Useful Recent Trends in Simulation Methodology

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Useful

Recent Trends in

Simulation Methodology

Shane Henderson Shane Henderson

Cornell University

Thanks: NSF DMI 0400287 Thanks: NSF DMI 0400287

e.g., Inbound call centre staffing

Min staffing costsMin staffing costs

s/t

s/t customers happy “enough”customers happy “enough”

Min Min ccTT _x_x

s/t

s/t gg_i_i(x(x) ) ≥≥ 0, i = 1, …, num periods_{0, i = 1, …, num periods}

Max minMax min_i_i gg_i_i(x(x)) s/t

s/t ccTT_x_x _B_B

From Simulation Model

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Inbound Call Centre Staffing

What is the arrival rate of customers?What is the arrival rate of customers?

Input Uncertainty

Conversations with HR folks suggests Conversations with HR folks suggests

dependence of call volumes between periods

Dependence Modeling

Outline

Part I: Simulation OptimizationPart I: Simulation Optimization

–

– Discuss, argue, argueDiscuss, argue, argue

Part II: Input UncertaintyPart II: Input Uncertainty

–

– Discuss, argue, argueDiscuss, argue, argue

Part III: Dependence ModelingPart III: Dependence Modeling

–

– Run out of time / discuss, argue, argueRun out of time / discuss, argue, argue

Caveat Emptor: I will be giving my views on these subjects, whicCaveat Emptor: I will be giving my views on these subjects, which h are diametrically opposed to common sense and good

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Part I

Simulation Optimization

The Generic Problem

MinMin f(f(xx)) s/t s/t g(g(xx) ) ≥≥ 0₀ HereHere f(f(xx) = E ) = E f(f(xx, , YY)) g( g(xx) = E ) = E g(g(xx, , YY))

Example 1: NewsvendorExample 1: Newsvendor

–

– YY = demand= demand

–

– xx = amount to stock= amount to stock –

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Example 2: Call Center

Min staff costs, Min staff costs, s/ts/t satisfactory servicesatisfactory service

xx_i_i = number of staff working shift i= number of staff working shift i

Y = Y = interarrivalinterarrival, service, abandonment , service, abandonment times etc for one day’s operation

times etc for one day’s operation

f(f(xx) = ) = ccTT_x_x _{= staffing cost}_{= staffing cost}

gg_i_i((xx, , YY) = ) = Num happy customersNum happy customers –

– 0.8 0.8 Num arrivalsNum arrivals

Example 3, 4

Example 3: Ambulance base locationExample 3: Ambulance base location

–

– xx = vector of ambulance base locations= vector of ambulance base locations –

– YY = call locations, times, scene times etc over = call locations, times, scene times etc over

time horizon

–

– f(f(xx, , YY) = fraction of calls reached in ) = fraction of calls reached in ₁₀₁₀ mins

mins

Example 4: Ambulance redeploymentExample 4: Ambulance redeployment

–

– xx = parameters of redeployment policy= parameters of redeployment policy –

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Sample Average Approximation

AKA external sampling, sample path opt, AKA external sampling, sample path opt,

stochastic counterpart. Strong links with

retrospective optimization

Replace E(…) with sample average of …Replace E(…) with sample average of …

Fix the random samples beforehandFix the random samples beforehand

Apply a deterministic optimization Apply a deterministic optimization

algorithm

Doesn’t always work … some weird Doesn’t always work … some weird

counterexamples

f(f(xx, , YY) = ) = ccxx -- s s min(min(YY, , xx) ) –– v v max(max(xx ––YY, 0), 0)

Generate demands YGenerate demands Y₁₁, …, , …, YY_n_n

ff_n_n is piecewise linear and convexis piecewise linear and convex

Can apply a convex optimization codeCan apply a convex optimization code

Example: Newsvendor

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SAA General Results

Lots known about Lots known about xx_n_n, the optimal solution , the optimal solution

Under reasonable conditions, including Under reasonable conditions, including

unique minimum:

If multiple optima, related results holdIf multiple optima, related results hold

Gradient Estimation

Derivatives of Derivatives of ff_n_n(x(x) boost computation) boost computation

Same thing as IPA derivativesSame thing as IPA derivatives

Sometimes need extra smoothing when fSometimes need extra smoothing when f_n_n

is not differentiable

Easily found for newsvendor. What about Easily found for newsvendor. What about

harder problems?

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Example 3: Base Location (Mason)

•f_n not continuous, but is piecewise differentiable •E f’_n ≠ f’

•But does that matter?

Stochastic Approximation

TimeTime--honouredhonoured approach. Dates to 50’sapproach. Dates to 50’s

Let Let G(xG(x) be estimate of ) be estimate of ∇∇f(x_f(x)₎

xx_n+1_n+1 = = xx_n_n –– aa_n_n G(xG(x_n_n))

{a{a_n_n} is gain (step length) sequence} is gain (step length) sequence

Essentially steepest descentEssentially steepest descent

Works well only if choose Works well only if choose aa_n_nss carefullycarefully

Can prove lots of theorems, but …Can prove lots of theorems, but …

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Selecting the Best System

Finite number of Finite number of x’sx’s

Allows us to make statements likeAllows us to make statements like

Use after search procedure is completeUse after search procedure is complete

“Clean up afterwards”“Clean up afterwards”

Additional runs will probably be neededAdditional runs will probably be needed P(x

P(x_n_n is truly best of is truly best of x’sx’s visited) visited) ≥≥ 0.950.95

Bird’s

-

Eye View

Apply SAA if good optimization algorithms Apply SAA if good optimization algorithms

available for SAA problem

–

– Usually need structured problemUsually need structured problem –

– IPA (true) gradients help a lot, not essentialIPA (true) gradients help a lot, not essential –

– Requires careful control of random number streamsRequires careful control of random number streams –

– Samples don’t need to be Samples don’t need to be iidiid… can apply variance … can apply variance reduction methods

reduction methods

If deterministic problems are tough, then If deterministic problems are tough, then

anything goes. (Random search,

anything goes. (Random search, metaheuristicsmetaheuristics))

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Where to Find Out More

SAASAA

–

– Alex Shapiro (GA Tech), esp. MC sampling methods Alex Shapiro (GA Tech), esp. MC sampling methods chapter in “Handbook of Stochastic Programming”, chapter in “Handbook of Stochastic Programming”, and

and LinderothLinderoth, Shapiro and Wright paper, Shapiro and Wright paper –

– AtlasonAtlason, , EpelmanEpelman, Henderson , Henderson --Annals of OR, Man Annals of OR, Man SciSci submission

submission ⇒⇒_{discrete problems hard!}_{discrete problems hard!}

Gradient estimation: Gradient estimation: GlassermanGlasserman, Fu and , Fu and HuHu books,

books, L’EcuyerL’Ecuyerpapers, reviews by Michael Fupapers, reviews by Michael Fu

Selecting the best: Chapter by Kim and Nelson Selecting the best: Chapter by Kim and Nelson

in forthcoming “Handbook of Simulation”

Part II

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The Bruce Lee bakery

Open from 6am till 3pm every dayOpen from 6am till 3pm every day

Service times i.i.d., exp(Service times i.i.d., exp(µµ))

How many staff are needed so that 90% How many staff are needed so that 90%

of customers wait

of customers wait 1 minute in queue?_{1 minute in queue?}

Poisson arrival process, rate Poisson arrival process, rate ΛΛ_i_i on day on day

i

((ΛΛ_i_i: :

i

≥≥≥≥≥≥≥≥ 0) i.i.d. _{0) i.i.d.}

G

(₍αα, _,ββ)₎

The Bruce Lee bakery

Similar for E Similar for E

S

₁₁ so standard approach so standard approach works

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The Steve Russell wine store

Open from 11am till 8pm every dayOpen from 11am till 8pm every day

Service times i.i.d., exp(Service times i.i.d., exp(µµ))

How many staff are needed so that 90% How many staff are needed so that 90%

of customers wait

of customers wait _{1 minute in queue?}_{1 minute in queue?}

Poisson arrival process, rate Poisson arrival process, rate λλ

Don’t know Don’t know λλ

Uncertainty in Uncertainty in λλ well modeled as well modeled as

G

((αα, , ββ))

The Steve Russell wine store

Performance measure still E Performance measure still E

S

₁₁ / E / E

N

11

E E

N

₁1 = 9 = 9 λ = ?λ = ?

Which value of Which value of λλ to use?to use?

Same problem with E Same problem with E

S

₁1

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Terminology

Simulation model needs input Simulation model needs input

distributions and parameters

Model uncertainty: Model uncertainty: which family?which family?

Parameter uncertainty: Parameter uncertainty: what parameters?what parameters?

Input uncertainty: Input uncertainty: Both problemsBoth problems

Why do we care?

M

//

M

/1 queue, arrival rate /1 queue, arrival rate λλ₀₀ = 9 = 9 (unknown), service rate

(unknown), service rate µµ₀₀ = 10= 10

Expected time in system is fExpected time in system is

f

((λλ₀₀) ) where

where

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Estimating

λ

₀₀

Assume this holds exactlyAssume this holds exactly

Possibly negative … Possibly negative … ignore

ignore

Possibly bigger than Possibly bigger than µµ₀₀

Assume perfect simulation that reportsAssume perfect simulation that reports

The induced distribution of

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Back to the wine store

By ignoring input uncertainty, we can By ignoring input uncertainty, we can make one of two errors:

make one of two errors:

–

– Overestimate Overestimate λλ₀₀: Hire too many staff: Hire too many staff –

– Super service, expensiveSuper service, expensive

–

– Underestimate Underestimate λλ₀₀: Hire too few staff: Hire too few staff –

– Poor service, cheapPoor service, cheap

Not so bad at a wine store, but what Not so bad at a wine store, but what about at a 911 call center?

about at a 911 call center?

What should we do about it?

Any method should beAny method should be

–

– TransparentTransparentto usersto users –

– Statistically Statistically validvalid –

– Easily Easily implementedimplemented –

– Computationally Computationally efficientefficient

Approach 0: Do “nothing”Approach 0: Do “nothing”

OK so long asOK so long as

–

– users understand that the simulation analyzes the users understand that the simulation analyzes the

chosen

chosen systemsystem –

– a sense of model bias is obtaineda sense of model bias is obtained

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Cheng and Holland ‘03

Suppose Suppose θθ = (= (θθ₁₁, , θθ₂₂) (e.g., arrival, service rates)) (e.g., arrival, service rates)

f increases θ₁ θ₂ Simulate here (L_b, U_b) And here (L_s, U_s) (L_s, U_b)

Bayesian methods

Chick (1997, 1999, 2000, 2001)Chick (1997, 1999, 2000, 2001)

Chick and Ng (2002)Chick and Ng (2002)

Ng and Chick (2001)Ng and Chick (2001)

ZouaouiZouaoui and Wilson (2001a, 2001b, 2003a, and Wilson (2001a, 2001b, 2003a, 2003b)

2003b)

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Induced distribution methods

Study Study

distribution

of of

Distribution function: Lee (1998), Lee and Distribution function: Lee (1998), Lee and

Glynn (1999, 2003)

Mean: Mean: AndradottirAndradottir and Glynn (2003)and Glynn (2003)

Density: Density: SteckleySteckley and Henderson (2003), and Henderson (2003), Steckley Steckley (2005)(2005) Helton: Helton:

G

((

x

; ;

m

, , θθ) = P() = P(

X

₁₁((

m

, , θθ) )

x

)₎ Look at distribution of Look at distribution of

Bird’s Eye View and More Info

Cheng and Holland for small numbers of Cheng and Holland for small numbers of

parameters

Induced distribution if you can get your Induced distribution if you can get your

head around the idea

Bayesian model average once the tools are Bayesian model average once the tools are

automated and in software

In the meantime: sensitivity and In the meantime: sensitivity and

uncertainty analysis

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Part III

Modeling Dependence

Dependence

Some of the random variables in the Some of the random variables in the

simulation are related

Can dramatically alter performance, Can dramatically alter performance,

decisions based on model

–

– # arrivals in morning # arrivals in morning vsvsafternoon in call afternoon in call centre

centre

–

– Wind fields in yacht match race simulationWind fields in yacht match race simulation –

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Modeling Philosophy

What What needsneeds to be modeled?to be modeled?

Use simple models to help decide.Use simple models to help decide. 1.

1. Capture the physicsCapture the physics 2.

2. Use reasonable joint distribution with Use reasonable joint distribution with well understood pluses and minuses

well understood pluses and minuses

3.

3. Use partially specified distributionsUse partially specified distributions

Call Centre Arrivals

PalmPalm--KhintchineKhintchine thmthm suggests arrivals suggests arrivals

follow a

follow a nonhomogeneousnonhomogeneous Poisson processPoisson process

–

– # arrivals in a period is Poisson# arrivals in a period is Poisson –

– different time intervals are independentdifferent time intervals are independent

Call centre HR folks and data:Call centre HR folks and data:

–

– # arrivals has a bigger variance than Poisson# arrivals has a bigger variance than Poisson –

– busier than usual at 9am busier than usual at 9am ⇒⇒ _{busier at 10am}_{busier at 10am}

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Randomly Varying Arrival Rates

Vijay Vijay MehrotraMehrotra, Sam , Sam SteckleySteckley

One explanation:One explanation:

–

– Arrival rate functionArrival rate function is randomis random –

– Once arrival rate function realized, arrivals Once arrival rate function realized, arrivals follow a NHPP with that rate function

follow a NHPP with that rate function

Why is this at least plausible?Why is this at least plausible?

–

– Mechanism generating arrivals can vary due Mechanism generating arrivals can vary due to weather, promotions, etc

to weather, promotions, etc

Trying to capture the unknown physicsTrying to capture the unknown physics

Yacht Match Race Simulation

Andy Andy PhilpottPhilpott, Hamish , Hamish

Sheild

Sheild, U of Auckland, U of Auckland

2 yachts2 yachts

Design & Design & tacticaltacticalQsQs

Model wind speed and Model wind speed and

direction on a grid direction on a grid Spatial/temporal Spatial/temporal dependence dependence

We use a vector ARWe use a vector AR

WellWell--understood understood joint distribution

joint distribution Wind direction

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Vector AR Process

Ambulance Travel Speeds

Call locations are randomCall locations are random

So are travel speeds!So are travel speeds!

–

– Traffic lightsTraffic lights –

– Weather (visibility, road conditions)Weather (visibility, road conditions) –

– CongestionCongestion

Should we explicitly model random travel Should we explicitly model random travel

speeds?

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Responding From Base

Randomness in

travel speeds

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What About

Queueing

Effects?

Service time of a call = travel + Service time of a call = travel + nontravelnontravel

Travel time = distance / speed and speed Travel time = distance / speed and speed

depends on

macro

and and

micro

factorsfactors

Macro factors affect all calls, so Macro factors affect all calls, so average

average

of service time changes

Micro factors influence individual calls, so Micro factors influence individual calls, so

variability

of service time changesof service time changes

Looked at M/G/1 using transform Looked at M/G/1 using transform

inversion and M/G/

inversion and M/G/∞ ∞ models_models

Size Matters

Fraction on time 5 ambulances 30 ambulances 80%

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Random Travel Speeds

Help a bit if calls are clustered near

the boundary of reachable calls

Dependence structure matters: One

form increases

variability.

The other

increases

mean

service time

Hurt heavily utilized fleets more

than lightly

than lightly-

-utilized fleets

utilized fleets

More Info

Henderson tutorial at WSC ’05 discusses Henderson tutorial at WSC ’05 discusses

these examples in more depth

BillerBiller and and GhoshGhosh chapter in forthcoming chapter in forthcoming “Handbook of Simulation”

“Handbook of Simulation”

Prediction: Spatial/temporal models will Prediction: Spatial/temporal models will

become more common

become more common –– need some need some convenient model classes

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Last Words

Are you a tourist wanting to hook up with Are you a tourist wanting to hook up with

simulation folks?

–

– INFORMS Simulation Society business INFORMS Simulation Society business meeting tonight 6:15pm in the Taylor room

meeting tonight 6:15pm in the Taylor room

Are you a colleague who thinks I Are you a colleague who thinks I

misrepresented things? Speak now!

Are you a researcher looking for a new Are you a researcher looking for a new

area?

area? SimSim Opt Opt

in an application setting

is is a great area!