OR OR and and 2 Simulation Simulation for for Health Care - - PowerPoint PPT Presentation

Simulation Simulation OR OR and and

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M Healthcare PW 2010 OR and Simulation

Health Care Optimization Health Care Optimization for for

Two parallel servers Two parallel servers

M Healthcare PW 2010 OR and Simulation

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λ λ µ µ

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2λ 2µ

Stochastic OR - group Stochastic OR - group Erik van der Sluis Jan van der Wal Nikky Kortbeek Jan van der Wal Nikky Kortbeek

M Healthcare PW 2010 OR and Simulation

Ivo Adan Nico van Dijk Sindo Núñez-Queija Michiel Jansen Ivo Adan Nico van Dijk Sindo Núñez-Queija Michiel Jansen

Simulation Simulation

Familiar

M Healthcare PW 2010 OR and Simulation

Great Tool

Operating Theatre

Purposes Purposes

• Organization / Scenario’s
• Process Insights
• What-If

M Healthcare PW 2010 OR and Simulation

• Capacities
• Uncertainties
• Process Times

Organizational Organizational

• Specialization
• Pooling of services
• Specialized Programs (Allin)

M Healthcare PW 2010 OR and Simulation

• Specialized Programs (Allin)
• Triage systems / Nurse practitioners
• Supply Chain and ICT Conceptualization

Capacities Capacities

• # OK
• # ICU beds

M Healthcare PW 2010 OR and Simulation

• # Nurses / beds

: :

Planning Planning

• OK’s
• Shifts
• Part-time

M Healthcare PW 2010 OR and Simulation

• Diagnosis Process
• Therapeutic Treatments

: :

Introduction Introduction

Simulation Most

M Healthcare PW 2010 OR and Simulation

Most Valuable Evaluation Tool But: No Optimization

OR OR

• Insights
• Scenarios

M Healthcare PW 2010 OR and Simulation

• Techniques for

Optimization

=

Combination Combination

Simple models Strict assumptions Real life complexity Real life uncertainties Disadvantages Advantages OR Simulation Advantages Disadvantages

M Healthcare PW 2010 OR and Simulation

Optimization By techniques Also buy insights Evaluation By scenarios By numbers only Advantages Disadvantages Advantages Advantages

Combination of

O R Simulation

M Healthcare PW 2010 OR and Simulation

O R Simulation

Application Examples Application Examples

I Pooling II ICU dimensioning

M Healthcare PW 2010 OR and Simulation

II ICU dimensioning III Blood Platelet Production

Summarizing Summarizing

Simulation Operations Research Optimization

M Healthcare PW 2010 OR and Simulation

Advantages Advantages Optimization

Challenges Challenges

For OR For HC-practitioners

M Healthcare PW 2010 OR and Simulation

For OR

• Language
• Dare to step in
• Not just conceptual

For HC-practitioners

• Confidence in OR
• Software Development
• Implementation

Simulation Simulation OR OR and and

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Wp = Wa w w w w w w w Pw ait>0 Pw ait>tau

M Healthcare PW 2010 OR and Simulation

Health Care Optimization Health Care Optimization for for

Questions and/or suggestions Questions and/or suggestions

M Healthcare PW 2010 OR and Simulation

I: Should we Pool or Not I: Should we Pool or Not

?

M Healthcare PW 2010 OR and Simulation

• N.M. van Dijk

University of Amsterdam Operations Research & Management

N.M. van Dijk & E. van der Sluis

Operations Research & Management University of Amsterdam

M Healthcare PW 2010 OR and Simulation

• Pooling?

Two parallel servers Two parallel servers

λ λ µ µ

M Healthcare PW 2010 OR and Simulation

2λ 2µ

A Pooling: An Instructive Example A Pooling: An Instructive Example

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A single server (M/M/1)-queueing model A single server (M/M/1)-queueing model

M Healthcare PW 2010 OR and Simulation

      1 D = - λ

Two parallel servers Two parallel servers

λ λ µ µ       1 D = - λ

M Healthcare PW 2010 OR and Simulation

2λ 2µ       1 D = 2 - 2λ

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Pooling (A): 1 ≠ 2 Pooling (A): 1 ≠ 2

?

M Healthcare PW 2010 OR and Simulation

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And yet?

Pollaczek-Khintchine’s Formula Pollaczek-Khintchine’s Formula

M / G / 1

M Healthcare PW 2010 OR and Simulation

+

exp 2 G

W = ½(1 )W c

Squared Coefficient of Variation

Example Example

Mix ratio k =10

M Healthcare PW 2010 OR and Simulation

Deterministic ρ = 0.83

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• Pooling or separate queue

Pooling or separate queue

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OR and Simulation

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W1 = 6.15 W2 = 6.15 WA = 6.15 W1 = 2.50 W2 = 25.0 WA = 4.55

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Can we do better? Can we do better?

?

M Healthcare PW 2010 OR and Simulation

1, 2

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?

Overflow scenario(s) Overflow scenario(s)

Type 1

1, 2

• M Healthcare PW 2010

OR and Simulation

Type 1

2, 1

• Type 2

Overflow Result Overflow Result

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OR and Simulation

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W1 = 6.15 W2 = 6.15 WA = 6.15 W1 = 2.50 W2 = 25.0 WA = 4.55 W1 = 3.66 W2 = 8.58 WA = 4.11

How? How?

Queueing Result

M Healthcare PW 2010 OR and Simulation

Simulation

Overflow scenario(s) Overflow scenario(s)

Type 1

1, 2

Simulation

• M Healthcare PW 2010

OR and Simulation

Type 1

2, 1

Simulation

• Type 2

Can we do better? Can we do better?

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2.50 4.55 25.0

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4.11 8.58 1.80 3.92 25.2 25.0

By OR and Simulation By OR and Simulation

60% 80% 100% 120% Pooled Thr(Opt) Unpooled

Rank No. Scenario WA

7 1 6 5 5 2 0.71 0.68 0.63

M Healthcare PW 2010 OR and Simulation

0% 20% 40% 60% 1 5 2 3 4 6 7 Two-way One-way Prio(1,N) Prio(1,P)

4 3 3 4 2 6 1 7 0.58 0.52 0.38 0.20

Average waiting time for s = 10

• B Pooling: MRI-scans

B Pooling: MRI-scans

U 10% W1 < 3 days

M Healthcare PW 2010 OR and Simulation

• R 90%

W2 < 9 days

U + R

• W < 3 days ?

M Healthcare PW 2010 OR and Simulation

U + R

• W < 3 days ?

Results Results Pool if: WP < W1

M Healthcare PW 2010 OR and Simulation

By Queueing (OR)

2 2 1 1 1

½ 8 ½ ρ ρ ρ ρ ≤ + −

0.6 0.8 1.0

Pooling or Not? Pooling or Not?

2

ρ ↑

No Pooling Pooling

M Healthcare PW 2010 OR and Simulation

0.0 0.2 0.4 0.0 0.2 0.4 0.6 0.8 1.0 Wp = Wa wwwwwww Pwait>0 Pwait>tau

WP = W1 P{WP > 0} = P{W1 > 0} P{WP > τ } = P{W1 > τ }

1

ρ →

RT – application RT – application (# consults per week) Capacity needed for SL-1

M Healthcare PW 2010 OR and Simulation

SL-1 6 days 5 days 4 days 3 days (spare) P 34.5 35.5 36.5 39 6.5 NP 35 35.5 36 36.5 4

1

C Pooling: Nurse Practitioner C Pooling: Nurse Practitioner

Overflow

M Healthcare PW 2010 OR and Simulation

2 Necessarily Simulation

Pooled scenario Pooled scenario

Next

τ1

M Healthcare PW 2010 OR and Simulation

Next

τ2

Overflow scenario Overflow scenario

Overflow

τ1

M Healthcare PW 2010 OR and Simulation

Necessarily Simulation

τ2

Comparison Comparison

τ1 = 1 τ = 1.4

Next Next

τ1 τ2 τ1 τ2 M Healthcare PW 2010 OR and Simulation

τ2 = 1.4

Pooled Overflow

WA = 2.59 WA = 2.37

To Pool or Not: To Pool or Not:

Remains to be a Question

• f Practical and Scientific Interest

( , , ) ( +1, , ) ( , , )

m o n m

• n

m o n h µ

∆ = − = + V V

M Healthcare PW 2010 OR and Simulation

Only answered by OR and Simulation

[ ]

1 ( 1, , ) 1 ( , 1, ) 1 ( , , 1) ( , , ) ( , , ) 1 ( 1, , ) 1

• k

h hm m

• n

h o m o n h n m o n h m o n m o n h m

• n

h µ µ µ µ µ λ λ

+ + ∆ − + ∆ − + ∆ − + − + ∆ + + V V

[ ] [ ]

( , , ) 1 ( , , ) ( , , ) 1 ( , , +1) 1 ( , , ) 1 ( ) ( , , )

• n

• n

• n

• n

N o n h N o n N o n h m o n h m o n h m

• h

hn h h m o n λ λ λ µ µ µ λ λ

∆ + − + ∆ + ∆ + − + − − − − ∆ V V

• . .

. .. .

• . .

. .. . . . . .. .

II OR and Simulation ICU optimization II OR and Simulation ICU optimization

EuroSim 2007 Ljubljana OR and Simulation

Nico van Dijk & Nikky Kortbeek

Background Background

• Limited ICU capacity (# beds)
• Rejection

M Healthcare PW 2010 OR and Simulation

• Rejection

=> Cancellation of surgery, Pre-discharge, Transfer

• Consequences

patients:

• Emotional effect
• Health damage

hospitals:

• Extra costs
• Empty OT

Seriousness Seriousness

• Yearly a couple of hundred unnecessary deaths
• Rejection probability R ~ 1-15 % (?)

M Healthcare PW 2010 OR and Simulation

• Rejection probability R ~ 1-15 % (?)
• Interest: R

Intensive Care Intensive Care

Planned direct, 26

• Em ergency first

aid post, 171 Planned surgery 200 M Healthcare PW 2010 OR and Simulation

Source Jeroen Bosch Hospital 2005

Em ergency ,nursing w ard 185 Em ergency other hospital, 53 Em ergency surgery 80

Simulation Simulation

But

• Confidence ?

M Healthcare PW 2010 OR and Simulation

• Data on distributions ??
• Insights + Dimensioning ???!!

A Queueing Modelling A Queueing Modelling

OT ICU

Reje ct

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Interactio n

Capacity ct

Reje ct

Tandem model Tandem model

1 2

( , ) n n π

OT ICU

2

M Healthcare PW 2010 OR and Simulation

OT ICU

1

Modelling Modelling

(1) Patients that do not require an ICU bed are not included. (2) An exponential service time for the surgery at the OT with rate 1. (3) An exponential sojourn time at the ICU with parameter 21 for OT- patients and for direct patients.

M Healthcare PW 2010 OR and Simulation

patients and 22 for direct patients. (4) OT-patients are rejected upon arrival at the OT. (5) An ongoing operation is continued. If upon its completion the ICU is still congested, the patient is kept in the recovery and its operating room is stopped; that is, no new patient is taken into operation at this operating room before an ICU bed has become available.

Simulation 1: Simulation 1:

• To justify assumptions

M Healthcare PW 2010 OR and Simulation

Assumption 1 Assumption 1

OT

• With

Non-ICU Patients .12834

• Without

Non-ICU Patients .12797

M Healthcare PW 2010 OR and Simulation

Assumption 2 Assumption 2

Surgery Time

Variation Coefficient Rejection probability

M Healthcare PW 2010 OR and Simulation

Variation Coefficient Rejection probability

Assumption 3 Assumption 3

ICU Sojourn Time

Variation Coefficient Rejection probability

M Healthcare PW 2010 OR and Simulation

Variation Coefficient Rejection probability

OR-Modification OR-Modification

Operations directly stopped if no ICU-bed available

2

M Healthcare PW 2010 OR and Simulation

ICU Rejection ~ Minor Effect

OT ICU

1

Consequence 1: Insensitive Product Form Consequence 1: Insensitive Product Form

• Balansvergelijkingen

Balansvergelijkingen.

M Healthcare PW 2010 OR and Simulation

Proof – non standard

Consequence 2 Consequence 2

• Justifies M |G | c | c (Erlang approximation)

M Healthcare PW 2010 OR and Simulation

• Insensitive

But

• Numerically appears Lower Bound
• Random arrivals

M Healthcare PW 2010 OR and Simulation

• Random arrivals
• Upper Bound ?

Result Result

B ( M / G / c / c ) ≤

≤ ≤ ≤ R ≤ ≤ ≤ ≤ B ( M / G / c-1 / c-1 )

M Healthcare PW 2010 OR and Simulation

Original model Upper bound

Proof Proof

(1) By simulation 2:

(1) By simulation 2:

• No counterexample found

No counterexample found

M Healthcare PW 2010 OR and Simulation

(2) Stochastic monotonicity

(2) Stochastic monotonicity

• (special case)

(special case)

• (still highly complicated)

(still highly complicated)

Proof (Exp + single OR) Proof (Exp + single OR)

M Healthcare PW 2010 OR and Simulation

Results Results

M Healthcare PW 2010 OR and Simulation

Application (Dimensioning) Application (Dimensioning)

R (≤) Lower Bound # Beds Upper Bound # Beds

• M Healthcare PW 2010

OR and Simulation

Simulation for Simulation for OR OR and and

M Healthcare PW 2010 OR and Simulation

Intensive Care Optimization Intensive Care Optimization

III Blood Platelet Production (BPP): Optimization by Dynamic Programming and Simulation

EuroSim 2007 Ljubljana OR and Simulation

René Haijema Jan van der Wal Nico M. van Dijk (presents) University of Amsterdam

• Voluntary
• Otherwise expensive
• Highly Perishable (5-7 days)

Blood Platelets Blood Platelets

M Healthcare PW 2010 OR and Simulation

Minimize Spill

Practice Practice

• USA

Western Europe

• Shortages ~ 1 %

M Healthcare PW 2010 OR and Simulation

• Shortages ~ 1 %
• Spill (Outdating) ~ 20 %
• Simple Order-up-to Rules

5-step procedure 5-step procedure

• Step 1: SDP formulation
• Step 2: Downsizing + Solve
• Step 3: Simulation table
• Step 4: Simple rule (search)

M Healthcare PW 2010 OR and Simulation

• Step 4: Simple rule (search)
• Step 5: Simulation for evaluation

OR (DP) OR (DP)

• Epochs: each morning
• Decision: production volume = k

(= 0 on Saturday and Sunday)

• States:

1 2

( , ) ( , , ,..., )

m

d d x x x = x

M Healthcare PW 2010 OR and Simulation

• States:

where: x = inventory state d = day of the week xr = # pools with residual shelf life of r days m = max. residual shelf life (= 6 days)

1 2

( , ) ( , , ,..., )

m

d d x x x = x

OR (DP) OR (DP)

{ }

, 1

( , ) max ( , , ) ( ) ( ) ( 1, ( , , , ) V x x V z x

R R R y a R k d i j n n d

d c d k p j p i d k i j

−

= + +

∑

( , ) : x V R

n d

minimal expected cost over planning horizon of n days when starting at day d with inventory x

M Healthcare PW 2010 OR and Simulation

{ }

n n-1

( , ), x d k

Complications Complications

• Complexity (intractable)
• Complicated structure of optimal policy

M Healthcare PW 2010 OR and Simulation

BUT BUT

No Simple (Practical) Optimal Strategy

Production Inventory (old,…, young) 7 (0, 0, 5, 0, 0, 9) 8 (0, 0, 6, 0, 0, 8)

M Healthcare PW 2010 OR and Simulation

8 (0, 0, 6, 0, 0, 8) 9 (0, 0, 8, 0, 0, 6) 10 (0, 6, 2, 0, 0, 6) 10 (5, 0, 3, 0, 0, 6)

Step 3: Simulation table Step 3: Simulation table

6 7 8 9 10 11 12 13 14 15 16 17 18 19 total 22 1 1 21 1 2 4 2 1 10 20 759 5481 19706 40627 50741 39344 18762 5391 837 56 3 181707 19 141 3402 35656 92771 165052 206142 174524 97942 34736 7208 708 818282 18 total 141 3402 35656 93530 170533 225848 215151 148683 74081 25972 6103 839 56 5 1000000 Stock Repl.

• M Healthcare PW 2010

OR and Simulation

total 141 3402 35656 93530 170533 225848 215151 148683 74081 25972 6103 839 56 5 1000000

1 50741 39344 97942 34736 148683 74081 Stock Repl. 21 20 19 13 14 total

Most frequent order-up- to level (82%) Optimal 5, 6 or 7 units Age plays a role

Results (Case Study) Results (Case Study)

M Healthcare PW 2010 OR and Simulation

Case study: Dutch blood bank Case study: Dutch blood bank

• The Netherlands: 4 regional blood banks
• North East region demand is about 7500 pools:
• Demand = 144 pools / week (=> 26, 20, 32, 20, 26, 7, 13)

M Healthcare PW 2010 OR and Simulation

• Demand = 144 pools / week (=> 26, 20, 32, 20, 26, 7, 13)
• 70% Hematology and Oncology (‘young’ pools),
• 30% Trauma and Surgery (‘any’ age).
• Production stop during weekend
• Lead time 1 day
• Shelf life = 6 – 1 =

5 days

Results Results

• 1. |C*(SDP) – C(Sim)| < 1%
• 2. Spill = 0.1% < 4% < 20%

while shortages < 0.1%

M Healthcare PW 2010 OR and Simulation

• 3. Formal computerized approach

⇒ Simple rule

By By

Combination of:

M Healthcare PW 2010 OR and Simulation

O R Simulation