Bdsm-ch8_linking Feedback With Stock Qand Flow Structure

Business Dynamics y and System y Modelingg

Chapter 8: Linking Feedback with Stockk & Flow l Structure Pard Teekasap Southern New Hampshire University

Outline 1. 2. 3. 4. 5 5.

First-order linear feedback systems Positive feedback and exponential growth Negative feedback and exponential decay Multiple-loop systems SS-Shaped Shaped growth

Quiz Take k an ordinary d sheet h off paper. Fold ld it in h half. lf Fold the sheet in half again. The paper is still less than a millimeter thick. • If yyou were to fold it 40 more times,, how thick would the paper be? times, how thick • If you folded it a total of 100 times would it be? ☺ Only O l iintuitive t iti estimate, ti t no need d ffor calculator l l t ☺ Give your 95% confidence interval

Paper Folding • 42 Folds = 440,000 kms thick More than the distance from the earth to the moon

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

First order Linear Feedback System First-order • 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 state variables and any exogenous inputs • dS/dt = Net Flow = a1S1+aa2S2+…+a … anSn+b b1U1+b b2U2+…+b … bmUm

Basic Structure and Behavior Goal

State of the System

State of the System

Time

Time + State of the System Net Increase Rate +

R

+ State of the S t System

B

Goall G (Desired State of System)

Discrepancy

Corrective Action +

+

Positive Feedback and Exponential Growth • First-order positive feedback loop • The state of the system accumulates its net inflow rate • The h new iinflow fl d depends d on the h state off the h system

Structure for first-order, linear positive feedback system

Solution for the linear first-order system Net inflow = gS = dS/dt

dS = gdt dt S dS = gdt ∫S ∫ ln(S ) = gt + C S(t) = S(0)exp(gt) S = State; g = fractional growth rate (1/time)

Phase plot diagram for the first-order, linear positive feedback dS/dt = Net Inflow Rate = gS

Net Inflow w Rate (units/ttime)

.

g 1

0 0

State of the System (units)

Unstable Equilibrium

Exponential growth: Phase plot VS Time plot • Fractional growth rate g = 0.7% St Structure t 10

Behavior

1024

t = 1000

6 t = 900

4 t = 800

2

t = 700

State of the System (Units)

8

768

0

128 256 384 512 640 768 896 1024 State of System (units)

7 68 7.68

640

6.4

512

5.12

384

3.84

256

2.56

128

Net Inflow (right scale)

0

0

8.96

State of the System (left scale)

0

200

400

Time

600

800

1.28

0 1000

Ne et Inflow (uniits/time)

Net Inflow w (units/time e)

896

10.24

Rule of 70 • Exponential growth has the property that the state of the system y doubles in a fixed period p of time • 2S(0) = S(0)exp(gtd) • td = ln(2)/g • td = 70/(100g) • E.g. E an iinvestment t t earning i 7%/ 7%/year d doubles bl iin value after 10 years

Misperception of Exponential Growth: it’s not linear Time Horizon = 0.1t d

2

State of the Sys stem (units)

State of the Sys stem (units)

2

Time Horizon = 1t d

0 0

2

4

6

8

Time Horizon = 10t d

0

20

40

60

80

100

8000

10000

Time Horizon = 100t d

30

1 10

State of the System (units)

State of the Sy ystem (units)

1000

0

10

0

0 0

200

400

600

800

1000

0

2000

4000

6000

Negative Feedback and Exponential Decay

Negative feedback • Net Inflow = - Net Outflow = -dS d = fractional decay rate (1/time). It is the average lifetime of units in the stock • S(t) S( ) = S(0) S(0)exp(-dt) (d) • This system y has a stable equilibrium. q Increasing the state of the system increases the decay rate rate, moving the system back toward zero

Phase plot for exponential decay Net Inflow Rate = - Net Outflow Rate = - dS

Net Inflo N ow Ratte (units s/time)

Stable Equilibrium 0

State of the System (units) 1 -d d

Exponential decay: Phase plot VS Time plot Structure

0 t = 40

Net Inflo ow (units/tim me)

t=30 t = 20

t = 10

Behavior

100 t=0

10

-5 0

20 40 60 80 State of System (units)

100

50

5

Fractional decay rate d = 5%

Net Inflow (right scale)

0 0

20

40

60 Time

80

0 100

Net Inflow (un nits/time)

State of the System (left scale)

Exponential decay with the goal not zero • In general, the goal of the system is not zero and should be made explicit p • Net Inflow = Discrepancy/AT = (S*- S)/AT • S* = d desired i d state off the h system, A AT = adjustment time or time constant • AT represents how quickly the firm tries to correct the shortfall

First-order linear negative feedback system with explicit goal General Structure dS/dt Net Inflow Rate +

S State of the System

S* Desired State of the System

B -

AT Adustment Time

Discrepancy (S* - S)

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

+

Examples Desired Inventory

Inventory Net Production Rate +

B -

AT Adustment Time

Inventory Shortfall

+

Desired Labor Force

Labor -

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

Net Hiring Rate +

AT Adustment Time

B Labor Shortfall

+

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

Phase plot for first-order linear negative feedback system with explicit g goal Net Inflow Rate = - Net Outflow Rate = (S* - S)/AT

Net Inflo ow Rate (units//time)

1 -1/AT 0 S*

Stable Equilibrium State of the System (units)

Exponential approach to a goal Sta ate of the e System m (units))

200

100

0 0

20

40

60

80

100

Time constants and half half-lives lives • S(t) = S* - (S* - S(0))exp(-t/AT) • 0.5 = exp( exp(-tth/AT) = exp( exp(-dt) dt) • th = ATln(2) = ln(2)/d ≈ 0.70AT = 70/(100d)

Goal seeking behavior Goal-seeking 2000

Labor Force (peo ople)

1750

1500 1250

1000

Net Hiriing Rate (people e/week)

1. AT = 4 weeks 2. AT = 2 weeks

Desired Labor Force

0

2

4

6

8

10

12

14

16

18

20

22

24

0

2

4

6

8

10 12 14 16 Time (weeks)

18

20

22

24

0

Goal seeking behavior Goal-seeking 2000

Labo or Force (pe eople)

1500 1000 Desired Labor Force 500 0

Net Hiriing Rate (people e/week)

AT = 4 weeks Does the workforce equal the desired workforce?

0

2

4

6

8

10

12

14

16

18

20

22

24

0

2

4

6

8

10

12

14

16

18

20

22

24

0

Time (weeks)

Solution

Solution

Multiple loop Systems Multiple-loop • 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 = bP – dP = (b-d)P b = fractional birth rate, d = fractional death rate

Phase plot for multiple linear firstorder loops Structure (phase plot)

1

Birth Rate

Net Birth Rate 1 b-d

0 Death Rate

E Exponential ti l Growth G th

b

1

Population

Birth and Death Rates

b>d

Behavior (time domain)

-d

0 0

Population

0 Equilibrium

Birth Rate Net Birth Rate 0

Popu ulation

Birth and Death Rates

b=d

Death Rate

0

Population

0

0

b
Time Exponential Decay

Birth Rate 0 Net Birth Rate

Popu ulation

Birth and Death Rates

Time

Death Rate 0 0

Population

0

Time

Nonlinear first-order systems: SShaped growth • No real quantity can grow forever. It will eventuallyy approach pp the carrying y g capacity p y of its environment • As the system approaches its limits to growth growth, it goes through a nonlinear transition from a regime where positive feedback f 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 +

R

B

+

Death Rate

+

+

B Fractional Birth Rate -

+ Population Relative to Carrying Capacity Carrying Capacity

B Fractional Death Rate +

Nonlinear birth and death rate

Fractiona al Birth an nd Death R Rates (1/tim me)

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

0 0

1

Population/Carrying Capacity (dimensionless)

Large

Fractiional Birrth and Death R Rates (1/time)

Nonlinear relationship between population density and the fractional ggrowth rate Fractional Birth Rate

Fractional Death Rate

0 0

1

Fractional Net Birth Rate Population/Carrying Capacity (dimensionless)

Phase plot for nonlinear population system Positive Feedback Dominant

Negative Feedback Dominant

Birrth and D Death Ra ates (individu uals/time e)

Death Rate Bi th Rate Birth R t 0

•

(P/C)inf

0

Unstable q Equilibrium

•1

Stable Equilibrium

Net Birth Rate Population/Carrying Capacity (dimensionless)