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This powerpoint is used in the Business Dynamics and System Modeling class at Southern New Hampshire University
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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
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)