Learning Curves-Modeling and Simulation-Lecture Handouts | Lecture notes Mathematical Modeling and Simulation

1CIS308‐ModelingandSimulation

Handout#9

LearningCurves

Sometimesaquantitygrowsveryrapidlyatfirst,butthengrowsslowerandslowerasthequantitygets

closerandclosertosomelimitingvalueL.Therearemanywaysthiscouldhappen.Onefamousexample

comesfrompsychology.Peopleinanexperimentareoftenaskedtomemorizealonglistofdigits,and

attimet,experimentersmeasurep(t)=thefractionofthetotallistlearnt.Theexperimentshowsthatp

startingat0,growsquicklyatfirst,andthengrowsmuchmoreslowlyaspeopleintheexperimentlearn

moreandmoreofthelist.Itisfromthisexperimentthatwegetthetermlearningcurves.Thisbehavior

canbethoughtofassimilartorestrictedgrowthmodel.

MathematicalModel



Thedifferentialequationdescribingarestrictedgrowthmodelisgivenby

)( QCk

dQ −= ‐‐‐‐‐‐‐‐‐‐‐‐‐‐(I)

wherebothkandCarepositiveconstants.Tosolvethisseparablefirst‐orderdifferentialequation,we

firstseparatethevariables,obtaining

kdt

dQ =

−

Integratingbothsides

dkdk

AeCtQ

eeeQC

dktQC

kdt

−

−−−−

−=

==−

−−=−

+=−−

−∫∫

)(

||ln



‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐(II)

Theaboveequationisknownastheequationoflearning

curve(asshown).



ExercisesinMATLAB



1.YieldofaWheatField:InanexperimentconductedbyresearchersoftheAgricultureDepartmentof

aUniversity,itwasfoundthatthemaximumyieldofwheatintheuniversity’sexperimentalfield

stationwas150bushelsperacre.Furthermore,theresearchersdiscoveredthattherateatwhichthe

yieldofwheatincreasedwasgovernedbythedifferentialequation

)150( Qk

dQ −= 

whereQ(x)denotestheyieldinbushelsperacreandxistheamountinpoundsofanexperimental

fertilizerusedperacreofland.Dataobtainedintheexperimentindicatedthat10poundsoffertilizer

peracreoflandwouldresultinayieldof80bushelsofwheatperacre,whereas20poundsof

fertilizerperacreoflandwouldresultinayieldof120bushelsofwheatperacre.Determinetheyield

if30poundsoffertilizerwereusedperacre.PlottheaffectoffertilizerontheyieldusingtheMATLAB.

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Handout#

Learning Curves Sometimes a quantity grows very rapidly at first, but then grows slower and slower as the quantity gets closer and closer to some limiting value L. There are many ways this could happen. One famous example comes from psychology. People in an experiment are often asked to memorize a long list of digits, and at time t, experimenters measure p(t) = the fraction of the total list learnt. The experiment shows that p starting at 0, grows quickly at first, and then grows much more slowly as people in the experiment learn more and more of the list. It is from this experiment that we get the term learning curves. This behavior can be thought of as similar to restricted growth model. Mathematical Model The differential equation describing a restricted growth model is given by

k ( C Q )

dt

dQ

= − ‐‐‐‐‐‐‐‐‐‐‐‐‐‐(I)

where both k and C are positive constants. To solve this separable first‐order differential equation, we first separate the variables, obtaining

kdt

C Q

dQ

Integrating both sides kt k d k d

Qt C Ae

C Q e e e

C Q kt d

kdt

C Q

dQ

− −− − −

ln| |

‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐(II)

The above equation is known as the equation of learning curve (as shown). Exercises in MATLAB

1. Yield of a Wheat Field: In an experiment conducted by researchers of the Agriculture Department of a University, it was found that the maximum yield of wheat in the university’s experimental field station was 150 bushels per acre. Furthermore, the researchers discovered that the rate at which the yield of wheat increased was governed by the differential equation

k ( 150 Q )

dt

dQ

where Q ( x ) denotes the yield in bushels per acre and x is the amount in pounds of an experimental fertilizer used per acre of land. Data obtained in the experiment indicated that 10 pounds of fertilizer per acre of land would result in a yield of 80 bushels of wheat per acre, whereas 20 pounds of fertilizer per acre of land would result in a yield of 120 bushels of wheat per acre. Determine the yield if 30 pounds of fertilizer were used per acre. Plot the affect of fertilizer on the yield using the MATLAB.

Solution: The given differential equation has the same form as eq(I) with C= 150. Solving it directly or using the result obtained in the solution of given by eq (II), we see that the yield per acre is given by

Q ( x )= 150 − Ae −^ kx

The first of the given conditions implies that Q (10) = 80; k k

A e

Ae

10 10

Therefore, we get ( 10 ) 10

− − −

k x k kx

e

Qx e e

The second of the given conditions implies that Q (20) = 120

10 10 ( 2010 )

− − − − k k k

e

Take log of both sides

ln 10 ln

k

e k

Therefore,

085 ( 10 )

− −

Qx e

In particular, if x=30, then

7
085 ( 3010 )

− − −

e

Qx e

The yield would be 137 bushels per acre if 30 pounds of fertilizer were used per acre.

At t=0, 20% of the students had contracted influenza. Therefore,

(ln 4 ln 9 ) 0. 0000312

50000 ln 4 ln 19

50000 50000 5000

− − −

k

e

Qt

k k kt Therefore, we get

= − t

e

Qt

In particular, the number of students who had contracted the flu by the 13 th^ day^ is^ given^ by

Learning Curves-Modeling and Simulation-Lecture Handouts, Lecture notes of Mathematical Modeling and Simulation

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Handout#

k ( C Q )

dt

dQ

= − ‐‐‐‐‐‐‐‐‐‐‐‐‐‐(I)

kdt

C Q

dQ

Qt C Ae

C Q e e e

C Q kt d

C Q kt d

kdt

C Q

dQ

ln| |

ln| |

‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐(II)

k ( 150 Q )

dt

dQ

Q ( x )= 150 − Ae −^ kx

A e

Ae

e

Qx e e

e

e

e

ln 10 ln

k

e k

Qx e

e

Qx e

(ln 4 ln 9 ) 0. 0000312

50000 ln 4 ln 19

k

k

e

e

e

Qt

= − t

e

Qt

e

Qt