OR r

The surgery team of the OR r completes its daily due before time T + δr with probability α

where

Τ = regular OR session time (T)

δr = allowable overtime

Chance constraint = End-of-duty guarantee

Examples: No overtime with proba 90% : δr = 0, α = 0.1

at most 1h overtime with proba 95% : δr = 1, α = 0.05

- 91 -

A chance constrained OR scheduling setting

An informal setting

Decisions: Surgeries-to-ORs assignment Constraints: For each chance constrained OR:

P(OR overtime ≤ δk) ≥ 1 - α

Objective: Minimize the expected overtime

A version of chance constrained extensible bin-packing problem

- 92 -

A stochastic programming formulation

Decision variables:

(1a) = Minimize total expected ovetime

(1b) = Assign each surgery to an OR

(1c) = Determine the overtime

(1d) = Chance constraints

I, R set of surgeries and set of ORs

di(ω) duration of surgery i under scenario ω

T regular OR session time

set of ORs of chance constraint k

xir binary var equal to 1 if surgery i is assigned to OR r

or(ω) overtime of OR r under scenario ω

Defining elements:

CkR

- 93 -

Solving Stoch. Prog. formulation: Branch-and-Price

Master problem

Decision variables

p column containing surgeries to be allocated in the same OR

λp binary var equal to 1 if the column p is selected

- 94 -

Solving Stoch. Prog. formulation

Key ideas of branch-and-price

1. Branch on constraints

• Select a pair: (i, j )

• Left side (in the same bin): yip = yjp

• Right side (in separate bins): yip + yjp ≤ 1

2. Enforcing the antisymmetry constraints due to identical ORs

- 95 -

Solving Stoch. Prog. formulation

Pricing problem

Decision variables

yip binary var equal to 1 if surgery i is in column p

ckp binary var equal to 1 if column p is type-k chance constrained

op(ω) overtime of column p

Stochastic knapsack problem

- 96 -

Solving Stoch. Prog. formulation

Pricing problem solution acceleration

• Tight lower bound by replacing the chance constraint by Cvar (Conditional Value at Risk) reformulation with convex recourse

• Tight upper bound with probabilistic covers and probabilistic packings (Song et al., 2014).

( ) ( ) ( )( )( ) ( ) ( )

( )

Chance constraint

1- inf 1-

1inf

= convex set

k k

k kz

k

P X T VaR X z P X z T

VaR X T CVaR X z E X z T

CVaR X T

δ α α δ

δ δα

δ

+

≤ + ≥ ↔ ≤ ≥ ≤ +

≤ + ← = + − ≤ +

≤ +

- 97 -

Robust optimization formulation

Assumptions:

A1. Given first two-moments (mi, σi) of surgery durations.

A2. Unknown probability distributions of surgery durations.

Chance constraints replaced by worst-case chance constraints:

where D is the set of all distributions matching the first two moments:

inf 1i ir kD i IP d x T δ α

∈∈

≤ + ≥ − ∑d

[ ]{ }2 2 2,i i i i iD E d m E d m σ = = + d

- 98 -

Robust optimization formulation: key result

Theorem: For any random variable X of mean m and standard deviation s, the worst-chance probability CP is reached by a three-point distribution such that

( )( )

( )

22 2

22

2 2

1, if

, if ,

, if ,

k

k kk

k kk

m T

CP m T m m TT m

m m T m m TT

δ

σ δ σ δσ δ

δ σ δδ

> += ≤ + + ≤ +

+ + − ≤ + + > +

+

Under the mild assumption CV≤ (α-1 – 1)0.5,

the robust optimization formulation can be converted into a deterministic mixed-integer-programming model.

- 99 -

Case study

• Experiments are based on real data of 21 surgical days.

• Number of ORs: m = 3 + 3 + 3; OR session time T = 10h; Overtime threshold dk ∈ {0.0; 2.0h; ∞}.

• Number of surgeries: dk ∈ [11; 37].

- 100 -

Performance of Branch-and-Price

Performance of different methods for the stochastic model

• Simple size: 500 • Computation time limit: 15,000 seconds. • Probability guarantee: 1 - α = 0.9.

- 101 -

Value of Robust Optimization

Worst-case probability

Experimental setting: 1- α = 0.9 (stochastic), 1- α = 0.9 (robust), n [21; 25]

• Extensive form of robust optimization can be solved by Cplex • The unachieved probability of stochastic solution could be 0.16 • The average overtime of robust solution could be 2 times higher

Average overtime

90%

- 102 -

Value of Robust Optimization

Worst-case probability

Experimental setting: 1- α = 0.9 (stochastic), 1- α = 0.7 (robust), n [21; 25]

• More robust solution with slighter higher overtime

Average overtime

90%

- 103 -

Conclusions

• The Branch-and-price can effectively solve the real-size problem instances

• The robust optimization problem can be much easier to solve than the stochastic problem

• The robust optimization can provide more robust solution with slightly higher overtime

- 104 -

Accounting for congestion behavior in appointment scheduling

“Appointment Scheduling Problem When the Server Responds to Congestion”

Zhang, Berg, Denton, Xie

Submitted

- 105 -

Evidences from the literature

• Outpatient clinic: physicians tend to speedup when they perceive congestion in the waiting area (Rising et al. 1973; Cayirli et al. 2008);

• Emergency department: triage-ordered testing and task reduction are used to reduce service time (Batt and Terwiesch 2012);

• ICU/ED: delays in receiving intensive care can result in longer lengths of stay in the ICU (Chan et al. 2015).

- 106 -

An outpatient procedure case

• Data for a one year period

• Samples are classified by surgeon and procedure type

• Specific records on patient waiting time, pre-procedure time, procedure time and post-procedure time.

We look at the impact of waiting time on different service times

- 107 -

A case in the context of outpatient procedures

• Negative correlation Between pre-procedure time and waiting time

• No correlation between procedure time, post-procedure time and waiting time

- 108 -

Related work

• Although there is a vast literature on appointment scheduling, none of the existing studies considered endogenous randomness.

• Congestion was incorporated in queuing models by Chan et al (2014), Vericourt and Jennings(2011), …

• However, appointment systems have a little number of customers and they need to determine arrival times.

- 109 -

Research questions

• Can the appointment scheduling problem be solved when the endogenous randomness is incorporated?

• How important is it to anticipate a congestion response from the server when scheduling appointments?

• Why is the dome shaped rule that is claimed "optimal", in practice, not widely implemented?

- 110 -

Problem setting

A1/A2

FCFS assignment

Appointment optimization

for a given sequence of customers

to a single server system with congestion response behaviour

in order to minimize the total cost related to

• Customer waiting (lower service quality)

• Service time reduction (lower quality service)

• Overtime.

- 111 -

Problem setting

A1/A2

FCFS assignment

Decision variables:

xi = customer-i allowance or interarrival time between i-1 and i

( ) ( ) ( )( ) ( )

( ) ( ) ( )( )( ) ( ) ( )( )

( ) ( ) ( )

2 2

1 1

2

min

, ,

, , , ,

,

n nw s oi i i i i

i i

i i i i

i i i i

n

i n ni

E c w c Z d c o

w w Z x i

Z f w d i

o x w Z T

ω ω ω ω

ω ω ω ω

ω ω ω ω ω

ω ω ω ω

= =

+

+ +

+

=

+ − +

= + − ∀

= ∀

= + + − ∀

∑ ∑

∑

( )( )( ) ( )

: waiting cost

: service reduction cost

: overtime cost: normal service time

: actual service time

, ,

wisio

i

i

i i i i

c

c

cd

Z

Z f w d

ω

ω

ω ω=

- 112 -

Problem setting : congestion behavior models

A1/A2

FCFS assignment

Linear response model

- 113 -

Problem setting : congestion behavior models

A1/A2

FCFS assignment

Logic Regression response model

( ) ( ) ( )2

11 i

ii i i w

Z de ω

θω ω θ

−

= + − +

- 114 -

Problem setting : congestion behavior models

A1/A2

FCFS assignment

Linear response model with customer no-show

( ) ( ) ( ) ( )

( )[ ] ( )

0, if no-show

1 , if show and

1 , if show and

ii i i i i

i

i i i i

Z d w w tt

d w t

θω ω ω ω

ω θ ω

= − <

− ≥

- 115 -

Solution approaches

Under mild continuity condition of the server reponse model,

• Stochastic-optimization with unbiased sample path gradients

Under linear response model

• Stochastic linear Mixed Interger Programming

- 116 -

Computational results : Comparison of SimOpt and SMIP

• Identical customers

• 500 samples are used for the SMIP, and 107 samples for the SimOpt.

• Costs are evaluated based on 106 samples.

• The SimOpt is much more efficient than solving the SMIP

• Across all instances, the SimOpt solved the global optimum

- 117 -

Computational results : Solution

• Allowances increase with variability and waiting cost

• Congestion reduces allowances

• Congestion makes allowances more flat

- 118 -

Computational results : comparison with heuristics

• Our method always finds the best solution

• Mean-value solution may outperform the Dome solution when congestion occurs

- 119 -

Computational results : comparison with heuristics

• Our method always finds the best solution

• Mean-value solution may outperform the Dome solution when congestion occurs

- 120 -

Conclusions

• Simulation-based Optimization can efficiently solve the congestion anticipated AS problems

• Variability and waiting coefficient affect the allowance and cost, while congestion behavior helps to lower the cost and smooth the allowances

• Ignoring the congestion is costly; the dome-shaped solution may perform worse than the mean-value solution

- 121 -

General conclusions

- 122 -

What next?

Joint optimization of surgery sequence and surgeon appointment times.

simulation-based discrete optimization + stochastic approximation

Chance constraints of surgery starts

Dynamic control of overtime allocation

Surgeon behavior

Joint scheduling of inpatient and day surgeries

- 123 -

Relevant previous work

Planning operating theatres with both elective and emergency surgeries

M. Lamiri, X.-L. Xie, A. Dolgui and F. Grimaud. "A stochastic model for operating room planning with elective and emergency surgery demands", European Journal of Operational Research, Volume 185, Issue 3, 16 March 2008, Pages 1026-1037

Mehdi Lamiri, Xiaolan Xie and Shuguang Zhang, "Column generation for operating theatre planning with elective and emergency patients," IIE Transactions, 40(9): 838 – 852, 2008

M. Lamiri, F. Grimaud, and X. Xie. “Optimization methods for a stochastic surgery planning problem,” International Journal of Production Economics, 120(2): 400-410, 2009