Business Dynamics and System Modelingy y g

Chapter 8: Linking Feedback with k & lStock & Flow Structure

Pard TeekasapPard Teekasap

Southern New Hampshire University

OutlineOutline

1. First-order linear feedback systems

2. Positive feedback and exponential growth2. Positive feedback and exponential growth

3. Negative feedback and exponential decay

4. Multiple-loop systems

5 S-Shaped growth5. S Shaped growth

QuizQuiz

k d h f ld h lfTake an ordinary sheet of paper. Fold it in half.Fold the sheet in half again. The paper is still less than a millimeter thick.

• If you were to fold it 40 more times, how thick y ,would the paper be?

• If you folded it a total of 100 times how thickIf you folded it a total of 100 times, how thick would it be?☺ O l i t iti ti t d f l l t☺ Only intuitive estimate, no need for calculator☺ Give your 95% confidence interval

Paper FoldingPaper Folding

• 42 Folds = 440,000 kms thickMore than the distance from the earth to the moon

• 100 Folds = 850 trillion times the distance from the earth to the sunfrom the earth to the sun

First order Linear Feedback SystemFirst-order Linear Feedback System

• Order of a system of loop is the number of state variables

• Linear systems are systems in which the rate equations are linear combination of the stateequations are linear combination of the state variables and any exogenous inputs

• dS/dt = Net Flow = a1S1+a2S2+…+anSn+b1U1+b2U2+…+bmUma1S1 a2S2 … anSn b1U1 b2U2 … bmUm

Basic Structure and BehaviorBasic Structure and BehaviorGoalGoal

State of theState of the

System

State of theSystem

TimeTime

G l

B

+

-

Goal(Desired

State of System)

State of theSystem

RNet

Increase State of theS t

+

CorrectiveAction

B Discrepancy +

+

RIncreaseRate System

+

Action +

Positive Feedback and Exponential Growth

• First-order positive feedback loop

• The state of the system accumulates its netThe state of the system accumulates its net inflow rate

h i fl d d h f h• The new inflow depends on the state of the system

Structure for first-order, linear positive feedback system

Solution for the linear first-ordersystem

Net inflow = gS = dS/dt

dtdS

dS

gdtS

=

gdtSdS

=∫ ∫CgtS

S+=)ln(

S(t) = S(0)exp(gt)

S = State; g = fractional growth rate (1/time)S = State; g = fractional growth rate (1/time)

Phase plot diagram for the first-order,linear positive feedback

.dS/dt = Net Inflow Rate = gSw

Rat

etim

e)et

Inflo

w(u

nits

/t

g

N 1

State of the System (units)00

UnstableEquilibrium

Exponential growth: Phase plot VS Time plot

• Fractional growth rate g = 0.7%

St t

8

10Structure

e) t 1000 768

896

1024

7 68

8.96

10.24Behavior

ts) N

eState of the System(left scale)

6

w (u

nits

/tim

e t = 1000

t = 900512

640

768

5.12

6.4

7.68

Syst

em (U

ni

et Inflow (uni

(left scale)

2

4

Net

Inflo

w t 900

t = 800

t = 700 128

256

384

1.28

2.56

3.84

Stat

e of

the

its/time)

Net Inflow(right scale)

00 128 256 384 512 640 768 896 1024

State of System (units)

0 00 200 400 600 800 1000

(right scale)

Time

Rule of 70Rule of 70

• Exponential growth has the property that the state of the system doubles in a fixed period y pof time

• 2S(0) = S(0)exp(gt )• 2S(0) = S(0)exp(gtd)

• td = ln(2)/g

• td = 70/(100g)

E i t t i 7%/ d bl i• E.g. an investment earning 7%/year doubles in value after 10 years

Misperception of Exponential Growth: it’s not linear

2Time Horizon = 0.1td

2Time Horizon = 1t d

stem

(uni

ts)

stem

(uni

ts)

Stat

e of

the

Sys

Stat

e of

the

Sys

1000Time Horizon = 10t d 1 1030

Time Horizon = 100td

00 2 4 6 8 10 0

0 20 40 60 80 100

1000

tem

(uni

ts)

yste

m (u

nits

)

Stat

e of

the

Sys

0

Stat

e of

the

Sy

00 200 400 600 800 1000

00 2000 4000 6000 8000 10000

Negative Feedback and Exponential Decay

Negative feedbackNegative feedback

• Net Inflow = - Net Outflow = -dS

d = fractional decay rate (1/time). It is thed fractional decay rate (1/time). It is the average lifetime of units in the stock

S( ) S(0) ( d )• S(t) = S(0)exp(-dt)

• This system has a stable equilibrium. y qIncreasing the state of the system increases the decay rate moving the system backthe decay rate, moving the system back toward zero

Phase plot for exponential decayPhase plot for exponential decayNet Inflow Rate = - Net Outflow Rate = - dSNet Inflow Rate Net Outflow Rate dS

StableEquilibrium

te

State of the System (units)0

ow R

ats/

time)

1

dNet

Inflo

(uni

ts

-dN

Exponential decay: Phase plot VS Timeplot

Structure0

Structure

me)

t = 3 0t = 40

ow (u

nits

/tim

t = 10

t = 20

Behavior

Net

Inflo

t = 0

100 10Behavior

Ne

State of the System(left scale)

-50 20 40 60 80 100

State of System (units)50 5

t Inflow (un

Fractional decay rated = 5%

nits/time)

Net Inflowd 5%0 0

0 20 40 60 80 100

(right scale)

Time

Exponential decay with the goal not zero

• In general, the goal of the system is not zero and should be made explicitp

• Net Inflow = Discrepancy/AT = (S*- S)/AT

S* d i d f h A• S* = desired state of the system, AT = adjustment time or time constant

• AT represents how quickly the firm tries to correct the shortfallcorrect the shortfall

First-order linear negative feedback system with explicit goal

dS/dt

General Structure

B

Net InflowRate

SState of

the System

S*Desired State of

the System

dS/dt

+-

+

-Discrepancy

(S* - S)

dS/dt = Net Inflow RatedS/dt = Discrepancy/ATdS/dt = (S* - S)/AT

Examples

ATAdustment Time

-

NetProduction

Rate

Inventory DesiredInventory

Examples

ATAdustment Time

BRate+

+

-InventoryShortfall

Net Production Rate = Inventory Shortfall/AT = (Desired Inventory - Inventory)/AT

Net HiringRate

Labor DesiredLabor Force

+

-

B

+

-Labor

Shortfall

Net Hiring Rate = Labor Shortfall/AT = (Desired Labor - Labor)/AT

+

ATAdustment Time

Phase plot for first-order linear negative feedback system with explicitnegative feedback system with explicit

goalgNet Inflow Rate = - Net Outflow Rate = (S* - S)/AT

1

-1/AT

ow R

ate

/tim

e)

0

StableEquilibrium

Net

Inflo

(uni

ts/ 0

S*State of the System

(units)

Exponential approach to a goalExponential approach to a goal

200

)m

(uni

ts)

100

e Sy

stem

ate

of th

e

00 20 40 60 80 100

Sta

0 20 40 60 80 100

Time constants and half livesTime constants and half-lives

• S(t) = S* - (S* - S(0))exp(-t/AT)

• 0.5 = exp(-th/AT) = exp(-dt)0.5 exp( th/AT) exp( dt)

• th = ATln(2) = ln(2)/d ≈ 0.70AT = 70/(100d)

Goal seeking behaviorGoal-seeking behavior2000

Desired Labor Force

1. AT = 4 weeks

2. AT = 2 weeks1750

1500Forc

eop

le)

2. AT 2 weeks 1500

1250Labo

r (p

eo

10000 2 4 6 8 10 12 14 16 18 20 22 24

0ing

Rat

ee/

wee

k)N

et H

iri(p

eopl

e

Time (weeks)0 2 4 6 8 10 12 14 16 18 20 22 24

Goal seeking behaviorGoal-seeking behavior2000

AT = 4 weeks

Does the workforce

1500

1000

or F

orce

eopl

e)

Desired Labor ForceDoes the workforceequal the desiredworkforce?

500

Labo (p

e Desired Labor Force

workforce? 00 2 4 6 8 10 12 14 16 18 20 22 24

0ing

Rat

ee/

wee

k)N

et H

iri(p

eopl

e

Time (weeks)0 2 4 6 8 10 12 14 16 18 20 22 24

SolutionSolution

SolutionSolution

Multiple loop SystemsMultiple-loop Systems

• Assuming that we disaggregate the net birth rate into a birth rate BR and a death rate DR

• Population = INTEGRAL(Net Birth Rate, Population (0)

• Net Birth Rate = BR DR• Net Birth Rate = BR - DR

• Net Birth Rate = bP – dP = (b-d)P

• b = fractional birth rate, d = fractional death rate

Phase plot for multiple linear first-order loops

Structure (phase plot) Behavior (time domain)

b d E ti l G th

0d

Dea

th R

ates

Net Birth RateBirth Rate 1

b

1 b-d

pula

tion

b > d Exponential Growth

Population

Birt

h an

0

Death Rate 1

0Time0

Po

-d

0Dea

th R

ates

Net Birth Rate

Birth Rate

ulat

ion

b = d Equilibrium

Population

Birt

h an

d

0

Death Rate

0Time0

Popu

0Dea

th R

ates

Birth Rate

ulat

ion

b < d Exponential Decay

Population

Birt

h an

d

0

Death RateNet Birth Rate

0Time0

Popu

Nonlinear first-order systems: S-Shaped growth

• No real quantity can grow forever. It will eventually approach the carrying capacity of y pp y g p yits environment

• As the system approaches its limits to growth• As the system approaches its limits to growth, it goes through a nonlinear transition from a

fregime where positive feedback dominates to a regime where negative feedback dominates

• It’s a S-Shaped growth

Diagram for population growth with a fixed environment

• Net Birth Rate = BR – DR = b(P/C)P – d(P/C)P

Population

Birth Rate DeathRateBR ++ +

PopulationBB +

+

PopulationRelative toCarryingCapacity

FractionalBirth Rate

FractionalDeath Rate

-- +

CarryingCapacity

Nonlinear birth and death rateNonlinear birth and death rate

• Sketch the graph showing the likely shape of the fractional birth and death rate

Rat

esnd

Dea

th R

me)

0

al B

irth

an(1

/tim

Large0 1

Frac

tiona

Population/Carrying Capacity(dimensionless)

Nonlinear relationship between population density and the fractionalpopulation density and the fractional

growth rategR

ates Fractional

Birth Rate Fractional

Dea

th R Birth Rate Fractional

Death Rate

0

rth

and

(1/ti

me)

0 1

iona

l Bir 0

Frac

ti

Fractional Net Birth Rate

Population/Carrying Capacity(dimensionless)

Phase plot for nonlinear population system

Positive Feedback Dominant

Negative FeedbackDominant

ates

e) Bi th R t

Death Rate

0Dea

th R

aua

ls/ti

me Birth Rate

••0

rth

and

D(in

divi

du

0 Stable EquilibriumUnstable

Equilibrium

•• (P/C)inf 1

Bir

Net Birth Rate

q

Population/Carrying Capacity(dimensionless)