bdsm-ch8_linking feedback with stock qand flow structure
DESCRIPTION
This powerpoint is used in the Business Dynamics and System Modeling class at Southern New Hampshire UniversityTRANSCRIPT
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Business Dynamics and System Modelingy y g
Chapter 8: Linking Feedback with k & lStock & Flow Structure
Pard TeekasapPard Teekasap
Southern New Hampshire University
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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
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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
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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
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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
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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 +
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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
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Structure for first-order, linear positive feedback system
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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)
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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
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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
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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
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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
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Negative Feedback and Exponential Decay
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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
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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
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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
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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
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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
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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)
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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
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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)
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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
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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
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SolutionSolution
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SolutionSolution
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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
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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
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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
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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
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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)
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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)
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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)